ACTEX Study Manual: SOA exam FM, CAS exam 2 - Hassett M.J., Ratliff M.I.

Module 1 - Interest Rates and the Time Value of Money
1.2 Present Value and Future Value with Compound Interest
1.3 Functions of Investment Growth Notation
1.4 Effective Rate of Interest for a Specified Period
1.9 Continuous Interest and Force of Interest
1.10 Relating Discount, Force of Interest and Interest Rate
1.11 Solving for PV, FV, n, and i with Compound Interest
2.4 Annuities with Level Payments Other Than 1
2.7 Basic Annuity Problems for Calculator Practice
2.9 Increasing Annuities with Terms in Arithmetic Progression
2.10 Decreasing Annuities with Terms in Arithmetic Progression
2.11 A Single Formula for Annuities with Terms in Arithmetic Progression
2.12 Annuities with Terms in Geometric Progression
2.13 Equations of Value and Loan Payments
2.14 Deferred Annuities and a Useful Annuity Identity
2.16 Annuity Problems with Interest Rate Variations
2.17 Rates Where Interest is Convertible More or Less Frequently than Paid
3.3 Formulas for Level Payment Loan Amortization
3.8 Capitalization of Interest and Negative Amortization
5.2 Time Weighted and Dollar Weighted Rates
5.3 The Investment Year and Portfolio Methods
Module 6 - Term Structure of Interest Rates
Module 7 - Asset Liability Management, Duration, and Immunization
7.4 Helpful Formulas for Duration Calculations
7.5 Using Derivatives to Approximate Change in Price
7.8 Stocks and Other Investment Opportunities
Module 8 - Review of Derivatives Markets, Chapter 1
8.2 What is a derivative security?
Module 9 - Review of Derivatives Markets, Chapter 2
9.12 Don't forget Appendix 2.A
9.13 Module 9 Summary
9.15 Module 9 Computational Review Problems
Module 10 - Review of Derivatives Markets, Chapter 3
10.2 Strategies Combining an Option and an Asset
10.3 Synthetic Forwards and Put-Call Parity
10.5 The Marshall & Illsley Corp. Equity Linked Note
10.8 Module 10 Computational Review Problems
Module 11 - Review of Derivatives Markets, Chapter 4
11.2 Using Derivatives to Manage the Risk of a Producer-Seller
11.3 Using Derivatives to Manage the Risk of a Producer-Buyer
11.4 Reasons Firms Might Want to Manage Risk
11.5 More Complex Strategies for Hedging
11.8 Module 11 Computational Review Problems
Module 12 - Review of Derivatives Markets, Chapter 5
12.4 Forward Contracts on Stock or Stock Indices
12.8 Module 12 Computational Review Problems
Module 13 - Review of Derivatives Markets, Chapter 7
13.2 New Notation for Spot and Forward Rates
Module 14 - Review of Derivatives Markets, Chapter 8
14.7 Module 14 Computational Review Problems
Автор: Hassett M.J. Ratliff M.I.
Теги: mathematics finance study guide exam preparation financial analysis
ISBN: 978-1-56698-680-9
Год: 2009
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Текст
                    ACTEX
STUDY MANUAL
SOA Exam FM
CAS Exam 2
Probability
Pension
Finance
i ;£ onomics
f^il^^l suiting
1,11
illlli
ill
2009 Edition
Matthew J. Hassett, Ph.D., ASA
Michael I. Ratliff, Ph.D., ASA
Toni Coombs Garcia
Amy C. Steeby, MBA
ACTEX Publications
Actuarial & Financial Risk
Resource Materials
Since 1972

Copyright © 2009, by ACTEX Publications, Inc.
No portion of this ACTEX Study Manual may be
reproduced or transmitted in any part or by any means
without the permission of the publisher.
Printed in the United States of America.
ISBN 13: 978-1-56698-680-9

SOA Exam FM and CAS Exam 2 have changed. For the past few years these exams
tested only the traditional material on interest theory. For Spring 2007, entirely new
material on financial mathematics was added. In this guide, the traditional interest
theory is covered in Modules 1-7 and the new material in financial mathematics is
covered in Modules 8-15.
Modules 8-15 contain lecture notes on the required chapters of the financial
mathematics textbook Derivatives Markets and solutions to odd-numbered homework
problems in that text. (Answers to even-numbered problems are available in the student
solution manual which you can purchase with the text.)
Contents of this guide:
Module 1
Module 2
Module 3
Module 4
Module 5
Module 6
Module 7
Module 8
Module 9
Module 10
Module 11
Module 12
Module 13
Module 14
Module 15
Practice Exams
Interest Rates and Time Value of Money
Annuities
Loan Repayment
Bonds
Yield Rate of an Investment
Term Structure of Interest Rates
Asset Liability Management, Duration and Immunization
Review of Derivatives Markets, Chapter 1
Review of Derivatives Markets, Chapter 2
Review of Derivatives Markets, Chapter 3
Review of Derivatives Markets, Chapter 4
Review of Derivatives Markets, Chapter 5
Review of Derivatives Markets, Chapter 7
Review of Derivatives Markets, Chapter 8
Supplemental Material on Currency Forward Contracts
Seven Practice Exams
A note about Errors:
If you find a possible error in this manual, please let us know at the "Customer
Feedback" link on the ACTEX homepage (www.actexmadriver.com).
We will review all comments and respond to you with an answer. Any confirmed
errata will be posted on the ACTEX website under the "Errata & Updates" link.
October, 2008
Matt Hassett, ASA, PhD
Toni Garcia, MS
Amy Steeby, MBA
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Study Tips
1) Come up with a schedule to complete your studying in time for the exam. Divide
your schedule into time for each section and time at the end to review and to do
final practice problems. This may vary depending on how much time you have
before the exam. A reasonable amount may be one chapter per week.
2) If possible, join a study g$>up of peers studying for thySf same exam.
3) For each chapter:
a) Read the chapter in the FM manual.
b) Make sure that you can compute the examples in the text correctly as you're
reading through them.
c) Recite or summarize each concept learned in the margins or in a notebook.
d) Understand the main idea of each concept and be able to summarize in your own
words. Imagine that you are trying to teach somebody else this concept.
e) While reading, create flash cards for formulas to start memorization.
f) Learn the calculator skills and know all of your calculator functions.
g) Do a review of the corresponding chapter in the recommended text,
h) Do the Basic Review Problems and review your solutions.
i) Do the Sample Exam Problems and review your solutions.
i) If you have been stuck on a problem for more than 20 minutes, it is OK if
you need to refer to the solutions. Just make sure that when you are finished
with the problem, you can recite the concept that you missed and summarize
it in your own words. If you get stuck on a problem, think about what
principles were used in this question and see if you could rewrite a different
problem with similar structure, as if you were the exam writer.
ii) Mark each sample exam problem as an Easy, Medium, or Hard problem.
4) After learning each chapter, it is a good idea to go back to previous chapters and do
a quick half-hour to hour review, so that information isn't forgotten.
5) Go back and redo the sample exam problems that you have marked as Medium or
Hard when you looked through them the first time.
6) After learning each chapter and reviewing past chapters, go to the practice exams.
a) Attempt the first three practice exams in a non-timed environment
b) Attempt the last four practice exams in a timed environment similar to the
timing structure of the formal administered exam.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

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Table of Contents
PageTOC- 1
Contents h
Module Topic
Module 1
Time Value of Money
Compound Interest
Simple Interest
Present Value
Future Value
Accumulation Functions
Effective Interest Rate
Nominal Interest Rate
Periodic Interest Rate
Convertible Interest
Discount Rate
Nominal Discount Rate
Conversion of Nominal Interest rate to Discount Rate
Accumulation Functions, Continuous Interest
Force of Interest
Constant Force of Interest
Relating Interest Rate and Force of Interest
Equation of Value
Module 2
Annuities
Annuity Immediate
Annuity Due
Unit Annuity
Timelines
Geometric Series
Future Value of Annuities
Perpetuities
Annuities with Level Payments
Continuous Annuities
Annuities with Varying Payments
Increasing Annuities
Decreasing Annuities
Annuities with Arithmetic Progression
Annuities with Geometric Progression
Deferred Annuities
Variable Annuities
Reinvestment Problems
Inflation
Page
Ml-l
MM
Ml-l
Ml-2
Ml-2
Ml-4
Ml-6
Ml-7
Mi-8
Ml-8
Ml-10
Ml-12
Ml-13
Ml-15
Ml-15
Ml-15
Ml-18
Ml-19
M2-1
M2-3
M2-8 "
M2-1
M2-2
M2-2
M2-4
M2-6
M2-7
M2-11
M2-15
M2-16
M2-18
M2-19
M2-20
M2-24
M2-26
M2-31
M2-32
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page TOC-2 Table of Contents
Module 3
Amortization M3-1
Amortization Table M3-2
Amortization with Variable Payments M3-4
Amortization with Level Payments M3-6
Prospective Method M3-7
Retrospective Method M3-8
Amortization with Arithmetic Payments M3-10
Amortization with Geometric Payments M3-9
Amortization with Monthly Payments M3-11
Installment Loan M3-14
Sinking Fund M3-15
Netlnterest M3-16
Sinking Fund Deposit M3 16
Sinking Fund Balance M3-17
Capitalization of Interest M3-19
Negative Amortization M3-19
Module 4
Bonds M4-1
Face Value M4-1
Par Value M4-1
Coupon Rate M4-1
Redemption Value M4-1
Premium Bond M4-2
Discount Bond M4-2
Bond Price M4-4
Premium-Discount Formula for Bonds M4-5
Makeham's Formula M4-5
Amortization of Premium M4-6
Amortization of Discount M4-6
Amount for Accumulation of Discount M4-7
Negative Amortization of Discount M4-8
Callable Bond M4-9
Call Provisions M4-9
Pricing Bonds between Payment Dates M4-11
Price-Plus Accrued M4-11
Flat Price \ ][ _/ ~ '[.'.]. ,[..[[ ..'/.['.[[. .'.'. ...,M4:11
Settlement Date M4-11
Market Price M4-12
Accrued Interest M4-12
True Price M4-12
Module S
Internal Rate of Return M5-1
Cash Flows M5-1
Modified Internal Rate of Return M5-5
Uniqueness of Internal Rate of Return M5-5
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Table of Contents
PageTOC- 3
Borrowing Projects M5-6
Time Weighted Rate M5-7
Dollar Weighted Rate M5-7
Investment Year Method M5-11
Portfolio Method M5-11
New Money Rate M5-12
Net Present Value M5-13
Module 6
Term Structure of Interest Rates M6-1
Zero Coupon Bond M6-1
Risk-Free Rates M6-1
Spot Rate M6-2
Yield Curve M6-2
Treasury STRIP bond M6-?
Inverted Yield Curve M6-3
Flat Yield Curve M6-3
Law of One Price M6-4
Forward Rate M6-5
Implied Forward Rate M6-5
Module 7
Assets M7-1
Liabilities M7-1
Liability Management M7-1
Matching Assets and Liabilities „M7"1
Duration M7-4
Interest Rate Risk \\ "\ *,..*'.. M7-4
Weighted Average M7-4
Macaulay Duration M7-4
Modified Duration M7-6
Volatility M7-6
Macaulay Duration of Coupon Bond M7-8
Taylor Series M7-10
Price Function, P(i) M7-10
Convexity M7-12
Change in Price M7-13
Duration of Portfolio M7-14
Parallel Shift in Yield Curve M7-15
Immunization M7-16
Present Value Matching M7-18
Duration Matching M7-18
Greater Convexity for Assets M7-18
Fully Immunized M7-19
Stocks M7-20
Dividends M7-20
Price of Stock M7-20
Mutual Funds M7-21
Certificate of Deposit M7-22
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page TOC-4 Table of Contents
Money Market Funds M7-22
Mortgage-Backed Securities M7-22
Module 8
Derivative Security M8-1
Hedging M8-2
Bid-ask spread M8-3
Short Sale of Stock M8-3
Long Position in Stock M8-3
Module 9
Forward Contract M9-1
Spot price M9-1
Stock index M9-1
Cash settlement M9-2
Long forward M9-2
Short Forward M9-2
Payoff for Forward M9-3
Profit for forward M9-3
Zero Coupon Bond Profit M9-7
Call Option M9-9
European Option M9-9
American Option M9-9
Bermudan Option M9-9
Premium M9-9
Written Call Option M9-12
PutOption [\ .....[„ ^ '_ [ ' ] ]tr ' '[M9-14.
Written PutOption ]'„[]["*""]] ".[]'.'.. ". ['." **.V. [[[,[ '. ... y//.^" M9-"l6 '
In the Money Option M9-17
At the Money Option M9-17
Out of the Money Option M9-17
Insurance, Options s M9-18
Equity Linked CD M9-19
Module 10
Floor Strategy M10-2
Cap Strategy M10-3
Covered Call M10-4
Covered Put M10-5
Parity, Put-Call M10-6
Synthetic Forward M10-6
Spread M10-9
Bull Spread MlO-10
Bear Spread M10-11
Box Spread M10-12
Collar M10-13
Collar, Hedging with M10-14
Zero Cost Collar M10-15
Straddle M10-16
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Table of Contents
PageTOC- 5
Strangle M10-18
Equity Linked Note, Marshall & Isley M10-21
Module 11
Hedging, Producer-Seller Ml 1-2
Hedging, Buyer Mll-5
Hedging, Reasons for Ml 1-7
Hedging with a Collar Mll:9
Paylater Strategy Mll-11
Module 12
Prepaid Forward Price M12-3
Arbitrage Pricing M12-3
Forward Contract on Stock, Pricing M12-7
Forward Premium M12-8
Synthetic Stock ' M1J2:9_
Hedging with a Synthetic Stock M12-10
Cash and Carry Hedge M12-11
Quasi Arbitrage M12-15
Cost of Carry M12-17
Lease Rate M12-17
Futures Contracts M12-18
Clearing House M12-18
Open Outcry M12-18
Mark to Market M12-19
S&P 500 Futures Contract M12:19
Margin M12-19
Forward and Futures Prices Compared M12-22
Quanto Index Contracts M12-24
Module 13
Spot Rate M13-2
Forward Interest Rate M13-2
Zero-coupon Bonds M13-3
Implied Forward Rate M13-3
FRA (Forward Rate Agreement) M13-4
Eurodollars M13-6
Module 14
Swap, Oil M14-2
Swap Payment M14-3
Dealer as Swap Counterparty M14-4
Swap, Market Value M14-5
Interest Rate Swap M14-7
Swap Rate R M14-8
Swap Curve M14-10
Accreting Swap M14-11
Amortizing Swap M14-11
Swap rate, general formula M14-13
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2
- Financial Mathematics

Module 1 - Interest Rates and the Time Value of Money
Page Ml- 1
Interest Rates and Time Value of Money
Section 1.1
Time Value of Money
Interest theory deals with the time value of money. For example, a dollar
invested at 6% per year is worth $1.06 one year from today. What happens after
the first year depends on whether you are earning compound interest or simple
interest. We illustrate this with an example. Suppose you invest 100 at 6%
interest for two years.
a) Compound interest. You earn interest on the total amount in your
account at the beginning of each year. The amounts in your account at
the end of year 1 and year 2 are:
Year 1: 100 + 0.06(100) = 100(1.06) = 106
Year 2: 106 + 0.06(106) = 106(1.06) = 100(1.06)2 = 112.36
b) Simple interest. You earn interest only on the original 100 each year. The
amounts in your account at the end of year 1 and year 2 are:
Yearl: 100+ 0.06(100) = 100(1.06) = 106
Year 2: 106 + 0.06(100) = 100*(1 + 2(0.06)) = 112
Compound interest is the most widely used method, especially for multi-period
investments. Simple interest is more commonly used for shorter term
investments. There are other ways to calculate interest, and we will see some of
these later.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Ml-2
Module 1 - Interest Rates and the Time Value of Money
Section 1.2
Present Value and Future Value with Compound Interest
We will start with a look at compound interest, since it is so widely used, The
value today is the present value [PV] and the value n periods from today is the
future value [FV]. If funds are invested at a periodic compound interest rate i
for n periods, the basic relationships are:
(1.1)
FV = PV(l + i)n
PV = -
FV
d + O"
Example (1.2)
Let n = 10 and i =
a) UPV =
b) IfFV =
= 0.06.
= 1,000, then FV --
■-1,000,
then PV =
= 1,000(1.06)10
= 1,790.85
1.000 _«8,9
O06F
PV
FV
N|
pmt|
i/yJ
The BA II Plus calculator has five time value of money keys.
Present value
Future value
Number of periods
Periodic payment
Periodic interest rate
In this module we will not look at any problems that involve periodic
payments. The JPMTJ key will be used starting in Module 2. The other four
keys can be used to solve compound interest problems like Example (1.2), as
we illustrate next.
To begin any new problem, it is wise to clear the time value of money [TVM]
registers to erase any numbers left over from prior problems. Note that the
legend CLR TVM appears above the [FV 1 key on the BA II Plus calculator. To
clear the TVM registers use the keystrokes 12ND 1CLR TVM.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 1 - Interest Rates and the Time Value of Money Page Ml- 3
Before we do the actual calculation, we must point out an important BA II
Plus convention for signs on answers: money that you receive is positive, but
money that you pay out is negative. Thus, if you put 1000 into an account now,
you should enter it into the calculator as -1000 to indicate that it is out of
pocket. You can make an e&itry negative using the 1+7-1 key.
Now, let's rework Example (1.2) using the calculator:
To find the future value of 1000 in 10 years at 6% compound interest per
year, use the keystrokes
10000 [PV| 6 jj/Y| 10 §[CPl3 [FV]
You will see the display FV = 1790.85. Note that the answer is positive since
this is money that is paid back to you.
To find the present value at 6% compound interest of 1000 paid 10 years in
the future use the keystrokes
1000 |fv| 6 jj/Y| 10 §|CPl3 |jv| .
You will see the display PV = -558.39. Note that the answer is negative for
money that you put into the account.
Exercise (1.3)
Using an interest rate of 5% compounded annually, find a) the present
value of 20,000 payable in 15 years and b) the future value of 5,000 in 6
years.
Answer a) -9620.34 b) 6700.48
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Ml-4
Module 1 - Interest Rates and the Time Value of Money
Section 1.3
Functions of Investment Growth Notation
A long-term investor might wish to plot growth of an invested amount over
time. There are two functions of interest:
a(t), which is the amount an initial investment of 1 grows to by time £,
- and-
A(t), the amount an initial investment of A(0) grows to by time t.
For compound interest applied on a per year basis, amounts change only at year
end when interest is paid. For positive integer values of n,
(1.4)
a(n) = (l + i)n
A(n) = A(Q)(l + i)n
For i = 0.06 compound interest per year paid at year end, the first four values of
a(n) are:
N
a(n)
0
1
1
1.060
2
1.1236
3
1.1910
The graph of a(n) is a step function:
a(n)
1.1910 —h
1.1236
1.0600 —|-
1.0000
Interest may also be paid on a continuous basis, which is an advantage for the
investor who wishes to get his money before year-end. The accumulated
amount under continuous compound interest at time t is:
(1.5)
a(t) = (l + i)t=etln(1+0
A(t) = A(0)(l + i)'
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 1 - Interest Rates and the Time Value of Money
Page Ml- 5
In this case, the graph of a(t) is a smooth, continuous function. For i = 0.06:
We will look at continuous interest in more detail later.
For simple interest the accumulation functions are
(1.6)
a(t) = (1 + it) A(t) = A(0) (1 + it)
For i = 0.06 simple interest per year paid at year end, the first four values of
a(n) are:
N
a(n)
0
1
1
1.06
2
1.12
3
1.18
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Ml-6
Module 1 - Interest Rates and the Time Value of Money
Section 1.4
Effective Rate of Interest for a Specified Period
We can use the accumulation functions to find an effective rate of interest for
any time period. For the time period [t,t +1], the beginning amount is A(t), and
the amount earned over the interval is A{t +1) - A(t). The effective rate of
interest over this period is
(1.7)
. amount earned _A(t + l)-A(t) a(t + l)-a(t)
beginning amount A(t) a(t)
Example (1.8)
Let the interest rate be 6% and the time interval be [1,2].
For compound interest
a(2)-q(l) .0636
12 = a(l) =T06"=°6
For simple interest
a(2)-q(l)sJ06_
a(l) 1.06
Exercise (1.9)
Let the interest rate be 6% and the time interval be [2,3]. Find i3 for a)
compound interest and b) simple interest.
Answer a) .06 b) .0536
Note that over multi-year periods compound interest gives a constant effective
rate of 6% over each year, while simple interest leads to declining effective
rates over time.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 1 - Interest Rates and the Time Value of Money
Page Ml- 7
Section 1.5
Nominal Rates of Interest
In many instances where payments are made for a period less than a year (e.g.,
monthly, quarterly, or semi-annually), the period interest rate is stated as a
nominal annual rate, which is the interest rate per period multiplied by the
number of periods per year.
For example, if you are to earn interest at 2% compounded quarterly, you could
multiply 2% by 4 and refer to a nominal rate of 8%, converted quarterly. This
gives a simple way of referring to the quarterly rate on an annual scale, but it is
not the rate you actually earn. In the example of a 2% rate compounded
quarterly, one actually earns more than 8%. One dollar actually accumulates to
(1.02)4 = 1.0824, so that the nominal rate of 8% actually leads to a true annual
earning rate of 8.24%. This true annual earning rate is referred to as the
effective rate.
Many students find this confusing, so we will go over again for reinforcement:
1. The given rate is your starting point
Example: 2% per quarter.
2. Calculate the annual nominal rate.
Nominal Rate (Rate/period)(Number of periods per year)
Example 2%x 4 = 8%
3. The effective rate is what you really earn with compound interest
Example. Compound accumulation is (1.02)4 = 1.0824
Effective rate is 8.24%.
The nominal rate is an artificial rate that gives you a way of
talking about a periodic rate (such as a quarterly or monthly) in
familiar annual terms. The effective rate is not artificial. It tells
you what you really earn with compounding in a year.
Exercise (1.10)
Suppose you are earning 1% interest compounded monthly.
a) What is your annual nominal rate?
b) What is your annual effective rate?
__ a) Nominal 12% b) Effective 12.6825%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Ml-8
Module 1 - Interest Rates and the Time Value of Money
In the general case for which there are m payment periods per year, we denote
i(m)
the nominal rate by i(m). The periodic interest rate is , and the effective rate
m
is:
(1.11)
This has the important consequence that
(1.12)
You will often see the terminology the interest is credited or convertible m
times per year.
Example (1.13)
Suppose interest is credited monthly and the nominal rate is
i(12) = 0.09 . Then, the effective rate is:
fl + M^I -1 = 1.007512 -1 = 0.0938.
I 12
This process can easily be reversed to find the nominal rate given the effective
rate
Example (1.14)
Interest is paid semi-annually and results in an effective rate of
10.25%. Find the nominal annual rate.
Solution:
m = 2, so we need to find i(2).
By (1.11),
( jW\2
1 + —
v 2y
( 7(2) A
1 + —
v 2y
:-2,
-1 = 0.1025
1.1025
= Vl.l025=1.05
Thus, i(2)= 10%, and the periodic interest rate is 5% per semi-annual
period.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 1 - Interest Rates and the Time Value of Money
Page Ml- 9
Note that you can derive a formula that solves for i(m) given iand m. It is
(l + i)m-l
i{m)=m
We did not use this formula above. Formula (1.11) is intuitive and easy to
remember, and we can always substitute given values into it to solve for i(m)
given iand m. This approach is what we used in Example (1.14), and leaves us
with one less formula to memorize.
The BA II Plus calculator has an interest conversion worksheet that can be
used to solve these problems. The legend above the § key is ICONV, which
stands for interest conversion. You can get into the worksheet using the
keystrokes. [2ND| ICONV. The worksheet has three variables:
NOM for nominal rate
EFF for effective rate
C/Y for number of conversion periods per year.
You can scroll between these variables using the t and I keys.
In (1.13) we found the effective rate corresponding to a nominal rate of 9%
credited monthly. To do this on the BA II Plus calculator, enter the ICONV
worksheet and scroll to the line for NOM. Key in 9 and hit the [Enter] key.
Then scroll t to the line for C/Y and key in 12 and hit the [Enterl key. Then
scroll t to the line for EFF and use the ICPTJ key to compute the effective
rate. The rate displayed is EFF = 9.38 (to two decimal places).
In (1.14) we found the nominal rate corresponding to an effective rate of
10.25% convertible semiannually. To do this on the BA II Plus calculator,
enter the ICONV worksheet and scroll to the line for EFF. Key in 10.25 and hit
the [Enterl key. Then scroll I to the line for C/Y and key in 2 and hit the [Enterl
key. Then scroll I to the line for NOM and use the JCPTJ key to compute the
effective rate. The rate displayed is NOM = 10.00.
To exit the ICONV worksheet, press the |C/E| key. This will also allow you to
exit any other BAH Plus worksheet.
Exercise (1.15)
a) Given i(12) = 6%, find the effective rate i.
b) Given an effective rate of i = 5%, find i(12).
Answers:
a) 6.168%
b) 4.889%
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page Ml-10
Module 1 - Interest Rates and the Time Value of Money
Section 1.6
Interest Rate v. Discount Rate
Investments can be structured in many ways. Consider an investor who would
like to earn 6% for one year. Two common approaches are:
a) Invest a given sum at the beginning of the year. If you invest $1,000
at the beginning of the year at 6% per year, you would require a
payment of $1,060 at year end.
b) Require a given sum at the end of the year, but take a discount on the
amount invested. Suppose that you require $1,000 at year end. The
present value of $1,000 at 6% is ^1,00° = $943.40.
You would invest $943.40 and be repaid $1,000. The difference of
$56.60 is referred to as a discount. This is really only a present value
problem, but the discount is quoted instead of the present value.
United States Treasury bills are quoted on a discount basis.
The rate of discount, d, is used extensively in interest theory and actuarial
mathematics. We can easily derive an expression for d in terms of i. If you
wish to obtain a future value of 1, the present value to invest is:
1
PV =
(1 + 0
Thus, the discount d is:
(1.16)
d = l
1
(1 + 0 (1 + 0
This yields the key relationship:
(1.17)
d =
(1 + 0
Example (1.18)
For i = 0.06, d = — = 0.0566
1.06
Exercise (1.19)
Given i = .10, find d.
Answer : 0.0909
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 1 - Interest Rates and the Time Value of Money
Page Ml-11
Section 1.7
Essential Interest Theory Notation
A critical notation for actuarial interest problems is
(1.20)
v = -
1 + i
From the definition of d in (1.17), we see:
(1.21) 1 d = iv
Note also that l-v=l--—- = _-L- = d. Thus
1 + i 1+i
(1.22)
d = l-v and l-d = v
The difference i - d simplifies nicely:
i-d=i-
(1 + 0 (1 + 0
= id
(1.23)
i-d = id
The preceding relationships are often used in actuarial examination solutions.
Example (1.24)
Given d = 0.07, find v and i.
Solution.
v = l-d = 0.93. Then 1 + i = - = 1.0753 . It follows that i = 0.0753.
v
Exercise (1.25)
Given d = 0.05, find v and i.
Answer v = 0.95, i = 0.0526:
Note that we can now write
FV
PV= =vnFV
(1 + 0"
The use of the v notation is common in actuarial texts and essential for the
actuarial exams. Many other financial professionals do not use it.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Ml-12
Module 1 - Interest Rates and the Time Value of Money
Section 1.8
Nominal Rates of Discount
You can also quote a discount rate per period as a nominal annual rate. If you
were using a discount rate of 2% per quarter, you could refer to this as a
nominal discount rate of 8% convertible quarterly. The effective annual rate of
discount would not be 8%, as we shall see below.
The nominal discount rate convertible m-thly is denoted by d(m). For example, a
nominal discount rate convertible quarterly would be denoted d(4). It is related
to the effective annual discount rate d by the equation
(1.26)
l-d =
1-
d(m)
\
m
y
v for a
mth-ly period
raised to mth power
This equation can be remembered by noting that the left side represents v and
the right side represents the v for an m-thly period raised to the m-th power.
Example (1.27)
Find the effective annual discount rate for a nominal rate of 8% \
convertible quarterly.
Solution.
»--(,J!rJ
= (0.98)4
= 0.9224
d = 0.0776
Example (1.28)
Find the nominal discount rate convertible semiannually
corresponding to an annual effective discount rate of 6%.
Solution.
(
1-0.06 = 0.94 =
1-
V094= 0.9695 =
d(2)= 0.0609
1-
2 J
d<2>A
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 1 - Interest Rates and the Time Value of Money
Page Ml- 13
If you use the ICONV worksheet with either EFF or NOM entered as a
negative number, the BA II Plus will interpret the negative number as a
discount rate and solve for the other discount rate as a negative.
For example, in (1.27) we found the effective rate discount rate
corresponding to a nominal discount rate of 8% credited quarterly. To do this
on the BA II Plus calculator, enter the ICONV worksheet and scroll to the line
for NOM. Key in 8 0] and hit the [Enter! key. Then scroll t to the line for C/Y
and key in 4 and hit the [Enter] key. Then scroll t to the line for EFF and use
the |CPT| key to compute the effective rate. The rate displayed is EFF = -7.76
(to two decimal places). That is the correct discount rate for (1.27).
You can clear out the computed values in the worksheet by keying in 2ND
CLR WORK (above the CE/C key.) The value of C/Y will remain but will be
changed as soon as you enter a new value for it.
Exercise (1.29)
Find
a) The effective discount rate for a nominal rate of 7.5%
convertible every 4 months (ra=3), and
b) the nominal discount rate convertible monthly corresponding to
an annual effective discount rate of 6%.
Answer a) 7.31% b) 6.17%
Occasional problems require conversion of a nominal interest rate convertible
m times per year to an equivalent nominal discount rate convertible p times per
year. The equation for this problem is
1 + -
;(m)
m
(1-^'p
-=l+i
In this equation the left hand side represents 1 + i and the right hand side
represents — = 1 + i,
v
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Ml-14
Module 1 - Interest Rates and the Time Value of Money
Example (1.30)
Find the rate of discount convertible semi-annually that is
equivalent to a nominal rate of interest of 8% convertible monthly.
Solution.
1 +
0.08
12
= 1.083= 1-
VT083 =1.0407= 1-
2
sn1
V2
d(2) =0.0782
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 1 - Interest Rates and the Time Value of Money
Page Ml- 15
Section 1.9
Continuous Interest and Force of Interest
We have already noted that interest may also be paid on a continuous basis. The
accumulated amount under continuous compound interest at time t is:
(1.3D
a(t) = (l + i)t=etln(1+0
A(t) = A(0)(l + i)
For example, if interest is paid continuously at 6%, we have
a(t) = (1.06)'=etln(106)
There is an important distinction here. Under continuous interest at 6% at time
2, a(2) = 1.062, which is the same amount you would have under annual interest.
However, there is difference at fractional times like t = 1.5. For continuous
interest, you would have 1.0615 at t = 1.5, but for annual interest, you still only
have 1.06 at t = 1.5, because interest is not compounded until t = 2.
The accumulation function above implies continuous growth at a continuously
compounded rate of ln(1.06) = 0.0583. The constant continuous growth rate is
denoted by S. In general, for the compound growth model a (t) = (1 + i)f, we can
write
(1.32)
a(t) = e5t
S = ln(l + i)
The next two formulas are very important variations of (1.32)
(1.33)
(l + i)n=en
(1.34)
vn=(l + i)-n=e-
In actuarial textbooks, S is referred to as a constant force of interest.
Continuous interest has been applied in practice - some banks paid continuous
interest in the 1980's.
We can define the instantaneous force of interest for any accumulation function
a(t) at time t by
(1.35)
S(t) =
am
a(t)
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Ml-16
Module 1 - Interest Rates and the Time Value of Money
This has a natural interpretation. The percentage rate at which you are earning
at time t is the rate of change of a(t) expressed as a percentage of a(t).
For the constant force model, a(t) = e5^, this definition yields:
fipst
Thus definition (1.35) is consistent with our original reference to 8 as the
constant force of interest.
The force of interest does not have to be constant as the following example
shows:
Example (1.36)
Let a(t) = (t + l)2.
(Note: This is an unrealistic accumulation function, but easy to compute.)
/x 2(t + l) 2
Then S(t)=-± t = -±t-
w (t + 1)2 t + 1
Note that as n, the number of compounding periods per year becomes large, the
resulting interest paid approximates continuous interest with a constant force
of interest equal to the nominal rate.
For example, if we have a constant force of interest, S = 8%, we have an
effective annual rate of (e°08 -1) = 8.32871%. The table below shows the
effective rate for a nominal rate of 8% with increasing numbers of
compounding periods per year.
Nominal Rate: 8%
Conversion
Semi-annual
Quarterly
Monthly
Daily
Hourly
Per Minute
n
2
4
12
365
8,760
525,600
Effective Rate
8.16000%
8.24322%
8.29995%
8.32776%
8.32867%
8.32871%
Note the result of compounding every minute matches the continuous effective
rate to five decimal places.
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 1 - Interest Rates and the Time Value of Money
Page Ml- 17
This approximation is based on the useful identity (1.37) below, which you
should memorize. The next few lines give a derivation which you may skip, if
you like.
(1.37)
lim|l + -N
There is another useful relationship which enables us to find a(t) if only S(t) is
given. Note that A lnteO] = ^v = *(*)•
dt a(t) v '
Thus, £ S(t)dt = ln[a(t)f0 = ln[a(fc)] - ln(l) = ln[a(fc)]
This implies that:
(1.38)
ik = a(t)
Example (1.39)
Given 8{t) =
Solution.
To find ait),
2
Finri n(f)
(t+i) v*"
we first need to integrate S(t):
(s(u)du= f 2 du = 21n(u + l)r=21n(t
Jo *(u + l) lo
Thus,
a(t) =
Note:
e21n(t+l)=(eln(t+l))2=(t + 1)2
\ If your calculus is rusty, you may need to review
+ D.
to do these
problems. \
Exercise (1.40)
Given fi(f} - FinH n(f}
1 v" (2t + l)" v*"
Answer:
3ln(2t + l
e
= (2t + l)'
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Ml-18
Module 1 - Interest Rates and the Time Value of Money
Section 1.10
Relating Discount, Force of Interest and Interest Rate
A very important relationship is:
(1.41)
d<d{m) <5<i{m) <i, i>0,m>l
This relationship is good to know for plausibility checking of results. It is not
hard to see that d<8<i, since for i > 0
7 < ln(l + i)<i
1 + i
For a concrete example, let i = 0.05 and m = 4. Then
S = 1n (1.05) = 0.0488 d = — = 0.0476
v } 1.05
i(4) = 0.049089 d(4) = 0.048494
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 1 - Interest Rates and the Time Value of Money Page Ml-19
Section 1.11
Solving for PV, FV, n, and / with Compound Interest
Time value of money calculations can be done easily with the BAII Plus. This
section will show you how to solve for PV, FV, n, and i,. First, we will solve the
problems without the calculator to establish the logic, and then show you how to
use the calculator's time value of money [TVM] keys to save time.
Example (1.42)
You want to have 80,000 in a college fund in 18 years,
should you deposit now into an account earning 6%?
Solution.
You need to have FV = 80,000. Thus
PV = 80,000 v18 = -^^ = 28,027.50
1.0618
For the BA II Plus the keystrokes are
80000 [FV| 6 |l/Yl 18 M£PT| |PV|
How much
The equation used in the last example is called an equation of value. It equates
the unknown PV with a mathematical expression for it. In each of the next
problems we will refer to the appropriate equation of value in order to solve.
Exercise (1.43)
How much should you deposit in the fund described in (1.42) if you
wanted 100,000 in 16 years?
Answer 39,364.63
Example (1.44)
You deposit 1,000 in an account earning 5.75%.
have in 5 years?
Solution.
The equation of value is
FV = 1000(1.0575)5 =
For the BA II Plus the keystrokes are
1 1000 1+H |PV| 5.75 |l/Yl 5 MlCPTl |FV|
= 1,322.52
How much will
you
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Ml-20
Module 1 - Interest Rates and the Time Value of Money
Exercise (1.45)
How much will be in the account in the last problem in 10 years?
Answer 1,749.06
Example (1.46)
You deposit 1000 in an account earning 6% compounded continuously.
How long will it take to double your money?
Solution.
Doubling your money gives FV
2000 = lOOOe06t
It follows that
2 = e06t
ln(2) = 0.06t
t = 11.5525
= 2000. The equation of value is
Exercise (1.47)
For the account in Example (1.46) how long would it take to triple your
money?
Answer 18.3102
Now, let's look at a variation on the preceding example that requires careful
thinking:
Example (1.48)
You deposit 1000 in an account earning 6% compounded annually.
How long will it take to have at least 2000 dollars?
Solution.
In this case interest is only paid at year end. Since 2000 would be
reached exactly with continuous interest in 11.8957 years, you will
have less than 2000 at the end of 11 years and more at the end of the
12th year. The answer here is 12 years.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 1 - Interest Rates and the Time Value of Money Page Ml- 21
Example (1.49)
1 You make
in 5 years.
Solution.
an investment where you pay 1000
What interest rate did you earn?
The equation of value is
Thus
1000 (1 + i)5 =1500
(1 + i)5 =1.5
51n(l + i) = ln(1.5)
ln(l + i) = .0811
1 + i = e0811 =1.0845 -► i = .0845
For the BA II Plus the keystrokes are
loooyj lEYl 1500 |fv|5[n] icpti
1 The calculator saves a bit of time here.
now and get 1500 back 1
ll/Yl
Exercise (1.50)
You make an investment where you pay 1000 now and get 2000 back in
12 years. What interest rate did you earn?
Answer 5.9463
The problems can be made more complex, as you will see when you move to the
exam problems at the end of this module. One way to make a problem a bit
more complex is to state it using a nominal interest rate.
Example (1.51)
You deposit 1,000 in an account earning 5.75% convertible
semiannually. How much will you have in 5 years?
Solution.
Now we have a interest rate of — = 0.02875 per semiannual
2
period for 2x5 = 10 periods. The equation of value is
FV = 1000 (1.02875)10 = 1327.70
The BA II Plus keystrokes are
1000 |+i] |>V| 2.875 g/Y| 10 §^Pg |fv|
The BA II Plus also has an option under which you can set the number
of payments per year to 2 using the P/Y option. We advise against this
since it is more complicated to use and it is easy to forget to reset the
number of payments per year which can cause trouble on later
problems.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Ml-22 Module 1 - Interest Rates and the Time Value of Money
Example (1.52)
You make an investment where you pay 1000 now and get 1500 back
in 5 years. What nominal interest convertible quarterly did you earn?
Solution.
Here we will give only the calculator solution. The interest is paid
over 20 quarters.
To find the quarterly interest rate, the BAII Plus the keystrokes are
10001+0 gvj 1500 |FV| 20 |n] [CPTJ |/Y|
The quarterly rate is 2.048%. The required nominal rate is
4 x 2.048% = 8.192% convertible quarterly.
Other problems may have a few different future amounts or an unknown
amount at some point. We see this in the next two examples.
Example (1.53)
How much should you deposit now in a bank account earning 5%
annually to be able to withdraw 1000 in 2 years and 2000 in 4 years?
Solution.
The equation of value is
PV = lOOOv2 + 2000v4 = -^- + -^ = 2552.43
1.052 1.054
Example (1.54)
You deposit 1000 in an account now and an amount X in one year. The
account pays 6% annually. What amount X is required to have 2000 in
the account at the end of two years?
Solution.
The equation of value is
1000(1.06)2+X(1.06) = 2000 -> X = 826.79
Another type of problem that requires more thought is one in which the interest
rates change over time.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 1 - Interest Rates and the Time Value of Money
Page Ml- 23
Example (1.55)
You deposit 5000 to an account that earns 5% compounded annually
for two years and 7% in all subsequent years. What has the account
grown to in 5 years?
Solution.
FV = 5000Q.05)2 (1.07)3 = 6753.05
You could easily do the problem on the BA II Plus in two steps.
Amount in 2 years 5000 @ [PVJ S^2§ |CPT| |FV|
(Answer 5512.50)
Amount in 5 years 5512.5 0 |PV| 7 |/Y| 3 InUcFII [fv|
(Answer 6753.05)
Example (1.56)
What constant rate of interest is equivalent to the 5 year return
above?
Solution.
The BA II Plus can be used to get this quickly. We accumulated FV =
6753.05 in 5 years from an initial investment of 5000. Solve for the
interest rate using
5000 0 |PV| 6753.05 |fv| 5 0 |CPg g/Y] (Answer 6.20%)
To solve mathematically, denote the unknown interest rate by i.
(l + i)5=1.052(l.073) = 1.3506
51n(l + i) = ln(1.3506)
ln(l + i) = .0601
l + i = e-0601= 1.062 -> i = .062
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page Ml-24
Module 1 - Interest Rates and the Time Value of Money
Example (1.57)
You deposit 5000 to an account that earns 5% compounded annually
2
for two years and continuous interest with S(t) = in
(t + 1)
subsequent years. What has the account grown to in 5 years?
Solution.
At the end of two years the account contains 5512.50. [See (1.55),
above]. Note that the force of interest must be applied from time 2
through time 5. The equation of value for the final amount in 5 years
is
5512.50
Note the limits on the integral. A common mistake is to integrate
from 0 to 3.
We now calculate
2
r^ = 21n(t + l)|25=2[ln(6)--ln(3)]
^T^ = e21n(6)-21n(3) = 4
The final answer is
5512.50
f f5 2
6
^\ = 5512.50 (4) = 22,050
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 1 - Interest Rates and the Time Value of Money
Page Ml- 25
Section 1.12
Formula Sheet
FV = PV(l + i)n
(1 + 0
i-d = id
PV =
FV
a+o"
v = -
1 + i
v = l-d
d = iv
d = l-v
a(t): the amount an initial investment of 1 grows to by time t
A(t): the amount an initial investment of A(0) grows to by time t
a(t) = (l + i)t=etln(1+i)
<5 = ln(l + i)
a'(t)
S(t)
a(t)
A{t) = A(0) (1 + i)' = A(0)etln(1+i)
a(t) = est vn=(X + i)'n=e'nS
e'>)du=a(t)
Effective interest rate with nominal rate i(m) convertible m-thly.
( \w\
1 + —
m
-1
Effective discount rate d with nominal rate d(m) convertible m-thly.
l-d
f sMY1
1--
m
Nominal rate equivalence
f jW
1 +
m
1-
d«
Note the negative exponent, -p, above.
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page Ml-26
Module 1 - Interest Rates and the Time Value of Money
Section 1.13
Basic Review Problems
1. Let the annual interest rate be 5% and the time interval be [3,4]. Find i4
for (a) annual compound interest and (b) simple interest.
2. (a) Given i(2) = 5%, find the effective rate i.
(b) Given an effective rate of i = 5.26%, find i(6).
3. Given d = 0.056, find v and i.
4. Given i(4) = 0.07. Find d(2).
5. Find the effective annual discount rate for a nominal discount rate of 9%
convertible monthly.
6. Find the rate of interest convertible quarterly that is equivalent to a
nominal rate of interest of 6% convertible semiannually.
7. Let a (t) = (t +1)3. Find S (t).
4
8. Given S(t) = . Find a(t).
it + 3)
9. You deposit 1800 in an account earning 5% compounded continuously.
How long will it take to accumulate 2,700?
10. You make an investment where you pay 10,500 now and get 12,500 back
in 3 years. What nominal interest convertible monthly did you earn?
11. You deposit 1,500 to an account that earns a nominal 6% convertible
monthly for one year and a nominal 8% convertible quarterly for the
next two years, a) How much is in the account in 3 years? b) Find an
equivalent level nominal rate convertible semiannually for this account.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 1 - Interest Rates and the Time Value of Money
Page Ml- 27
Section 1.14
Basic Review Problem Solutions
Calculator solutions will be given whenever possible.
1. (a) a(t) = 1.05*. U =
a (4)-a (3) l.Q54-1.053
= 0.05.
a (3) 1.053
Note that for compound interest the periodic rate is the always the
effective rate.
(b) a(t)-l + 0.0». U-a(4)7.?(3) = 120-.115 =0-0435
a (3)
1.15
2. Calculator solutions using the ICONV feature:
(a) Set NOM = 5 and C/Y = 2. CPT EFF = 5.0625
(b) Set EFF = 5.26 and C/Y = 6. CPT NOM = 5.1483
3. v = l-d = 0.944, - = l + i = 1.0593->i = 0.0593
v
4. 1 +
0.07^
= 1.07186 =
1-
d<2n
/
0.9659 =
1-
d<2>
d{2) = 0.0682
5. We are trying to solve
l-d =
1-
0.09
12
but we think it is easier to use the calculator's ICONV feature. Set
NOM = -9 and C/Y = 12. CPT EFF = -8.6379. Answer 8.6379%.
6. We want to solve
(i+0=
( i(4)V
4
1 +
0.06
You can do this by hand, but we think it is easier to use the calculator in
steps:
First find the annual effective rate using the given nominal semiannual
rate. Set NOM = 6, C/Y = 2 and CPT EFF = 6.09. Then use this effective
rate to find the quarterly nominal rate. You already have EFF = 6.09. Set
C/Y = 4 and CPT NOM = 5.9557. Answer 5.9557%.
7. 5(f)-
a\t) 3(t + l)2
ait) (t + iy t + 1
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page Ml-28
Module 1 - Interest Rates and the Time Value of Money
4 „ ., . „j' ., ft + 3}
8. [5{u)du=l—-—du = 41n(u + 3)f =
* * (u + 3) lo
4 In
a(t) = e41n((t+3)/3) =
rt+3
9. A(t) = 1800e05t. We need
2700 = 1800eost -> eost = 1.5 -> .05t = ln(1.5), thus
ln(1.5)
t = -
0.05
= 8.1093
10. First, we need to get the annual effective rate, then we can use this to
solve for the nominal rate. Formulaic version:
(i+0 =
i(m) \m
1 + -
m
Using the calculator: 10500 tt!d EY) 12500 [FVj3 N |CPT| H/Y|
Answer 5.984
Now use ICONV to get the nominal rate convertible monthly. Set EFF =
5.984 , C/Y = 12 and CPT NOM = 5.826.
U. (a)
(b)
1500
1 +
0.06 ^2'
12
1 +
0.08
\«(2)
= 1,865.89
We are trying to find i(2).
1 +
0.06
1 +
0.08
\0(2)
12 ,
1.24391/6= 1.037
i(2) = 0.074
l+l—
2
= 1.2439 =
(2) A
1+^
2
\6
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 1 - Interest Rates and the Time Value of Money
Page Ml- 29
Section 1.15
Sample Exam Problems
1. (2005 Exam FM Sample Questions #1)
Bruce deposits 100 into a bank account. His account is credited interest at a
nominal rate of interest of 4% convertible semiannually.
At the same time, Peter deposits 100 into a separate account. Peter's account is
credited interest at a force of interest of 5.
After 7.25 years, the value of each account is the same. Calculate 8.
(A) 0.0388 (B) 0.0392 (C) 0.0396 (D) 0.0404 (E) 0.0414
2. (2005 Exam FM Sample Questions #3)
Eric deposits 100 into a savings account at time 0, which pays interest at a
nominal rate of i, compounded semiannually.
Mike deposits 200 into a different savings account at time 0, which pays simple
interest at an annual rate of i.
Eric and Mike earn the same amount of interest during the last 6 months of the
8th year. Calculate i.
(A) 9.06% (B) 9.26% (C) 9.46% (D) 9.66% (E) 9.86%
3. (2005 Exam FM Sample Questions #12)
Jeff deposits 10 into a fund today and 20 fifteen years later. Interest is credited
at a nominal discount rate of d compounded quarterly for the first 10 years, and
at a nominal interest rate of 6% compounded semiannually thereafter. The
accumulated balance in the fund at the end of 30 years is 100.
Calculate d.
(A) 4.33% (B) 4.43% (C) 4.53% (D) 4.63% (E) 4.73%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page Ml-30
Module 1 - Interest Rates and the Time Value of Money
4. (2005 Exam FM Sample Questions #13)
Ernie makes deposits of 100 at time 0, and X at time 3. The fund grows at a
force of interest
The amount of interest earned from time 3 to time 6 is also X. Calculate X.
(A) 385 (B) 485 (C) 585 (D) 685 (E) 785
5. (2005 Exam FM Sample Questions #20)
David can receive one of the following two payment streams:
(i) 100 at time 0, 200 at time n, and 300 at time In
(ii) 600 at time 10
At an annual effective interest rate of i, the present values of the two streams
are equal. Given vn = 076, determine i.
(A) 3.5% (B) 4.0% (C) 4.5% (D) 5.0% (E) 5.5%
6. (2005 Exam FM Sample Questions #27)
Bruce and Robbie each open up new bank accounts at time 0. Bruce deposits
100 into his bank account, and Robbie deposits 50 into his. Each account earns
the same annual effective interest rate.
The amount of interest earned in Bruce's account during the 11th year is equal
to X. The amount of interest earned in Robbie's account during the 17th year is
also equal to X. Calculate X.
(A) 28.0 (B) 31.3 (C) 34.6 (D) 36.7 (E) 38.9
7. (May 05, #13)
At a nominal interest rate of i convertible semi-annually, an investment of 1000
immediately and 1500 at the end of the first year will accumulate to 2600 at the
end of the second year. Calculate i.
(A) 2.75% (B) 2.77% (C) 2.79% (D) 2.81% (E) 2.83%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 1 - Interest Rates and the Time Value of Money
Page Ml- 31
8. (May OS, #18)
A store is running a promotion during which customers have two options for
payment.
Option one is to pay 90% of the purchase price two months after the date of
sale.
Option two is to deduct X% off the purchase price and pay cash on the date
of sale.
A customer wishes to determine X such that he is indifferent between the two
options when valuing them using an effective annual interest rate of 8%.
Which of the following equations of value would the customer need to solve?
UOOA 6 ) { 100A 6 J
C)f^1(1.08)"'-.90 D)pUr^Wo
E) (l-^)(1.08)1/6=.90
9. (May 05, #19)
Calculate the nominal rate of discount convertible monthly that is equivalent to
a nominal rate of interest of 18.9% per year convertible monthly.
(A) 18.0% (B) 18.3% (C) 18.6% (D) 18.9% (E) 19.2%
10. (Nov 05, #7)
A bank offers the following choices for certificates of deposit:
Term (in years)
1
3
C/l
Nominal annual interest rate convertible quarterly
4.00%
5.00%
5.65%
The certificates mature at the end of the term. The bank does NOT permit early
withdrawals. During the next 6 years the bank will continue to offer
certificates of deposit with the same terms and interest rates.
An investor initially deposits 10,000 in the bank and withdraws both principal
and interest at the end of 6 years. Calculate the maximum annual effective rate
of interest the investor can earn over the 6-year period.
(A) 5.09% (B) 5.22% (C) 5.35% (D) 5.48% (E) 5.61%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page Ml-32
Module 1 - Interest Rates and the Time Value of Money
11. (Nov OS, #25)
The parents of three children, ages 1, 3, and 6, wish to set up a trust fund that
will pay X to each child upon attainment of age 18, and Y to each child upon
attainment of age 21.
They will establish the trust fund with a single investment of Z.
Which of the following is the correct equation of value for Z ?
(A) "17—^ IT+ 20 Ts Is" (B)3[Xv18+Yv21l
v17+v15+v v +v +v L J
(C) 3Xv3+Y[v20+v18+vls] (D) (X + Y)
(E) X[v17+v15+v12] + Y[v20+v18+vls]
v20+v18+vls
V3
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 1 - Interest Rates and the Time Value of Money
Page Ml- 33
Section 1.16
Sample Exam Solutions
We look at the future value in 7.25 years for each person.
Bruce. He is credited interest for 29 quarters. We are given that his interest
rate per semiannual period is 2%. Thus his interest rate per quarter is
VL02 -1, and his future value is FV = (VL02)29100 = 133.26
Peter. He earns continuous interest at a rate of 5 for 7.25 years. His future
value is FV = 100e725s.
To finish the problem we equate the two future values and solve.
133.26 = 100e7 25'
1.3326 = e725s
ln(1.3326) = 7.25£
._ In (1.3326) _
7^25
S = -
= .0396
Answer C
2.
The last 6 months of the eighth year are in the time interval from time 7.5 to
time 8. For each of the two savers we will find the find the half year interest on
that interval.
Eric. At time 7.5 he has a balance of 100 1 +
\15
His interest on this balance
over the next half year is 100
( 7 ^5
1 + -
2y
Mike . Since Mike only earns simple interest on the original amount, his
interest earned in any half year is 200 - .
Since these interest amounts are equal
100
1 + -
v
15m
= 200
as
1 + -
v
= 2
i
= .0473
i = .0946.
Answer C
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page Ml-34
Module 1 - Interest Rates and the Time Value of Money
First we will deal with the initial 10 years during which a discount rate d was
quoted. For each of these 10 years the relevant values of v and i are
v=l-d=
(1_d^_
and l + i = — =
v
1-
d<4n
Thus after 10 years the initial deposit of 10 grows to
10(l + i)10=10
V^°
The accumulated balance on this 10 after 20 more years (or 40 semiannual
periods) at 3% per semiannual period is
10
1-
d(4>
s-*0
(1.03) = 32.62
(. d(4)
v-M
1-
4
The second deposit of 20 accumulates after 30 semiannual periods at a rate of
3% to a value of
20(1.03) =48.55
The total accumulated balance is
d<4>
/
100 = 32.62
1--
Thus
v " 4
= 1.577-
+ 48.55
v " 4y
= .634-
1-
d«'
= .98867^ d(4)=.0453
Answer C
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 1 - Interest Rates and the Time Value of Money Page Ml- 35
4.
The amount of interest earned from time 3 to time 6 is the difference between
the ending amount at time 6 and the starting amount in the account at time 3.
We will start by looking at the original deposit of 100. At time 3 it has grown to
j.3 i>3 t2 d 27
100eJo ' x = 100eJo 10° = lOOe300 «109.42.
At time 3 deposit of X is made, so that the beginning amount at time 3 is
A (3) = 109.42 + X
At time 6 the account grows to
A(6) = (109.42 + X)e]*Stdt =(109.42 + X)eh™> * =.
(109.42 + X)1.8776 = 205.45 + 1.8776X
The interest earned between time 3 and time 6 is
A(6) - A(3) = 96.03 + .8776X.
This interest must equal X, so that
X = 96.03 + .8776X -> X = 784.56
Answer E
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page Ml-36 Module 1 - Interest Rates and the Time Value of Money
S.
Present value of stream (i):
100 + 200vn + 300v2n = 100 + 200 (76) + 300 (762) = 425.28
Present value of stream (i):
600v10
Since the present values are equal
600v10 =425.28 — v = .9662 -> 1 + i = 1.035
Answer A
6.
The interest earned during a year equals
(Balance at the start of the year)x(Interest Rate)
Let i denote unknown interest rate.
For Bruce the interest during year 11 is X = il00(l + i)
For Robbie the interest during year 17 is X = i50(l + i)
It follows that
i50(l + i)16=il00(l + i)10 -» 50 (1 + i)16 =100 (1 + i)10
(1 + i)6 =2 -» (l + i) = 1.12246
X = il00(l + i)10 = .12246 (100) (1.12246)10 = 38.88
Answer E
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 1 - Interest Rates and the Time Value of Money
Page Ml- 37
The equation of value is
1000 1 +
i(2)l
;(202
+ 1500 1 + —
= 2600
This is a problem that can be reduced to a quadratic -a common exam trick. Set
L i(2)
H1+T
Then the above equation becomes
lOOOx2 + 1500x = 2600 or x2 + 1.5x - 2.6 = 0
The positive root of the quadratic (quadratic formula) is x = 1.0283 = 1 + i. Thus
(2)
\2
1 + —
= 1.0283
= 1.0141 ->i(2) =.0281
Answer D
8.
The customer has two options. Let P be the purchase price.
Pay cash on the date of the sale with X% taken off the price. The amount paid is
r X \
1 immediately.
100 J
Pay 90% of the purchase price in two months. The amount paid in two months
(1/6 of a year) is .90P . The present value on the date of sale is
.90P
1.08
1/6 *
The equation of value is P
X
.90P
100 J 1.081'6
This is equivalent to
^iw
.90.
Answer E
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Ml-38
Module 1 - Interest Rates and the Time Value of Money
1 +
Answer C
0.189
12 j
r.i.
2063= 1-
12
-> 0.9845 = 1-
d<12>
12
d(12)= 0.186
10.
Since the one year rate of 4% is the lowest, we can immediately eliminate the
possibility of investing in six consecutive one year CDs or three consecutive
one year CDs coupled with a three year CD. The two possible choices for
maximum yield are A) two consecutive 3 year CDs or B) a 5 year CD coupled
with a one year CD. Note that our CDs will earn at quarterly rates.
Term
1
3
C/l
Nominal Annual Rate
4.00%
5.00%
5.65%
Quarterly Rate
1.0000%
1.1250%
1.4125%
The total accumulation factors under the two options are:
Option A) Two consecutive 3 year CDs for n=12 quarters each.
(1.0125)24 = 1.34735
Option B) A 5 year CD coupled with a one year CD.
(1.014125)20 (1.01)4 = 1.377575
Option B) is better. It gives a total accumulation of 1.377575 over 24 quarters
(which is six years). Therefore, the maximum annual effective rate an investor
can earn over the six-year period is
1.3775751/6-l = .0548
Answer D
11.
Below we tabulate the years remaining to ages 18 and 21 for each child.
Age now
Years to age 18 (X)
Years to age 21 (Y)
17
20
15
18
12
15
The present value of the age 18 payments is X(v17 + v15 + v12)
The present value of the age 21 payments is Y(v20 + v18 + v15)
The present value of the total fund required is
Z = X(v17 + v15 + v12) + Y(v20 + v18 + v15)
Answer E
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 1 - Interest Rates and the Time Value of Money Page Ml- 39
Section 1.17
Supplemental Exercises
1. Given d(4) = 0.05, find i(6), v and 8.
2. Money accumulates at a simple interest rate of 6.5% per year. For the
interval [4, 5], find i5.
3. If a(t) = (2t + l)4, find 8(t).
4. A deposit is made into a fund. For the first 5 years interest is credited at
an annual nominal rate of 6% convertible quarterly. For the next 5 years
interest is credited at an annual discount rate of 7% convertible
semiannually. What is the equivalent constant force of interest for 10
year period?
5. A man deposits 500 into an account. At the end of 5 years the account has
grown to 650. Find the annual nominal rate of interest convertible
quarterly for this account.
6. Tom and Jerry deposit money into accounts at the same time. Tom's
account earns at an annual effective rate of r. Jerry's account earns at a
simple rate of r. For year 8, Tom's effective rate of interest is 1.5 times
Jerry's effective rate for year 8. Find r.
7. An amount X is deposited into an account that pays 8% simple interest.
At the same time — is deposited into an account that accumulates at a
constant force of interest 8. The total interest earned in each account
after 10 years is the same. Find 8.
8. A bank pays an annual effective interest rate of i. A man deposits 1000
today and 1500 in one year. At the end of two years his account is at 2800.
Find i.
9. A woman makes deposits into an account of 100 today and 300 12 years
later. For the first 12 years interest is credited at an annual nominal rate
of 6% convertible quarterly. For the next 8 years the account earns at a
force of interest of 8. At the end of 20 years the accumulated amount is
802. Find 8.
10. Elmer deposits 1000 into a bank account. The bank credits interest at an
annual nominal rate of i convertible quarterly for the first 8 years and an
annual nominal rate of 1.5i convertible bimonthly thereafter. The amount
in his account at the then end of 5 years is 1516. What is the amount in
his account at the end of 10 years?
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Ml-40
Module 1 - Interest Rates and the Time Value of Money
Section 1.18
Supplemental Exercise Solutions
1. We have the following equivalences:
(1 - d»V4)-* = (1 + i(6)/6)6 = 1 + i = v"1 = es
The common value is (1 - 0.05/4)"4 = 1.05160.
i<« = 6(1.05161/6 - 1) = 0.0505.
v = 1/1.0516 = 0.9509
£=ln(1.0516) = 0.0503
2. is = [a(5) - a(4)]/a(4) aft) = 1 + 0.065t a(4) = 1.260 a(5) = 1.325.
is = (1.325 - 1.26)/1.26 = 0.0516
3. S(t) = d(t)la{t) = 4(2r + l)3(2)/(2t + l)4 = 8/(2t + 1)
4. The accumulation factor a(10) for the 10-year period is
(1.015)20(1 - 0.035)10 = 1.9233 = ewS
5 = (l/10)ln(1.9233) = 0.0654
5. 500(1 + i(4)/4)20 =650
i(4> = 4[(650/500)1'20 - 1] = 0.0528
6. Tom's effective rate of interest for year 8 is r.
Jerry's effective rate of interest for year 8 is
(1 + 8r -1 - 7r)/(l + 7r) = r/(l + 7r).
Therefore r = 1.5r/(l + 7r) => 1 + Ir = 1.5 => r = 0.5/7 = 0.0714
7. The total interest earned on the first account is (0.08)(10)X = 0.8X.The total
interest earned on the second account is (X/2)(ewS-1)
0.8X = (X/2)(e10*-1) => 1.6 = eloS-1
10«5=ln(2.6) => S =0.0956
8. The accumulated amount in the account is
1000(1 + i)2 + 1500(1 + i) = 2800. Let x = 1 + i.
This yields the quadratic equation 10x2 + 15jc - 28 = 0.
The positive root of the equation is x = 1 + i = 1.084. => i = 0.084
9. The accumulated amount is [100Q.015)48 + 300]e8,s = 802.
Thus 504.35e8,s= 802 => 8£=ln(1.59) => £=0.058
10. At the end of 5 years the accumulation is 1000(1 + i/4)20 = 1516.
At the end of 10 years the accumulation is
1000(1 + i/4)32(l + 1.5i/6)12 = 1000(1 + i/4)44 (i/4 = l.Si/6)
(1 + i/4)44 = 1.516*"20 = 2.49769
Accumulation is 2497.69
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2-Annuities
Annuities
Section 2.1
Introduction
Many financial obligations require a regular series of periodic payments.
Mortgage and car payments are made at the end of every month. My pension
plan pays me a set amount at the start of every month, and deducts another set
amount for health insurance. Series of regular payments such as these are
called annuities. Annuities are so widely used that calculators for business
professionals are programmed to do annuity calculations. A deeper
understanding requires knowledge of the mathematics behind the calculator
automation.
A unit annuity is one for which each regular payment is 1. As we saw above,
annuity payments can be made at the beginning or the end of the time period.
An annuity is immediate if payments are made at the end of the period, and due
if the payments are made at the beginning. Below are diagrams for unit
annuities with four payments.
Annuity Immediate
payments 1111
I 1 1 1 1 —
time t 0 1 2 3 4
/
First payment /
made at the end of
the first year (t=l)
Annuity Due
i i i
12 3 4
PageM2- 1
payments 1
time t *0
First payment made at the
beginning of the first year
(t=0)
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-2
Module 2 - Annuities
The preceding diagrams are called timelines. You will find them to be a very
important tool in solving many future problems.
Geometric Series
To find the present value or future value of an annuity, we will need to use the
formula for the sum of a geometric series. Geometric series are very important
for Exam FM.
(2.1)
1 _ rn+1
1 + r + r +... + r = ,r *1
1-r
Note that if \r\<l, the infinite geometric series converges:
(2.2)
1 + r + r +...:
1-r
©ACTEX 2009
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Exam FM / Exam 2 - Financial Mathematics

Module 2-Annuities
PageM2- 3
Section 2.2
Annuity Immediate Calculations
The present value of an immediate annuity with n payments of 1 and interest
rate i is denoted by a^i9 or a^ if the value of the interest rate is clear and does
not need to be specified . The basic formula for a^ t is so important that we will
derive it here:
The present value of the unit annuity immediate is the sum of the individual
present values of the payments of 1.
payments
time t
0
At t=0, the first payment is worth v.
At t=0, the second payment is worth v
At t=0, the n-th payment is worth v
Present value =a^
= v + v +... + v
= v(l + v + ... + vn_1)
(l-vn) (l-vn)
= V- - = V-
1-v d
(1-y-)
= v
IV
l-vn
i
Thus, we obtain the important formula:
(2.3)
l-vn
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M2-4
Module 2 - Annuities
Example
1 Ifi
(2.4)
= 0.05 and n =
i-m10
_ u.05j
0.05
10,
= 7.7217
Not surprisingly, a financial calculator can be used for the above problem.
The PMT key is used for the periodic payment of 1. On the BAII Plus
Professional, the following entries give the result PV = - 7.7217:
0
1
5
10
|CPT}
FV|
pmt|
vy\
n|
pv)
Note the sign convention. Positive amounts represent money paid to you, and
negative amounts represent cash that you must pay out. If the applicable
interest rate is 5%, you would need to pay out - 7.7217 to receive ten
subsequent payments of+1.
On exams most students use the calculator instead of the formula whenever
possible to save time. You must still know the formula, since some questions
are designed so that the calculator cannot be used directly and formula
knowledge is required for solution.
Exercise (2.5)
Find the a2oi.05 using the formula and then check it using the calculator.
Answer -12.4622
The future value of the unit annuity immediate with n payments is denoted by
s^. It is the sum of the future values of the individual payments of 1.
Sfl = (1 + i)n_1 + (1 + i)n~2 +.... + (1 + i) +1
Note that since the immediate annuity has year end payments the first payment
earns interest for only n-1 periods and the last payment of 1 earns no interest.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 2-Annuities
PageM2- 5
payments
time t °
2 • • • n
At t=ny the n-th payment is worth 1.
At t=n, the second payment is worth (l + i)
At t=n, the first payment is worth (l + i)" .
We could use geometric series summation to find s-%, but we can also find it
quickly by multiplying a^ by (l + i)n:
(2.6)
s^=(l + i)na^ =
(1 + 0" -1
This approach helps to avoid excessive memorization. If you know a^ you can
easily get s^.
Example (2.7)
If i = 5% and n = 10,
sja = (L05)10 (7.7217) = i^l = 12.5779
This could be done on the financial calculator as above.
Set PMT =1, N=10,1/Y = 5 and CPT FV.
Exercise (2.8)
If n = 15 and i = 6%, find a^ and s^.
Answers:
«£1 = 9712 > siil = 23-276
To get another very helpful relationship, divide both sides of (2.6) by (1 + i)n:
(2-9) laa-v-sa
The relationships between a^ and s^ in (2.6) and (2.9) are intuitive.
©ACTEX 2009
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Exam FM / Exam 2 - Financial Mathematics

Page M2-6
Module 2 - Annuities
Section 2.3
Perpetuities
A perpetuity is an annuity in which payments are promised forever. The British
government once sold securities called consols which would pay interest in
perpetuity. The present value of a perpetuity immediate that pays 1 per period
is denoted by a^.
(2.10)
2 3 1
a-i = v + v^ +v +... = -
-' i
Note: If we write a^ as a limit, we obtain
(2.11)
a3=limas=hm
l-vn 1
n->ro i
I
Example (2.12)
If i = 5% ,
aa=-i- = 20
"' 0.05
Exercise (2.13)
Find the present value of a unit perpetuity immediate with i = 8%.
Answer: 12.5
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2-Annuities Page M2- 7
Section 2.4
Annuities with Level Payments Other Than 1
Note that the present or future value of any immediate annuity can be found
using a^\ and s^. If an immediate annuity has payment P, its present and
future value are given by
PV = Pc^
FV = Ps^
Example (2.14)
Find the present value of an annuity of n = 10 payments of P = 100
if i = 5%:
100 am = 100(7.7217) = 772.17
Before the electronic computing age, mathematicians compiled tables of values
of a^ for ranges of n and i. Present values of annuities were calculated by
multiplying tabular values of a^ by the relevant P, as above. At present, the
problem in Example (2.14) is more likely to be solved using a financial
calculator and computing PV with |PMT| = 100, |/Y] = 5, |FV| = 0 and |n] = 10.
Exercise (2.15)
Find the present value of an
i = 8%
annuity of n =
= 30
payments of P =
Answer:
= 500 if
5,628.89 |
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-8
Module 2 - Annuities
Section 2.5
Annuity Due Calculations
The present value of an n-period unit annuity due is denoted by a^.
payments 1 1 • • • 1
I 1 1 1 1
time t 0 1 "• n-1
0 1
At t=0, the first payment is worth 1.
At t=0, the second payment is worth v.
At t=0, the n -th payment is worth v
Notice the n-th payment occurs at t=n-l.
Since payments are made at the beginning of the period
, n_x l-vn l-vn
1-v
Thus,
(2.16)
<*H1
1-v"
This is easy to remember, since it is obtained by taking the equation for a^ and
replacing the i in the denominator by d. This pattern persists in the future value
and perpetuity formulas and is most helpful in keeping memorization to a
minimum.
(2.17)
s^
_(i+jT-i
(2.18)
a^=-
Another way to look at this relationship is to say that we could get a^ by
multiplying a~} by —. Since — = 1 + i, we have
a d
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2-Annuities
PageM2- 9
(2.19)
(2.20)
(2.21)
aa:
= 1^
=(i
+0as
s'sM
i
= (!■
^)s^
a=i =
-ia*
= (1
+0a^
This was useful in the days of tables, since it meant that it was not necessary to
show a^i in the tables.
Calculator technology has replaced the old table methods for
interest theory in most disciplines, but actuarial mathematics is
still table-based. You need to understand how to use the tables to
tackle actuarial mathematics.
Example(2.22)
Given i = 5% and n = 10, find a^ directly and check it using (2.19).
Solution.
1-
1
,10
1.05
Q^~"f0.05
U-05.
Check:
From Example(2.4), am = 7.7217
= 8.1078
aiol
(0.05)
ro.05^
U.05J
(7.7217) = 1.05(7.7217) = 8.1078
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M2-10
Module 2 - Annuities
Calculator Note
Annuity due calculations are done with the calculator reset to the BGN
(begin) mode for payment made at the beginning of the payment period. Note
that the letters BGN appear above the PMT key. If you key in 2ND BGN you
will see either BGN or END. You can change to another mode by keying 2ND
SET and ENTER. Remember that you can leave this menu by pressing the
CE/C key.
It is most important on actuarial exams to be aware of your present mode.
The majority of problems require END mode. If you do a BGN mode problem
and do not set your mode back to END, you will have trouble on subsequent
problems.
Exercise (2.23)
If n = 15 and i = 6%, find a^ and s^.
Answers: a^\ = 10.295 , s^\ = 24.673
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 2-Annuities
PageM2-ll
Section 2.6
Continuous Annuities
A continuous unit annuity pays a total of 1 per year, but spreads the payment
out continuously by paying ldt in each small time interval of length dt. The
present value of a continuous unit annuity paying from time 0 to time n is
denoted by a^.
The present value of a continuous annuity is found by integration:
-f
■f
eto(v)tdt
-e
-St
-e-5n+l
8
1-v"
In the above, we use the important identity 8 = ln(l + i) [from (1.32)] and the
fact that
ln(v) = In
— ~to(l + <)~*
The final result is:
(2.24)
_ l-vn i
Note that this shows a pattern similar to that observed for d^. We can find a^\
by replacing i by S in the denominator of a^. This is equivalent to multiplying
an\ by —. Similarly,
0
(2.25)
©ACTEX 2009
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Exam FM / Exam 2 - Financial Mathematics

Page M2-12
Module 2 - Annuities
(2.26)
Example (2.27)
_ 1 i
For i = 5%, n = 10, and S = In (1.05) = 0.0488, find a^.
Solution.
1-
a = ,105/ =79132
161 ln(1.05)
This can be checked approximately using the — relationship and
S
the fact that am = 7.7217:
. _ (0.05)
(0.0488)
(7.7217) = 7.9116
The slight discrepancy above is due to rounding. If the above
calculation is done by storing am and ln(1.05)in the calculator
memory directly, more significant figures are used and we see that
ajoj is 7.9132. This leads to a useful principle, below.
ssNf Calculator Note
It is best to store needed numbers in calculator memory with full calculator
precision. Writing down a three or four place approximation is a useful
record of your work, but re-entering the approximation in the calculator in
place of the original number takes time and loses accuracy.
Exercise (2.28)
If n = 15 and i = 6%, find a^ and s^.
Answers: , a"^ = 10.0008, s^] = 23.9675
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2-Annuities
Section 2.7
Basic Annuity Problems for Calculator Practice
Note that we can solve for each of the variables PV, FV, PMT, N and I/Y using
the BA II Plus. In this section we give an example of each.
Example (2.29)
I PMT problem
A loan for 20,000 must be repaid with 5 year end payments at an
annual rate of 12%. What is the annual payment?
Solution.
Set PV = 20000, N=5, I/Y = 12 and CPT PMT = -5,548.19.
| Your annual payment is $5,548.19
Exercise (2.30)
A loan for 20,000 must be repaid with 5 year end payments at an
annual rate of 10%. What is the annual payment?
| Answer: $5,275.95
Example (2.31)
PV problem
You wish to make a deposit now in an account earning 5% annually
I so that that you can get a payment of 1000 at the end of each of the
next 15 years. How much should you deposit today?
Solution.
Set PMT=1000, N=15, I/Y = 5 and CPT PV = -10,379.66.
You should deposit $10,379.66.
Exercise (2.32)
What would the required deposit be in (2.31) if you wanted 20
years of payments instead of 15?
Answer: $12,462.21
Example (2.33)
FV problem
You want to accumulate 20,000 in an account earning 4.5% per year
by making a level deposit at the beginning of each of the next 12
years. Find the required level payment.
Solution.
The calculator needs to be put in BGN mode. Once this is done set
FV=20,000, N=12, I/Y = 4.5 and CPT PMT = -1,237.63
The level payment is $1,237.63.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-14
Module 2 - Annuities
Exercise (2.34)
What would the required level deposit be in (2.33) if the interest
rate were 6%?
I Answer: $1,118.43
At this point, be sure to reset the calculator to END mode for the next problem.
Example (2.35)
I/Y problem
You have borrowed 15,000 and agreed to repay the loan with 5 level
payments of 4000, with the first payment occurring one year from
today. What interest rate are you paying?
Solution.
Set PV=15,000, N=5, PMT = -4000 and CPT I/Y = 10.42
You are paying 10A2% interest per year.
Note that the PV is positive since it represents cash given to you
and the PMT is negative because it is cash that you must pay. If
you forget the minus sign the BA II Plus will give an error message
1 when you hit CPT.
Exercise (2.36)
What would the interest rate be in (2.35) if the payment were 4300?
Answer: 13.34%
Note: Usually, it is not possible to solve for the exact interest rate, so the
calculator uses a numerical approximation method to find it.
Example (2.37)
N problem
You want to accumulate at least 20,000 in an account paying 4.5%
annually by making a level deposit of 1000 at the beginning of the
year for as long as necessary. Find the required number of
deposits.
Solution.
The calculator needs to be put in BGN mode. Once this is done set
FV=20,000, I/Y = 4.5, PMT = -1000 and CPT N = 14.11. This means
that 14 payments are not enough, and you must make a 15th
I payment to have at least 20,000
Exercise (2.38)
How many payments would be needed in (2.37) if the interest rate
were 6%?
Answer: 13
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2-Annuities
PageM2-15
Section 2.8
Annuities with Varying Payments
Not all series of payments are level. In practice, it's quite possible to encounter
varying series of payments such as those below:
Series of Payments:
500, 0, 200, 300
1,2,3,4
4, 3, 2,1
1,1.05, 1.1025= (1.05)2
Payments Made:
At end of period
At end of period
At end of period
Beginning of period
Type of Annuity Sequence:
—
Arithmetic increasing
Arithmetic decreasing
Geometric annuity
In interest theory, there are complicated formulas for the last three sequences
presented here. But your calculator will do any of them, and do them faster
than using formulas if there are only four or five terms to input.
If i = 0.05, you could use the BAII Plus to find the present value of the
increasing annuity {1,2,3,4} using the CF and NPV keys:
Hit the |CF] key to activate the cash flow menu. You will see a prompt for the
value of CF0, the cash flow at time 0. In this case there is no payment until time
1. Use your arrow keys to scroll down and you will see a prompt for C01, the
cash flow at time 1. Enter the number 1. Scroll down again, and there will be a
new prompt - "F01=" . This is a request for the number of times (frequency)
that this value is repeated. The default value is 1, and if you scroll past, the
value of 1 will be assumed with no entry. Scroll down again, and you will be
prompted for the value of C02. Enter 2. Repeat this process until all values are
entered. Then calculate the NPV with the keystrokes.
ENTER
NEV§
CPT
The display will show the answer 8.6488.
Another example: if i = 0.05, we can use the BAII Plus to find the present value
of the first series {500, 0, 200, 300} using the NPV function with 1=5 and the cash
flows provided. The present value (NPV) is 895.77.
Note that CF0=0y because CF0 is the initial payment at the beginning time t=0y
and the payment of 500 is at the end of the period (t=l). Thus, we have CF0=0>
CF 1=500, and CF2=0.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-16
Module 2 - Annuities
Section 2.9
Increasing Annuities with Terms in Arithmetic Progression
Calculator knowledge will help on some problems, but to prepare for Exam FM
we must review the specialized interest theory formulas for increasing and
decreasing annuities. The use of these formulas is often required on exam
problems.
An annuity whose n payments are 1, 2,..., n is called a unit increasing
immediate annuity. If payments are made at the end of each period, the
annuity is immediate and is denoted by (Ia)^. Clearly,
(Ia)^ =v + 2v2+3v3 +... + nvn.
payments ►
h
time t ► 0
It can be shown that
(2.39)
('").=
_flfl-w
Example (2.40)
Let i = 5% and n = 4. Then the annuity payments are 1, 2, 3, 4 and
(la) -^-4y4= 8.6488
v ^ 0.05
This can be checked on the BAII Plus Professional by using the
NPVfunction on the cashflow sequence 1, 2, 3, 4 (where CF1=1 and
1=5).
Exercise (2.41)
Find (la)^ for i = 0.06.
Answer:
10.295 - 6.259
(Ia)^\ = = 67.2668
0.06
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 2-Annuities
PageM2-17
As with level annuities, the formulas for the increasing unit annuity due can be
obtained by multiplying by -- = 1 + i.
a
(2.42)
(^h4(lah=(1+i)(Iah=^T-
Example (2.43)
Let i = 5% and n = 4.
Then, (Id)^ = 1.05(8.6488) = 9.0812
Exercise (2.44)
Find (la)^ for i = 0.06.
Answer:
(7a)^i = 71.3028
The future value of an increasing unit annuity immediate is denoted by (Is)^.
One can avoid excessive memorization by using the relationship
(Is)^ = (1 + i)n (Ia)^. Below, we show the commonly used expressions for (Is)^.
(2.45)
(Js)a=(l + i)"(Jfl)a =
_Sfl-rc
i
(2.46)
(ls),=(l + ir(la),=^ = l(ls).
The number of formulas here appears overwhelming, but the
situation is quite simple. If you can calculate (la)^ all of the
other values discussed can be obtained by multiplication by
(1 + i)n and — = 1 + i
d
Extensive memorization is not required!
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page M2-18
Module 2 - Annuities
Section 2.10
Decreasing Annuities with Terms in Arithmetic Progression
The unit decreasing immediate annuity has n payments: n, n-1, . . . , 1. Its
present value is denoted by (Da)^.
payments
time t
n-1
As before, you really need to know only one formula. It can be shown that
(2.48)
n-fla
(Da),=^ = (l + i)(Da),
(2.49)
(2.50)
(D8)a=a + 0"(Da)a
(Ds)a=(l + i)-(Da)a
Each of the last three values can easily be obtained from (2.47)
Example (2.51)
Given i = 5% and n = 4,
41 0.05 0.05
This can be checked on the BAII Plus Professional using the NPV
function on the sequence 4, 3, 2,1 with 1=5.
Exercise (2.52)
Find (Da)^ for i = 0.06.
15 - 9.7122
Answer: (Da)^\ = = 88.1292
0.06
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 2-Annuities
PageM2-19
Section 2.11
A Single Formula for Annuities with Terms in Arithmetic Progression
Suppose the first payment in an annuity immediate is P > 0 and the subsequent
payments change by Q per period, where Q can be either positive or negative.
If the annuity has n payments, the sequence of payments is
P,P + Q,P + 2Q,...,P + (n - 1)Q. It can be shown that the present value of this
annuity is
(2.S3)
P*a + Q
'a^-nvn
Note that (Ia)^ is the special case where P=l and Q=l and (Da)^ is the special
case where P = n and Q = -1. Note that multiplication of (2.S3) by (1 + i)n shows
that the future value of the annuity at time n is Ps^\ + ——. -.
Some students prefer to memorize only this single more general PQ formula.
Our own recommendation is to know it in addition to (Da) and (la), since each
formula has time saving features in different problems.
Note that the limit of the above expression as n becomes infinite gives the
present value of an increasing perpetuity immediate of the form
P,P + Q,P + 2Q,...,P + nQ,... as
(2.S4)
?♦*
This has been used in past exam problems.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-20
Module 2 - Annuities
Section 2.12
Annuities with Terms in Geometric Progression
Consider the sequence of payments 1, 1.05, 1.1025 = (1.05)2, made at the end of
the period.
payments
time t
1.05
(1.05)2
Suppose that we wish to find the present value of this series at the rate i = 10%.
From first principles,
1.10 (1.10)2 (1.10)3 110
, 1.05 fl.05
1 + +
1.10 11.10
This is a geometric series with
n=2andr=H5
1.10
1.10
1-
1.05
1.10
1-
1.05
1.10
= 2.6052
Payments increased geometrically with a rate of growth of g= 0.05.
In general, we can consider a geometrically increasing sequence with a rate of
growth g and n terms:
l^l + ^^l + g)2,...^^^-1
Suppose that these payments are made at the end of the period. If we wish to
find the present value at some interest rate i, we have
(i+t) (l+o (i+i)
•\3
+ ...+
1 + i
i+ii±iyi+s
l + i
■\2
l + i
+ ...+
(1 + g)"
1+g
l + i
n-l
The quantity in parentheses is a geometric series with ratio r =
(Hi)
(l + i)"
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 2-Annuities
PageM2-21
Thus:
(2.SS)
Some students memorize this, but they have to be careful to adjust for
modifications such as payments at the beginning of the period instead of the
end. Others feel that problems are best attacked by simply recognizing the
pattern and applying geometric series formulas.
Example (2.56)
Given i = 10%, find the present value of the sequence of payments
1.05,(1.05)2,...,(1.05)10
Payments are made at the beginning of the period.
Payments 1.05 (l.05)2 (l.05)3 (l.05)4 (l.05)S
I 1 1 1 1—
Time, t =0 1 2 3 4
4-
(1.05)9 (1.05)"
+
8
10
Solution.
Note that this series starts with 1.05, not 1, and that the payments
are made at the beginning of the period. (2.S7) does not directly
apply. We will need to factor to get a geometric series since a
geometric series must begin with 1:
10
PV = 1.05 + ± L" + 77T7T2"f' •• +777^9
= L05
factor
-1.05
(1.10)'
1 1.05 (1.05
1 + +
i.io U-io
+...+
(1.10)"
1.05
1.10
geometric series—begins with 1
,10'
1.05V
l.ioj
1.05
1.10
= 8.59
The perpetuity version of (2.55) is easier to remember:
If g is the rate of growth, i is the interest rate, and g<i, the present value is
(2.S7)
i-g
If g > i the present value of the perpetuity is infinite.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M2-22 Module 2 - Annuities
Note that (2.S7) applies to end-of-period payments and must be adjusted for
beginning-of-period problems. This will be illustrated in the next example.
Example (2.58)
Given i = 10%, find the present value of the perpetuity
1.05,(1.05)2,...,(1.05)\...
if a) payments are made at the end of the period, and
b) payments are made at the beginning of the period.
Solution.
a) END
PVEND= — = 21
0.10-0.05
b) BEGIN
PVBEGIN = I.IPVend = 1.1(21) = 23.1
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2-Annuities
PageM2-23
Section 2.13
Equations of Value and Loan Payments
We already looked at the problem of finding the payment on a loan using a
financial calculator. Now we will discuss how this is handled using unit annuity
notation. Suppose that you borrow $10,000 at an interest rate of i = 8% with
level payments at the end of each year for 10 years. How do you find the
payment P?
The principle that is used to find P is that the present value of payments must
equal the value of the loan: 10,000 = Pam. Thus, P =
10,000 10,000
fliol
6.7101
= 1490.29.
The computation is simple, but the key point here is the principle
involved: the two present values must be the same.
Also note: the payment above could easily be calculated from a financial
calculator with inputs [|v| =10,000, | = 8 and |nJ = 10 for a calculation of |PMT|,
Example (2.59)
You will deposit 10,000 in a bank at the beginning of this year and
the following two years. At the end of two years, you will retire
and want to withdraw a level payment P starting at the beginning
of year 4 and continuing for five years. The bank pays interest at
a rate of i = 8%. What is P?
Solution.
The diagram below illustrates the problem.
Deposit 10,000 10,000 10,000
Time, t =0
Withdrawals
3
P
4
P
5
P
6
P
7
P
We will use the value equation for t = 3:
[value of account at t = 3] = [PV of withdrawals t=3]
10,000(s^)= Pd^.
Thus,
P = 10,000
lfisl.
= 10,000
3.5061
4.3121
= 8,130.82
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-24 Module 2 - Annuities
Section 2.14
Deferred Annuities and a Useful Annuity Identity
There are cases in which you may want an annuity to begin in some future
period. For example, you might plan to retire in 5 years and want to purchase
an annuity immediate that pays 10,000 per year for ten years starting 5 years
from now. The present value of this annuity would be v5 (10,000)a^.
An annuity like this is called a deferred annuity. In general the present value of
an n-year unit annuity immediate deferred for k years is vka^.
There is a nice identity that breaks down the present value of an annuity
immediate into the sum of a shorter term annuity and a deferred annuity. We
will illustrate this by looking at an example with n = 5.
The present value of a five-period immediate annuity is
da = v + v2 + v3 + v4 + v5 = v + v2 + v3 + v3(v + v2) = (Z3J + v3a^\
Payment 11111
I 1 1 1 1 1
Time, t =0 1 2 3 4 5
V _ SK w J
V V
Three payment annuity; jwo payment annuity; PV = aj\ at t=3.
PV = (Z31 at t=0 To obtain py t=ot you must discount
3
three periods: PV [at t=0]= V a^.
PV@t=0: ^3l+V3a2i
Thus the present value of a five-period immediate unit annuity can be broken
down into the present value of a three period annuity and the present value of a
two period annuity to start in three periods.
This reasoning works in general.
The present value of an annuity for n+k periods is the sum of the present value
of an n-period annuity starting immediately and a fc-period annuity deferred for
n periods.
We can rewrite this identity as
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2-Annuities
PageM2-25
It is typical for actuarial examination questions to give pieces of this identity
when you really need other pieces:
Example (2.60)
Given a% = 3.5460 and v4 = 0.8227. Find a^.
Solution.
<*8l = <*3i + v4aa = 3.5460 + .8227(3.5460) = 6.463
This is a pre-calculator era interest theory problem. Using the BA
II Plus calculator, one could set N=4, PMT = 1 and PV = -3.5460 and
solve for the interest rate -it is 5%. Then change N to 8 and solve for
I PV. This gives the answer 6.463. |
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-26 Module 2 - Annuities
Section 2.15
Variable Annuities
We pointed out previously that the financial calculator NPV function could be
used to evaluate increasing and decreasing annuities. In some problems, the
calculator approach may require steps, as we shall see in the next problem
where the interest rate is not given directly.
Example (2.61)
An annuity pays 1 at the end of each of the next four years and 2 at the
end of each of the four following years. Given a^ = 3.5460 and
v4 = 0.8227, find the present value of the annuity.
Solution.
We can break this annuity into two pieces: an 8 year unit annuity and a
second 4 year deferred annuity.
8-Year 11111111
4-Year 00001111
Total Received 11112222
I 1 1 1 1 1 1 1 1
Time, t =0 1 2 3 4 5 6 7 8
We have already used the given information to find a^ = 6.463. Thus
a$ + v4a^ = 6.4630 + (0.8227)(3.5460) = 9.380
Exercise (2.62)
An annuity pays 100 at the end of each of the next 10 years and 200
at the end of each of the five following years. If i = .08, find the
present value of the annuity.
J Answer: 1,040.89
The variability of an annuity can take many different forms, which you will see
as you look at the examination problems at the end of this module. The next two
examples illustrate this.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 2-Annuities PageM2-27
Example (2.63)
An annuity immediate has a first payment of 100 and increases by 100
each year until payments reach 500. There are 10 further payments of
^OH TTinH thf* nrpspnt v^Iiia at f\ 53k
500. Find the present value at 6.5%
Solution.
The equation of value is
PV = 100 (la)^ + v5 500a^ = 100 (11.9445) + 0.7299 (500) (7.1888) = 3817.95
100 (la)^ 500 a^
^ ^ ^ A
r ^^ ^
Payments 100 200 ••• 400 500 ••• 500 500
I 1 1 1 1 1 1 1 1
Time, t =0 1 2 ••• 4 5 •■■ 14 15
Example (2.64)
An annuity immediate has 5 initial payments of 100 followed by a
perpetuity of 200 starting in the 6th year. Find the present value at 8%.
Solution.
There are a number of ways to attack this problem. Perhaps the
simplest is to think of this annuity as a perpetuity immediate of 100
starting now augmented by a second perpetuity immediate of 100
starting in 5 years.
Payments:
PV 100/0.08 100 100 100 100 100 100 100
PV vs(l00/0.08) o 0 0 0 0 100 100
h 1 1 1 1 1 1 1
Time, t=0 1 2 3 4 5 6 7 '
The present value of a single perpetuity of 100 is = 1250
Thus the total present value is 1250 + v51250 = 2100.73
Exercise (2.65)
An annuity immediate has a first payment of 100 and increases by
100 each year until payments reach 500. The remaining payments
are a perpetuity immediate of 500 beginning in year 6. Find the
present value at 6.5%.
Answer: 6,808.92
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-28 Module 2 - Annuities
Section 2.16
Annuity Problems with Interest Rate Variations
We saw in Module 1 that interest rates may be specified in many different ways
through use of nominal rates, discount rates, continuous rates and rate
equivalents. In this section we will look at a number of variations that you may
see on the exams.
Use of nominol rotes
The first and most useful to know in practice is the direct use of nominal rates,
since this is the way mortgage rates are quoted in the United States. If a lender
tells you that he can give you a mortgage rate of 6%, he probably means a
nominal rate of 6% convertible monthly, or 6% -5-12 = 0.5% per month.
Example (2.66)
Find the monthly level payment for a 6% thirty year mortgage loan
of 300,000.
Solution.
The calculator solution is direct. Note that mortgage payments are
made at the end of the month, so that you should be in END mode.
The loan is for 360 months at 0.5% per month. Set N=360,
PV=300000,1/Y=.5 and CPT PMT = -1,798.65.
Exercise (2.67)
Find the monthly payment for the loan above if it is made for 15
years.
I Answer: 2,531.57
Any of the problem types we have seen so far can be re-stated in terms of a
nominal rate.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2-Annuities PageM2-29
Example (2.68)
An annuity immediate has twenty initial quarterly payments of 25
followed by a perpetuity of quarterly payments of 50 starting in the
sixth year. Find the present value at 8% convertible quarterly.
Solution.
We can think of this annuity as a quarterly perpetuity immediate of 25
starting now augmented by a second quarterly perpetuity immediate of
25 starting in 5 years.
Payments:
PV
PV
25/0.02
v20 (50/0.02)
1
25
0
1
25
T°
25
0
1
T°
25
-ho
25
25
1
25
25
1
1
Quarters t =0 1 2 3 ••• 19 20 21
The quarterly interest rate is 2%.
25
The present value of a single perpetuity of 25 is — = 1250
Thus the total present value is 1250 + 1 5,n = 2091.21
1.0220
There is an actuarial notation that was used in conjunction with compound
interest tables in the past to solve problems. The notation a^ was used for the
present value of an annuity which paid 1/m at the end of each m-th of a year.
Thus a^2) stood for the present value of an annuity which paid 1/12 at the end of
each month for 30 years. It can be shown that
n\ j(m) n|
This notation was helpful when problems had to be solved using compound
interest tables. We will only mention it in passing here, since it is now less
widely used and we can solve practical problems without it.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-30 Module 2 - Annuities
Section 2.17
Rates where Interest is Convertible More or Less Frequently than Paid
The next cases involve instances where interest is convertible either more or
less frequently than it is paid. We will illustrate what to do with examples, since
the basic approach is intuitively obvious.
Example (2.69)
An annuity immediate has semiannual payments of 100 for 10 years
at a rate of 6% convertible monthly. Find its present value.
Solution.
There are 20 semiannual payments of 100. We are given a monthly
rate of 6% -s-12 = 0.5%, but we need a semiannual rate. Compound
the monthly rate 6 times to get the semiannual rate.
i = (1.005)6-l = .0304.
Now we have a calculator problem.
Set N=20, PMT=100,1/Y=3.04 and CPT PV = -1,482.57
Exercise (2.70)
An annuity immediate has quarterly payments of 200 for 15 years
at a rate of 9% convertible monthly. Find its present value.
Answer: 6,523.84
Example (2.71)
An annuity immediate has monthly payments of 100 for 10 years at
a rate of 6% convertible semiannually. Find its present value.
Solution.
There are 120 monthly payments of 100. We are given a semiannual
rate of 6% + 2 = 3%, but we need a monthly rate. Take the 6th root of
the semiannual interest factor 1.03 to obtain a monthly rate.
i = (1.03)1/6-l = .004939.
Now we have a calculator problem.
Set N=120, PMT=100,1/Y=0.4939 and CPT PV = -9,037.42
Exercise (2.72)
An annuity immediate has quarterly payments of 200 for 15 years
at a rate of 9% convertible semiannually. Find its present value.
Answer: 6,588.05
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2-Annuities Page M2-31
Section 2.18
Reinvestment Problems
In some cases where payments are made to you, you might reinvest the
payments. The next examples illustrate first a basic reinvestment problem and
then a more complex problem of a type that has appeared on exams.
Example (2.73)
You lend a relative 1000 and he agrees to pay you 6% interest on the
original $1,000 at the end of every year for 10 years and then return the
1000. You can reinvest the interest payments at 5%. How much will you
have in total in 10 years? What is your overall interest earnings rate?
Solution.
At the end of ten years you will have:
a) The return of the original 1000.
b) The future value of 10 payments of 60 (the interest at 6% on
$1000). Set PMT=-60,1/Y=5, N=10 and CPT FV = 754.67
The total is 1000+754.67 = 1754.67.
To find the overall interest rate earned note that you invested 1000 and
had a total of 1754.67 in 10 years. Set PV = -1000, FV = 1754.67, N=10 and
CPT I/Y = 5.78. It makes sense that your rate is between the two interest
| rates of 5% and 6%.
Exercise (2.74)
How much would you have in (2.73) if your reinvestment rate was 4%?
Answer: 1,720.37
Example (2.75)
You invest payments of 1,000 per year at the beginning of each year for 5
years. The original payments earn 10% interest, but the interest received
on the payments must be reinvested at 8%. How much will you have at the
end of 5 years?
Solution.
The table below shows the relevant payments. I
Time
1 Payment invested
1 Total payments to date
1 Interest on payments at 10%
0
1,000
1,000
0
l
1,000
2,000
100
2
1,000
3,000
200
3
1,000
4,000
300
4
1,000
5,000
400
5 1
0
5,000
500 |
Note that the numbers in last row are the deposits to the 8% reinvestment
account, and these are an arithmetically increasing annuity. At the end of
5 years you will have 5000 in payments to date plus the amount in the
reinvestment account. 5000 +100 (Is)^ 08 = 5000 +100 (16.6991) = 6669.91
Note that the answer pattern in this type of problem is
(# of payments)(payment)+(payment)(interest rate on payments) (7s)^,
where (Js)^ is the reinvestment rate.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-32 Module 2 - Annuities
Section 2.19
Inflation
Price inflation is a constant topic in our daily news. Everyone is worried about
increases in the price of gas and food. Inflation also affects the interest rates
that lenders charge -if a lender expects high inflation she will raise the interest
rate charged to borrowers.
We will start our discussion of inflation with a simple example of the effect of
inflation on purchasing power. Suppose that you like to have wine with dinner,
and buy an annual supply of 52 liters of wine (one per week) each January. If
the price today is $10 per liter, you will spend $520 this January. If you want to
put money aside for next year's purchase, you might decide to invest another
$520 to provide for a wine purchase next year. If the current interest rate is
5%, in one year you will have.
$520(1.05) = $546
Hopefully this investment will enable you to buy more wine next year than this
year, but inflation has to be considered. Suppose that next January the price of
wine has inflated by 3% to $10.30 per liter. Then the number of liters you can
buy next January is
546 ,53.01
10.30
This certainly gives you more wine, but not 5% more. Your increase in wine
consumption is
^1-1*0.0194
52
Thus your 5% investment gives only a 1.94% real increase in purchasing power.
We will denote the 5% rate at which you can lend by i. This is called the
nominal rate. We will denote the inflation rate by r. The rate at which you can
really increase purchasing power is referred to as the real rate, and will be
denoted by j. In general
(l + i) = (l + j)(l + r)«r (1 + J) = j^
In practice, you will generally know i and r and will solve for the real rate of
return j using the second equation above.
The first equation gives us the relation; -i-r -jr.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2-Annuities
Page M2-33
It is common to omit the jr term and approximate the real rate of return by
i-r.
In our first example an analyst might say that the real rate of return is
5% - 3% = 2%, but the real rate is truly 1.94%. This is a small difference, but
the approximation can work badly in countries which are experiencing
hyperinflation and high interest rates. For example if i = 0.50 and r = 0.30, then
i-r = 0.20 andj = — -1*0.154.
1.3
Economists provide a number of different indices to measure inflation. The
most commonly quoted index is the Consumer Price Index (CPI), which covers
the average price of a typical market basket of goods and services needed for
daily life. The details of CPI calculation are not tested on exam FM/2.
Inflation estimates are used in two different ways:
1. The first is use of the historical inflation index to see what has already
happened.
In the wine buying example, we found that with a nominal earnings rate
of 5% and actual past inflation of 3% we had a real increase in
purchasing power of 1.94%.
2. The second way is to use a projected inflation rate to make a decision
about the future.
Suppose that you are a lender who wants to earn a real year rate of 3%
over the next year and believes that inflation will be 2%. Then you will
want to lend at the nominal rate i defined by
1 + i = 1.03(1.02) = 1.0506 or i = 5.06%.
In one year you can look back at actual inflation and see if you actually
did earn at the required real rate.
You can buy bonds which are designed to adjust over time so as to protect you
against inflation risk. The United States government sells Treasury Inflation
Protected Securities (TIPS) and I-Bonds for this purpose. The details of TIPS
and I-Bond calculation are not tested on exam FM/2.
A bond issuer can create a bond that is made more attractive by having
payments adjusted to account for anticipated inflation. This is a source of
possible test problems that look like growing annuity problems. The next
example illustrates this type of problem.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-34
Module 2 - Annuities
Example (2.76)
A corporation issues a ten year bond that is designed to compensate for
expected inflation of 3% per year. Instead of using a coupon of 5% on a
face value of $1000, the company offers a coupon series starting at
50(1.03) = 51.50 and increasing each year by 3%. The payment at
maturity will be adjusted to 1000 (1.03)10 = 1343.92.
If investors are willing to buy this bond at a nominal yield of 5%, the
price of the bond is the present value
50(1.03) 50(1.03)2 50(1.03)10 1000(1.03)
10
1.05
1.052
= 50
1.03
1.05
1.0510 ' 1.0510
50 50 ^
, (1.03) (1.03)2 (1.03)9
1 + - - + - t- + ... + ■
1.05 1.052
1.05*
1000(1.03)
1.051
10
50
U-05
1.03
1.05
10 A
1-
1.03
1.05
,100("1:°3''°, 450.50, 825.05 = 1275.55
1.05
10
Note that since the real rate 1 + j is given by ——, this present value is
equal to SOa^j +
1000
(i+i)
10 '
Thus we can evaluate this present value at the real rate on the BAII Plus
calculator using the key strokes:
1.05 01.03 = - 1 = x 100= I/Y
50PMT
ION
1000 FV
CPTPV
In court awards for personal injury the final award may be an annuity intended
to replace the income of a disabled person. The payments can be indexed so
that they increase with anticipated inflation, just as the person's earnings
would have increased with inflation. The next example illustrates such a
payment scheme.
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 2-Annuities
Page M2-35
Example (2.77)
A court award is intended to replace an individual's current annual
salary of $50,000. Inflation is anticipated to be 3%. The award will consist
of a series of 25 end-of-year payments starting at 50,000(1.03) and
increasing at 3% per year until the final payment.
At a nominal rate of 6%, the present value of this award is
50000(1.03) 50000(1.03)2 50000(1.03)
25
1.06
1.06l
= 50000 f—,
[1.06J
50
(1.03) (1.03)2
1.06 1.062
1.0625
50 N
(1.03)24
1.06
24
= 50000
ri.03
U-06J
V
/1_ri.o3^
2S\
1.06J
1-
1.03
1.06
= 879,195.31
Note that since the real rate factor 1+j is given by
value is equal to SOjOOOa^j.
1.06
1.03
,this present
Thus we can evaluate this present value at the real rate on the BA II Plus
calculator using the key strokes:
1.06 S 1.03 = - 1 = x loo = I/Y
50000 PMT
25 N
0FV
CPTPV
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-36 Module 2 - Annuities
Section 2.20
Formula Sheet
Geometric Series
l-rn+1 1
1 + r + r2 +... + rn = ,r*l 1 + r + r2 +... = , Irkl
1-r 1-r
Annuities
Immediate a^=—7— ss=(l + i) as=^ j a^=vns^
Due a^=——- ss=(l + i) a^ = aia=vnsia
Perpetuities a^ = v + v2 + v3 +... = - a-^ = —
i a
Relation 1 = 1 + i-> a^=±a^ = (l + i)a^ s^ =lsa =(l + i)sa
Continuous as = —— = -aa sa = i j = -sn
Increasing Payments are 1, 2,..., n
('«), -^ («)a ->), -<i+0(H. -^
(JS)a = (1 +1)" (la), = S!fi (S)a = (1 + 0" (M)a = ^ = i(JS)a
Decreasing Payments are n, n-1,...,2,1
Present value of the annuity with terms PyP + QyP + 2Q,...,P + (n - 1)Q
Finite n Pa^ + QI a^~ny Perpetuity — + ^»
\ l J l l
A useful identity
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2-Annuities
Section 2.21
Basic Review Problems
1. Find a25i>06 ,S25i.06 , aMo6, a^ and a^06
2. A loan for 8,000 must be repaid with 6 year end payments at an annual rate
of 11%. What is the annual payment?
3. You wish to make a deposit now in an account earning 6% annually so that
you can get a payment of 250 at the end of each of the next 8 years. How
much should you deposit today?
4. You want to accumulate 12,000 in a 5% account by making a level deposit at
the beginning of each of the next 9 years. Find the required level payment.
5. You have borrowed 10,000 and agreed to repay the loan with 5 level
payments of 2500. What interest rate are you paying?
6. For i = 0.06 find (la)^ (Is)^ and (Da)^.
7. Given i = 8%, find the present value of the perpetuity
1.04,(1.04)2,...,(1.04)n,... for a) the immediate case, and b) due case.
8. An annuity pays 100 at the end of each of the next 5 years and 300 at the end
of each of the five following years. If i = .06, find the present value of the
annuity.
9. An annuity immediate has a first payment of 200 and increases by 100 each
year until payments reach 600. There are 5 further payments of 600. Find
the present value at 5.5%.
10. An annuity immediate has 40 initial quarterly payments of 20 followed by a
perpetuity of quarterly payments of 25 starting in the eleventh year. Find
the present value at 4% convertible quarterly
11. An annuity immediate has semiannual payments of 1000 for 25 years at a
rate of 6% convertible quarterly. Find its present value.
12. An annuity immediate has quarterly payments of 500 for 6 years at a rate of
4% convertible semiannually. Find its present value.
13. You lend 10,000 and the borrower agrees to pay you 8% interest at the end
of every year for 5 years and then return the 10,000. You can reinvest the
interest payments at 6%. How much will you have in total in 5 years?
14. You invest payments of 2000 per year at the beginning of each year for 8
years. The original payments earn 8% interest, but the interest received on
the payments must be reinvested at 5%. How much will you have at the end
of 8 years.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-38
Module 2 - Annuities
Section 2.22
Basic Review Problem Solutions
Calculator solutions will be given whenever possible. For problems 1-7, the
calculator was used; keystrokes can be found in similar problems in the text.
1. 12.78, 54.86,12.16,11.81,16.67
2. 1,891.01
3. 1552.45
4. 1036.46
5. 7.93%
6. 67.2668,161.2088, 88.1292
7. a) Payments made at the end of the period (immediate):
PF»°=ra=26
b) Payments made at the beginning of the period (due):
PVbegin = 1.08PVEnd = 1.08(26) = 28.08
8. We can break this annuity into two parts -a ten year annuity with
payment of 100 and a 5 year deferred annuity with payment of 200.
5-year annuity with payment of
$200; PV = 200asi at t=5- To
obtain PV t=0y you must discount 5
periods:
PV[att=0]= V5(200)a5i.
5-year deferred:
10-year annuity:
r
A
100
—h-
100
—h-
200
100
Time, t =0
26c
100
H
10
10 year annuity with payment of $100; PV = lOOa^ at t=0
pv@t=0: I00a^+v5(200)a^
The present value is lOOa^ + v5 (200)^ = 736.01 + (0.7473)(842.47) = 1,365.59
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 2-Annuities PageM2-39
9. Consider the payment streams as three separate annuities:
1 Total payment:
Which equals:
lOOdsi
100(la)j,
v5(600)aJ{
l
1
| Time, t =0
200
100
100
1
1
1
300
100
200
i
l
2
600
100
500
i
l
5
600
600
i
I
6
600
600
i
l
7
600
600
i
I
10 |
The equation of value is
PV = lOOaji +100 (la)^ + v5600a^
= 100 (4.270) + 100 (12.3542) + 0.7651 (600) (4.270)
= 3622.61
10. We can think of this annuity as a quarterly perpetuity immediate of 20
starting now augmented by a second quarterly perpetuity immediate of 5
starting in year 11. The quarterly interest rate is 1%.
20
The present value of a single perpetuity of 20 is = 2000.
0.01
The present value of a single perpetuity of 5 is = 500.
Thus the total present value is 2000 + ^ = 2,335.83.
(1.01)40
11. There are 50 semiannual payments of 1000. We are given a quarterly rate of
6% + 4 = 1.5%, but we need a semiannual rate. Compound the quarterly rate
twice to get the semiannual rate.
i = (1.015)2-l = .0302.
Years
alf-years
Quarters
0
0
0
V
1
1
2
J
Two quarterly compounding periods in
one semi-annual compounding period:
i = (1.015)2-l = .0302
Now we have a calculator problem.
Set N=50, PMT=1000,1/Y=3.02 and CPT PV = -25,620.20.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-40
Module 2 - Annuities
12. There are 24 quarterly payments of 500. We are given a semiannual rate of
4% + 2 = 2%, but we need a quarterly rate. Take the square root of the
semiannual interest factor 1.02 to obtain a quarterly rate.
i = (1.02)1/2-l = . 00995.
Years 0 1
Half-years 0 12
Quarters 0 12 3 4
One half of a semi-annual compounding
period in one quarterly compounding period:
i = (1.02)1/2-1 = 0.00995
Now we have a calculator problem.
Set N=24, PMT=500,1/Y=0.995 and CPT PV = -10,627.96
13. At the end of five years you will have:
• The return of the original 10,000.
• The future value of payments of 800 (the interest at 8% on 10,000).
Set PMT=-800,1/Y=6, N=5 and CPT FV = 4,509.67
The total is 10,000+4509.67 = 14,509.67.
14.
1 Time
1 Payment invested
1 Total payments to
date
1 Interest on
1 payments at 8%
0
2000
2000
1
2000
4000
160
2
2000
6000
160(2)
3
2000
8000
160(3)
4
2000
10,000
160(4)
5
2000
12,000
160(5)
6
2000
14,000
160(6)
7
2000
16,000
160(7)
8 1
16,000
160(8)
8 (2000) +160 (Is)q 05 = 16,000 + 160 (40.5313) = 22,485.01
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2-Annuities
Section 2.23
Sample Exam Problems
1. (2005 Exam FM Sample Questions #2)
Kathryn deposits 100 into an account at the beginning of each 4-year period
for 40 years. The account credits interest at an annual effective interest
rate of i.
The accumulated amount in the account at the end of 40 years is X, which is
5 times the accumulated amount in the account at the end of 20 years.
Calculate X.
(A) 4695 (B) 5070 (C) 5445 (D) 5820 (E) 6195
2. (2005 Exam FM Sample Questions #6)
A perpetuity costs 77.1 and makes annual payments at the end of the year.
The perpetuity pays 1 at the end of year 2, 2 at the end of year 3,...., n at
the end of year (n+1). After year (n+1), the payments remain constant at n.
The annual effective interest rate is 10.5%. Calculate n.
(A) 17 (B) 18 (C) 19 (D) 20 (E) 21
3. (2005 Exam FM Sample Questions #7)
1000 is deposited into Fund X, which earns an annual effective rate of 6%.
At the end of each year, the interest earned plus an additional 100 is
withdrawn from the fund. At the end of the tenth year, the fund is depleted.
The annual withdrawals of interest and principal are deposited into Fund Y,
which earns an annual effective rate of 9%.
Determine the accumulated value of Fund Y at the end of year 10.
(A) 1519 (B) 1819 (C) 2085 (D) 2273 (E) 2431
4. (2005 Exam FM Sample Questions #11)
A perpetuity-immediate pays 100 per year. Immediately after the fifth
payment, the perpetuity is exchanged for a 25-year annuity-immediate that
will pay X at the end of the first year. Each subsequent annual payment will
be 8% greater than the preceding payment. The annual effective rate of
interest is 8%.
Calculate X.
(A) 54 (B) 64 (C) 74 (D) 84 (E) 94
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-42
Module 2 - Annuities
5. (2005 Exam FM Sample Questions #14)
Mike buys a perpetuity-immediate with varying annual payments. During
the first 5 years, the payment is constant and equal to 10. Beginning in year
6, the payments start to increase. For year 6 and all future years, the
current year's payment is K% larger than the previous year's payment.
At an annual effective interest rate of 9.2%, the perpetuity has a present
value of 167.50.
Calculate K, given K < 9.2.
(A) 4.0 (B) 4.2 (C) 4.4 (D) 4.6 (E) 4.8
6. (2005 Exam FM Sample Questions #17)
To accumulate 8000 at the end of 3n years, deposits of 98 are made at the
end of each of the first n years and 196 at the end of each of the next 2n
years. The annual effective rate of interest is i. You are given (1 + i)n = 2
Determine i.
(A) 11.25% (B) 11.75% (C) 12.25% (D) 12.75% (E) 13.25%
7. (2005 Exam FM Sample Questions #18)
Olga buys a 5-year increasing annuity for X. Olga will receive 2 at the end
of the first month, 4 at the end of the second month, and for each
month thereafter the payment increases by 2. The nominal interest rate is
9% convertible quarterly.
Calculate X.
(A) 2680 (B) 2730 (C) 2780 D) 2830 (E) 2880
8. (2005 Exam FM Sample Questions #21)
Payments are made to an account at a continuous rate of (8k + tk)y where
0 < t < 10. Interest is credited at a force of interest 8t = . After 10
8 + t
years, the account is worth 20,000.
Calculate Jc.
(A) 111 (B) 116 (C) 121 (D) 126 (E) 131
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2 - Annuities
Page M2-43
9. (2005 Exam FM Sample Questions #25)
A perpetuity-immediate pays X per year. Brian receives the first n
payments, Colleen receives the next n payments, and Jeff receives the
remaining payments. Brian's share of the present value of the original
perpetuity is 40%, and Jeffs share is K.
Calculate K.
(A) 24% (B) 28% (C) 32% (D) 36% (E) 40%
10. (2005 Exam FM Sample Questions #29)
At an annual effective interest rate of i, i > 0%, the present value of a
perpetuity paying 10 at the end of each 3-year period, with the first
payment at the end of year 3, is 32. At the same annual effective rate of i,
the present value of a perpetuity paying 1 at the end of
each 4-month period, with first payment at the end of 4 months, is X.
Calculate X.
(A) 31.6 (B) 32.6 (C) 33.6 (D) 34.6 (E) 35.6
11. (2005 Exam FM Sample Questions #31)
An insurance company has an obligation to pay the medical costs for a
claimant. Average annual claims costs today are $5,000, and medical
inflation is expected to be 7% per year. The claimant is expected to live an
additional 20 years. Claim payments are made at yearly intervals, with the
first claim payment to be made one year from today.
Find the present value of the obligation if the annual interest rate is 5%.
(A) 87,932 (B) 102,514 (C) 114,611
(D) 122,634 (E) Cannot be determined
12. (2005 Exam FM Sample Questions #48)
A man turns 40 today and wishes to provide supplemental retirement
income of 3000 at the beginning of each month starting on his 65th
birthday. Starting today, he makes monthly contributions of X to a fund for
25 years. The fund earns a nominal rate of 8% compounded monthly. On his
65th birthday, each 1000 of the fund will provide 9.65 of income at the
beginning of each month starting immediately and continuing as long as he
survives.
Calculate X.
(A) 324.73 (B) 326.89 (C) 328.12 (D) 355.45 (E) 450.65
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-44
Module 2 - Annuities
13. (2005 Exam FM Sample Questions #49)
Happy and financially astute parents decide at the birth of their daughter
that they will need to provide 50,000 at each of their daughter's 18th, 19th,
20th and 21st birthdays to fund her college education. They plan to
contribute X at each of their daughter's 1st through 17th birthdays to fund
the four 50,000 withdrawals. If they anticipate earning a constant 5%
annual effective rate on their contributions, which the following equations
of value can be used to determine X, assuming compound interest?
(A) xfvJs + v2os +... + vZ] = 50,000^05 +... + vJs]
(B) x[(1.05)16+(1.05)ls+... + 1.051] = 50,000[l + ... + v30s]
(C) x[(1.05)17+(1.05)16+... + l] = 50,000[l + ... + v30S]
(D) x[(1.05)17 + (1.05)16 +... + (1.05)1] = 50,000[l +... + v305]
(E) x[vj» + v2os +... + v&] = 50,000[vS +... + vS]
14. (May 05 #1)
Which of the following expressions does NOT represent a definition for
(A) vn
(i+O"-i
(B) ^—— (C) v + v2+... + v"
(D) v
1-v"
1-v
(E)
ssi
(1-M)-1
15. (May 05 #4)
An estate provides a perpetuity with payments of X at the end of each year.
Seth, Susan, and Lori share the perpetuity such that Seth receives the
payments of X for the first n years and Susan receives the payments of X
for the next m years, after which Lori receives all the remaining payments
of X. Which of the following represents the difference between the present
value of Seth's and Susan's payments using a constant rate of interest?
(A) X[a^-vna^] (B) x[dj-v"d^] (C) x[aa -V+1a^]
(D) xfaa-v-1^] (E) X[va^-vn+1a^]
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2-Annuities
PageM2-45
16. (May 05 #9)
The present value of a series of 50 payments starting at 100 at the end of
the first year and increasing by 1 each year thereafter is equal to X. The
annual effective rate of interest is 9%. Calculate X.
(A) 1165 (B) 1180 (C) 1195 (D) 1210 (E) 1225
17. (May 05 #12)
Which of the following are characteristics of all perpetuities?
I. The present value is equal to the first payment divided by
the annual effective interest rate.
II. Payments continue forever.
III. Each payment is equal to the interest earned on the principal.
(A) I only
(B) II only
(C) III only
(D) I, II, and III
(E) The correct answer is not given by (A), (B), (C), or (D).
18. (May 05 #14)
An annuity-immediate pays 20 per year for 10 years, then decreases by 1
per year for 19 years. At an annual effective interest rate of 6%, the
present value is equal to X. Calculate X.
(A) 200 (B) 205 (C) 210 (D) 215 (E) 220
19. (May 05 #17)
At an annual effective interest rate of i, the present value of a perpetuity-
immediate starting with a payment of 200 in the first year and increasing
by 50 each year thereafter is 46,530. Calculate i.
(A) 3.25% (B) 3.50% (C) 3.75% (D) 4.00% (E) 4.25%
20. (May 05 #20)
An investor wishes to accumulate 10,000 at the end of 10 years by making
level deposits at the beginning of each year. The deposits earn a 12%
annual effective rate of interest paid at the end of each year. The interest is
immediately reinvested at an annual effective interest rate of 8%.
Calculate the level deposit.
(A) 541 (B) 572 (C) 598 (D) 615 (E) 621
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-46
Module 2 - Annuities
21. (May 05 #21)
A discount electronics store advertises the following financing
arrangement: "We don't offer you confusing interest rates. We'll just
divide your total cost by 10 and you can pay us that amount each month for
a year." The first payment is due on the date of sale and the remaining
eleven payments at monthly intervals thereafter.
Calculate the effective annual interest rate the store's customers are
paying on their loans.
(A) 35.1% (B) 41.3%
22. (May 05 #24)
(C) 42.0% (D) 51.2% (E) 54.9%
An annuity pays 1 at the end of each year for n years. Using an annual
effective interest rate of i, the accumulated value of the annuity at time
(n +1) is 13.776 . It is also known that (1 + i)n = 2.476.
Calculate n.
(A) 4
23. (Nov 05 #3)
(B) 5
(C) 6
(D) 7
(E) 8
An investor accumulates a fund by making payments at the beginning of
each month for 6 years. Her monthly payment is 50 for the first 2 years,
100 for the next 2 years, and 150 for the last 2 years. At the end of the 7th
year the fund is worth 10,000. The annual effective interest rate is i, and
the monthly effective interest rate is j.
Which of the following formulas represents the equation of value for this
fund accumulation?
(A) S2it(l + i)[(l + i)4+2(l + i)2+3] = 200
(B) S24t(l + j)[(l-fj)4+2(l-fj)2+3] = 200
(C) s^.(l + i)[(l + i)4+2(l + i)2+3] = 200
(D) s^(l + i)[(l + i)4 + 2(l + i)2+3] = 200
(E) s^ (1 + i) [(1 + j)4 + 2 (1 + j)2 + 3] = 200
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2 - Annuities
PageM2-47
24. (Nov OS #8)
Matthew makes a series of payments at the beginning of each year for 20
years. The first payment is 100. Each subsequent payment through the
tenth year increases by 5% from the previous payment. After the tenth
payment, each payment decreases by 5% from the previous payment.
Calculate the present value of these payments at the time the first payment
is made using an annual effective rate of 7%.
(A) 1375 (B) 1385 (C) 1395 (D) 1405 (E) 1415
25. (Nov 05 #9)
A company deposits 1000 at the beginning of the first year and 150 at the
beginning of each subsequent year into perpetuity. In return the company
receives payments at the end of each year forever. The first payment is
100. Each subsequent payment increases by 5%.
Calculate the company's yield rate for this transaction.
(A) 4.7% (B) 5.7% (C) 6.7% (D) 7.7% (E) 8.7%
26. (Nov 05 #12)
Megan purchases a perpetuity-immediate for 3250 with annual payments of
130. At the same price and interest rate, Chris purchases an annuity-
immediate with 20 annual payments that begin at amount P and increase by
15 each year thereafter. Calculate P.
(A) 90 (B) 116 (C) 131 (D) 176 (E) 239
27. (Nov 05 #13)
For 10,000, Kelly purchases an annuity-immediate that pays 400 quarterly
for the next 10 years. Calculate the annual nominal interest rate
convertible monthly earned by Kelly's investment.
(A) 10.0% (B) 10.3% (C) 10.5% (D) 10.7% (E) 11.0%
28. (Nov 05 #14)
Payments of X are made at the beginning of each year for 20 years. These
payments earn interest at the end of each year at an annual effective rate
of 8%. The interest is immediately reinvested at an annual effective rate of
6%. At the end of 20 years, the accumulated value of the 20 payments and
the reinvested interest is 5600. Calculate X.
(A) 121.67 (B) 123.56 (C) 125.72 (D) 127.18 (E) 128.50
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-48
Module 2 - Annuities
29. (Nov 05 #23)
The present value of a 25-year annuity-immediate with a first payment of
2500 and decreasing by 100 each year thereafter is X. Assuming an annual
effective interest rate of 10%, calculate X.
(A) 11,346 (B) 13,615 (C) 15,923 (D) 17,396 (E) 18,112
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2-Annuities
Page M2-49
Section 2.24
Sample Exam Solutions
We solve this problem by putting things in terms of 4-year periods. For each 4-
year period the interest rate j is given in terms of the annual rate by the
equation (1 + i)4 = (1 + j).
We can think of 40 years as 10 4-year periods. Thus the accumulated value at
the end of 40 years is FV40 = s'i^IOO = X.
X
Similarly the accumulated value at the end of 20 years is FV2o = S^-lOO = —.
We can solve the problem easily if we know what j is. To find j we use the fact
the accumulated value at time 40 is 5 times the accumulated value at time 20.
51 — 1 = 55151,100 —> Ss^=sm
Now we use the definition of s^ to create an equation which can be solved for j.
•\S
(i+Jy-i
(i + ;) -i
5[(l + ;)s-l] = (l + ;)10-l
(l + j)10-5(l + j)5+4 = 0
The standard trick here is to reduce this to a quadratic by making the
substitution x = (1 + j)5. Then we have the quadratic x2 - 5x + 4 = 0 with roots
x = l,4.
The root of 1 would give j = 0, which is not a valid solution. Thus we have
x = (l + j)5=4 -* j = .31951.
Using the financial calculator with a rate of 31.951% and PMT = 1 to find s^,
we have X = ^100 = 6,194.72.
Answer E
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-50
Module 2 - Annuities
We will start by looking at a timeline for the perpetuity.
payment
Time, t =0
n
-H—
n+1
n
—I—
n+2
n+3
The perpetuity is the sum of an arithmetic increasing annuity immediate
deferred to time 1 and a perpetuity immediate of n starting at time n+1. Thus
its present value (which is its cost) is:
Increasing annuity ' wJfT^
deferred one period gffES*
*a-™>
+ V
n+i n _ va^ _ a^
i
i
of n dollars
deferred
n+1 years
We are given the cost is 77.1 and i = 0.105. Thus -^Li2L = 77.1 _> a = 8.0955 .
0.105
We can solve for N on the BA II Plus calculator by using:
PV = -8.0955,1/Y = 10.5, and PMT = 1. The computed solution is N = 19.
Answer C
3.
First we will diagram the cash flow pattern.
1 Time
1 Principal withdrawn
1 Interest withdrawn
| Balance
0
-
-
1000
1
100
.06(1000)=60
900
2
100
.06(900)=54
800
9
100
.06(200)=12
100
10
100
.06(100)=6
0
The principle and interest payments are both deposited in Fund Y at 9%. The
principle payments are a level annuity and the interest payments are a
decreasing annuity. The accumulated value is lOOs^ +6(Ds)m.
Using the financial calculator, s^ = 15.1929 and
(Ds)m = 1.09"(Da)m=1.09
10
10-do'
.09
-94.23
Thus the accumulated value is 100(15.1929)+6(94.23) = 2084.67.
Answer C
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2-Annuities
PageM2-51
4.
The annuity exchange is made at time 5, immediately after the 5th payment on
the original perpetuity. At the point the value of the original perpetuity is
= 1250. The new 25 year annuity has geometrically increasing payments.
.08
Its present value is X
1 1.08 1.08
- + —- + ...+
> 24
1.08 1.082
1.08
X
25
1.08
For an exchange to be made, the present values must be equal.
X = 54.
2S
X^- = 1250
1.08
Answer A
The first 5 payments are a level annuity of 10. The present value lOa^ can be
found on the financial calculator by entering PMT = 10, N = 5 and I/Y = 9.2 and
computing the PV of 38.70.
The first increased payment occurs at time 6. The present value of the entire
perpetuity is then
38,7,i°(i+f>tio(i+y,io<i+y+,„
1.0926 1.0927 1.0928
= 38.7 +
10(1 + K)
1.0926
1 +
1 + K
1.092
1 + K
1.092
1 + K
1.092
+ ...
The trick here was to rearrange the final terms involving K into an infinite
geometric series with ratio r =
1 + K
1.092
1-
.092-K
1.092
and sum
1.092 J 1-r
1.092
.092-K
In terms of K that sum is
This simplifies the present value to
38.7+'"l1^
1.0926
1.092
.092-K
= 38.7 + -
10
1.0925
1 + K
.092-K
= 38.7 + 6.44
1 + K
.092-K
Since we are given that this present value is 167.50, we have
1 + K
167.50 = 38.7 + 6.44
1 + K
.092-K
Answer A
= 20
..092-K
K = .04
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page M2-52
Module 2 - Annuities
We can look at each payment in the last In years as the sum of 2 payments of
98. Then the total annuity is the sum of two level payment annuities -the first
starting at time 1 and continuing to time 3n and the second starting at time n+1
and continuing to time 3n.
Time, t =0
98
-f-
98
98
98
n+1
98
98
+
3n
Then, the accumulated value of 8000 is 98s^ + 98s^ = 8000.
In order to get an equation in i we use the annuity formulas:
98
(1 + i) -1
+ 98
(1 + if-l
= 8000.
Now we can use the given value of (1 + i)n = 2
~(2f-l
98
980
(2)-l
i
= 8000
+ 98
= 8000
U0.1225
Answer C
Olga has an increasing annuity extending over 60 months. The nominal interest
rate convertible quarterly is 9%, giving a rate of 2.25% per quarter. We first
solve for the implied monthly rate i.
Exponent is 3 because there are three
' months to a quarter year. An easy mistake
is to put 4 instead.
(1 + i)3 = 1.0225 1 + i = 1.007444
Thus the rate per month is 0.7444%.
The present value of the increasing annuity is
2(la)
601.007444
= 2
Q6o1-60v
.007444
60
= 2
48.61-38.45
.007444
: 2729.71
Answer B
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2-Annuities Page M2-53
8.
We will look at the accumulation of the amount deposited in a small interval of
length dt starting at time t, and then we will integrate to sum the accumulations
of all amounts contributed from t=0 to t=10.
Amount deposited at time t: (8k + tk) dt = k (8 +1) dt
Accumulation factor for growth on [£,10] of amount deposited at t:
eCMu = jX^)dU = eln(18)-ln(8+t) = 18
8 + t
18
Amount deposited at time t accumulates to: k (8 +1) dt = 18kdt
8 +1
Total of all accumulations: (°18kdt = 180k
Since we are given that the account is worth 20,000 in 10 years
180k = 20,000 k = 111.11
Answer A
9.
I 1 1 1 1 1 1 1 1
Time, t =0 1 • • • n ■ ■ ■ 2n
V v 'v v " v '
Brian Colleen Jeff
Brian has an immediate annuity of X for n periods. His present value is
B = Xa^. The total perpetuity has a present value of —. Thus Brian's share of
= l-vn
the total is y—^ = ia^ = i
V l J
I
Since Brian's share is 40%, we have 0.4 = 1 - vn vn = 0.6.
Colleen has an immediate annuity of X for n periods, but it is deferred for n
periods. Her present value is C = vnXa^ = .6Xa^ = .6B
Since Brian's share is 40%, Colleen's share is .6(40%) = 24%. Thus Jeff's share
is 100% - (40% + 24%) = 36%
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-54
Module 2 - Annuities
10.
Let j denote the effective rate per three year period for the perpetuity that pays
10 at the end of each three year period. Using the basic formula for the present
value of a perpetuity immediate we see that
Present value = 32 = — j = 0.3125
A four month period is one ninth of a three year period. Thus the effective rate
for a four month period is
(1 + j)9 -1 = (1.3125)9 -1 = 0.030676
The present value of a perpetuity immediate with payments of 1 at this rate is
1
.030676
Answer B
= 32.6
11.
The present value is calculated here under the assumption that the claimant
survives for exactly the expected 20 years. His payments will be
5000(1.07), 5000(1.07)2... 5000(1.07)20
payments
time t
5000(1.07) 5000 (1.07)2
5000 (l.07) 2
20
The present value of these payments at 5% is
5000
1.07 1.072 1.0720
+ Z- + ...+
1.05 1.052
1.05'
_ 5000(1.07)
1.05
' 1.07 1.0719
1+ + ...+ 7T-
1.05 1.0519
The final term in square brackets is a geometric series with ratio r =
1.07
1.05'
ml_ L , . 5000(1.07)
Thus the present value is - -
1.05
H
1-
1.07N
1.05,
'1.07
,1-05
20"
)
= 122,634
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2-Annuities
Page M2-55
12.
Let k be the number of 1000s of dollars in the fund at the 65th birthday. Each
1000 of the fund will provide 9.65 of income, so the total income of the fund will
be 9.65k. The desired income is 3000 so 9.65fc = 3000 k = 310.88083.
Thus the value of the fund at the 65th birthday must be 1000k = 310,880.83.
The man must accumulate an FV = 310,880.83 with unknown beginning of month
payments PMT for n = 12(25) = 300 months at a rate of i = 8 +12 per month.
Using a financial calculator set to BEGIN mode for the unknown payment, we
find that PMT = -324.725.
Answer A
13.
The correct answer is D, which is based on equating current values on the
daughter's 18th birthday. At that point in time the present value of the 4 future
payments of 50,000 is
50000[l + v + v2 + v3] = 50000[l + ... + v.o53].
The accumulated value of the 17 end-of-year payments of X is
X[l.0517+1.0516+... + 1.05].
Unfortunately, one must review all of the choices to see if they are correct -and
this can be a bit time consuming.
14.
E is not a correct expression for a^ . If the denominator were (1 + i)n it would
be correct, since a^ is the present value of s^. Note that C is the definition of
a^|, B is the computational formula most of us memorize for a^ and A is
clearly equivalent of B. Thus the only real thinking here involves D and E.
15.
Since Seth gets the first n payments, he has an n period annuity immediate of
X. His present value is Xa^.
Since Lori gets the next m payments after the first n years, she has a deferred
annuity of X. Her present value is vnXa^.
The difference is Xa^ - vnXa^ = X[a^ - vna^l
Answer A
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-56
Module 2 - Annuities
16.
We can think of this annuity as the sum of a level annuity of 99 and an
increasing annuity which starts at 1 and increases by one. The equation of value
is PV = 99am + (Ia)Mo9 = 99 (10.962) +125.287=1210.525
Answer D
17.
I. Definition of level perpetuity—but perpetuities do not have to be
level.
II. True for all perpetuities.
III. True for level perpetuities because
principal = *—¥-. > payment = (principal^
i
Answer B
Again, perpetuities may not be level.
18.
Here we have a level annuity of 20 for 10 years. Then payments drop to 19 in
year 11 and by one each year thereafter -so that we have a deferred decreasing
annuity for 19 years.
20 20 ••• 20 19 18 ••• 1
I 1 1 1 1 1 1 1 1
Time, t=0 1 2 -■• 10 11 12 •■• 29
V _ J K ^ ^ J
v v
Level annuity of 20 Deferred decreasing
for 10 years annuity for 19 years
The equation of value is
X = 20a^ + v10 (Da)^ = 147.20 + 0.5584 (130.6981) = 220.18
Answer E
19.
In this problem we will use the fact that the present value of the increasing
perpetuity immediate with firs payment P and periodic increase Q is
?4
i r
In this problem P=200 and Q=50, but the interest rate is unknown and we must
find it. The present value of the perpetuity is 46,530 so the equation of value is
200 50 A. „n
-^- + — = 46,530
i r
This gives us the quadratic equation 46,530i2 - 200i - 50 = 0.
The quadratic formula give us the roots i = .035 and i = -.0307. The correct
answer is the positive root i = .035.
Answer B
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2-Annuities
Page M2-57
20.
We will first create a table to illustrate the pattern of payments.
1 Time
1 Payment invested
1 Total payments to
date
1 Interest on payments
I at 12%
0
D
D
1
D
2D
0.12D
2
D
3D
0.12(2D)
• • •
...
9
D
10D
0.12(9D)
10
10D
0.12(10D)
Note that the numbers in last row are the deposits to the 8% reinvestment
account, and these are an arithmetically increasing annuity. At the end of 10
years the reinvestment account plus total deposits will reach 10,000, so the
equation of value is
10,000= 10D +0A2D(Is)m
Total
deposits Increasing annuity of
0.12D for 10 years
= 10D + 0.12D (70.5686)
= 18.4682D
D = 541.47
Answer A
21.
For each dollar of cost, the consumer makes 12 monthly payments of 0.10
starting immediately. This is an annuity due with unknown interest rate. The
monthly interest rate can be found directly on the BA II Plus in BGN mode.
Set PV = 1, N= 12, PMT = -0.10 and CPTI/Y = 3.5032. In actuarial notation, we
j(12)
have found = 0.035032
12
In this question we are asked for the effective rate i, which is given by
(i+0 =
1+ —
12
= (1.035032)12 = 1.5116 -> i = .5116
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-58
Module 2 - Annuities
22.
Here we have unknown n and i, which indicates that we cannot find the answer
directly with the calculator. The annuity in this question is a unit annuity. We
are given the accumulated at time n+1 of an n-period annuity. That
accumulated value is s^ (1 + i) =13.776. With some algebra we can use this to
find i.
,„,, ,. .v (l + i)"-lH .v 2.476-1... .v
13.776 = s^ (1 +1) = - '- (1 +1) = : (1 +1)
In the last step we used the given fact that (1 + i)n = 2.476. The above gives the
linear equation 13.776i = 1.476(1 + i) ->i = .12
Now we can find n using given information.
(1 + i)n = 2.476
n ln(1.12) = ln(2.476) -> n = 8.
Answer E
23.
Each separate block of 24 identical payments P accumulates to Ps^j at the end
of its 24 months, and then accumulates under an annual rate of i for the
remaining years. Since the total accumulation is 10,000 we have
50sj4|, (1 + i)S + lOOs^ (1 + i)3 + 150sj4|, (1 + i)1 = 10,000
*m (i + 0s + ^'sb (i + O3 + ®m (i + 01 = 200
(1 + i) sjg, [(1 + i)4 + 2 (1 + if + 3] = 200
Answer C
24.
Payment 100 100(1.05)
+
100(1.05)
= 155.13
1
.95(155.13) .952(155.13)
Time, t =0 1
The present value at 7% is
100
, 1.05 (1.05
1 + + ...+
= 100
1.07 U-07
fl.05Y°
+
10
155.13 (.95)
1.07J
1.05
,1.07
: 919.95+ 464.71 = 1384.66
+ 74.92
1-
1.0710
,1.07
.95
1.07
.9510(155.13)
11
. .95 ( .95
1 + + ...+
1.07
H
19
1.07
Answer B
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 2-Annuities
Page M2-59
25.
The pattern of payments is illustrated in the following table:
1 Time
Company deposits
1 Company receives
0
1000
1
150
100
2
150
100(l.05)
3
150
100(l.05)2
...
The yield rate i is unknown (the yield rate is just the rate of interest the
company earns). At this rate the present values of deposits and receipts are
PV of deposits = 1000 + ^°-
PV of receipts
100 100(1.05)
T+7 + (1 + i)2 +'":
(—)
[l + ij
1-
1.05
1 + i
f100l
100
1 105 I
1 + +
[ 1 + i 1
'1.05
i-.05
At the rate i the present values of deposits and receipts are equal. This gives us
*u *• i«™ 150 100
the equation 1000 + =
i i-.05
Solving for i we obtain
1000 (i2 - .05i) +150 (i - .05) = lOOi
lOOOi2 = 7.5
i = V-0075 = .0866
Answer E
26.
First we will find the unknown interest rate i .The value of Megan's perpetuity
immediate of 130 is
130
= 3250 i = 0.04.
Let P be the unknown initial payment for Chris. The annuity that is purchased
by Chris has payments P, P+15, P+2(15),..., P+19(15).
The present value of these year-end payments at 4% is
15 ^
Pa2oi+T^rMm
1.04
Thus 3250 = 13.59P +1673.51
13.59P +
1.04
116.03 = 13.59P +1673.51
P = 116
Answer B
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-60
Module 2 - Annuities
27.
This is a simple financial calculator problem. The effective quarterly yield can
be found directly using N = 40 (quarters), PMT = 400, and PV = -10,000 (price).
The computed quarterly yield is I/Y = 2.524%. This converts to a monthly yield
of 1.025241/3-l = .00834.
The nominal annual yield convertible monthly is 0.00834(12) = 0.10008.
Answer A
28.
The table below shows the pattern of payments and interest earned at 8%
Time
Payment
Total payments
Interest earned
0
X
X
1
X
2X
.08X
2
X
3X
.08(2X)=.16X
18
X
19X
.08(18X)=1.44X
19
X
20X
.08(19X)=.1.52X
20
20X
.08(20X)=1.60X
The total value of 5600 at time 20 is equal to
20X + .08X (Is)Mo6 = 20X + .08X (316.5454) = 45.3236X =5600
Thus X = 123.56
Answer B
29.
This is a straightforward decreasing annuity problem.
X = 100(Da)^ = 100[25"a^ ] = 15,922.96
Answer C
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 2-Annuities
Section 2.25
Supplemental Exercises
1. A man has a loan of 15,000 for 5 years with quarterly end of quarter
payments at a nominal interest rate of 6.8% convertible quarterly. What are
his payments?
2. A man plans to work for 25 years, during which time he will set a retirement
fund by making end of month payments of 100 per month. The fund earns
interest at a nominal rate of 6% convertible monthly. He will use the
accumulated funds to purchase a 20-year retirement annuity. Assuming he
can get 6% converted monthly on the annuity, what will his monthly end of
month payments be?
3. A woman purchases an annuity that makes annual payments at the
beginning of each year for 20 years. The first 10 payments are 1000 and the
last 10 are 1500. The annuity earns 6.5% annually. Find the cost of the
annuity.
4. A 20-year annuity immediate with annual payments earns 6.2%. The first
payment is 500 and each subsequent payment is increased by 4% over the
previous one. Find the present value of this annuity.
5. A man borrows 25,000 for 15 years. He repays the loan with quarterly end of
quarter payments of 650. What is his nominal rate of interest convertible
quarterly?
6. A perpetuity immediate makes quarterly payments and earns 6%
convertible quarterly. The first 20 payments are 10. Starting with the 21st
payment all subsequent payments are 15. Find the present value of this
perpetuity.
7. A perpetuity immediate makes annual payments and earns 5% annually. The
first payment is 100 and each subsequent payment is increased by 3% over
the previous one. Find the present value of this perpetuity.
8. You set up a retirement fund by making annual payments at the end of each
year for 30 years. The first payment is 1000 and each subsequent payment is
increased by 100 over the previous one. The fund earns 5.8% annually. How
much has accumulated in the fund at the end of the 30 years?
9. You invest 1000 at the beginning of each year for 20 years. The investment
makes interest payments to you at the end of each year at the rate of 7%.
You invest the interest payments in a fund that pays 6% annually. What is
your total accumulation at the end of the 20 years?
10. A man makes fixed end of year payments into a fund that earns at an annual
rate of r. He determines that the accumulation at the end of 20 years will be
triple the accumulation at the end of 10 years. Find r.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M2-62
Section 2.26
Supplemental Exercise Solutions
1. The number of payments is 20 and the periodic interest is 1.7%. Using the BA II
Plus calculator, set N = 20,1/Y =1.7, PV = 15,000 and FV = 0.
Then CPT PMT = - 891.01. So the payment is 891.01.
2. There are 300 payments with monthly rate of 0.5%. The accumulation is
100s^0005 = 69,299.40. To get monthly benefit payment set
N = 240,1/Y = 0.5, PV = -69.299.40 and FV = 0.
Then CPT PMT = 496.48.
3. To find the cost of the annuity, first set calculator to BGN mode. The cost is
1000a^006S + 500 v10 ajoiaow = 1000(11.73471) + 500(0.53273)(7.65610) = 13,774.03
4. The present value of the annuity is
500/1.062 + 500(1.04)/1.0622+ ... +500(1.0419)/1.06220
= (500/1.062)[l + (1.04/1.062) + ... + (1.04/1.062)19]
= 500[1 - (1.04/1.062)20]/(1.062 - 1.04) = 7,774.43
5. Using the BA II Plus set N = 60, PV = 25,000, PMT = -650 and
FV = 0. Then CPT I/Y = 1.592. Annual nominal rate is 6.40%.
6. This can be viewed as a perpetuity with payments 10 plus a deferred perpetuity
with payments 5.
The present value is
10/0.015 + v20(5/0.015) = 914.16.
7. The present value of the perpetuity is
100/1.05 + 100(1.03)/1.052 + ... +100(1.03n)/1.05n+1 + ...
= (1.00/1.05)[1 + 1.03/1.05 + .... + (1.03/1.05)n + ...]
= 100/(1.05-1.03) = 5,000.
8. The total accumulation A is given by
A = lOOOs^ + 100(s^- 30)/0.058
s^= 76.3298
A = 76,329.80 + 100(46.3298)/0.058 = 156,208.77
9. The yearend totals of the original investments are 1000, 2000, 3000,...., 20,000.
The interest amounts invested in the second fund are 70,140, 210,...., 1400. The
total accumulated in this fund is
70(Isho.os = 70(316.5454) = 22,158.18.
The overall total accumulation is 20,000 + 22,158.18 = 42,158.18
10. The accumulation at the end of 10 years is P[(l + r)10 - l]/r.
The accumulation at the end of 20 years is P[(l + r)20 - l]/r.
Then P[(l + r)20 - l]/r = 3P[(1 + r)10 - l]/r, and
(1 + r)10 + 1 = 3 => (1 + r)10 = 2 => r = 0.072.
[Recall that x2 - 1 = (x + l)(x - 1)]
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 3 - Loan Repayment
PageM3- 1
Now we understand how to find the payment on a level payment loan. However,
there are loans whose payments are not level, and there are ways of repaying
loans that we have not introduced yet. This module will go into more detail on
how loans can be structured and repaid.
Section 3.1
The Amortization Method of Loan Repayment
The amortization method is the most common method of loan repayment. The
fundamental principle behind it is simple. When a pavment is made, it must be
first applied to pav interest due and then any remaining part of the pavment is
applied to pav principal. We will illustrate this with an example.
Consider a loan for 30,000 with level payments to be made at the end of each
year for 5 years at an annual rate of 8%. We already know how to find the
annual payment. Set N=5,1/Y=8, PV=30000 and CPT PMT = -7513.69. We will
look at the first two payments to show how the method is applied.
Pavment 1. Beginning Balance = 30,000
Interest due = 30,000 (.08) = 2,400
Payment made = 7513.69
Interest paid =2400
Principal paid = 7,513.69 -2,400.00 = 5,113.69
Balance after payment = 30,000-5,113.69 = 24,886.31
Pavment 2. Beginning Balance = 24,886.31
Interest due = 24,886.31 (.08) = 1,990.90
Payment made = 7513.69
Interest paid = 1,990.90
Principal paid = 7,513.69-1,990.90 = 5,522.79
Balance after payment = 24,886.31 - 5,522.79 = 19,363.52
The following table shows the result of amortizing the loan over all 5 years.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M3-2
Module 3 - Loan Repayment
Table (3.1)
Year
0
1
2
3
4
5
Payment
7,513.69
7,513.69
7,513.69
7,513.69
7,513.69
Interest
2,400.00
1,990.90
1,549.08
1,071.91
556.57
Principal
5,113.69
5,522.79
5,964.61
6,441.78
6,957.12
Balance
30,000.00
24,886.31
19,363.52
13,398.90
6,957.12
0.00
You should check the detail on a few more lines to verify your understanding of
the method. There are some key points here:
a) The final balance is 0. The level payment pays off the loan as
intended.
b) As the balance declines over time the amount of interest due in each
period decreases.
c) As the interest due goes down, the amount of principal paid in each
period increases.
It is important to add a note on rounding here. The amortization table was
generated in EXCEL with amounts calculated to 10 decimal places. The
payment was actually 7513.6936370051, but was displayed as rounded to 2
places. Thus if you re-do the table with amounts rounded as they are in
practice, you will find discrepancies of a few pennies in the table.
The BA II Plus has amortization features that are helpful with amortization of
level payment loans. To demonstrate these features, we will look again at the
example loan for 30,000 at 8% for 5 years. (That loan is probably still in your
calculator. If not, compute the payment again to follow the discussion below.)
The two features that will help you are:
a) The FV key will give you the loan balance for period N. To see this,
set N=3 and CPT FV = -13,398.90.
b) Above the PV key you will see AMORT for the amortization
worksheet. To start the amortization use the keystrokes |2nd|
AMORT
The amortization routine allows you to find principal and interest paid over
the time span starting at a first period PI and ending at a second period P2. If
you wish to focus on just one period, set both PI and P2 equal to the value of
that period. Thus to see what happens in period 1, set PI =1 =P2.The first
display asks you to enter the value of PI. Use the keystrokes 1 |ENTER| and
then scroll down and repeat this for P2. Scroll down three more times and you
will see the displays
BAL= 24,886.31
PRN =-5,113.69
INT = -2,400.00
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 3 - Loan Repayment PageM3- 3
Exercise (3.2)
Use the AMORT worksheet to check balance, principal and interest for
period 2 for the loan in table (3.1).
Answer: 19,363.52, -5,522.79, -1,990.90
Note that all loan payments are assumed to be made at the end of
the period under the amortization method.
The ability to do amortization calculations easily is of great practical value. For
example the amount of interest paid on your mortgage is needed for your taxes.
Or, if you wish to pay off your loan early, the balance is the amount you must
pay.
The author was an expert witness in a case in which the plaintiff wished to pay
off his loan and could calculate the balance. He sued the loan company after he
paid the correct balance and was told that he owed a few thousand dollars
more.
Some exam FM questions can be done directly and rapidly with the financial
calculator, although many questions are designed to force use of formulas that
we will introduce in a few pages.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M3-4
Module 3 - Loan Repayment
Section 3.2
A Variable Payment Loan
Note that the financial calculator features we have covered so far apply only to
level payment loans. Many loans have payments that vary over time. In the
next example we will demonstrate how to set up a particular kind of variable
payment loan.
Example (3.3)
A borrower would like to borrow 30,000 at 8% for 5 years, but would like
to pay only 5,000 for the first two years and then catch up with a higher
payment for the final three years. What is the payment for the final 3
years?
Solution.
First we will use the calculator to find the loan balance in 2 years if the
payment each year is 5000:
Set PV=30,000, PMT=-5,000 , N=2 , I/Y=8 and CPT FV=-24,592,00.
The balance is 24,592.
After the second payment of 5000, the borrower owes 24,592. He can pay
this off in 3 years with a higher payment.
To find it, set PV=24,592, N=3, FV=0,I/Y=8 and CPT PMT =-9542.52.
The payment for the final 3 years is 9542.52.
Below we show the amortization table for the loan. As before, you should
check a few of the rows to assure that you understand the process.
Table (3.4): Variable payment loan
Year
0
1
2
3
4
5
Payment
5,000.00
5,000.00
9,542.52
9,542.52
9,542.52
Interest
2,400.00
2,192.00
1,967.36
1,361.35
706.85
Principal
2,600.00
2,808.00
7,575.16
8,181.17
8,835.67
Balance
30,000.00
27,400.00
24,592.00
17,016.84
8,835.67
0.00
Exercise (3.5)
What would the payment for the final three years be if the borrower in
the above loan wanted to pay only 4000 in each of the first two years?
I Answer: 10,349.63
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 3 - Loan Repayment
PageM3- 5
Some Notation
We can summarize the amortization using the following notation. For a loan
with periodic interest rate i:
Loan Payment at time fc Pmtk
Loan Balance after Pmtk is made: Balk
Principal paid in period fc PRink
Note that the loan amount is Bal0. For fc > 1 the amortization method is
described by the recursive relations:
Interest paid in Pmtk+i Intk+1 -i(Balk)
Principal paid in Pmtk+1: PRink+1 = Pmtk+1 - i (Balk)
Balance after Pmtk+1: Balk+1 = Balk - PRink+1
= Balk-[Pmtk+1 - i(Balk)]
= (l + i)Balk -Pmtk+i
Note that reference Mathematics of Investment and Credit uses the notation
OBt for the balance at time i
Ki for the payment at time i
L for the original loan amount.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M3-6
Module 3 - Loan Repayment
Section 3.3
Formulas for Level Payment Loan Amortization
It can be shown that for a level-payment loan with payment PMT,
(3-6> I Interest paid in Pmtt : lntt = PMT(l-v"-'+1)
Principal paid in Pmtt: PRint = PMTvn'M
Example (3.7)
1 For the loan in table (3.1) we had i = .08 and PMT =
Then t = 4, n -
PRin*
= 7,513.69. 1
-1 +1 = 2 and the principal in the 4th payment is
= 1™™.-. 6441.78
1.082
\ This matches the value given in table (3.1).
Exercise (3.8)
An annual loan for 10 years has interest rate 6% and level payment 1000.
Find the amount of principal and interest in the 6th payment.
Answer : Principal 747.26 Interest 252.74
Exam questions can be designed to be solved conveniently using these
formulas, as we shall see in the next example. You should know these formulas.
Example (3.9) '
For an 8% level payment loan, the amount of principal in the second
payment is 5,522.79. Find the amount of principal in the 4th payment.
Solution.
We are not given the total payment amount PMT or the term of the loan.
Using (3.6) we have
PRin2 = v{n~2+1)PMT = 5,522.79
PRin* = v{n-M)PMT = (1 + i)2 PRin2 = 1.082 (5522.79) = 6441.78
This problem was taken from the loan in table (3.1), and you can check
the answer of 6441.78 there.
Exercise (3.10)
For a 6% level payment loan, the amount of principal in the first
payment is 5321.89. Find the amount of principal in the 4th payment.
Answer: 6338.46
Note that for a level payment loan one can say in general that
(3.11) I ~ '
1 PRin„+fc=(l + i)fcPRin„
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 3 - Loan Repayment PageM3- 7
Section 3.4
Looking Forward and Looking Back
To this point we have concentrated on showing how a level payment loan is
amortized period by period. Now we discuss another valuable perspective on
loan balance.
Looking Forward: the Prospective Method.
Let us look again at the example loan from Table (3.1). The loan was for 30,000
at 8% annually for 5 years. The annual payment was 7,513.69. We already know
how to find the loan balance after the third payment in more than one way.
Here is another.
The loan balance after a payment is the present value of the remaining
payments at the time of payment. So the loan balance after the third payment
should be the present value of the remaining two payments at time 3.
payments 7513.69 7513.69 7513.69 7513.69 7513.69
time t 0 1 2
/
Loan balance at t= 3 is
equal to PV of
remaining two
payments at t=3.
To check this, find the present value of the remaining two payments after the
third payment using the BA II Plus. Set PMT = -7513.69, N=2,1/Y = 8 and CPT
PV = 13,398.90. This is the same number that we obtained for the balance after
the third payment in Table (3.1).
This makes some sense. Once you have made a payment you still have an
obligation to make the remaining payments. The loan balance is the present
value of your remaining obligation.
Exercise (3.12)
A loan made at an annual rate of 6.5% has 7 remaining payments of 950.
What is the loan balance?
Answer: 5210.29
In actuarial notation, for a level payment loan with periodic payment PMT at a
rate i for n periods, the balance after payment k is
(3.13)
Prospective balance = PMTa—^\ {
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M3-8
Module 3 - Loan Repayment
Looking Back: the Retrospective Method.
We just saw that you can find a loan balance by looking at the value of your
remaining future obligations. What about the past? It will also tell you what
your balance is.
Think about this by looking at what is owed on a loan and what has already been
paid at any point in time. If the loan was for an amount PV and no payments
were made by time k, you would owe PV(1 + i)k
If you have actually made payments at times l,...,/c, then you have reduced the
amount of that obligation by the future value at time k of those payments
FV( payments made at times l,..,k).
The loan balance is
(3.14)
Retrospective Balance = PV(1 + i)k - FV(payments made at times l,...,k)'
For a level payment loan with payment PMT, the balance can be expressed in
actuarial notation as
(3.1S)
PVd + ir-PAfTSfc]
In the next example we replicate this reasoning for the omnipresent loan of
30,000 at 8% for 5 years.
Example (3.16)
Use the retrospective method to check the balance after the third
payment for the loan in Table (3.1)
Solution.
The loan was for 30,000 at 8% annually for 5 years. The annual payment
was 7,513.69. If no payments had been made you would owe
30,000(1.08)3 =37,791.36
You actually made 3 payments of 7,513.69. The future value of those
payments at time 3 is 24,392.44. The total obligation (balance) is
37,791.36 - 24,392.44 = 13,398.92
Note two cent rounding error
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 3 - Loan Repayment
PageM3- 9
Exercise (3.17)
A loan for 40,000 at a 7% annual rate has an annual payment of 9,755.63.
Find the balance after the 4th payment.
Answer: 9117.40
More Challenging Questions
On the exams, questions are made tougher by introducing loans whose terms
fall in arithmetic or geometric series. The prospective and retrospective
methods can be used to find loan balances for these, as the next examples
indicate.
Example (3.18)
A loan at 10% annually has an initial payment of 100, and 9 further
payments. The payment amount increases by 2% each year. Find the
loan balance immediately after the fourth payment.
Solution.
The payments are 100, 100(1.02), 100(1.02)2,..., 100(1.02)9. Immediately
after the fourth payment the remaining payments
arel00(1.02)4,..., 100(1.02)9. Using the prospective method, the balance
immediately after the fourth payment is the present value of those
remaining payments.
100(1.024) 100(l.029)
Bah =
1.10
100(l.024)
- + .... + -
1.106
1.10
, 1.02
1 + —— +
1.10 11.10
1.02
\2
1.02
1.10
100(1.024)
1.10
1-
1.02
1.10
1-
1.02
1.10
= 492.93
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M3-10
Module 3 - Loan Repayment
Example (3.19)
A loan at 10% annually has an initial payment of 100, and 9 further
payments. The payment amount increases by 10 each year. Find the loan
balance immediately after the fourth payment.
Solution.
The payments are 100, 110, 120,..., 190. Immediately after the fourth
payment the remaining payments are 140, 150,..., 190. Using the
prospective method, the balance immediately after the fourth payment
is the present value of those remaining payments. Using P = 140 and Q =
10 we have
BaU=Pa^Q^^
140^+10
(a«r-6v
6\
0.1
f
140 (4.355)+ 10
706.57
4.355-6(0.5645)
0.1
Exercise (3.20)
A loan at 8% annually has an initial payment of 1000, and 9 further
payments. The payment amount decreases by 2% each year. Find the
loan balance immediately after the third payment.
Answer: 4644.38
Exercise (3.21)
A loan at 8% annually has an initial payment of 100, and 9 further
payments. The payment amount decreases by 5 each year. Find the loan
balance immediately after the sixth payment.
Answer: 208.6
©ACTEX 2009 ' Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 3 - Loan Repayment Page M3-11
Section 3.5
Monthly Payment Loans
The exams have questions with scenarios like nominal rates convertible every 4
years applied to semiannual payments, but what you will see most in your own
life are monthly payment loans with a quoted rate nominal rate convertible
monthly. We will take this opportunity to review what we have done so far in
the context of the familiar monthly payment loan.
Example (3.22)
A thirty year monthly payment mortgage loan for 250,000 is offered at a
nominal rate of 6% convertible monthly.
Find the
a) monthly payment
b) the total principal and interest that would be paid on the loan over 30
years
c) the balance in 5 years
d) the principal and interest paid over the first 5 years.
Solution.
a) In the US, quoted rates on mortgages are nominal rates convertible
monthly (although Exam FM may use other periods for conversions).
Thus the loan has a monthly rate of 6% +12 = 0.5%.
Find the payment using the BA II Plus calculator.
Set PV = 250,000,1/Y=.5, N=360 and CPT PMT = -1,498.88.
b) The total principal paid is just the amount of the loan -i.e.
250,000. The total interest can be found as
Total payments-Principal Paid = 1498.88(360) - 250,000 = 289,596.80
You could also do this with the AMORT worksheet. The answer will
be slightly different due to rounding.
c) The balance is easy on the calculator. Set N=60 (for 5 years) and CPT
FV =-232,635.89
d) This can be done in two ways, just as part b). First note that the
amount of principal paid in 5 years is just the original amount of the
loan less the balance in 5 years or 250,000 - 232,635.89 = 17,364.11.
The total interest paid in 5 years can be found as
Total payments-Principal Paid = 1498.88(60) - 17364.11 = 72,568.69
With the AMORT worksheet the answer will again be slightly
different.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M3-12
Module 3 - Loan Repayment
Exercise (3.23)
A fifteen year monthly payment mortgage loan for 250,000 is offered at
a nominal rate of 6% convertible monthly.
Find
a) the monthly payment
b) the total principal and interest that would be paid on the
loan over 15 years
c) the balance in 5 years and d) the principal and interest paid
over the first 5 years.
Answer: a) 2109.64
b) Principal 250,000, Interest 129,735.57
c) 190,022.75
d) Principal 59,977.25 Interest 66,601.27
Example (3.24)
A thirty year monthly payment mortgage loan for 250,000 is offered at a
rate of 6%. The borrower would like to have graduated payments where
the first year's monthly payment is P , the second year's monthly
payment is P+100 and all subsequent monthly payments are P+200.
a) Find the initial payment P
b) Find the balance at the end of one year.
Solution.
a) Use the equation of value
250,000 = Pamms + v12 (100)a^005 + v24 (100) a^^
= P166.7916 + .9419 (100) (164.7434) + .8872 (100) (162.5688)
= P166.7916 +29,940.28
This gives P=1319.37
b) Using the prospective method,
Bal2 = (P + 100) a^^ + v12 (100) a^^
= 1419.37(164.7434) + .9419(100)(162.5688) = 249,144.19
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 3 - Loan Repayment Page M3-13
Exercise (3.25)
A 15 year monthly payment mortgage loan for 250,000 is offered at a
rate of 6%. The borrower would like to have graduated payments where
the first year's monthly payment is P , the second year's monthly
payment is P+100 and all subsequent monthly payments are P+200.
Find
a) Find the initial payment P.
b) Find the balance at the end of one year.
Answer: a) 1938.49 b) 241,507.13
Lenders may charge percentage fees called points on a loan to raise their yield.
The next example illustrates the effect of this.
Example (3.26)
A thirty year monthly payment mortgage loan for 300,000 is offered at a
nominal rate of 7.2% convertible monthly. Thus the monthly interest
rate is 0.6% and the calculated monthly payment is 2036.36. (Calculate
the payment on your calculator and leave it there for the moment.) When
the loan closes the lender includes a fee of 3 points for which no service
is performed. He is taking away 3% of the loan or 9,000 as a fee that
raises his yield. In effect, the borrower is really getting a loan of only
291,000. This raises the borrower's interest rate. To see this, modify the
loan in your calculator by setting PV = 291,000 and CPT I/Y. The result is
a monthly rate of 0.63%. Multiply this by 12 and you will see a nominal
rate Of 7.51%. That is the borrowers true nominal annual rate.
The true nominal annual rate is called the annual percentage rate or APR in the
United States. Lenders are required by law to reveal this rate to the borrower.
The intent of the law is to provide information that would prevent abuse of
borrowers by lenders quoting lower rates and then charging a very large point
fee.
Exercise (3.27)
A fifteen year monthly payment mortgage loan for 200,000 is offered at
a nominal rate of 7.2% convertible monthly. The lender charges a fee of
2% for which no services are provided. Find the APR.
Answer : 7.53%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M3-14
Module 3 - Loan Repayment
Section 3.6
An Installment Loan Example
With an installment loan, you pay regular payments of principal plus interest
due at the end of each period. The next example illustrates this.
Example (3.28)
You have a 30,000 loan at 8% annually for 5 years. You agree to pay off
the principal in installments of 6,000 per year, and to pay interest on the
outstanding balance each year. The amortization table is below.
1 Year
1 Payment
1 Principal
1 Balance
0
30,000
l
8,400
2,400
6,000
24,000
2
7,920
1,920
6,000
18,000
3
7,440
1,440
6,000
12,000
4
6,960
960
6,000
6,000
5 1
6,480
480
6,000
0 1
Here you find the interest due first and add it to the principal installment
to get the total payment. For example at time 2 you would first calculate
8% interest on the previous outstanding balance of 24,000
0.08(24,000) = 1,920.
Then you would add that to the required principal of 6,000 for a total
payment of 7,920. Note that now the principal is constant but the interest
due and total payment decrease.
It is easy to find the interest due for a period without constructing the
table. Suppose that you wanted to find the interest due in the 4th payment.
Note that
Bal3 = 30,000 - 3 (6000) = 12,000
InU =i(Bal3) = (.08)12,000 = 960
The interest due in the 4th payment is 960, and the total payment is 6,960. |
Exercise (3.29)
You have a 30,000 loan at 8% annually for 30 years. You agree to pay off
the principal in installments of 1,000 per year, and to pay interest on the
outstanding balance each year.
Find
a) the interest due in the 11th payment
b) b) the actual 11th payment.
Answer : a) 1600 b) 2600
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 3 - Loan Repayment Page M3-15
Section 37
Sinking Fund Repayment of a Loan
When you use a sinking fund, you only pay the lender the interest at his stated
rate i on the loan each period. In addition you make level deposits to an account
called a sinking fund that earns interest at a rate j.
The goal is to make a deposit into the sinking fund that will cause the fund to
grow to the amount of the loan at the end of the loan term. Thus you can pay off
the loan when it is due using the sinking account funds. We will use the notation
SFD for the required sinking fund deposit.
Example (3.30)
A 100,000 annual payment loan is made for a term of 10 years at 10%
interest. The lender wants only payments of interest until the end of
year 10 when the 100,000 must be repaid. The borrower will make level
annual year-end payments to a sinking fund earning 8%.
Find the sinking fund deposit and the balance in the sinking fund at
times 3 and 4.
Solution.
The first task is to find the required sinking fund deposit. This is easily
done using the TI BA II Plus. Set FV=100,000,1/Y=8, N=10 and CPT
PMT= -6,902.95. Note that each year the borrower will also pay
100,000 (.10) = 10,000 to the lender resulting in a total loan payment of
16,902.95.
Next we will look at the balance in the sinking fund. The balance at time
k is the future value of k payments of 6,902.95 to the fund. To get the
balance at time 3, set N=3 and CPT FV = -22,409.73. To get the balance
at time 4, set N=4 and CPT FV = -31,105.46.
Exercise (3.31)
A 70,000 annual payment loan is made for a term of 10 years at 8%
interest. The lender wants only payments of interest until the end of
year 10 when the 70,000 must be repaid. The borrower will make level
annual year-end payments to a sinking fund earning 6%.
Find the sinking fund deposit and the balance in the sinking fund at
times 5 and 6.
| Answer : SFD=5310.76; Balances: 29,937.23, 37,044.22
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratiiff, Garcia, & Steeby

Page M3-16
Module 3 - Loan Repayment
The sinking fund balance is important, and we will refer to the balance at time
k as SFBalk. We have looked at the sinking fund balances at two successive
time periods in the preceding example and exercise for a reason. We need them
to find the amount of principal in a sinking fund payment For example,
Principal in 4th payment = SFBaU - SFBah = 31,105.46 - 22,409.73 = 8,695.73
The sinking fund is where you accumulate principal to repay the original
100,000. Thus the change in the sinking fund from time 3 to time 4 is the amount
of principal accumulated during that time period to repay the loan.
The general rule is that: Principal in fcth payment = SFBah - SFBah-i.
Once you have the principal in a payment, you can find the interest in that
payment too.
Interest in 4th payment=Total PMT-Principal Paid=16,902.95-8695.73=8,207.22
There is an alternative way to find the interest in a sinking fund payment. In
each period you pay interest to the lender but also earn interest on the sinking
fund. The difference between what you pay and what you earn is called your
net interest.
At time 3, the balance in the sinking fund was 22,409.73 and the interest
earned was 0.08(22,409.73) = 1,792.78.
For the 4th payment we have: Net Interest in 4th PMT=10,000-1,792.78 = 8,207.22
Exercise (3.32)
For the sinking fund loan in (3.31), find the principal and interest paid in
the 6th payment.
I Answer : Principal=7106.99; Interest=3803.77 |
We have proceeded intuitively and relied on the calculator so far. Now we will
introduce some formulas that are commonly found in actuarial texts. We will
denote the loan amount by L and the term by n. Recall that the loan interest rate
is i and the sinking fund rate is j .The sinking fund deposit satisfies the
equation SFD(s^) = L. Thus
SFD = —
The interest payable to the lender each period is Li, so the total loan payment
each period is given by
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 3 - Loan Repayment
PageM3-17
(3.34)
Total sinking fund loan payment = + Li = L
S^J
- + i
The balance in the sinking fund at time k is given by:
(3.35)
SFBalk=SFDsxi=L^L
The principal paid in payment k is
(3.36)
SFBah -SFBah-x^SFDSftj -SFDs^j
= SFD(sxJ-s^j)
= SFD(l + j)fc_1
Thus we have a formula that is useful for exam problems
(3.37)
Principal paid in sinking fund payment k = SFD (1 + j)
fc-i
The interest paid in payment k can be given in two ways
(3.38)
Net interest = Total Payment - Principal Paid
= (SFD + Li)-SFD(l + j)
k-i
(3.39) Interest to lender - Interest on sinking fund = Li- SFBalk.i (j)
This is a large assortment of formulas. A crucial one for exam problems is
(3.37). In the next examples we will give a simple application and then an
example of a tougher problem that is easy to solve if you know (3.37).
Example (3.40)
For the sinking fund loan in (3.30) we found that SFD = 6,902.95 and the
principal paid in the 4th payment was 8695.73. We can check the principal
paid amount using (3.37).
SFD (1 +j) =6,902.95(1.08) =8,695.73
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M3-18
Module 3 - Loan Repayment
Exercise (3.41)
Use (3.37) to verify the principal paid amount in Exercise (3.32).
Example (3.42)
For a sinking fund loan SFD = 5310.76 and the amount of principal in the
third payment is 5967.17. What is the interest rate?
Solution.
Principal Paid = SFD (1 + jf'1
= 5310.76 (1 + j)2
= 5967.17
, . /5967.17 1 „ . „
1 + j = J = 1.06 -» j = .06
J V 5310.76
Exercise (3.43)
For a sinking fund loan SFD = 5066.43 and the amount of principal in the
fourth payment is 6206.59. What is the interest rate?
Answer : 7%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 3 - Loan Repayment Page M3-19
Section 3.8
Capitalization of Interest and Negative Amortization
Capitalization of interest and negative amortization occur when the payment
made is less that the interest on the loan. Let's look at an example:
Example (3.44)
A borrower would like to borrow 30,000 at 8% for 5 years, but would
like to pay only 2,000 for the first two years and then catch up with a
higher payment for the final three years. What is the payment for the
final 3 years?
Solution.
First we will use the calculator to find the loan balance in 2 years if the
payment each year is 2000.
Set PV = 30,000, PMT=-2,000 , N=2 and CPT FV= -30,832,00.
The balance is 30,832.
After the second payment of 2000, the borrower owes 30,832.. He can
pay this off in 3 years with a higher payment.
To find it, set PV=30,832, N=3, FV =0,1/Y=8 and CPT PMT=-11,963.85.
The payment for the final 3 years is 11,963.85.
Below we show the amortization table for the loan. I
1 Year
1 Payment
1 Interest
1 Principal
1 Balance
0
30,000.00
1
2,000.00
2,400.00
-400.00
30,400.00
2
2,000.00
2,432.00
-432.00
30,832.00
3
11,963.85
2,466.56
9,497.29
21,334.71
4
11,963.85
1,706.78
10,257.07
11,077.64
5 1
11,963.85
886.21
11,077.64
0.00 |
Note what happened in years 1 and 2. The total payment was less than
the interest required so the principal paid amount was negative. When
the negative "principal paid" was subtracted from the prior balance,
the effect was to add the shortfall to the balance of the loan. In period
1 the borrower has a shortfall of 400, and this means that he now owes
1 30,000+400 = 30,400. |
Such an increase in balance is called negative amortization because the amount
of principal amortized is negative. It is also referred to as capitalization of
interest since the unpaid interest is turned into capital when it is added to the
balance of the loan.
Negative amortization is now present in many United States mortgage loans
which were structured to have low initial payments. This facilitates the sale of
home, but is raising concerns about the soundness of the loans. What happens if
the borrower cannot make the payment when it increases?
Exercise (3.45)
Find the principal paid, interest paid and balance in year 1 if the initial
payment were 1500 instead of 2000.
I Answer : Interest=2400; Principal=-900; Balance=30,900
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M3-20
Module 3 - Loan Repayment
Section 3.9
Formula Sheet
• Amortization Method:
When a payment is made, it must be first applied to pay interest due and
then any remaining part of the payment is applied to pay principal
•
•
•
•
•
•
Loan Payment at time k:
Loan Balance after Pmtk
Loan amount:
Interest paid in Pmtk+1:
Principal paid in Pmtk+i:
Balance after Pmtk+1:
is made:
Pmtk
Balk
Bal0
Intk+1 =i(Balk)
PRink+1= Pmtk+i-i(Balk)
Balk+1 = Balk - PRink
= Balk-[Pmtk+1 - i(Balk)]
= (l + i)Balk -Pmtk+1
• For a level payment loan with payment PMT
o Interest paid in Pmtt : Intt = PMT(l - vn"t+1)
o Principal paid in Pmtt: PRint = PMTvn~t+1
• PRinn+k = (1 + i)kPRinn
• Prospective Method: The loan balance after a payment is the
present value of the remaining payments at the time of payment.
• Level Payment Loan Prospective Balance = PMTa^\ t •
• Retrospective Balance: PV(1 + i)k - FV( payments made at times l,..,k)
• Level Payment Loan Retrospective Balance = PV(1 + i)k - PMTs^
Sinking Fund Loon
L
• sfd = -
s^j
• Total sinking fund loan payment = + Li = L
(
s^
■ + i
• SFBalk=SFDsklj=L
• Principal Paid: SFBah - SFBal^ = SFD (s^ - s^) = SFD i1 + J)
• Net interest = (SFD + Li)- SFD(1 + jf'1 = Li-SFBal^ (j)
.xfc-l
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 3 - Loan Repayment Page M3- 21
Section 3.10
Basic Review Problems
1. A loan for 50,000 has level payments to be made at the end of each year for 10 years
at an annual rate of 9%. Find a) the balance at the end of 3 years and b) the principal
and interest paid in the third payment.
2. A borrower would like to borrow 50,000 at 7.5% for 5 years, but would like to pay
only 6,000 for the first two years and then catch up with a higher payment for the
final three years. What is the payment for the final 3 years?
3. You have a 20,000 loan at 7.2% annually for 8 years. You agree to pay off the
principal in installments of 2,500 per year, and to pay interest on the outstanding
balance each year. Find the interest due in the 6th payment.
4. For a 6.3% level payment loan, the amount of principal in the third payment is
845.28. Find the amount of principal in the 7th payment.
5. A loan made at an annual rate of 6% has 10 remaining payments of 1000. What is the
loan balance?
6. A loan at 7% annually has an initial payment of 250, and 9 further payments. The
payment amount increases by 3% each year. Find the loan balance immediately
after the 7th payment.
7. A loan at 6.5% annually has an initial payment of 300, and 9 further payments. The
payment amount increases by 20 each year. Find the loan balance immediately after
the 6th payment.
8. A thirty year monthly payment mortgage loan for 500,000 is offered at a nominal
rate of 8.4% convertible monthly. Find the a) monthly payment, b) the total
principal and interest that would be paid on the loan over 30 years c) the balance in
5 years and d) the principal and interest paid over the first 5 years.
9. A thirty year monthly payment mortgage loan for 325,000 is offered at a rate of
6.6%. The borrower would like to have graduated payments where the first year's
monthly payment is P and all subsequent monthly payments are P+500. a) Find the
initial payment P. b) Find the balance at the end of one year.
10. A thirty year monthly payment mortgage loan for 250,000 is offered at a nominal
rate of 6.3% convertible monthly. The lender charges a fee of 2.5% for which no
services are provided. Find the APR.
11. A 65,000 annual payment loan is made for a term of 10 years at 7.3% interest. The
lender wants only payments of interest until the end of year 10 when the 65,000
must be repaid. The borrower will make level annual year-end payments to a
sinking fund earning 4.8%. Find the level sinking fund deposit and the balance in
the sinking fund at time 5.
12. For the loan in problem 11, find the total payment and the principal in the 6th
payment.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M3-22
Module 3 - Loan Repayment
Section 3.11
Basic Review Problem Solutions
1. Use the AMORT worksheet. First input the loan and find the annual
payment of 7,791.00. Then key 2ND AMORT and enter 3 for both PI and P2.
The remaining balance after the 3rd payment is 39,211.77. The principal paid
is 3910.04 and the interest is 3880.96.
2. First we will use the calculator to find the loan balance in 2 years if the
payment each year is 6000.
Set PV = 50,000, PMT=-6,000 , N=2,1/Y = 7.5 and CPT FV=-45,331.25.
The balance is 45331.25.
After the second payment of 6000, the borrower owes 45331.25. He can pay
this off in 3 years with a higher payment.
To find it, set PV=45331.25, N=3, FV=0,1/Y = 7.5 and CPT PMT=-17,431.57.
The payment for the final 3 years is 17,431.57.
3. This is an installment loan
The interest due on the 6th payment is 540 and the total payment is 3040.
4. The level payment implies the amortization method.
We will use the formula PRinn+k = (1 + i)kPRinn to solve.
PRm3+4 = (1 + i)*PRin3 = (1.063)4 (845.28) = 1,079.28
5. This is the prospective method.
On the BA II Plus set PMT=-1000,1/Y=6, N=10 and CPT PV=7360.09.
6. The payments are 250, 250(1.03), 250(1.03)2,..., 250(1.03)9.
Immediately after the 7th payment the remaining payments are
250(1.03)7,250(1.03)8, 250(1.03)9. Using the prospective method, the
balance immediately after the 7th payment is the present value of those
remaining payments.
250(l.037) 250(l.038) 250(l.039)
Bal7 = —^ l + —A 2 } + V - ; = 830.24
1.07 1.072 1.073
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 3 - Loan Repayment
PageM3-23
7. First, we will show the (much faster) calculator solution, then the traditional
increasing annuity method, if you want to practice it.
Calculator method:
Use the CF worksheet with CF1=420, CF2=440, CF3=460 CF4=480 and find
the NPV with 1=6.5 CPT NPV=1536.22.
Traditional method:
The payments are 300, 320, 340,..., 480. Immediately after the 6th payment
the remaining payments are 420, 440, 460, 480. Using the retrospective
method, the balance immediately after the fourth payment is the present
value of those remaining payments. Using P = 420 and Q = 20 we have
Bal6=Pa^+Q
= 420^+20
ra*-nv^
v l j
/
= 420 (3.426) + 20
= 1,536.40
065
( 3.426 -4 (0.7773)
.065
8. 8.4% convertible monthly => 0.7% per month. The number of compounding
periods is 360 (30 years x 12 months).
a. Set PV = 500,000,1/Y=.7, N=360 and CPT PMT = -3809.19.
b. The total principal paid is just the amount of the loan -500,000.
The total interest can be found as
Total payments-Principal Paid=3809.19(360)-500,000=871,307.79
c. (Prospective method)
The balance is easy on the calculator.
Set N=60 (for 5 years) and CPT FV = -477,043.37.
d. The AMORT worksheet with Pl=l and P2=60 gives principal of
22,956.75 and interest of 205,594.65.
9.
a. The monthly rate is 0.55%. Use the equation of value
325,000 = Pamoo5S + v12 (500)
^3481.0055
= P156.5781 + .9363 (500) (154.861)
= P156.5781 + 72,498.19
This gives P=l,612.63
b. Use the prospective method. We can do this one on the calculator
because there is no step after the end of the first year. All
payments will be 2,112.63.
Set N=348,1/Y=0.55, PMT=-2112.63 and CPT PV=-327,163.90.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M3-24
Module 3 - Loan Repayment
10. First find the monthly payment. The monthly rate is 0.525%
Set N=360,1/Y=.525, PV=250,000 and CPT PMT=-1547.43. The 2.5% fee is
6,250, so the actual loan is for 250,000-6,250 = 243,750.
Set PV = 243,750 and CPT I/Y=.5452%. Multiply by 12 to get the nominal
APR rate of 6.5422%.
11. Sinking fund deposit.
Set FV=65,000,1/Y=4.8, N=10 and CPT PMT=-5216.23.
Balance at time 5.
Set N=5 and CPT FV=-28,708.06.
12. The annual interest payment is 65,000(.073) = 4,745.
Thus, the total payment is 4745 + 5216.13 = 9961.13
The principal in the 6th payment is (1 + j)5 SFD = 1.048s (5216.23) = 6,594.22 .
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 3 - Loan Repayment Page M3- 25
Section 3.12
Sample Exam Problems
1. (2005 Exam FM Sample Questions #4)
John borrows 10,000 for 10 years at an annual effective interest rate of 10%.
He can repay this loan using the amortization method with payments of
1,627.45 at the end of each year. Instead, John repays the 10,000 using a
sinking fund that pays an annual effective interest rate of 14%. The deposits
to the sinking fund are equal to 1,627.45 minus the interest on the loan and
are made at the end of each year for 10 years.
Determine the balance in the sinking fund immediately after repayment of
the loan.
(A) 2,130 (B) 2,180 (C) 2,230 (D) 2,300 (E) 2,370
2. (2005 Exam FM Sample Questions #9)
A 20-year loan of 1000 is repaid with payments at the end of each year.
Each of the first ten payments equals 150% of the amount of interest due.
Each of the last ten payments is X. The lender charges interest at an annual
effective rate of 10%. Calculate X.
(A) 32 (B) 57 (C) 70 (D) 97 (E) 117
3. (2005 Exam FM Sample Questions #15)
A 10-year loan of 2000 is to be repaid with payments at the end of each year.
It can be repaid under the following two options:
(i) Equal annual payments at an annual effective rate of 8.07%.
(ii) Installments of 200 each year plus interest on the unpaid balance at an
annual effective rate of i.
The sum of the payments under option (i) equals the sum of the payments
under option (ii).
Determine i.
(A) 8.75% (B) 9.00% (C) 9.25% (D) 9.50% (E) 9.75%
4. (2005 Exam FM Sample Questions #16)
A loan is amortized over five years with monthly payments at a nominal
interest rate of 9% compounded monthly. The first payment is 1000 and is to
be paid one month from the date of the loan. Each succeeding monthly
payment will be 2% lower than the prior payment.
Calculate the outstanding loan balance immediately after the 40th payment
is made.
(A) 6751 (B) 6889 (C) 6941 (D) 7030 (E) 7344
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M3-26
Module 3 - Loan Repayment
5. (2005 Exam FM Sample Questions #24)
A 20-year loan of 20,000 may be repaid under the following two methods:
i) amortization method with equal annual payments at an annual effective
rate of 6.5%
ii) sinking fund method in which the lender receives an annual effective
rate of 8% and the sinking fund earns an annual effective rate of j
Both methods require a payment of X to be made at the end of each year for
20 years.
Calculate j.
(A) j < 6.5% (B) 6.5% < j < 8.0% (C) 8.0% < j < 10.0%
(D) 10.0% < j < 12.0% (E) j > 12Wo
6. (2005 Exam FM Sample Questions #26)
Seth, Janice, and Lori each borrow 5000 for five years at a nominal interest
rate of 12%, compounded semi-annually.
• Seth has interest accumulated over the five years and pays all the
interest and principal in a lump sum at the end of five years.
• Janice pays interest at the end of every six-month period as it accrues
and the principal at the end of five years.
• Lori repays her loan with 10 level payments at the end of every six-
month period.
Calculate the total amount of interest paid on all three loans.
(A) 8718 (B) 8728 (C) 8738 (D) 8748 (E) 8758
7. (2005 Exam FM Sample Questions #28)
Ron is repaying a loan with payments of 1 at the end of each year for n
years. The amount of interest paid in period t plus the amount of principal
repaid in period t + 1 equals X.
Calculate X.
(A) 1 + — (B) 1 + — (C) l + vn-fi (D) l + vn-td(E) l + v^
i d
8. (2005 Exam FM Sample Questions #46)
Seth borrows X for four years at an annual effective interest rate of 8%, to
be repaid with equal payments at the end of each year. The outstanding loan
balance at the end of the third year is 559.12.
Calculate the principal repaid in the first payment.
(A) 444 (B) 454 (C) 464 (D) 474 (E) 484
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 3 - Loan Repayment
Page M3-27
9. (May OS #8)
A loan is being repaid with 25 annual payments of 300 each. With the 10th
payment, the borrower pays an extra 1000, and then repays the balance over
10 years with a revised annual payment. The effective rate of interest is 8%.
Calculate the amount of the revised annual payment.
(A) 157 (B) 183 (C) 234 (D) 257 (E) 383
10. (May 05 #2)
Lori borrows 10,000 for 10 years at an annual effective interest rate of 9%.
At the end of each year, she pays the interest on the loan and deposits the
level amount necessary to repay the principal to a sinking fund earning an
annual effective interest rate of 8%.
The total payments made by Lori over the 10-year period is X.
Calculate X.
(A) 15,803 (B) 15,853 (C) 15,903 (D) 15,953 (E) 16,003
11. (May 05 #25)
A bank customer takes out a loan of 500 with a 16% nominal interest rate
convertible quarterly. The customer makes payments of 20 at the end of
each quarter.
Calculate the amount of principal in the fourth payment.
(A) 0.0 (B) 0.9 (C) 2.7 (D) 5.2
(E) There is not enough information to calculate the amount of principal.
12. (Nov 05 #18)
A loan is repaid with level annual payments based on an annual effective
interest rate of 7%. The 8th payment consists of 789 of interest and 211 of
principal.
Calculate the amount of interest paid in the 18th payment.
(A) 415 (B) 444 (C) 556 (D) 585 (E) 612
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M3-28
Module 3 - Loan Repayment
Section 3,13
Sample Exam Solutions
1.
John owes interest of 10,000 (.10) = 1000 at the end of each year. He puts the
remaining 627.45 in the sinking fund at 14%.
We can find the accumulated value of the sinking fund in 10 years using the
financial calculator with PMT = 627.45, N=10 and I/Y = 14. Computing FV gives
the accumulated value of 12,133.19.
John must use this accumulated value to pay the loan amount of 10,000. The
balance in his account is 12,133.19-10,000.00=2,133.19.
Answer A
2.
The loan has varying payments for the first 10 years and then a level payment
X for the final 10 years. We will look at the first few payments to examine the
pattern of the first 10 payments.
1 Time
1 Interest due
1 Payment
1 Principal paid
| Balance
0
1000.00
1
100.00
150.00
50.00
950.00
2
95.00
95(1.5)=142.50
47.50
902.50
Note that the balance is decreasing, but not arithmetically. A good guess would
be that the decrease is geometric, and this is borne out by the observation that
each entry in the table is 95% of the previous. If we denote the interest due, the
payment, the principal paid and the balance immediately after the payment for
time k < 10 by INTk, PMTkyPRINk and BALK we can see this relationship more
formally.
INTk = .10BALk.x PMTk = .15BALk_!
PRINk = PMTk - INTk = .05BALk_i
BALk =BALk-! -PRINk = BALk_i - .05BALk_i = .95BALk_i
This means that the balance at time 10 immediately after the tenth payment is
1000 (.95)10 =598.74.
At this point we have a level payment loan with payments of X for 10 more
periods at a rate of 10%. The level payment X can be found using the financial
calculator with PV=-598.74,1/Y=10 and N=10. The computed PMT is 97.44 = X.
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 3 - Loan Repayment
PageM3-29
3. We will first find the total payments for each of the two options.
Option (i).
We can calculate the payment for option (i) using the calculator with PV=2000,
N = 10 and 1= 8.07. The payment is 299, so that the total payments are 2990.
Option (ii).
Note that if full interest is paid on the unpaid balance each year, the additional
payment of 200 is a payment of principal. We do not know the unknown rate i,
but we can derive an expression for the total payments in terms of i. A partial
table is helpful to visualize the payments.
1 Time
1 Balance after
payment
1 Interest paid
| Principal paid
0
2000
1
1800
2000 i
200
2
1600
1800 i
200)
• • •
9
200
400 i
200)
10
0
200 i
200
Total Principal Paid =2000
Total interest paid=i[2000 +1800 + ... + 200] = 200i[10 + 9 + ...1] = 200i[55] = ll,000i
Total Payments = 2000 + llOOOi
Since total payments for both options are the same, 2990 = 2000 + llOOOi i = .09
Answer B
4.
With a nominal rate of 9%, the monthly rate is .75%. The sequence of 60
payments is 1000,1000(.98),1000(.98)2,...,1000(.98)59.
The loan balance after the 40th payment is made is the present value of the
remaining 20 payments at time 40.
" .9840 .9841 .98s9
1000
1000
1.0075 1.00752
( _9840 A
1,1.0075
1 +
1.007540
.98
.98 \
1.0075 ){1.0075
\2
..+
.98
-f
1.0075 J
1000
r .9840 "|
v 1.0075 J
U-0075,
i-f—
U-0075
20"
)
= 6889.11
Answer B
S.
For method (i), we have PV = 20,000, N=20 and I/Y = 6.5. Use the financial
calculator to compute the annual payment PMT = -1815.13. Therefore, the X
referred to in the problem is 1815.13.
For method (ii), the total payment made is also X=1815.13. The 8% interest paid
each year is .08(20,000) = 1600. This leaves 215.13 for deposit to the sinking
fund. We need the sinking fund to accumulate to FV = 20,000.over N = 20 years
with an annual payment of PMT = -215.13. We can find the yield by using the
BAII Plus calculator to compute I/Y = 14.18.
Answer E
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page M3-30
Module 3 - Loan Repayment
6.
Total Interest Paid = Total of All Payments - Loan Amount
Seth.
Seth has a rate of 12% -r 2 = 6% per semiannual period for 5 years (10 periods).
At the end of five years he will make a single total payment of
5000 (1.06)10 = 8954.24 . His total interest is 8954.24 - 5000 = 3954.24.
Janice pays 6% interest at the end of each semiannual period and repays the
principal at the end. Thus she pays .06x5000 = 300 interest ten times. Her total
interest is 10(300) = 3000.
Lori has a level payment loan with N = 10, PV = 5000 and I/Y = 6. Use the
financial calculator to find PMT = -679.34. Her total interest is
10 (679.34) - 5000 = 1793.40 .
Total Interest Paid on all loans is the sum 3954.24 + 3000 +1793.40 = 8747.64
Answer D
7.
The key identities for amortization of a level payment loan with payment of 1
are stated below for period t, immediately after the payment is made
Period t: Interest paid = 1 - vn~t+1 Principal paid = vn_t+1
Thus for Ron's loan
Interest pd in period t + Principal pd in period t+1 = (1 - vn_t+1) + vn_(t+1)+1
= (l-vn"t+1) + vn-t
= l + vn-t(l-v) = l + vn"td
Answer D
8.
Let P be the unknown monthly payment. We will use the loan amortization
schedule formulas here. The outstanding balance at the end of the third year is
Parm = Pa* = Pv = — .= 559.12. -+ P = 604.85.
1.08
The principle paid in the first payment is
1
Pv4"1+1=Pv4= 603.85
Answer A
f 1 A4
1.08
443.85
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 3 - Loan Repayment
PageM3-31
9.
We will first find the amount of the original loan using the BA II Plus.
Set PMT=-300, N=25,1/Y = 8 and CPT PV = 3,202.43 That is the amount of the
original loan.
Next we look at what happens at the time of the 10th payment. The balance after
the regular payment of 300 is made can be found by setting N=10 and
computing FV. The balance is 2,567.84. The additional payment of 1,000 reduces
the balance to 1,567.84.
The loan is now revised to pay off the remaining balance over 10 years and we
need to find the annual payment on that revised loan.
Set PV=1567.84, N=10,1/Y=8, FV=0 and CPT PMT = -233.65.
Answer C
10.
Lori pays interest of 900 (9% of 10,000) each year. She also makes a sinking
fund deposit to accumulate 10,000 in an 8% account in 10 years. On the BA II
Plus, set FV=10,000, N=10,1/Y=8 and CPT PMT= -690.29. Thus each year she
pays a total of 900 + 690.29 =1,590.29. In ten years she pays 15,902.90.
Answer C
11.
The nominal rate of 16% convertible quarterly translates to a rate of 4% per
quarter. Note that the interest on 500 at 4% is 20. Thus the customer's
payments of 20 are covering interest only, and no principal is paid in any
payment.
Answer A
12.
The principal paid in the kth payment is PMTvn~k+1, where PMT is the full level
payment. Note that we are not given n. We do know that the total level payment
is 789 + 211 = 1000. We are given the principal paid in the 8th payment is
PR8= 211 = 1000vn"8+1.
The principal paid in the 18th payment is
PR1S = 1000vn"18+1 = v101000vn-8+1 = v-10211 = 1.0710 (211) = 415.07 .
Since all payments are 1000, the interest paid in the 18th payment is
1000-415.07 = 584.93
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Has sett, Ratliff, Garcia, & Steeby

Page M3-32
Module 3 - Loan Repayment
Section 3.14
Supplemental Exercises
1. A man has a 200,000 home loan for 30 years at a nominal annual rate of
7.5% convertible monthly. Find
(a) His monthly payment
(b) His balance after 12 years
(c) The amount of interest paid in the 40th payment.
2. Suppose that at the end of 10 years the man in Problem 1 is able to
refinance the balance of his loan at 6% convertible monthly. What is his
new monthly payment?
3. A woman has a loan of 55,000 at an annual rate of 6.8% for 10 years. She
makes annual end of year payments on the principal plus interest on the
unpaid balance. Her principal payments start at 1000 and increase by
1000 each year thereafter. What is her total payment for year 6?
4. A loan at 5.8% has level annual end of year payments. The principal
repaid in the 8th payment is 1234.08. What is the amount of the principal
repaid in the 15th payment?
5. A fixed rate loan has level annual payments. The principal repaid in the
5th payment is 1489.40. The principal repaid in the 15th payment is
2795.81. What is the annual interest rate on the loan?
6. A man has a loan of 50,000 for 10 years at 6.5% annually with annual
payments. His payments are 4500 for the first 5 years and X for the next
5 years. Find X.
7. A company has a loan of 80,000 to be paid in 20 annual level payments.
The interest and the principal repayment in the 13th payment are the
same. Find the amount of principal repaid in the 6th payment.
8. On a loan of 50,000 for 20 years at 6.2% annually the lender wants the
interest paid annually and the principal repaid at the end of the 20 years.
The borrower makes annual level payments into a sinking fund to raise
the 50,000. The fund earns 5.8% annually. What are the borrowers total
annual payments?
9. A man borrows 250,000 for 30 years at 6.8% annually. He agrees to make
annual payments of 15,000 for the first 10 years, 15,000 + P for the next
10 years and 15,000 + 2P for the last 10 years. Find P.
10. For Problem 9, find the balance at the end of year 20.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 3 - Loan Repayment Page M3-33
Section 3.15
Supplemental Exercise Solutions
1. (a) Using the BA II Plus calculator set N = 360,1/Y = 0.625, PV =
200,000 and FV = 0. Then CPT PMT = 1398.43. The payment is 1398.43.
(b) To get the balance after 12 years (144 months) reset N = 216, the
number of months remaining. Then CPT PV = 165,499.78.
(c) The interest in the 40th is PMT(1 - v360A0+1) =
1398.43[1 - (1/1.00625)321] = 1209.17.
2. To get his balance at the end of 10 years (240 months left) set
N= 240,1/Y = 0.625, PMT = -1389.43 and FV = 0.
Then CPT PV = 173,589.97. To get his new payment reset I/Y = 0.5.
Then CPT PMT = -1243.65. New payment is 1243.65.
3. At the end of year 5 the woman has paid 15,000 towards the principal, so her
balance is 40,000. The interest due at the end of year 6 is 40,000(0.068) = 2720.
Total payment for year 6 is 8720.
4. If PRink is the principal repaid in the Jeth payment,
then PRink+n = (1 + i)nPRinn. Hence PRin15 = (1.0587)PRin8.
PRin15 = 1.4839(1234.08) = 1831.23.
5. PRin15 = (1 + i)10PRin5 => 2795.81 = (1 + i)10(1489.40)
Hence i = (2795.81/1489.40)1/10 - 1 = 0.065
6. Using the BA II Plus calculator, set N = 5, I/Y = 6.5, PV = 50,000 and
PMT = -4500. Then CPT FV = -42,882.95 is the outstanding balance. To find X
change PV = 42,882.95 and FV = 0.
Then CPT PMT = -10,319.12. X = 10,319.12.
7. PRiriu = PMTv2013+1 = PMT/2 => v8 = Vi =>i = 0.0905. To get the payment set
N= 20, I/Y = 9.05, PV = 80,000 and FV = 0. Then CPT PMT = -8794.97.
The principal repaid in the 6th period is
PMTv20-6+1 = 8794.97Q/1.0905)15 = 2398.00.
8. The annual interest on the loan is 50,000(0.062) = 3100. To find the payments into
the sinking fund on the BA II Plus set N = 20, I/Y = 5.8,
PV = 0 and FV = 50,000. Then CPT PMT = -1388.72.
The total annual payments are 1388.72 + 3100 = 4488.72.
9. The present value of the man's payments is
15,000a^0t068 +Pv10a^0.068 +Pv20a^0,68 =250,000
a3oio.o68 = 12.6625, a2oio.068 = 10.7607, amMS = 7.0890
v10 = 0.51795 and v20 = 0.26827.
15,000(12.6625) + P[0.51795(10.7607) + 0.26827(7.0890)] = 250,000
P = 8,034.83
10. For the last 10 years the payments are 15,000 + 2P = 31,069.66.
The balance due is 31,069.66 amo6s = 31,069.66(7.0890) = 220,252.82.
(Note: for the first 10 years there was negative amortization.)
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 4 - Bonds
PageM4- 1
Section 4,1
Introduction to Bonds
Corporations need to find money to start projects. One way to get funding is to
sell stock in the company, but that expands the ownership of the company -
every stockholder is an owner of a proportional share of the company. Another
way to get money is to borrow it. Corporations can and do borrow money from
banks and other lenders, but a more widely used method of borrowing is to
issue bonds. Bonds are also issued by national governments, municipalities,
school districts and various other entities. We will illustrate how bond
borrowing works with a simple example.
Example (4,1)
Company A needs $100,000,000 to build new manufacturing facilities.
The company creates a bond issue consisting of 100,000 bonds in
denominations of $1,000. This amount is called the face value or par
value of the bond.
The bond will pay a nominal interest rate of interest of 10% convertible
semiannually for 10 years and then return the face value of 1000 along
with the final interest payment. The 10% rate is called the coupon rate
of the bond. The $1000 return of the face value is called the redemption
value. Most bonds are issued so that the final redemption value equals
the face value, although it is possible to have a redemption value that
does not equal the face value.
Individual investors who wish to earn interest can buy these bonds. The
investors are making a loan to Company A. If the Company A can sell all
100,000 bonds at $1000 each they will have the $100,000,000 they need.
In the above example, Company A's bond pays 10%. As you are probably aware,
interest rates change daily. If the current rates an investor can earn on a
similar investment are higher than 10%, an investor will not want to pay full
price for Company A's bond. Conversely, if rates on similar investments are
less than 10%, investors will be willing to pay more than the $1,000 face value.
This is demonstrated in the following exercise and example.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M4-2 Module 4 - Bonds
Example (4,2)
Company A puts its bonds up for sale on a day when investors in the
marketplace are demanding an earnings rate of 10.2% convertible
semiannually or 5.1% per semiannual period. They want to buy each
bond at a price that gives that yield. This price is easy to find on the BA
II Plus. In 10 years each 1000 bond will give 20 semiannual payments of
$50 and a final payment of $1000 at the same time as the last payment of
$50. The price is the present value of this series of payments. Set
PMT=50,FV=1000, N=20,1/Y=5.1 and CPT PV=-987.64. The investors will
only pay $987.64 for each bond, and Company A will collect a total of
$98,764,000, which is $1,236,000 short of the amount needed.
Exercise (4,3)
Suppose the bonds were sold on a day when the desired interest rate was
9.8% convertible semiannually. What is the price of an individual $1,000
bond?
Answer: 1012.57
Note that when interest rates demanded by investors went up, the price of the
bond went down below the redemption value of 1000. In this case the bond is
said to have sold at a discount of 12.36. When interest rates demanded by
investors went down, the price of the bond went up above the redemption value
of 1000. In this case the bond is said to have sold at a premium of 12.57.
It is important to remember that the price of the bond moves in
the opposite direction from interest rates.
The examples above are simplified to illustrate the basics. In practice, there
are also expense and risk issues associated with issuing bonds. We will not
review these issues here, since they are covered in finance courses. Instead, we
will focus on the mathematics of bonds.
Note that in the above examples we stated coupon rate and yield as nominal
rates convertible semiannually. This is the most common practice in Canada
and the United States. However an issuer (or an exam question) can use any
conversion period. Annual rates have been used in Europe and are used in some
Exam FM questions.
Also in the above examples we assumed that the borrower would pay back the
face value at the maturity of the bond. It is possible for the borrower to offer a
different final payment.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 4 - Bonds Page M4- 3
Example (4.4)
The company issuing the bonds of Example (4.1) decides to make the
bonds more attractive by offering to pay $1100 at the time of the final
payment of $50. If the bonds are purchased to yield 10.2%, we can find
the price of this new bond on the BA II Plus.
Set PMT=50, FV=1100, N=20,1/Y=5.1 and CPT PV=-1,024.62.
Exercise (4.5)
What is the price of the bond in (4.4) if the required yield is 9.8%
convertible semiannually?
Answer: 1050.98
The final amount of 1100 in example (4.4) is called the redemption value.
If no separate redemption value is specified, you can assume that the bonds will
be redeemed at par.
Note that raising the redemption value to 1100 resulted in a price above 1000. It
is natural to ask what redemption value would give a price of 1000.
Example (4.6)
Suppose that the company issuing the bonds of Example (4.1) wants to
raise the redemption value to keep the price at 1000 if there is a raise in
the required nominal rate to 10.2% We can find the required redemption j
value on the BA II Plus.
Set PMT=50, PV =-1000, N=20,1/Y=5.1 and CPT FV= 1033.42.
| That is the required redemption value. |
Exercise (4.7)
What redemption value would assure a price of 1,000 for the above
bonds if the required nominal rate were 10.1%
Answer: 1016.62
Investors are often offered a bond at a price. In this case, they would like to
find the yield that they would earn with the offered price.
Example (4.8)
A 1000 par value bond with a term of 10 years and a coupon of 10%
convertible semiannually is offered at a price of 990. We can find the
yield per semiannual period on the BA II Plus.
Set PMT=50, PV = -990, N=20, FV=1000 and CPT. I/Y=5.08.
| This gives a nominal yield convertible semiannually of 10.16%.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M4-4
Module 4 - Bonds
Exercise (4.9)
A 1000 par value bond with a term of 10 years and a coupon of 10%
convertible semiannually is offered at a price of 1020. Find the implied
yield.
Answer: 9.68
You have certainly observed by now that we have solved all problems to this
point on the financial calculator, without a single mathematical notation or
formula. It is important to recognize when exam problems can be done so easily
and directly. As always, there are notations and formulas to learn and problems
that require their use. The key variables are:
F = par value
r = coupon rate
Fr = coupon amount
C = redemption value (usually = F)
n = number of periods to redemption
P = price
i = yield per period to investor at price P
1
l + i
The most basic formula for the price P is
(4.10)
P = PV(coupons) + PV (redemption payment)
-or-
P = (Fr)a^+Cv?
If the bond is redeemed at par, we have F = C, and then P = (Fr) a^ t + Fv?.
This can be used to derive a formula in terms of premium or discount.
When F = C
(4.11)
P = F+ F(r~i)a^
price face
value premium or discount
We can illustrate this best with another example.
Example (4.12)
Consider again a 1000 bond redeemable at par in 10 years with a nominal
rate of 10% convertible semiannually. This bond has F=1000, r=.05 and n
= 20. We have already shown in example (4.2) that the price of this bond
at a yield of 10.2% is 987.64. We can check this using the above formula
with i = .051.
P = F + F(r-i)a^li= 1000 +1000 (.05 - .051) am 051
= 1000 + (-1) a^ 051 = 1000 -12.36 = 987.64
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 4 - Bonds
Page M4- 5
Note that the term F{r-i)a^\i gave us the discount of 12.36 on the bond. The
discount is the present value of the difference between the actual coupon and
what the coupon would have been if the coupon rate was i. This formula is
referred to as the premium-discount formula.
Exercise (4.13)
Use (4.11) to verify the price of the premium bond in Exercise (4.3)
There are a number of other possible formulas for the price of a bond, but we
have found that most problems for which these formulas were used historically
can now be solved using the BA II Plus directly.
Mathematics of Investment and Credit has another formula worth mentioning,
Makeham's Formula:
(4.14)
If we let K = Fvni9 then
P = K + ?r(F-K)
Example (4.15)
Consider again the 1000 bond redeemable at par in 10 years with a
nominal rate of 10% convertible semiannually to be purchased at a
nominal rate of 10.2% convertible semiannually. For this bond
1000
K = -
1.0512
= 369.78
r OS
P = K + ^(F-K) = 369.78 + -^-(1000 - 369.78) = 987.64
The specialized formulas in (4.11) and (4.14) are based on the
assumption that the bond is redeemable at par and F-C.
If F *C, you must analyze the problem from first principles (see
#12 in the Sample Exam Problems for this module for an example
of a problem of this type,)
Formula (4.17) and the amortization formulas in the next section
all have this same disclaimer.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M4-6
Module 4 - Bonds
Section 4.2
Amortization of Premium or Discount
Given the fluidity of interest rates, bonds are usually sold at a premium or
discount. In this case the premium or discount must be amortized for
accounting and valuation purposing. The method is similar to the method used
in the last module for amortizing loans. We will illustrate this with an example
of amortization of premium.
Example (4.16)
1 A three year $1000 par bond has a coupon rate of 6% convertible 1
semiannually. It is sold at a yield of 5% convertible semiannually. We can
see immediately that this will be a premium bond. We find the price using
the BA II Plus.
Set FV = 1000,1/Y = 2.5, PMT = 30, N=6 and CPT PV=-1027.54.
The price is 1027.54 and the premium is 27.54.
Thus, the buyer of the bond has an investment of 1027.54 which pays
interest at the true yield rate of 2.5%. Now we will break down the first
payment using the amortization method.
First payment: 30
Interest Paid: 1027.54 (.025) = 25.69
Principal Paid: 30-25.69 = 4.31
The table below shows the result of continuing this process over the life
of the bond. I
| Period
| Coupon
| Redemption Value
| Interest Paid
| Principal Paid
| Balance
| Premium
| Amortized Premium
0
1027.54
27.54
1
30
25.69
4.31
1023.23
23.23
4.31
2
30
25.58
4.42
1018.81
18.81
4.42
3
30
25.47
4.53
1014.28
14.28
4.53
4
30
25.36
4.64
1009.64
9.64
4.64
5
30
25.24
4.76
1004.88
4.88
4.76
6 1
30
1000
25.12
4.88
1000.00
0
4.88 |
Under the amortization method for this premium bond, part of the coupon
is a payment of principal. As the principal amount is reduced in each
period, the premium is lowered by the amount of principal paid. Thus the
amount of principal paid is referred to as the amount of amortization of
premium. When the final coupon is paid, the balance owed is equal to the
| original amount of 1000 which is paid off by the redemption payment. |
The table in the example above illustrates how the method works, but for exam
questions you do not want to build the entire table. Fortunately, the
amortization method and the premium discount formula can be used to derive a
simple formula that is used on exam problems.
For a bond that is redeemable at face value:
(4.1 /; | Amortization of premium in period k = F(r-i)vn~k+1
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 4 - Bonds
PageM4- 7
You can read the derivation of (4.17) in Mathematics of Investment and Credit
(page 240, Table 4.3). We find this to be simple to remember since it uses the
same power of v as the loan amortization formula in Module 3.
Example (4.18)
For the bond in, (4.16), the amortization of premium in period 5 is found
using (4.17) with F = 1000.
r-i = .03-.025 = .005, n-k + 1 = 6-5 + 1 = 2.
The result is identical with the number shown in the preceding table.
f 1 \2
F(r-i)vn-k+1= 1000 (.005)
1.025
= 4.76
Exercise (4.19)
Verify the amortization of premium in period 3 in the preceding table
using (4.17).
In the next example we will look at ariortization of discount.
Example (4.20)
1 A three year
semiannually
see immedia
the BA II Pli
Set FV = 100
The price is
Thus the buy
interest at th
payment usii
First i
Intere
Princi
The table be]
| Period
| Coupon
| Redemption Value
| Interest Paid
| Principal Paid
| Balance
| Discount
| Amortized Discount
$1000 par bond has a coupon rate of 6% convertible 1
y. It is sold at a yield of 7% convertible semiannually. We can
tely that this will be a discount bond. We find the price using
is.
0,1/Y = 3.5, PMT = 30, N=6 and CPT PV=-973.36.
973.36 and the discount is 26.64.
rer of the bond has an investment of 973.36 which pays
e true yield rate of 3.5%. Now we will break down the first
lg the amortization method.
)ayment: 30
st Paid: 973.64 (.035) = 34.07
pal Paid: 30-34.07 = -4.07
low shows this continuing process over the bond's life: |
0
973.36
26.64
1
30
34.07
-4.07
977.43
22.57
4.07
2
30
34.21
-4.21
981.63
18.37
4.21
3
30
34.36
-4.36
985.99
14.01
4.36
4
30
34.51
-4.51
990.50
9.50
4.51
5
30
34.67
-4.67
995.17
4.83
4.67
6 1
30
1000
34.83
-4.83
1000
0
4.83 1
Under the amortization method for this discount bond, the actual coupon
payment is less than the interest due This is called negative amortization
in which the interest shortfall is added to the principal balance. Thus the
amount of principal added back is sometimes referred to as the amount
for accumulation of discount. When the final coupon is paid, the balance
owed is equal to the original amount of 1000 which is paid off by the
1 redemption payment. |
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M4-8
Module 4 - Bonds
Formula (4.17) works here too.
For a bond that is redeemable at face value:
(4.21)
Negative amortization of discount in period k = F(r-i)v
n-k+l
Example (4.22)
For the bond in (4.20) the amortization of discount in period 5 is found
using (4.21) with F = 1000.
r-i = .03 -.035 = -.005 n-/c + l = 6-5 + l = 2.
The result is identical with the number shown in the preceding table.
= -4.67
F(r-i)y"-fc+1 =1000(-.005)(^r^j
Exercise (4.23)
Verify the amortization of discount in period 3 in the preceding table
using (4.21).
As we have seen in Module 3, the amortized amount increases geometrically.
Amortized amount in period k F(r-i)vn~k+1
Amortized amount in period k+m. F(r-i) v*-(fc+™>+1 = (i + i)mF(r -i)vn~k+1
This leads to questions similar to those seen in Module 3.
Example (4.24)
A premium bond is purchased to yield 4% convertible semiannually. The
amount of premium amortized in the second payment is 8.37. Find the
amount of premium amortized in the 7th payment.
Solution. 8.37 (1.02)5 = 9.24
Exercise (4.25)
A premium bond is purchased to yield 4% convertible semiannually. The
amount of premium amortized in the third payment is 4.10. Find the
amount of premium amortized in the 6th payment.
Answer: 4.35
You can also use the AMORT feature of the BAII Plus to find the numbers in
the preceding examples. To apply AMORT to the premium bond in Example
(4.16), re-enter the bond information. Set FV = 1000,1/Y = 2.5, PMT = 30, N=6
and CPT PV=-1027.54. Then go to 2ND AMORT. To check the first entry, enter
Pl=l and P2=l. Scroll down and you will see PRIN=4.31, the same result given
in the table. Check the final entry by entering Pl=6 and P2=6. Scroll down and
you will see PRIN=4.88, the same result given in the table.
If you apply AMORT to the discount bond in (4.20) you will see the same
numbers again. This can be a valuable time-saving tool.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 4 - Bonds PageM4- 9
Section 4.3
Callable Bonds
Sometimes it is advantageous to pay off a loan early. In recent years when
mortgage rates dropped,many Americans re-financed their home loans by
taking out a new loan at lower rates and using the proceeds to pay off the old
higher-rate loan. Corporations often have similar motivations to pay off their
bonds early. Thus, some bonds are designed with call provisions that allow
them to be paid off or "called" at some specified future date before maturity.
Any bond that does not have call provisions must pay coupons until maturity.
When you buy a callable bond, the price is based on either the call date or the
final maturity date. The rule is given below and is based on the worst case of
the two choices.
Maturity to use in pricing
Type of Bond
Premium Bond
Discount Bond
a callable bond:
Find N using
Earliest Possible Redemption Date
Latest Possible Redemption Date
If the bond is a 10-year semiannual bond, the full maturity is N=20. If the bond
is also callable in 5 years, the call period gives N=10. Thus if the latest possible
maturity is N=20 and the earliest is N=10, the investor will price the bond using
N=20 for a discount bond and N=10 for a premium bond.
We will illustrate this further in the next examples.
Example (4.27)
A 10 year 1000 bond has a 10% coupon rate convertible semiannually. It
is callable in 6 years. An investor wishes to buy the bond to yield 8%
convertible semiannually. This means this is a premium bond.
Using (4.26), the investor would price the bond using N=12. Using the BA
II Plus set N=12,1/Y=4, PMT=50, FV=1000 and CPT PV = -1093.85.
To understand the reasoning behind this, take a look at what the price would be
if the investor used the latest maturity date instead:
Set N=20 and CPT PV = -1135.90. That gives a higher price.
The investor is protected by choosing the maturity that gives the lowest price.
Another way to think of this is to remember that a premium bond pays a rate of
interest that is above the desired yield. If this high interest payment is cut off
early, there is a loss of value to the investor.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M4-10
Module 4 - Bonds
Example (4.28)
A 10 year 1000 bond has a 10% coupon rate convertible semiannually. It
is callable in 6 years. An investor wishes to buy the bond to yield 12%
convertible semiannually. This means this is a discount bond.
Using (4.26), the investor would price the bond using N=20. Using the BA
| II Plus set N=20,1/Y=6, PMT=50, FV=1000 and CPT PV = -885.30. |
To see the reasoning behind this discount strategy, take a look at what the price
would be if the investor used the earliest maturity date.
Set N=12 and again CPT PV = -916.16. That gives a higher price.
The investor is protected choosing the maturity that gives the lowest price.
Another way to think of this is to remember that a discount bond earns interest
by recapturing discount. Early recapture of discount at a call raises yield, but
later recapture lowers it.
Exercise (4.29)
A 5 year 1000 bond has a 6% coupon rate convertible semiannually. It is
callable in 2 years. At what price should the investor buy the bond to
yield 6.2% convertible semiannually?
Answer: 991.51 J
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 4 - Bonds Page M4-11
Section 4.4
Pricing Bonds Between Payment Dates
Every problem we have done so far used an integer for N, implying that every
calculation was either on the bond's origination date or on a coupon date
immediately after the coupon was paid. However bonds are bought and sold
daily, and we need to discuss how to handle pricing a bond between payment
dates. If a bond is priced between payment dates the number of periods would
be fractional. In general the fractional period t is defined by
number of days from last coupon date to settlement date
t = -
number of days in the bond period
As usual, we will give an example to make this concrete:
Example (4.30)
In this problem, we will give you the number of days between dates, as an exam
problem might Later, we will show how to use the BAII Plus to find the number of days
between dates.
A bond with par value of 1000 has payment dates of January 21 and July
21. The nominal coupon rate convertible semiannually is 6%. The bond
matures on January 21, 2009. On January 21, 2007 a coupon payment of
30 was made. The bond is sold 45 days later on the settlement date of
March 7, 2007 to yield 8% convertible semiannually. There are 181 days
between the coupon payment dates of January 21, 2007 and July 21,
2007. Thus the fraction of the bond period that the seller of the bonds
45
owned them was = 0.24862
181
A timeline is helpful for visualization:
Seller: 45 days Buyer: 136 days Coupon to buyer
,—A—v ~ v
I 1 1
1/21 3/7 7/21
On January 21, immediately after the coupon payment was made, there
were 4 coupon payments of 30 remaining. At that point, we can find the
price of the bond using the BA II Plus. Set N=4, PMT=30, FV = 1000,1/Y = 4
and CPT PV = -963.70.
In the above example, the seller of the bond is entitled to interest at 4% per
semiannual period, and will require the value of 963.70 plus compound interest
for the fractional period. This total sale price includes accrued interest for the
fractional period, so it is referred to as the price-plus accrued of the bond. It is
also referred to as the flat price.
45
Price-plus-accrued = 963.70 (1.04) isi =973.14
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M4-12
Module 4 - Bonds
Part of the total sale price of 973.14 is accrued interest in addition to the
principal value. The buyer needs to establish a true price for the bond. The true
price is the total price less the accrued interest.
45
In this case the buyer will note that a fraction of -— of the next coupon
lol
payment represents an interest amount that should be subtracted from the total
sale price to get book value. This gives
/45^
Accrued interest = 30
181
:7.46
Price = (Price-plus-accrued) - (Accrued interest) = 973.14 - 7.46 = 965.68
The price is also referred to as the market price.
Following Mathematics of Investment and Credit, we use P0 to denote the price
immediately after the last coupon date and Pt to denote the price at the date of
sale (settlement date). If iis the required yield per period for the buyer, then
(4.31)
Price-plus-accrued = P0 (1 + i)
If F is the face value of the bond and r is the coupon rate, then
(4.32)
Accrued interest =t(Fr)
(4.33)
Price = (Price plus accrued) - (Accrued interest)
= P0(l + i)t-t(Fr)
Notice that we have used compound interest in finding the price-plus-accrued
and simple interest in finding the accrued interest. This is the convention that
is used in section 4.1.2 of the official Exam FM reference text Mathematics of
Investment and Credit. However problems need to be read carefully to assure
that some other convention is not being specified. Problems 4.1.25, 4.1.26 and
4.1.32 of Mathematics of Investment and Credit explore alternate conventions.
Exercise (4.34)
Find the market price and accrued interest of the bond in Example (4.30)
if it is purchased on March 11, 2007 to yield 7%.
Answer:
Price=982.70;
Accrued interest=8.12
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 4 - Bonds
PageM4-13
The BA II Plus has a bond worksheet that will find prices on a given settlement
date. We will go through Example (4.30) to illustrate this. There are a number
of entries in the process, but they are natural and easy to learn. We like the
bond worksheet as a time saver.
Example (4.35)
The BOND legend appears above the § key. Enter the worksheet by
keying 2ND BOND. You will see the display SDT=. This is where you
enter the settlement date. To enter March 7, 2007, key in the number
3.0707 and press the ENTER key. Note: the method of date entry is to enter a
number which has the month before the decimal point and then two digits for
the day and two digits for the year after the decimal point. Years from 00 to 49
are read as 21st century years and years from 50 to 99 are read as 20th century.
Now scroll down and you will see CPN. Enter the value of 6 given in the
problem. Scroll down again and you will see RDT=. This is where you
enter the redemption date. Enter the number 1.2109. Next you scroll to
RV=. Here you enter the redemption value as a percent of the face value.
Enter 100 for a par bond.
At the next scroll down you will see either ACT or 360. Bonds can be
priced using either actual days (ACT) or the mortgage convention that a
year is composed of 12 months of 30 days each (for 360 days). Use 2ND
SET to select ACT and press ENTER.
Scroll down again and you will see either 2/Y for semiannual coupons or
1/Y for annual coupons. Use 2ND SET to select 2/Y and press ENTER.
Scroll down again and you will see YLD=. Enter the required yield of 8
from the problem. Scroll down again and you will see PRI= . Hit CPT and
I you will see the price of 96.569. This is given as a percent of the face
value, and translates to 965.69 for a $1000 bond.
Scroll down again and you will see AI= . Hit CPT and you will see the
accrued interest of 0.746. This is given as a percent of the face value,
and translates to 7.46 for a $1000 bond. Note that these numbers match
the answers we obtained in Example (4.28).
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M4-14
Module 4 - Bonds
Calculator Note
Note that the BOND worksheet automatically finds the days between the
dates involved, although it does not display them. There is also a DATE
worksheet which will find days between dates. The DATE legend appears
above the [l] key.
As a working example, we will show you how to find the number of days from
January 21, 2007 to July 21 2007:
Enter the worksheet by keying 2ND DATE. You will see the display DT1=.
This is where you enter the first date. To enter January 21, 2007, key in the
number 1.2107 and press the ENTER key. Scroll down and you will see the
display DT2=. Key in the number 7.2107 and press the ENTER key. Scroll
down again and you will see DBD =. This is where you can compute the days
between dates, but you first scroll down again to see the display where you
choose between ACT and 360. Make sure that you have chosen ACT and the
scroll back to DBD and CPT DBD = 181. Note that if you are in 360 mode the
answer would be 180.
Exercise (4.36)
Calculate the number of days until January 1.
You could also calculate the number of days that you have been alive if
you were born in 1950 or later. This author is out of luck, since I was born
in 1941.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 4 - Bonds Page M4-15
Section 4,5
Formula Sheet
F = par value
r = coupon rate
Fr = coupon amount
C = redemption value (usually = F)
n = number of periods to redemption
P = price
i = yield per period to investor at price P
1
Vi=—7
l + i
P = PV(coupons) + PV(redemption payment) or P = (Fr)a^\ t + Cv"
Assuming F = C
Basic Formula P = (Fr) a^ t + Fvf
Premium-Discount Formula: P = F + F(r-i)a^i
Makeham Formula: P = K + ^(F-K), where K = Fvni
Bond Amortization
• Amortized amount in period k: F(r- i) vn~k+1
• Amortized amount in period k+m: F(r-i) vn-(k+m)+1 = (1 + i)m F (r - i) vn"fc+1
Maturity to use in pricing a callable bond:
Type of Bond
Premium Bond
Discount Bond
Find N using
Earliest Possible Redemption Date
Latest Possible Redemption Date
Price Between Payment dates
number of days from last coupon date to settlement date
number of days in the bond period
Price-plus-accrued = P0 (1 + i)'
Accrued interest =t(Fr)
Price = (Price-plus-accrued) - (Accrued interest) = P0 (1 + i)f - t(Fr).
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M4-16
Section 4.6
Basic Review Problems
1. A 1000 par value bond with a term of 5 years and a coupon of 6% convertible
semiannually is purchased to yield 8% convertible semiannually. Find the
purchase price.
2. 1000 par value bond with a term of 5 years and a coupon of 6% convertible
semiannually is offered at a price of 975. Find the yield.
3. Use the premium discount formula (4.11) to verify the price of the discount
bond in problem #1.
4. Find the (negative) amortization of discount in period 4 for the bond in
problem #1.
5. A premium bond is purchased to yield 5% convertible semiannually. The
amount of premium amortized in the third payment is 4.10. Find the amount
of premium amortized in the 8th payment.
6. A 5 year 1000 bond has a 6% coupon rate convertible semiannually. It is
callable in 2 years. An investor wishes to buy the bond to yield 5.5%
convertible semiannually. Find the purchase price of the bond.
7. A bond with par value of 1000 has payment dates of April 15 and October 15.
The nominal coupon rate convertible semiannually is 7%. The bond matures
on October 15, 2009. On April 15, 2007 a coupon payment of 35 was made.
The bond is sold 80 days later on the settlement date of July 4, 2007 to yield
6% convertible semiannually. There are 183 days between the coupon
payment dates of April 15, 2007 and October 15, 2007. Find the price-plus
accrued, the accrued interest and the price.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 4 - Bonds Page M4-17
Section 4.7
Basic Review Problem Solutions
1. BA II Plus. Set N=10, PMT=30, FV=1000,1/Y=4 and CPT PV= -918.89
2. BA II Plus. Set N=10, PMT=30, FV=1000, PV= -975 and CPT I/Y=3.30 yield
per semiannual period. Answer 6.6%
3. P = F + F(r-i)aiai= 1000 +1000(.03-. 04)a^ 04 =1000+ (-10) (8. Ill) = 918.891
4. F = 1000, r-i = .03-.04 = -.01,n-k + l = 10-4 + l = 7.
F(r-i)v"-'c+1=1000(-.01)
f 1 ^7
1.04
= -7.60
5. 4.10(1.025)5=4.64
6. Use the earliest possible redemption date for BA II Plus. Set N=4, PMT=30,
FV=1000,I/Y=2.75 and CPT PV= -1009.35
7. Price on last coupon date.BA II Plus.
Set N=5, PMT=35, FV=1000, I/Y=3 and CPT PV= -1022.90. P0 = 1022.90
183
80
Price-plus-accrued = 1022.90 (1.03)w3 =1036.20
f 80 ^
Accrued interest = 35 = 15.30
U83J
Price = (Price-plus-accrued) - (Accrued interest) = 1036.20 - 15.30 = 1020.90
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M4-18
Section 4.8
Sample Exam Problems
1. (200S Exam FM Sample Questions #10)
A 10,000 par value 10-year bond with 8% annual coupons is bought at a
premium to yield an annual effective rate of 6%.
Calculate the interest portion of the 7th coupon.
(A) 632 (B) 642 (C) 651 (D) 660 (E) 667
2. (200S Exam FM Sample Questions #2)
You have decided to invest in Bond X, an n-year bond with semi-annual
coupons and the following characteristics:
• Par value is 1000.
• The ratio of the semi-annual coupon rate to the desired semi-annual yield
rate, - is 1.03125.
i
• The present value of the redemption value is 381.50.
Given vn = 0.5889, what is the price of bond X?
(A) 1019 (B) 1029 (C) 1050 (D) 1055 (E) 1072
3. (2005 Exam FM Sample Questions #30)
As of 12/31/03, an insurance company has a known obligation to pay
$1,000,000 on 12/31/2007. To fund this liability, the company immediately
purchases 4-year 5% annual coupon bonds totaling $822,703 of par value.
The company anticipates reinvestment interest rates to remain constant at
5% through 12/31/07. The maturity value of the bond equals the par value.
Under the following reinvestment interest rate movement scenarios
effective 1/1/2004, what best describes the insurance company's profit or
(loss) as of 12/31/2007 after the liability is paid?
(A)
(B)
(C)
(D)
(E)
Interest Rates Drop by ¥2%
+6,606
(14,757)
(18,911)
(1,313)
Breakeven
Interest Rates Increase by ¥2%
+11,147
+14,418
+19,185
+1,323
Breakeven
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 4 - Bonds
PageM4-19
4. (200S Exam FM Sample Questions #47)
Bill buys a 10-year 1000 par value 6% bond with semi-annual coupons. The
price assumes a nominal yield of 6%, compounded semi-annually. As Bill
receives each coupon payment, he immediately puts the money into an
account earning interest at an annual effective rate of i.
At the end of 10 years, immediately after Bill receives the final coupon
payment and the redemption value of the bond, Bill has earned an annual
effective yield of 7% on his investment in the bond. Calculate i.
(A) 9.50% (B) 9.75% (C) 10.00% (D) 10.25% (E) 10.50%
5. (200S Exam FM Sample Questions #50)
A 1000 bond with semi-annual coupons at i(2) = 6% matures at par on
October 15, 2020. The bond is purchased on June 28, 2005 to yield the
investor i(2) = 7%. What is the purchase price?
Assume simple interest between bond coupon dates and note that:
Date Day of the Year
April 15 105
June 28 179
October 15 288
(A) 906 (B) 907 (C) 908 (D) 919 (E) 925
6. (2005 Exam FM Sample Questions #54)
Matt purchased a 20-year par value bond with semiannual coupons at a
nominal annual rate of 8% convertible semiannually at a price of 1722.25.
The bond can be called at par value X on any coupon date starting at the end
of year 15 after the coupon is paid. The price guarantees that Matt will
receive a nominal annual rate of interest convertible semiannually of at
least 6%. Calculate X.
(A) 1400 (B) 1420 (C) 1440 (D) 1460 (E) 1480
7. (2005 Exam FM Sample Questions #55)
Toby purchased a 20-year par value bond with semiannual coupons at a
nominal annual rate of 8% convertible semiannually at a price of 1722.25.
The bond can be called at par value 1100 on any coupon date starting at the
end of year 15.
What is the minimum yield that Toby could receive, expressed as a nominal
annual rate of interest convertible semiannually?
(A) 3.2% (B) 3.3% (C) 3.4% (D) 3.5% (E) 3.6%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M4-20
Module 4 - Bonds
8. (2005 Exam FM Sample Questions #56)
Sue purchased a 10-year par value bond with semiannual coupons at a
nominal annual rate of 4% convertible semiannually at a price of 1021.50.
The bond can be called at par value X on any coupon date starting at the end
of year 5. The price guarantees that Sue will receive a nominal annual rate
of interest convertible semiannually of at least 6%.
Calculate X.
(A) 1120 (B) 1140 (C) 1160 (D) 1180 (E) 1200
9. (2005 Exam FM Sample Questions #57)
Mary purchased a 10-year par value bond with semiannual coupons at a
nominal annual rate of 4% convertible semiannually at a price of 1021.50.
The bond can be called at par value 1100 on any coupon date starting at the
end of year 5.
What is the minimum yield that Mary could receive, expressed as a nominal
annual rate of interest convertible semiannually?
(A) 4.8% (B) 4.9% (C) 5.0% (D) 5.1% (E) 5.2%
10. (May 05 #5)
Susan can buy a zero coupon bond that will pay 1000 at the end of 12 years
and is currently selling for 624.60. Instead, she purchases a 6% bond with
coupons payable semi-annually that will pay 1000 at the end of 10 years. If
she pays X she will earn the same annual effective interest rate as the zero
coupon bond.
Calculate X.
(A) 1164 (B) 1167 (C) 1170 (D) 1173 (E) 1176
11. (May 05 #11)
A 1000 par value bond pays annual coupons of 80. The bond is redeemable at
par in 30 years, but is callable any time from the end of the 10th year at
1050. Based on her desired yield rate, an investor calculates the following
potential purchase prices, P:
• Assuming the bond is called at the end of the 10th year, P = 957
• Assuming the bond is held until maturity, P = 897
The investor buys the bond at the highest price that guarantees she will
receive at least her desired yield rate regardless of when the bond is called.
The investor holds the bond for 20 years, after which time the bond is called.
Calculate the annual yield rate the investor earns.
(A) 8.56% (B) 9.00% (C) 9.24% (D) 9.53% (E) 9.99%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 4 - Bonds
PageM4-21
12. (Nov OS #4)
A ten-year 100 par value bond pays 8% coupons semiannually. The bond is
priced at 118.20 to yield an annual nominal rate of 6% convertible
semiannually.
Calculate the redemption value of the bond.
(A) 97 (B) 100 (C) 103 (D) 106 (E) 109
13. (Nov OS #11)
An investor borrows an amount at an annual effective interest rate of 5%
and will repay all interest and principal in a lump sum at the end of 10 years.
She uses the amount borrowed to purchase a 1000 par value 10-year bond
with 8% semiannual coupons bought to yield 6% convertible semiannually.
All coupon payments are reinvested at a nominal rate of 4% convertible
semiannually.
Calculate the net gain to the investor at the end of 10 years after the loan is
repaid.
(A) 96 (B) 101 (C) 106 (D) 111 (E) 116
14. (Nov OS #16)
Dan purchases a 1000 par value 10-year bond with 9% semiannual coupons
for 925. He is able to reinvest his coupon payments at a nominal rate of 7%
convertible semiannually.
Calculate his nominal annual yield rate convertible semiannually over the
ten-year period.
(A) 7.6% (B) 8.1% (C) 9.2% (D) 9.4% (E) 10.2%
15. (Nov OS #22)
A 1000 par value bond with coupons at 9% payable semiannually was called
for 1100 prior to maturity. The bond was bought for 918 immediately after a
coupon payment and was held to call. The nominal yield rate convertible
semiannually was 10%.
Calculate the number of years the bond was held.
(A) 10 (B) 25 (C) 39 (D) 49 (E) 54
16. (Nov OS #24)
A 30-year bond with a par value of 1000 and 12% coupons payable quarterly
is selling at 850. Calculate the annual nominal yield rate convertible
quarterly.
(A) 3.5% (B) 7.1% (C) 14.2% (D) 14.9% (E) 15.4%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M4-22 Module 4 - Bonds
Section 4.9
Sample Exam Solutions
1.
The coupon payment is .08(10,000) = 800. The interest paid in the 7th coupon is
just 6% of the value of the bond at time 6. We can use the financial calculator to
find the value at time 6, when there are only 4 coupon payments of 800 and the
redemption value of 10,000 left to be paid. Set PMT = 800,1/Y = 6, FV = 10,000
and N =4. The computed value of PV is 10,693.02. The interest portion is
.06(10,693.02) = 641.58.
Answer B
2.
This problem cannot be done directly using the financial keys on the calculator.
You must set up equations and do some algebra. The price (i.e., present value)
of the bond is given by:
Present value of coupons + Present value of redemption value
We are given the present value of redemption value -it is 381.50.
The coupons equal lOOOr, so the present value of the coupons is
1000r{am) = 1000r[ Lj^L ] = 1000W(l- v2n)
= 1000(1.03125)(l - .58892) = 673.61
Thus the present value of the bond is
381.50 + 673.61 = 1055.11
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 4 - Bonds
Page M4-23
3.
i) First we will look at why the purchase will cover the obligation of 1,000,000 in
four years if reinvestment rates remain at 5%. This will illustrate how the
strategy works. On an exam you would start directly at ii) which follows and
this is background for your reference only.
We first need to determine the coupon amount. The key here is that the bonds
purchased have "822,703 of par value". The coupon is a percent of par value
(which is another name for face amount), so that the total coupon payment each
year is .05(822,703) = 41,135.15.
We also need to know the payment at maturity, but we are told that "the
maturity value of the bond equals the par value". Thus there is a payment at
maturity of 822,703.
When the bonds mature and the obligation of 1,000,000 must be paid, the
company wishes to pay it using the accumulated value of the coupons
(reinvested at 5%) plus the maturity value of 822,703. We can calculate the
accumulated value of the coupons on the financial calculator with PMT = -
41,135.15, N=4 and I/Y=5. The computed value of FV is 177,297.64, and this is the
accumulated value of the reinvested coupons. The total available at time 4 from
the bonds with reinvestment at 5% is
Accumulated value of reinvested coupons + Maturity value
= 177,297.64 + 822,703 = 1,000,000.64
ii) Now we can look at what happens if reinvestment rates drop by 1/2%. In
this case the reinvestment rate drops to 4.5%. We can calculate the
accumulated value of the coupons (reinvested at 4.5%) on the financial
calculator with PMT = -41,135.15, N=4 and I/Y=4.5. The computed value of FV is
175,984.03, and this is the accumulated value of the reinvested coupons.
The total available at time 4 from the bonds with reinvestment at 4.5% is
175,984.03 + 822,703 = 998,687.03
The company has a loss of
998,687.03 -1,000,000 = -1312.97.
This matches the loss of (1313) in choice D, and no other answer has this value
for the result of a V2% drop in interest rates. Thus D is the only possible answer,
(Note: the second part of choice D is correct.)
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M4-24
Module 4 - Bonds
4.
The coupon on the nominal 6% semiannual bond for 1000 is 3% of 1000, or 30.
Since the bond is a par bond priced at the same nominal rate, the price of the
bond was 1000. Thus Bill's initial investment was 1000.
For Bill to earn an effective annual yield of 7% over 10 years on his investment
of 1000, at the end of 10 years he must have 1000 (1.07)10 = 1967.15.
At the end of 10 years Bill actually has a) the 1000 repayment of the face value
of the bond and b) the total future amount of the reinvestment account, which
we shall denote by FVreinv. This tells us that
1967.15 = 1000 + FVreinv FVreinv = 967.15.
We know that the semiannual payment to the reinvestment account was 30,
made for a total of 20 semi-annual periods. Thus we can calculate the
semiannual yield of the reinvestment account on the BA II Plus using
PMT = -30, N = 20 and FV = 967.15. The calculated semi-annual yield is 4.7596%.
The annual effective yield is 1.0475962 -1 = 0.097458
Answer B
S.
Note: there is some potential for confusion here, since the problem does not specify
whether purchase price means price-plus-accrued or market price. The authors of the
posted answer key take the words purchase price to imply the full price-plus-accrued.
The bond has semiannual payment dates of April 15 and October 15 with
coupons of 30. It is purchased between coupon dates on June 28, 2005. There are
183 days between April 15 and October 15, and 74 days between April 15, 2005
and June 28, 2005. Thus fractional period to time of purchase is
t-H.
183
After the preceding April 15, 2005 coupon payment, the price of the bond can be
found using the financial calculator with PMT = 30, FV = 1000, yield i = 3.5 and n
= 31 (one payment in 2005 and 2 each in the remaining 15 years from 2006 to
2020). The value is PV = 906.32. This is P0 .The total amount paid at purchase
time is the price-plus-accrued, but the instructions say to use simple interest
between bond coupon dates so we calculate.
P0 (1 + ti) = 906.32^ 1 + -^t (.035)1 = 919.15
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 4 - Bonds
PageM4-25
6.
Since the semiannual yield rate of 3% is less than the semiannual coupon rate
of 4%, this is a premium bond. Since the bond is callable in 15 years, it is priced
as if it will be redeemed in 15 years. The problem does not directly give the par
value X or the coupon amount .04X, so the financial calculator cannot be used
directly. Instead we must set up an equation of value for the price of 1722.25.
1722.25 = (.04X) aMo3 + Xv30 = .784X + .412X = 1.196X
X = 1440
Answer C
7.
The statement that "The bond can be called at par value 1100 on any coupon
date starting at the end of year 15" shows that the face value F is also 1100.
Since the price is 1722.25, the bond is a premium bond. The minimum yield
would be obtained if the bond is called and redeemed in 15 years or 30 bond
periods.
We can obtain this (semiannual) yield from BA II Plus calculator using n = 30,
PV = -1722.25, FV = 1100 and PMT = 44. The semiannual yield is 1.608 leading to
a nominal annual purchase yield of 3.216%.
Answer A
8.
This is like Problem 6, but this time we have a discount bond since the
semiannual yield rate of 3% is greater than the semiannual coupon rate of 2%.
The discount bond is callable in 5 years, but it is priced as if it will be redeemed
as late as possible in 10 years. The problem does not directly give the par value
X or the coupon amount .02X, so the financial calculator cannot be used
directly. Instead we must set up an equation of value for the price of 1021.50.
1021.50 = (.02X) aMo3 + Xv20 = .2975X + .5537X = .8512X
X = 1200.07
Answer E
9.
Since the par value of 1100 is greater than the price of 1021.50, this is a
discount bond and its minimum yield is obtained when it is held to maturity for
10 years (or 20 semiannual periods). The semiannual coupon is 1100(.02) = 22.
Thus we can find the semiannual yield on the BA II plus using
PV = -1021.50, PMT = 22, FV = 1100 and n = 20.
The semiannual yield is 2.456%. Thus the minimum nominal annual yield to
Mary is 4.912%.
Answer B
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M4-26
Module 4 - Bonds
10.
First we find the effective rate on the zero coupon bond using the BA II Plus.
Set N=12, FV=1000, PV=-624 and CPT I/Y = 4. The effective annual rate is 4%,
and Susan should buy the bond for the price X that yields 4% effective
annually. The semiannual yield corresponding to an effective annual rate of 4%
is VL04-1 = .0198
Thus Susan will buy the bond at a semiannual yield of 1.98%. We assume that
the redemption value given in the problem is also the face value of the bond, so
that the semiannual coupon payment is 30.
Using the BA II Plus, set N = 20, I/Y = 1.98, FV = 1000, PMT = 30 and then CPT
PV = -1167.04.
Answer B
11.
The investor will buy at the lower price of 897 to assure the desired yield. To
find the yield on the BA II Plus, set PV=-897, PMT=80, N=20, FV=1050 and CPT
I/Y = 9.24.
Answer C
12.
This is an extremely simple financial calculator problem, since the redemption
value is the future value FV. On the BA II Plus, use PMT = 4 (for the 8%
semiannual coupon payment), PV = -118.20 (the price), N=20 and I/Y=3 (for the
yield of 6% nominal convertible semiannually). This will enable you to compute
FV = 106.
Answer D
13.
The purchase price of the bond can be obtained by using the financial
calculator with FV = 1000, PMT = 40, N = 20 and 1 = 3. The price is PV = 1148.77.
Thus the investor will borrow 1148.77 at 5%, and repay in 10 years the amount
1148.77(1.05)10 =1871.23.
The future value of the reinvested coupons can be obtained from the financial
calculator with PMT = 40, N = 20,1 = 2, PV=0. The computed sum is FV = 971.89.
The redemption value of the bond is 1000, so the investor will have 1971.89 at
maturity. After repayment of the loan, the investor will have a net gain of
1971.89 - 1871.23 = 100.66
Answer B
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 4 - Bonds
PageM4-27
14.
The semiannual coupons provide 20 semiannual payments of 45. These are
reinvested at a nominal 7% convertible to 3.5% semiannually, and we can use
the financial calculator to find their future value in ten years. Set PMT = -45,
I/Y = 3.5 and N=20. The computed FV is 1272.59. Dan also gets the 1000
redemption value of the bond in ten years, for a total of 2272.59. His original
investment was 925, so his semiannual yield on the investment is
( 2272.59
1 = .046
(, 925
The nominal yield convertible semiannually is 2 (.046) = .092.
Answer C
15.
This can be solved using a financial calculator to solve for the number of
periods n. The values to enter are PV = -918 (the price paid), PMT = 45 (the
semiannual coupon payment), FV = 1100 (the redemption value) and I/Y = 5 (the
5% semiannual yield derived from 10% convertible semiannually). The
computed value of N is 49.35 semiannual periods. This must be converted to
49.35/2 = 24.675 years. (Note that choice D will trap the student who does not
convert back to years.)
Answer B
16.
This can be solved using a financial calculator. The values to enter are PV = -
850 (the price paid), PMT = 30 (the quarterly coupon payment), FV = 1000 (the
redemption value) and N=120 (quarters in 30 years). The computed value of I/Y
is 3.539% per quarter. This must be converted to the nominal annual yield
convertible quarterly of 4(3.539) = 14.156
Answer C
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M4-28
Section 4,10
Supplemental Exercises
1. A 15-year 1000 par bond with 7% semiannual coupons is priced to yield 6%
convertible semiannually. Find the price.
2. Suppose the bond in Problem 1 is offered at a price of 975. What is the
nominal yield convertible semiannually?
3. The company offering the bond in Problem 1 decides to make it more
attractive at that price by increasing the redemption value to 1050. What is
the nominal yield convertible semiannually for this bond with the new
redemption value?
4. A 10-year bond with a face value of 1000 and 5% semiannual coupons is sold
for 980. What should the redemption value be if the bond is to yield 5.4%
convertible semiannually?
5. A 10-year 1000 par bond with 6% semiannual coupons is priced to yield 6.5%
convertible semiannually. Find the discount for this bond.
6. A 5-year 1000 par bond with 7% semiannual coupons is purchased to yield
6.4% convertible semiannually. The coupon payments are reinvested in a
fund that earns 7.2% convertible semiannually. What is the annual effective
yield on the total accumulation at the end of the 5-year period?
7. A 10-year 1000 par bond with 6% semiannual coupons is priced to yield 5.6%
convertible semiannually. How much of the premium is amortized in the 8th
period?
8. A 1000 par bond with 6.5% semiannual coupons is priced to yield 5.8%
convertible semiannually. If the amount of the premium amortized in the 4th
period is 2.12, how much of the premium is amortized in the 9th period?
9. A 10-year 1000 par bond with 5% semiannual coupons is priced to yield 5.6%
convertible semiannually. How much of the discount is amortized in the 6th
period?
10. A 10-year 1000 par bond with 8% semiannual coupons is callable in 7 years.
At what price should an investor buy the bond to yield 7.2% convertible
semiannually?
11. A 1000 par bond with 8% semiannual coupons has payment dates of May 31
and November 30. The bond matures on November 30, 2010. On May 31, 2007
the coupon payment of 40 is paid. The bond is sold 70 days later on the
settlement date of August 9. The bond is sold to yield 7.4% convertible
semiannually. Find the price plus accrued interest, the accrued interest and
the price.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 4 - Bonds Page M4- 29
Section 4,11
Supplemental Exercise Solutions
1. Using the BAII Plus set
N = 30,1/Y = 3, PMT = 35 and FV = 1000. CPT PV = -1098. Price is 1098.
2. To get the new yield set PV = -975. CPT I/Y = 3.6383.
The nominal yield = 7.2766%
3. For the bond with new redemption value set
N = 30, PMT = 35, PV = -1098 and FV = 1050. CPT I/Y = 3.097
Nominal yield = 6.194%
4. To find the redemption value set
N = 20, I/Y = 2.7. PMT = 25 and PV = -980. CPT FV = 1018.05
Redemption value should be 1018
5. To find the discount we must first find the price. Set
N = 20, I/Y = 3.25, PMT = 30 and FV = 1000. CPT PV = -963.65
Discount = 1000 - 963.65 = 36.35
6. The accumulation of the reinvested coupon payments is 35Sioi0.o36 = 412.50.
The total accumulation is 1000 + 412.50 = 1412.50. The total invested is the
price of the bond which is found by setting
N = 10, I/Y = 3.2, PMT = 35 and FV = 1000. CPT PV = -1025.33.
To find yield, (1 + j)5 = 1412.50/1025.33 = 1.3776. Then j = 6.6%
7. The amount of premium amortized in the 8th period is
1000(r- i)vn8+1 = 1000(0.03 - 0.028)(1/1.028)13 = 1.40.
8. The amount of premium amortized in the 4th period is 2.12.
The amount of premium amortized in the 9th period is 2.12(1.0295) = 2.45.
9. The amount of principal paid in the 6th period is
1000(0.025 - 0.028X1/1.028)15 = -1.98.
The amount of discount amortized in the 6th period is 1.98
10. This is a premium bond so it is priced at the earliest redemption date,the
end year 7. To get the price using the BA II Plus set
N = 14, I/Y = 3.6, PMT = 40 and FV = 1000. CPT PV = -1043.39.
The price is 1043.39.
11. Immediately after the coupon is paid on Mar 31, there are 7 coupon
payments remaining. At this point the price of the bond can be obtained
using the BA II Plus calculator.
N = 7, I/Y = 3.7, PMT = 40 and FV = 1000. CPT PV = -1018.21.
The price plus accrual is 1018.21(1.037)70/183 = 1032.46.
The accrued interest is 40(70/183) = 15.30.
The price is 1032.46 - 15.30 = 1017.16.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 5 - Yield Rate of an Investment
PageM5- 1
Yield Rate of an Investment
The yield rate for an investment is just the interest rate that the investor
ultimately earns. We have already found yield rates for investments. In this
module we will look at investment yield in more depth, and will begin by
looking at the internal rate of return.
Section 5,1
IRR: Internal Rate of Return
An investor is interested in what must be paid out to invest and what is paid
back in return. Suppose an investor is asked to invest 1,000 and is promised in
return a payment of 600 in one year and 550 in the second year. Using the
convention that money paid out is negative and money paid back is positive, the
investor would describe this investment as the sequence
-1000, 600, 550.
The payments made each period are called cash flows. The cash flow at time
k is denoted by Ck. In this two period investment the initial cash flow is
Co = -1000, and the investment returns are d = 600 and C2 = 550. The investor
is interested in answering the question: "What interest rate (i.e., yield) am I
earning on my invested money?" The internal rate of return answers that
question. First we will define IRR and show how to calculate it, and then we will
talk about why the IRR measures the investment yield.
Definition.
Suppose an investment for n periods has cash flows C0,Ci,...,Cn. An internal
rate of return for the investment is a solution for i of the equation
(5.1)
C0 +
cx
+ ... + -
cn
(i+0 (i+i)2 (i+i)n
0
We can also write (5.1) in the form C0 + Civ + C2v2 +... + Cnvn = 0.
If we find a root for v, we can immediately find / = (1/v) -1 .Thus an IRR
problem with n periods is really the problem of finding a root of a polynomial of
degree n.
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page M5-2
Module 5 - Yield Rate of an Investment
There is one additional constraint that is applied in IRR problems. With a
sequence of cash flows C0,Ci,...,Cn the worst that can happen is that you invest
an amount C0 > 0 and then end up getting nothing back with Cx = C2 =... = Cn = 0.
In that case the return is -100%, which would be -1 in decimal form. Thus in
IRR problems we ignore all extraneous roots that are less than -1.
Example (5.2)
Consider the investment discussed above where the cash flows are. -
1000, 600, 550. The internal rate of return is a solution for iof the
quadratic equation
_1000 + J^L + 550 = -1000 + 600v + 550v2 = 0
(1 + 0 (1 + i)2
Using the quadratic formula we see that there are two solutions for
v = 1/(1 + i). The solutions are given below along with the corresponding
values of i.
v = — ->i = .10 v = -2->i = -1.50
11
As in many applied problems we have one realistic solution of 10% and
one unrealistic solution of -150%.We discard the root of -1.5<-1 The true
earning rate is an IRR of 10%. To see why 10% is the true earning
yield, let us compare this investment to a bank account earning 10%
annually. The timeline below shows the result of this bank account over
2 years:
time 0 12
i 1 1
1,000 1,100 -600=500 550-550=0
Initikdeposit ETT / / ?f* /Withdrawal
(includes 107, Withdrawal balance i n
interest) fne|- bf <™- £f
10% interest) ^al.
In one year, the original 1000 grows to 1100 at 10% interest. The
withdrawal of 600 lowers the balance to 500. Then this 500 grows to 550
and that is withdrawn to close the account. The investment we just
reviewed behaves exactly like an account earning 10%.
Another way to describe the internal rate of return is that it is the rate
of interest at which the present value of all amounts invested is equal to
the present value of all the amounts paid back to the investor. For
example, in (5.2) we discovered that for i = .10
innn 600 550 n
-1000 + ^ + r- = 0
(1 + .10) (1 + .10)2
This implies that the present value of the payments of 600 and 550 is
equal to the original investment of 1000.
iooo=^+-550
(11) (1.1)2'
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 5 - Yield Rate of an Investment Page M5- 3
Exercise (5.3)
An investor is asked to invest 1,000 and is promised in return a payment
of 450 in one year and 630 in the second year. Find the IRR.
Answer: 5%
Note that the cash flow Ck represents net income at time fc. You may be given
information about revenue and expense at time k instead of being given Ck
directly. Then Ck = Revenue at k - Expense at fc.
In the text Mathematics of Investment and Credit revenue at k is denoted by Ak
and expense at k is denoted by Bk. Thus Ck = Ak - Bk
For example, in (5.2) we were directly given the cash flows -1000, 600, 550. This
problem could have been alternatively described as follows:
An investor spends 1000 to set up a mining operation for 2 years.
In his first year he has revenues of 800 and expenses of 200. In his
second year he has revenues of 800 and expenses of 250. Find the
IRR of this investment.
This is the same problem as (5.2), since the cash flows are -1000,
800-200=600 and 800-250=550.
The internal rate of return is widely used to evaluate investments. For this
reason modern financial calculators like the BA II plus have a worksheet for
entering the cash flows and an IRR key which will provide the calculation of
i. We have already discussed entering cash flows in module 2. Here we will
go through the steps again for the investment in (5.2).
Entering cash flows: Press |CF] to enter the worksheet. Key in 12ND1 CLR
WORK to remove any numbers left over from prior work. You will see a
prompt for the value of CF0, the cash flow at time 0. Enter the value -1000
(don't forget to press ENTER after keying in -1000). Scroll down and you will
see a prompt for C01, the cash flow at time 1. Enter the number 600. Scroll
down again, and there will be a new prompt - "F01=" . This is a request for
the number of times (frequency) that this value is repeated. The default value
is 1, and if you scroll past, the value of 1 will be assumed with no entry. Scroll
down again, and you will be prompted for the value of C02. Enter 550.
Calculate the IRR with the keystrokes.
The display will show the answer 10.00
IRR CPT
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M5-4
Module 5 - Yield Rate of an Investment
IRR problems with more than two periodic payments will be difficult to solve
because they involve higher degree polynomials instead of quadratics. The
reader who knows about Galois theory knows that there is no general quadratic
formula type method to find exact roots of all polynomials of degree > 5.
Higher degree polynomials need to be solved approximately using iterative
methods like Newton's method, which is how modern financial calculators like
the BAH Plus find IRR. Fortunately we can solve problems for investments
with a large number of cash flows using the BA II Plus. Microsoft EXCEL has
an IRR function which is extensively used for cash flow analysis of real
investments. This author used EXCEL spreadsheets to create problems and
check answers for this guide.
Exercise (5.4)
An investor:
of 380 in one
his IRR.
is asked to invest 1,000 and is
year, 256 in the second year
promised
and 540 in
in return a payment |
the third year.
Answer:
Find
8% |
Remember that the term IRR is synonymous with investment
yield. We could have phrased the last problem to ask for the yield
on the investment or the true interest rate earned. Many financial
professionals use the terms yield and IRR interchangeably.
Note also that we previously found interest rates for investments
with level payments using the TVM keys.
Example (5.5)
A lender invests 15,000 to make a loan which will be repaid with 4 annual
end of year payments of 5,000. What is her yield on this investment?
Solution.
Set PV = -15000, N=4, PMT=5000, FV=0 and CPT I/Y = 12.59.
I Her yield is 12.59%.
In the above problem we could have also asked for the IRR, as it means same
thing. Many of our students find this confusing. Due to the calculator key
structure they think of IRR as something that applies only when cash flows are
not level—but this is not true. For example, you can use the BAII Plus CF
worksheet for this problem by entering CF0 = -15,000, COl = 5000, FOl = 4 and
then pressing the IRR key and CPT. The answer of 12.59 is the same.
Exam problems often make yield questions a bit tougher by asking for nominal
or annual effective yield. In the next exercise we review a type of problem that
we have already done in Module 2.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 5 - Yield Rate of an Investment Page M5- 5
Exercise (5.6)
A lender
payments
makes a
of 3,500.
loan of
What is
24,000 to be repaid
a) her nominal
annually and b) her effective annual yield?
Answer:
with 10
semiannual
yield convertible semi- |
a) 15.04%
b) 15.61% 1
Why is the Rate of Return Internal?
The investment yields are called internal because they do not apply to money
after it is paid out. Consider our original problem of finding the IRR for the
investment with cash flows -1000, 600, 550.
Suppose the investor is trying to build a fund for use in two years. Then when
she is paid the first payment of 600 she will re-invest it. If she can only reinvest
it at 5% one year from now she will have 1.05 (600) + 550 = 1,180 at the end of the
second years. The yield on an investment of 1000 which pays 1,180 in two years
is an annual 8.628%, less than 10%.
Some analysts prefer to use a modified IRR which adjusts for reinvestment.
What method to use really depends on the investor's objectives. Most investors
we know use the IRR as the primary tool for evaluating an investment. We will
focus on it as the primary yield tool here.
However, the reader should remember that there are reinvestment problems in
Module 2 in the section entitled Reinvestment Problems and you are required to
know them for the exam.
Please note: Section 5.1.4 of Mathematics of Investment and Credit discusses
several alternatives to IRR. This section is not on the Exam FM syllabus at the
time of writing this guide.
Uniqueness of the Internal Rate of Return
A polynomial of degree n can have anywhere from 0 to n real roots. In all of the
previous problems, there was only one meaningful IRR solution. Confusing
situations involving multiple roots can arise as we will see in the next example.
Also, the text Mathematics of Investment and Credit has a section (5.1.2) on the
uniqueness of the internal rate of return. The section points out that multiple
IRRs can occur and gives some examples. It also mentions that in the very
common situations where a) C0 > 0 and Ci,...,Cn are all negative or b) C0 < 0
and Ci,...,Cn are all positive there is a unique IRR >-l.
Further results which guarantee the desired unique IRR are given in the
exercises. For example, if you create a sample bank account at a given IRR and
find that borrowing never occurs then that IRR is unique.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M5-6
Module 5 - Yield Rate of an Investment
Example (5.7)
An investor can invest 10,000 for a mining operation. In one year he will
get a payout of 23,000, but at the end of the second year he must pay
13,200 for cleanup cost. His cash flow sequence is
-10,000, 23,000, -13,200.
The internal rate of return is a solution for i of the quadratic equation
-10,000 + 23,OOOv -13,200v2 = 0 .
Using the quadratic formula we see that there are two solutions for
v = 1/(1 + i). The solutions are given below along with the corresponding
values of i. v = .909 -> i = .10 v = .833 -> i = .20
This is confusing. The project appears to earn realistic rates of 10% or
20%. To see why it can happen, we will give savings account tables for
the project at both rates.
IRR: 20%
time 0 12
I 1 1
100
Initial deposit
120-230= -110
-132+132=0
Beginning ^
balance ^
(includes 20%
interest)
\
Ending
balance
(debt)
. bat \
Beg. bal
(inch 20%
interest)
End.
Bal.
Deposit
Withdrawal
IRR: 10%
time
100
110 -230= -120
-132+132=0
Initiataeposit
Beginning ^
balance
(includes 10%
interest)
\
Withdrawal
Ending
balance
(debt)
/
*\
End.
Bal.
Beg. bal.
(incl.10%
interest)
Deposit
In either case, if this were a true bank account, the withdrawal of 230
would be more than is in the account, so that the investor would be in
debt to the bank. The investor would then pay back this loan with
interest at time 2. Such projects are called borrowing projects, and
they can lead to multiple rates of return because the IRR is used both
as a payout rate and a borrowing rate.
Neither IRR in this example is valid for investment purposes. Only use
IRR when situations like this do not happen. For investments of this
nature, there are modified IRR calculations that are used instead.
These modified versions are not part of the Exam FM material.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 5 - Yield Rate of an Investment Page M5- 7
Section 5.2
Time Weighted and Dollar Weighted Rates
Two methods are given in this section to measure the rate of return on an
investment fund and evaluate how well the investment manager is performing.
They are the time weighted and dollar weighted methods -and they do not
generally give the same answer. We will illustrate these methods next.
Example (5.8)
An investment manager had a fund of 100,000 at the start of the year
2006. On June 30th that fund had dropped to 90,000 and new deposit of
110,000 was made. At year end the account balance was 220,000. We will
measure the return on this fund using both methods.
Time -weighted rate.
Let ji and j2 represent the earnings rates for the first and second
halves of the year. Then for the first half where 100,000 dropped to
90,000
J 100,000 J
After the new deposit of 110,000 the second half started off with 200,000
in the fund and that grew to 220,000.
220,000
1+h = = 1.1 —> Ji =.10
J 200,000 .
To get the time weighted yield j for the entire year we use the time
weighted yield relationship.
l + j = (l + j1)(l + j2) = 0.9(l.l) = 0.99 -+ j = -0.01
Using the time weighted yield the manager's returns over different
periods of the year are compounded to get a compound return for the
entire year. Under this method he has a loss of 1%.
Dollar weighted rate.
Here we are looking for i, the rate of simple interest that would cause
the invested dollars to result in a fund of 220,000 at year end if it had
been in effect for the entire year.. This rate i should satisfy the equation
100,000(1 + i) + 110,000[1 + -1 = 220,000
This is easily solved for i.
155, OOOi = 10,000 -> i = .0645.
Under the dollar weighted method the investment manager's yield is
6.45%. It is not hard to see why performance looks better under this
method. The fund had more in it in the second half of the year when
performance was better. The dollar weighted method takes account of
how many dollars were in the fund during each period of the year, and
the time weighted method does not.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M5-8
Module 5 - Yield Rate of an Investment
Exercise (5.9)
Find the time weighted and dollar weighted yields if the original deposit
of 100,000 dropped to 90,000 at mid-year but the deposit made at that
point was 10,000 and the final amount in fund was 110,000.
Answer; Time-weighted: -1% Dollar-weighted: 0%
Note that the dollar weighted method used simple interest. A similar
computation could be performed to find a dollar weighted yield using
compound interest. With modern computer and calculator tools that is not a
hard problem to solve. It is like an IRR problem. Historically the dollar
weighted method evolved using simple interest because the necessary
computer tools for compound interest were not available when it was first used.
We used only the starting amount and one deposit in the fund year in the
preceding example to keep it simple. Next we will summarize the general
methods used for measurement where the fund may have many deposits or
withdrawals. As we introduce these general methods we will refer to Example
(5.8) at some points to make the general formula concrete.
Time Weighted Rates of Interest
Suppose that contributions are made at times ti,t2,...,tm_i with the fund year
starting at time t0 = 0 ending at time tm = 1. In Example (5.8) ra=2 and there is
one contribution at time U = .5. We will use the notation
CJc = Contribution at time tk, where a negative amount is a withdrawal
B'k = fund value at time tk before the contribution Ck is made.
In our example the single contribution is C[ = 110,000 while B'0 = 100,000,
B[ =90,000 and B'2 =220,000.
We use jfcto denote the effective rate over [tfc_i,tfc] where
(5.10)
i B'k Current Balance
J- + Jk =
Bfc_i + Cfc_i Last Balance + Last Contribution
The time weighted rate j is found by calculating (as we did in the previous
example)
(5.11)
l + j = (l + ji)(l + j2)...(l + jm)
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 5 - Yield Rate of an Investment
PageM5- 9
Dollar Weighted Rate of Interest
A = initial fund balance B = final fund balance.
I = interest earned
In our Example (5.8), A = 100,000 and B = 220,000. The interest earned will be
calculated below.
We use Ct to denote the contribution or withdrawal at time t. Then C = ^Ct
represents the total cash contribution (net). In Example (5.8) there was one
contribution of 110,000 at time t=.5. Thus C = CS= 110,000.
Note that in dollar weighted problems time is represented as if the year had 12
months of equal length. The calculation of interest is based on the observation
that interest income should account for the difference between the ending
amount B and the sum of the starting amount A and total net contributions C.
Thus
(5.12)
I=B-A-C
In Example (5.8) 1 = 220,000 -100,000 -110,000 = 10,000
Then the dollar weighted yield is defined by
(5.13)
i =
I
A + ]Tct(l-t)
In Example (5.8), this would give i =
10,000 = 10,000
100,000 +110,000(.5) 155,000
= .0645.
Note that the denominator of (5.13) is the sum of the initial amount and the
midyear contribution of 110,000 applied for the remaining half of the year. The
answer is identical with the answer in (5.8). Formula (5.13) simply summarizes
the result of the reasoning that was used in (5.8).
The next example and exercise will apply this method to a slightly more
involved problem.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M5-10
Module 5 - Yield Rate of an Investment
Example (5.14)
An investment manager had a fund of 100,000 at the start of the year
2006. On April 1st that fund had risen to 112,000 and a new deposit of
30,000 was made. On October 1st the fund balance was 125,000 and a
withdrawal of 42,000 was made. At year end the account balance was
100,000. We will show a timeline of contributions and fund balances and
then calculate both rates
100,000
+30,000
112,000
-42,000
125,000
100,000
J F M A M
Time weighted return
1 +j = (1 +Ji)(l +J2)(l +J3) =
-» j = .1878
O
112,000
100,000 J
r 125,000
1142,000
100,000
83,000
N
1.1878
D
Dollar weighted return
B = 100,000 A = 100,000 and C = 30,000 - 42,000 = -12,000.
I = B-A-C = 100,000 -100,000 - (-12,000) = 12,000
12,000
100,000 +11 —y 30,000 + (1 - ^-] (-42,000)
12
12
( 12,000
112,000
0.1071
Exercise (5.15)
An investment manager had a fund of 100,000 at the start of the year
2006. On May 1st that fund had risen to 108,000 and a new deposit of
20,000 was made.
On December 1st the fund balance was 130,000 and a withdrawal of
12,000 was made. At year end the account balance was 110,000. Find the
time weighted yield and the dollar weighted yield.
Answer: Time weighted: 2.25%, Dollar weighted 1,78%
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 5 - Yield Rate of an Investment Page M5-11
Section 5.3
The Investment Year and Portfolio Methods
Once you have invested money in a fund, there are different ways that your
return can be calculated:
I. All investors are pooled together in the same overall portfolio, and every
investor in the fund gets the same return. This is called the portfolio
method.
II. Segregate the money of all individuals who started in a given year and give
them all the return on that segregated fund that is unique to them. This is
called the investment year method.
In the next example we will illustrate how these methods work.
Example (5.16)
Clearly we first need information on the rates earned for the entire
portfolio and the separate accounts, which is typically displayed in a table
like the one below.
1 Calendar
1 Year of
Original
Investment
y
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
Investment Year Rates (in %)
il
8.25
8.5 '
9.0
9.0
9.25
9.5
10.0
10.0
9.5
9.0
il
8.25
8.7
9.0
9.1
9.35
9.5
10.0
9.8
9.5
il
8.4
8.75
9.1
9.2
9.5
9.6
9.9
9.7
il
8.5
8.9
9.1
9.3
9.55
9.7
9.8
il
8.5
9.0
9.2
9.4
9.6
9.7
Portfolio
Rates
(in %)
iy+s
8.35
8.6
8.85
9.1
9.35
Example continued on following page
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M5-12
Module 5 - Yield Rate of an Investment
Example continued from previous page.
The notation i£ stands for the segregated account return in fcth year for a
person who started in year y. For example a person who started in 1992
had a first year earnings rate of i\992 = 8.25% and a second year earnings
rate of if92 = 8.25%. The starting rate for any year is called the new
money rate. The new money rate in 1993 was 8.5%.
At the end of 5 years investors are moved to an appropriate portfolio rate.
At the end of the line for 1992 is the portfolio rate 5 years later,
i1997 = 8.35%. There is no subscript because the portfolio rate is a return
on the entire portfolio for a year and does not depend on the year of
startup. We will illustrate how accumulation factors are computed for an
investor who began in 1997 and left money in for two years.
Investment year method. The investment year rates needed for this
problem are il997 = 9.5% and i\997 = 9.5%. The two year accumulation
factor is 1.095(1.095) = 1.199.
Portfolio method. The portfolio rates for 1997 and 1998 are
i1997 = 8.35% and i1998 = 8.6%. The two year accumulation factor is
1.0835(1.086) = 1.1767.
Exercise (5.17)
Using the table in example (5.16) find the two year accumulation factors
for an investor starting 1998 under a) the investment year method and b)
the portfolio method.
Answer: a) 1.210 b) 1.1821
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 5 - Yield Rate of an Investment Page M5-13
Section 5.4
Net Present Value
The net present value (NPV) of a series of cash flows C0,Ci,...,Cn .at a rate iis
just the present value of the cash flows.
(s,18) I Npy = Co+7^T+-^T+...+-^- I
(i+») (i+i) (i+i)n
Your calculator has a very useful NPV key and you should at least be familiar
with it.
Example (5.19)
Find the net present value of the annual cash flow series from Example
(5.2) at the rates i = 0.09, i = 0.10, i = 0.11.
Solution.
This is a direct calculator problem for the BA II Plus. The cash flows
were -1000, 600, 550.
Go to the CF worksheet and enter C0 = —1000, Cl= 600 and C2=550. Then
press the NPV key and you will see a prompt for the interest rate. Enter
9 for the interest rate, scroll down to NPV and hit CPT.
The NPV at 9% is 13.38. Similarly, the NPV at 10% is 0 and the NPV at
11% is -13.07.
The above problem could easily have been phrased to ask for the present value
instead of the net present value. Thus most old exam FM problems which
discuss net present value could be rephrased to simply say present value and
be legitimate for the current syllabus. Don't skip them.
Note that we found in Example (5.2) that the IRR of this investment was 10%.
Now we have seen that the NPV of this investment at 10% is 0. This relationship
always holds. In fact some texts define an IRR as a solution of the equation
NPV = 0.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M5-14
Module 5 - Yield Rate of an Investment
Section 5.5
Formula Sheet
Internal Rate of Return
Given investment cash flows C0,Ci,...,Cn, an internal rate of return is a solution
for i of the equation
Co+t^t- + °2 -+-.+ Cn =0 or C0 + CiV + C2v2+... + CnvB=0.
(1 + i) (1 + i)2 (l + i)n
The internal rate of return is the rate of interest at which the present value of
all amounts invested is equal to the present value of all the amounts paid back
to the investor.
There may be multiple IRR solutions if the investment is a borrowing project.
Time Weighted Rates of Interest
Cfc = Contribution at time tk
Bk = fund value at time tk before the contribution Ck is made.
jfcis the effective rate over [tfc_i,tfc]
l + jfc= , Ek ,
-Bit-i + Cfc-i
The time weighted rate i is found by calculating
l + i = (l + ji)(l + j2)...(l + jm)
Dollar Weighted Rate of Interest
A = initial fund balance B = final fund balance. I = interest earned
Ct = contribution or withdrawal at time t. C = Y,Ct=1 tota^ cas^ contribution
(net)
B = A + C + I -+ I = B-A-C
I
''A^Ctd-t)
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 5 - Yield Rate of an Investment Page M5-15
Section 5.6
Basic Review Problems
1. An investor is asked to invest 1,100 and is promised in return a payment of
500 in one year and 700 in the second year. Find the IRR.
2. An investor is asked to invest 11,000 and is promised in return a payment of
4000 in one year, 5000 in the second year and 4500 in the third year. Find his
IRR.
3. A lender invests 20,000 to make a loan which will be repaid with 3 annual
end of year payments of 8,000. What is her yield on this investment?
4. An investment manager had a fund of 100,000 at the start of the year 2006.
On February 1st that fund had dropped to 98,000 and a withdrawal of 10,000
was made. On September 1st the fund balance was 100,000 and new deposit
of 10,000 was made. At year end the account balance was 105,000. Find the
time weighted and dollar weighted rates of return.
5. Using the table in Example (5.16) find the three year accumulation factors
for an investor starting 1998 under a) the investment year method and b) the
portfolio method.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M5-16
Module 5 - Yield Rate of an Investment
Section 5.7
Basic Review Problem Solutions
1. Calculator. Use the CF worksheet with C0 = -1100, CI = 500 and C2 = 700.
Then key IRR CPT. The yield is 5.67%. (You could also do this one as a
quadratic but it will take more time.)
2. Calculator. Use the CF worksheet with C0 = -11,000, CI = 4000. C2 = 5000
and C3 = 4500.. Then key IRR CPT. The yield is 10.75%.
3. Set PV = -20000, N=3, PMT=8000, FV=0 and CPT I/Y = 9.70. Her yield is
9.70%.
4. Time weighted return
1 + J = (l + Ji)(l + J0(l + J3) =
98,000 y 100,000
100,000A 88,000 )
(105,000
110,000
= 1.063^ j = .063
Dollar weighted return.
B = 100,000 A = 105,000and C = -10,000 + 10,000-0.
J = B-A-C = 105,000-100,000-0 = 5,000
U , ,, 5>000 , , J 5>000 I = .0531
100,000 + f 1 - —) (-10,000) + (1 - — ] (10,000) ^ 94,166,77
v 12J v 12y
5. Investment year method. The investment year rates for needed are
if8 = 10.0% , i21998 = 10.0% and U998 = 9.9%. The two year accumulation
factor is 1.10(1.10)(1.099) = 1.3298.
Portfolio method. The portfolio rates for 1998,1999 and 2000 are
i1998 = 8.6%, i1999 = 8.85% and i2000 = 9.1% The three year accumulation
factor is 1.086 (1.0885) (1.091) = 1.2897.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 5 - Yield Rate of an Investment Page M5-17
Section 5.8
Sample Exam Problems
1. (2005 Exam FM Sample Questions #5)
An association had a fund balance of 75 on January 1 and 60 on December
31. At the end of every month during the year, the association deposited 10
from membership fees. There were withdrawals of 5 on February 28, 25 on
June 30, 80 on October 15, and 35 on October 31.
Calculate the dollar-weighted (money-weighted) rate of return for the year.
(A) 9.0% (B) 9.5% (C) 10.0% (D) 10.5% (E) 11.0%
2. (2005 Exam FM Sample Questions #8)
You are given the following table of interest rates:
Calendar
Year of
Original
Investment
y
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
Investment Year Rates (in %)
il
8.25
8.5
9.0
9.0
9.25
9.5
10.0
10.0
9.5
9.0
n
8.25
8.7
9.0
9.1
9.35
9.5
10.0
9.8
9.5
il
8.4
8.75
9.1
9.2
9.5
9.6
9.9
9.7
il
8.5
8.9
9.1
9.3
9.55
9.7
9.8
iys
8.5
9.0
9.2
9.4
9.6
9.7
Portfolio
Rates
(in %)
p+5
8.35
8.6
8.85
9.1
9.35
A person deposits 1000 on January 1,1997. Let the following be the
accumulated value of the 1000 on January 1, 2000:
P: under the investment year method
Q: under the portfolio yield method
R: where the balance is withdrawn at the end of every year and is reinvested
at the new money rate
Determine the ranking of P, Q, and R.
(A) P>Q>R (B) P>R>Q (C) Q>P>R
(D) R >P>Q (E) R >Q>P
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M5-18
Module 5 - Yield Rate of an Investment
3. (2005 Exam FM Sample Questions #19)
You are given the following information about the activity in two different
investment accounts:
Account K
Date
January 1,1999
July 1,1999
October 1,1999
December 31,1999
Fund Value Before Activity
100.0
125.0
110.0
125.0
Activity
Deposit
2X
Activity
Withdrawal
X
Account L
Date
January 1,1999
July 1,1999
December 31,1999
Fund Value Before Activity
100.0
125.0
105.8
Activity
Deposit
Activity
Withdrawal
X
During 1999, the dollar-weighted (money-weighted) return for investment
account K equals the time-weighted return for investment account L, which
equals i. Calculate i.
(A) 10% (B) 12% (C) 15% (D) 18% (E) 20%
4. (2005 Exam FM Sample Questions #23)
Project P requires an investment of 4000 at time 0. The investment pays
2000 at time 1 and 4000 at time 2.
Project Q requires an investment of X at time 2. The investment pays 2000
at time 0 and 4000 at time 1.
The net present values of the two projects are equal at an interest rate of
10%. Calculate X.
(A) 5400 (B) 5420 (C) 5440 (D) 5460 (E) 5480
5. (2005 Exam FM Sample Questions #32)
An investor pays $100,000 today for a 4-year investment that returns cash
flows of $60,000 at the end of each of years 3 and 4. The cash flows can be
reinvested at 4.0% per annum effective.
If the rate of interest at which the investment is to be valued is 5.0%, what is
the net present value of this investment today?
(A) -1398 (B) -699 (C) 699 (D) 1398 (E) 2629
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 5 - Yield Rate of an Investment
PageM5-19
6. (2005 Exam FM Sample Questions #45)
You are given the following information about an investment account:
Date
January 1
July 1
December 31
Value Immediately Before
Deposit
10
12
X
Deposit
X
Over the year, the time-weighted return is 0%, and the dollar-weighted
(money weighted) return is Y.
Calculate Y.
(A) -25% (B) -10% (C) 0% (D) 10% (E) 25%
7. (May 05 #7)
Mike receives cash flows of 100 today, 200 in one year, and 100 in two years.
The present value of these cash flows is 364.46 at an annual effective rate of
interest i.
Calculate i.
(A) 10% (B) 11% (C) 12% (D) 13% (E) 14%
8. (May 05 #16)
At the beginning of the year, an investment fund was established with an
initial deposit of 1000. A new deposit of 1000 was made at the end of 4
months. Withdrawals of 200 and 500 were made at the end of 6 months and 8
months, respectively. The amount in the fund at the end of the year is 1560.
Calculate the dollar-weighted (money-weighted) yield rate earned by the
fund during the year.
(A) 18.57% (B) 20.00% (C) 22.61% (D) 26.00% (E) 28.89%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M5-20
Module 5 - Yield Rate of an Investment
9. (May 05 #21)
A discount electronics store advertises the following financing
arrangement:
"We don't offer you confusing interest rates. We'll just divide your total cost
by 10 and you can pay us that amount each month for a year."
The first payment is due on the date of sale and the remaining eleven
payments at monthly intervals thereafter.
Calculate the effective annual interest rate the store's customers are paying
on their loans.
(A) 35.1% (B) 41.3% (C) 42.0% (D) 51.2% (E) 54.9%
10. (Nov 05 #1)
An insurance company earned a simple rate of interest of 8% over the last
calendar year based on the following information:
Assets, beginning of year
Sales revenue
Net investment income
Salaries paid
Other expenses paid
25,000,000
X
2,000,000
2,200,000
750,000
All cash flows occur at the middle of the year.
Calculate the effective yield rate.
(A) 7.7% (B) 7.8% (C) 7.9% (D) 8.0% (E) 8.1%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 5 - Yield Rate of an Investment Page M5-21
Section 5.9
Sample Exam Solutions
1.
We have A = initial fund balance = 75 B = final fund balance = 60
C = ^Ct = total cash contribution (net)= 10 (12) - 5 - 25 - 80 - 35 = -25
I = B-A-C =60-75-(-25) =10.
We will use the standard convention of counting time in even months. Thus for
1 11
example, for the end of month payment in January, t = — and 1 -1 = —. The
calculation of i requires us to find
ii + £Ct(l-t)
'2.5
__ ... 11 10 1
75 + 10 — + — + ... + —
12 12 12
-<]§M£
-80
12
-351-
. 75 + »(!L^l-«2 = 90.83
12i, 2 J 12
Thus
i. ' .-1°~.H01
A + £Ct(l-t) 90.83
Answer E
2.
Under the investment year method we use the investment year rates in the 1997
line of the table. P = 1000 (1.095) (1.095) (1.096) = 1314.13
Under the portfolio method we use the portfolio rates in the last column of the
table. The rate for 1997 is the first rate in the last column, 8.35%.
Q = 1000 (1.0835) (1.086) (1.0885) = 1280.82
When the money is withdrawn and reinvested at the next year's starting rate,
we have the new money rate each year.
R = 1000(1.095)(1.10)(1.10) = 1324.95
Thus R>P>Q.
There was no need to actually calculate the values ofR, P and Q since their
ordering is obvious. This could save some time.
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M5-22
Module 5 - Yield Rate of an Investment
For the dollar weighted account K we have:
A = initial fund balance = 100
B = final fund balance = 125
C = total cash contribution (net) = 2X-X = X
I = B-A-C= 25-X
25-X
i
100-X
/
U2;+2XU2
25-X
100
1 + i
125-X
100
For the time weighted account L we have only 2 time periods to consider.
(125^ . . ( 105.8 ^
l + j
1= 100
1 +J2
125-X
The time weighted rate is given by
1 . „ • V1 • , (125Y 105.8 A
1 + i = (1 + j1)(1 + ;2) =
Uoo.
125-X
132.25
125-X
Since the value of i is the same for both accounts we have:
132.25 125-X (i25-X)2=13,225-X = 10
125-X
100
It follows that
, . 125-10 , 1c . . 1e
l + i = = 1.15 and i = .15.
100
Answer C
The net present value of P at 10% is -4000 + ^^ + -^ = 1123.97
(This can also be found on the BA II Plus using the NPV function on the
calculator with CF0 = -4000, C01 = 2000, C02 = 4000 .and I = 10.)
The net present value of Q is 2000 +
4000 X
Thus 2000 +
4000 X
1.1 l.l2
= 1123.97
1.1 l.l2 '
X = 5460
Answer D
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 5 - Yield Rate of an Investment
Page M5-23
The timeline for the investment is below.
-100,000
60,000
60,000
0
There is only one cash flow to reinvest, the amount of 60,000 at time 3. At a 4%
rate it grows to 60,000(1.04) = 62,400 at time 4. The total amount returned to the
investor at time 4 is then 122,400. The resulting situation for the investor is that
he invests 100,000 and gets back 122,400 in 4 years.
-100,000
122,400
122 400
The net present value is -100,000 + ' , = 698.78.
1.05"
Answer C
6.
We can use the fact that the time weighted yield is 0% to find X.
(12Y X
1 + 0 =
12 + X
,10
120 + 10X = 12X
X = 60
We can use the value of X to find the dollar weighted yield. For this calculation
we need
A = initial fund balance = 10
B = final fund balance=X
C = total cash contribution (net) = X
Then we can find I = interest earned using the relation
Z = B-A-C = X-10-X = -10.
The deposit of X = 60 was made at time t= Vi Thus the dollar weighted return is
given by
-10
Y = -
10 +
>
= -.25.
Answer A
©ACTEX 2009
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Exam FM / Exam 2 - Financial Mathematics

Page M5-24
Module 5 - Yield Rate of an Investment
7.
The equation of value here is 364.46 = 100 + 200v + 100v2.
Thus you only need to solve the quadratic 0 = -264.46 + 200v + 100v2
The root v = .90908 gives i = .10
Answer A
Calculator note: the quadratic we solved is the IRR equation for the cashflow
sequence -264A6, 200,100.
If you recognize this you can enter these values in the CF worksheet and
compute the IRR. Of course, the answer is still 10%
8.
This is a standard dollar weighted yield question.
A = 1,000, B = 1,560, C=1000-200-500=300
7 = 1560-1000-300 = 260
260 26°= 0.1857
l,0W + (l-Aj(i^
1400
Answer A
9.
No price is specified. We will find the monthly yield assuming a price of 100
and monthly payments of 10 at the beginning of each month for a year. On the
BA II Plus in BGN mode, set PMT=-10, N=12, PV=100 and CPT I/Y = 3.5032. The
monthly interest rate is 3.5032%, Then the annual effective rate is
(1.035032)12-1 = 0.512
Answer D
10.
The company earns 8% interest on its initial assets of 25,000,000 for a full year
and on the net amount added at midyear for half of a year. The amount added at
midyear is X - 2,200,000 - 750,000 = X - 2,950,000.
Net investment income is 2,000,000. Thus
.08 (25,000,000) + .5 (.08) (X - 2,950,000) = 2,000,000
.5 (.08) (X-2,950,000) =0
X = 2,950,000
The net amount added at midyear is 0. Hence investment income consists only
of earnings of 2,000,000 on the original 25,000,000 invested for one year at 8%
and the effective yield must be 8%. (Note that 2,000,000 is 8% of 25,000,000.)
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 5 - Yield Rate of an Investment Page M5-25
Section 5.10
Supplemental Exercises
1. For an investment of 15,000 an investor is promised return payments of
6,000 in one year, 7000 in two years and 7000 in three years. Find the IRR
for these cash flows.
For Problems 2 and 3 use the following account summary.
Balance
Date Before Activity Deposits Withdrawals
January 1
March 1
Julyl
November 1
December 31
1000
1020
990
1100
1050
70
50
120
2. Find the time-weighted yield for this account.
3. Find the dollar-weighted yield for this account.
4. You are given the following account summary.
Balance
Date Before Activity Deposits Withdrawals
January 1
April 1
September 1
December 31
2000
2060
2010
2405
300
X
The time-weighted yield is 11.11%. Find X.
5. You are given the following account summary.
Balance
Date Before Activity Deposits Withdrawals
January 1
March 1
T
December 31
1000
1020
1110
1050
60
100
The dollar-weighted yield is 8.852%. Find the date T.
6. The following two investment projects have the same net present value
at i = 8%.
(1) Invest 5000 now and receive 3000 in one year and 4000 in two years.
(2) Invest 2500 now and receive 2000 in one year and K in two years.
FindK
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M5-26
Module 5 - Yield Rate of an Investment
7. An investment of 20,000 now is projected to return 5000 in one year, 6000
in two years, 7000 in three years and 10,000 in four years. What is the net
present values of these cash flows at i = 10%?
8. Using the table in Example 5.16, find the three year accumulation factor
for an investor who began in 1999 using the investment year method.
9. Using the table in Example 5.16, find the three year accumulation factor
for an investor who began in 1999 using the portfolio method.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 5 - Yield Rate of an Investment Page M5-27
Section 5.11
Supplemental Exercise Solutions
1. To solve using the BA II Plus hit the CF key then enter:
Co = -15,000, Ci = 6,000, C2 = 7,000 and C3 = 7,000.
Then IRR CPT = 15.44%.
2. The time-weighted yield is found by setting
1 + i = (1020/1000)(990/970)(1100/1060)(1050/980) = 1.157
i = 15.7%
3. The interest earned i - 1050 - 1000 - 70 + 50 + 120 = 150.
i = 150/[1000 - 50(5/6) + 70(1/2) - 120(1/6)] = 0.154 or 15.4%
4. 1 + i = 1.1111 = (2060/2000)[2010/(2060 - X)](2405/2310)
2060 - X = 1940 (to nearest dollar) => X = 120
5. The interest earned i = 1050 - 1000 -60 + 100 = 90.
Let x be the fraction of a year for which the 100 loses interest.
i = 0.08852 = 90/[1000 + 60(5/6) - lOOx]
1050 - lOOx = 90/0.8852 = 1016.72 => x = 0.333
The 100 loses interest for 0.333 years so it was withdrawn on Sept. 1.
6. The net present value on the first project is
NPV = -5000 + 3000/1.08 + 4000/1.082 = 1207.13
The net present value of the second project is
NPV = 1207.13 = -2500 + 2000/1.08 + K/1.082
K/1.082 = 1855.28 => K = 2164
7. To find the NPV using the BA II Plus, first hit the CF key and then enter
Co = -20,000, Ci = 5,000, C2 = 6,000, C3 = 7,000 and C4 = 10.000. Then hit the
NPV key and enter I = 10. Scroll down to NPV and hit CPT.
This gives 1,593.47.
8. The investment year rates needed here are ii1999 = 10%, i21999 = 9.8% and
i31999 = 9.7%. The three year accumulation factor is
1.10(1.098X1.097) = 1.325.
9. The portfolio rates for 1999, 2000 and 2001 are i1999 = 8.85%,
J2000 _ 9 1% and poi _ 9,35%, xhe three year accumulation factor is
1.0885(1.091X1.0935) = 1.299.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 6 - Term Structure of Interest Rates
PageM6- 1
rm Structure of Interest Rotes
Section 6.1
Spot Rates and the Yield Curve
Interest rates on loans depend in part on the time to maturity of the loan.
Anyone who has invested money in a certificate of deposit has observed that
the interest rate paid depends on the term of the CD. Typically, the longer the
term the higher the rate, although this does not have to be the case.
For example, individuals looking for mortgage loans will generally find that
they can get a lower rate on a 15 year loan than on a 30 year loan. The
dependence of yield on maturity is referred to as the term structure of interest
rates. The term structure is established by looking at the rates of zero coupon
bonds based on United States government bonds. This requires a little
clarifying discussion.
If a bond has a zero coupon, this means there are no coupon payments to the
bond holder. The only payments involved are the original investment and the
final repayment of the redemption value at maturity. A zero coupon bond for
two years with redemption value of 1000 and an annual yield of 3% would have
n 1000
price P = -
1.032
: 942.6.
In practice, investors buy bonds at prices which give them the yield they
desire. Thus, if investors were willing to pay 942.60 for a two year zero coupon
bond, we could look at this price and calculate the implied two year annual
interest rate:
942.60 = 100°, -> (1 + if = 1.0609 -> (1 + i) = 1.03 -> i = 3% .
(l + i)
Investors require higher interest rates on bonds issued by firms considered
risky, because there is a greater chance of default (i.e., not paying the bond)
from a risky firm. Thus if we looked at market prices for two year zero coupon
bonds we would find different annual interest rates for different borrowing
firms, based on risk.
The United States government is regarded as having a lower risk of default
than any other borrower, so U.S. government bonds are used as the base to
which all other bonds are compared. In fact, the interest rates on short term
Treasury bills are sometimes referred to as risk-free rates.
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page M6-2
Module 6 - Term Structure of Interest Rates
A bond's riskiness is captured in its interest rate, so if a company's bonds yield
8%, and U.S. government bonds yield 5 Vi%y the 2 Vi% difference in yield is
what investors demand to account for the difference in risk.
The United States government does not directly issue zero coupon bonds.
Investment bankers buy U.S. government coupon bonds and break them down
in single payment components called Treasury STRIPS. To get a two year
Treasury STRIP zero-coupon bond they buy a large dollar amount of a Treasury
coupon bond and resell the coupon payment due in 2 years as a 2-year Treasury
STRIP. The annual interest rate on the n-year Treasury STRIP is called the n-
year spot rate, and the series of spot rates over time is called the yield curve.
There is much more detail on the mechanics of determining the yield curve in
the Mathematics of Investment and Credit reference. On exams you will often
simply be given a yield curve for use in problems. The next table gives a set of
spot rates for years 1 through 5.
(6.1) Yield Curve Example
Year Spot Rate
1 2.00%
2 3.00%
3 3.50%
4 4.00%
5 4.50%
The Wall Street Journal has a daily graph of the previous day's actual yield
curve. Below we give the graph of the example yield curve above for years 1-5.
late
Spot F
6.00%
5.00%
4.00%
3.00%
2.00%
1.00%
0.00%
1
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Yield Curve

Module 6 - Term Structure of Interest Rates
PageM6- 3
In normal times, lenders demand higher rates of interest for longer term loans,
and the increasing vield curve above might be referred to as a normal vield
curve. In times when current rates are high but lenders anticipate the rates will
drop in the future you might see an inverted vield curve or a flat vield curve.
6.00%<
5.00%-
1 <D
IS 4.00%-
^ 3.00%-
o. 2.oo%-
(/)
1.00% -
0.00% -
Inverted Yield Curve
N=^— —•■ ■
^^^» »
♦ ^
1 2 3
Year
4 5
t
6.00%
5.00%
O 4.00% f
OS
2: 3.00%
o
a
CO 2.00%
1.00%
0.00%
Flat Yield Curve
3
Year
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M6-4 Module 6 - Term Structure of Interest Rates
The yield curve rates can be used to price a bond as the following example
shows.
Example (6.2)
A four year annual $1000 par bond has a coupon rate of 3%. Thus its
payments are
Year
Payment
1
30
2
30
3
30
4
1030
To value the bond, take the present value of each payment at the
appropriate yield curve rate and sum the present values. Using the
example yield curve in table (6.1), the price P is
D 30 30 30 1030 0„_
P = + T + =- + T = 965.20 .
1.02 1.032 1.0353 1.044
The law of one price is a financial principle which says that if you can
calculate the value of a financial instrument in two ways they must both
give the same answer. This means that if you calculate the price of this
bond using a single yield-to-maturity i, that method must give the same
price of 965.20. This in turn means that once we have found the price
using the yield curve we can find the yield to maturity by finding the
single yield rate for the cash flow sequence
-965.20, 30, 30, 30,1030.
This can be performed on the BA II Plus.
Set PV=-965.20, PMT = 30, N=4, FV=1000 and CPT I/Y=3.9578.
The yield to maturity is 3.9578%.
Exercise (6.3)
A three year annual $1000 par bond has a coupon rate of 3.2%. Use the
yield curve in Table (6.1) to find the price P and then use this price to
find the yield to maturity.
Answer: Price=992.34 Yield to maturity=3.4729%
We will use the notation sn for the spot rate of a zero coupon STRIP maturing in
n years.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 6 - Term Structure of Interest Rates Page M6- 5
Section 6.2
Forward Rates
The n-year forward rate is the rate agreed upon today for a one year loan to be
made n years in the future. For example, the one year forward rate is the rate
that would be agreed upon now for a one year loan to start one year from now.
Mathematics of Investment and Credit uses the notation in_i,nfor the n-1 year
forward rate, since that rate begins at time n-1 and after one year ends at time
n. The one year forward rate would be written i1)2.
The yield curve implies certain values for the forward rates -which are also
called implied forward rates. We can calculate the n-1 -year forward rate if we
are given the spot rates sn_i and sn . We illustrate this in the next example.
Example (6.4)
We will calculate the one and two year forward rates using the spot rates
from the yield curve in (6.1).
One year forward rate. We are given sl = .02 and s2 = .03. There are two
ways to get an accumulated value for a two year investment.
a) Invest for the entire two years at the known rate s2 = .03. The
accumulation factor is 1.032.
b) Invest for one year at the first year rate of Si = .02 and then reinvest
the first year accumulation at the one year forward rate il2. The
accumulation factor is 1.02(1 + Ut2).
Equating the two accumulation factors we have
1.02(l + i12) = 1.032->(l + ii2) = :!^^-->ii2=.0401
Two year forward rate. We are given s3 = .035 and s2 = .03. There are
two ways to get an accumulated value for a three year investment.
a) Invest for the entire three years at the known rate s3 = .035. The
accumulation factor is 1.0353.
b) Invest for two years at the rate of s2 = .03 and then reinvest the
accumulation at the one year forward rate i2,3. The accumulation
factor is 1.032 (1 + 12,3).
Equating the two accumulation factors we have
1.032 (1 + i2,3) = 1.0353 -»(1 + i2i3) = ^g- -> i2f3 = .0451.
I
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M6-6
Module 6 - Term Structure of Interest Rates
The general pattern should be fairly clear from this example.
(6.5)
1 + in_x n = -A ?z_ or equivalently
(1 + Sn)n =(l-fSn.1)n-1(l + in-l,n)
Exercise (6.6)
Find the 3 year forward rate i3>4 for the yield curve in (6.1).
Answer:
0.0551
Note that it is also possible to recover the spot rates if you are given the
sequence of forward rates. If we define the 0-year forward rate i0,ito be Si,
then we have
(l + i0,i)(l + ii,2) = (l + s2)2
(l + i0,i)(l + ii,2)(l + i2,3) = (l + s3)3
(l + i0,l)(l + il,2)...(l + in-l,n) = (l + Sn)n.
The result of compound accumulation at the first n single period forward rates
is the same as compound accumulation at the n-year spot rate for n years.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 6 - Term Structure of Interest Rates Page M6- 7
Section 6.3
Formula Sheet
Spot rates s„
The annual interest rate on the n-year Treasury STRIP is called the n-year spot
rate, and the series of spot rates over time is called the yield curve.
To value a bond, take the present value of each payment at the appropriate
yield curve rate and sum the present values.
Once we have found the price of a bond using the yield curve we can find the
yield to maturity as the constant yield on the bond at that price.
Forward rates in-\,n
1 + in-M = , T^T or equivalents (1 + sn )n = (1 + s„_i)n_1 (1 + in-i,n)
(1 + Sn-l)
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M6-8
Module 6 - Term Structure of Interest Rates
Section 6.4
Basic Review Problems
The problems in this section use the yield curve table.
YearSpot Rate
1 5.00%
2 4.50%
3 4.00%
4 4.00%
5 4.00%
1. A three year annual $1000 par bond has a coupon rate of 4%. Use the yield
curve above to find the price P and then use this price to find the yield to
maturity.
2. Find the one year forward rate.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 6 - Term Structure of Interest Rates
PageM6- 9
Section 6.5
Basic Review Problem Solutions
1. Thus its payments are
Year
Payment
1
40
2
40
3
1040
To value the bond, take the present value of each payment at the
appropriate yield curve rate and sum the present values. Using the given
yield curve the price P is
p = J0_+_40_^ + 1040==99928
1.05 1.0452 1.043
We can find the yield to maturity by finding the single yield rate for the
cash flow sequence
-999.28, 40, 40,1040.
This can be done on the BA II Plus. Set PV=-999.28, PMT = 40, N=3, FV=1000
and CPT I/Y=4.026. The yield to maturity is 4.026%.
(1 + sA1 1-05
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M6-10
Module 6 - Term Structure of Interest Rates
Section 6.6
Sample Exam Problems
1. (Fall OS Sample Problems #33)
You are given the following information with respect to a bond:
par amount: 1000
term to maturity 3 years
annual coupon rate 6% payable annually
Term
1
2
3
Annual Spot Interest Rate
7%
8%
9%
Calculate the value of the bond.
(A) 906 (B) 926 (C) 930 (D) 950 (E) 1000
2. (Fall OS Sample Problems #34)
You are given the following information with respect to a bond:
par amount: 1000
term to maturity 3 years
annual coupon rate 6% payable annually
Term
1
2
3
Annual Spot Interest Rate
7%
8%
9%
Calculate the annual effective yield rate for the bond if the bond is sold at a
price equal to its value.
(A) 8.1% (B) 8.3% (C) 8.5% (D) 8.7% (E) 8.9%
3. (May OS #10)
Yield rates to maturity for zero coupon bonds are currently quoted at 8.5%
for one-year maturity, 9.5% for two-year maturity, and 10.5% for three-year
maturity. Let i be the one-year forward rate for year two implied by current
yields of these bonds.
Calculate i.
(A) 8.5% (B) 9.5% (C) 10.5% (D) 11.5% (E) 12.5%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 6 - Term Structure of Interest Rates
PageM6-ll
4. (Nov OS #6)
Consider a yield curve defined by the following equation:
ifc= 0.09 +0.002k-0.001k2
where ik is the annual effective rate of return for zero coupon bonds with
maturity of k years.
Let j be the one-year effective rate during year 5 that is implied by this yield
curve.
Calculate j.
(A) 4.7% (B) 5.8% (C) 6.6% (D) 7.5% (E) 8.2%
5. (Nov 05 #15)
You are given the following term structure of spot interest rates:
Term (in years)
1
2
3
4
Spot interest rate
5.00%
5.75%
6.25%
6.50%
A three-year annuity-immediate will be issued a year from now with annual
payments of 5000. Using the forward rates, calculate the present value of
this annuity a year from now.
(A) 13,094 (B) 13,153 (C) 13,296 (D) 13,321 (E) 13,401
6. (Nov 05 #19)
Which of the following statements about zero-coupon bonds are true?
I. Zero-coupon bonds may be created by separating the coupon payments
and redemption values from bonds and selling each of them separately.
II. The yield rates on stripped Treasuries at any point in time provide an
immediate reading of the risk-free yield curve.
III. The interest rates on the risk-free yield curve are called forward rates.
(A) I only
(B) II only
(C) III only
(D) I, II, and III
(E) The correct answer is not given by (A), (B), (C), or (D).
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M6-12
Module 6 - Term Structure of Interest Rates
Section 6.7
Sample Exam Solutions
1.
Recall that the spot rate at term n is the interest rate that is used to find the
present value of the payment at time n. The annual 6% coupon on the bond is
60, and there is a final payment of 1060. The present value of the bond
payments is
60 +_60 +1060 =926m
1.07 1.082 1.093
Answer B
2.
We have already seen that the value of this bond is 926.03. Thus we need to find
the yield i for an investor who buys it at this price.
We can solve for i using the financial calculator (and we really did not need to
write down the equation above.) On the BA II Plus enter PMT=60, FV=1000,
N=3, and PV=-926.06 and compute I/Y. The result is 8.92.
Answer E
3.
We are given s2 = .095 and S\ = .085. We are asked to find U>2.
l + i12=fil^l =1^1 = 1.1051->i12-.1051
(1 + sX 1.085
(1 + Sl)
Answer C
4.
Year 5 extends from times n=4 to n=5. We are looking for the implied forward
rate i4>5. This is defined by (l + s5)5 = (l + s4)4(l + i4>5).
We are given that sk = 0.09 + 0.002k -0.001k2. (The problem uses the notation ik
where we used sfc). Thus
s4 =0.09 + 0.002(4)-0.00l(42) = .082
s5 = 0.09 + 0.002 (5) - 0.001 (52) = .075
(1.075)5 = (1.082)4 (1 + i4fS) -+ i4fs = .047
Answer A
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Module 6 - Term Structure of Interest Rates
PageM6-13
The idea here is to use forward rate reasoning to create an implied yield curve
for one year from now and then use that new yield curve to find the present
value of the annuity to be issued then. The one year spot rate ji a year from will
be today's one year forward rate.
- . - . 1.05752 1 n„
1 + h = 1 + h,2 = = 1.065.
The new two year spot rate j2 can be obtained by using both the one and two
year forward rates.
(l + j2) =(l + il,2)(l + J2,3) =
Similarly
1.0575
1.05
2 V
1.06253
1.05752
1.0625
1.05
3\
(l + j3)3=(l + i,2)(l + i2,3)(l + i3,4) = [^|-4'
The implied present value of the annuity in a year is
5000
l + ji (l + j2)2 (1-fja)3
= 5000
1.05 1.05 1.05
■ + =- + -
1.05752 1.06253 1.0654
13,152.50
Answer B
6.
I is true. Treasury STRIPS.are created in this fashion.
II is true.
III is false. The interest rates on the yield curve are called spot rates.
Answer E
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Exam FM / Exam 2 - Financial Mathematics

Page M6-14
Module 6 - Term Structure of Interest Rates
Section 6.8
Supplemental Exercises
For Problems 1, 2 and 3 use the following yield curve.
Year Spot Rate
1
2
3
4
5%
6%
7%
8%
1. For a 4-year 1000 par bond with 5% annual coupons, calculate the price
of the bond.
2. If the bond in Problem 1 is sold at its price, what is its annual effective
yield?
3. For the above yield curve, find the three year forward rate.
For Problems 4 and 5, use the following yield curve.
sk = 0.085 + 0.003k - 0.0015k2
4. For a 3-year 1000 par bond with 6% annual coupons, calculate the price
of the bond.
5. Find the three year forward rate implied by this yield curve.
6. You are given the following n-year forward rates:
Year Forward Rate
0
1
2
3
3.0%
4.4%
4.8%
5.6%
Find s4.
7. For a four-year 1000 par bond with 5% annual coupons, find the price of
the bond using the spot rates implied by the forward rates in Problem 6.
8. Find the yield to maturity of the bond in Problem 7.
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Module 6 - Term Structure of Interest Rates Page M6-15
Section 6.9
Supplemental Exercise Solutions
1. The price of the bond is
50/1.05 + 50/1.062 + 50/1.073 + 1050/1.084 = 904.72
2. To find the yield using the BA II Plus calculator, set
N = 4, PMT = 50, PV = -904.72 and FV = 1000.
Then CPT I/Y = 7.868%
3. The three year forward rate is
1 + 13,4 = (1 + s4)4/(l + s3)3 = 1.08V1.073 = 1.111
i3>4 = 11.1%
4. We first need to find the one-year effective rates for years 1, 2 and 3.
St = 0.085 + 0.003 - 0.0015 = 0.0865
s2 = 0.085 + 0.006 - 0.006 = 0.0850
s3 = 0.085 + 0.009 - 0.0135 = 0.0805
The price of the bond is
60/1.0865 + 60/1.0852 + 1060/1.08053 = 946.49
5. For this problem we also need s4.
Sa = 0.085 + 0.012 - 0.024 = 0.073
The three year forward rate is
1 + i3A = 1.0734/1.08053 = 1.0508
i3.4 = 5.08%
6. (1 + s4)4 = (1 + io.iXl + ii,2)(l + i2,3)(l + i3>4)
= (1.03)(1.044)(1.048)(1.056) = 1.1900
1 + s4 = 1.0445 ^ s4 = 4.45%
7. The price of the bond is given by
50/(1 + si) + 50/(1 + s2)2 + 50/(1 + S3)3 + 1050/(1 + s4)4
From Problem 6 we know that (1 + s4)4 = 1.1900
1 + si = 1.03, (1 + s2 )2 = (1.03X1.044) = 1.0753
(1 + S3)3 = (1.03)(1.044)(1.048) = 1.1269
P = 50/1.03 + 50/1.0753 + 50/1.1269 + 1050/1.1900 = 1021.77
8. To find the yield to maturity using the BA II Plus calculator set
N = 4, PMT = 50, PV = -1021.77 and FV = 1000.
Then CPT I/Y = 4.395
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 7 - Asset Liability Management, Duration, and Immunization
PageM7- 1
Asset Liability Management,
Duration, and Immunization
Section 7.1
Introduction to Matching Assets and Liabilities
Insurance companies collect premiums from their customers and then invest
these premiums. These premiums and the interest earned on them are the
insurance company's assets. The assets are used to pay claims as they occur.
The claims are liabilities. Insurance companies are required to make sure that
the assets are matched to the liabilities to assure that the cash will be available
to pay claims as they occur.
In this chapter, we will first give some examples of asset liability management
for some very simple situations to illustrate the basic ideas. Then we will move
to the tools of duration and immunization which are used for the more realistic
complex situations that occur in reality. The examples used here to illustrate
asset liability management are based on actuarial examination problems.
Example (7.1)
A company must pay liabilities of 2000 and 3000 at the end of years 1
and 2, respectively. The only investments available to the company are
the foil
owing two zero-coupon bonds:
Maturity (years)
1
2
Effective Annual Yield
5%
6%
Par
1000
1000
The company today can cover its liabilities exactly by buying two of
the 5% one-year zero coupon bonds and three of the 6% two year zero
coupon bonds. This is called an exact match. Next we will find the cost
of the necessary bonds.
For year 1 the company is investing in $2000 worth of 5% one-year
zero coupon bonds. The cost is
2000
1.05
= 1904.76.
For year 2 the company is investing in $3000 worth of 6% two-year
zero coupon bonds. The cost is
3000
1.062
= 2669.99.
The total invested to match liabilities is 1904.76 + 2669.99 = 4574.75.
©ACTEX 2009
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Exam FM / Exam 2 - Financial Mathematics

Page M7-2
Module 7 - Asset Liability Management, Duration, and Immunization
Exercise (7.2)
Suppose that liabilities in the above problem were 1000 in one year and
2000 in two years. Find the cost of exactly matching those liabilities.
Answer: 2732.37
The matching process is a bit more complicated when the bonds are coupon
bonds, as the next example shows.
Example (7.3)
Joe must pay liabilities of 1,000 due 6 months from now and another 1,000 due
one year from now. There are two available investments:
1) 6-month bond with face amount of 1,000, a 6% nominal annual coupon
rate convertible semiannually, and a 5% nominal annual yield rate
convertible semiannually
2) 1-year bond with face amount of 1,000, a 7% nominal annual coupon rate
convertible semiannually, and a 8% nominal annual yield rate
convertible semiannually.
We will first look at the amount of each bond to buy. (Note: problems like this
assume that you can purchase fractions of bonds.)
First note that in 12 months only the one-year bond will remain. The total
payments for the 1-year bond at month 12 consist of a coupon of 35 and the
redemption value of 1000 for a total of 1035. To cover a liability of 1000, the
required percentage Joe must buy of the 1-year bond is = .96618 .
Once you have purchased .96618 of the 1-year bond, it will provide a coupon
payment of (.96618)35 = 33.82 at month 6. To fund a total liability of 1000 at
month 6, the additional amount needed from the 6-month bond is 1000-33.82 =
966.18.
The total payments for the 6 month bond at month 6 consist of a coupon of 30
and the redemption value of 1000 for a total of 1030. To cover a liability of
966.18, the required percentage Joe must buy is :— = .93804 .
1030
Now we can look at the cost of the bonds required to match the liabilities. We
have seen that Joe must purchase .93804 of the 6-month bond and .96618 of the
1-year bond. We can find the prices of the separate bonds using the financial
calculator.
6-month. N=l, PMT = 30, FV=1000, yield i = 2.5% per semiannual period. This
gives a price of PV = -1004.88.
1-vear. N=2, PMT = 35, FV=1000, yield i = 4% per semiannual period. This gives
a price of PV = 990.57.
The total cost of purchasing the required bonds is
.93804(1004.88)+.96618(990.57)=1899.69.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 7 - Asset Liability Management, Duration, and Immunization Page M7- 3
Exercise (7.4)
Joe must pay liabilities of 1,000 due 6 months from now and another
2,000 due one year from now. There are two available investments:
1) 6-month bond with face amount of 1,000, a 4% nominal annual
coupon rate convertible semiannually, and a 5% nominal annual
yield rate convertible semiannually
2) 1-year bond with face amount of 1,000, a 6% nominal annual
coupon rate convertible semiannually, and a 8% nominal annual
yield rate convertible semiannually.
Find the amount of each bond to purchase and the total cost of the bonds.
Answer: Buy 1.94175 of the one year bond and 0.92328 of the six month bond. Cost 2823.90
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M7-4
Module 7 - Asset Liability Management, Duration, and Immunization
Section 7.2
Duration
The reality of investments for an insurance company or a bank is much more
complex that the previous exact match examples. There are thousands of claim
liabilities, thousands of accounts and thousands of bonds and other investments
to buy. The company may have to sell bonds to meet unexpected liabilities at
various times, and it also faces interest rate risk. Interest rate risk occurs
because the value of its investments decreases when interest rates go up and
increases when interest rates decline.
The concept of duration gives an investment manager a way of calculating what
his interest rate risk is so as to control that risk and match assets and liabilities
for his entire portfolio. There are two closely related types of duration,
Macaulay duration and modified duration. There is a simple way to describe the
Macaulay duration of an investment-it is the weighted average time at which
the investment pays.
It is worthwhile to review the concept of weighted average. Let
Xi9...fxn be a set of n real numbers and Wi,...,wn be a set of n
positive real numbers such that Wi +... + wn = 1. The weighted
average of Xi,...,x„ with the weights Wi,...,wn is the sum
W = JtiWi + ... + Jt„W„.
For example, the weighted average of the numbers 1,2,3 with
weights .5, .3, .2 is .5(l)+.3(2)+.2(3) = 1.1.
For an investment which has cash flows CFi,...,CF„at times 1, 2, ...,n, the
duration D is a weighted average of the times of payment 1,2,...,n.
The weights are based on the terms of the present value sum. The present value
or price of this investment is P = vCFi + v2CF2 +... + vnCFn.
The weight for the ith payment is just its term in the expression for P divided
by P.
(7.S)
Wi
vlCFt
vlCFt
vCFl + v2CF2+... + vnCFn
It is clear that wx +... + wn = 1.
The Macaulay duration D is defined using the weights wt from (7.5) by
(7.6)
D = (l)wi +... + (n)w„
"(DvCFx +(2)v2CF2 +... + (n)vnCFn
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Exam FM / Exam 2 - Financial Mathematics

Module 7 - Asset Liability Management, Duration, and Immunization Page M7- 5
We illustrate (7.6) in the next example.
Example (7.7)
An investment pays 1000 in one year, 2000 at the end of the second year
and 3000 at the end of the third year. An investor has purchased it to
yield the annual rate i = .10. The present value is
p = 1000 2000 3000 = =
1.1 l.l2 l.l3
The weights for the duration are
909 09
yvy.vy 188768
4815.92
1652,89 =343214
4815.92
2253.94 = 468Q18
4815.92
The Macaulay duration is the weighted average time
D = .188768 (1) + .343214 (2) + .468018 (3) = 2.27925
Exercise (7.8)
An investment pays 1000 in one year, 2000 at the end of the second year
and 3000 at the end of the third year. An investor has purchased it to
yield the annual rate i = .08. Find the Macaulay duration.
Answer: 2.28983
A comment on notation:
The above example illustrates that the Macaulay duration depends
on the interest rate i, and can be thought of as a function D(i).
We have used the notation D since that is used in Mathematics of
Investment and Credit. The other official reference, Financial
Mathematics, uses the notation MacD for Macaulay duration.
Many actuaries will use the notation d for Macaulay duration,
since that notation is used in the classic text The Theory of Interest
by Kellison (the Kellison text is no longer part of the syllabus).
You have seen that the Macaulay duration is a weighted average payoff time
for an investment. It may not be immediately obvious why this tells you
something about interest rate risk (although it does). We will see how this
works in the next section, where we will study modified duration.
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Page M7-6
Module 7 - Asset Liability Management, Duration, and Immunization
Section 7.3
Modified Duration
The interest rate risk that worries an investment manager is the change in
value that occurs when interest rates change. We can study the rate of change
in price when interest rates change by looking at the derivative of price P with
dP
respect to interest rate i, —. We will illustrate this in the next example.
di
Example (7.9) _^^____
We return to the investment of (7.7). The investment pays 1000 in one year,
2000 at the end of the second year and 3000 at the end of the third year. The
price P at a rate iis
P(i) = i22o+^22_+Jooo_=iooo(i+i)-1+2ooo(i+i)-2+3ooo(i+i)-3
1 + i (1 + i) (1 + i)
Thus 4L = p'(i) = (-1) 1000 (1 + i)'2 + (-2) 2000 (1 + i)'3 + (-3) 3000 (1 + i)^.
In Example (7.7) the investor purchased this investment to yield i = .10 For
U.10,
^r = (-1)1000 (1.1)~2 + (-2)2000(1.1)"3 + (-3) 3000 (1.1) ~" = -9978.83
Note that the derivative is negative, since the price of this investment is a
decreasing function of i.
The modified duration DM ( also referred to as the volatility) is the negative of
the derivative divided by the price -representing the rate of change as a
percent of price.
(7.10)
Example (7.11)
For the investment in (7.7) and (7.9), the price was P = 4815.92. Thus the
dPN
modified duration was
DM = -
di
-9978.83
4815.92
= 2.07205.
Note that the modified duration above is close to the actual Macaulay duration
of 2.27925. There is a nice relationship between DM and D which follows:
(7.12)
DM =
D
1 + i
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Module 7 - Asset Liability Management, Duration, and Immunization
PageM7- 7
We will discuss why this is true, but first let's verify the answer from (7.11):
D 2.27925
DM
= 2.07205
1 + i 1.1
It is important to memorize the relationship in (7.12) and be able to use it.
The following derivation shows why the relationship in (7.12) holds. You may
skip it you like.
dvk d /„ .x-fc 7 /„ .\-(k+i)
First note that
= -k-
di di ' v / 1 + i
Since the price P is given by P = vCfi + v2CF2 +... + vnCFn,
^r = -Kh-l)vCFi + (-2) v2CF2 +... + (-n)vnCFn 1.
di l + iL v ' v ' J
-^t [(-DvCFi + (-2) v2CF2 +... + (-n) vnCFn]
DM =
dP^
di
1 + i
(DvCFj + (2) v2CF2 +... + (n) vnCF„
1 + i
D
Exercise (7.13)
In Exercise (7.8) we found the Macaulay duration D for the investment
when i = .08. Find the modified duration DM directly and verify it using
(7.12).
Answer: 2.12022
The relationship (7.12) gives a nice insight into the behavior of assets under
interest rate risk. The change in value due to changes in interest rates is
greater for longer duration assets. All the needed risk information can be
obtained from the duration.
A comment on notation:
The notation DM is used for modified duration in Mathematics of
Investment and Credit, Financial Mathematics uses the notation
ModD instead. Kellison uses the notation v and refers to the
modified duration as the volatility.
The price P can also be expressed in terms of the continuous rate S = ln(l + i)
In this form, v = e~5 and
P = vCFi + v2CF2 +... + vnCFn = e~5CFx + e~2SCF2 +... + enSCFn.
It can be shown that DM = - —
©ACTEX 2009
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Exam FM / Exam 2 - Financial Mathematics

Page M7-8
Module 7 - Asset Liability Management, Duration, and Immunization
Section 7.4
Helpful Formulas for Duration Calculations
When payments are level (CFi = CF2 =... = CFn) it can be shown that
(7.14)
Duration of level payment investment: D =
Ma
We can see why this works by looking at an example.
Example (7.15)
An investment pays 1000 at the end of each year for the next 3 years.
Then at a rate ithe price is P = lOOOa^ Macaulay duration is given by
l(1000)v + 2(1000)v2 + 3 (1000) v3 _ y + 2v2 + 3v3 _ jla)^
lOOOa^
a3t
At the rate i = .06, we have
D
(Ja)3n = 5.242
a^ 2.673
1.96.
Exercise (7.16)
An investment pays 2000 at the end of each year for the next 5 years.
Find the Macaulay duration D at rate i = 0.06.
Answer: 2.88
There is also a similar simplifying formula for the duration of a coupon bond.
(7.17)
Macaulay duration of a coupon bond with face value F and
coupon Fr for n periods and redemption value C
Fr (Ia)^ + nCvn Fr (Ia)^ + nCvn
Fr(a^) + Cvn ~ Bond Price
Note: when F-C and r=i in the above formula, you can prove
that D = a^t.
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Module 7 - Asset Liability Management, Duration, and Immunization
PageM7- 9
Example (7.18)
An annual par bond has face value of 1000, a coupon rate of 5% and
three years to maturity. At a rate of i = .06,
50 (Ia)^Q 06 + 3(1000) v3 _ 50(5.242247) + 3000(.839619)
50(a^006) + (1000)v3 "50 (2.673012)+ 1000 (.839619)"
Exercise (7.19)
An annual par bond has face value of 1000, a coupon rate of 6% and 5
years to maturity. Find D at a rate of i = .05.
Answer: 4.47
Formula (7.17) gives an intuitively obvious result for zero coupon bonds. If the
coupon rate r is zero, we have
Fr(Ia)^ + nCvn nCvn
D =—-—— = = n .
Fr(a^) + Cvn Cvn
The above formula says that duration of a zero coupon bond
payable in n periods is n.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M7-10
Module 7 - Asset Liability Management, Duration, and Immunization
Section 7.5
Using Derivatives to Approximate Change in Price
Once we know the duration or modified duration we can find the derivative
from it, and vice-versa. Derivatives can be used to find the Taylor series for a
function f(x). The basic Taylor series formula is
x «./ x , / x f"(x)AX2 f{n)(x)AXn
/(x + ax) = /(x) + /'(x)ax + ^-^ + .... + i_U + ...
This gives a formula for the change in f(x) from x to x + Ax
(7.20)
Af = f(x + Ax)-f(x)
= / (x) Ax + —M + •— + — + •
2!
n!
If we just use the first term of the above series we have the familiar
approximation
(7.21)
Af = f(x + Ax)- f(x)* f'(x)Ax
Using the first two terms to improve the approximation we have
(7.22)
Af = f{x + Ax)- f{x)« f(x)Ax +
f"(x)Ax2
2!
The price function, P(i), is a function of i, and we can use the above formula to
approximate change in price as i changes.
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Module 7 - Asset Liability Management, Duration, and Immunization Page M7-11
Example (7.23)
We return again to the investment of (7.7). The investment pays 1000 in one
year, 2000 at the end of the second year and 3000 at the end of the third year.
The price P at a rate iis
p(.) = iooo+^ooo_+jooo_ = 1000(1+ 1+200 1+ 2 + 3000(1+.r
1 + i (1 + 0 (1 + 0
The first two derivatives of the price function are
^ = P'(i) = (-1) 1000 (1 + i)"2 + (-2) 2000 (1 + i)"3 + (-3) 3000 (1 + i)"4
fL£ = p»(i) = (2) 1000 (1 + i)~3 + (6) 2000 (1 + i)~* + (12) 3000 (1 + i)"5
Suppose this asset is purchased to yield i = .10. We have already seen that
the price is
p(10) = 1000 + 20^+3^ = 48159279
Now suppose that the yield changes by Ai = .001 to i + Ai = .101. The actual
new price is
P(.101) = 15~+^ + J5^ = 480S.96Sl.
v } 1.101 1.1012 1.1013
The actual change in price is
AP = P (.101) - P(.10) = 4805.9651 - 4815.9279 = -9.9628
Now we will use the two approximation formulas. First we need to evaluate
the derivatives involved.
P'(.10) = (-l)1000(l.l)"2+(-2)2000(l.l)"3+ (-3)3000(1.1)^ =-9978.8266
P"(.l) = (2)1000(l.l)"3 +(6)2000(1.1)^ +(12)3000(1.1)"5 =32051.9587
Using the first derivative approximation (7.21) we have
AP = P (.101) - P (.10) * P' (.10) .001 = -9.9788
AP = P(.101)-P(.10)«P'(.10).001 + P (10)(001) =_9.9628
The first approximation to the true change in price is reasonable and the
second is accurate to 4 decimal places.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M7-12
Module 7 - Asset Liability Management, Duration, and Immunization
Exercise (7.24)
An investment pays 3000 in one year, 2000 at the end of the second year
and 1000 at the end of the third year. It is priced to yield 8% annually.
Find
a) the current price
b) the exact change in value if rates go down to 7.95%
c) the approximate change in price using (7.21)
d) the approximate change in price using (7.22).
Answer:
a) 5286.29
b) 3.9789
c) 3.9762
d) 3.9789
Recall that we defined modified duration as DM =
In financial mathematics, the second derivative is used to define the convexity.
Recall from basic calculus that the second derivative is used to study maxima
and minima.
At a local minimum, the second derivative is non-negative and the curve is concave up.
r>o
+ f"±o
At a local maximum, the second derivative is negative or 0 and the curve is concave
down.
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Module 7 - Asset Liability Management, Duration, and Immunization
Page M7- 13
(7.25)
Convexity =
P(i)
Using this terminology we can write the approximation formulas for change in
price either in terms of P(i) and its derivatives or in terms of duration and
convexity.
(7.26)
AP = P(i + Ai)-P(i)*P'(i)Ai:
P(i)
(P(i)M) = -(DM)P(i)M
(7.27)
AP*P'(i)Ai +
P"(0(A*)2
2!
= -DM(P(i)M) + (Convexity) "^ l'
We developed our approximation formulas in terms of derivatives rather than
duration and convexity because the Taylor series is the mathematical base for
these approximations. However portfolio managers think more in terms of
duration, and are more likely to estimate price changes due to interest rate
changes using the simple estimate
(7.28)
AP = -(DM)P(i)M =
f D
1 + i
P(i)Ai
Example (7.29)
An annual corporate bond is priced to yield 6.5% annually and has a
price of 969.56 and a Macaulay duration of D=6.5772. You could estimate
the change in price if rates increase by 0.10% as
(D)P(i)Ai _ 6.5572 (969.56) (.001)
AP:
1 + i
1.065
= -5.9693
Exercise (7.30)
An annual corporate bond is priced to yield 6% annually and has a price
of 965.35 and a Macaulay duration of D=3.7177. Use (7.28) to estimate the
change in price if rates increase by 0.10%.
Answer: -3.3857
The clear implication of (7.28) for portfolio managers is that
longer duration investments undergo greater price changes than
do shorter duration investments when interest rates change.
Longer duration is viewed as a sign of greater volatility.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M7-14
Module 7 - Asset Liability Management, Duration, and Immunization
Section 7.6
The Duration of a Portfolio
Up to this point we have concentrated on finding the duration for a single asset.
It is more common for investors to own a portfolio containing a number of
different investments. Since duration is used to estimate interest rate
sensitivity, a portfolio investor would like to know the duration of his portfolio.
We will begin by looking at a simple example where the portfolio in question
has only two assets, both priced at par.
Example (7.31)
An investor can buy two annual payment bonds.
a) A $1000 annual bond for four years with coupon of 5% priced at
$1000 to yield 5%. Its modified duration is DMa = 3.5460.
b) A $1000 annual bond for eight years with coupon of 7% priced
at $1000 to yield 7%. Its modified duration is DMb = 5.97130 .
The investor buys 3 of bond a) for $3000 and 2 of bond b) for $2000. She
could estimate the change in price if rates on both bonds increase by
0.10% by doing separate duration estimates and adding them.
a) Pa = 3000 and
APa * -(DMa)Pa(i)Ai = -(3.5460)(3000)(.001) = -10.6380
b) Pb = 2000 and
APb * -(DMb)Pb(i)Ai = -(5.9713)(2000)(.001) = -11.9426
For the entire portfolio with original value P = Pa+Pb = 5000, we have
AP = APa+APb= -10.6380 -11.9425 = -22.5806
We could look at this sum in a slightly different way.
AP = APa + APb » -(DMa)Pa(i)Ai + (-(DMb)Pb(i)Ai)
= -P
(DM^I^ + iDM,)^^
Al
= -50001 (3.546)^0 +(5.9713)^
V 5000 5000
= -5000 (4.5161) .001 = -22.5805
.001
The value of 4.5161 in parentheses in the last line above is the weighted
average of the durations of the two bonds, weighted according to each
bond's percent of total price. It functions as a single duration for the
entire portfolio, and can be used to evaluate interest sensitivity for the
entire portfolio if the interest rates on each bond change bv the same
amount. (The calculation above was based on assuming that the same Ai
applied to the interest rate on each bond.)
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Module 7 - Asset Liability Management, Duration, and Immunization
Page M7- 15
This reasoning works in general. Suppose that there are m investments with
present values of Xi,X2,...,Xm in the portfolio, and that the modified durations
of these investments are DM1,DM2,...,DMm . Then the present value (price) of
the entire portfolio at the rate i is P(i) = Xi + X2 +... + Xm. Note that we are
assuming that each bond has the same change in i.
The modified duration of the portfolio is the weighted average of the modified
durations of the investments with each investment Xk having weight equal to
its percent of total portfolio value:
(7.32)
In words, the modified duration of a portfolio in which all investments have the
same interest rate shift, Ai, is the weighted average of the individual
investment modified durations, with the weight for each investment equal to its
percent of total portfolio value.
Example (7.33)
An investor has a portfolio containing $30,000 worth of a two year bond
with a modified duration of 1.96, $20,000 worth of a three year bond with
a modified duration of 2.88, and $50,000 worth of a 5 year bond with a
modified duration of 4.59. Thus her portfolio has weights of 30% in two
year bonds, 20% in three year bonds and 50% in 5 year bonds. The
modified duration of the entire portfolio is
.30(1.96) + .20(2.88) + .50(4.59) = 3.46
Exercise (7.34)
An investor has a portfolio containing $10,000 worth of a two year bond
with a modified duration of 1.95, $40,000 worth of a four year bond with
a modified duration of 3.71, and $50,000 worth of a 6 year bond with a
modified duration of 5.50. Find the modified duration of the entire
portfolio.
Answer: 4.43
When all investments have the same interest rate shift, Ai, we say that there is
a parallel shift in the yield curve. Yield curve shifts are not always parallel, and
when they are not this weighted average approach to portfolio duration will be
less accurate than desired in determining interest rate sensitivity for the
portfolio. There is an extensive discussion of this problem in Mathematics of
Investment and Credit (pages 354-357) for the reader who wants more detail.
However you should be aware that it is common to use such value weighted
averages of modified duration or Macaulay duration to make a quick estimate
of portfolio volatility.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M7-16
Module 7 - Asset Liability Management, Duration, and Immunization
Section 7.7
Immunization
In Section 7.1 we dealt with the simple situation where assets and liabilities
could be matched exactly, but we pointed out that in many cases exact
matching is not possible. Immunization is a method designed to protect against
adverse interest rate changes in these more complex cases.
Suppose that the current interest rate for valuation of assets and liabilities is
i0. The idea behind immunization is to look at the present value of assets and
the present value of liabilities both using rate i0. We will denote these by
PVA (i0) and PVL(U).
To begin, we need the present value of assets and liabilities to be equal. We
know our liabilities, so we want to choose assets that match.
(7.35)
PVA(io) = PVL(i0)
An interest rate fluctuation means that there is a small change in the interest
rate from i0 to a value i near i0. We would like this change to cause the present
value of assets to be greater than the present value of liabilities, leaving us
better off.
(7.36)
PVA(i)>PVL(i)
Mathematics of Investment and Credit uses the notation
(7.37)
h(i) = PVA(i)-PVL(i)
The function h(i) represents the difference between the present value of
assets and the present value of liabilities. We want h(i) to be 0 at the rate i0
and positive when rates change by a small amount to i.
(7.38)
h(i0) = 0 and h(i)>0 for i near i0
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Module 7 - Asset Liability Management, Duration, and Immunization
Page M7-17
The kind of graph we would desire for h(i) near i0 should look like the one
below, in which i0 = 0.05.
The graph illustrates clearly that we need to have a local minimum for h (i) at
i0 to achieve immunization. This means that we would like to have the first
derivative equal to 0 and the second derivative positive.
(7.39)
h'(i0) = 0 and h"(i0)>0
These derivative conditions are usually stated in terms of duration and
convexity, since duration is related to the first derivative and convexity to the
second. Since h(i) = PVA (i)-PVL (i), we can re-state (7.38)) and (7.39) in terms
of duration and convexity as follows:
(7.40)
Present Value Matching:
PVA(io) = PVL(i0)
(7.41)
Duration Matching:
di
PVA (i)
di
PVL(i)
(7.42)
Greater Convexity for Assets:
d2
J>40|
di
-PVL(i)
Remember that portfolio managers typically think in terms of duration and
convexity. Mathematics of Investment and Credit gives a nice working version
of the above equations in terms of the asset and liability amounts at time t, At
and Lt.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M7-18
Module 7 - Asset Liability Management, Duration, and Immunization
(7.43)
Present Value Matching:
(7.44)
Duration Matching:
%tAtvl =2,tLtvl
(7.45)
Greater Convexity for Assets:
Xt2Atv!0>£t2Ltv[0
The idea for a portfolio manager is to look at the known liabilities Lt and then
find assets At that satisfy these equations and thus "immunize" the combined
portfolio of assets and liabilities. In real world cases where there can be
thousands of liabilities and possible asset choices, computers would be used to
perform this task. However we can illustrate the basic ideas of the computation
with a simple example in which there is only one liability and only two assets to
work with. (This is a level of problem that might be possible on exam FM.)
Example (7.46)
You have a single liability of 120,000 payable at time 6. The valuation
interest rate is iQ = .05 .You wish to attempt to immunize this portfolio by
buying two zero coupon bonds with maturities at times 2 and 12. Thus
you know that L6 = 120,000. You need to find the amounts of the two
bonds, A2 and Au. You can develop a system of equations to find A2 and
A12 using present value and duration matching.
120,000 _ A2 A12
Present Value Matching:
1.056
. „ u. (6)120,000
Duration Matching: -^ 2—
1.056
1.052 1.05
2A2 12A12
12
1.052 1.05
12
This reduces to a system of two equations in two unknowns.
89,545.8476 = 0.90703A2 + 0.55684Ai2
537,275.0856 = 1.81406A2 +6.68205A12
The solution rounded to dollars and cents is
A2 = 59,234.58 A12 = 64,324.59
Now that we have found A2 and Au, we can check the convexity
condition to see if we have immunized the portfolio.
Y,t2Ltvi=62
St2Atv[0=22
f 120,000^
1.056
59,234.58
1.052
= 3,223,650.51
64,324.59^
+ 122
1.05
12
5,372,750.80
We see that ^t2Atv^ >YJt2Ltvtio , so we have immunized the portfolio.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 7 - Asset Liability Management, Duration, and Immunization Page M7- 19
Exercise (7.47)
You have a single liability of 100,000 payable at time 5. The valuation
interest rate is i0 = .06. You wish to attempt to immunize this portfolio
by buying two zero coupon bonds with maturities at times 3 and 10. Find
the amounts of the two bonds, and verify that the portfolio is immunized.
Answer: A3 =63,571.17, A10 =38,235.02 It2Ltv?0 =1,868,145.43 Y.t2Atv\Q =2,615,403.61
Note that the conditions for a local minimum only guarantee that immunization
protects us for a small change in the interest rate to a value i near i0. Thus
portfolio managers talk about the possibility of needing to re-structure the
portfolio again after a substantial shift in interest rates.
In some cases we get lucky and our portfolio is protected against any change in
interest rates. In this case, the portfolio is fully immunized and PVA (i) > PVL (i)
for any for any positive rate i * i0
The portfolios in our exercise and example here were fully immunized. This is
proved in Mathematics of Investment and Credit (page 366) where it is shown
that when a single liability Ls is immunized by two assets of longer and shorter
maturity Atland At2, ti<s<t2y full immunization always results.
Mathematics of Investment and Credit also gives an example of a portfolio that
is immunized but not fully immunized in Example 7.6 (c). In that case, there are
15 liabilities and two assets. The calculations are similar to those of the
preceding exercises.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M7-20
Module 7 - Asset Liability Management, Duration, and Immunization
Section 7.8
Stocks and Other Investment Opportunities
Bonds are widely used as investment vehicles for insurance companies.
However, there are many more places for insurers to invest premium income.
Chapter 8 of Mathematics of Investment and Credit contains sections which
give brief descriptions of stocks, mutual funds, CDs, money market funds and
mortgage backed securities. These sections are on the Exam FM syllabus, and
we will review them here.
Stocks
If you own a share of a company's stock you are a partial owner of the
company. Stockholders are typically paid a share of the company's profits
called dividends.
There are many ways to value a stock. The valuation method discussed in
Mathematics of Investment and Credit for Exam FM values the price of a stock
as the present value of all future expected dividends dk at a valuation interest
rate i.
(7.48)
Z-ffc=i
(i+0*
A very common model is based on constant percentage growth in dividends. If
the dividend expected at the end of the current period is D and the constant
percentage growth rate is g, the price is given by
p_ D D(l + g) D(l-fg)2
1 + i (1 + i)2 (1 + i)3
= D
1 , 1 + g (1 + g)2
1 + i (1 + i)2 (1 + i)3
D
1 + i
D
i+ii±*ui+*
1 + i
1-
D
1 + i
1 + g
1 + i
1 + i
+ ...
©ACTEX 2009
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Exam FM / Exam 2 - Financial Mathematics

Module 7 - Asset Liability Management, Duration, and Immunization Page M7- 21
Thus the price of a stock can be obtained using the dividend growth model:
(7.49)
Example (7.50)
p. °
<*-*)
A stock is expected to have a dividend of 5 in one year. The valuation
interest rate is i = .05. If each subsequent annual dividend is expected to
be 3% larger than the preceding one, the value of the stock now is
P=»-l— = 250.
(i-g) .05-.03
Stock valuation is also performed using spot rates or forward rates instead of a
single valuation rate. Recall that Mathematics of Investment and Credit uses
the notation sn for the n-year spot rate and in_i,n for the forward rate in year n,
which extends from time n-1 to time n. Using these notations, the valuation
formulas are
dk
Spot rate: p = E°=i
(1 + s*)*
Forward rate P = 7—-—- + ^ r- +....
(l + i0,i) (l + io,i)(l + ii,2)
These two formulas are equivalent. The valuation rate i for the model in (7.48)
should be chosen so as to give the same answer as the two preceding formulas.
The dividend growth model is widely used, and you will see it on some of the
sample exam problems at the end of this chapter.
Mutual funds
A mutual fund allows you to invest in a pool of stocks selected by professional
managers. Investors pay a unit price to the managers and receive a unit share
of the fund in return. The managers use the total amount of cash received from
all the unit shares to buy selected stocks. An investor can sell his units back to
the mutual fund and be paid at the current value per unit. The managers may
sell stock to buy back units or buy stock when new units are purchased.
There are a wide variety of mutual funds with differing strategies. A fund
might specialize in the stocks of an industry ( e.g., an energy stock fund), a
particular type of stock (e.g, a growth stock fund) or a geographical region
(e.g., a Pacific Rim fund.)
Mutual funds give the investor the advantage of diversification, since they can
invest in many more stocks than most investors could afford to buy
individually. The ultimate in diversification comes from an index fund, which
buys shares in all of the stocks of a particular index such as the S&P 500 with
purchases weighted so as to have the fund track the index.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M7-22
Module 7 - Asset Liability Management, Duration, and Immunization
CDs
CD is an abbreviation for certificate of deposit. CDs are offered by banks,
credit unions and savings and loan associations. When you buy a CD you are
making a deposit that will pay a stated rate of interest at a specified time. A
minimum deposit amount will be required. On the day that this paragraph was
written, my own bank was advertising two "special offers" - a 5.25% CD
maturing in 7 months and a 5.40% CD maturing in 23 months. The minimum
deposit was 10,000.
The CD has the advantage of coverage by Federal Deposit Insurance, which
protects deposits up to $100,000 in the event of a bank failure.
Money Market Funds
A money market fund is a mutual fund that invests in short term secure
investments like Treasury bills. It functions very much like a bank savings
account, paying interest and possibly allowing the investor to write checks. The
fund is managed in an attempt to pay a higher interest rate than a bank account.
Although the fund is managed to be secure, there are no government
guarantees backing it.
Mortgage-Backed Securities
Interest rates on mortgages are typically higher than the rates you can earn in
bank accounts or on Treasury bonds, making mortgages attractive for some
investors. Mortgage lenders gather large numbers of mortgages into pools of
loans and create mortgage-backed securities (MBS). For example, a lender who
has just made 100 loans of $200,000 each can combine these into a $20,000,000
MBS. An insurance company with a large amount of money to invest can buy
this security and receive mortgage rates of interest without having to manage
all of the individual loans.
A mortgage borrower may default. To give security to MBS investors,
mortgage loans can be insured by the Federal Housing Authority (FHA). In
addition, many MBS pools are put into GNMA securities, which are guaranteed
by the Government National Mortgage Association.
Analysis of MBS is a bit complex, since borrowers can prepay their mortgages
and this makes the timing of payments uncertain. Analysts use standard
prepayment models to value MBS, but the assumptions of the analysis may fail
to hold, leading to surprise losses or gains on the MBS.
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Module 7 - Asset Liability Management, Duration, and Immunization Page M7- 23
Section 7.9
Formula Sheet
Investment cash flows CFi,...,CFn
Investment price P = vCFi + v2CF2 +... + vnCFn
v{CFi vlCFi
Weights for Macaulay duration: wt = ■
Macaulay duration: D = (1) Wi +... + (n) wn =
P vCFi + v2CF2 +... + vnCFn
"(l)vCF1+(2)v2CF2+... + (n)vnCFn
Modified Duration: DM = -
dp;
di ) D
1 + i
(la)-.
Duration of level payment investment:
Macaulay duration of a coupon bond with face value F and coupon Fr for n
. , , , . , Fr(Ia)-, + nCvn Fr(Ia)-]^nCvn
periods and redemption value C: D = — , n[ = —-——
Fr(a^) + Cvn Bond Price
The duration of a zero coupon bond payable in n periods is n.
Approximations of change in price:
P'd)
AP = P (i + Ai) - P (i) * P' (i) Ai = —V^ (P(0 Ai) = -(DM)P(i)M.
AP * P' (i) Ai + P"^Al) = -DM (P (i) Ai) + (Convexity) PMAl)
Modified duration of a portfolio is the weighted average of the durations of the
investments with each investment Xk having weight equal to its percent of
total portfolio value:
( Xfc ^
DM = W1DM1 + w2DM2 +... + wmDMm where wk =
\X\ +X2 +... + Xm J
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Page M7-24
Module 7 - Asset Liability Management, Duration, and Immunization
Immunization
h(i) = PVA(i)-PVL(i).
For immunization, we need ft(i0) = 0, h'(i0) = 0 and ft"(i0)>0.
In terms of duration and convexity, we need
Present Value Matching: PVA (i0) = PVL (i0)
Duration Matching: —PVA (i)\
di v 7|
■aw'('«
Greater Convexity for Assets —TPVA (i)\
dr v 7
>£«*«
In terms of the asset and liability amounts at time t, At and Lt.
Present Value Matching: ^AtVio = ^LtvJo
Duration Matching: ^tAtv[0 =^]tLtv[0
Greater Convexity for Assets ^t2Atv\0 >^jt2Ltv\Q
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Module 7 - Asset Liability Management, Duration, and Immunization Page M7- 25
Section 7.10
Basic Review Problems
1. A company must pay liabilities of 3000 and 5000 at the end of years 2 and 4,
respectively. The only investments available to the company are the
following two zero-coupon bonds:
Maturity (years)
2
4
Effective Annual Yield
5.5%
6.8%
Par
1000
100
Find the cost of exactly matching those liabilities.
2. John must pay liabilities of 1,000 due 6 months from now and another 1,000
due one year from now. There are two available investments:
a. 6-month bond with face amount of 1,000, a 4% nominal annual
coupon rate convertible semiannually, and a 3% nominal annual
yield rate convertible semiannually
b. 1-year bond with face amount of 1,000, a 5% nominal annual
coupon rate convertible semiannually, and a 6% nominal annual
yield rate convertible semiannually.
Find the amount of each bond to purchase and the total cost of the bonds.
3. An investment pays 1000 in three years and 3000 at the end of the fourth
year. An investor has purchased it to yield the annual rate i = .075. Find the
Macaulay duration and the modified duration.
4. An annual corporate bond is priced to yield 7% annually and has a price of
940.29 and a Macaulay duration of D= 6.5317. Estimate the change in price if
rates increase by 0.10%.
5. An investor has a portfolio containing $1,000 worth of a three year bond with
a modified duration of 2.7, $4,000 worth of a five year bond with a modified
duration of 4.6, and $5,000 worth of a 6 year bond with a modified duration
of 5.50. Find the modified duration of the entire portfolio.
6. You have a single liability of 200,000 payable at time 7. The valuation
interest rate is i0 = .06. You wish to attempt to immunize this portfolio by
buying two zero coupon bonds with maturities at times 4 and 10. Find the
amounts of the two bonds, and verify that the portfolio is immunized.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M7-26
Module 7 - Asset Liability Management, Duration, and Immunization
Section 7.11
Basic Review Problem Solutions
1. For year 2 the company is investing in $3000 worth of 5.5% two-year zero
coupon bonds. The cost is
3000
1.0552
= 2695.36
For year 4 the company is investing in $5000 worth of 6.8% four-year zero
coupon bonds. The cost is
-^ = 3843.13
1.0684
The total invested to match liabilities is 2695.36 + 3843.13 = 6538.49.
2. We look at the longest term asset and liability first.
The total payments for the 12 month bond at month 12 consist of a coupon of
25 and the redemption value of 1000 for a total of 1025. To cover a liability
of 1000, the required amount of the 12-month bond required is = .9756.
Once you have purchased .9756 of the 12 month bond, it will provide a
coupon payment of (.9756)25 = 24.39 at month 6. To fund a total liability of
1000 at month 6, the additional amount needed from the 6-month bond is
1000-24.39 = 975.61
The total payments for the 6 month bond at month 6 consist of a coupon of
20 and the redemption value of 1000 for a total of 1020. To cover a liability
of 975.61, the amount of the 6-month bond required is :— = .9565.
1020
John must purchase .9756 of the 6-month bond and .9565 of the 12-month
bond. We can find the prices of the separate bonds using the financial
calculator.
6-month. N=l, PMT = 20, FV=1000, yield i = 1.5% per semiannual period.
This gives a price of PV = -1004.93
12-month. N=2, PMT = 25, FV=1000, yield i = 3% per semiannual period. This
gives a price of PV = 990.43.
The total cost of purchasing the required bonds is
.9756(990.43)+.9565(1004.93)= 1927.48
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Module 7 - Asset Liability Management, Duration, and Immunization
Page M7- 27
3. The present value is
1.0753 1.0754
The weights for the duration are
804.96 _„_ 2,246.40 _.„
w3 = = .2638 w4 = — = .7362
3,051.36 3,051.36
The Macaulay duration is the weighted average time
D = .2638(3) + .7362(4) = 3.7362
The modified duration is Modified Duration: DM = = —'■ = 3.4755.
1 + i 1.075
4 ap- (P)P(OAi 6.5317(940.29)(.001) ^
1 + i 1.07
5. The modified duration of the entire portfolio is
.10(2.7) + .40(4.6) + .50(5.50) = 4.86
6. We know that L6 = 120,000 and we need to find A4 and A10.
r, mm n/r * u- 200,000 At Aio
Present Value Matchmg: '-^— = T + tt-
e 1.067 1.064 1.0610
t^ .- „ * u- (7)200,000 4A, 10A
Duration Matching: ±-£ =— = r +
1.067 1.064 1.06
The solution to this system is A, = 83,961.93, A10 = 119,101.60.
Y^fLtvi = 6,517,559.71<7,714,662.52 = £t2A,vf0, so the portfolio is
immunized.
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Page M7-28
Module 7 - Asset Liability Management, Duration, and Immunization
Section 7.12
Sample Exam Problems
1. (Fall 05 Sample Problems #35)
The current price of an annual coupon bond is 100. The derivative of the
price of the bond with respect to the yield to maturity is -700. The yield to
maturity is an annual effective rate of 8%.
Calculate the duration of the bond.
(A) 7.00 (B) 7.49 (C) 7.56 (D) 7.69 (E) 8.00
2. (Fall 05 Sample Problems #36)
Calculate the duration of a common stock that pays dividends at the end of
each year into perpetuity. Assume that the dividend is constant, and that the
effective rate of interest is 10%.
(A) 7 (B) 9 (C) 11 (D) 19 (E) 27
3. (Fall 05 Sample Problems #37)
Calculate the duration of a common stock that pays dividends at the end of
each year into perpetuity. Assume that the dividend increases by 2% each
year and that the effective rate of interest is 5%.
(A) 27 (B) 35 (C) 44 (D) 52 (E) 58
The following information applies to questions 4 thru 6.
Joe must pay liabilities of 1,000 due 6 months from now and another 1,000 due
one year from now. There are two available investments:
1) a 6-month bond with face amount of 1,000, a 8% nominal annual coupon
rate convertible semiannually and a 6% nominal annual yield rate
convertible semiannually, and
2) a one-year bond with face amount of 1,000, a 5% nominal annual coupon
rate convertible semiannually, and a 7% nominal annual yield rate
convertible semiannually
4. (Fall 05 Sample Problems #51)
How much of each bond should Joe purchase in order to exactly (absolutely)
match the liabilities?
Bond I Bond II
(A) 1 .97561
(B) .93809 1
(C) .97561 .94293
(D) .93809 .97561
(E) .98345 .97561
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 7 - Asset Liability Management, Duration, and Immunization
Page M7- 29
5. (Fall OS Sample Problems #52)
What is Joe's total cost of purchasing the bonds required to exactly
(absolutely) match the liabilities?
(A) 1894 (B) 1904 (C) 1914 (D) 1924 (E) 1934
6. (Fall OS Sample Problems #53)
What is the annual effective yield rate for investment in the bonds required
to exactly (absolutely) match the liabilities?
(A) 6.5% (B) 6.6% (C) 6.7% (D) 6.8% (E) 6.9%
7. (May 05 #3)
A bond will pay a coupon of 100 at the end of each of the next three years
and will pay the face value of 1000 at the end of the three-year period. The
bond's duration (Macaulay duration) when valued using an annual effective
interest rate of 20% is X. Calculate X.
(A) 2.61 (B) 2.70 C) 2.77 (D) 2.89 (E) 3.00
8. (May 05 #6)
John purchased three bonds to form a portfolio as follows:
Bond A has semi-annual coupons at 4%, a duration of 21.46 years, and was
purchased for 980.
Bond B is a 15-year bond with a duration of 12.35 years and was purchased
for 1015.
Bond C has a duration of 16.67 years and was purchased for 1000.
Calculate the duration of the portfolio at the time of purchase.
(A) 16.62 years (B) 16.67 years (C) 16.72 years
(D) 16.77 years (E) 16.82 years
9. (May 05 #15)
An insurance company accepts an obligation to pay 10,000 at the end of each
year for 2 years. The insurance company purchases a combination of the
following two bonds at a total cost of X in order to exactly match its
obligation:
1-year 4% annual coupon bond with a yield rate of 5%
2-year 6% annual coupon bond with a yield rate of 5%.
Calculate X.
(A) 18,564 (B) 18,574 (C) 18,584 D) 18,594 (E) 18,604
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M7-30
Module 7 - Asset Liability Management, Duration, and Immunization
10. (Nov OS #2)
Calculate the Macaulay duration of an eight-year 100 par value bond with
10% annual coupons and an effective rate of interest equal to 8%.
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8
11. (Nov OS #10)
A company must pay liabilities of 1000 and 2000 at the end of years 1 and 2,
respectively. The only investments available to the company are the
following two zero-coupon bonds:
Maturity (years)
1
2
Effective annual yield
10%
12%
Par
1000
1000
Determine the cost to the company today to match its liabilities exactly.
(A) 2007 (B) 2259 (C) 2503 (D) 2756 (E) 3001
12. (Nov OS #21)
Which of the following statements about immunization strategies are true?
I. To achieve immunization, the convexity of the assets must equal the
convexity of the liabilities.
II. The full immunization technique is designed to work for any change in
the interest rate.
III. The theory of immunization was developed to protect against adverse
effects created by changes in interest rates.
(A) None (B) I and II only
(C) I and III only (D) II and III only
(E) The correct answer is not given by (A), (B), (C), and (D).
13. (May 2005 #23)
The stock of Company X sells for 75 per share assuming an annual effective
interest rate of i. Annual dividends will be paid at the end of each year
forever. The first dividend is 6, with each subsequent dividend 3% greater
than the previous year's dividend. Calculate i.
(A) 8% (B) 9% (C) 10% (D) 11% (E) 12%
14. (November 2005 #20)
The dividends of a common stock are expected to be 1 at the end of each of
the next 5 years and 2 for each of the following 5 years. The dividends are
expected to grow at a fixed rate of 2% per year thereafter. Assume an
annual effective interest rate of 6%.Calculate the price of this stock using
the dividend discount model.
(A) 29 (B) 33 (C) 37 (D) 39 (E) 41
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 7 - Asset Liability Management, Duration, and Immunization Page M7- 31
Section 7.13
Sample Exam Solutions
1.
In this problem we will use the relationship between duration D and the
modified duration DM.
P(i) 1 + i
We are given P'(i) = -700, P(i) = 100 and i = .08. Thus we have
-P'(i) 700 _ D
P(i) " 100 " 1.08
D = 7.56.
Answer C
2.
The series of dividends is a level perpetuity of the dividend D. Thus the price of
the stock at the rate i is
Thus
From this we can get the modified duration and the duration
-P' 1 (l + i\
MD = — = ± and D = (l + i)DM = ± '-
Pi y ' i
When i = .10
D.H.u
.1
Answer C
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M7-32
Module 7 - Asset Liability Management, Duration, and Immunization
3.
In this problem we will also use the relationship between duration D and
modified duration DM.
P(i) 1 + i
First we denote the beginning dividend by Div and note that the price of the
stock at the rate i is and growth rate r is given by the constant growth model
w (i-.02)
Next we take the derivative of the expression above with respect to i and
obtain
W (i-.02)2
It follows that
-P'(i)_ 1 _ D
DM =
P(i) i-.02 1 + i
D = 35
For i = .05 this tells us that
1 1 _ D
i-.02".03 "1.05
Answer B
4.
In 12 months, the 6-month bond will be gone and only the 12-month bond will be
available to pay. The total payments for the 12 month bond at month 12 consist
of a coupon of 25 and the redemption value of 1000 for a total of 1025. To cover
a liability of 1000, the required amount of the 12-month bond required is
™° =.97561.
1025
Once you have purchased .97561 of the 12 month bond, it will provide a coupon
payment of (.97561)25 = 24.39 at month 6. To fund a total liability of 1000 at
month 6, the additional amount needed from the 6-month bond is
1000-24.39 = 975.61.
The total payments for the 6 month bond at month 6 consist of a coupon of 40
and the redemption value of 1000 for a total of 1040. To cover a liability of
975.61, the required amount of the 6-month bond required is
975.61
.93809.
1040
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 7 - Asset Liability Management, Duration, and Immunization
Page M7- 33
5.
We have seen in the last problem that Joe must purchase .93809 of the 6-
month bond and .97561 of the 12-month bond. We can find the prices of the
separate bonds using the financial calculator.
6-month. N=l, PMT = 40, FV=1000, yield i = 3% per semiannual period.
This gives a price of PV = 1009.71.
12-month. N=2, PMT = 25, FV=1000, yield i = 3.5% per semiannual period.
This gives a price of PV = 981.00.
The total cost of purchasing the required bonds is
981 (.97561) +1009.71 (.93809) = 1904.27
Answer B
6.
Joe will pay PV = -1904.27 in order to receive n = 2 payments of PMT= 1000. The
financial calculator shows that the yield per semiannual period for this
sequence is i = 3.333%. The effective annual yield is 1.03332 -1 = .0677
Answer D
7.
Fr (la)-, + nCvn Frila)-, + nCvn
We use D = —V^ = —-—-r1 • The price of the bond can be
Fr (a^) + Cvn Bond Price
obtained using the BA II Plus with N=3, PMT=100, FV=1000 and I/Y=20. It is
P = 789.35. The coupon is Fr = 100 and C = 1000. For an interest rate of 20%,
,r x .^oo ™ ~ 100(3.9583) + 3(1000)/1.23 „
(la)-, = 3.9583. Thus D = * '- * '- = 2.70
v ^ 789.35
Answer B
Since there are only three cashflows, this problem could also have been worked
directly from the definition.
8.
The duration of the portfolio is the weighted average of the individual
durations. The total purchase price of the portfolio is
980 + 1015 + 1000 = 2,995
Thus the duration is
\2,99S) {2,995) 1,2,995 J
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M7-34
Module 7 - Asset Liability Management, Duration, and Immunization
9.
Let x and y represent the required amounts of the one and two year bonds
respectively. The total amount paid by the bonds should be 10,000 at times 1
and 2. The amounts paid can be given in terms of x and y.
Time 2 1.06y = 10,000
Timel 1.04* + .06y = 10,000
This system of equations solves for x = 9071.12 and y = 9433.96 . To solve, we
must assume that the face value of the bonds is the redemption value. With this
assumption we can analyze each of the bonds using the BA II Plus.
One year bond
Coupon = 9071.12(.04) = 362.84 = PMT, FV = 9071.12, N=l, I/Y = yield = 5.
This gives a price of PV = -8984.73.
Two year bond
Coupon = 9433.96 (.06) = 566.04 = PMT, FV = 9433.96, N=2, I/Y = yield = 5.
This gives a price of PV = -9609.38
The total cost X is 8984.73 + 9609.38 = 18,594.11
Answer D
10.
The duration of a bond with face value F and coupon Fr for n periods and
Fr(Ia)n+nCvn
redemption value C is — , n! ,
y Fr(an) + Cvn '
where the denominator is the price of the bond. In this case the coupon Fr = 10,
F=C=100 and n=8.
The price of the bond can be obtained from the financial calculator with PMT =
10, n = 8,1/Y=8 and FV=100. The price (PV) is 111.49.
mL u J . . 10 (la)-, +8 (100) v8 10 (23.553)+ 8 (100) (.5403) _„
Thus the duration is —^—^ - '-— = —* '- * ^ 1 = 5.99
111.49 111.49
Answer C
11.
Since there are separate zero coupon bonds for each year, the company can
invest an amount in each bond to cover the liability for that year only.
For year 1 the company can invest the present value of 1000 in one year at 10%
^°°= 909.09
1.1
For year 2 the company can invest the present value of 2000 in two years at
12% per year.
^1594.39
1.122
The total invested to match liabilities is 909.09 + 1594.39 = 2503.48.
Answer C
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 7 - Asset Liability Management, Duration, and Immunization Page M7- 35
12.
Statement I is false. The assets must have greater convexity for immunization.
Statement II is true.
Statement III is true.
Answer D
13.
This can be done directly with the constant growth model
Using P = 75, D = 6 and g = .03, we have
75 = ,. 6 . i = .ll.
(i-.03)
Answer D
14.
This stock will pay dividends in three different phases.
Phase 1. A level annuity of 1 for 5 years.
Phase 2. After a deferral period of 5 years, a level annuity of 2 for 5 years.
Phase 3. After a deferral period of 10 years, a constant growth rate
perpetuity for which we can use the stock pricing model P = .
i-r
Care must be taken with the final piece. It is clear that the growth rate is 2%.
The value of D to use is 2(1.02) = 2.04, since the first expected dividend at the
end of year 11 has already experienced one year of growth. (A common mistake
is to inadvertently use D=2).
Thus the price is
laa+v^aa+v10!" 2'04
51 5I L06-.02
. 0i0 2(4.212) 51
= 4.212 + —^ =-^ +
1.065 1.0610
38.985
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M7-36 Module 7 - Asset Liability Management, Duration, and Immunization
Section 7.14
Supplemental Exercises
1. A company has liabilities of 3000 and 6000 due at the end of years one and three
respectively. The investments available to the company are the two zero-
coupon bonds:
Maturity Effective
(years) Annual Rate Par
1
3
4.8%
5.6%
1000
1000
Find the cost of exactly matching these liabilities.
2. A company has liabilities of 1000 and 2000 due at the end of years two and three
respectively. It can purchase two zero-coupon bonds to match these liabilities.
The first has a par value of 1000 and matures in two years. The second has a par
value of 1000 and matures in three years with an effective annual rate of 6%. If
the cost of matching these liabilities is 2,586.27, what is the effective annual
yield on the first bond?
3. An investment pays 1000 at the end of year 2, 2000 at the end of year 3 and 4000
at the end of year 4. It was purchased to yield an annual rate of 6.5%. Find the
Macaulay duration for this investment.
4. An investor has a portfolio containing 2000 worth of a three-year bond with a
modified duration of 2.85, 5000 worth of a six-year bond with a modified
duration of 5.24 and 8000 worth of a ten-year bond with a modified duration of
9.13. Find the modified duration of the entire portfolio.
5. An annual corporate bond is priced to yield 7.5% annually and has a price of
972.18. Its Macaulay duration is 5.8215. Estimate the change in price if rates
decrease by 0.10%.
6. An annual par bond has a face value of 1000, a coupon rate of 4.5% and matures
in 3 years. Find the Macaulay duration of this bond at a rate of i = 0.04.
Problems 7 and 8 use the following:
A company has liabilities of 2000 due in 6 months and another 2000 due in one year.
It has two available investments:
1) A 6-month bond with a face value of 1000, a 4% nominal annual coupon
convertible semiannually and a 5% nominal annual yield rate convertible
semiannually,
2) A 1-year bond with a face value of 1000, a 7% nominal annual coupon
convertible semiannually and a 6% nominal annual yield rate convertible
semiannually.
7. How much of each bond must the company purchase to exactly match its
liabilities?
8. What is the total cost of these bonds?
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 7 - Asset Liability Management, Duration, and Immunization Page M7- 37
Section 7.15
Supplemental Exercise Solutions
1. For year 1 the company buys 3000 worth of 4.8% zero coupon bonds. The cost is
3000/1.048 = 2826.60
For year 3 the company buys 6000 worth of 5.6% zero coupon bonds. The cost is
6000/1.0563 = 5095.18
The total invested to match liabilities is
2862.60 + 5095.18 = 7957.78
2. Let i be the effective annual rate of the first bond. The cost of matching
liabilities by purchasing 1000 worth of bond 1 and 2000 worth of bond 2 is
1000/(1 + i)2 + 2000/1.063 = 2586.27
(1 + i)2 = 1-1025 => i = .05
3. The present value of the payments is
P = 1000/1.0652 + 2000/1.0653 + 4000/1.0654
= 881.66 + 1655.70 + 3109.29
= 5646.65
The weights for the duration are
Wi = 881.66/5646.65 = 0.1561, w2 = 1655.70/5646.65 = 0.2932 and
w3 = 3109.29/5646.65 = 0.5507.
D = 0.1561(2) + 0.2932(3) + 0.5507(4) = 3.3946
4. The price of the entire portfolio is 15,000. The weights for the
modified durations are Wi = 2/15, w2 = 1/3 and w3 = 8/15.
The modified duration for the entire portfolio is
DM = (2/15X2.85) + (l/3)(5.24) + (8/15)(9.13) = 6.996
5. AP= -(D)P(i)M/(l + i) = -(5.8215)(972.18)(-0.001)/1.075
= 5.2647
6. The Duration of the bond is
D = [Fr (Ia)^ + nCv"]/(Bond Price)
F = C = 1000, r = 0.045, n = 3 and i = 0.04.
v3 = 0.8890, as = 2.8861 (calculator in BGN mode), (ia)^ = 5.4775
Bond Price = 1013.88.
(N = 3,1/Y = 4, PMT = 45, FV = 1000. Then CPT PV = - 1013.88)
D = [45(5.4775) + 3000(0.889)]/ 1013.88 = 2.8736
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M7-38
Module 7 - Asset Liability Management, Duration, and Immunization
7. The total of the 1-year bond at redemption is 1035. The liability is 2000, so the
company needs 2000/1035 = 1.9324 of this bond.
The six-month coupon on this amount of the bond is 35(1.9324) = 67.73.
The amount needed to cover the liability due at 6 months is 2000-67.73 = 1932.37.
The total of the 6-month bond at redemption is 1020.
The company needs 1932.37/1020 = 1.8945 of this bond to cover the liability at 6
months.
8. The price of the 6-month bond is 995.12.
(N = 1, PMT = 20,1/Y = 2.5, FV = 1000. CPT PV = -995.12)
The price of the one-year bond is 1009.57
(N = 2, PMT = 35,1/Y = 3, FV = 1000. CPT PV = -1009.57)
Cost of bonds is 1.9324(1009.57) + 1.8945(995.12) = 3836.15
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 8 - Review of Derivatives Markets, Chapter 1 Page M8- 1
Section 8.1
Overview
Chapter 1 of Derivatives Markets has very little quantitative analysis. It
introduces terminology, discusses why people use derivatives, and how
derivatives are traded. In this lecture note, we have not discussed every detail
of the chapter as it can be read fairly easily. It would be a mistake to skip
Chapter 1, though. Despite the lack of math, it could be a source of true-false
problems on terminology.
We will restate some of the text discussion in our own words here to give you a
chance to rethink the text material. Be sure to read the book's wording
carefully, since that is the wording that is more likely to appear on the exam.
Section 8.2
What is a derivative security?
It may be helpful to have a preliminary discussion of the most basic ideas of
derivative securities before beginning study. A derivative is defined to be a
"financial instrument that has value determined by the price of something
else." This basic idea is easily explained with an example of a farmer planning
to sell his corn crop.
Suppose that you are a farmer with corn that is not yet ready to harvest, but
you would like the security of knowing that you will get a definite price for it
when it is harvested in a month. In the town near you, there is a cereal company
that wants to assure a supply of corn at a definite price in a month. The two of
you could agree on a contract that specifies that you will deliver corn to the
cereal company on an agreed upon date in a month for $2.44 per bushel. These
types of contracts are called forward contracts, and are really used by farmers
in this way.
How does the term derivative come to be applied to instruments like forwards?
The answer is that their value is derived from the value of something else, like
corn. If you are the farmer with the forward contract to sell corn for $2.44, the
contract is valuable to you in a month if the price of corn has dropped below
$2.44.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M8-2
Module 8 - Review of Derivatives Markets, Chapter 1
Section 8.3
Uses of derivatives
When a farmer uses a derivative to protect the price of his corn, he is engaging
in risk management. (He is also said to be hedging.) There are other reasons
that people use derivatives. Suppose that you think that the price of corn will be
$3.10 per bushel in a month but someone else is willing to make a forward
contract to sell it for $2.44 in a month. Then you could enter into the forward
agreement, get the corn for $2.44 in a month and immediately resell it for a
higher price. This is a bet. If you are wrong and the price in a month is low, you
can lose money. You are using the forward contract for speculation on the price
of corn.
There are other more complex reasons to use derivatives, and these will only
become clear as you look at the ways derivative securities can be structured in
later chapters. The simplest thing to say at this point is that derivatives can be
structured to avoid taxes or transaction costs.
Derivatives are so useful that they are used for risk management by banks,
insurance companies and other corporations. There are institutions such as the
Chicago Board of Trade which enable standardized trading of derivatives.
Derivatives should be of interest to actuaries, since they are a form of
insurance. The farmer with a forward contract in corn is insuring himself
against price drops.
There are a few points in Chapter 1 that might require some clarification: bid-
ask spreads and short sales. We discuss these in the following sections.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 8 - Review of Derivatives Markets, Chapter 1 Page M8- 3
Section 8.4
Bid-ask spreads
The text uses the example of buying a stock to illustrate that this is more
complicated than you might think. To begin, there is always a commission cost
added to the price. In addition, the stock really has two prices. Market makers
are the individuals who get stock from sellers and provide it to buyers. They
make a living by buying for a lower price called the bid price and selling for a
higher price called the ask price or offer price. The difference between the two
is called the bid-ask spread. The text gives the example of a stock which you
can buy (ask price) for $50 and sell (bid price) for $49.75. Note that the terms
bid and ask mean different things to the market maker and the individual
buying or selling. The table below summarizes this.
Price
Bid
Ask
Magnitude
Lower
Higher
For market maker
Buy price
Sell Price
For investor
Sell Price
Buy Price
Example 1.1 illustrates a simple cost calculation, and you should read it.
Section 8.5
Short sales
Suppose a stock can now be purchased for $45 and you believe that it will drop
to $40 in a month. You can bet on this by borrowing a share of the stock now
and selling it for $45. If you are right, in a month you can buy a share of the
stock for $40, giving it back to the lender and pocketing a profit of $5. There
are some possible complications. If the stock pays a dividend before you
replace it, you have to pay the missing dividend to the stock owner. And of
course, if the stock goes up to $50 you still have to buy it back, replace it and
lose money.
This is referred to as selling the stock short. (In contrast, if you buy the stock
today and hold onto it you have a long position in the stock.)
The pattern in short selling is:
Today: Borrow stock —> Sell stock for cash —> Hold cash
Future: Buy stock -> replace borrowed stock and any dividend —> Profit or loss?
Students who have studied for exam FM/2 in 2006 may have found the initial
description of a short sale confusing (short sales were covered in the 2006
syllabus). For example, short sales were covered in section 8.2.2 of
Mathematics of Investment and Credit. That section was included in the 2006
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M8-4
Module 8 - Review of Derivatives Markets, Chapter 1
exam syllabus, but was not included in the 2007 syllabus. The confusing part is
the fact that section 8.2.2 included return calculations that assumed that the
short seller had to leave a deposit in a margin account and earned interest on
that account, but the text Derivatives Markets does not specifically mention the
margin account in the initial description of a short sale.
This issue is handled implicitly in the following section on risk and scarcity in
short selling:
Credit risk
The lender may be concerned that you will not be able to pay to replace the
borrowed asset when the time comes to do so. To feel more comfortable, the
lender could insist on holding all or part of the money from the short sale. That
is essentially the margin account of the old exam material. If the lender
perceives a serious risk that the asset will go up in value and cause a loss to the
short seller, he might require the short seller to put up additional cash called a
haircut.
Scarcity risk
If the lender holds your money in a margin account, he will pay you a rate of
interest on it. This section makes the point that if the asset borrowed is scarce
and hard to replace, the lender might pay you little or no interest. The interest
rate that is paid on collateral is called the repo rate in bond markets and the
short rebate in the stock market.
We recommend that students focus on short sales as described in Derivatives
Markets, since Section 8.2.2 of Mathematics of Investment and Credit is no
longer in the syllabus.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 8 - Review of Derivatives Markets, Chapter 1 Page M8- 5
Section 8,6
Solutions to odd-numbered problems
1.1.
a) Soft drink manufacturers sell less on abnormally cold days, and would
like a futures contract that provides extra cash on such days.
b) Ski resort operators lose business on unseasonably hot days, and would
like a future contract that paid extra cash on such days.
c) Electric utilities have peak demand for air conditioning on abnormally
hot days and for electric heating on abnormally cold days. They would
buy contracts to protect against both abnormal heat and cold.
d) Amusement park operators have reduced sales demand for on
abnormally hot days and abnormally cold days. They would buy
contracts to protect against both abnormal heat and cold.
a) 41.05(100) + 20 = 4,125
b) 40.95(100)-20 = 4,075
c) Cost = 4,075-4,125 = 50
1.5.
The market maker will buy at the lower price of 100 and sell at the higher
price of 100.12, thus making a spread of 0.12 per share. When 100 shares are
traded he will make $12.
1.6
This problem is in the solutions manual, but in the solution the number 29.87
is changed to 29.875. Using the numbers given in the problem the answer is
slightly different:
Cash from short sale: 300(30.19)(l-.005) = 9,011.715
Cost to return stock: 300(29.87)(l+.005) = 9,005.805
Profit = 9,011.715 - 9,005.805 = 5.91
1.7
a) Cash from sale of borrowed shares: 25.12(400) = 10,048
Cost to replace stock: 23.06(400) = 9,224
Profit = 10,048 - 9,224 = 824
With commission:
b) Cash from sale of borrowed shares: 10,048(1 -.003) = 10,017.856
Cost to replace stock: 9,224(l+.003) = 9,251.67
Profit = 10,017.86 - 9,251.67 = 766.19
c) 10,017.86(.03) = 300.54
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M8-6
Module 8 - Review of Derivatives Markets, Chapter 1
1.9.
You must pay the dividend of 3 to the registered owner of the stock, and this
will be tax deductible for you. Since your planning was based on a dividend
cost of 3, an increase to 5 will increase your anticipated dividend
replacement expense -but that expense is tax deductible. If nothing has
changed in the company except for dividend policy, the stock price will fall
by the amount of the dividend on the ex-dividend date thus increasing the
value of your short position by an amount that will replace the unanticipated
dividend expense. In this case you would not care about the increased
dividend. However if the increased dividend is a result of a major positive
change in the company, the stock value is going to increase and this.will
damage your short position.
1.11.
You have a short position in cash.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 9- Review of Derivatives Markets, Chapter 2
Page M9- 1
Review of Derivatives Markets}—-—■
Chapter 2
Section 9.1
Forward Contracts
A forward contract is a contract to buy or sell a specified asset at a designated
future time. The contract binds both buyer and seller and obligates the buyer to
purchase the asset even if the future value is less than the market price
The forward contract should specify:
• Underlying asset The type and quantity of the asset to deliver.
• Expiration date. Time, place and date of delivery.
• The sale price.
Futures contracts are a type of forward contracts that will be studied further
when we reach chapter 5 of Derivatives Markets.
The current price of an asset at any time is called its spot price. The value of a
forward contract at a future time will depend on the relation of the forward
price to the spot price.
The text gives additional simplified examples of forward contracts in Chapters
2 and 3. A fictional stock index is used in the examples.
A stock index is an average of the prices of stocks in a specified group. Widely
used real-life indices include the Dow-Jones Industrial Average, the S&P 500
index, the NASDQ index and the Russell 2000 index. The fictional index used in
the text is called the S&R 500 index. It is simpler to study by design because it
does not pay any dividends.
You might question how a person can own an "average". In practice there are
funds called index funds that invest in a representative sample of an index. You
can buy a share in an S&P 500 index fund and own the average in some sense.
In the examples in Chapter 2, the text does not worry about the practical issues
of ownership or delivery. It uses the convention that if you have a forward
commitment to buy the average for 1020 and the average is worth a spot price
of 1040 on the expiration day, you have made a profit of 20. In practice, when
you buy or sell a stock there are commission expenses that would reduce your
20 profit. Thus the situation is a bit complicated if there is actually to be a
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M9-2
Module 9- Review of Derivatives Markets, Chapter 2
delivery of a stock index share. However, forward contracts can be written for
cash settlement in which a net payment is made for the difference between the
spot price at expiration and the forward price, so that no stock actually changes
hands. In this case, you truly can say that you made a profit of 1040 -1020 = 20.
Cash settlement is available through marketplaces called futures exchanges.
A forward contract must have a buyer and a seller. The buyer is said to have a
long forward and the seller is said to have a short forward. Note that the buyer
and seller have credit risk, since neither one can be sure that the opposite party
will have the required cash or stock at the time of expiration. Futures
exchanges attempt to lower credit risk by requiring collateral from the parties
to the contract.
The text looks at forward contracts on the fictional S&R index beginning on
page 23. The continuing example is one in which the S&R spot price today is
1000 and the six month forward price is 1020. In this chapter, the forward price
of 1020 is not derived -it is simply given to you.
There is still something important to observe about this price. The text makes
the statement (page 27) that "the 6-month interest rate is 2%". This rate of 2%
is really the rate at which the investors involved could buy or sell a 6-month
zero coupon bond -the working 6-month interest rate for the buyer or seller.
That is, you could go to a bank and borrow or deposit and earn at this rate. (In
practice, banks lend to you at a higher rate than they will pay to you, but this
analysis uses the simplifying assumption that they are the same.)
Note that
(9.1) 1 Forward Price = 1000(1.02) = Spot Price (1+interest rate)
In Chapter 5 of the text, it is shown that this is how the forward price should be
determined for a stock that does not pay dividends. Thus the forward price of
1020 is not a rabbit pulled out of a hat. It is what the forward price should be for
this fictional index. It also makes some identities work nicely later.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 9- Review of Derivatives Marketsf Chapter 2
PageM9- 3
Section 9.2
Payoff and profit for forwards
On page 23 there is a definition of the payoff on a contract -it is the value of the
financial result at expiration. For a forward contract:
(9.2) 1 Payoff to long forward = Spot price at expiration - forward price
(9.3) I Payoff to short forward = Forward price -spot price at expiration
Thus if the forward price is 1020 and the spot price at expiration is 1040,
Payoff to long forward = 1040 - 1020 = 20
Payoff to short forward = 1020 -1040 = -20.
It is easy to see that
(9.4)
Payoff to long forward = - Payoff to short forward
In words, the short has the opposite position from the long. The text gives both
a table and a graph of the long and short forward. We will do the same thing
here, with a slightly different scaling so that it is clear that 1020 is the spot
price that gives 0 payoff to both the long and the short.
S&R Index
960
980
1000
1020
1040
1060
1080
1100
Forward Payoff
Long
-60
-40
-20
0
20
40
60
80
Short
60
40
20
0
-20
-40
-60
-80
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page M9-4
Module 9- Review of Derivatives Markets, Chapter 2
The table and graph here were created as a spreadsheet and graph in EXCEL.
Even though they are quite simple, we recommend making your own
spreadsheet copy of each of the text examples. You will find that you can often
re-use the spreadsheets for end-of-chapter problems. Of course EXCEL will not
be available on Exam FM/2, but we find that it is still a helpful study aid.
The text does not look at payoff alone. It distinguishes payoff from profit,
which is payoff less the future value of any expenses incurred in setting up the
financial structure involved.
(9.5)
Profit = Payoff - Future value of expenses incurred
For a long forward, the profit is the same as the payoff, because the cost of the
contract is 0. (You will see an example where expenses are not 0 on the next
page) The forward situation is illustrated in the table below.
S&R
Index
960
980
1000
1020
1040
1060
1080
1100
Long
Forward
Payoff
-60
-40
-20
0
20
40
60
80
FV of Long
Forward
Cost
0
0
0
0
0
0
0
0
Long
Forward
Profit
-60
-40
-20
0
20
40
60
80
The text points out that if you buy the asset (an S&R share) instead of entering
a forward contract to get it, the payoff and the profit are not the same. In our
continuing example, the cost to buy the S&R index at time 0 was 1000. Suppose
the index is valued at 1040 in 6 months. Then the payoff in 6 months would be
the value of the asset at expiration.
Payoff on S&R share = 1040
However the asset at time 0 had a cost of 1000. The profit analysis in the text
assumes that you will borrow the funds needed to buy the asset. If you borrow
the cost of 1000, you would have to repay the loan with 2% interest in 6 months.
Thus the future value of expenses would be 1000(1.02) = 1020 and
Profit on S&R share = payoff - future value of expenses =1040 - 1020 =20.
Note that this is the same as the profit on the long forward when the value of
the asset is 1040. Since the forward price is the same as the required loan
payment, the profit on the forward is the same as the profit on the asset.
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 9- Review of Derivatives Markets, Chapter 2
PageM9- 5
In general
(9.6)
Profit on S&R share
= value of index - future value of loan
= value of index - 1020
= value of index - forward price
= profit on forward
We see this illustrated concretely in the table below:
S&R
Index
Index
Share
Payoff
Less
Future
Value of
Cost
Index
Share
Profit
960
980
1000
1020
1040
1060
1080
1100
960
980
1000
1020
1040
1060
1080
1100
-1020
-1020
-1020
-1020
-1020
-1020
-1020
-1020
-60
-40
-20
0
20
40
60
80
The payoff vs. profit distinction is important for problems in Chapters 2 and 3
of Derivatives Markets, and thus it is fair game for exam problems.
The author of the text notes that profit is really the more important concept on
page 28, where he states: "Because this calculation accounts for differing initial
investments in a simple fashion, we will primarily use profit rather than payoff
diagrams throughout the book." Thus, for exam study, you may be asked to find
a payoff, but most analysis will deal with profit.
It is important to remember that buying the asset and entering a
long forward contract have the same profit function.
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page M9-6
Module 9- Review of Derivatives Markets, Chapter 2
Section 9.3
A more mathematical approach
You must have observed by now that Derivative Mathematics is not written for
mathematicians. It is designed for MBAs and other business majors. In the
early chapters you will generally see analysis using spreadsheet tables and
word equations, similar to what we have done so far in this module. At this
point we will introduce a more mathematical approach to the material above,
which many actuarial students will find easier to work. The notation we use
here is eventually introduced in Derivatives Markets in Chapter 5.
Given:
T = The time of expiration of an S&R index forward contract written
at time t = 0.
So = Asset prices at the contract origination
ST = Asset prices at the contract expiration
F0,r = Forward price at time t =0 for a forward contract expiring at
timeT
r = The continuously compounded interest rate1 for the forward
buyer and seller.
Since the index pays no dividends, we can use the result to say that
(9.7)
Fo,T = Sq6
This is also the future value of the expense of borrowing S0 to buy an index
share. The word equation (2.1) from the text then becomes
(9.8)
Profit on S&R share = ST - S0en = ST - F0>T = Profit on forward
In our lectures on Chapters 1-4 we will use this notation to write some of the
verbal arguments of the text more compactly.
1 Your interest theory comes in handy here. Derivatives Markets uses per period rates like 2% for
6 months in Chapters 1-4 and then switches to a continuously compounded rate in Chapter 5. In
any case, you get to the same place, since 1000(1.02) = 1000 eh(102).
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 9- Review of Derivatives Markets, Chapter 2 Page M9- 7
Section 9.4
Zero coupon bonds have 0 profit
Suppose that you buy a zero coupon bond that costs 1000 and pays 2% interest
with payoff in 6 months of 1020. Since the expense of buying the bond was 1000,
we have
Bond Profit = Payoff - Future value of expenses incurred = 1020 -1020 = 0.
In general, if r is the continuously compounded interest rate for a zero coupon
bond that costs a price of P at time 0 and matures at time T, then
Bond Profit = Payoff - Future value of expenses incurred = Pe77 - PeTT = 0 .
Note that this does not mean that you did not earn anything on your investment
in the bond. You earned $20 in interest (and the IRS will make you pay taxes on
that). However, what you earned is what anyone can earn by depositing the
money in a bank account at the rate 2%, and as such does not satisfy the
textbook's definition of profit.
This means that if you have a long forward contract for the S&R index and a
zero coupon bond today at time 0, your profit at time T for the combined
position will be the same as the profit from holding only the forward. The bond
does not contribute any profit.
Profit(Bond+Forward)=Profit(Bond)+Profit(Forward)=0+Profit(Forward)
Page 29 of Derivatives Markets has additional discussion with graphs and word
equations.
It is important to remember that holding a forward contract and a bond today at
time 0 does not generate the same payoff as the forward contract alone. In the
table below we show the payoff and profit for the combined position of a zero
coupon bond and a long forward in our continuing example. Here, the S&R
index is currently at 1000, and you pay 1000 for a zero coupon bond paying 2%
interest in six months and simultaneously enter into a long forward contract for
six months at a forward price of 1020.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M9-8
Module 9- Review of Derivatives Markets, Chapter 2
Less Future Net Profit
S&R Long Forward Bond Payoff for Value of for
Index Payoff & Profit Payoff Forward + Bond Cost Combined
960
980
1000
1020
1040
1060
1080
1100
-60
-40
-20
0
20
40
60
80
1020
1020
1020
1020
1020
1020
1020
1020
960
980
1000
1020
1040
1060
1080
1100
-1020
-1020
-1020
-1020
-1020
-1020
-1020
-1020
-60
-40
-20
0
20
40
60
80
Note that the payoff in column 4 for the combined position of the bond and the
forward is the same as the value of the index in column 1. This is important for
discussion of tax issues, because it means you can act like the owner of the
index by using this combination instead of actually buying the index. Thus, you
might be able to get the return on the index using this new combination which
could help you to pay less tax.
The text discusses tax issues on page 58, and states that if you buy zero coupon
bonds and a forward contract, you effectively mimic a similar investment in
stock.
There is another way to look at the bond + forward combination relative to the
stock. If you buy the bond, it will give you 1020 in cash, and you can then
immediately buy the stock for that 1020 using the forward contract.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 9- Review of Derivatives Markets, Chapter 2 Page M9- 9
Section 9.5
Purchased call options
A long forward contract requires you to buy an asset at a specified price. If you
have a 6-month forward contract to buy the S&R index at a price of 1000, you
must buy for 1000 at expiration whether the spot price is higher than 1000
(good for you) or less than 1000 (bad for you.)
An alternative investment is a call option which would give you option of
buying for 1000 in 6 months if the spot price were higher than 1000, but would
not oblige you to buy if the spot price were lower that 1000. Of course, you must
pay a price or premium for a call option, since it has less risk than a forward
for the buyer. This is different from a forward, since a forward has an initial
investment of $0.
Before we talk further about call options, we need to introduce some
terminology:
The value of 1000 for which you can buy the S&R index in 6 months is called the
strike price or exercise price. If you do use the call to buy the stock, you
exercise the option. The date 6 months in the future is called the expiration
date. If you do not exercise your option by that date it expires worthless. There
are three different styles of exercise possible for options:
1. European option Can be exercised only on the expiration date.
2. American option Can be exercised on any date from its creation to
expiration.
3. Bermudan option Can be exercised only during specified periods
The options studied analytically in the exam FM/2 chapters will primarily be
European, but you need to know what the others are.
In the example above we described a European S&R call option with a strike
price of 1000 and expiration in 6 months (this option is analyzed on pages 33-37
of the text). We have not discussed yet how much you will need to pay for this
option. The textbook tells you that the premium is 93.81, but does not derive
that number. A footnote points out that the premium is actually computed using
the Black-Scholes formula, which is discussed in Chapter 12. You will study
that formula for Exam M. It is a simple matter to create a spreadsheet that
calculates Black-Scholes premiums, but for exam FM/2 you do not need to do
this. You will just be given the premium on questions here.
Most options are standardized, exchange traded and cash settled. Under such a
system, if you had an S&R European call with strike price of 1000 and the spot
price on the expiration date was 1100, you would exercise the option to receive
a cash settlement of 1100-1000 = 100. If the spot price on expiration day was
900, you would simply not exercise the option. The payoff function of this call is
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M9-10
Module 9- Review of Derivatives Markets, Chapter 2
Purchased Call Payoff = max(0, Spot price at expiration - 1000)
The future value of the premium expense is 93.81(1.02) = 95.69. So the profit is
Purchased Call Profit = max(0, Spot price at expiration - 1000) - 95.69.
The future value of 95.69, above, is rounded. The text shows the future value of this premium as
95.68, most likely because the author started with more decimal places on the 93.81 figure.
Next we show a table and graph of the payoff and profit.
Less FV
S&R Call of Call
Index Payoff premium Profit
900
925
950
975
1000
1025
1050
1075
1100
1125
1150
0
0
0
0
0
25
50
75
100
125
150
-95.69
-95.69
-95.69
-95.69
-95.69
-95.69
-95.69
-95.69
-95.69
-95.69
-95.69
-95.69
-95.69
-95.69
-95.69
-95.69
-70.69
-45.69
-20.69
4.31
29.31
54.31
The profit graph is simply the payoff graph shifted down by the
amount of the future value of the premium.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 9- Review of Derivatives Markets, Chapter 2
PageM9- 11
We can write this more compactly using the notations previously introduced.
Let:
T = The time of expiration of the call.
So = Asset prices at the contract origination
ST = Asset prices at the contract expiration
r = The continuously compounded interest rate for the forward
buyer and seller.
K = Exercise price
PCaii = Premium
Then
Purchased Call Payoff = max(0,ST - K)
Purchased Call Profit = max(0,Sr - K) - PcaiierT
A call has lower profit than a forward when spot prices are high (due to
premium cost) but loses less than a forward when spot prices are low. See page
36 of Derivatives Markets for a graph of this.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M9-12
Module 9- Review of Derivatives Markets, Chapter 2
Section 9.6
Written call options
You cannot buy a call unless someone is willing to sell one to you. The person
selling the call is said to write a call. For a 6-month S&R written European call
with strike price of 1000, the call writer is obliged to sell the stock for a price of
1000 in 6 months if the buyer exercises the option. In return, the call writer is
paid the premium of 93.81 (which we have already discussed). The call writer's
payoff and profit are the negative of the call buyer's payoff and profit, since
the call writer must make the payoff but gets the premium.
Written Call Payoff = - max(0, Spot price at expiration - 1000)
The future value of the premium expense is 93.81(1.02) = 95.69. Thus the profit
is
Written Call Profit = - max(0, Spot price at expiration - 1000) + 95.69.
The profit and payoff table and graph are given below for a 6-month S&R
written European call with strike price of 1000.
S&R
Index
1 900
925
950
975
1000
1025
1050
1075
1100
1125
1150
Written Call Plus FV of
Payoff premium
0
0
0
0
0
-25
-50
-75
-100
-125
-150
95.69
95.69
95.69
95.69
95.69
95.69
95.69
95.69
95.69
95.69
95.69
Written
Call Profit
95.69
95.69
95.69
95.69
95.69
70.69
45.69
20.69
-4.31
-29.31
-54.31 |
Written call
100 m
50
04
-50
-100
-150
-200
-♦—Payoff
H*~ Profit
900
950
1000
1050
1100
1150
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 9- Review of Derivatives Markets, Chapter 2
Page M9- 13
In the more compact notation,
Written Call Payoff = -max(0,ST - K)
Written Call Profit = -max(0,ST - K) + PCaiierT
You can lose a lot of money on a written call if the spot price increases
dramatically above the strike price. So, why would anyone write a call? We
have seen portfolio managers write calls on stock they own to generate extra
income. They typically are forecasting that the price of the stock will not go up
by a large amount and they wish to collect the premium income.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M9-14
Module 9- Review of Derivatives Markets, Chapter 2
Section 9.7
Purchased put options
A call option is valuable to someone who wishes to profit from appreciation in
the price of an asset. A put option is valuable to someone who wishes to profit
from a decline in the price of an asset.
A 6-month European S&R put option with a strike price of 1000 gives you the
right, but not the obligation, to sell the index for 1000 in six months. If the index
has declined to 600 in 6 months, you can cash settle for 1000 - 600 = 400. If the
index increases to 1100, you would let the option expire.
The put and call premiums for the same strike price and expiration are not
necessarily equal. For example, the textbook gives a premium of 74.20 for this
put option, with a future value of 74.20(1.02) = 75.68. Note that this is not the
same as the call option premium for the same expiration and strike price. We
will discuss this further when we cover Chapter 3 of Derivatives Markets.
The payoff function of the above put is
Purchased Put Payoff = max(0,1000-Spot price at expiration)
The profit is
Purchased Put Profit = max(0,1000-Spot price at expiration ) - 75.68.
Next we show a table and graph of the payoff and profit.
Less FV
S&R
Index
900
925
950
975
1000
1025
1050
1075
1100
1125
1150
Put
of
Payoff premium
100
75
50
25
0
0
0
0
0
0
0
-75.68
-75.68
-75.68
-75.68
-75.68
-75.68
-75.68
-75.68
-75.68
-75.68
-75.68
Put
Profi
24.32
-0.68
-25.68
-50.68
-75.68
-75.68
-75.68
-75.68
-75.68
-75.68
-75.68
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 9- Review of Derivatives Markets, Chapter 2
Page M9- 15
-100
900
Purchased Put
-#—mr—-«—-»»—~~»~—*
Payoff
Profit
950
1000
1050
1100 1150
In the more compact notation, we will denote the put premium by Pput
Purchased Put Payoff = max(0,K-ST)
Purchased Put Profit = max(0,K-ST)- PputerT
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page M9-16
Module 9- Review of Derivatives Markets, Chapter 2
Section 9.8
Written put options
Every purchaser of a put option must be paired with a seller who is willing to
write the put. For a 6-month S&R written European put with strike price of
1000, the put writer is obliged to buy the stock for a price of 1000 in 6 months if
the put purchaser exercises the option. In return, the put writer is paid the
premium of 74.20 (which we have already discussed). The put writer's payoff
and profit are the negative of the put buyer's payoff and profit, since the put
writer must make the payoff but gets the premium.
Written Put Payoff = - max(0,1000-Spot price at expiration )
The future value of the premium revenue is 74.20(1.02) = 75.68 . The profit is
Written Put Profit = -max(0,1000-Spot price at expiration) + 75.68.
The profit and payoff table and graph are given below for a 6-month S&R
written European put with strike price of 1000.
S&R
Index
900
925
950
975
1000
1025
1050
1075
1100
1125
1150
Put
Payoff
-100
-75
-50
-25
0
0
0
0
0
0
0
Plus FV
of
premium
75.68
75.68
75.68
75.68
75.68
75.68
75.68
75.68
75.68
75.68
75.68
Put
Profit
-24.32
0.68
25.68
50.68
75.68
75.68
75.68
75.68
75.68
75.68
75.68
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 9- Review of Derivatives Markets, Chapter 2
Page M9- 17
-150
900
Written Put
-m m m-
~* ■
-♦—Payoff
-♦—Profit
950
1000
1050
1100
1150
In our standard notation
Written Put Payoff = -max(0,K-ST)
Written Put Profit = -max(0,K - ST) + Pput e71*
Section 9.9
In and out of the money
Page 43 the text introduces some important terminology that is based on the
possible payoff if an option were to be exercised immediately:
An option is
• In the money if the payoff would be positive if the option were
exercised immediately.
• At the money if the payoff would be 0 with immediate exercise
• Out of the money if the payoff would be negative with immediate
exercise.
Thus if you had a six month call with a strike of 1000 on the S&R index, the
option would be in the money if the spot rate was 1001, at the money if the spot
rate was 1000 and out of the money if the spot rate was 999. Note that this
terminology is based on payoff, not profit.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M9-18
Module 9- Review of Derivatives Markets, Chapter 2
Section 9.10
Viewing options as insurance
Page 45 of the text discusses how homeowner's insurance can be viewed as a
put option. This is done using an example which is worked with a graph. We will
discuss the illustrative example in the text in our own words to give you
another view of it.
The text example deals with a $200,000 house which is insured under a policy
with a $25,000 deductible for a premium of $15,000. Since the house is only
worth $200,000, the largest possible payment after deductible is $175,000, which
would be paid in the event of a total loss. For a loss less than 25,000, the policy
pays nothing. To see what happens when there is a loss that is larger than the
deductible but not total, we will look at a $55,000 loss. In that case the payment
is
Payment = Loss - deductible = 55,000 - 25,000 = 30,000
There is another way to look at this insurance payment. The value of the house
is reduced by the 55,000 loss to
Value = 200,000 - 55,000 = 145,000
The insurance policy covers the value up to $175,000. Thus the insurance
payment will restore the value to that level.
Payment = 175,000 - Value = 175,000 - 145,000 = 30,000
Using this reasoning, you can show that the insurance payment is
Payment = max(0,175,000 - Value)
This is the payoff on a put option with strike price of 175,000 on the value of the
house. Thus the insurance company has written the home owner a put option
with a strike price of 175,000 on the value of the house.
This is an important connection to make, and it is worth thinking about this
explanation and reading Section 2.5 carefully to see it in a slightly different
way.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 9- Review of Derivatives Marketsy Chapter 2 Page M9- 19
Section 9.11
The equity-linked CD example
In section 2.6 of Derivatives Markets there is a nice example of how to use call
options to create a CD whose return is linked to an index like the S&P 500. To
understand why an investor would like such a product, we need to review some
basic finance.
The long term return on the S&P 500 is higher than the long term return on
safer investments like government bonds. Ibbotson Associates, Inc. reported in
their 1998 Yearbook that from 1927 to 1996 the average annual nominal return
was 13% for the S&P 500 and only 5.6% for government bonds. So, you can see
why an investor would rather have the long run S&P return.
However investing in the S&P 500 can be problematic for an investor such as a
retiree who needs the return on investment now to pay the bills. The long term
return on the S&P 500 results from a long period in which there were some very
good years and some very bad years. The retiree does not want to invest in a
decade that has too many bad years. He wants to get something like the S&P
return if that return is positive, but not lose his money in the bad years. That is,
he wants the option to get positive return when that is available but not be
obliged to accept negative return.
The equity-linked CD described in the text meets this need. It is originally
structured when the S&P 500 index is at a current value of 1300. The investor
will invest 10,000 and be paid in 5.5 years. The final payment depends on
whether the S&P 500 index is above or below 1300 in 5.5 years, as the table
below indicates.
Index in 5.5 years
Index < 1300
Index> 1300
Payoff
10,000
10,000(1+ .7 (percentage gain on index))
In other words -if the index goes down, the investor gets his money back, but if
the index goes up he gets the money back and an additional return equal to 70%
of the percentage gain on the S&P 500. Suppose, for example, that the S&P
index in 5.5 years has gone up by 40% to 1820. Then the investor will be paid
10,000 (1 + .7 (.40)) = 10,000 (1.28) = 12,800
(Note, the interest rate per semiannual period here is 2.27)
The text notes that if Sfinai is used to represent the S&P index in 5.5 years, the
percentage gain is
Sfinai -J
1300
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M9-20
Module 9- Review of Derivatives Marketsy Chapter 2
Thus the CD pays 10,000 1 + .7
max| 0,^-1
1 1300
. We will write this in a
different way to give you a different look at it. We will write the percentage
gain on the S&P 500 index as
JL(S«-1300).
Then we write the payment on the CD in 5.5 years as
lO.OOO + .y^^max^.S^ai -1300) = 10,000+ 5.3846 max (0,S/ma, -1300)
JLoUU
The final two terms represent i) a payoff of the original 10,000 plus ii) 5.3846
European call options on the S&P 500 index with strike price of 1300 and
expiration in 5.5 years. Thus what the CD really gives the investor is
Return of Original Amount + Payoff of 5.3846 S&P 1300 strike call options
The first component lets the investor get his money back, and the second links
his return to the S&P 500 index.
Note that an investor does not have to get this payoff from a bank CD. The
author points out that the investor could buy a zero coupon bond paying 10,000
in 5.5 years and also buy the call options on his own. (The text says this a
different way, saying that the investor will buy 7.69 units of a package for
which a single unit consists of a zero coupon bond for 1300 and .7 of an index
call option.)
However, this is all too complicated for most investors to do. The service the
bank provides with this CD is to do all that investing and buying for the
investor. As an actuary, you may do this kind of work. One of my previous
employers had an equity linked annuity that promised a return on investment
linked to an index, and it was structured using options. It was designed by one
of the actuaries.
The text also points out that this CD does have a cost. If, for example, the
investor had an interest rate of 6% per year, a deposit of 10,000 would grow to
10,000(1.06)55 =13,777.88
Thus he would have earned 3,777.88 in interest, and he has forgone that interest
to get the S&P linked return. The present value of that interest accumulation is
3,777.88
1.06
5.5
2,741.99
Thus as he invests today he has foregone interest with a present value of
2,741.99, and that is the implied cost today of investing in the CD.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 9- Review of Derivatives Markets, Chapter 2
Page M9- 21
Section 9.12
Don't forget Appendix 2.A
Appendix 2.A discusses practical issues related to dividends, exercise, margins
and taxes. It is clearly written, and we will not go over it here. It is in the exam
syllabus.
Section 9.13
Module 9 summary
Forward contract -a contract to buy or sell a specified asset at a designated
future time.
The current price of an asset at any time is called its spot price.
Settlement options: 1) actual delivery of the asset
2) cash settlement in which a net payment is made for
the difference between the spot price at expiration
and the forward price.
The forward buyer is said to have a long forward and the seller is said to have a
short forward.
Payoff on a contract -the value of the position at expiration.
Profit - payoff less the future value of any expenses
Payoff to long forward = Spot price at expiration - forward price
Payoff to short forward = Forward price - spot price at expiration
Since forward expenses are 0, forward payoff and forward profit are the same.
Graph of forward pay off/prof it:
80 •
60 |
40 ■
20 •
0 -
-20 -
-40 -
-60 4
-80 -
96
I
^ _
— "^ ■
■**.
0
""^^ fl
*■* ^
1010
~^"
■
1060
►
1
♦
Long
— Short J
Profit on S&R share = Profit on forward
Buying the asset and entering a long forward contract at the correct arbitrage-
free forward price have the same profit function.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M9-22 Module 9- Review of Derivatives Markets, Chapter 2
Mathematical notation
T Time of expiration of an S&R index forward contract which was written
at time 0.
So asset price at the contract origination
ST asset price at the contract expiration.
F0,t Forward price at time 0 for a forward contract expiring at time T.
r Continuously compounded interest rate for the forward buyer and seller.
Forward results re-stated
Index pays no dividends —► F0,t = S0erT = FV expense of borrowing S0
Profit on S&R share = ST - SQen = ST - F0>T = Profit on forward.
Zero Coupon Bonds Have 0 Profit
Bond Profit = Payoff - Future value of expenses incurred = PerT - PerT = 0.
Buying zero coupon bonds and a forward contract mimics a stock investment.
Options
Purchased call option gives the right but not the obligation to buy an asset at
the specified strike price or exercise price. You must pay a price or premium
for it. The option expires after the expiration date.
Styles of exercise possible for options:
A European option can be exercised only on the expiration date.
An American option can be exercised on any date from its creation to
expiration.
A Bermudan option can be exercised during specified periods, but not on any
date.
The options studied in the exam FM/2 chapters will be European only.
Denote the exercise price by K and the premium by PCflM.
Purchased Call Payoff = max(0,ST - K)
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 9- Review of Derivatives Markets, Chapter 2
Page M9- 23
Purchased Call Profit = max(0, ST-K)- PcaiieTT
Graph of purchased put option payoff and profit.
200 -i
150
100
50 ■
-50-
-100 !
9C
Purchased Call
4^
J*^'^ A
. . . r**"^ B-^""
^mimmMtmammMam,*
)0 950
►
1
1000 1050 1100 1150
| » Payoffl
|~H»—Profit |
Written Coll Options
Written Call Payoff = -max(0,ST - K)
Written Call Profit = -max(0,ST - K) + PcaiierT
Graph of written call payoff and profit.
Purchosed Put Options
A put gives the right but not the obligation to sell at the strike price.
Purchased Put Payoff = max(0, K-ST)
Purchased Put Profit = max(0,K -ST) -Pme
rT
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page M9-24
Module 9- Review of Derivatives Markets, Chapter 2
Purchased Put Graph
Written Put Options
Written Put Payoff = -max(0,K-ST)
Written Put Profit = -max(0,K-ST) + Pput erT
Written put graph
50 •
-50 -
-100 i
9C
Written Put
~~"w m »"""
-~~~*~~»
Wl
tm.
""Wt"
J
%
M^^m
)0
950
1000
1050
1100
11
§
50
♦ Payoff I
—Wk~~- Profit i
©ACTFX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 9- Review of Derivatives Markets, Chapter 2 Page M9- 25
Section 9.14
Solutions to odd-numbered problems
Please note: Some problems request a graph, but we give only the table that
produces the graph to save space.
2.1.
We will assume that the long position is for one period, since that gives the
requested result of 0 profit at a price of 55 in one year. If you borrow 50 to
buy the asset now, the future value of expenses in one year is 55. The payoff
and profit functions are displayed in the table below.
Less
Future
Stock Value of
Stock Payoff Cost Profit
1 50
51
52
53
54
55
56
57
58
59
60
50
51
52
53
54
55
56
57
58
59
60
-55
-55
-55
-55
-55
-55
-55
-55
-55
-55
-55
-5
-4
-3
-2
-1
0
1
2
3
4
5 1
2.3.
A purchased call has
Purchased call payoff = max(0,ST -K).
If we take opposite to mean negative, the opposite is a written call with
Written call payoff = -max(0,ST-K)..
Similarly, the opposite of a purchased put is a written put.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M9-26
Module 9- Review of Derivatives Markets, Chapter 2
2.5.
The following table shows the payoff for the short forward sale at 50 and the
put purchase with a strike of 50. The put gives the same gains but no losses.
It should cost more.
Short
Forward Put
Stock Payoff Payoff
40
45
50
55
60
10
5
0
-5
-10
10
5
0
0
o
2.7.
The table below summarizes the profit diagrams for parts a-c
Profit on Profit on
St Long Forward FV of Cost to Stock without Stock with
Stock Profit buy stock dividend dividend
45
50
55
60
65
-10
-5
0
5
10
55
55
55
55
55
-10
-5
0
5
10
-8
-3
2
7
12
a) The short forward profit is ST = 55, and is displayed in column 2.
To buy the stock, borrow $50 now at 10% and repay $55 in one
year. The profit on the stock is also ST - 55, as displayed in
column 4.
b) There is no advantage to owning the stock if there are no
dividends.
c) However, there is a profit advantage to owning the stock if there
is a dividend of $2, since the stock owner gets that $2 in addition
to the asset profit.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 9- Review of Derivatives Markets, Chapter 2
Page M9- 27
2.9.
a) If the interest rate is 10% you can borrow 1000 at 10% to buy the index
and repay 1100 as the future value expense in one year. The table below
illustrates that the profit functions of the long forward and the index
share are the same when the forward price is 1100.
1100 Forward Price
S&R Forward Index Share Less Future Index Share
Index Profit Payoff Value of Cost Profit
950
1000
1050
1100
1150
1200
1250
1300
-150
-100
-50
0
50
100
150
200
950
1000
1050
1100
1150
1200
1250
1300
-1100
-1100
-1100
-1100
-1100
-1100
-1100
-1100
-150
-100
-50
0
50
100
150
200
b) If the forward price is 1200 there is an advantage to buying the stock, as
the next table indicates.
1200 Forward Price
S&R Forward Index Share Less Future Index Share
Index Profit Payoff Value of Cost Profit
950
1000
1050
1100
1150
1200
1250
1300
-250
-200
-150
-100
-50
0
50
100
950
1000
1050
1100
1150
1200
1250
1300
-1100
-1100
-1100
-1100
-1100
-1100
-1100
-1100
-150
-100
-50
0
50
100
150
200
The forward price is 100 too high, and the long forward will make 100
less profit than you would if you bought the index. You should be paid
the present value at 10% of the lost 100
M = 90.90
1.1
c) That contract is priced 100 too low and you would make 100 in excess
profit. You would pay the present value
M = 90.90
1.1
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M9-28
Module 9- Review of Derivatives Markets, Chapter 2
2.11.
a) For ST < 1000:
Purchased Put Profit = (1000 - ST) - 75.68
The put diagram intersects the x-axis when
0 = (1000 -ST)- 75.68 — ST = 924.32
b) For ST > 1000:
Purchased Put Profit = -75.68
Short Forward Profit = 1020 - ST
The intersection occurs when 1020 - ST =-75.68. Thus ST = 1095.68
2.13.
a) The payoff table and diagram are:
Strike price
Stock
35
40
45
30
35
40
45
50
55
60
65
0
0
5
10
15
20
25
30
0
0
0
5
10
15
20
25
0
0
0
0
5
10
15
20
30
25
20
15
10
0 4-
30
35
40
45
50
Payoff 35
Payoff 40
Payoff 45
55
60
65
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 9- Review of Derivatives Markets, Chapter 2 Page M9- 29
The profit table and diagram are:
Strike price
Premium
FV Premium
35
9.12
9.85
40
6.22
6.72
45
4.08
4.41
Stock Profit Profit Profit
35
40
45
50
55
60
65
-9.85
-9.85
-4.85
0.15
5.15
10.15
15.15
20.15
-6.72
-6.72
-6.72
-1.72
3.28
8.28
13.28
18.28
-4.41
-4.41
-4.41
-4.41
0.59
5.59
10.59
15.59
b) Lower strike price leads to a higher possible payoff. Intuitively, an
investor should pay more for a higher payoff.
2.1S.
Part of the cash from the short sale would be left on margin and earn
interest. Thus you would need to short more than 1000 in IBM stock and
account for interest on margin. In addition you would need to pay any
dividends that occurred.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M9-30 Module 9- Review of Derivatives Markets, Chapter 2
Section 9.15
Module 9 Computational Review Problems
1. (1 pt) 1) Suppose you enter into a long 6-month forward
position at a forward price of $ 30. What is the payoff in 6
months for prices of $ 20,$ 30,$ 40 ?
When price is $ 20, the payoff is $ ?
When price is $ 30, the payoff is $ ?
When price is $ 40, the payoff is $ ?
2) Suppose you buy a 6-month call option with a strike price
of $ 30. What is the payoff in 6 months for prices of the
underlying asset of $ 20, $ 30,,$ 40 ?
When price is $ 20, the payoff is $ ?
When price is $ 30, the payoff is $ ?
When price is $ 40, the payoff is $ ?
3) Comparing the payoffs of parts a) and b), which contract
should be more expensive (i.e. the long forward, or the long
call? Enter 1, or 2, respectively.) ?
ANSWER1: -10
ANSWER2: 0
ANSWER3: 10
ANSWER4: 0
ANSWER5: 0
ANSWER6: 10
ANSWER7: 2
2. (1 pt) An off-market forward contract is a forward where
either you have to pay a premium or you receive a premium for
entering into the contract. (With a standard forward contract,
the premium is zero.) Suppose the effective annual interest rate
is 14 % and the S-R index is 1000. Consider 1-year forward
contracts.
a) Suppose you are offered a long forward contract at a forward
price of $ 1390. How much would you need to be paid to enter
into this contract $ ?
b) Suppose you are offered a long forward contract at a forward
price of $ 990. How much would you need be willing to pay to
enter into this contract $ ?
ANSWER1: 219.3
ANSWER2: 131.58
3. (1 pt) Suppose a security has a bid price of $ 116 and an
ask price of$ 116.35.
At what price can the market-maker purchase the security $
At what price can a market-maker sell the security $
What is the spread in dollar terms when 100 shares are traded $
9
ANSWER1: 116
ANSWER2: 116.35
ANSWER3: 35
4. (1 pt) 1) Suppose you enter into a short 6-month forward
position at a forward price of $ 20. What is the payoff in 6
months for prices of $ 10,$ 20,$ 30 ?
When price is $ 10, the payoff is $ ?
When price is $ 20, the payoff is $ ?
When price is $ 30, the payoff is $ ?
2) Suppose you buy a 6-month put option with a strike price
of $ 20. What is the payoff in 6 months for prices of the
underlying asset of $ 10,$ 20,$ 30?
When price is $ 10, the payoff is $ ?
When price is $ 20, the payoff is $ ?
When price is $ 30, the payoff is $ ?
3) Comparing the payoffs of parts a) and b), which contract
should be more expensive (i.e. the long forward, or the long
call? Enter 1, or 2, respectively.) ?
ANSWER1: 10
ANSWER2: 0
ANSWER3: -10
ANSWER4: 10
ANSWER5: 0
ANSWER6: 0
ANSWER7: 2
5. (1 pt) Suppose a stock is priced at $ 70 at expiry and the
annual effective interest rate is 8 %. Determine the profit at
expiry for the following one-year european call options:
A $ 65-strike call with premium $ 9.11 ?
A $ 70-strike call with premium $ 6.87 ?
A $ 75-strike call with premium $ 4.44 ?
ANSWER1: -4.84
ANSWER2: -7.42
ANSWER3: -4.8
6. (1 pt) Suppose a stock is priced at $ 25 at expiry and the
annual interest rate is 6 %. Determine the profit at expiry for the
following one-year european put options:
A $ 20-strike put with premium $ 4.51 ?
A $ 25-strike put with premium $ 6.33 ?
A $ 30-strike put with premium $ 9.56 ?
ANSWER 1: -4.78
ANSWER2: -6.71
ANSWER3: -5.13
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 9 - Review of Derivatives Markets, Chapter 2 Page M9- 31
Section 9.16
Supplemental Exercises
1. The S&R index currently has a price of 1000. The price of a six month
long forward contract is 1025. What is the profit (or loss) on a six month
forward purchase if the spot price of the S&R index is 1020 at expiration
in six months?
A)-5 B)-2.25 C)0 D) 2.25 E) 5
2. The S&R index currently has a price of 1000. The price of a six month
forward contract is 1025. The annual interest rate is 4.94% compounded
continuously. A buys the index and B enters a forward purchase
agreement. What is the difference between the profit for A and the profit
for B if the spot price of the S&R index is 1020 at expiration in six
months?
A)-5 B)-2.25 C)0 D) 2.25 E) 5
3. The S&R index currently has a price of 1000. The price of a six month
forward contract is 1025. What annual interest rate (compounded
continuously) is implied by this forward price?
A) .02482 B) .02500 C) .02543 D) .0494 E) .0500
4. The S&R index currently has a price of 1000. The price of a six month
1010-strike call is 93.93. In six months the index price is 1025. The annual
interest rate is 4.94% compounded continuously. What is the profit on the
call?
A)-96.28 B)-93.93 C) -81.28 D)-75.93 E)-74.08
5. The S&R index currently has a price of 1000. The price of a six month
1010-strike call is 93.93. In six months the index price is 1025. The annual
interest rate is 4.94% compounded continuously. What is the difference
between the payoff and the profit on the call?
A) -96.28 B) -93.93 C) -81.29 D) -75.93 E) -74.08
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M9-32
Module 9 - Review of Derivatives Markets, Chapter 2
6. The S&R index currently has a price of 1000. The price of a six month
1010-strike put is 74.08. In six months the index price is 1025. The annual
interest rate is 4.94% compounded continuously. What is the profit on the
put?
A)-96.28 B)-93.93 C) -81.29 D)-75.93 E)-74.08
7. The S&R index currently has a price of 1000. The price of a six month
1010-strike put is 74.08. The annual interest rate is 4.94% compounded
continuously. A buys this put, and B enters into a long forward contract.
In six months A and B have the same profit. What is the price of the
index in six months?
A) 979.53 B) 1000 C) 1025 D) 1037.92 E) 1097.32
8. The S&R index currently has a price of 1000. The price of a six month
1010-strike put is 74.08. The annual interest rate is 4.94% compounded
continuously. What is the profit on this put in six months if the spot price
then is 980?
A) -94.35 B) -45.93 C) 0 D) 30 E) 104.53
9. Your home has a value of 300,000. Your annual insurance premium is
5,000 and your deductible is 20,000. If you look at your insurance as a put
option, what is the strike price?
A) 300,000 B) 295,000 C) 280,000 D) 275,000 E) 270,000
10. The stock of ABC company pays no dividends and has a current price of
40. The forward price for delivery in one year is 42. If there is no
advantage to buying either the stock or the forward contract, what is the
continuously compounded one year interest rate.
A) .0488 B) .0494 C) .05 D) .0506 E) .0512
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 9 - Review of Derivatives Markets, Chapter 2
Page M9- 33
11. An insurance company sells single premium deferred annuity contracts
with return linked to a stock index, the time-t value of one unit of which
is denoted by S(t). The contracts offer a minimum guarantee return rate
of g = 2.5%. At time 0, a single premium of amount n is paid by the
policyholder, and n x y% is deducted by the insurance company.
In one year the insurance company will pay the policyholder
n X (1 - y%) x Max[S(T)/S(0), (1 + g%)], where ) S(0) =100
You are given the following information:
(i) Dividends are incorporated in the stock index. That is, the stock
index is constructed with all stock dividends reinvested,
(ii) The price of a one-year European put option, with strike price of
$102.50, on the stock index is $16.
Determine y%, so that the insurance company does not make or lose
money on this contract.
A) 13.2% B) 13.35% C) 13.5% D) 13.65% E) 13.80%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Has sett, Ratliff, Garcia, & Steeby

Page M9-34
Module 9 - Review of Derivatives Markets, Chapter 2
Section 9.17
Supplemental Exercise Solutions
1) Long Forward Profit = ST - F0,T = 1020 -1025 = -5.
Answer A
2) We have already noted that Profit on S&R share = Profit on forward
The difference is 0.
Answer C
3) F0>T = Soe77 -> 1025 = lOOOe5r -> r = .0494
Answer D
4) 1025 -1010 - 93.93e0494(5) = -81.28
Answer C
5) -93.93e0494(5) =-96.28
Answer A
6) 0-74.08e0494(5) =-75.93
Answer D
7) The forward price is F0(T = S0erT = lOOOes(0494) = 1025. The long forward
profit is ST - F0,T = ST -1025.
The put profit is
max (0,1010 - ST )-74.08e0494(5)= max (0,1010-ST)-75.93.
Assume that ST < 1010 . Then the equality of prices implies that
ST -1025 = 1010 -ST- 75.93 -> ST = 979.53
Answer A
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 9 - Review of Derivatives Markets, Chapter 2
Page M9- 35
8) The put profit is
max (0,1010 -ST) - 74.08e0494(5) = max (0,1010 -980) - 75.93 = -45.93
Answer B
9) Let Vt be the value of the house at time T. The payoff has value
max (0,300,000 - 20,000 - VT) = max (0,280,000 - VT).
This is the payoff of a put with K = 280,000.
Answer C
10) The correct theoretical price of the forward should apply, Thus
Fo,T = Soe^ -* 42 = 40er(1) -> r = .0488
Answer A
ll)Using g = .025,T = 1,S0 = 100, the total payoff is
Si
-,±.UZO I
=^(l-y)[sl+m*x(102.so-sl,o)]
^(l-y)max ^,1.025l = I^(l-y)max(S1,102.50)
100
The expression in square brackets is the payoff of a single share of the
index and a put, while the two lead terms give the number of units of this
combination the company needs to buy to pay off the single premium
deferred annuity. The company wants to use the premium n to buy the
shares and the options needed. The cost of those shares and options
today is
I^(l-y)[So + putcost] = I^(l-y)116 = 1.16^(l-y)
To break even this cost must equal the premium collected.
1.16^(l-y) = ^->y = .138
The required percentage is 13.8%.
Answer E
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Module 10 - Review of Derivatives Markets, Chapter 3
Page M10- 1
Section 10.1
Overview
In this chapter, the textbook studies various useful combinations of assets,
forwards, puts and/or calls. A very useful relationship called put-call parity is
also derived and applied. The analysis is intuitive and generally based on
looking at payoff tables and graphs for specific examples to make general
conclusions. We will discuss the various combinations in a slightly different
way to give you additional insight as you read the text. To simplify discussions,
we will denote the payoff and profit of a combination by
Payoff [Combination] and Profit[Combination].
For example, we will look at the combination consisting of buying an asset and
a put. Then we would write the payoff and profit as
Payoff [Index + Put] and Profit[Index + Put].
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Page M10-2
Module 10 - Review of Derivatives Markets, Chapter 3
Section 10.2
Strategies Combining an Option and an Asset
A basic strategy is to use options as insurance against changes in the value of
assets.
Using Puts to Create Floors for Asset Value
If you own an asset (such as the S&R index), your main concern is that the
value of the asset might decrease. You can use a put to create a floor (basically,
a guaranteed lowest price) for that value. Suppose that you own the S&R index
and it is currently at 1000 and you buy a 6-month put with strike price of 1000.
Then, if within that six month period, the S&R index has dropped below 1000,
you can use the put to sell it for 1000 anyway. You have put a floor of 1000 on
the value of your position, but can benefit from any increase of the index above
1000 because you can sell at any price above that if the value of the index rises.
Note that if you have a call option with the same strike and expiration, the
payout has a floor of 0, and enables you to benefit from any increase of the
index above 1000.
You might expect that there is a relation between the floor and the call. There
is:
(10.1)
Payoff[Index+Put with strike K]
= Payoff [Call with Strike K+Zero-Coupon Bond for K]
(10.2)
Profit[Index + Put with strike K]
= Profit[Call with Strike K]
On pages 60 and 61 McDonald goes through an example of this for the S&R
index.
As you work through the examples on pages 60-61, you should understand that
the purpose of the exercise is to get you to conclude that (10.1) and (10.2) are
valid for the S&R index in general. The idea is that you compare the results
here to results in Chapter 2. For 10.2, you would compare the profit column in
Table 3.1 to the profit column for the call in Table 2.2 on page 35.
As an actuarial student, you might like a more mathematical approach. We can
derive (10.1) using our standard notation. Let T be the time of expiration of the
put and call. Denote the asset prices at the contract origination and expiration
by So and ST respectively. Then
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Module 10 - Review of Derivatives Markets, Chapter 3
Page M10- 3
Pay of f [Index+Put with strike K] = ST + max(0, K-ST) = \ ' T
\K, ST<K
Payoff [Call with Strike K+Zero-Coupon Bond for K] = K + max(0, ST-K) = l
The two payoff functions are identical. You would expect (10.2) to follow from
(10.1) because the profit of a zero-coupon bond is 0. We could derive (10.2)
using the put-call parity relation that comes in a later section, but the text is not
holding you responsible for a derivation here. You are just supposed to
convince yourself of (10.2) by example and general reasoning.
In the initial discussion of floors in the textbook, the underlying asset is the
S&R index. The text points out that a homeowner with insurance owns an asset
consisting of the home and has a put on the value of the house with his
insurance. Thus the homeowner has a floor for the value of his home, and his
combined position has the same profit diagram as a purchased call.
Using Calls to Create Caps for Short Positions
If you sell short an asset such as the S&R index, your main concern is the value
of the asset might increase. You can use a call to create a cap (i.e., guaranteed
highest price) for that value.
Suppose that the S&R index is currently at 1000 and you sell the index short and
buy a 6-month call with strike price of 1000. Then if the S&R index increases
above 1000 in 6 months, you can exercise the call to get the index for 1000 and
cover the short sale. Thus you can benefit from the short sale if the price of the
index goes below 1000, but you are protected against loss if the index increases
above 1000. Note that if you have a put option with the same strike and
expiration, the payout has no loss if the index is above 1000, and enables you to
benefit from any decrease of the index below 1000. You might expect that there
is a relation between the cap and the put. There is:
(10.3)
Payoff [Short Index+Call with strike K]
=Payoff[Put with Strike K+Sale of a Zero-Coupon Bond for K]
The text refers to the bond sale as borrowing. This is because the bond issuer is both
borrowing money from and paying interest to the bond buyer.
(10.4)
Profit[Short Index + Call with strike K]
= Profit[Put with Strike K]
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Page M10-4
Module 10 - Review of Derivatives Markets, Chapter 3
On page 63 McDonald goes through an example for the S&R index.
As you work through the example on page 63, you should understand that the
purpose of the exercise is to get you to conclude that (10.3) and (10.4) are valid
for the S&R index in general. Note also that the cash proceeds from the short
sale are taken to be a negative cost (gain).
We can derive (10.3) using our standard notation.
Payoff [Short Index+Call with strike K] = -ST + max(0,Sr - K) =
St y St < -K
-K, ST Z K
Payoff [Put with Strike K+Sale of Zero-Coupon Bond for K] = max(0, K-ST)-K =
-K, ST>K
The two payoff functions are identical. As in the previous section, (10.4) can be
expected to follow from (10.3) because the profit of a zero-coupon bond sale is
0.
Covered Calls
We mentioned in Module 9 that a call might be written by an owner of an asset
who wishes to earn the call premium income. You have a covered call (aka
covered writing or option overwriting) on the S&R index if you buy the index
and write a call on it. Table 3.3 on page 65 of the text enables you to see by
example that:
(10.5)
Profit[covered call] = Profit[written put]
A covered call is the opposite side of a cap.
Combination
Cap
Covered Call
Strategy
Sell Index & Buy Call
Buy Index and Sell Call
Profit equivalent
Purchased Put
Written Put
Note that writing a covered call is less risky than writing a call without owning
the asset. The loss on a written call is unlimited if you do not own the asset and
its price increases, but if you own the asset you can use it to satisfy the call and
avoid a large loss. Writing a call without owning the asset is called naked
writing.
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Exam FM / Exam 2 - Financial Mathematics

Module 10 - Review of Derivatives Markets, Chapter 3
Page M10- 5
Covered Puts
You have a covered put on the S&R index if you sell the index short and write a
put on it. See Figure 3.5 on page 67 of the text for a visual interpretation.
(10.6)
Profit[covered put] = Profit[written call]
Problem 3.2 on the text asks you to build the table behind the graph. This is a
good exercise.
A covered put is the opposite side of a floor.
Combination
Floor
Covered Put
Strategy
Buy Index & Buy Put
Sell Index and Sell Put
Profit equivalent
Purchased Call
Written Call
Note that writing a covered put is less risky than writing a put without owning
the asset. If you have sold the index short you can use the stock that is put to
you to cover the short sale.
©ACTEX 2009
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Exam FM / Exam 2 - Financial Mathematics

Page M10-6
Module 10 - Review of Derivatives Markets, Chapter 3
Section 10.3
Synthetic Forwards and Put-Call Parity
Synthetic Forwards
A synthetic forward is a combination of puts and calls that acts just like a
forward.
In this section and the next we will use the following notation.
Call(K,T): Price of a call with expiration date T time units in the
future and strike price K
Put(KyT): Price of a put with expiration date T time units in the future
and strike price K.
If you buy a call option for the S&R index with strike price K and sell a put with
the same strike and expiration, you pay Call(K,T)for the cost and are paid
Put(K,T) for the written put. Thus your net cost is:
Net cost of synthetic forward = Call(K,T) - Put(K,T).
The combination of options means that at time T you will get the index for a
price of K If the ST > K, you can call the asset for a price of K. If ST < K, you
will be required to buy the asset for a price of K. Thus you have really
constructed a synthetic forward for the index at the forward price K You are
guaranteed the possession of the asset at time T.
Put-Call Parity
We now have two ways to guarantee ownership of the S&R index at time T:
1) Enter into a forward contract, with forward price F0tT = S0en.
To assure having the money to purchase the forward at time T, you
would need to set aside now the present value of F0tT, denoted by
PV(F0,T). Note that for the S&R index this is just S0. Your total cost
today is PV(F0>T) = S0.
2) Pick any forward price K and construct a synthetic forward that enables
you to buy the index for K at time T as above.
To assure having the money to purchase the forward at time T, you
would need to set aside now the present value of K, which the text
denotes by PV(K). Thus you would have a total cost now of the net cost
of the options plus the amount set aside for the future purchase. Your
total cost today is
Call(KyT) - Put(K,T) + PV(K).
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 10 - Review of Derivatives Markets, Chapter 3
Page M10- 7
The put-call parity relation is obtained by noting that the costs for the two
different ways of guaranteeing ownership of the index at time T should be the
same.
(10.7)
PV(F0,t) = Call(KyT) - Put(KyT) + PV(K)
On page 70 of the text, McDonald notes that this equality is based on the principle that if
two investments give the same payout they must have the same cost. If you want to
think more deeply about this, you can do so in Chapter 5 when the text discusses no-
arbitrage pricing. For the moment, lefs keep it simple
(10.7) is a very important identity. When we later use the Black-Scholes pricing
model, we will use it to find the value of a call, and then use parity to find the
value of the corresponding put. To illustrate the identity, we look back at the
text example that was used for illustration of synthetic forwards on page 67.
The values there were:
K = 1000 Fo.r = 1020 Call(KyT) = 93.81 Put(KyT) = 74.20
The interest rate for 6 months (not continuously compounded) was r = .02. Thus
PV(K) = 1^ = 980.39 and PV(F0,T) = ^^ = 1000
In this case the identity (10.7) becomes:
1000 = (93.81-74.20) + 980.39 = 19.61 + 980.39.
It should not be too surprising that the numbers match, since the value of the
put was computed from (10.7).
We can rewrite (10.7) in a version that tells us more about the net cost of a
synthetic forward.
(10.8)
PV(F0,t)- PV(K) = PV(F0,t-K)= Call(K,T) - Put(KyT)
For the S&R contract, a standard zero-cost forward contract should have the
forward price 1020. If you construct a synthetic forward for 1000 you must pay
a premium to get that lower forward price. That premium is the net cost of the
synthetic forward. Equation (10.8) says that the net cost is the present value of
the difference between 1020 and the lower price of 1000. Note that (10.8) has
some natural consequences:
(10.9)
Fo.t >K -+ Call(KyT) > Put(KyT)
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M10-8
Module 10 - Review of Derivatives Markets, Chapter 3
(10.10)
(10.11)
Fo,T
= K
-
Call(K,T)
= Put(K,T)
Fo,T
<K
-
Call(K,T)
< Put(K,T)
Another simple rearrangement of terms in (10.7) enables us to relate that
equation to the prior analysis of floors in this chapter.
(10.12)
PV(F0,t)+ Put(K,T) = Call(KyT) + PV(K)
Recall that a floor consisted of buying the index and a put. The cost of that is
the left hand side of (10.12)
S0+ Put(KyT) = PV(K) + Call(KyT)
We showed that a floor has the same payout as the combination of a call with
strike K and a zero coupon bond which pays K at expiration. The cost of that is
Call(K,T)+ PV(K).
Thus (10.12) says that the two combinations with the same payout must have
the same cost.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 10 - Review of Derivatives Markets, Chapter 3 Page M10- 9
Section 10.4
Spreads
Think of a spread as a combination of puts and calls, each one possibly long or
short. There are many possibilities here, and the text gives a survey of some
basic possible combinations, an example of each important spread type, and
gives reasons to use that particular combination. We will summarize the basics
here. You should remember that the reasons for creating some of these spreads
are based in tax and other regulatory law issues that are not part of Exam FM/2,
so try to learn the basic combinations for testing purposes.
In this section of the text the author moves away from the S&R index examples
and studies combinations of options for a fictional stock. The 3-month option
prices for this stock at various strike prices are given in the following table.
Strike Call Put
35.00
40.00
45.00
6.13
2.78
0.97
0.44
1.99
5.08
The motivating examples for the spreads in Section 3.3 all deal with
combinations of these 3-month options . The underlying interest rate is 8.3%
annually, the future value factor for future value of expenses is 1.0833025 for a
3 month period, and S0 = 40.
Problem 3.20 of the text challenges you to create a spreadsheet that will
provide profit or payout diagrams for combinations of options like the above.
The creation and use of the spreadsheet is very helpful in understanding the
spreads studied in this section.
For each section, we will assume that you have worked through each profit
table or reproduced it in a spreadsheet. We will review the strategy, the graph
and why an investor would use the strategy.
Details on Spreads
A spread results when you buy a call at one price and sell another call at a
different price. You also have a spread when you buy a put at one price and sell
another put at a different price. The text says "A spread is a position consisting
of only calls or only puts, in which some options are purchased and some
written." There are three different common spreads discussed here -bull
spreads, bear spreads, and ratio spreads. In addition, we will review something
called a box spread, which is not technically a spread since it consists of both
puts and calls
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M10-10
Module 10 - Review of Derivatives Markets, Chapter 3
Bull Spread
A bull is an investor who is betting on an increase in market value of an asset.
Do you think that the stock above will be at 45 in 6 months? Then you could buy
calls with at strike of 40 for 2.78 and make some money. However there is a
cheaper way to profit from an increase to 45. If you buy a call for 40 and sell a
call for 45, you will receive a premium of .97 for the sold call. Now your net
cost is only 2.78 - .97 = 1.81. However if the stock price goes above 45 you will
lose some profit due to using the written call.
Bull spread strategy. Buy a call at a lower price (40) and sell a call at a higher
price (45). You can achieve the same result with puts - Buy a put at a lower
price (40) and sell a put at a higher price (45).
Profit
$4 j 1
$3 j P ♦ ♦ ♦ ♦ * ♦
$2 j 4-
$1 J J-
$0 | ■ , , 1—+- ■ , ,
-$1 | -/
.$2 ♦ ♦♦♦♦»♦ *=d.
-$3 -I 1
20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00
Why do this? You think that there will be a price increase to a range around the
price of 45. This is a cheaper way of profiting from such an increase than
simply buying a call with a strike of 40.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 10 - Review of Derivatives Markets, Chapter 3
Page M10-11
BeorSpread
A bear is an investor who thinks the asset price will go down. A bear spread
profits from a decrease in the stock, and is the opposite of a bull spread. A bear
might think that this stock, currently at 40, might drop to 35.
Bear spread strategy. Sell a call at a lower price (35) and buy a call at a higher
price (40).
Profit
$4-r 1
$3 T \
$2 j V
$11 V
$0 j , : , X—, ■ , ■
-$i | V 1
.$2 1 _ . — *—-- ——T
20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00
Why do this? You think that there will be a price decrease to a range around the
price of 35.
Bull and bear spreads are known as vertical spreads due to the vertical
increase or decrease from one level to another.
Ratio Spread
Ratio Spread Strategy The strategy here is a variation on the bull or bear
spread. Instead of buying one call at one price and selling one call at another,
you would buy m calls at one price and sell n calls at another. More on this in
the next module.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M10-12
Module 10 - Review of Derivatives Markets, Chapter 3
Box Spread
The intention here is to create a mix of options that pays off like a zero-coupon
bond.
Box spread strategy Use options to create two synthetic forwards, one for a
forward purchase at a lower price and the other for a forward sale at a higher
price. The book example is
1. Create a synthetic forward purchase at 40 in three months. Buy a call
with a strike of 40 for 2.78 and sell a put with a price of 40 for 1.99. The
total cost is 2.78-1.99=0.79.
2. Create a synthetic forward sale at 45 in three months. Sell a call with a
strike of 45 for 0.97 and buy a put with a price of 45 for 5.08 . The total
cost is 5.08-.97 = 4.11
Box spread results. At time 0 you have a total cost of 4.11 + .79 = 4.90. In three
months you can buy for 40 and sell for 45, giving a certain profit of 5.00. So you
are investing 4.90 to get 5.00 in three months.
Cash -4.90 5.00
Time 0 12 3
This looks like a zero coupon bond. The author points out that if you bought a 3
month zero coupon bond paying 5 in 3 months at your annual interest rate of
8.33%, the price would be
- = 4.90
1.0833025
Thus the box spread built a zero coupon bond out of puts and calls.
Why do this? This has been done in an attempt to lower taxes. Read the box on
page 74 of the text for discussion of the use of box spreads to create risk free
capital gains to offset capital losses for tax purposes. Laws have been passed to
eliminate this loophole, but those laws are hard to enforce in practice.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 10 - Review of Derivatives Markets, Chapter 3
Page Ml 0-13
Collars
We have already seen that if you sell a call option for the S&R index with strike
price K and buy a put with the same strike and expiration, you have created a
synthetic forward sale for the index at a forward price of K. A purchased
collar results if you make a slight variation and sell a call option for the S&R
index with strike price K and buy a put with a lower strike and the same
expiration.
Purchase Collar Strategy. Buy a put option and sell a call with the same
expiration and a higher price. [If you reverse the process to sell a put option
and buy a call with the same expiration and a higher price, you have a written
collar.]
The example in the text is obtained by buying a 3-month put with a strike of 40
and selling a three month call with a strike of 45. Note that if the call also had a
strike of 40 you would have a synthetic forward sale for 40. The collar strategy
separates the two option strike prices. The difference in strikes is called the
collar width.
$25
$20
$15
$10
$5
$0
-$5
-$10
-$15
-$20
20.00
Profit for collar
25.00 30.00 35.00 40.00 45.00 50.00
55.00
60.00
Note that the profit graph for the collar looks very much like the profit graph
for a short forward, but has a flat spot between 40 and 45.
Why do this? The collar can be used to hedge the price of a purchased asset,
since it is similar to a short forward sale.
©ACTEX 2009
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Exam FM / Exam 2 - Financial Mathematics

Page M10-14
Module 10 - Review of Derivatives Markets, Chapter 3
Use of collar for hedging the price of a purchased asset
Strategy. Borrow to buy the asset and use the collar to stabilize profit. In the
text Table 3.6 shows the profit from borrowing to buy the stock at 40 and
hedging the profit with a collar resulting from a 3-month purchased put with
strike of 40 and a 3-month written call with a strike of 45.
Net Profit
4.00 -r
3.00 4
2.00 1
1.00 |
0.00 J
-1.00 I
-2.00 4
-3.00 J-
20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00 65.00 70.00
The profit is identical to the profit of a bull spread.
Why do this? The asset by itself can have large losses or gains from the original
price of 40. The combined position of asset and covered call has a maximum
loss of 1.85 and a maximum gain of 3.15. Thus the owner of the asset has hedged
a portion of his price risk.
i
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 10 - Review of Derivatives Markets, Chapter 3
Page M10-15
Zero-Cost Collars
Note that the cost of a collar made up of a purchased 40-strike put and a written
K-strike call is:
Call(K,T) - Put(40,T).
You could make this cost zero by solving for K in the equation
Call(KyT) = Put(40,T)
This turns out to be simple to do with a Black-Scholes spreadsheet model and
the EXCEL Solver. The strike price that gives a collar with zero-cost is
K = 41.72. In course FM level work you will need to be given the value of 41.72
without derivation. In Exam M level work you will learn how to derive this.
Zero cost collar strategy. This is the usual collar strategy, with the additional
step of choosing strike prices so that the cost is zero.
This is the standard collar graph for the strike prices in the deal.
Why do this? To create a hedging tool with zero initial cost. We will look at the
result of the hedge next.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M10-16
Module 10 - Review of Derivatives Markets, Chapter 3
Use of a zero cost collar for hedging the price of a purchased asset.
Strategy. Borrow to buy the asset and use a zero cost collar to stabilize profit.
In Figure 3.9, the text shows the result of a zero cost collar on XYZ with the
zero cost collar using strike prices of 40 and 41.79. The text does not give the
table for this, so we will display that as well as the graph of the profit for the
hedged position.
Stock Price Asset Profit
1 20.00
22.50
25.00
27.50
30.00
32.50
35.00
37.50
40.00
41.72
45.00
47.50
50.00
52.50
55.00
57.50
60.00
-20.8082
-18.3082
-15.8082
-13.3082
-10.8082
-8.30818
-5.80818
-3.30818
-0.80818
0.911825
4.191825
6.691825
9.191825
11.69182
14.19182
16.69182
19.19182
Option ProfitNet Profit
20.00
17.50
15.00
12.50
10.00
7.50
5.00
2.50
0.00
0.00
-3.28
-5.78
-8.28
-10.78
-13.28
-15.78
-18.28
-0.81
-0.81
-0.81
-0.81
-0.81
-0.81
-0.81
-0.81
-0.81
0.91
0.91
0.91
0.91
0.91
0.91
0.91
0.91 |
Net Profit
1.00
0.80
0.60
0.40
0.20 1
0.00
-0.20
-0.40
-0.60
-0.80 «
-1.00
20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00 65.00 70.00
Note that the gain and loss is much more effectively restricted than with the
first collar hedge that we looked at.
Why do this? This is often done by corporate executives who wish to hedge the
price risk of stock that they own in their own companies.
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 10 - Review of Derivatives Markets, Chapter 3
Page M10-17
Additional discussion of zero cost collars. Note that the zero-cost collar does
not eliminate all price risk. If it did, the net profit function above would be
identically 0. In fact, the profit analysis assumes that the money to buy the
stock is borrowed. Thus there is an interest payment of 0.81 (interest at 8.33%
on 40 for 3 months) in the above calculation. The author discusses this same
interest rate cost on page 77.
We have already observed the put price and call price are identical for options
with the same expiration and strike price K = F0,t . If we buy a put and write a
call at this price we have constructed a zero-cost collar that is actually a
forward sale at the price K. This zero-cost collar is less likely to be used to
hedge, since corporate executives do not want to be seen selling their stock.
Using Options to Speculate on Volatility
Bull spreads and collared stock positions profit only when the stock price
increases. Bear spreads profit only when the stock price decreases. If you want
to speculate on high volatility, you would like to profit when there is either a
large increase or a large decrease. You could do that with straddles or
strangles, which are reviewed next. Writing straddles or strangles could also be
used to speculate on low volatility.
Purchased Straddle
Strategy Buy a put and a call with the same strike price. The strike would
probably be the current price of the stock. In our graph below we look at the
straddle for XYZ with a 40-strike. Note that the cost of the straddle is 4.77,
since you pay 2.78 for the call and 1.99 for the put.
Profit
$20 -j j
.$10 -I — —-—— —— 1
20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00
Why do this? You think that the stock value will make a large move either
above 45 or below 35. The graph is misleading: the profit at 35 and 45 looks like
zero, but it is really 0.13.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M10-18
Module 10 - Review of Derivatives Markets, Chapter 3
Purchased Strangle
Strategy Lower the cost of buying the options by purchasing a put with a lower
price and a call with a higher price. The text gives the example of buying a 35-
strike put and a 45-strike call. The total cost of 1.41 is much less than the
straddle cost of 4.77.
The text shows the straddle and strangle graphs superimposed on page 80. The
strangle has lower losses if volatility is low, but makes less profit when
volatility is high.
Why do this? You think that the stock value will make a large move in either
direction but want to pay less to make the bet.
Written Straddle
Strategy Sell a put and a call with the same strike price. The strike would
probably be the current price of the stock. In our graph below we look at the
straddle for XYZ with a 40-strike. Note that the premium income from the
straddle is 4.77, since you are paid 2.78 for the call and 1.99 for the put.
Why do this? You think that the stock value will be relatively stable and will
not make a large move either above 45 or below 35.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 10 - Review of Derivatives Markets, Chapter 3
Page M10-19
Written Strangle
Strategy Sell a put with a lower price and a call with a higher price. The text
gives the example of selling a 35-strike put and a 45-strike call. The total
premium income of 1.41 is much less than the written straddle income of 4.77.
However the strangle is profitable over a wider range than the straddle.
Why do this? You think that the stock value will be relatively stable and will
not make a large move.
Butterfly Spreads
If you look back at the written straddle and strangle on the previous pages, you
will see that there can be large losses if there is a drastic increase or decrease
in XYZ price. The butterfly spread takes a written straddle and adds to it a
purchased strangle that covers some of the risk.
Strategy Write a straddle and purchase a strangle. The text gives the example
of writing a straddle with a 40-strike and then buying a strangle consisting of a
purchased put with a 35-strike and a purchased call with a 45 strike.
The butterfly spread limits losses at high volatility and has lower profit with
low volatility. In Figure 3.14, the text superimposes the written straddle and
corresponding butterfly spread to show this.
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page M10-20
Module 10 - Review of Derivatives Markets, Chapter 3
Asymmetric butterfly spread
The previous butterfly spread was symmetric, with maximum profit occurring
at a price of 40, the midpoint of the interval (35,45). The text gives an example
of how to create a butterfly spread that peaks at 43 instead of 40 by selling 10
calls with a 43-strike, and buying two 35-calls and eight 45 strike calls. (The text
has to tell you that the premium for a 43 strike call is 1.525.) The profit graph
for this asymmetric butterfly is displayed on page 83.
There is a formula on page 83 that can be used to get the numbers 10, 2 and 8
used above. It is better to use the intuitive reasoning used on page 82. The
number 43 is 80% of the way from 35 to 45. To get a peak at 43, for every 10
calls written at strike-43 buy .80(10) = 8 calls at strike-45 and .20(10) = 2calls at
35.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 10 - Review of Derivatives Markets, Chapter 3 Page M10-21
Section 10.5
The Marshall & llsley Corp. Equity Linked Note
A convertible bond is a bond which can, in some circumstances, be redeemed at
maturity in stock instead of cash. If the bond must be redeemed in stock it is
mandatorily convertible.
This section discusses a manditorily convertible bond issued by Marshall &
llsley Corp in July 2004 with maturity of August, 2007. The text refers to the
issuer using its ticker symbol MI and refers to the MI stock price at maturity
as Smi-
Face Value of the bond.
The bond was designed to sell for $25 at issue. Since the stock share price was
37.32 on the day of issue, the $25 was the price of 25/37.32 = .6699 shares of MI
stock. The bond investor could have purchased .6699 shares of MI stock instead
of the bond.
Bond Coupon.
The coupon was 6.5% of 25, or 1.625. Note that this is higher than the dividend
on the stock, which had been running at 2%. At a price of 37.32, the dividend on
.6699 shares of the stock would be .02(37.32)06699) = .50.
Bond payment at maturity.
The purchaser of the bond got a specified number of shares at maturity. That
number of shares depended on the stock price at maturity.
MI price at maturity Number of shares paid in redemption
Price down from 37.32, SMi < 37.32
Price in range 37.32-46.28,
37.32 <SMI< 46.28
Price above 46.28, SMi > 46.28
Original proportion of .6699 shares
$25/SMi, or $25 worth of shares
Lower proportion of .5402 shares
In figure 3.16, the text shows a graph comparing this to the payoff from just
holding 0.6699 shares of MI. The bond payoff is the same as the payoff on
0.6699 shares for prices below 37.32, but then is lower at higher price ranges.
However the bond payoff mimics the payoff on 0.6699 shares of the stock fairly
well without behaving exactly like the stock.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M10-22
Module 10 - Review of Derivatives Markets, Chapter 3
The bondholder can still profit from appreciation in the stock. The bondholder
really gets:
1) A payoff that increases with gains in the stock, but pays off slightly less
that the stock at MI prices above 37.32.
2) A coupon that is higher than the stock dividend. This higher coupon
compensates for the lower payout.
In the end the bondholder gets a derivative that looks a lot like a stock
investment but is not the same as a stock investment.
Why do this? McDonald notes that the MI deal will be discussed further in
Chapter 15. Since Chapter 15 is not required for either Exam FM or M, we will
give you the highlights now:
On page 494 McDonald notes that: "Under US tax law, interest payments
on corporate debt are tax-deductible, while dividends on equity are not"
and "In practice it is possible to design financing vehicles that have a
significant equity component, yet for which the payments are at least
partially tax-deductible for the firm." Marshall & Ilsley Corp. was doing
what the last sentence describes and looking for a tax deduction.
The text notes that the graph in Figure 3.16 looks somewhat like a written
collar. A written collar consists of selling a put option and buying a call with at
higher strike. Thus it is reasonable to look for a way to model the MI bond
payoff using puts and calls. This is not easy for the beginner to do, but the text
gives you the answer. It is useful to remember when reading that answer that
the stock percentage when the SMi > 46.28 is 0.5402 = — .
Using that information you can check the formula on page 85 which states that
the bond redemption value payoff is given by
Payoff =0.6699
SMi -max(0,SM/ -37.32) + (max(0,SM/ -46.28))
1.24
This really says that you can get the same payoff by owning 0.6699 shares of
the stock, selling 0.6699 calls with a 37.32 strike and buying 0.5402 calls with a
46.28 strike. That is easier to see if you rewrite the above as
.6699SM/ -.6699max(0,Sm/ -37.32) + .5402(max(0,SM/ -46.28))
This takes some thought to work through, but has a nice consequence. Since
puts and calls can be priced using Black-Scholes and parity, we can use those
methods to attach a fair price to the stock payoff on the bond.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 10 - Review of Derivatives Markets, Chapter 3
Page M1053
Section 10.6
Module 10 summary
Floor at price K: Buy index and put with strike K.
Payoff[Index+Put with strike K] = Payoff[Call with Strike K+Zero-Coupon Bond for K]
Profit[Index + Put with strike K] = Profit[Call with Strike K].
Cap at price K: Sell the index short and buy a call with strike K.
Payoff [Short Index+Call with strike K]
=Payoff[Put with Strike K+Sale of Zero-Coupon Bond for K]
Profit[Short Index+Call with strike K] = Profit[Put with Strike K]
A covered call is the opposite side of a cap.
Combination
Cap
Covered Call
Strategy
Sell Index & Buy Call
Buy Index and Sell Call
Profit equivalent
Purchased Put
Written Put
A covered put is the opposite side of a floor.
Combination
Floor
Covered Put
Strategy
Buy Index & Buy Put
Sell Index and Sell Put
Profit equivalent
Purchased Call
Written Call
Synthetic forward at price K:
Buy a call option with strike price K and sell a put with the same strike
and expiration.
Net cost of synthetic forward: Call(K, T) - Put(K, T).
Put-call parity. PV(F0,T) = Call(K,T) - Put(K,T) + PV(K)
Or PV(F0,t)- PV(K) = PV(F0lT -K)= Call(K,T) - Put(K,T)
F0,t > K -> Call{K,T) > Put(K,T)
Fo,r = K -»• Call(K,T) = Put(K,T)
FolT<K -» Call(K,T) < Put(K,T)
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page M10-24 Module 10 - Review of Derivatives Markets, Chapter 3
Review of Insurance Strategies 3.1-3.2 of text
[Name
Floor
Cap
Covered Call
Covered Put
Strategy
Buy index at S0 and buy put with X=S0
Short index at S0 and buy call with
X=S0
Own index at S0 and sell call with X=S0
Short index at S0 and write put with
X=S0
Synthetic Forward |Buy call at S0 and sell put with X=S0
Profit Equivalent
Buy Call with X=S0
Buy Put with X=S0
Sell Put with X=S0
Sell Call with X=S0
Forward for X with
premium
Review of option strategies. 3.3-3.4 of text.
[Name
Bull Spread
Bear Spread
Box Spread
Ratio Spread
Collar
Straddle
Strangle
Write Straddle
Write Strangle
Butterfly
Strategy
Buy call at lower price and sell call at
higher
Write call at lower price and buy call
at higher
Buy synthetic forward at lower price
and sell synthetic forward at higher
Buy m calls at one strike and sell n at
another.
Buy a put and sell a call at a higher
price.
Buy a call and a put with same strike
Buy a put with a lower exercise price
and a call with a higher exercise price.
Sell a call and a put with same strike
Sell an out of the money put and an in
the money call
Write straddle and buy strangle
Comment
Speculation on increase in a
range
Speculation on decrease in a
range
Bond-like investment
intended to look like a
capital gain and offset
capital losses.
Pay later strategies. See ch.
4
Zero cost collars are used to
hedge a stock position
Bet on high volatility
Bet on high volatility with
lower cost
Bet on low volatility
Bet on low volatility
Bet on low volatility with
lower cost |
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 10 - Review of Derivatives Markets, Chapter 3 Page M10-25
Section 10.7
Solutions to Odd-Numbered Problems
Borrow 980.39
Buy Index 1000.00
Payoff = Index value + put payoff
Strike Price 1000
S&R Index Put Payoff Index Payoff Repay Loan FV(OP) Profit
900
950
1000
1050
1100
1150
1200
100
50
0
0
0
0
0
1000
1000
1000
1050
1100
1150
1200
-1000.00
-1000.00
-1000.00
-1000.00
-1000.00
-1000.00
-1000.00
-95.69
-95.69
-95.69
-95.69
-95.69
-95.69
-95.69
-95.69
-95.69
-95.69
-45.69
4.31
54.31
104.31
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby
Out of Pocket (OP)
Buy Index 19.61
Buy Put 74.20
Total OP 93.81

Page M10-26
Module 10 - Review of Derivatives Markets, Chapter 3
[Buy index for 1000 and 950 put
Cost
Put Premium 51.777
Buy Index 1000.00
Total 1051.78
S Put Payoff FV(Cost) Option Profit
800
850
900
950
1 1000
1050
1100
1150
1200
150
100
50
0
0
0
0
0
0
950
950
950
950
1000
1050
1100
1150
1200
1072.81
1072.81
1072.81
1072.81
1072.81
1072.81
1072.81
1072.81
1072.81
-122.81
-122.81
-122.81
-122.81 1
-72.81
-22.81
27.19
77.19
127.19
[invest 931.37, buy index and buy a 950 call
Cost
Buy Call 120.41
Invest 931.37
Total 1051.78
The investment is needed for equal payoffs. It has 0 profit.
Profit is equal to call profit
Strike Price 950
S Call Payoff FV(Cost) Profit
900
950
1000
1050
1100
1150
1200
0
0
50
100
150
200
250
950
950
1000
1050
1100
1150
1200
1072.81
1072.81
1072.81
1072.81
1072.81
1072.81
1072.81
-122.81
-122.81
-72.81
-22.81
27.19
77.19
127.19
Note that put-call parity is involved here.
PV (F0,r) + Put(K, T) = Call(K, T) + PV(K)
1000 + 51.78 = 120.41 + 931.37
= 1051.78
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 10 - Review of Derivatives Markets, Chapter 3
Page M1057
|Short for 1000 and buy 1050 call Call Premium 71.802
S&R Index Call Payoff Cost Profit
900
950
1000
1050
1100
1150
1200
0
0
0
0
50
100
150
-900
-950
-1000
-1050
-1050
-1050
-1050
-946.76
-946.76
-946.76
-946.76
. -946.76
-946.76
-946.76
46.76
-3.24
-53.24
-103.24
-103.24
-103.24
-103.24
[Borrow 1029.41 Buy 1050 put 101.214
S&R Index Put Payoff Cost Profit
900
950
1000
1050
1100
1150
1200
150
100
50
0
0
0
0
-900.00
-950.00
-1000.00
-1050.00
-1050.00
-1050.00
-1050.00
-946.76
-946.76
-946.76
-946.76
-946.76
-946.76
-946.76
46.76
-3.24
-53.24
-103.24
-103.24
-103.24
-103.24
Note that this problem does not ask for a spreadsheet table or a graph. In
fact, the solution to 3.6 in the solutions manual is analytic instead of table or
graph oriented. We will proceed in the same fashion here.
a) Short the S&R index for 1000.
Pavoff = -ST to replace the stock.
Profit = Payoff + 1000 cash with interest = -ST +1020
b) Sell a 1050 strike S&R call, buy a 1050-strike put and borrow 1029.41.
Note that the repayment on the loan is 1029.41(1.02) = 1050
Pavoff = -max(ST -1050,0) + max(1050-ST,0)-1050 = -ST
Note that at time 0 you receive the amount of 71.802 - 101.214 +
(1029.41) = 1000. This is regarded as a negative expense. Thus
FV(expense) = -1020.
Profit = Payoff - FV(expense) = -ST +1020
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M10-28
Module 10 - Review of Derivatives Markets, Chapter 3
3.9
Even though the text asks for a table, we will look at it analytically.
A) Buy a 950 call and sell a 1000 call. (This is a bull spread.)
(0, ST < 950
PayoffA = max(Sr - 950,0) - max(ST -1000,0) = j ST - 950, 950 <ST< 1000
[50, ST > 1000
Premium = 120.405 - 93.809 = 26.596 FV(Premium) = 27.13
ProfitA = PayoffA -27.13
B) Buy a 950 put and sell a 1000 put.
f-50, Sr<950
Payof fB=max(950-Sr,0)-max(1000- ST, 0) = <
ST -1000, 950 <Sr< 1000
0, ST >1000
= PayoffA - 50
Premium = 74.201-51.777 = 22.434 FV(Premium) = 22.87
ProfitB = PayoffB + 22.87 = PayoffA - 50 + 22.87 = PayoffA - 27.13
= ProfitA
In words, the payoff for B is 50 less than the payoff for A, but for A you pay
premium and for B you earn premium. The relation between the premiums
causes the profits to be the same.
Note that 3.10 is a very similar problem, and the solution given in the
manual goes through the problem graphically.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 10 - Review of Derivatives Markets, Chapter 3
Page Ml0-29
3.11
This is a stock position hedged with a collar.
Invest 1000 Buy Index
Buy Put 950 Premium 51.777
Sell Call 1050 Premium 71.802
Net -20.025 Negative cost (gain)
60 -1 .
AC\ -
nr\ .
Profit
o c
-90 -
-40 -
<
-fin -
♦
—♦—
♦
>
800
♦
850
♦
900
950
1000
S&R Index
1050
1100
1150
1200
You would need to raise the call strike (thus lowering the price) to get a zero
cost collar. The next problem shows that you very nearly get a zero cost
collar with a call strike of 1107.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M10-30
Module 10 - Review of Derivatives Markets, Chapter 3
3.13
a)
$100.00
-$200.00
Profit
800 850 900 950 1000 1050 1100 1150 1200 1250 1300
b)
$200.00
$150.(
$100.00
-$50.(
-$100.00
-$150.00
-$200.00
Profit
^
800 850 900 950 1000 1050 1100 1150 1200 1250 1300
C)
$150.00
$100,00 4
$50.00
$0.00
-$50.00
-$100.00
-$150.00
Profit
800 850 900 950 1000 1050 1100 1150 1200 1250 1300
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 10 - Review of Derivatives Markets, Chapter 3
Page M1031
c) For 0 premium we would need n (120.405) = m (71.802).
rp. n 71.802 endZO
Thus — = = .5963
m 120.405
3.17
The peak value will be at a strike of 1020.1020 is 70% of the way from 950 to
1050. Write ten strike-1020 calls and buy seven strike-1050 calls and three
strike-950 calls.
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page M10-32
Module 10 - Review of Derivatives Markets, Chapter 3
Section 10.8
Module 10 Computational Review Problems
1. (1 pt) Suppose the premium on a 6-month S-R call is $
110 and the premium on a put with the same strike price is $
57.6. Assuming that the effective 6-month interest rate is 2 %,
and that today's price for the S-R index is $ 1,000, what is the
strike price ?
ANSWER: 966.55
2. (1 pt) For the following problem assume the effective 6-
month interest rate is 2 %, the S-T 6-month forward price is $
1020, and use the premiums listed below for S-T options with 6
months to expiration.
Strike
950
Call
120.405
Put
51.777
1) Suppose you buy the S-T index for $ 1000 and buy a 950-
strike put. Determine the profit for the following S-T index spot
prices at expiry.
When price is $ 950, the profit is $ ?
When price is $ 1000, the profit is $ ?
When price is $ 1050, the profit is $ ?
2) Suppose you buy a 950-strike call and invest $ 931.37 in
zero-coupon bonds. Determine the profit for the following S-T
index spot prices at expiry.
When price is $ 950, the profit is $ ?
When price is $ 1000, the profit is $ ?
When price is $ 1050, the profit is $ ?
ANSWER1: -122.81
ANSWER2: -72.81
ANSWER3: -22.81
ANSWER4: -122.81
ANSWER5: -72.81
ANSWER6: -22.81
3. (1 pt) For the following problem assume the effective 6-
month interest rate is 2 %, the S-T 6-month forward price is $
1020, and use the premiums listed below for S-T options with 6
months to expiration.
Strike
950
Call
120.405
Put
51.777
1) Suppose you short the S-T index for $ 1000 and buy a
950-strike call. Determine the profit for the following S-T index
spot prices at expiry.
When price is $ 950, the profit is $ ?
When price is $ 1000, the profit is $ ?
When price is $ 1050, the profit is $ ?
2) Suppose you buy a 950-strike put and borrow $ 931.37.
Determine the profit for the following S-T index spot prices at
expiry.
When price is $ 950, the profit is $ ?
When price is $ 1000, the profit is $ ?
When price is $ 1050, the profit is $ ?
ANSWER1: -52.81
ANSWER2: -52.81
ANSWER3: -22.81
ANSWER4: -52.81
ANSWER5: -52.81
ANSWER6: -52.81
4. (1 pt) For the following problem assume the effective 6-
month interest rate is 2 %, the S-T 6-month forward price is $
1020, and use the premiums listed below for S-T options with 6
months to expiration.
Strike
950
1050
Call
120.405
71.802
Put
51.777
101.214
1) Suppose you buy a 1050-strike S-T straddle, determine the
profit for the following S-T index spot prices at expiry.
When price is $ 950, the profit is $ ?
When price is $ 1000, the profit is $ ?
When price is $ 1050, the profit is $ ?
2) Suppose you write a 950-strike S-T straddle, determine
the profit for the following S-T index spot prices at expiry.
When price is $ 950, the profit is $ ?
When price is $ 1000, the profit is $ ?
When price is $ 1050, the profit is $ ?
3) Suppose you simultaneously buy a 1050-strike S-T straddle
and write a 950-strike S-T straddle, determine the profit for the
following S-T index spot prices at expiry.
When price is $ 950, the profit is $ ?
When price is $ 1000, the profit is $ ?
When price is $ 1050, the profit is $ ?
ANSWER1: -76.476
ANSWER2
ANSWER3:
ANSWER4;
ANSWER5
ANSWER6:
ANSWER7:
ANSWER8
ANSWER9:
-126.476
-176.476
175.626
125.26
75.26
99.1493
-0.85068
-100.851
5. (1 pt) For the following problem assume the effective 6-
month interest rate is 2 %, the S-T 6-month forward price is $
1020, and use the premiums listed below for S-T options with 6
months to expiration.
Strike
950
1107
Call
120.405
51.873
Put
51.777
137.167
Suppose you buy the S-T index for $ 1000 and buy a 950-
strike put, and sell a 1107-strike call. Determine the profit for
this position at the following S-T index spot prices at expiry.
When price is $ 950, the profit is $ ?
When price is $ 1000, the profit is $ ?
When price is $ 1050, the profit is $ ?
ANSWER1: -69.9021
ANSWER2: -19.9021
ANSWER3: 30.0979
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 10 - Review of Derivatives Markets, Chapter 3
Page M10- 33
Section 10.9
Supplemental Exercises
Use the following option prices for the S&R index in problems 1- 8. All options
have an expiration time of T = .25. The current value of the index is S0 = 1000,
and the index has dividend yield 0.
! Strike K
975
1000
1025
Call Price
77.716
64.595
53.115
Put Price
43.015
67.916
1. Find the continuously compounded annual interest rate r.
A) 1.98% B) 2% C) 3.96% D) 4% E) 7.93%
2. Find the put price for K = 1000.
A) 54.645 B) 55.466 C) 56.912 D) 57.254 E) 57.893
3. What is the cost at time 0 for a long forward contract with forward price
975?
A) 35.050 B) 34.701 C) 30.76 D) 20
E)0
4. Investor A buys the index at time 0 and sells a 1025 strike call with
T = .25. Investor B writes a 1025 strike put and lends x. The two
investors have the same payout functions. What is x?
A) 1000 B) 1007.40 C) 1014.80 D) 1025 E) 1037.40
5. Investor C buys the index at time 0 and buys a 975 put with T = .25.
What is his minimum profit (loss)?
A) -18.015 B) -43.015 C) -57.64 D) -78.50
E) There is no minimum
6. Investor D buys a 975-strike call and sells a 1025-strike call. What is his
maximum profit?
A) 24.45 B) 25
C) 25.15
D) 50 E) There is no maximum
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M10-34
Module 10 - Review of Derivatives Markets, Chapter 3
7. Investor E buys the index, buys a 975-strike put and sells a 1025-strike
call. What is his maximum profit?
A) 24.85 B) 25 C) 25.1 5 D) 50 E) There is no maximum
8. Investor F buys a 975-strike put and a 975-strike call. What is his
maximum profit?
A) 120.73 B) 121.94 C) 132.18 D) 150
E) There is no maximum
9. Near market closing time on a given day, the European call and put
prices for a stock are available as follows:
Strike Price
40
50
55
Call Price
11
6
3
Put Price
3
8
11
The options have expiration time T = .5. The continuously compounded
annual interest rate is r = .04. Mary constructs the following portfolio:
Long one call option with strike price 40; short three call options with
strike price 50; lend $1; and long some calls with strike price 55. The
dollar she lends is obtained from the sale and purchase of the options.
What is her profit at T = .5 if the price of the stock is 52 at that time?
A)l B)1.02 C)2 D)2.02 E) 7.02
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 10 - Review of Derivatives Markets, Chapter 3 Page M10- 35
Section 10.10
Supplemental Exercise Solutions
1) By put-call parity
C-P = S0-Ke-TT
77.716 - 43.05 = 1000 - 975e"r( 25) -» r = .04
Answer D (Note that this rate will be used in questions 2-8).
2) By put-call parity
C-P = So-Ke"rT
64.595 - P = 1000 - lOOOe"04( 25) -» P = 54.645
Answer A
3) This is the price of a synthetic long forward constructed by buying a call
and selling a put, each with strike K = 975. The cost is
C-P = 77.716 - 43.015 = 34.701
Answer B
4) Buying the index and selling a call creates a covered call. You obtain the
same payoff function if you write a put for the same exercise price K
and lend the present value of K.
Thus the amount loaned here must be Ke~rT = 1025e"01 = 1014.80
Answer C
5) Buying the index and buying a put with strike 975 creates a floor. The
floor has the same profit function as a long call with strike 975. The
minimum profit on the floor is the (negative) loss of the future value of
the call premium when the call expires unexercised.
-77.716e01 = -78.50
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M10-36
Module 10 - Review of Derivatives Markets, Chapter 3
6) This is a bull spread which attains its maximum when S = 1025. At that
point the profit is 50 + 53.115e01 - 0 - 77.716e01 = 25.15
Answer C
7) In this case the investor has purchased the index and a collar with
strikes of 975 and 1025. The combination of index and collar has a graph
similar to that of a bull spread, with maximum profit at S = 1025.
The cost of the index at time 0 is 1000, and the collar gives a positive
cash flow of 53.115-43.015 = 10.10 at time O.The profit at time .25 is
Index profit + Call Profit + Put Profit
= 1025 - lOOOe01 + 0 + 53.115e01 + 0 - 43.015e01 = 25.15
Answer C
8) This is a purchased straddle. There is no maximum.
Answer E
9) For Mary's portfolio the number of long calls at K = 55 is not given.
However you can quickly figure out what it is. The arbitrage lends $1, so
in order to have 0 outlay at the beginning there must be $1 of excess cash
obtained from the sale and purchase of calls. It there are n long calls at
K = 55 we have the following proceeds from options.
Strike
Position
Proceeds
40
Long 1
-11
50
Short 3
+6(3)
55
Long n
-3n
Since total proceeds are 1 to lend, we have -ll + 18-3n = l-*n = 2
Mary has no out-of-pocket cost at time 0. She earns $1 and invests it at
the continuous rate r = .04. Her profit at time .5 is the future value of the
invested $1 + the sum of the payoffs of the options in the portfolio.
le02 + (52 - 40) - 3 (52 - 50) + 2 (0) = 7.02
Answer E
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 11 - Review of Derivatives Markets, Chapter 4
Page Mil- 1
Section 11.1
Using Derivatives to Manage Risk
In Module 8, we gave the simple example of a farmer using a forward
agreement to guarantee a future price for his corn by entering into an
agreement with a cereal company that wanted to get corn at a set price in the
future. Both the farmer and the cereal company were managing their price risk
using a derivative.
Now that we have studied puts and calls and their various combinations, we can
see many other ways to manage price risk. In Chapter 4 of Derivatives Markets,
the author applies the various derivative strategies we have learned to two
fictional companies. The first, Golddiggers, is a gold-mining firm that has
pricing risk when it sells. The second, Auric, is a manufacturer of golden
widgets and thus has price risk when it buys gold to make widgets. The
fictional companies are very much simplified, but this gives us a chance to
focus clearly and directly on the analytics of risk management.
In this module, you will find the tables and graphs used for analysis to be
similar to those in the last chapter. The only thing that is new in the analysis is
that the tables go beyond the profit analysis of the derivatives to include a final
column that shows the overall profit of the company after derivatives are used.
In this chapter we will concentrate on overview, not on the basic computations
that are in the text.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Mll-2
Module 11 - Review of Derivatives Markets, Chapter 4
Section 11.2
Using Derivatives to Manage the Risk of a Producer-Seller
The text first studies Golddiggers, the humorously-named gold-mining firm.
Golddiggers will mine and sell 100,000 ounces of gold over the next year. The
firm has a fixed cost of $330 per ounce and a variable cost of $50 per ounce, for
a total cost of $380 per ounce. Note that the fixed cost covers all the needs of
the business that would be there whether or not gold is mined and sold -the cost
of land, buildings, equipment, administrative salaries and similar items. The
only way to avoid fixed costs is to shut down the business.
There is a point related to fixed and variable cost that you might find
confusing. In a footnote, the text says that if the price of gold is above $50 per
ounce, the company will still mine it even though it loses money. For example,
if the price is $55, the firm can still earn 5 more than the variable cost of $50,
and use the $5 to offset a small part of the fixed cost. There would still be a loss
of $325 per ounce, but that is better than the loss of $330 that would occur if no
gold were mined.
The text states that Golddiggers would like to manage its price risk. The price
of gold today is $405 per ounce. Golddiggers' total cost of production is $380 per
ounce. However the gold will be sold in one year, not today. If Si is the spot
price in one year and Golddiggers does not hedge, the net income per ounce of
gold will be: Unhedged Net Income = Si - 380.
However, Golddiggers can hedge. They can enter a forward contract for sale at
$420 in one year or buy a $420-strike put for $8.77 per ounce. The tables and
graphs in the text derive the firm's profit using each of these alternatives. Our
next graph reviews the final profit results for no hedge, a forward contract
hedge and a hedge with a put.
200 -
150 -
100 -
I * 50 |
s I
1 ft- 0
-50
-100
_i^n -
2e
Profit for Golddiggers with and without hedges
A
i=-
—^m^,„—^m-^^^^^fe^^—^^^~~^^~^~~~.~,~%
50
300 350 400 450 500 5£
Gold Price
|
l
• Profit Unhedged 1
«H&~~ With Forward Hedge
—A—With Put Hedge |
>0
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 11 - Review of Derivatives Markets, Chapter 4
Page Mil-3
The results are as expected:
• With no hedge there is substantial risk from a price drop.
• With the forward hedge profit is constant, since Golddiggers always gets
the forward price of $420 with a cost of $380 for a constant profit of $40.
• With the put hedge the cost of the put provides insurance for low prices
and permits higher profits as the spot price increases.
On page 96, the text shows the results of hedging by writing a $420-strike call.
That alternative gives lower profit at most future price levels than buying a
put, and probably would not be considered by Golddiggers.
In actuality, an analyst at Golddiggers would probably show the graph of
alternatives to management to allow them to decide what should be done. The
final selection is based on management preferences for risk, and there is no
single best answer (the text notes this on page 95 in the third paragraph).
The text also makes a point to answer a question that an actuarial student might
ask. Actuaries are trained to make decisions based on probabilities, but we see
no probability in this analysis. However the fact that is hidden from us is that
the pricing of the put option does rely on the probability distribution of the
asset price, since the Black-Scholes model has the standard deviation of the
return on the stock as an input.
The most likely choice would probably be to use the $420-strike put, although
there are even more alternatives. If the $420-strike put is viewed as too
expensive, a manager might want to use a cheaper $400-strike put to manage
price risk. (Remember that it costs less to require someone to buy at a lower
price.). A manager who is very conservative might want to buy a more
expensive $440-strike put to assure a higher profit at low price levels. In Figure
4.5, the text gives a summary graph that compares these alternatives.
180
160
140
120
100
80
60
40
20
0
250
Net Profit for Golddiggers with varying put strikes
-400-strike put
-420-strike put
-440-strike put
300
350
400
450
500
550
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page Ml 1-4
Module 11 - Review of Derivatives Markets, Chapter 4
The final decision about which put price to use will vary with
different managers.
The first paragraph on page 98 requires some review of prior work. In Module
9, we noted that insurance is really a put option for the insured party. Buying
insurance on an asset gives the insured an option whose strike price is:
Value of asset - Amount of deductible.
This put option (insurance) has a lower strike price for a higher deductible, so
it will be cheaper to buy the insurance with a high deductible. Similarly, the
option (insurance) has a higher price for a low deductible. This is not really
new. Most of us are familiar with the fact that a high deductible insurance on
our car is cheaper than a low deductible insurance.
The text makes a special point here. The price of the insurance on your car is
adjusted by the insurance company based on your driver category. You pay
more if you have a bad driving record and can pay less if you have a good one.
Thus in this case, the price of the put option depends partly on who is buying it.
For the Golddiggers, the price of the put option does not depend on who the
buyer is. This point could have been made in Module 9, but it is here instead.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 11 - Review of Derivatives Markets, Chapter 4
Page Mil-5
Section 11.3
Using Derivatives to Manage the Risk of a Producer-Buyer
The next fictional firm in the text is Auric, a manufacturer of gold widgets.
Auric will sell a set number of widgets next year for a fixed price of $800. Their
fixed cost per widget is $340, and they have no variable costs other than the
costs of gold they purchase. If Auric does not hedge, their net income is
Unhedged Net Income = 800 - 340 - Si = 460 - Si
Auric can also hedge, the main choices being a forward contract for purchase
at a price of 420 in one year or the purchase of a 420-strike call with premium
of 8.77. [Note that the $420-strike put for Golddiggers and the $420-strike call
for Auric both have strikes equal to the one year forward price, and thus have
the same premium.]
The alternatives are graphed below for comparison.
Net Income for Auric with various hedging strategies
- Profit Unhedged
- With Forward Hedge
-With Call Hedge
350 400 450
Spot Price
As before, there is no single best answer for Auric. Management will choose
some strategy based on risk tolerance and personal estimates of what the spot
price will be in a year. The discussion for Auric is basically a repetition of the
discussion for Golddiggers.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Ml 1-6
Module 11 - Review of Derivatives Markets, Chapter 4
Section 11.4
Reasons Firms Might Want to Manage Risk
Overview
Chapter 1 of Derivatives Markets gave four basic reasons to manage risk:
• Hedging
• Speculation
• Regulatory arbitrage (including tax avoidance)
• Reduction of transaction costs.
The text notes on page 101, that there is still some question as to why firms
hedge, since public companies are owned by their stockholders and the
stockholders can hedge their own risk. For example, a stockholder could buy
stock in both Golddiggers and Auric, and thus balance his portfolio so that it is
not affected by gold price changes. As Golddiggers loses from a price decrease,
Auric will benefit.
Firms do hedge, so there must be some reason why managers do not leave
hedging up to the stockholders. If the managers of Golddiggers choose the
forward hedge, they always have a profit of $40 per ounce. This means that
they sacrifice profit in good years and avoid loss in bad years: they are willing
to pay to avoid loss. The text says this in a different way, stating "we can
describe the hedging strategy as shifting dollars from more profitable states
(when gold prices are high) to less profitable states (when gold prices are low).
This shifting of dollars from high gold price states to low gold price states will
have value for the firm if the firm values the dollar more in a low gold price
state than in a high gold price state."
However you word this, a basic question remains. Why are managers willing to
sacrifice profits to avoid loss if the stockholders could hedge for themselves?
An Example of the Need to Hedge
Page 102 the text answers our question by giving an example where you can
clearly see why avoiding loss is a priority when tax effects are considered.
In this example, the firm has only two possible future spot prices, $9 and
$11.20, each with probability .5. The firm has manufacturing cost of $10. It
must pay a 40% tax on profits but has no tax deduction for losses. The text
gives a table which summarizes the firm's situation.
Price Price
9.00 11.20
1.00 1.20
0.00 1.20
0.00 0.48
1.00 0.72
Pre-tax operating income
Taxable Income
Tax @ 40%
After Tax Income
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 11 - Review of Derivatives Markets, Chapter 4
Page Mil-7
Note that the tax difference makes pre-tax dollars of loss more painful than
pre-tax dollars of gain. In fact the expected value of the firms after-tax income
is
Expected value of unhedged after-tax income = .5(-l) + .5(72) = -.14
This firm can enter a forward contract to sell for a price of $10.10. In that case
the firm always sells for 10.10, and its after tax income is always 0.06 (See
Table 4.7 of the text.)
Expected value of hedged after-tax income = 0.06
In this case, hedging stabilizes after-tax income and avoids an expected loss, so
it is desirable. That's why a manager might want to hedge in this situation.
Reasons for Hedging
The preceding example is intended to give an example that clearly and simply
outlines a situation where the need to hedge is obvious. Most companies are not
this simple, so the text follows the example with a list of reasons that real
managers hedge.
• Lower taxes. We have already seen an example of this in case of the
Marshall and Illsley bonds. The text notes a number of other ways that
derivatives can be used to shift income for tax purposes.
• Avoid bankruptcy and distress costs. If your company has a large loss,
people may be afraid to buy from it since it might go bankrupt. I have
avoided flying on airlines that were close to bankruptcy out of fear that
my ticket for next month might not take me anywhere.
• Costly external financing. If your company has a big loss, it looks riskier
to banks and other lenders. If they lend to you they will charge a higher
interest rate.
• Protect debt capacity. Debt capacity is the amount that a firm can
borrow. If a loss puts your company in debt, you have used up part of
that amount and can now borrow less.
• Managerial risk aversion. Managers are people, and some people try to
avoid risk as much as possible. They are risk-averse, and some managers
are naturally risk-averse. In addition, many managers have
compensation that is tied to the performance of the company and will be
paid less if there is a loss.
• Non-financial risk management Some problems can be solved by
changing your business to avoid the need for hedging. If you make a
product in the US and sell it in Germany, you have currency exchange
risk. If you build a factory in Germany and produce there you can avoid
some of that risk.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Ml 1-8
Module 11 - Review of Derivatives Markets, Chapter 4
Reasons Not to Hedge
Many firms do not hedge. The text gives a few possible reasons not to hedge.
• There are transaction costs (like commissions).
• The strategy is complex and might require the firm to hire expensive
experts.
• The execution of a hedge involves trading transactions that require
substantial managerial control.
• Accounting and tax become much more complicated when you hedge.
One firm I worked for decided not to hedge because they knew that their
present staff could not handle the accounting.
How much hedging is there?
The text points out that it is hard to know how much hedging really goes on.
Part of the reason for this is a regulatory change. In 2000, the accounting
standard SFAS 133 required that derivatives be recognized as either assets or
liabilities reported at market value. Unfortunately, the reported value doesn't
tell you what is really happening -e.g., forward contracts have zero value.
Research on use of derivatives thus relies more on data from the 1990s when
the reporting standard was more informative. The text reviews a number of
studies on use of hedging, and the list of results is a bit confusing to sort out.
Hov/ever, there is an overview summary on the bottom of page 107. "The varied
evidence suggests that some use of derivatives is common, especially at large
firms, but the evidence is weak that economic theories explain hedging."
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 11 - Review of Derivatives Markets, Chapter 4
Page Mil-9
Section 11.5
More Complex Strategies for Hedging
Hedging with only a forward, put, or only a call is straightforward, but hedges
can also be constructed using collars, zero cost collars, synthetic forwards and
other strategies that we have not seen yet. The text returns to analysis of
Golddiggers in order to illustrate the use of these more complex alternatives.
Hedging with a Collar
A collar is a modified version of a forward sale, so you might expect the result
of hedging company profit with a collar to look a bit like the result of the
forward hedge previously studied here.
The text examines the results for Golddiggers with a collar consisting of a
purchased $420-strike put and a sold $440 strike call. Next, we give the detail
table and graph for that hedged position. (The text does not give the
spreadsheet table for this one.)
Price
300
320
340
360
380
400
420
440
460
480
500
Sale Income
-80.00
-60.00
-40.00
-20.00
0.00
20.00
40.00
60.00
80.00
100.00
120.00
Buy put
110.79
90.79
70.79
50.79
30.79
10.79
-9.21
-9.21
-9.21
-9.21
-9.21
Sell Call
2.61
2.61
2.61
2.61
2.61
2.61
2.61
2.61
-17.39
-37.39
-57.39
Total Profit
33.41
33.41
33.41
33.41
33.41
33.41
33.41
53.41
53.41
53.41
53.41
70.00
60.00
50.00
£ 40.00
£ 30.00
20.00
10.00
0.00
300
Profit with 420-440 collar
320 340 360 380
400
Price
420 440 460 480
500
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page Ml MO
Module 11 - Review of Derivatives Markets, Chapter 4
Recall that the $420 forward hedge gave a constant profit of $40. The collar
gives a lower constant profit of $33.41 for prices lower than $420, and then
increases linearly to a constant profit of $53.41 for prices above $440.
Golddiggers could also create a zero cost collar. The text gives a zero cost
collar with the purchased put at a 400.78 strike and the written call at 440.78.
Note that this collar is wider than the last one. The detail and graph for this
collar are:
Price Sale Income Buy put Sell Call Total Profit
1 300
320
340
360
380
400.78
420
440.78
460
480
500
-80.00
-60.00
-40.00
-20.00
0.00
20.78
40.00
60.78
80.00
100.00
120.00
98.31
78.31
58.31
38.31
18.31
-2.47
-2.47
-2.47
-2.47
-2.47
-2.47
2.47
2.47
2.47
2.47
2.47
2.47
2.47
2.47
-16.75
-36.75
-56.75
20.78
20.78
20.78
20.78
20.78
20.78
40.00
60.78
60.78
60.78
60.78 |
Profit with 400.78-440.78 Collar or Zero Cost Collar
/ u.uu -
fin nn -i
^n nn
.** /nnn .
>rof
D C
D C
20.00 <
m nn -
0.00 -
A A
A k
► ^ ♦
A
A M
| , ( ! ! ! ( J
300 320 340 360
380
400
Price
420 440 460 480
500
The zero cost collar offers less protection at lower price levels but more profit
at higher price levels.
There is no simple single answer as to which collar is best here. As before, the
company reviews its options and management makes a choice based on risk
preference and market outlook.
We have already looked at a $420 forward hedge. The text points out that you
can create a synthetic hedge that is exactly the same (and zero cost) with a
purchased $420-strike put and a written $420-strike call. That was already
established in Chapter 3 of Derivatives Markets , so this is just an application of
something we knew already. The text also discusses applying synthetic
forwards at other prices, and this too was studied in Chapter 3.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 11 - Review of Derivatives Markets, Chapter 4
Page Ml 1-11
The Paylater Strategy
The zero cost collar hedge we saw before was inferior to a put hedge at high
price levels, since the profit on a put hedge increases without limit as prices
increase. However, Golddiggers has to pay the price of a put up front, while the
zero-cost collar requires no advance payment.
This raises the question of whether you can create something that works like a
put at high price levels, but has zero initial cost.
The text demonstrates that this can be done for Golddiggers by establishing a
hedge consisting of one sold 434.6 put and two purchased 420 strike puts. (This
is a ratio spread.) This has 0 cost, since the premium received for the sold put is
17.55 while the premium paid for each purchased put is 8.775, leading to
Total Premium = 17.55 - 2(8.775) = 0.
Clearly, Golddiggers is not paying for this hedge now. It is called paylater for
reasons which will be discussed on the next page. For completeness, we first
give the worksheet table for the Golddigger profit hedged by the paylater.
Price
300
320
340
360
380
400
420
434.6
440
460
480
500
Sale Income
-80.00
-60.00
-40.00
-20.00
0.00
20.00
40.00
54.60
60.00
80.00
100.00
120.00
Sell Put
-134.60
-114.60
-94.60
-74.60
-54.60
-34.60
-14.60
0.00
0.00
0.00
0.00
0.00
Buy 2 puts
240.00
200.00
160.00
120.00
80.00
40.00
0.00
0.00
0.00
0.00
0.00
0.00
Total Profit
25.40
25.40
25.40
25.40
25.40
25.40
25.40
54.60
60.00
80.00
100.00
120.00
You can see from the table that at prices of $434.6 and higher, the paylater
hedge gives the full profit from sale due to its zero cost. Below is the graph that
compares the put hedge result to the paylater result.
The graph follows on the next page.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

PageMll-12
Module 11 - Review of Derivatives Markets, Chapter 4
2
CL
Golddiggers profit with put and paylater
Hedge 420 put
Paylater Hedge
300 320 340 360 380 400 420 440 460 480 500
Price
For the paylater hedge, nothing is paid to start but if there is a price below $420
the paylater hedge provides less protection. This loss of protection is
Golddiggers payment for not paying in advance -it is the paylater amount.
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 11 - Review of Derivatives Markets, Chapter 4
Page Ml 1-13
Section 11.6
Module 11 summary
Basic hedging for seller: Sell forward, buy put.
180 -
160 -
140 -
120 -
100 -
80 -
60 -
40 j
2£
Net Profit for Golddiggers with varying put strikes
J
^J
yy
yy
jTS
*^/
-sW/
I ;—~m in m~/9
\ 4 4 |T
♦ 400-strike put
~~«— 420-strike put
A 440-strike put |
>0 300 350 400 450 500 550
Hedging for buyer: Long Forward, Buy call
Net Income for Auric with various hedging strategies
-150
♦ Profit Unhedged
-HI—With Forward Hedge
—A-With Call Hedge
250 300 350 400 450 500 550
Spot Price
Reasons for hedging
• Lower taxes..
• Avoid bankruptcy and distress costs..
• Costly external financing.
• Protect debt capacity..
• Managerial risk aversion..
• Nonfinancial risk management..
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page Ml 1-14
Module 11 - Review of Derivatives Markets, Chapter 4
Reasons not to hedge
• There are transaction costs (like commissions).
• The strategy is complex and might require the firm to hire expensive
experts.
• The execution of a hedge involves trading transactions that require
substantial managerial control.
• Accounting and tax become much more complicated when you hedge.
One firm I worked for decided not to hedge because they knew that their
present staff could not handle the accounting.
Hedging with a collar for a seller
7n nn
60.00 -
50.00 -
£ 40.00 -
£ 30.00 <
20.00 -
m nn -
n nn -
Profit with 420-440 collar
>
A
W
/
r
▲
w
A
W
300
320
340 360 380 400 420
Price
440
460
480
►
500
Pa/later hedge:
Use zero-cost ratio spread.
Golddiggers profit with put and paylater
140.00
120.00
100.00
80.00
60.00
40.00
20.00 f—-&—&—i-~
0.00
^
^
-Hedge 420 put
Paylater Hedge
30
0
32
0
34
0
36
0
3 40
0
Price
42
0
44
0
46
0
48
0
50
0
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 11 - Review of Derivatives Markets, Chapter 4
Page Ml 1-15
Section 11.7
Solutions to Odd-Numbered Problems
Note to students:
This section includes problems 4.1 through 4.19. Problems 4.21 and beyond are
from a section that is not on the Exam FM syllabus.
4.1.
XYZ unhedged
Price
0.80
0.90
1.00
1.10
1.20
1.30
1.40
Fixed
Cost
0.50
0.50
0.50
0.50
0.50
0.50
0.50
Variable
Cost
0.40
0.40
0.40
0.40
0.40
0.40
0.40
Unhedged
Profit
-0.10
0.00
0.10
0.20
0.30
0.40
0.50
Forward hedge
Price
0.80
1.00
1.20
1.40
1.60
1.80
2.00
Fixed
Cost
0.50
0.50
0.50
0.50
0.50
0.50
0.50
Variable
Cost
0.40
0.40
0.40
0.40
0.40
0.40
0.40
Forward
Sale Price
1.00
1.00
1.00
1.00
1.00
1.00
1.00
Hedged
Profit
0.10
0.10
0.10
0.10
0.10
0.10
0.10
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

PageMll-16
Module 11 - Review of Derivatives Markets, Chapter 4
4.3
Put Strike
Put Cost
A
0.95
0.0178
B
1.00
0.0376
C
1.05
0.0665
Price
0.80
0.90
0.95
1.00
1.05
1.15
1.25
Profit A
0.0311
0.0311
0.0311
0.0811
0.1311
0.2311
0.3311
Profit B
0.0601
0.0601
0.0601
0.0601
0.1101
0.2101
0.3101
Profit C
0.0795
0.0795
0.0795
0.0795
0.0795
0.1795
0.2795
0.80 0.90 1.00 1.10 1.20 1.30 1.40
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 11 - Review of Derivatives Markets, Chapter 4
Page Ml 1-17
4.S
Call
Strike
Call
Premium
Cost
1.0000
0.0376
0.0198
1.0250
0.0274
0.0009
1.0500
0.0194
0.0471
Price
0.700
0.725
0.750
0.775
0.800
0.825
0.850
0.875
0.900
0.925
0.950
0.975
1.000
1.025
1.050
1.075
1.100
profit A
0.0710
0.0710
0.0710
0.0710
0.0710
0.0710
0.0710
0.0710
0.0710
0.0710
0.0710
0.0960
0.1210
0.1210
0.1210
0.1210
0.1210
profit B
0.0760
0.0760
0.0760
0.0760
0.0760
0.0760
0.0760
0.0760
0.0760
0.0760
0.0760
0.0760
0.1010
0.1260
0.1260
0.1260
0.1260
profit C
0.09999
0.09999
0.09999
0.09999
0.09999
0.09999
0.09999
0.09999
0.09999
0.09999
0.09999
0.09999
0.09999
0.09999
0.09999
0.09999
0.09999
0.25
0.00 4
0.20 ^ * A " * A * A A a A A A A * * * ^
0.15
0.10
0.05
0.700
0.800
0.900
1.000
~f—g—f
- profit A
- profit B
- profit C
1.100
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

PageMll-18
Module 11 - Review of Derivatives Markets, Chapter 4
Telco unhedged and hedged profit
Wire Unhedged Forward Hedged
Price Cost Revenue Profit Profit Profit
1 0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
5.70
5.75
5.80
5.85
5.90
5.95
6.00
6.05
6.10
6.15
6.20
6.25
6.30
6.20
6.20
6.20
6.20
6.20
6.20
6.20
6.20
6.20
6.20
6.20
6.20
6.20
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.20
0.20
0.20
0.20
0.20
0.20
0.20
0.20
0.20
0.20
0.20
0.20
0.20 |
Price .95 -strike 1-strike 1.05 -strike
1 0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
0.27
0.27
0.27
0.27
0.27
0.27
0.22
0.17
0.12
0.07
0.02
-0.03
-0.08
0.24
0.24
0.24
0.24
0.24
0.24
0.24
0.19
0.14
0.09
0.04
-0.01
-0.06
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.17
0.12
0.07
0.02
-0.03
«*•—
*
.95-strike
1-strike j
1.05-strike
0.70 0.90 1.10 1.30
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 11 - Review of Derivatives Markets, Chapter 4
Page Ml 1-19
4.11
Paylater profit for Telco
Call Strike
Call Premium
Call Strike
Call Premium
Cost
A
0.9750
0.0500
1.0340
0.0243
0.0014
B
1.0000
0.0376
1.0340
0.0243
0.0023
Price Buy 2, sell 1 Buy 3, sell 2
1 0.700
0.725
0.750
0.775
0.800
0.825
0.850
0.875
0.900
0.925
0.950
0.975
1.000
1.034
1.059
1.084
1.109
1.134
1.159
1.184
1.209
0.5015
0.4765
0.4515
0.4265
0.4015
0.3765
0.3515
0.3265
0.3015
0.2765
0.2515
.0.2265
0.1765
0.1085
0.1085
0.1085
0.1085
0.1085
0.1085
0.1085
0.1085
1 0.5024
0.4774
0.4524
0.4274
0.4024
0.3774
0.3524
0.3274
0.3024
0.2774
0.2524
0.2274
0.2024
0.1004
0.1004
0.1004
0.1004
0.1004
0.1004
0.1004
0.1004 |
—♦— Buy 2, selM
~~*~~ Buy 3, sell 2
0.700 0.800 0.900 1.000 1.100 1.200
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Mll-20
Module 11 - Review of Derivatives Markets, Chapter 4
4.13
Problem 4.12 establishes that the profits of Wirco do not depend on the cost
of copper. Thus any derivative strategy can increase variability in profits.
This is what happened with the forward strategy in the last problem.
4.1S
With a tax deduction for losses, the alternative profits are calculated below:
Price
Price
9.00
-1.00
-1.00
-0.40
-0.60
11.20
1.20
1.20
0.48
0.72
Pre-tax op income
Taxable Income
Tax @ 40%
After Tax Income
The expected value is 0.5(-0.60) + 0.5(0.72) =0.06.
With a forward sale at 1.10, the profits are 0.06 in each case with and expected
value of 0.06. Thus the forward sale gives the same expected value but reduces
variability of profit.
4.17
/
a) What is the expected pre-tax profit?
Firm A: E[pre-tax profit] = .5 * $1000 + .5 * (-$600) = $200,
Firm B: E[pre-tax profit] = .5 * $300 + .5 * ($100) = $200.
Notice that both firms have the same expected pre-tax profit.
b) What is the expected after-tax profit?
Firm A: E[pre-tax profit] = .5 * ($1000*.6) + .5 * (-$600) = $0,
Firm B: E[pre-tax profit] = .5 * ($300*.6) + .5 * ($100*.6) = $120.
Notice that the after-tax profits of Firm B stay the same as they were in
Question 4.16, while those of Firm A changed. This is because they no
longer receive tax credit on the loss.
c) Firm B would not pay anything, because it always makes positive profits,
which means that the lack of a tax credit does not affect them.
Firm A would be willing to pay the discounted difference between its
after-tax profits calculated in Question 4.16 b), and its new after-tax
profits, $0 from Question 4.17. It is thus willing to pay: =$109.09.
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 11 - Review of Derivatives Markets, Chapter 4
Page Ml 151
4.19
When we buy the call, we buy it at the ask price, which is $0.25 above the
Black-Scholes price, and we sell the put at the bid price, which is $0.25
below the Black-Scholes price. So we must have,
C + $0.25-(P-$0.25) = 0,orP-C = $0.50.
We also need the call strike to be 30 higher than the put strike.
This problem can be solved by trial and error (which is what the instructor's
manual says to do, but that does not make it a reasonable exam problem.
We solved it using a spreadsheet implementing the Black-Scholes model and
MS Excel's Solver menu to obtain the proper answer. This too, is not exam
material.
If you are curious about the problem anyway, the answer is a call strike of
436.53, and a put strike of 406.53. The Black-Scholes call premium is $3.1938,
and the put has a premium of $3.6938.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Mll-22 Module 11 - Review of Derivatives Markets, Chapter
Section 11.8
Module 11 Computational Review Problems
1. (1 pt) The 1-ycar forward price of copper is $ l/lb.
Suppose CDE mines copper, with fixed costs of $ 0.50/lb and
variable cost of $ 0.40/lb. If CDE does nothing to manage
copper risk:
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 0.90 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.00 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.10 $ ?
If on the other hand CDE sells forward its expected copper
production:
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 0.90 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.00 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.10$ ?
ANSWER1:0
ANSWER2: 0.1
ANSWER3: 0.2
ANSWER4: 0.1
ANSWER5:0.1
ANSWER6: 0.1
2. (1 pt) The 1-ycar forward price of copper is $ 0.80/lb.
Suppose CDE mines copper, with fixed costs of $ 0.50/lb and
variable cost of $ 0.40/lb. If CDE does nothing to manage
copper risk:
What is its profit 1 year from now, per pound of copper, if the
copper price in I year is $ 0.80 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 0.90 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.00 $ ?
If on the other hand CDE sells forward its expected copper
production:
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 0.80 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 0.90 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.00 $ ?
ANSWER1: -0.1
ANSWER2: 0
ANSWER3: 0.1
ANSWER4: -0.1
ANSWER5:-0.1
ANSWER6: -0.1
3. (1 pt) The 1-year continuously compounded interest rate
is 6
Strike
0.95
1
1.05
Call
0.0649
0.0376
0.0194
Put
0.0178
0.0376
0.0665
Suppose CDE mines copper, with fixed costs of $ 0.50/lb and
variable cost of $ 0.40/lb.
If CDE buys a put option with a strike of $ 0.95:
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 0.90 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.00 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.10$ ?
If CDE buys a put option with a strike of $ 1.00:
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 0.90 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.00 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.10 $ ?
If CDE buys a put option with a strike of $ 1.05:
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 0.90 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.00 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.10 $ ?
ANSWER!: 0.03
ANSWER2: 0.08
ANSWER3: 0.18
ANSWER4: 0.06
ANSWER5: 0.06
ANSWER6:0.16
ANSWER7: 0.08
ANSWER8: 0.08
ANSWER9: 0.13
4. (1 pt) The 1-year continuously compounded interest rate
is 6
Strike
0.95
1
1.05
Call
0.0649
0.0376
0.0194
Put
0.0178
0.0376
0.0665
Suppose CDE mines copper, with fixed costs of $ 0.50/lb and
variable cost of $ 0.40/lb.
If CDE sells a call option with a strike of $ 0.95:
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 0.90 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.00 $ ?
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 11 - Review of Derivatives Markets, Chapter 4
Page Ml 1-23
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.10$ ?
If CDE sells a call option with a strike of $ 1.00:
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 0.90 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.00 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.10 $ ?
If CDE sells a call option with a strike of $ 1.05:
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 0.90 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.00 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.10 $ ?
Answers can be found in the table below:
Sell Call Sell Call Sell Call
K 0.95 1 1.05
Price Profit Profit Profit
0.9 0.06891 0.03993 0.02060
1 0.11891 0.13993 0.12060
1.1 0.11891 0.13993 0.17060
5. (I pt) The I-year continuously compounded interest rate
is 6
Strike
0.95
0.975
1
1.025
1.05
Call
0.0649
0.05
0.0376
0.0274
0.0194
Put
0.0178
0.0265
0.0376
0.0509
0.0665
Suppose CDE mines copper, with fixed costs of $ 0.50/Ib and
variable cost of $ 0.40/lb.
If CDE buys collars with a strike of $ 0.95 for the put and $
1.00 for the call:
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 0.90 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.00 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.10$ ?
If CDE buys collars with a strike of $ 0.975 for the put and $
1.025 for the call:
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 0.90 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.00 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.10$ ?
If CDE buys collars with a strike of $ 1.05 for the put and $
1.05 for the call:
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 0.90 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.00 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.10 $ ?
Answers can be found in the table below:
Buy Collar(Buy put, sell call)
0.95 1 1.05
Kput 0.95 0.975 1.05
Kcall 1 1.025 1.05
Price Profit Profit Profit
0.9 0.07102 0.07596 0.09999
1 0.12102 0.10096 0.09999
1.1 0.12102 0.12596 0.09999
6. (1 pt) The 1-year continuously compounded interest rate
is 6
Strike
0.975
1
1.025
1.034
Call
0.05
0.0376
0.0274
0.0243
Put
0.0265
0.0376
0.0509
0.0563
Suppose CDE mines copper, with fixed costs of $ 0.50/lb and
variable cost of $ 0.40/lb.
If CDE sells one 1.025-strike put and buys two 0.975-strike
puts:
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 0.90 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.00 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.10$ ?
If CDE Isells two 1.034-strike puts and buys three 1.00-strike
puts:
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 0.90 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.00 $ ?
What is its profit 1 year from now, per pound of copper, if the
copper price in 1 year is $ 1.10 $ ?
ANSWER1:0.02277
ANSWER2: 0.07277
ANSWER3: 0.19777
ANSWER4: 0.031788
ANSWER5: 0.031788
ANSWER6: 0.199788
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Mll-24 Module 11 - Review of Derivatives Markets, Chapter 4
Section 11.9
Supplemental Exercises
In problems 1-5, we will look at profit for a farmer who grows corn. For all of
these problems, the current (spot) rate for corn is 1.60 per bushel. The 6 month
forward price is $1.50 per bushel. The continuously compounded annual rate is
r = .04.
The farmer, Farmer Inadel, has total fixed and variable costs of 1.45 per
bushel, and plans to produce 100,000 bushels for $145,000.
1. What are the minimum and maximum profits for Farmer Inadel in six
months if he is not hedged?
A) minimum = 145,000, no maximum
B) minimum = -145,000, maximum = 145,000
C) minimum = -145,000, no maximum
D) none of the above
2. What are the minimum and maximum profits for Farmer Inadel six
months if he is hedged with a short forward contract?
A) minimum = maximum = 1450
B) minimum =maximum = 5000
C) minimum= 1450, no maximum
D) minimum = -145,000, no maximum
E) none of the above
3. A six month (T = .5) put with a strike price of 1.50 per bushel is available
at a price of 0.11. What are the minimum and maximum profits for
Farmer Inadel in six months if he is hedged with a purchase of this put?
A) minimum = -6000, maximum = 18778
B) minimum =-6222, maximum = 18778
C) minimum= -6000, no maximum
D) minimum = -6222, no maximum
E) none of the above
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 11 - Review of Derivatives Markets, Chapter 4
Page Mil- 25
4. A six month (T = .5) call with a strike price of 1.55 per bushel is available
at a price of .10. What are the minimum and maximum profits for
Farmer Inadel in six months if he is hedged with a sale of this call?
A) minimum = -134800, maximum = 28,022
B) minimum =-134800, maximum = 20,202
C) minimum= -134,800, no maximum
D) no minimum , maximum = 28,022
E) none of the above
5. What are the minimum and maximum profits for Farmer Inadel in six
months if he hedges by buying the 1.50-strike put for .11 and sells the
1.55 call for .10?
A) minimum = 5000, maximum = 5000
B) minimum =-4800, maximum = 5000
C) minimum= 3980, maximum = 8980
D) minimum = 3980, no maximum
E) none of the above
6. Company AOK makes an aircraft which costs 90,000,000 to manufacture.
It will be completed in 6 months. At that time it will sell either for
102,500,000 with probability .5 or 80,000,000 with probability .5. The
company has a 40% tax rate, and has no tax benefit for losses. What is
the company's expected profit before tax?
A) -1,250,000 B) -1,000,000 C) 0
D) 1,000,000 E) 1,250,000
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Page Mll-26
Module 11 - Review of Derivatives Markets, Chapter 4
7. Company AOK makes an aircraft which costs 90,000,000 to manufacture.
It will be completed in 6 months. At that time it will sell either for
102,500,000 with probability .5 or 80,000,000 with probability .5. The
company has a 40% tax rate, and has no tax benefit for losses. What is
the company's expected profit after tax?
A) -1,250,000 B) -1,000,000 C) 0
D) 1,000,000 E) 1,250,000
8. Company AOK makes an aircraft which costs 90,000,000 to manufacture.
It will be completed in 6 months. At that time it will sell either for
102,500,000 with probability .5 or 80,000,000 with probability .5.
The company decides to enter into a forward contract to sell the aircraft
for 90,800,000 in six months The company has a 40% tax rate, and has no
tax benefit for losses. What is the company's expected profit after tax?
A) -1,000,000 B) -480,000 C) 0
D) 480,000 E) 1,000,000
9. Which of the following is not a good reason for a producer of widgets to
start a hedging program?
A) The board of directors is concerned about prices of widgets
dropping.
B) The company may face bankruptcy risk if widget prices drop.
C) Lenders will be reluctant to make loans to the company if widget
prices drop.
D) The president of the company wants to demonstrate to the board
that his accounting department can learn to handle the
complexities of accounting for hedges.
E) None of the above.
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Module 11 - Review of Derivatives Markets, Chapter 4 Page Mil- 27
Section 11,10
Supplemental Exercise Solutions
1. The unhedged profit function is P = 100,000* -145,000 where x is the
spot price of corn in 6 months and 0 < *.
Answer C
2. The profit from the forward contract in six months is
150,000 -100,000*. Thus the farmers hedged profit is the sum of the
unhedged profit and the forward profit
100,000* -145,000 + (150,000 -100,000*) = 5000.
Answer B
3. The profit from the put option is
100,000[max(0,1.5-*) -.lie 04(5)] = 100,000max(0,1.5-*)-11,222.21.
The total profit for the hedged position is
100,000*-145,000 + (100,000max(0,1.5-*)-11,222.21)
_ J-6,222.21, *<1.5
" [100,000* -156,222.21, * > 1.5
Answer D
4. The profit from the written call option is
100,000[-max(0,* -1.55) + .le04(5)] = -100,000max(0,* -1.55) +10,202.
The total profit for the hedged position is
100,000* -145,000 + (-100,000 max(0, * -1.55) +10,202)
_ f 100,000* -134,798 * < 1.5
" [20,202 *>1.5
Answer B
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Page Ml 1-28
Module 11 - Review of Derivatives Markets, Chapter 4
5. The total profit for the hedged position here has a profit graph that looks
like the graph of a bull spread. We can look at two cases to find the
minimum and maximum
a) x < 1.50. The call expires worthless, but the hedger received
10,000 for the 100 calls sold, and that has a future value of 10,202.
The 100,000 puts are worth
100,000(1.5-jc)-11,222.21 = 138,778-100, OOOx.
The unhedged sale yields P = 100,000* -145,000. The total of the
three components is 3,980.
b) x > 1.55. The put expires worthless but 100,000 puts required a
purchase expense of -11000 with a future value of -11,222. The
100,000 written calls have a value of
-100,000* +155,000 +10,202
The unhedged sale yields P = 100,000* -145,000. The total of the
three components is 8980.
Answer C
6. The calculations for problems 6 and 7 are in the table below. Values are
given in millions.
Firm manufactures for
Pre-tax operating income
Taxable Income
Tax @ 40%
After Tax Income
90
Price
80.00
-10.00
0
0
-10.00
Price
102.50
12.50
12.50
5
7.50
Probability
Expected Value
0.5
0.5
Before TaxAfter Tax
1.250 -1.250
Answer E
7. See the table above
Answer A
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Exam FM / Exam 2 - Financial Mathematics

Module 11 - Review of Derivatives Markets, Chapter 4
Page Mil- 29
8. The calculations for problems 8 are in the table below. Values are given
in millions.
With Short Forward at
Pre-tax op income
Income from Forward
Taxable Income
Tax @ 40%
After Tax Income
90.80
Price
80.000
-10.000
10.800
0.800
0.320
0.480
Answer D
9. Accounting complexity is given as a reason for not hedging. The other
choices correspond to reasons given for hedging in the text.
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Price
102.500
12.500
-11.700
0.800
0.320
0.480

Module 12 - Review of Derivatives Markets, Chapter 5
Page M12- 1
Section 12.1
Financial Forwards and Futures
This chapter studies forward contracts for stocks and stock indices like the
fictional S&R. It also discusses futures, which are standardized forward
contracts traded on exchanges. Section 4 of the chapter is devoted to the
mechanics of futures, with special emphasis on futures for the S&P stock index.
We have seen some of the results from this chapter already. For example, the
forward price for the S&R index is calculated using Equation 5.5 from the text,
and we quoted that result in Module 9 to justify the price used in Chapter 2 of
Derivatives Markets.
Here we will show how to derive Equation 5.5 using a no-arbitrage method.
That method is widely used in finance and it is very important to master it.
Please remember that the results in this chapter are derived for stocks and
stock indices. They do not work for commodities like corn and gold. The pricing
of forward contracts for commodities is covered in Chapter 6 of Derivatives
Markets, and that chapter is not on the exam FM/2 syllabus.
There is an important prerequisite review for the actuarial student here. When
we deal with continuous interest, we denote the continuously compounded rate
by 8, and the annual effective rate by i. The two are related by the equations
J = ln(l + i) l + i = es.
Derivatives Markets does not use this notation. In this chapter the symbol 5 is
used to represent the continuously compounded dividend rate. In addition, the
letter r is used differently here than in Chapter 2. Recall that in Chapter 2 the
S&R index had a current price of S0 = 1000 and the effective interest rate was
r = .02 for a semiannual period. The six month forward price was
So (l + r) = 1000(1.02) = 1020
In this chapter the author does most derivations using continuous interest. The
forward price is written in equation 5.5 as S0erT.
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Page M12-2
Module 12 - Review of Derivatives Markets, Chapter 5
Thus here r represents the continuously compounded rate, not the effective
rate. The continuously compounded rate that gets you an effective rate of 0.02
is ln(1.02) = .0198. In this module, we will use r to represent the continuously
compounded rate unless otherwise specified.
The previous chapters were very example oriented with little theoretical
notation. The situation is reversed here, and these notes will provide additional
examples to make the theory concrete.
Section 12.2
Four Ways to Buy a Stock
The text lists four different ways to purchase a stock with current price S0
• Outright purchase. You have the price in cash and buy the stock for S0 in
cash today.
• Fully leveraged purchase. You borrow the price in cash and buy the
stock today. The loan will be repaid at time T, and you must pay S0erT at
that time.
• Prepaid forward contract. You pay for the stock now at time 0 but
receive it at a specified time T in the future.
• Forward contract. You receive the stock and pay for it at a specified
time T in the future.
We have not yet derived the price that must be paid for a prepaid forward or a
forward, although we have used the correct price for the forward in the
examples of Chapters 2-4.
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Module 12 - Review of Derivatives Markets, Chapter 5 Page M12- 3
Section 12.3
Pricing Prepaid Forward Contracts
The price of a prepaid forward for receipt at time T is denoted by F0ptT. This
price depends on whether or not dividends are to be paid on the stock. The
easier case occurs when no dividends are paid.
Pricing Prepaid Forward Contracts for a Stock with no Dividends
There are three ways to derive the correct price for a stock with no dividends.
• Pricing by analogy. If there are no dividends and you are only interested
in owning the stock at time T, you will have the same position at time T
as someone who buys the stock now for S0 and holds it until time T. Thus
you should pay S0 now for the prepaid forward. If the S&R index is at
1000 today, you should pay 1000 for either an immediate purchase of the
index or a prepaid forward for receipt in 6 months.
• Pricing by discounted present value. A new rate variable is introduced
here -the variable a, which represents the expected rate of return on
the stock or index. We have previously noted that stocks have a higher
average rate of return than bonds (which involve borrowing and lending
at the rate r.) Thus an S&R prepaid forward contract could be written at
time when the continuous interest rate is r = .04 but the expected rate of
return on the index is a continuous rate of a = .12.
Suppose that we are pricing a prepaid S&R forward for time T = .5 and
the index has value S0 = 1000. At time 0 the expected future value of the
S&R index is
(12.1)
E0 (ST) = S0eaT = lOOOe06 = 1061.84.
An investor who uses discounted present value would want to prepay the
discounted present value of that expectation today. Since the investor
looks at a as the appropriate rate of return for the stock, he should
discount at that rate. Thus his price would be
(12.2)
F0pt = e-aTE0 (ST) = e~aTS0eaT =S0= 1000
This is the same price we arrived at by analogy. You pay the current
stock price for a prepaid forward on a stock with no dividends.
• Pricing by arbitrage. You have an arbitrage if you can make money with
no risk by simultaneously buying and selling related assets. The theory
applied here is that arbitrages result from incorrect pricing and if
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Page M12-4
Module 12 - Review of Derivatives Markets, Chapter 5
traders see such incorrect pricing they will immediately try to benefit
from it and thus drive the price to its correct level and eliminate the
arbitrage. We will illustrate this by returning to the S&R index example
where the current price is S0 = 1000 and the interest rate is r = .04. We
will look at how to use arbitrage pricing in two cases.
a) The prepaid forward price is higher than the current price. (F<£r > S0)
Suppose that the prepaid forward for the S&R is priced at F<£r = 1001.
Then you can buy the index for S0 = 1000 and simultaneously sell a
prepaid forward for 1001. This gives an immediate profit of 1 for no
expenditure, and you own the stock that is necessary to deliver
forward at time T. Since arbitrages should not exist, it is not possible
thatF0Pr>So.
b) The prepaid forward price is lower than the current price. (FotT < S0)
Suppose that the prepaid forward for the S&R is priced at Fo)T = 999.
Then you can sell the index short for S0 = 1000 and simultaneously
buy a prepaid forward for 999. This gives an immediate profit of 1 for
no expenditure, and the prepaid forward will provide stock at time T
to replace the stock borrowed for the short sale. Since arbitrages
should not exist, it is not possible that F0pr < S0.
Since it is not possible that F0Pr > S0 or F0Pr < S0, the price must
be Foj = So.
Arbitrages do occur in practice, but investors called arbitrageurs are
constantly on the lookout for arbitrage opportunities and will indeed
trade rapidly on any such opportunity. This will generate price pressure
that will eliminate the arbitrage. Think of the above case where
F0Pr = 1001. This will cause a flurry of offers to sell prepaid forwards at
1001, and that over supply of offers will drive the price down toward
1000.
Arguments such as the above are sometimes called "no-arbitrage"
arguments. Such arguments justify a price by showing that any other
price would lead to an arbitrage.
Pricing Prepaid Forward Contracts for a Stock with Dividends.
Suppose that you want to purchase a prepaid forward contract for delivery at
time T for a stock that does pay dividends. In this case you will not get any
dividends that are paid before time T. To adjust for this loss, you would adjust
the price of the prepaid forward by subtracting the present value of the
missing dividends from the current price of the stock. The text states on page
131 "In general, the price paid for a prepaid forward contract will be the stock
price less the present value of the dividends to be paid over the life of the
contract."
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Module 12 - Review of Derivatives Markets, Chapter 5
Page Ml2- 5
For individual stocks, dividends are paid at discrete time intervals -e.g.,
dividends might be paid quarterly. A stock index contains a large number of
stocks which pay dividends at different times, so it is common to model the
index as if dividends are paid continuously.
Thus, there are two possible calculation methods depending on whether
discrete or continuous dividends are assumed. Note that all prepaid forward
calculations will rely on an estimate of what the expected future dividends will
be. So, the validity of derivative strategies may depend on the accuracy of the
estimated dividends.
Discrete dividends.
If n dividends di,...,d„are paid over n periods at times ti,...,t„and the
continuously compounded interest rate per period is r, the price of a prepaid
forward for delivery at time T will be
(12.3)
F(Tr=So-X^"rti
i=l
Example 5.2 of the text looks at a one year prepaid forward for the stock of
XYZ. The stock price is currently 100, and it is expected that quarterly
dividends of 1.25 will be paid, with the last dividend occurring just before the
delivery of the stock. The continuously compounded interest rate is .025 per
quarter. Then the price of the prepaid forward contract is
F0Pi =100- Jl.2Se-025' =95.30
i=l
Continuous dividends.
Suppose that dividends are paid on a stock index at the continuous rate 8. The
continuous dividend model assumes that all dividends are reinvested in the
stock index. Thus if you buy one unit of the stock today, the growth due to
continuous dividend reinvestment will leave you with e6T shares at time T (i.e.,
more than one share).
To have only one share at time T, you would buy e~6T shares now and that
fractional share would grow to one share at time T. If the current price of the
index is S0, you could be assured of having one share of the index by paying
e~STS0 for e~6T shares of the index now. Pricing by analogy says that the prepaid
forward should have the same price. Thus the price of the prepaid forward is
(12.4)
Fq,T =6 So
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Page M12-6
Module 12 - Review of Derivatives Markets, Chapter 5
The purchase of e~ST shares to assure 1 share at time T is called tailing your
position. This terminology is used again in the text, so it is useful to remember
it.
To illustrate what happens under this continuous model, we will invent our own
fictional dividend paying stock index: the S&Q index. The S&Q index has:
• A continuous dividend yield of S = .02
• Current value of 1000
• Continuous interest rate of r = .04.
To create a tailed position today for time T = 0.5, you would buy a fractional
share of e"02(5) = 0.99005 or 99.005% of a share. The cost of this tailed fractional
share would equal the prepaid forward price.
F0pt = e~STS0 = e011000 = 990.05.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 12 - Review of Derivatives Markets, Chapter 5
Page M12- 7
Section 12.4
Forward Contracts on Stock or Stock Indices
Finding the Forward Price
The forward price is paid T time units in the future instead of today. It is equal
to the future value at the continuous rate r of the prepaid forward price which
would be paid today. (We could also get the forward price using a no-arbitrage
argument, but the answer would be the same and this is simpler.)
Recall that the forward price is denoted by F0tT while the prepaid forward price
is Fqj . The relationship is
(12.5)
Fnr = & Fr
"o/r
o/r
This gives us the following forward prices for the basic pricing cases:
(12.6)
No dividends F0j = S0e
rT
(12.7)
Discrete dividends
Fo,T=erT
( " ^
= erTS0-
V i=l J
-±dte«T-"
i=l
(12.8)
Continuous dividends
F0>T=erTe-STSo=S0e{r-s)T
For a concrete example, recall that for the S&Q index when r = .04,8 = .02,
T = .5 and S0 = 1000 the prepaid forward price was
F0pt = e~STS0 = e-011000 = 990.05.
The forward price is then
Fo.t = Soe{r~s)T = lOOOe01 = 1010.05.
Derivatives Markets notes that the forward contract has a zero cost and thus
can be considered to have no premium payment initially. Since the prepaid
forward is paid for immediately, it does have a premium payment initially.
However the "zero cost" forward does have a price to pay in the future, and
that is the future value of the initial prepaid forward price. In other words,
each contract has a cost but you have to be careful about how you refer to it.
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Page M12-8
Module 12 - Review of Derivatives Markets, Chapter 5
There is a brief discussion on page 134 about how to use the forward price to
get the market price. The text points out that this is relevant overnight when
the New Your Stock Exchange is closed but traders are still trading the futures
contract on the S&P 500 index. Overnight, you can see futures prices but not the
actual index value. (Both change from hour to hour.) However, you can
calculate the implied value of the index using the equations above.
For example, suppose that the S&Q index has a futures price of F0tT = 1012 for
T = .5. Then we can use Equation 12.8 with r = .04 and S = .02 to find S0.
1012 = Fo,t = S0e{r-S)T = S0e01 -* S0 = 1001.93.
When this is calculated, the rate r that is used is the risk-free rate, which is the
yield on a United States Treasury of the same duration as the forward contract.
In previous chapters the rate r was typically given without being identified as
the risk-free rate, but as the book goes on you will see calculations that specify
r as such.
The index price that is calculated in this way is referred to as the fair value for
the index.
The text defines the forward premium to be the ratio of the forward price to
the stock price:
Fqt
Forward premium =
So
For the S&Q index with current price of 1000 and six month forward price of
1010.50, the forward premium is :— = 1.01005.
1000
This is not a premium in the sense of the price paid for a derivative like an
option. It is simply a relative value factor, which says in the example above that
the forward price is 1.005% above the current index price. The 1.005% just
quoted is not a continuous rate, nor is it an annual rate. The text defines a
measure that is basically the periodic continuous rate converted to an annual
rate.
j /p \
Annualized Forward Premium = — In ——
T [So
For the S&Q index with current price of 1000 and six month forward price of
1010.50, the annualized forward premium is —In (1.011) = 2(0.0109) = 0.0218.
This answer should not be surprising. The correct futures price used above was
F0>r = S0e(r~^r. Thus if the correct price is used, the forward premium is
F0,T _ 6 *JQ _ {r-S)T
So So
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Module 12 - Review of Derivatives Markets, Chapter 5
Page M12- 9
Thus the annualized forward premium is
T [So J T \ ) { T )
= r-S
Derivatives Markets states that: "For the case of continuous dividends,
Equation (5.7), the annualized forward premium is simply the difference
between the risk-free rate and the dividend yield."
Creating Synthetic Stocks, Forwards and Bonds
Creating a synthetic stock.
The simplest way to understand and remember this is to use the word equation
(12.9)
Stock at time T = Long forward + zero-coupon bond
In words, you invest money in a zero-coupon bond which will pay the forward
price at maturity and you also enter into a long forward contract with the same
duration. Then at the time of maturity of the bond you collect the proceeds, pay
the forward price and you have the stock. To make this more precise, we must
first point out that the zero coupon bond is for the amount of a tailed position in
the stock. The steps are:
1) At time 0, invest S0e in a bond with yield rate r and maturity at T.
2) At time 0, enter into a zero cost forward for the forward price of
3) At time T, collect the bond proceeds of erTS0e-5T = S0eir-S)T
4) At time T, use the bond proceeds to buy the stock for the forward price
of Soe™.
The text has tables displaying the process, and you should read
those. We are purposely describing the process differently so that
you have another way to look at it.
You have invested S0e ST at time 0 and have the stock at time T. Note that you
can achieve the same result by buying a prepaid forward. If nobody is selling a
prepaid forward you can create one this way.
The word equation might be re-stated as:
To have the stock at time T:
Buy a forward and a zero coupon bond for the amount of a tailed position.
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Page M12-10
Module 12 - Review of Derivatives Markets, Chapter 5
We can derive two other synthetic instruments by manipulating the word
equation. Anything that has a minus sign will indicate a short position or sale.
For example, "- zero-coupon bond" means sell a zero-coupon bond for the
amount of a tailed position, which means that you are borrowing that amount.
(12.10)
Forward = Stock - zero coupon bond
Restated:
To create the equivalent of a forward contract for time T:
Buy the stock and sell a zero-coupon bond for the amount of a tailed position.
The steps this implies are:
1) Borrow S0e~ST at time 0.
2) Use the borrowed amount S0e~ST to buy a tailed position in the stock at
zero cost.
3) At time T, you will have the stock worth ST
4) Repay the loan by paying S0e(r~S)T = F0tT. This will leave with
ST - S0e(r~s)T =ST - F0)T, the payoff on a long forward contract.
(12.11)
Zero-coupon bond = Stock - forward
Restated:
To create a zero-coupon bond with maturity T:
Buy a tailed position in the stock and sell a forward contract.
The steps this implies are:
1) Invest S0e~ST to buy the tailed position in the stock at time 0.
2) Sell a forward obligating you to sell the stock at time T
torS0e™T=F0tT.
3) At time T, you will sell the stock for S0e(r_w = erS0e"*r
4) Thus you have invested S0e~ST at time 0 and been paid erSoe~ST at time
T. This is a zero-coupon bond paying the risk-free rate r..
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Module 12 - Review of Derivatives Markets, Chapter 5
Page M12-11
The text summarizes the steps of all of the preceding synthetics in Tables 5.3-
5.5. We find that the simplest approach here is to know the word equation (12.9)
and manipulate it to get the others. You must remember that amounts at time 0
are for a tailed position in the stock, and that a minus sign implies a short or
selling position.
There is another way to use the word equations -multiply through by -1,
reversing the signs and recall that a negative denotes a short position. For
example, we have the word equation
Forward = Stock - zero coupon bond
This changes to
-Forward = -Stock + zero coupon bond.
In other words, to create a synthetic short forward, sell the stock short and
invest the sale proceeds in a zero coupon bond.
Hedging and Arbitrage with Synthetic Forwards
Cash and Carry Hedge:
Suppose that you have sold a stock forward, agreeing to sell it at time T for
F0T = S0e(r~s)T. To hedge this position you create a synthetic long forward
agreeing to buy it for the same price at time T. You are hedged, since at time T
you net the sale price paid to you less the purchase price you pay:
S0e(r-OT-S0e(r-OT=0.
The procedure is common sense. To hedge a (short) forward sale, offset it with
a synthetic forward purchase.
This is simple in practice. Recall that the six month forward price when
r = .04, S = .02 and S0 = 1000 is F0>T = S0e{r-5)T = lOOOe01 = 1010.05.
Suppose that you have entered a forward contract to sell the stock in six
months for 1010.05. To hedge this position you create a synthetic long forward
for the same price.
Forward = Stock - zero coupon bond
Buy a tailed position in the stock today for S0e"JT = 1000e"01 = 990.05, and
borrow that 990.05 at the risk free rate r = .04. In six months you will have the
stock, and owe the amount 1000e"01e02 = 1010.05 on the borrowing. Sell the stock
under the forward contract that you are hedging and you will have 1010.05 to
pay off the loan. The payoff is 0.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M12-12
Module 12 - Review of Derivatives Marketst Chapter 5
Cash and Carry Arbitrage:
The cash and carry hedge assumed that the forward sale price was correct, or
Fo,T=S0e^T
Suppose you believe that the forward price offered is too high, or F0,r > S0e(r"J)T
Remember that the forward price depends on estimates of the correct 5 and r,
and could be wrong.
You can then arbitrage this error by the classic strategy of buying low and
selling high: sell a forward at the higher forward price. Then create a synthetic
purchase at the lower correct price.
Buy low and pay : S0e~ST.
Borrow: S0e{~s)T
Sell high and receive: F0>T.
Repay loan: S0e(r"')T
Profit: F0,T-S0e{r-S)T.
In the previous example, if the forward price was F0)t = 1011 while the correct
theoretical price is S0e(r"*)T = 1010.05, the arbitrage would be to borrow to
purchase a tailed position in the stock today. At time T
Repay loan: 1010.05
Sell high and receive: 1011
Profit: 0.95
This profit at time T had 0 cost at time 0. It is an arbitrage.
Reverse Cash and Carry Hedge:
Suppose that you have purchased a stock forward, agreeing to buy it at time T
for F0)T = S0e(r-<5)T. To hedge this position you create a synthetic short forward
agreeing to sell it for the same price at time T. You are hedged, since at time T
you net the sale price paid to you less the purchase price you pay:
S0e(r-OT-S0e(r-OT=0.
The procedure is again common sense. To hedge a (long) forward purchase,
offset it with a synthetic forward sale.
This too is simple in practice. Recall that the six month forward price when
r = .04, S = .02 and S0 = 1000 is F0)T = S0e(r~S)T = lOOOe01 = 1010.05.
Suppose that you have entered a forward contract to buy the stock in six
months for 1010.05. To hedge this position you create a synthetic short forward
for the same price.
-Forward = -Stock +zero coupon bond
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Module 12 - Review of Derivatives Markets, Chapter 5
Page M12-13
Sell a tailed position in the stock today for S0e"JT = lOOOe"01 = 990.05, and invest
that 990.05 at the risk free rate r = .04. In six months you will have to deliver
the stock that you have sold short, and be paid the amount 1000e"01e02 = 1010.05
from the zero coupon bond. Buy the stock under the forward contract that you
are hedging using the amount of 1010.05 from the loan and deliver the stock to
cover the short sale. The payoff is 0.
Reverse Cash and Carry Arbitrage:
The cash and carry hedge assumed that the forward sale price was correct, or
F0,t = S0e(r~*)T. And, suppose you believe that the forward price offered is too
low, or F0)T < S0e{r~5)T.
Then, you can once more arbitrage this error by the classic strategy of buying
low and selling high. Buy a long forward at the lower forward price. Then
create a synthetic sale at the higher correct price.
Buy low and pay : F0>T
Sell high (synthetic) and receive: S0e(r"^T
Profit: S0e(r-')T-Fo,T.
Suppose that the forward price was F0)T = 1009 while the correct theoretical
price is S0e(r"J)T = 1010.05. The arbitrage would be
Buy low and pay : 1009
Sell high and receive: 1010.05
Profit: 1.05
This profit at time T had 0 cost at time 0. It is also an arbitrage.
The Implied Repo Rate
Recall that (12,11) stated: zero-coupon bond = Stock - forward
In words, if you buy a tailed position in a stock and sell the stock forward, you
have the equivalent of a zero coupon bond at the risk free rate. If the forward is
delivered at time T, the precise results are:
Time 0: Invest S0e~*T to purchase the tailed position.
Time T: Sell the purchased stock and receive
Fo,T=Soe{r-5)T=erTS0e-ST for it.
The continuously compounded return is r, the risk free rate. In some cases, you
may not know r and wish to estimate it. If F0)T is theoretically correct, then
-^ = erT and lnf-^O = rT so that ilnfJ^l = r.
S0e~ST [Soe-^J T [s0e-5T)
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Page M12-14
Module 12 - Review of Derivatives Markets, Chapter 5
Thus we can estimate the interest rate r implied by the values of F0,t and 8.
This estimate is referred to as the implied repo rate. For example, if you see
that So = 1000 and F0,o.5 = 1010.25, if you believe that 8 = .02 then when T = .5
1, (1010.25^ n/m*o*
r = "^ln t^t^—or = .040396.
.5 l,1000e-01J
The implied repo rate is discussed briefly on page 338 of Derivatives Markets.
The term repo is short for repurchase. A repurchase agreement occurs when
the owner of an asset puts up the asset as collateral for a loan and agrees to buy
the collateral back in the future at a higher price. In the above example, the
owner of a tailed position in the stock could give a lender custody of stock
worth lOOOe"01 = 990.05 in return to a loan of 990.05 for 6 months at the
continuous rate r = .040396. At the end of 6 months the borrower pays back
990.05e040396( 5) = 1010.25 and gets his collateral back. This is equivalent to
selling the stock at time 0 for 990.05 and then buying it back for 1010.25, so it is
called a repurchase agreement.
The text points out the implied repo rate alerts you to arbitrage opportunities.
Suppose that you can borrow money at the continuous rate of 4%. You could
then borrow funds at 4% to buy a tailed position in the stock, and use the stock
to earn a repo rate of 4.0396% over the next 6 months. If the implied repo rate
is higher than your borrowing rate, there is an arbitrage opportunity.
Arbitrage: More Complex Examples
All of the previous discussions have been simplified by assuming no transaction
costs, a single interest rate for borrowing and lending and a single stock price
instead of a bid-ask spread. This helps you to learn the basic ideas without
being overwhelmed by details. However, in practice, when you buy or sell a
stock you pay a brokerage fee and borrowing and lending rates vary. If I
borrow from my bank with a home equity loan this week, I must pay an annual
rate of 7%. That is my borrowing rate. However a 10 year CD for 100,000 pays
the lower rate of 4.45%. That is my lending rate. Finally, stocks and forwards
have bid-ask spread spreads.
On page 138 the text illustrates what happens if transaction costs, different
borrowing and lending rates and a bid-ask spread are all present. Let's start
with a simple case where a stock pays no dividends. The text goes through a
general discussion with no numerical example. We will introduce the notation
along with a numerical example for the S&R index, which pays no dividends.
Bid Price on Stock: Sj = 999
Ask Price on Stock: S§ = 1001
Borrowing Rate: rb = .041
Lending Rate: rl = .039
Transaction cost: k = 1.
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Module 12 - Review of Derivatives Markets, Chapter 5
Page M12-15
Recall that with a single price S0, a single, continuous rate r and no transaction
costs, the correct forward price for time T is S0erT .There will be an arbitrage if
the forward price does not equal S0erT. The results in the more complicated
situation with differing rates and prices and a transaction cost look similar, but
now to avoid arbitrage the forward price quoted must be in a range. In our
example, the forward price must be between two values:
F+ =(SS +2k)erbT =(1001 + 2) e041(5) =1023.77
F~ = (Sb0 - 2k) er'T = (999 - 2) e039( 5) = 1016.63
Thus the forward price should not be above 1023.77 or below 1016.63, since if it
were there would be an arbitrage. The text discusses how to set up the
arbitrages that validate these bounds, so we will not repeat that here.
The formulas are not hard to remember. They look like S0erT, but in the upper
bound formula you use the higher ask price and the higher borrowing rate , and
add two transaction costs to the stock price. In the lower bound formula you use
the lower bid price and the lower lending rate , and subtract two transaction
costs from the stock price.
Quosi-Arbitroqe
We have seen that you could earn the rate r by investing in a tailed position in a
stock and entering a forward contract for the same stock. This means that you
can always earn the implied repo rate r in this fashion.
If the rate you can earn on your bond investments is less than r, it would be
better to switch from what you are doing with bonds and instead use the
implied repo method to earn r.
For example, if you could earn 7% interest on bond investments but the implied
repo rate was 7.5%, it would be better to skip the bond investment and earn the
implied repo rate instead. This is not a true arbitrage, but is does give an
alternative that will allow you to increase you interest earnings rate.
The Relation of the Forward Price to the Price Expected in the Future
On page 129 the text introduces a, the continuous expected rate of return on a
stock, and says that at time T the expected value of the stock is E0 (ST) = S0eaT.
Since the forward price of the stock is S0erT, the forward price is not the same
as the expected price in the future. On page 140 in Derivatives Markets this
observation is repeated, but page 140 may be a bit confusing because the
meanings of a and r are changed there. On page 140 these variables are used
for annual effective rates, while on page 129 they stood for continuous rates.
Here the text denotes the expected value at time 1 as Ex (S0), which is also a
slight change in notation. The discussion is simple to follow once you resolve
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Page M12-16
Module 12 - Review of Derivatives Markets, Chapter 5
the notational issue. The one year expected value of the stock, the one year
forward price and their difference are given by:
Expected value of stock: S0 (1 + a)
Forward price: S0(l + r)
Difference: S0 (1 + a) - S0 (1 + r) = S0 (a - r).
For example if S0 = 1000, a = .13 and r = .04, the expected stock price in one
year is 1130 and the forward price in one year is 1040. The difference is
90 = 1000(.13-.04) = 1000(.09). However the forward price is certain, while the
expected price in the future is the mean of a distribution of future prices which
may be higher or lower than 1040. The forward purchaser has a certain price,
while the stock owner has risk. On the average the risk-taking stock owner will
earn (a - r) = .09 or 9% more than the forward purchaser who has eliminated the
risk. The percentage given by (a - r) = .09 is referred to as the risk premium for
the stock.
The Cost of Carry in the Forward Pricing Formula
The general forward price formula is for a continuous interest rate r and a
continuous dividend rate 8 is F0tT = S0e(r"^T. In most (but not all) applications
that we will see, the interest rate r will be greater than the dividend yield 8.
Thus we will have F0,T = S0e{l"5)T > S0
For example, with r = .04,8 = .02 and S0 = 1000 the forward price is
Fo,T = S0e{r-S)T = lOOOe01 = 1010.05 .> S0 = 1000
The difference of 10.05 in the forward and current prices is referred to as the
cost of carry.
Cost of Carry = F0,t - S0
To justify this terminology, we will first review a bit of calculus. For small
values of x,
ex «1 + x and ex -1« x.
Using this approximation, we can see that the difference between the forward
and current prices is
Foj - So = S0e{r~6)T - So = So (e{r-5)T -1) * S0 (r - 8) T.
Cost of Carry *S0(r-8)T
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Module 12 - Review of Derivatives Markets, Chapter 5 Page M12-17
In the preceding example
Fo.t - S0 = 10.05 « So (.04 - .02) .5 = 10
Recall that one way to have the stock at time T is to buy it now by borrowing
the current price at the rate r and buying the stock now. If the stock pays
dividends, you can buy a tailed position instead of a full share because you will
earn dividends at the rate 8. Thus your net paid interest rate to buy the stock
now and hold (carry) it to time T is r - 8, and the total amount that you pay to
carry the stock to time T is approximately S0(r-8)T, the cost of carry.
The text notes that if you borrow the stock you must pay the dividends back to
the owner of the stock. This payment for use of the stock is like a lease
payment on a property, so the dividend rate is referred to as the lease rate. The
text has a word equation for the cost of carry.
Cost of Carry = Interest to carry asset - Asset lease rate.
This is equivalent to saying
Cost of carry = rS0T - 8S0T.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M12-18
Module 12 - Review of Derivatives Markets, Chapter 5
Section 12.5
Futures Contracts
Futures contracts are exchange traded forward contracts. Section 5.4 of
Derivatives Markets discusses the real world of futures contracts, with special
emphasis on the S&P 500 Futures Contract.
Terminology
Exchanges like the Chicago Board Options Exchange and the Chicago
Mercantile Exchange provide the opportunity for investors to trade futures
contracts without seeing their counterparties. The forward contracts that are
traded as futures are standardized. Most of our readers have probably seen
pictures of traders on the floor of an exchange, trading by open outcry. Futures
can also be traded electronically. The exchange has a clearinghouse which
matches buys and sales and keeps track of trading details. The clearinghouse
will simplify things by serving as the counterparty for both the buyer and the
seller.
Derivatives Markets lists five ways in which futures can differ from forward
contracts.
• Forward contracts are settled at expiration. Futures are tracked daily
and marked to market to establish the daily status of buyer and seller.
• Futures contracts are liquid. If you have a contract to sell in December,
you can effectively cancel it by entering an offsetting futures contract to
buy in December. This may have a cost. If you offer this week to sell at
1000 and want to offset next week, you will have to offer to buy at next
week's price, which most likely will not be the same. You can make or
lose money in the process.
• Futures contracts apply only to certain assets and have specified terms.
Forward contracts can be customized to fit any asset and any
specifications agreed upon by buyer and seller. One of a kind forwards
are referred to as traded over the counter.
• With a forward contract the counterparty could fail to perform, leading
to credit risk. Futures contracts are designed to keep credit risk to a
minimum.
• Futures contracts have built in price limits which stop trading when
prices suddenly move by a large amount. The text notes that the S&P 500
has a 5% limit on down moves and further limits on subsequent moves.
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Module 12 - Review of Derivatives Markets, Chapter 5
Page M12-19
The S&P 500 Futures Contract
This contract is based on the value of the S&P 500 index. On September 26, 2006
(the day before this section was written) that index had a closing value of
1336.34, published (among other places) in the Wall Street Journal The dollar
amount of the contract, referred to as the notional amount, is defined to be
250 x Index value = 250 x 1336.34 = 334,085.
This contract is cash settled, and no stocks are delivered. If you have entered a
futures contract to sell at 1336. 34 and the index goes up by 1 to 1337.34 at
expiration, you can cash settle for
1x250 = 250.
The text has more detail on the standard features of the S&P 500 contract on
page 144. It would be useful to look at futures reports in papers like the Wall
Street Journal The values for the December S&P contract were reported on
September 27, 2006 as
Open High Low Settle Open Interest
1335.70 1 1347.10 | 1333.50 1 1346.70 1 1,379,880
Open interest refers to the total number of open positions pairing one buyer to
one seller. Note that the futures price varied considerably during the day.
Futures trading can be exciting. Note also that the above futures prices
occurred on a day in September when the closing value of the actual index was
1336.34. Settlement for the December contract will be based on the opening
price of the S&P 500 on the third Friday of December, with trading ending the
day before.
Margins and marking to market
A buyer or seller of a futures contract must make a deposit referred to as
margin to protect the counterparty from risk. The margin account earns
interest, so it is not a premium cost. However gains or losses on the futures
contract are applied to it daily.
At expiration of the contract, the adjusted margin account is returned to the
depositor -reflecting the overall gain or loss. There is additional protection for
the counterparty. If the margin account gets too low, a new deposit is required
to maintain the account at a minimum level called the maintenance margin. The
text states that the maintenance level is often set at 70% to 80% of the original
margin requirement.
The request for margin is referred to as a margin call. If you fail to meet a
margin call, the broker will close your account by taking an offsetting position
and giving back to you whatever is left in the margin account after the closing.
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Page M12-20
Module 12 - Review of Derivatives Markets, Chapter 5
Example on page 144-147:
Derivatives Markets next goes through an example of an S&P 500 futures
position, and we will review it here. It is worth noting in advance that
calculations are done here using continuous interest at 6% for the margin
account. McDonald notes on page 148 that in practice continuous interest is not
used, but differences are small enough for the example to be a good illustration.
The example deals with an investor who wishes to acquire long futures
contracts with a notional value of 2.2 million dollars when the futures price (not
the current value of the index) is 1100. The notional value of a single contract is
250x1100 = 275,000
The number of contracts needed to have a notional value of 2.2 million is
2'200'000= 8 contracts.
275,000
If the margin requirement is 10%, we need
Initial Margin = 2,200,200 (.1) = 220,000
The text looks at the effect of changes in the index by first considering the
effect of a one point drop in the index from 1100 to 1099. The investor is long
and thus now obligated to buy the index for 1100 when is worth 1099. This is a
loss of 1 x 250 on each contract. For the position with 8 contracts we have
Loss for drop of 1 in index = 8 x 250 = 2000.
If the index changes from the futures price of 1100 by an amount AS instead of
1, the change in the notional amount is
Change in notional value= AS (2000).
The text makes this example simpler by using weekly instead of daily
settlement. With continuous interest, at the end of one week the initial margin
account of 220,000 would have grown (at 6% continuous interest for 1/52 of a
year) to
e .06/52 (220,000) = 220,253.99
The text looks at what would happen if at the end of one week the index had
dropped by 72.01 points. In this case
Change in notional value= AS(2000) = -72.01(2,000) = -144,020.
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Module 12 - Review of Derivatives Markets, Chapter 5
Page M1251
Thus the margin account has dropped to
Margin account in one week = 220,253.99 -144,020 = 76,233.99
Suppose that the maintenance margin is 70%. Then the required margin
account is
Required margin in one week = 220,000 x .70 = 154,000.
There will be a margin call. The investor has two options:
Option 1
Respond to the call by putting 154,000 - 76,233.99 = 77,766.01 in the
account.
Option 2
Close the contract by entering an offsetting futures contract to sell at
1099. The broker returns the 76,233.99 that is still in the account to the
investor. The investor has a loss, since he deposited 220,000 a week ago
and only received 76,233.99 in a week.
The text does not discuss Option 1, but we have included it to make the choices
more explicit.
The text follows with a table following the futures contract in our example
through 10 weeks of hypothetical index price changes. The purpose of the table
is to show that there can be a slight difference in the payoff of a forward and a
futures contract.
If you are not reading carefully, you may find the table a bit confusing. We
have just looked at the maintenance margin concept, but the table ignores
maintenance margin in its analysis. The header of the table states: "The last
column does not include additional margin payments." We will first review the
table and discuss its logic. Then we will look at the pricing difference that the
table illustrates. The table follows on the next page.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M12-22 Module 12 - Review of Derivatives Markets, Chapter 5
Price Margin
Week Multiplier Price Change AS Balance
1 °
1
2
3
4
C/1
6
7
8
9
10
2,000
2,000
2,000
2,000
2,000
2,000
2,000
2,000
2,000
2,000
2,000
1,100.00
1,027.99
1,037.88
1,073.23
1,048.78
1,090.32
1,106.94
1,110.98
1,024.74
1,007.30
1,011.65
-72
10
35
-24
42
17
4
-86
-17
4
220,000.00
76,233.99
96,102.01
166,912.96
118,205.66
201,422.13
234,894.67
243,245.86
71,046.69
36,248.72
44,990.57 |
The margin account balance for any week is given by
(Margin Balance for Prior Week ) x (e06/52) - AS(2000).
This is just an extension of what we did for the first week. Now we can compare
the profit of this futures contract to the profit of a forward contract for the
same amount.
Futures Contract Profit
With a futures contract, all your gains and losses are marked into the margin
account each week. What you get back in 10 weeks is the account balance of
44,990.57, which contains all your profit and loss. You initially deposited
220,000, but in 10 weeks at 6% continuous interest that deposit should have
grown to 220,000e06(10)/52 = 222,553.16. Thus you have a loss of
44,990.57 - 222,553.16 = -177,562.59 (the text differs from this by a penny)
Forward Contract Profit
With a forward contract, there is no margin account and no interest. Every
dollar loss still costs you 2000, but you only look at the final value of the index
(1011.65) to find your cash settlement. You have a loss of
(1011.65 - 1100) x 2000 = -176,700.
The final conclusion is that futures and forward prices can be different due to
the effect of interest earned on mark to market proceeds. However, there is
some tricky discussion following this conclusion in the text. (Do keep in mind
that this general statement follows from looking at only one example.)
In the earlier part of this chapter we stressed using tailed positions for pricing.
In Appendix 5B the text shows that if you adjust the position in this example to
a tailed position every week, the profits from the future and forward will be the
same. This all gets trickier in the next section, because the example here had
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 12 - Review of Derivatives Markets, Chapter 5
Page M12-23
the same interest rate of 6% throughout the 10 weeks. In practice the interest
rate on the margin account is a short term rate that can reset randomly from
week to week. The next section looks at this case.
How do the forward and futures prices compare if the interest rate varies
randomly? The text gives an answer based on general reasoning. The key factor
is the correlation between the index and the interest rate. Positive correlation
means that the interest rate is more likely to go up when the futures price goes
up. Negative correlation means the interest rate is more likely to go down when
the futures price goes up. (Stock market followers tend to think that stocks will
go up as interest rates go down). The text states:
1. "...when the interest rate is positively correlated with the futures
price, the futures price will exceed the price on an otherwise
identical forward contract."
2. "...when the interest rate is negatively correlated with the futures
price, the futures price will be less than an otherwise identical
forward price."
The reasoning behind the statement for positive correlation is that increases in
the index will raise the account balance and simultaneously give you higher
interest on it. That works in your favor. Similar statements are made for
negative correlation. There is no mathematical analysis of these assertions.
Actual Arbitrage Opportunities
On page 147, Derivatives Markets has a section which analyzes the actual
futures contract situation on August 30, 2004. On that day you could observe the
following values:
Closing S&P 500 Index Value: 1099.15
Closing S&P 500 December Futures Price: 1099.30
To look for arbitrages you need the theoretical price S0e(r"J)T. That price
depends on the risk-free rate r and the dividend yield 8. The text gives one
estimate for 8 and two possible rates for r.
Dividend yield 8 = .0175
Yield r on a U.S Treasury bill maturing in December 1.56%
London Interbank Offer Rate r (LIBOR)1 1.86%
The December futures contract expires on the third Friday of the expiration
month. The text gives the date of December 17, 2004 for expiration. This is 109
109
days from August 30, 2004. Thus the futures contract has a time of T = .
1 This yield is inferred from values of Eurodollar futures. The method for this is in section 5.7 which is not on
the Exam FM syllabus.
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Page M12-24
Module 12 - Review of Derivatives Markets, Chapter 5
Now you have all the numbers you need to calculate the theoretical futures
price and look for an arbitrage. There will be two calculations because there
are two possible interest rates.
Treasury Bill price: S0e{r~s)T = i099.15e(0156-0175)(109/365) = 1098.53
LIBOR price: S,e[r~5)T = i099.15e(0186-0175)(109/365) = 1099.51
Note that the actual futures price of 1099.30 is in between the Treasury Bill
price and the LIBOR price. The Treasury Bill price has a negative cost of
carry, and the LIBOR price has a positive cost of carry.
The ultimate conclusion is that you don't know if there is an arbitrage and more
analysis is needed (page 149). The text makes some additional observations.
• The dividend yield is a forecasted number and may not turn out to be
what you have predicted. There is risk in this.
• In the real world there are transaction costs which lead to no-arbitrage
intervals instead of no-arbitrage prices. We have already reviewed this.
• The interest rate used depends on issuer credit risk. In practice LIBOR
is more commonly used.
• If you arbitrage the S&P index, you will only buy a sample of the stocks
in the index as your assets. Your sample may not track the index as
desired.
Quanto Index Contracts
There is a brief discussion of this topic on page 149 of Derivatives Markets.
Quanto index contracts address the problem of currency risk for investors who
use dollars but want to invest in foreign stock indices like the Japanese Nikkei
225 index. If you buy Japanese stocks, you must convert your dollars to yen and
buy the stocks with yen. When you later sell your Japanese stocks, you will be
paid in yen and then need to convert the yen back to dollars. If the stock values
do not change but the value of the dollar has risen between purchase and sale,
you will lose money on the currency exchange even though the stocks have the
same value.
The Chicago mercantile exchange has a Nikkei index futures contract that is
dollar denominated to eliminate the currency risk. A single contract has
notional value of $5 x Nikkei Index and the contract is cash settled in dollars.
Thus the need to exchange currencies is eliminated.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 12 - Review of Derivatives Markets, Chapter 5 Page Ml2-25
Section 12.6
Module 12 Summary
Pricing Prepaid Forward Contracts for a Stock with no Dividends
• Pricing by analogy. Same position at time T as someone who buys the
stock now for S0 and holds it until time T. Pay S0 now for the prepaid
forward.
• Pricing by discounted present value. Interest rate r, expected stock
appreciation a.
E0(ST) = S0eaT
Foj — & -Eo \StJ = So
• Pricing by arbitrage.
a) The prepaid forward price is higher than the current price.
(FofT > So). Buy the index for S0 and simultaneously sell a prepaid
forward for F0pfT > S0.
b) The prepaid forward price is lower than the current price. (F0P,T <S0)
Sell the index short for S0 and simultaneously buy a prepaid forward
for F0pt < So.
Pricing Prepaid Forward Contracts for a Stock with Dividends
Discrete dividends.
If n dividends di,...,dnare paid over n periods at times l,...,nand the
continuously compounded interest rate per period is r, the price of a prepaid
forward for delivery at time T will be
FoPT=So-Y,die-ri
Continuous dividends.
Suppose that dividends are paid on a stock index at the continuous rate 5. The
continuous dividend model assumes that all dividends are reinvested in the
stock index.
Fqj — e So
Finding the Forward Price
No dividends F0,t = S0e
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Page M12-26
Module 12 - Review of Derivatives Markets, Chapter 5
Discrete dividends
Fo,T=erT
So-J d^ =erTS0-]>>e-
i=l
r(T-i)
i=l
Continuous dividends
F0tT=erTe-STSo=Soe{r-s)T
Forward premium =
Fq,t
So
Annualized Forward Premium = — In
T
Creating Synthetic Stocks, Forwards and Bonds
Ho,r
V ^o J
Stock at time T =long forward + zero-coupon bond
The steps are:
1) At time 0, invest S0e~ST in a bond with yield rate r and maturity at T.
2) At time 0, enter into a zero cost forward for the forward price of
Fo,T=Soe{r-s)T.
3) At time T, collect the bond proceeds of erTS0e~5T = S0e(r^)T
4) At time T, use the bond proceeds to buy the stock for the forward price
of S0e(r"OT.
Long Forward = Stock - zero coupon bond
The steps are:
1) Borrow S0e~ST at time 0.
2) Use the borrowed amount S0e~ST to buy a tailed position in the stock at
zero cost.
3) At time T, you will have the stock worth ST
4) Repay the loan for by paying S0e(r~s)T = F0>T. This will leave with
ST - S0eir~s)T =ST - F0,T, the payoff on a long forward contract.
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 12 - Review of Derivatives Markets} Chapter 5 Page M12-27
Zero-coupon bond = Stock - forward
Restated:
To create a zero-coupon bond with maturity T:
Buy a tailed position in the stock and sell a forward contract.
The steps are:
1) Invest S0e~ST to buy the tailed position in the stock at time 0.
2) Sell a forward obligating you to sell the stock at time T
torS0e™T=F0J.
3) At time T, you will sell the stock for S0e(r^)T = erS0e-5T
4) Thus you have invested S0e~ST at time 0 and been paid erS0e_JT at time
T. This is a zero-coupon bond paying the risk-free rate r..
Hedging and Arbitrage with Synthetic Forwards
Cash and Carry Hedge:
To hedge a (short) forward sale, offset it with a synthetic forward purchase.
Cash and Carry Arbitrage:
The forward price offered is too high, or F0>T > S0e(r"^T. Arbitrage
Buy low and pay : S0e(r"*)T.
Sell high and receive: F0,t .
Profit: Fo,T-S0e{r-s)T.
Reverse Cash and Carry Hedge:
To hedge a (long) forward purchase, offset it with a synthetic forward sale.
Reverse Cash and Carry Arbitrage:
The forward price offered is too low, or F0>T < S0e(r"J)T.
Buy low and pay : F0>T
Sell high and receive: S0e{r~s)T
Profit: S0e{r-S)T -F0,T.
The Implied Repo Rate
1 i / For I
—In n ' =r.
T {S0e-STJ
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M12-28
Module 12 - Review of Derivatives Markets, Chapter 5
The Relation of the Forward Price to the Price Expected in the Future
a, the continuous expected rate of return on a stock.
Expected value of stock: S0 (1 + a)
Forward price: S0(l + r)
Difference: S0 (1 + a) - S0 (1 + r) = S0 (a - r).
(a - r) is referred to as the risk premium for the stock.
The Cost of Carry in the Forward Pricing Formula
Cost of Carry = For — So
Cost of Carry *S0(r-S)T
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 12 - Review of Derivatives Markets} Chapter 5 Page M12-29
Section 127
Solutions to Odd-Numbered Problems
s.i
New table 5.1 from point of view of seller.
Description Receive Payment Deliver Security Payment
at Time at Time Received
Outright Sale
Sale to fully
leveraged buyer
Prepaid Forward
Sale
Forward
Contract
0
T
0
T
0
0
T
T
So
S0en
PV(Dividends)
S0e^T
S.3.
a) Prepaid forward: S0e"^ = 50e"08 = 46.16
b) Forward contract: S0e{r-S)T = 50e(06"08)1 = 49.01
S.S
a) Forward price for T=.75: S0e(r^)T = 1100e(05-°)75 =1142.03
b) Your opposite position is to hedge the short forward is a long forward at
1142.03. To hedge long this position you create a synthetic short forward
for the same price.
-Forward = -Stock + zero coupon bond
Short sell the stock today for S0 = 1100 and invest that 1100 at the risk
free rate r = .05. In nine months you will have to deliver the stock, and be
paid the amount HOOe05(75) = 1142.03. Buy the stock under the long
forward contract that you are hedging using the amount of 1142.03 from
the bond and deliver the stock to cover the short sale. The payoff is 0.
This is a reverse cash and carry hedge of a long forward.
c) You have entered a short forward contract to sell the customer the stock
in nine months for 1142.03. To hedge this position you create a synthetic
long forward for the same price.
Forward = Stock - zero coupon bond
Buy the stock today for S0 = 1100, and borrow that 1100 at the risk free
rate r = .05. In nine months you will have the stock, and owe the amount
lOOOe05(75) = 1142.03 on the borrowing. Sell the stock under the forward
contract that you are hedging and you will have 1142.03 to pay off the
loan. The payoff is 0.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M12-30
Module 12 - Review of Derivatives Markets} Chapter 5
S.7
Note that T = .5 and the no-arbitrage price is 1100e05(5) = 1127.85.
a) The six month forward price is 1135, higher than the no-arbitrage price.
You will buy the index low today and sell forward high. Sell the index
forward for 1135, and borrow 1100 at 5% to buy an index share
immediately. Your investment is 0. In 6 months you will deliver the
index share to the forward buyer, who will pay you 1135. You must pay
1100e05(5) =1127.85 to pay off the loan. This leaves you with a riskless
profit of 1135-1127.85=7.15.
b) The six month forward price is 1115, lower than the no-arbitrage price.
You will sell the index short today, and buy it forward at a low value.
Enter into a forward purchase contract to buy the index for 1115. Then
sell the stock short for 1100 today, and invest the 1100 in a zero coupon
bond at 5%. The bond will pay you back 1100e05(5) =1127.85 in six
months. Use the forward purchase contract to buy the index for 1115.
This leaves you with a riskless profit of 1127.85 - 1115 = 12.85.
5.9
a) The money manager could travel to 1981, invest money, travel back to
1982 and immediately collect the final amount and then take the
increased funds to 1981 and invest them. Do this n times and you will
increase your original amount by 1.125n. The sky is the limit.
b) Too many trades at 12.5% will begin to drive the rates down in a
competitive market. This won't last.
c) Time travel may come to pass, but free unlimited accumulation of money
will not. The time travel cannot be costless.
S.ll
a) The notational value of four contracts is: 4(250)(1200) = 1,200,000
because each index point is worth $250, and we buy four contracts.
b) This is 10% of the notational value, or $120,000
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 12 - Review of Derivatives Markets, Chapter 5
Page M1231
S.13
This question really asks us to verify the word equation.
Stock = long forward + zero coupon bond for tailed value of stock.
It is a proof type question, and not typical of exam questions -but we will
discuss it.
This verification has already been done for the continuous case with interest
rate r and dividend rate 8 in table 5.4 of the text. This gives the answer to
part c), and to part a for the special case where 8 = 0. Part b) is all that
needs to be done. If the stock pays discrete dividends, the value at time 0 of
a tailed position in the stock is the current value of the stock less the present
value of anticipated future dividends. This is
i=l
You lend this amount. The long forward has not cost initially. The forward
price is
Fo,T=erTSo-Jd^(T-^.
i=l
At time T you will have the value of a single share of stock, as we see below
Forward profit - repayment of loan
= ST -F0,T - erT(Loan amount)
= ST-erTS0 + J die"*™- erTfs0-J^e""'!
i=l V i=l J
= St
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M12-32
Module 12 - Review of Derivatives Markets, Chapter 5
Section 12.8
Module 12 Computational Review Problems
1. (1 pt) A $ 75 stock pays $ 4.5 every 3 months, with the
first dividend coming 3 months from today. The continously
compounded risk-free rate is 8 %.
a) What is the price of a prepaid forward contract that expires 1
year from today, immediately after the fourth-quarter dividend?
$ ?
b) What is the price of a forward contract that expires at the
same time?
$ ?
ANSWER1: 57.87
ANSWER2: 62.69
2. (1 pt) A $ 85 stock pays an 5 % continous dividend. The
continously compounded risk-free rate is 11 %.
a) What is the price of a prepaid forward contract that expires 1
year from today, immediately after the fourth-quarter dividend?
$ ?
b) What is the price of a forward contract that expires at the
same time?
$ ?
ANSWER1: 80.85
ANSWER2: 90.25
3. (1 pt) Suppose the stock price is $ 50 and the continously
compounded interest rate is 10 %.
a) What is the price of a 4 - month forward price, assuming
dividends are zero?
$ ?
b) If the 4 - month forward price is $ 51.1, what is the annualized
forward premium?
%?
c) If the 4 - month forward price is $ 51.1, what is the annualized
continous dividend yield?
%?
ANSWER1: 51.69
ANSWER2: 6.52844753445381
ANSWER3: 3.47155246554619
4. (1 pt) Suppose the S and P 500 index futures price is
currently 1135. You wish to purchase 6 futures contracts on margin,
a) What is the notational value of your postition?
$ ?
Assuming a 10 % initial margin, what is the value of the initial
margin?
$ ?
ANSWER1: 1702500
ANSWER2: 170250
ANSWER3:
5. (1 pt) Suppose the A and T index spot price is 1175 and
the continuously compounded risk-free rate is 9 %. You observe
a 6 - month forward price of 1176.95.
What
dividend
%?
yield is implied by this forward price?
ANSWER1: 8.67
©ACTEX 2009
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Exam FM / Exam 2 - Financial Mathematics

Module 12 - Review of Derivatives Markets, Chapter 5 Page M12- 33
Section 12,9
Supplemental Exercises
1. A stock has current price S0 = 50. The annual continuous interest rate
and dividend yield are r = .03 and 8 = .01. If the expiration time for a
forward contract is T = .25, what is the difference between the forward
price and the prepaid forward price.
A) 0.12 B)0.25 C)0.37 D) 0.49 E) 0.62
2. A stock has current price S0 = 50. The annual continuous interest rate is
r = .03. Semiannual dividends of $1 will be paid in six months and one
year. What is the price of a one year prepaid forward?
A) 50 B) 49.03 C) 49.02 D) 48.04 E) 48
3. A stock has current price S0 = 50. The annual continuous interest rate is
r = .04. If the expiration time for a forward contract is T = .25 and the
forward price is 50.30, what is the continuous dividend yield 8 ?
A) 0.01 B) 0.016 C) 0.020 D) 0.025 E) 0.03
4. A stock has current price S0 = 46. The annual continuous interest rate is
r = .035 and the continuous dividend yield is 8 = .01. You observe a one
year prepaid forward price of 45.60. Which of the following is true?
A) No arbitrage is possible.
B) You can create an arbitrage by buying one prepaid forward and
selling one share of the stock short
C) You can create an arbitrage by selling the prepaid forward and
buying one share of the stock.
D) You can create an arbitrage by buying the prepaid forward and
selling e"01 shares of the stock short
E) You can create an arbitrage by selling the prepaid forward and
buying e"M shares of the stock .
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M12-34
Module 12 - Review of Derivatives Markets, Chapter 5
5. A stock has current price S0 = 50. The annual continuous interest rate is
r = .035 and the continuous dividend yield is 5 = .01. You observe a one
year prepaid forward price of 49.50. Which of the following is true?
A) No arbitrage is possible.
B) You can create an arbitrage by buying one prepaid forward and
selling one share of the stock short
C) You can create an arbitrage by selling the prepaid forward and
buying one share of the stock.
D) You can create an arbitrage by buying the prepaid forward and
selling e"01 shares of the stock short
E) You can create an arbitrage by selling the prepaid forward and
buying e"01 shares of the stock
6. The S&R index has a spot price of S0 = 1000 . The continuous interest
rate is r = .04 and the continuous dividend yield is 8 = 0 You observe a six
month forward price of 1050. What arbitrage profit can be made in 6
months?
A) 0 B) 10.06 C) 15.73 D) 29.80 E) 30.17
7. The S&R index has a spot price of S0 = 1000 . The continuous interest
rate is r = .03 and the continuous dividend yield is 8 = 0 The one year
forward price is 1030.45. You enter into a forward sale contract and buy
the index. Which of the following positions is this equivalent to:
A) A short sale of the index.
B) Purchase of a one year zero-coupon bond with r = .03
C) A reverse cash and carry hedge.
D) A cash and carry arbitrage
E) None of these.
8. The S&R index has a spot price of S0 = 1000. The continuous interest
rate is r = .03 and the continuous dividend yield is 8 = 0 The one year
forward price is 1030.45. Which of the following positions results in a
synthetic long forward contract:
A) Sell the index short for 1000 and lend the proceeds at r = .03
B) Sell the index short for 1000 and borrow 1000 at r = .03
C) Borrow 1000 at r = .03 and buy the index.
D) Borrow 1000 at r = .03 and sell the index short
E) None of these.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 12 - Review of Derivatives Markets, Chapter 5
Page M12- 35
9. The S&R index has a spot price of S0 = 1000 . The continuous interest
rate is r = .03 and the continuous dividend yield is 8 = 0 The one year
forward price is 1030.45. Which of the following positions results in a
synthetic purchase of a share of the index:
A) Enter into a long forward contract.
B) Enter into a long forward contract and borrow 1000 at r = .03
C) Buy a zero-coupon bond for 1000 at r = .03 and enter into a long
forward contract.
D) Borrow 1000 at r = .03 and sell the index short
E) None of these.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M12-36 Module 12 - Review of Derivatives Markets, Chapter 5
Section 12.10
Supplemental Exercise Solutions
1. The forward price and prepaid forward price are
F0,o.2s =50e(03-01)-25 =50.25 , F^o.25 = 50e(-01)-25 =49.88
The difference is 50.25-49.88 = 0.37.
Answer C
2. The present value of dividends is
PV(div) = le-03(5) + le"03 = 1.96
The prepaid forward price is
S0 - PV (div) = 50 -1.96 = 48.04
Answer D
3.
F0tT = S0e{r~s)T -> 50.30 = 50e(04^25
In
( 50.30
V
50
£ = .016
.01 -.25 J
Answer B
4. The correct prepaid forward price is S0e~ST = 46e"01 = 45.54. Thus the
forward price of 45.60 is too high. You can sell the forward for 45.60 and
buy a tailed position in the stock for a price of S0e~ST = 46e"01 = 45.54.
This gives a profit of .06 at time 0. In one year the tailed position in the
stock will have grown to a full share, and that can be delivered to satisfy
the forward contract.
Answer E
5. The correct forward price is S0e"*T = 50e~01 = 49.50. Thus the market
price is correct and there is no arbitrage.
Answer A
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 12 - Review of Derivatives Markets, Chapter 5
Page M12- 37
6. The forward price should be lOOOe04( 5) = 1020.20. Thus you can create an
arbitrage by entering a forward sale contract at the price of 1050, and
borrowing 1000 to buy the stock today.
In six months you will deliver the share of stock and receive the forward
price of 1050. The loan repayment due is lOOOe04(5) = 1020.20. Thus there
is a profit of 1050 - 1020.20 = 29.80
Answer D
The next three problems all make use of the identity below or variants of it.
STOCK = LONG FORWARD + ZERO COUPON BOND
7. Your position is - LONG FORWARD + STOCK. This is equivalent to the
purchase of a zero-coupon bond at r = .03
The forward price is the correct theoretical price.
Answer B
8. STOCK-ZERO COUPON BOND = LONG FORWARD
Thus you buy the index for 1000 and sell a zero coupon bond for 1000
(borrow the money to buy the stock.).
Answer C
9. STOCK = LONG FORWARD + ZERO COUPON BOND
Thus you enter into a long forward at the price of 1030.45 and invest 1000
(lend) at r = .03.
Answer C
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 13 - Review of Derivatives Markets, Chapter 7
Page M13- 1
Review of Derivatives Markets,
Chapter 7
Section 13.1
Why this Chapter is Necessary
Chapter 7 is not included in the syllabus for Exam FM/2, but discussion of
Sections 7.1 and 7.2 will make it easier to understand section 8.2, which is
included on the syllabus. The particular items of interest are:
a) In section 7.1 Derivatives Markets covers spot rates and implied forward
rates. The notation used in section 7.1 is completely different from the
notation from the Broverman text used previously in Module 6 of this
guide. However this new notation is used again in section 8.2. Thus it is
helpful to review this notation. In fact, this new notation is also used in
later chapters which are required for the next exam, MFE.
b) In section 7.2 Derivatives Markets covers forward rate agreements for
interest rates. This material is needed for complete understanding of
interest rate swaps in section 8.2.
The coverage here is just a quick review, and that is all you will need. No
homework problems are needed.
We have also included a review of Eurodollar futures and LIBOR (from section
5.7). This material is not on the FM/2 syllabus, but it is used in section 8.2.
©ACTEX 2009
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Exam FM / Exam 2 - Financial Mathematics

Page M13-2
Module 13 - Review of Derivatives Marketsy Chapter 7
Section 13,2
New Notation for Spot and Forward Rates
In Module 6 of this guide, we used the notation sn for the n-year spot rate and
the notation in_lin for the implied forward rate in year n. The spot and forward
rates were related by
1 + *"-i." = 7^ T^T or equivalents (1 + sn)n = (1 + sn_i)n_1 (1 + in_i,n).
(1 + Sn-l)
Derivatives Markets uses the notation P(ti,t2) for the price of a zero coupon
bond that is purchased at time ti and pays 1 at time t2. The special case
P (0, n) gives the spot rate sn, since
(13.1)
In Derivatives Markets the implied forward rate at time 0 for the time interval
(ti,£2) is denoted by r0 (tlft2). Thus we have
(13.2)
in-i,n =r0(n-l,n)
The spot rate sn is denoted by r(0, n).
(13.3)
P(0,n-l) = P(0,n)(l + r0(n-l,n))
On page 206 in Table 7.1, the text gives a yield curve example which provides a
good review of these concepts.
Zero- Zero- One Year Continuously
Years to coupon coupon Implied Par Compounded
Maturity Bond Yield Bond Price Forward Rate Coupon Zero Yield
1
2
3
6.00%
6.50%
7.00%
0.943396
0.881659
0.816298
6.00000%
7.00236%
8.00705%
6.00000%
6.48423%
6.95485%
5.82689%
6.29748%
6.76586%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 13 - Review of Derivatives Markets, Chapter 7
Page M13- 3
We will work through the final row for year 3 to illustrate the calculations and
review both notations.
Zero-coupon Bond Yield s3 = 7.00%
Zero-coupon Bond Price P(0,3) = =- = r- = .816298
V ' ; (l + s3)3 1.073
One Year Implied Forward Rate
(l + M.(l + MW)).fg.l«^.1J8M705
Par Coupon This is the percentage coupon c which would give a bond with
redemption value of 1 an initial price of 1. We obtain the par coupon by solving
the bond pricing present value equation below for c.
^ c c c 1
1 = + =- + ;- + -
1.06 1.0652 1.073 1.073
1-1/1 073
c = ' , =- = 0.0695485
1/1.06 + 1/1.0652+1/1.073
In the notation of Derivatives Markets the value of c would be written as
1-P(0.3)
P(0,l) + P(0,2) + P(0,3)'
This is a special case of equation (7.6) on page 210 of the text.
The concept of par coupon bond rate will come up when interest rate swaps are
discussed in section 8.2 of Derivatives Markets.
Continuously Compounded Zero Yield.
Derivatives Markets denotes the continuously compounded yield on the
interval (0,n) by rcc (0,n). For n=3, we have
P(0,3) = e-rCC(0'3)3 -> -JL_ = e-cW
v ; 1.073
Taking logs of both sides of the above equation we get
-31n(1.07) = -3rcc(0,3) -> rcc (0,3) = ln(1.07) = .0676586
In general,
rcc(0,n) = ln(l + sn) = V V ' }).
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M13-4 Module 13 - Review of Derivatives Markets, Chapter 7
Section 13.3
Forward Rate Agreements
Section 7.2 introduces forward rate agreements, which are used to hedge
interest rate risk. As we indicated at the beginning of this chapter, the syllabus
material in section 8.2 requires a basic understanding of what a forward rate
agreement is. Fortunately, we need only cover some very basic material that is
on pages 214 and 215 of Derivatives Markets .It is not necessary to master the
remaining material in Section 7.2.
The text says that "A forward rate agreement (FRA) is an over the counter
contract that guarantees a borrowing or lending rate on a given notional
principal amount." The text clarifies this with an example, and we will discuss
that example in a slightly different way here.
The example concerns a firm that will need a 91 day (one quarter) loan for 100
million dollars in 120 days. The firm will pay whatever the current quarterly
loan rate is in 120 days, but cannot guarantee what that rate will be. The text
denotes the unknown rate that will be put on the loan in 120 days as rquarteriy.
However there is a future quarterly rate that is known -the implied forward
rate today for the quarter beginning in 120 days is 1.8%.
Today
h-
Implied
forward
1.8%0
If the company could find a lender who would commit today to a loan at the
implied quarterly forward rate of 1.8%, it would eliminate uncertainty and have
a guaranteed rate of 1.8%.
An FRA would achieve that guarantee in a different way. The necessary FRA
for this company covers a loan of 100m, so the FRA would be given a notional
amount of 100m. The company would like to assure a rate of 1.8% for the
quarter, so it will set up a rate for the FRA of rFRA = 1.8%.
Under the FRA the company and its counterparty would agree today that on the
repayment day in 211 days the company would be guaranteed a payment of
(rquarteriy ~ rFRA ) X notional amOUnt = (rauarterly - 1.8%) X 100,000,000.
If the above quantity is positive the company gets the payment, and if it is
negative the company makes a payment of that size. Thus the company gets or
makes a payment that brings its quarterly interest rate back to 1.8%. The text
looks at two examples to illustrate how this works:
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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120 days 91 days later (211 days total)
Borrow Repay loan
100mm
Loan rate?

Module 13 - Review of Derivatives Markets, Chapter 7
Page M13- 5
1) Interest rate in 120 days is higher than 1.8% and the loan is made at 2%.
Then at repayment time the company receives a payment of
(Quarterly - 1-8%) X 100,000,000 = (2% - 1.8%) X 100, 000, 000 = 200,000,
This lowers the net interest rate from 2% to 1.8%.
2) Interest rate in 120 days is lower than 1.8% and the loan is made at
1.5%. Then at repayment time the company makes a payment of
(Quarterly - 1.8%) X 100,000,000 = (1.5% - 1.8%) X 100,000,000 = -300,000,
This raises the net interest rate from 1.5% to 1.8%.
The text has further detail on FRAs, but the simple ideas above are all you
need to know for understanding section 8.2.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M13-6
Module 13 - Review of Derivatives Markets, Chapter 7
Section 13,4
LIBOR and Eurodollar Futures
The material here is covered in Chapter 5 and is intended to help with
understanding the material in section 8.2 starting on page 258 (titled The Swap
Curve). You do not need to know it in detail, but it is used to get implied
forward interest rates for use with interest rate swaps. You can read it for
general understanding, but it is not on the syllabus and does not require
homework problems.
A Eurodollar is a dollar deposited in an account outside of the United States.
Banks borrow and lend with other banks through Eurodollar account deposits,
and most of this activity takes place in London. LIBOR stands for the London
Interbank Offer Rate, and is described in Derivatives Markets as "the average
borrowing rate faced by large international London banks." LIBOR is a key
benchmark for floating rate loans, which typically state that the loan rate will
be a specified percent above LIBOR -e.g., LIBOR plus 0.25%.
A Eurodollar Future is a standardized futures contract which enables you to
hedge interest rate risk just as you might do with an FRA. On pages 158-160
Derivatives Markets looks at a Eurodollar futures contract based on 3 month
LIBOR. This contract has a notional amount of 1 million dollars. We will go
through an example from Derivatives Markets to show how it works.
The price of the LIBOR futures contract at expiration is quoted as
(100 - Annualized 3 month LIBOR).
The annualized number is computed as if the year had 360 days and each
quarter had 90 days. Thus if 3 month LIBOR is 1.5%
Annualized LIBOR = — x 1.5% = 6%
90
Futures Price = 100 - 6 = 94
Derivatives Markets gives the example of a borrower who will need to borrow 1
million dollars for 90 days beginning on the date 7 months from now. Today the
futures price is 94, implying a quarterly rate of 1.5% as above.
This borrower will sell the Eurodollar futures contract for seven months from
now at a price of 94. Now suppose that 7 months have gone by and quarterly
LIBOR is actually 2%. The borrower will still borrow at LIBOR for 2%, but on
the day that he takes out the loan the short the futures price is now 92. The
borrower receives a cash settlement for
1,000,000 x (94% - 92%) x - = 5,000
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 13 - Review of Derivatives Markets, Chapter 7
Page M13- 7
Note that the extra interest that the borrower will need to pay because interest
rates are 2% quarterly instead of 1.5% quarterly is
1,000,000(2% -1.5%) = 5000
The futures payment offsets the extra interest, and it is received in advance
while the interest payment itself will not be due for a quarter.
Since the Eurodollar futures contract is based on a future value of 3 month
LIBOR, it is often used in finding forward rates. That is how it will be used in
Section 8.2. One additional adjustment is used in that calculation. The futures
price F is based on a hypothetical 360 day year. To adjust for a 91 day quarter,
the implied 91 day interest rate is calculated as
1 1 Q1
r9i = (100-F)x —x-ix —
v ; 100 4 90
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 14 - Review of Derivatives Markets, Chapter 8
PageM14- 1
Review of Derivatives Markets
Chapter 8
Section 14.1
Swaps
In Chapter 8 of this guide, we looked at the example of a farmer who wanted to
guarantee a sale price of 2.44 per bushel for his corn in one month. He could do
this by entering a forward contract with a cereal company that wanted to
guarantee a purchase price of 2.44 per bushel in one month. In this chapter we
will look at a generalization of such a single forward agreement.
Suppose that the farmer wanted to guarantee a fixed price per bushel for sales
in each of the next two months, and the cereal manufacturer wanted to buy at
the same fixed price in each of the next two months. An agreement covering
forward sales over multiple time periods is called a swap. Note that the farmer
could guarantee prices for two months in a row by buying two separate forward
contracts. However it is simpler to have a single swap agreement that covers
multiple periods with the same price.
In Sections 8.1 and 8.2, the text Derivatives Markets illustrates how swaps work
by looking at two examples of swaps - a commodity swap to guarantee the price
of oil for two years and an interest rate swap to guarantee a set interest rate for
three years. The text points out that through these examples you will ultimately
see that "swaps are nothing more than forward contracts coupled with
borrowing and lending money."
©ACTEX 2009
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Exam FM / Exam 2 - Financial Mathematics

Page M14-2
Module 14 - Review of Derivatives Markets, Chapter 8
Section 14.2
An Oil Swap Example
Example 8.1 deals with a company that plans to buy 100,000 barrels of oil in
each of the next two years. The table below shows the forward prices and spot
rates that are in place today.
Year
Forward price of a barrel of oil
Spot Rate
1
20
6%
2
21
6.5%
The company can guarantee oil for two years by entering into two separate
forward contracts. The cost per barrel now to buy oil for 20 in one year and 21
in two years is
20 ■ + -* 37.38277.
1.06 1.0652
The text points out that the company could pay a single supplier 37.38277 per
barrel today for a forward agreement to deliver oil at 20 in one year and 21 in
two years. Payment in advance for multiple deliveries is called a prepaid swap.
A prepaid swap is risky: the company would face the risk that the supplier
encountered problems and could not deliver even though they had been paid in
advance.
A better solution would be to pay separately each year, since then the company
would only pay if the supplier was in business and able to deliver at that time.
In either case, the company is really buying the physical oil from a single
supplier. This is called physical settlement.
Note that the above buyer who is doing physical settlement does not have a
fixed price. The buyer could determine an appropriate fixed price x by offering
to make a level payment that gives the same present value of 37.38277. To find
x we solve the following present value equation.
: 37.38277 -* x = 20.48309
1.06 1.0652
Cash Settlement of a Swap
A simpler solution than physical settlement is to buy at market prices each year
but contract for cash settlement payments each year that compensate for price
changes. The current market price of oil at time n is referred to as the spot
price of oil at time n. We will denote it by Spotn.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 14 - Review of Derivatives Markets, Chapter 8
PageM14- 3
The idea behind cash settlement is to find a counterparty who will make an
adjusting payment to the buyer after the buyer has paid the spot price. The
adjusting payment that would take the company back to the desired fixed price
is
Spotn- 20.48309.
If spot prices went up in one year to 21.48309, the company would pay that
amount but receive +$1 per barrel from the counterparty to reduce the net
expenditure to 20.48309. If spot prices went down in one year to 19.48309, the
company would pay that amount but would be required to pay -$1 to the
counterparty to bring the net expenditure to 20.48309.
If we denote the level swap price by Swpr and the buyer's swap payment at
time n by Swpmtn, the general expression for the swap payment received by
the buyer at time n is
(14.1)
Buyer payment = Swpmtn =Spotn -Swpr
If we denote the net cost to the company at time n by Netn, we have
(14.2) , _
Net to buyer
= Netn = Swpmtn - Spotn = (Spotn - Swpr) - Spotn = -Swpr
The company always ends up with a net payment equal to the swap price. This
is derived as a word equation on page 249 of Derivatives Markets. The text
points out that the same level price of 20.48309 applies whether the swap is
structured using physical settlement or financial settlement.
Note that the seller or counterparty to the buyer has a payment equal to the
negative of the buyer's payment.
(14.3)
Counterparty payment = -Swpmtn = Swpr - Spotn
The next section examines the position of the counterparty in more detail.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M14-4
Module 14 - Review of Derivatives Markets, Chapter 8
The Dealer as Counterparty
The textbook Analysis of Derivatives for the CFA Program states (page 271)
that: "The swap market is almost exclusively an over-the-counter market, so
swaps contracts are customized to the parties' specific needs." Thus the buyer
of a swap is most likely to have a dealer as counterparty. (The dealer will most
likely be associated with a large bank or a major investment bank.)
Derivatives Markets analyzes the dealer's role as counterparty on page 250.
There are two ways that the dealer can function as a swap counterparty.
1) The dealer can find a seller to match with the buyer and simply act as a
go-between. The dealer will make his money by charging the buyer a
higher price than the seller gets. This creates a bid-ask spread that
serves as the dealer's fee. This kind of transaction is referred to as a
back-to-back or a matched book transaction.
The structure is diagrammed in Figure 8.3 on page 251. The swap
counterparty in the middle of Figure 8.3 is the dealer. In that diagram
the swap price is the same for buyer and seller, so there is no spread.
Spread is apparently not included to keep the example simple.
2) The dealer can provide the swap to the buyer and hedge it using long
forward or futures contracts. This is illustrated by reference to the
original example of oil forward prices over 2 years.
Year
Forward price of a barrel of oil
Spot Rate
1
20
6%
2
21
6.5%
If the dealer provides the two year swap with Swpr = 20.48309, here is
what the dealer gets per barrel:
Yearl.
Counterparty payment 20.48309 - Spotn
Forward profit Spotn - 20
Total 0.48309
Year 2.
Counterparty payment 20.48309 - Spotn
Forward profit Spotn - 21
Total -0.51691
Thus the dealer ultimately gets .48309 at time 1 but must pay back -
.51691 at time 2. This is a loan. The rate on it is — -1 = .07 \
.48309
1 The loan rate is 7.00%, and it is actually the implied forward rate for the time interval (1,2)
, . P(0,1) 1.06"1
which we denoted in the previous chapter as 1 + r0 (1,2) = = = 1.07
P(0,2) 1.065"2
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby^

Module 14 - Review of Derivatives Markets, Chapter 8
PageM14- 5
This example illustrates two important points.
1) The commodity swap is a combination of a loan and a series of
forwards. That is exactly how it played out for the dealer.
2) The dealer still has interest rate risk. The dealer has a 7% loan, and
will invest the amount loaned for a year to accumulate an amount to
repay the loan. If interest rates are above 7% in that year, he makes a
profit. If interest rates are below 7% in that year, he loses money.
The text points out that he will probably hedge his interest rate risk
using an FRA. Remember that the dealer really makes his money as a
fee obtained from a bid-ask spread -even though the example here
does not include that spread in order to keep things simple. So the
dealer's strategy is to hedge all risks and collect the fee income.
The Market Value of a Swap
A swap contract is like a forward contract. The forward agreement is made, but
no money changes hands when this happens. The initial value of the swap is 0.
However conditions can change. Suppose that you have the previous swap
which entitles you to buy oil for 20. 43809 in each of the next two years, and that
oil prices suddenly rise. Then you have the right to buy oil below market, and
someone will pay for that. We review these important points below.
1) The initial value of a swap is 0.
2) A swap can have a non-zero market value if market conditions change.
The text gives a simple example to illustrate this. In this example, as
soon as the swap contract is made forward prices go up by $2 for each of
year 1 and year 2. Thus we have
Year
Original Forward price of a barrel of oil
New Forward price of a barrel of oil
1
20
22
2
21
23
The text assumes that the interest rate yield curve does not change.
Year
Spot Rate
1
6%
2
6.5%
The new price for a two year swap can be calculated using the new
forward prices. The present value of a prepaid forward is now
22
23
= 41.03288
1.06 1.0652
The level swap payment x is found from the present value equation
x x
1.06 1.0652
: 41.03288
x = 22.48309
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M14-6
Module 14 - Review of Derivatives Markets, Chapter 8
Then the buyer in the original swap with price of 20.48309 could become
a counterparty and sell a swap at the price of 22.48309. The result for
any year is given below.
Buyer payment Spotn - 20.48309
Counterparty payment 22.48309 - Spotn
Total 2.00
The buyer of the original swap can net a total of $2 per year with this
strategy. The present value of these payments of $2 is
: 3.65011
1.06 1.0652
Thus the market value of the swap is 3.65011, the present value of the
payments you can net if you have that swap.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 14 - Review of Derivatives Markets, Chapter 8 Page M14- 7
Section 14.3
Interest Rate Swaps
Section 7.2 of Derivatives Markets shows how to guarantee an interest rate in
the future using a forward rate agreement (FRA). We reviewed that section in
Module 13 of this guide because it was needed as a prerequisite for interest
rate swaps. Section 8.2 of Derivatives Markets discusses interest rate swaps,
which are like a series of FRAs extending over a number of periods. An interest
rate swap might be entered by a company that has floating rate debt for a
number of years and would like the floating rate debt to be converted to fixed
rate debt. We will discuss how this works in the next section.
An Example of an Interest Rate Swap
On page 254, Derivatives Markets looks at the example of an interest rate swap
for XYZ company, which will borrow 200 million dollars at LIBOR for the next
three years, starting now. The example assumes the same yield curve that we
studied in the previous module.
It is useful to have the table of information that was presented there in Table
7.1. Note that the text takes this to be a LIBOR based yield curve, so that the
year one spot rate of 6% is the one year LIBOR rate and the implied forward
rates for years 2 and 3 are the implied one year LIBOR forward values.
Zero-
coupon Zero- One Year Continuously
Years to Bond coupon Implied Par Compounded
Maturity Yield Bond Price Forward Rate Coupon Zero Yield
1
2
3
6.00%
6.50%
7.00%
0.943396
0.881659
0.816298
6.00000%
7.00236%
8.00705%
6.00000%
6.48423%
6.95485%
5.82689%
6.29748%
6.76586%
The XYZ company would like to pay a fixed rate instead of a floating rate. The
company could pay off the floating rate debt and issue fixed rate debt (a bond)
in its place. This would have transaction costs, and the company does not want
to do that.
Instead the company enters into an interest rate swap contract with a notional
amount of 200 million. The contract is based on a fixed interest rate of
6.95485%. Note that the fixed swap rate is the par coupon for the full term of
the swap. (This is derived in Derivative Markets immediately following the
example.) The term of the swap is called the swap term or swap tenor. Under
the swap contract XYZ is guaranteed a payment with the counterparty at the
rate of LIBOR - 6.95485% applied to the notional amount, where the payment is
made:
a) by the counterparty to XYZ if LIBOR - 6.95485% > 0,
b) by XYZ to the counterparty if LIBOR - 6.95485% < 0.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M14-8
Module 14 - Review of Derivatives Markets, Chapter 8
Then XYZ will borrow at LIBOR each year, and have a net interest payment at
a rate of
-LIBOR + (LIBOR - 6.95485%) = - 6.95485%.
Note that although XYZ obtains a fixed rate with the swap, the counterparty's
payments of LIBOR - 6.95485% are floating, since they depend on LIBOR each
year.
Finding the Swap Rate R
On page 255 the text continues the example to show how the swap rate is
determined and illustrate why it turns out to be equal to the par bond coupon.
This is done by looking at how the counterparty to the swap will hedge his
interest rate risk. The counterparty is described as a market maker (dealer),
and has the risk of payments that vary (float) with LIBOR.
Suppose that R is the unknown fixed swap rate. The market maker has a
(positive or negative) payment each year of LIBOR-R. The text takes the one
year spot rate in the yield curve table to be one year LIBOR. Thus in year one
LIBOR = 6% and the payment is 6% - R. In the remaining two years the future
realized values of LIBOR , n and f2, are currently uncertain.
Thus the market maker faces payments at the rates in the table below. Here a
positive value means that the market maker receives a payment and a negative
value means that he makes one.
Year
1
2
3
Swap Payment Rate
R-6%
R-n
R-f2
Status
Certain
Uncertain
Uncertain
We know from the previous chapter that an uncertain future rate can be hedged
using an FRA. The market maker will then enter into FRAs for year 2 and year
3 to fix those rates. The text takes the FRA rate for a year to be the implied
forward rate for that year. Thus the market maker has hedge positions with the
FRA payments given below.
Year Forward Rate FRA Payment Rate
2
3
7.0024%
8.0071%
fx- 7.0024%
f2-8.0071%
©ACTEX 2009
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Exam FM / Exam 2 - Financial Mathematics

Module 14 - Review of Derivatives Markets, Chapter 8
PageM14- 9
Table 8.2 of Derivatives Markets shows the final position of the market maker
after swap payments and FRA hedging payments are made.
Year FRA Payment Rate Swap Payment Rate Net Rate
1
2
3
h -7.0024%
f2-8.0071%
R-6%
R-h
R-f2
R-6%
R-7.0024%
R-8.0071%
Now the uncertainty of fx and f2 has been eliminated. The final question is how
to set the value of R . The answer to this is that the market maker should set his
rate R so that the present value of all net payments made is 0. Each net
payment is discounted at the zero-coupon bond yield for its year. Thus we have
R-.06 R-.070024 R-.080071 A
• + — „— + ; = 0
1.06
R
1.0652
1
1.073
.1.06 1.0652 1.073
R = .069548
.06 .070024 .080071
■ + =- + ■
1.06 1.0652
1.073
The final value of R above was found using forward rates calculated to full
precision, and rounded answers may vary slightly. The point the example
attempts to make is that when you solve for R, you find that the theoretically
correct value of R is the par coupon bond rate as we predicted.
The text works to derive this on page 257 in a section entitled Computing the
Swap Rate in General. We can see the reasoning behind this derivation
concretely if we rewrite the right side of the second equation above with the
implied forward rates written using their exact definition.
.06
1.06 1.0652
.070024 .080071
■ + •
1.0652
1.06
1.073
.06
1.06
.06
1.06 + 1.06
1
-1
1.0652
1
1.073
1.0652
-1
1.073
1
1
1.0652 1.0652 1.073
= 1-
1.073
Thus the equation that we solved for R is equivalent to
R
1.06 1.0652 1.073
= 1 —
1
1.073
1-
This gives R = -
1.073
1-P(0,3)
111
■+ . „ +-
1.06 1.0652 1.073
P(0,l) + P(0,2) + P(0,3)
This is the formula for a par bond coupon given in Chapter 7 of Derivatives
Markets.
©ACTEX 2009
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Exam FM / Exam 2 - Financial Mathematics

Page M14-10
Module 14 - Review of Derivatives Markets, Chapter 8
Some Observations about the Swap
It is instructive to take a further look at the preceding swap from the point of
view of the market maker. Once we know that the correct swap rate is 6.9548%,
we can look at his specific net payments after hedging.
Year Net Rate
1
2
3
R-6% = 0.9548%
R-7.0024% = -0.0476%
R-8.0071% = -1.0523%
With an upward sloping yield curve, the market maker will be paid money in
year 1 and then begin paying money later. This is really a loan. The market
maker gets money in early years and pays it back later. Remember that the
same thing held for the oil swap in the previous section. The hedged dealer still
had cash flows that amounted to a loan. See the solution to question 8.9 for the
analysis of the loan and its implicit balance at each period.
Note that the value of the market maker's position changes over time. After
year one there is no further payment to the market maker, since all subsequent
cash flows are negative.
Derivatives Markets notes that if the XYZ company did not care about having a
fixed rate, it could have done what the dealer did and used FRA agreements to
lock in separate forward rates.
The Swap Curve
To find the swap rate in practice, we need to have a real LIBOR yield curve
which can be used to create LIBOR forward rates and par bond coupons for R.
On page 258 in the section entitled The Swap Curve Derivatives Markets
illustrates how implied forward quarterly rates can be obtained from published
prices of 90 day Eurodollar futures and then used to find the par bond rate that
will be the swap rate R. We have reviewed the Eurodollar basics in the previous
chapter of this guide. The steps in the swap curve calculations are clearly
outlined on page 259, and we will not repeat them. However they should be
reviewed.
It is worth looking again at the 3 year example yield curve from Table 7.1 to
note relationships among the rates in the table. With an upward sloping yield
curve, the forward rate for each year is greater than the spot rate for the same
year. Note also that the 3-year swap rate is less than the forward rate for year 3
but greater than the spot rate for year 3. For an upward sloping yield curve we
would expect
Spot rate for year n < Swap rate for n years < Forward rate for year n..
In a graph on page 260 the text shows a 10 year graph of the spot rate on
Treasuries, the swap rate and the forward rate by year for values computed
using Eurodollar futures prices. Not surprisingly, the spot rate curve is below
the swap curve, which is in turn below the forward curve. The difference
between the swap rate and the spot rate is called the swap spread.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 14 - Review of Derivatives Markets, Chapter 8
Page M14-11
Deferred Swaps
There are also swap contracts that start at some time in the future. These are
called deferred swaps, and their swap rate is also set using the same present
value reasoning. Derivatives Markets looks at a one year deferred swap lasting
for two subsequent years, again based on the yield curve in Table 7.1. The
equation to solve for R here is
R-.070024 [ R-.080071
1.0652 + 1.073
This equation contains terms for the final two hedged years of the full three
year swap that we previously analyzed. The answer is R = 7.4854%.
Reasons to enter into a swap
On page 262 of Derivatives Markets there is section entitled "Why Swap
Interest Rates?". The reasoning outlined here is:
a) Some companies would like to borrow at short term rates which are
typically lower.
b) Short term lenders dislike lending very large amounts to one borrower,
since they would rather diversify.
c) Long term lenders are willing to issue large amounts of fixed rate debt.
d) Thus companies may borrow long term at a fixed rate and swap into the
short term rate they desire.
Other texts give additional reasons that firms might wish to swap, but only this
text is required for exam FM/2.
Amortizing and Accreting Swaps
In our previous examples the notional amount of the swap was fixed through
the swap term. It is possible to set up a swap in which the notional amount
changes over time. If the notional amount increases over time the swap is
called an accreting swap. If the notional amount decreases over time the swap
is called an amortizing swap.
On page 263 of Derivatives Markets there is the formula for the swap price of a
swap with varying notional amounts. We will go over that in the next section.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M14-12
Module 14 - Review of Derivatives Markets, Chapter 8
Section 14.4
A General Formula for All Swaps
The reasoning used for oil swaps and interest rate swaps can be handled by a
single general formula, which is given as equation 8.13 in Derivatives Markets.
If you use the notation f0 (ti) to stand for the forward price at time U, the swap
price is given by
(14.4)
£p(0,tt)/o(ti)
R = ^-n
lP(0,t.)
i=l
We can see this concretely by reviewing our swap examples. For the oil swap
we solved the present value equation
X - + -^- = 30.38277 = — + - 21
1.06 1.0652 "" 1.06 ' 1.0652
Note that the forward prices were 20 and 21, so that the equation is of the form
x[P(0,l) + P(0,2)] = /0(l)P(0,l) + /o(2)P(0,2)
This follows (14.4)
For the interest swap we solved the equation
R\ + =- + ■
.06 .070024 .080071
+ =- + ■
1.06 1.0652 1.073J 1.06 1.0652 1.073 '
The rate for year 1 is the known rate of 6%, and the rates Of 7.0024% and
8.0071% are the forward rates for years 2 and 3. This is also of the form of 14.4
x[P(04) + P(0,2) + P(0,3)] = /o(l)P(0,l) + /o(2)P(0,2) + /o(3)P(0,3).
When the text analyzes interest rate swaps the above equation is modified to
read
x [P (0,1) + P (0,2) + P (0,3)] = r(0,1)P (0,1) + r(l, 2)P (0,2) + r(2,3)P (0,3).
In the discussion on page 257 Derivatives Markets uses the expression
R
_ i=l
Ii»(o,t«)
i=l
©ACTEX 2009
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Exam FM / Exam 2 - Financial Mathematics

Module 14 - Review of Derivatives Markets, Chapter 8
Page M14-13
This is just another way to write (14.4). However, familiarity with this notation
is very helpful in understanding formula (8.7) of Derivatives Markets. That
formula gives the swap price of an interest rate swap with varying notional
amounts. If Qt is the notional amount at time t, the swap price is given by
XQt«P(0,ti)r(tw,ti)
R=^—n •
ZQt,P(0,t,)
i=l
This is the basic formula modified by multiplying each term of the numerator
and denominator by the corresponding notional amount. This is used in
textbook problem 8.10.
There is also a nice formula for a deferred swap price. If a deferred swap
starts in period j and ends in period fc, the swap price is the usual formula but
with only the time periods from j to k included.
£p(0,ti)r(ti-i,ti)
g -hi
£p(o,t«)
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page M14-14 Module 14 - Review of Derivatives Markets, Chapter 8
Section 14.5
Module 14 Summary
An agreement covering forward sales over multiple time periods is called a
swap.
Payment in advance for multiple deliveries is called a prepaid swap.
Commodity Swap
If the buyer is really buying the swapped commodity from a single supplier, the
swap has a physical settlement.
The idea behind cash settlement is to find a counterparty who will make an
adjusting payment to the buyer after the buyer has paid the spot price.
Denote the level swap price by Swpr and the buyer's swap payment at time n
by Swpmtn.
Buyer payment = Swpmtn = Spotn -Swpr.
Denote the net cost to the company at time n by Netn.
Net to buyer = Netn = Swpmtn - Spotn = (Spotn - Swpr) - Spotn = -Swpr
Counterparty payment = -Swpmtn = Swpr - Spotn.
Two ways that the dealer can function as a swap counterparty.
The dealer can find a seller to match with the buyer and simply act as a go-
between: back-to-back or a matched book transaction.
The dealer can provide the swap to the buyer and hedge it using long forward
or futures contracts. Then
• The commodity swap is a combination of a loan and a series of forwards.
• The dealer still has interest rate risk due to the loan.
Interest Rate Swap
A floating rate loan can be converted to a fixed rate loan by a series of
counterparty payments. Theoretically correct value of the fixed swap interest
rate R is the par coupon bond rate. If the floating rate is LIBOR, the payment is
LIBOR - R. It is made:
a) by the counterparty to the floating rate borrower
if LIBOR - R > 0,
b) by the floating rate borrower to the counterparty
if LIBOR - R < 0.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Module 14 - Review of Derivatives Markets, Chapter 8 Page M14-15
The market maker should set his rate R so that the present value of all net
payments from the swap and forward hedges is 0. This will lead to the fixed
rate R being the par coupon bond rate.
Implied forward quarterly rates can be obtained from published prices of 90
day Eurodollar futures and then used to find the par bond rate that will be the
swap rate R..
The difference between the swap rate and the spot rate is called the swap
spread.
Deferred swap contracts start at some time in the future. Their swap rate is
also set using present value reasoning.
Companies may enter into swaps to avoid restrictions by short term floating
lenders.
• If the notional amount increases over time the swap is called an
accreting swap.
• If the notional amount decreases over time the swap is called an
amortizing swap.
If you use the notation f0 (ti) to stand for the forward price at time tiy the swap
price is given by
£p(o,tO/o(ti)
ZP(0,ti)
If Qt is the notional amount at time t for a variable notional amount swap, the
swap price is given by
£Q«.P(0,t.)r(ti-i,ti)
K = ^ •
£Q.,P(o,t«)
i=l
If a deferred swap starts in period j and ends in period k, the swap price is the
usual formula but with only the time periods from j to k included.
J^P(09ti)r(ti.uti)
£p(o)tj)
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M14-16
Module 14 - Review of Derivatives Markets, Chapter 8
Section 14.6
Solutions to Odd-Numbered Problems
8.1
The present value of forward payments for the individual years is
22 23
1.06 + 1.0652
= 41.03288
The level swap payment x should satisfy the present value equation
X ■+ , _*_, =41.03288 -♦ x = 22.483086
8.3
1.06 1.0652
The dealer will hedge his fixed price of 20.9519 with separate oil forwards
for years 1, 2 and 3 at the forward prices of 20, 21 and 22.
The dealer's position is summarized in the following table.
Year Swap Payment Short Oil Forward payment Net payment
1
2
3
Spotj-20.9519
Spot2 -20.9519
Spot3-20.9519
20-Spoti
21 -Spot 2
22-Spots
-0.9519
0.0481
1.0481
The present value of net cash flows is
-0.9519 .0481 1.0481
1.06 1.0652 1.073
:0.00
8.5
The dealer gets the same cash flows as shown in Problem 8.3 in any case,
since his payments are contracted by the swap and do not change. The
present value of his payments changes if the yield curve changes.
Rates up 0.5%. The present value of swap payments is
-0.9519 .0481 1.0481 nAOi
+ T + r- = -.0081
1.065 1.072 1.0753
Rates down 0.5%. The present value of swap payments is
-0.9519 .0481 1.0481
1.055 1.062 1.0653
= .008203
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Module 14 - Review of Derivatives Markets, Chapter 8
Page M14-17
This is really a spreadsheet problem, where you apply the pricing formula 8
times, once for each quarter. Our spreadsheet for this is below.
Quartern P(0, n) Forward/0(n) P(0, n)/0(n) Price R
1
2
3
4
5
6
7
8
0.9852
0.9701
0.9546
0.9388
0.9231
0.9075
0.8910
0.8763
21.0
21.1
20.8
20.5
20.2
20.0
19.9
19.8
20.6892
20.4691
19.8557
19.2454
18.6466
18.1500
17.7309
17.3507
21.0000
21.0496
20.9677
20.8536
20.7272
20.6110
20.5146
20.4305
The pricing formula applied is
£p(o,tO/o(*0
i=l
JTP(0,tO
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M14-18
Module 14 - Review of Derivatives Markets, Chapter 8
8.9
This problem covers the dealer's implicit loan and its balance at each
quarter. Recall that because of his forward hedging, the dealer's payment
rate is R - /0 (n), the difference between the swap price and the forward
rate applied in the FRA. Our spreadsheet for the problem is below. We will
explain the remaining steps in it below the spreadsheet.
Quarter n
1
2
3
4
5
6
7
8
P(0, n)
0.9852
0.9701
0.9546
0.9388
0.9231
0.9075
0.8919
0.8763
1+r (n-1, n)
1.0150
1.0156
1.0162
1.0168
1.0170
1.0172
1.0175
1.0178
R=
Mn)
21.0
21.1
20.8
20.5
20.2
20.0
19.9
19.8
20.4304
fo(n)-R
0.5696
0.6696
0.3696
0.0696
-0.2304
-0.4304
-0.5304
-0.6304
Balance
0.5696
1.2477
1.6367
1.7329
1.5316
1.1272
0.6162
-0.0034
In periods 1-4 the dealer collects positive amounts of cash. In periods 5-8 the
dealer must pay cash out. Thus he must earn interest on the cash from
periods 1-4 to accumulate funds to cover his negatives in periods 5-8. The
assumption is that he will invest at the implied forward rate in each period.
Recall that
P(0,1) = \—7 and P^°;n"1^l + r(n-l,n) forn>l.
v ; l + r(0,l) P(0,n) V ;
Thus we can find the rates r(n-l,n) for reinvestment from the given values
of P(0,n). The balance on the loan at time n will be the prior period balance
increased by interest earned plus the payment for the dealer
Balancen = Balancen-i (l + r (n - l,n)) + (/0 (n) - R).
For example
Balance2 = .5696 (1.015) + .6696 = 1.248
The final balance should be 0 with a full precision calculation, but shows a
small non-zero value due to rounding.
Note that this theoretical calculation assumes that the dealer can actually
earn at today's implied forward rate in the future. There is actually a risk
here, and the dealer may hedge that too.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 14 - Review of Derivatives Markets, Chapter 8
Page M14-19
8.11
This problem reverses the order of questioning and asks you to find the
forward price if you are given the swap price. Recall that the general
formula for the price is
SP(0,t,)/o(ti)
This gives us a starting point for n=l, since in that case
R = /0P(M),1)=/O^sothat/0^ = 2-25
Now we can find an equation that will take us recursively to find the
remaining f0 (n). Note that
R Zp(°>0 =ZP(0,i)/o(i) + P(0,n)/o(n)
1 ■ - J t=i
fo(n)
R
Vi=l
i=l
P(0,n)
This is again a spreadsheet problem, and our spreadsheet for it is below.
Quartern P(0, n) R f„(n) P(0,n)f0(n)
1
2
3
4
5
6
7
8
0.9852
0.9701
0.9546
0.9388
0.9231
0.9075
0.8919
0.8763
2.2500
2.4236
2.3503
2.2404
2.2326
2.2753
2.2583
2.2044
2.2500
2.5999
2.2002
1.8998
2.2001
2.4998
2.1501
1.8002
2.2167
2.5222
2.1003
1.7835
2.0309
2.2686
1.9176
1.5775
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page M14-20
Module 14 - Review of Derivatives Markets, Chapter 8
8.13
Recall that there is a simple way to find the swap price for a deferred swap.
If a deferred swap starts in period j and ends in period k, the swap price is
the usual formula but with only the time periods from j to k included.
XP(0,ti)r(ti_1,ti)
ip(°»*')
Below we give the spreadsheet that would be used to find R for a full 8
period swap.
Quarter n P(0, n) r(n-l, n) P(0, n) r (n-1, ri)
1
2
3
4
5
6
7
8
0.9852
0.9701
0.9546
0.9388
0.9231
0.9075
0.8919
0.8763
0.0150
0.0156
0.0162
0.0168
0.0170
0.0172
0.0175
0.0178
0.0148
0.0151
0.0155
0.0158
0.0157
0.0156
0.0156
0.0156
A 5 quarter swap starting with first settlement in quarter 2 starts in quarter
j=2 and ends in quarter k=6. Thus we have
£p(0,i)r(i-l,i)
R = M—_ = .0166.
ZP(<M)
Problems 8.15 and 8.17 rely on sections that are not in the syllabus for Exam
FM, so they are omitted here.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 14 - Review of Derivatives Markets, Chapter 8 Page M14-
Section 14.7
Module 14 Computational Review Problems
1. (1 pt) Consider the oil swap example in Section 8.1 of
Derivative Markets, by McDonald, with the 1- and 2- year forward
prices of $ 24.5 / barrel and $ 25.5 /barrel. The 1- and 2- year
interest rates are 11 % and 11.5 %, respectively.
Determine the 2-year swap price $ ?
ANSWER1: 24.972
2. (1 pt) Suppose the oil forward prices for 1 year, 2 years,
and 3 years are $ 22 / ban-el, $ 23 /barrel, and $ 24 /barrel. The
1- year effective annual interest rate is 11 %, the 2- year interest
rate is 11.5 %, and the 3- year interest rate is 12 %.
a) Determine the 3-year swap price $ ?
b) What is the price of a 2 - year swap beginning in one year
$ ?
ANSWER1: 22.922
ANSWER2: 23.469
3. (1 pt) Quarter-by-quarter zero-coupon bond prices and oil
forward prices.
Quarter
1
• 2
3
4
5
6
7
8
Zero-bond
0.9852
0.9701
0.9546
0.9388
0.9231
0.9075
0.8919
0.8732
Oil forward
21
21.1
20.8
20.5
20.2
20
19.9
19.8
Using the information about zero-coupon bond prices and oil
forward prices given in the table above, construct the set of oil
swap prices for quarters 1 through 8.
Oil swap price for quarter 1 is $ ?
Oil swap price for quarter 2 is $ ?
Oil swap price for quarter 3 is $ ?
Oil swap price for quarter 4 is $ ?
Oil swap price for quarter 5 is $ ?
Oil swap price for quarter 6 is $ ?
Oil swap price for quarter 7 is $ ?
Oil swap price for quarter 8 is $ ?
ANSWER1:21
ANSWER2: 21.0496
ANSWER3: 20.9677
ANSWER4: 20.8536
ANSWER5: 20.7272
ANSWER6: 20.611
ANSWER7: 20.5146
ANSWER8: 20.4305
4. (1 pt) Quarter-by-quarter zero-coupon bond prices and oil
forward prices.
Quarter
1
2
3
4
5
6
7
8
Zero-bond
0.9852
0.9701
0.9546
0.9388
0.9231
0.9075
0.8919
0.8732
Oil forward
21
21.1
20.8
20.5
20.2
20
19.9
19.8
Using the information about zero-coupon bond prices and oil
forward prices given in the table above, what is the price of an
8-period swap for which the number of barrels of oil delivered
in quarter j is 17-2j barrels.
Price of this special oil swap is $ ?
ANSWER1: 20.69185
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page M14-22 Module 14 - Review of Derivatives Markets, Chapter 8
Section 14.8
Supplemental Exercises
In Problems 1-5, use the following table of quarterly oil forward prices and
zero-coupon bond prices.
Quarter
Oil Forward Price
Zero-coupon bond price
1
21
.985
2
21.2
.971
3
20.9
.954
4
20.7
.933
1. Find the price of a four quarter oil swap.
A) 21.15 B) 21.12 C) 20.95 D) 20.83 E) 20.78
2. Suppose you enter a three quarter oil swap. What payment per barrel
will be made to you in the second quarter if the spot rate for the second
quarter is 21.1
A) .15 B).l Q.07 D).05 E) .01
3. What is the guaranteed quarterly rate on a four quarter interest rate
swap?
A) .011 B).013 Q.015 D).017 E) .019
4. Suppose you enter a three quarter interest rate swap. What net interest
payment will be made to you in the second quarter if the spot interest
rate for the second quarter is .017?
A) .0010 B) .0012 C) .0015 D) .0017 E) .0019
5. Suppose the forward oil price increases immediately by 1 for each of the
four quarters, but the zero-coupon bond values are unchanged. What is
the market value of a four quarter oil swap?
A) 1.00 B)1.72 C)2.95 D) 3.27 E) 3.84
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 14 - Review of Derivatives Markets, Chapter 8 Page M14- 23
Section 14.9
Supplemental Exercise Solutions
1. We will use the general formula
n
„ _Zp(°»t')^'(t0 21(.985) + 21.2(.971) + 20.9(.954) + 20.7(.933)
±P(0,ti) .985+ .971+ .954+ .933 =
i=l
Answer C
2. This is a longer problem, since we need to find the swap price first and
then find the payment. The swap price is
n
gP(0,tt)/0(t,) 21 (.985) + 21.2 (.971) + 20.9 (.954)
±P(Q,tl) -985+ .971+ .954 =
i=l
The spot price in the second quarter is 21.1, and the payment is
21.1-21.03 = .07
Answer C
3. The guaranteed interest rate is the four year par coupon bond rate.
i-Pfr4) - 1-M3 0174
P(0,l) + P(0,2) + P(0,3) + P(0,4) .985+ .971+ .954+ .933 '
Answer D
4. The guaranteed interest rate is the three year par coupon bond rate.
^P(°>3) 1-954 _ 015g
P(0,l) + P(0,2) + P(0,3) .985+ .971+ .954
The net rate paid to you will be .017 - .0158 = .0012
Answer B
5. The new swap price is 21.95, so you could sell a new swap, hold the old
one and net 1 per quarter. The present value of this series of payments
of 1 is 1 (.985 + .971 + .954 + .933) = 3.84
(This is just like the example on the bottom of page 253 in the text.)
Answer E
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Module 15 - Currency Forward Contracts
Page M15- 1
This brief chapter introduces currency forward agreements as covered in
section 5.6 of Derivatives Markets. This material is not included in the exam
FM/2 syllabus, but it is a prerequisite for exam MFE. The chapters required for
exam MFE often have sections with applications to currencies. You can read
this material after you have take exam FM/2 and are preparing for MFE. It will
only take a few minutes.
The test motivates the need for currency futures with the example of a United
States company that imports consumer electronic products purchased from a
manufacturer in Japan for sale in the United States. The importer must pay the
manufacturer 150 million yen in one year. The current exchange rate is
.009 dollar
One yen can be purchased today for $.009. If the importer buys yen today he
can reinvest at the current Japanese interest rate. That rate is 2% annually. To
have 1 yen in one year the importer needs only the present value of one yen
today. Thus, today he will pay .009e"02.
The amount invested today is the price of a prepaid forward. The example
illustrates the reasoning to apply to get the general formula for the prepaid
forward price. If the yen interest rate is ry, the price of a prepaid forward
purchase of 1 yen for time T will be F0Pr = x0e~ryT.
As before, the price of a forward contract for one yen will be the future value
of the prepaid forward price. This future value is taken at the dollar interest
rate r. Thus the forward price is F0,t = x0eir~ry)T
This reasoning can be used for forward currency agreements between any two
currencies. Suppose that you will use the currency of country A to make a
forward purchase of the currency of country B for time T. If the interest rates
of the two countries are rA and rB and the current exchange rate is x0 then the
prepaid forward price and the forward price per unit of the currency of
country B are given by
FoPT=x0e-rBT
o,t - Xoe
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Practice Exam 1 - Exam FM / Exam 2
PagePEl- 1
Exam FM
Questions
1. Consider the following yield curve:
Year
1
2
3
4
c/i
Spot Rate
5.5%
5.0%
5.0%
4.5%
4.0%
Find the four year forward rate.
A) 1.8% B) 1.9% C) 2.0% D) 2.1% E) 2.2%
2. Find the Macaulay duration of a 10-year 1000 par value bond with 8%
annual coupons and an effective annual interest rate of 6.5%.
A) 7.2 B) 7.4 C) 7.6 D) 7.8 E) 8.0
3. At an effective annual rate of interest i, a person can pay off a loan of K in
two ways:
1) 475 now and 475 in 1 year, or
2) 570 in 2 years and 570 in 3 years.
Calculate K
A) 893 B) 901 C) 909 D) 917 E) 925
4. A 10-year annuity-immediate pays 100 quarterly for the first year. In each
subsequent year, each payment is increased by 5% over the payment for the
previous year. There is a nominal annual interest of 8% convertible
quarterly. Find the present value of this annuity.
A) 2997 B) 3075 C) 3108 D) .3225 E) 3333
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page PE1-2
Practice Exam 1 - Exam FM / Exam 2
5. The present value of a 10-year annuity-immediate with level annual
payments and interest rate i is X. The present value of a 20-year annuity-
immediate with the same payments and interest rate is 1.5X. Find i.
A) 7.2% B) 7.4% C) 7.6% D) 7.8% E) 8.0%
For Problems 6 and 7 use the following account summary:
Date
January 1
March 1
September 1
December 31
Balance Before
Activity
100,000
105,000
112,000
95,000
Deposits
10,000
Withdrawals
30,000
6. Find the time-weighted yield for this account.
A) 17.2% B) 17.5% C) 17.9% D) 18.1% E) 18.5%
7. Find the dollar-weighted yield for this account.
A) 14.9% B) 15.3% C) 15.6% D) 16.1% E) 16.4%
8. An investor has 3000 worth of 5-year bonds with a modified duration of
4.615, 7000 worth of 10-year bonds with a modified duration of 9.323 and
10,000 worth of 20-year bonds with a modified duration of 19.085. What is
the modified duration of this entire portfolio?
A) 13.5 B) 13.7 C) 13.9 D) 14.1 E) 14.3
9. A company has liabilities of 1000, 3000 and 5000 payable at the end of years
1, 2 and 3 respectively. The investments available to the company are the
following zero-coupon bonds:
Maturity
(years)
1
2
3
Effective Annual
Yield
7%
8%
9%
Par
1000
1000
1000
Determine the cost for matching these liabilities exactly.
A) 6918 B) 7024 C) 7165 D) 7368 E) 7522
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Practice Exam 1 - Exam FM / Exam 2
Page PE1- 3
10. A man creates a retirement fund by depositing payments at the end of each
month for 20 years. For the first 10 years the deposits are 100 per month
and for the last 10 years the deposits are 200 per month. The fund earns
interest at a nominal rate of 6% per year converted monthly. Upon
retirement he uses the proceeds to purchase a 30-year annuity-immediate
with monthly payments. The annuity earns at a nominal rate of 8%
converted monthly. What are monthly payments from this annuity?
A) 408 B) 425 C) 437 D) 441 E) 459
11. An annuity pays annual payments at the beginning of each year for 20 years.
For the first 10 years the payments are 100. Starting with payment 11 each
payment is increased by 6% over the previous payment. The annuity earns
at an annual effective rate of 8%. Find the present value of this annuity.
A) 1177 B) 1190 C) 1202 D) 1213 E) 1225
12. A corporate bond is priced to yield 7.2% and has a price of 972.48. The
Macaulay duration is D = 7.1245. Estimate the change in price if rates
increase by 0.10%.
A) -6.463 B) -6.685 C) -6.814 D) -7.012 E) -7.163
13. A 40-year loan is paid with level annual payments at the end of each year.
The principal paid in the 20th payment is 166.59 and the principal paid in the
25th payment is 244.78. Find the interest rate for this loan.
A) 7.7% B) 8.0% C) 8.2% D) 8.5% E) 8.8%
14. Linus deposits 100 into an account at the end of each year for 20 years. This
account earns interest at an annual effective rate of 5%. Lucy deposits
money into an account at the end of each year for 20 years. Her account also
earns interest at an annual effective rate of 5%. Her deposits are:
P, 2P,...., 20P. At the end of 20 years the accumulated amounts are the
same. Find P.
A) 10.93 B) 11.05 C) 11.12 D) 11.23 E) 11.35
15. Schroeder borrows money to buy a new piano. He agrees to pay back the
loan with level annual payments at the end of each year for 30 years. The
annual interest rate is 7%. The interest in his 10th payment is 366.74. What is
the interest in his 20th payment?
A) 221.86 B) 229.64 C) 244.18 D) 250.72 E) 253.80
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page PE1-4
Practice Exam 1 - Exam FM / Exam 2
16. A woman makes a deposit into an account. For the first 5 years the account
accumulates with a force of interest of 0.05. For the next 10 years the fund
accumulates with an annual nominal discount rate of 6% convertible
quarterly. For the 15 year period, what is the annual nominal interest rate
convertible monthly?
A) 5.59% B) 5.71% C) 5.83% D) 5.96% E) 6.04%
17. Violet purchases a 10-year 1000 par bond with 8% semiannual coupons. The
bond is priced to yield 7.5% convertible semiannually. She reinvests the
coupon payments in a fund that pays a nominal rate of 7% convertible
semiannually. What is her nominal annual yield convertible semiannually?
A) 7.36% B) 7.41% C) 7.48% D) 7.56% E) 7.63%
18. You are given the following yield curve:
Year
1
2
3
4
5
Spot Rate
4.0%
4.2%
4.6%
—
5.1%
If i4,5 = 6.1%, find s4.
A) 4.81% B) 4.83 C) 4.85% D) 4.87% E) 4.89%
19. A 20-year annuity-immediate has annual payments. The first payment is 100
and subsequent payments are increase by 100 until they reach 1000. The
remaining payments stay at 1000. The annual effective interest rate is 7.5%.
What is the cost of this annuity?
A) 6201 B) 6372 C) 6413 D) 6584 E) 6700
20. A woman buys a 1000 par 5-year zero coupon priced to yield 6%. At the
same time she buys a 5-year 1000 par bond with 8% semiannual coupons
which is priced to yield 7% convertible semiannually. The coupon payments
are reinvested at 6.5% convertible semiannually. What is her annual
effective yield for the combined investment?
A) 6.0% B) 6.2% C) 6.4% D) 6.6% E) 6.8%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Has sett, Ratliff, Garcia, & Steeby

Practice Exam 1 - Exam FM / Exam 2
Page PE1- 5
21. The S&R index currently has a price of 1300. The price of a six month
forward contract is 1320. What annual interest rate (compounded
continuously) is implied by this forward price? Note that the S&R has no
dividend.
A) .02481 B) .02500 C) .0305 D) .0355 E) .0411
22. The S&R index currently has a price of 1300. The price of a three month
1320-strike put is 81.41. The annual interest rate is 4% compounded
continuously. A buys this put, and B enters into a long forward contract. In
three months A and B have the same profit. What is the price of the index in
three months?
A) 1310 B) 1297 C) 1289 D) 1291 E) 1275
23. The current value of the a stock isS0 = 25, and the continuously compounded
risk free rate is r = .04. The price of a six month (T = .5) 26-strike call is
1.7152 and the price of a six month (T = .5) 26-strike put is 2.5726. Find the
continuously compounded dividend yield 8.
A) 1% B) 2% C) 3% D) 4% E) 5%
24. Investor C buys the S&R index at time 0 for 1100 and buys an 1100-strike
put with T = .25 for a price of 81.51.If the interest rate is r=.04, what is his
minimum profit (loss)?
A) -93.38 B) -63.015 C) -57.64
D) -48.50 E) There is no minimum
25. The current (spot) rate for corn is 1.60 per bushel. The 6 month forward
price is $1.50 per bushel. The continuously compounded annual rate is
r = .035. Farmer Brown, has total fixed and variable costs of 1.44 per bushel,
and plans to produce 100,000 bushels for $144,000.
A six month (T = .5) put with a strike price of 1.52 per bushel is available at
a price of 0.12. What are the minimum and maximum profits for Farmer
Brown in six months if he is hedged with a purchase of this put?
A) minimum = -4,212, maximum = 19,678
B) minimum =-6222, maximum = 19,678
C) minimum= -4,212, no maximum
D) minimum = -6,242, no maximum
E) none of the above
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page PE1-6
Practice Exam 1 - Exam FM / Exam 2
26. Company XYZ makes an aircraft which costs 80,000,000 to manufacture. It
will be completed in six months. At that time it will sell either for 90,000,000
with probability .5 or 74,000,000 with probability .5. The company decides to
enter into a forward contract to sell the unit for 85,000,000 in six months The
company has a 40% tax rate, and has no tax benefit for losses. What is the
company's expected profit after tax?
A) -1,000,000 B) 0 C) 1,000,000
D) 2,000,000 E) 3,000,000
27. A stock has current price. S0 = 25 The annual continuous interest rate is
r = .03. If the expiration time for a forward contract is T = .25 and the
forward price is 25.15, what is the continuous dividend yield 81
A) 0.003 B) 0.006 C) 0.010 D) 0.015 E) 0.018
28. The S&R index has a spot price of S0 = 1100. The continuous interest rate is
r = .03 and the continuous dividend yield is 8 = 0 The one year forward price
is 1133.50. Which of the following positions results in a synthetic long
forward contract?
A) Sell the index short for 1100 and lend the proceeds at r = .03
B) Sell the index short for 1100 and borrow 1000 at r = .03
C) Borrow 1100 at r = .03 and buy the index.
D) Borrow 1000 at r = .03 and sell the index short
E) None of these.
In Problems 29-30, use the following table of quarterly oil forward prices and
zero-coupon bond prices.
Quarter
Oil Forward Price
Zero-coupon bond price
1
20.9
.984
2
21.2
.969
3
20.8
.953
4
20.7
.935
29. Find the price of a four quarter oil swap.
A) 21.18 B) 21.62 C) 20.90 D) 20.83 E) 20.78
30. Suppose you enter a three quarter interest rate swap. What net interest
payment will be made to you in the second quarter if the spot interest rate
for the second quarter is .018?
A) .0010 B) .0012 C) .0016 D) .0018 E) .002
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 1 - Exam FM / Exam 2
PagePEl- 7
Solutions
1. The four year forward rate i4,s is given by
1 + 14,5= (1 + s5)s/(l + s4)4 = 1.0471.045" = 1.020
i4,s = .02
Answer C
2.
D_[80(Ia)m+10q000)v10]
Bond Price
v10 = 1.065-10 = 0.532726
cijoi = 7.6561 (be sure calculator is in BGN mode)
(Ha =35.8284
Bond prices 1,107.83
(Reset calculator to END mode. N = 10, PMT = 80,1/Y =6.5, FV = 1000.
CPT PV = -1.107.83)
D = [80(35.8284) + 5,327.26]/l,107.83 = 7.396
Answer B
3. K = 475 + 475v = 570v2 + 5703
v2= [475(1+ v)]/[570(l + v)] = 0.8333 => v = 0.91287
K = 475(1.91287) = 908.61
Answer C
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page PE1-8
Practice Exam 1 - Exam FM / Exam 2
4. The accumulated amount at the end of year one is 412.16.
(N = 4,1/Y = 2, PMT = 100, PV =0. CPT FV = - 412.16)
We can view the annuity as a 10-year annuity-immediate with annual
payments, the first being 412.16 and subsequent payments are increase by
5% each year. The effective annual rate is i = (1.02)4 - 1 = 0.08243.
The present value of this annuity is
(412.16/1.08243)[1 + (1.05/1.08243) + ... + (1.05/1.08243)9]
= 412.16[1 - (1.05/1.08243)10]/(1.08243 - 1.05)
= 3333.30
Answer E
5. X = (l-v10)/i, 1.5X = (l-v20)/i
Hence 1 + v10 = 1.5, v10 = 0.5, i = 0.072
Answer A
6. For the time-weighted yield
1 + j = (105,000/100,000)(112,000/115,000)(95,000/82,000) = 1.185
j = 0.185
Answer E
7. For the dollar-weighted yield,
J = 95,000 - 100,000 - (10,000 - 30,000) = 15,000
i = 15,000/[100,000 + (1 - 1/6)(10,000) - (1 - 2/3)(30,000)]
= 0.153
Answer B
8. The weights are 3/20, 7/20 and 1/2 respectively for the 5-year, 10-year and
the 20-year bonds. The modified duration is
DM = (3/20)(4.615) + (7/20)(9.323) + (1/2)(19.085) = 13.498
Answer A
9. The company must invest the present values of 1000 in one year at 7%, 3000
in 2 years at 8% and 5000 in 3 years at 9%. The cost is
1000/1.07 + 3000/1.082 + 5000/1.093 = 7367.51
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 1 - Exam FM / Exam 2
Page PE1- 9
10. The deposits can be viewed as payments of 100 into a 20-year annuity-
immediate and 100 into a 10-year deferred 10-year annuity-immediate.
The accumulated amount in the first annuity is 46,204.09.
(N = 240,1/Y = 0.5, PMT = -100, PV = 0. CPT FV = 46,204.09)
The accumulated amount in the second annuity is 16,387.94.
(Reset N = 120. CPT FV = 16,387.94)
Total accumulation is 62,592.02.
The monthly payments from the 30-year annuity are 459.30.
(N = 360,1/Y =0.6667, PV = - 62,592.02, FV = 0. CPT PMT = 459.30)
Answer E
11. The present value of this annuity is
100a^ + (106/1.0810)[1 + (1.06/1.08) + ... + (1.06/1.08)9]
To get the value of the first term set the BA II Plus to BGN mode.
Set N = 10,1/Y = 8, PMT = -100, and FV = 0. CPT PV = 724.69.
The value of the second expression is
(106/1.0810)[1 - (1.06/1.08)10]/[1- (1.06/1.08)] = 452.03
Present value is 724.69 + 452.03 = 1,176.72
Answer A
12. The change is
AP = - (D)P(i)Ai/(l + i)
= - (7.1245)(972.48)(0.001)/(1.072)
= - 6.463
Answer A
13. PRinfc is the amount of principal repaid in the fcth period.
Prinfc+n = (l + i)nPRinfc.
Let k = 20 and n = 5.
244.78 = (1 + i)5 (166.59).
i = (244.78/166.59)1/5 - 1 => i = .08
Answer B
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE1-10
Practice Exam 1 - Exam FM / Exam 2
14. The accumulation in Linus's account is lOOs^i = 3,306.60.
(N = 20,1/Y = 5, PV = 0, PMT = -100. CPT FV = 3,306.60)
20l '
The accumulation is Lucy's account is P(Is)
^s^=1%ol2l=294-385
p_ 3,306.60 _n23
294.385
Answer D
15. Let P be the annual payment. The interest paid in the 10th payment is
P(l _ v3o-io+i) = p(1 _ V2i) = p(1 _ 0.24151) = 366.74
P = 483.51
For the 20th payment the interest is 483.51(1 - v") = 483.51(0.52491)= 253.80
Answer E
16. If Y is the amount deposited, then the accumulation is
A = Ye00S(S)(l - 0.015)"40 = 2.3503Y.
There are 180 months in the 15 year period. If j is the monthly interest then
j = 2.35031/18°-1 = 0.00476
i = 12(0.00476) = 0.0571
Answer B
17. The price of the bond is 1034.74.
(N = 20,1/Y = 0.375, PMT = 40 and FV = 1000. CPT PMT = -1034.74)
The accumulated amount of reinvested coupon payments is 4OS251 -1131.19.
The total accumulation is 2131.19.
The semiannual yield on the investment is
j = (2131.19/1034.74)1'20 - 1 = .0368.
The annual yield is 2(.0368) = 0.0736.
Answer A
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 1 - Exam FM / Exam 2
Page PE1-11
18. 1 + i4,5 = (1 + S5)5/(l + S4)4
(1 + s4)4 = (1 + s5)5/(l + ks) = (1.051)5/(1.061) = 1.20864
1 + s4 = 1.0485, s4 = 0.0485
Answer C
19. This can be viewed as a 10-year increasing annuity and a 10-year deferred
10-year annuity.
The present value of the 10-year deferred annuity is
lOOOv10 a^ = 1000 (0.48519) (6.8641) = 3,330.39.
The present value of the increasing annuity is 100 (Ia)^.
(J v (<^-10v10) 7.3789-4.8519 „ ,Q„
(la)-, = -^ '- = = 33.6933
v m i 0.075
Total cost is 3,330.39 + 3,369.33 = 6,699.72
Answer E
20. The price of the zero-coupon bond is 1000/1.065 = 747.26.
To find the price of the second bond with the BA II Plus set
N = 10,1/Y = 3.5, PMT = - 40 and FV = -1000. CPT PV = 1041.58.
The accumulation of the reinvested coupon payments is 463.87.
(N = 10,1/Y =3.25, PMT = -40 and PV =0. CPT FV = 463.87)
Total investment is 747.26 + 1041.58 = 1788.84.
Total accumulation is 1000 + 1000 + 463.87 = 2463.87.
Annual effective yield is
f2463.87^
5
U788.84,
Answer D
21. Fo,T = S0erT -> 1320 = 1300e5r -> r = .0305
Answer C
-1 = 0.066.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE1-12
Practice Exam 1 - Exam FM / Exam 2
22. The forward price is F0,T = S0erT = 1300e25(04) = 1313.07. The long forward
profit is ST - F0,T = ST-1313.07.
The put profit is max(0,1320 -ST) - 81.41e04{25) = max(0,1320 -ST) - 82.23.
Assume that ST < 1320. Then the equality of prices implies that
ST -1313.07 = 1320 - ST - 82.23 -> ST = 1275.42
Answer E
23. By put-call parity
C-P = S0e-*T-Ke-rT
1.7152 - 2.5726 = 25e"5* - 26e"04{ 5) -> r = .03
Answer C
24. Buying the index and buying a put with strike 1100 creates a floor. The floor
has the same profit function as a long call with strike 1100.
The minimum profit on the floor is the (negative) loss of the future value of
the call premium when the call expires unexercised.
By parity the value of the call premium is 92.4554. The minimum profit is
-92.4554 e01 =-93.38
Answer A
25. The profit from the put option is
100,000[max(0,1.52-jc)-.12e035(5)] = 100,000max(0,1.52-x)-12,211.85.
The total profit for the hedged position is
100,000x-144,000 + (100,000 max(0,1.52-*)-12,211.85)
_ J-4,211.85, jc<1.52
~ [100,OOOjc-156,211.85, x>1.5
Answer C
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Practice Exam 1 - Exam FM / Exam 2
Page PE1-13
26. The calculations are in the table below. Values are given in millions.
[With Short Forward at
[Pre-tax op income
Income from Forward
Taxable Income
Tax @ 40%
After Tax Income
85
Price
90
10
-5
5
2
3
Price
74
-6
11
5
2
3
Answer E
27.
Fo.r = S0e(r-')T -> 25.15 = 25e(03^-2S
lnf^^l = .0075 -.255
25
8 = .006
Answer B
28. STOCK - ZERO COUPON BOND = LONG FORWARD
Thus you buy the index for 1000 and sell a zero coupon bond for 1000
(borrow the money to buy the stock.).
Answer C
29. We will use the general formula
£p(<Mi)/o(t«)
P =
i=l
lP(o,t.)
i=l
Answer C
_ 20.9(.984) + 21.2(.969) + 20.8 (.953) + 20.7(.935)
" .984 + .969 + .953 + .935
: 20.90
30. The guaranteed interest rate is the three year par coupon bond rate.
1-P(0,3) _ 1-.953
c =
P(0,l) + P(0,2) + P(0,3) .984+ .969+ .953
The net rate paid to you will be .018- .0162 = .0018
Answer D
0162
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Practice Exam 2 - Exam FM / Exam 2 Page PE2- 1
Exam FM
Questions
1. A man borrows 1000 for 2 years at an annual effective rate of i. He has two
payment options:
1. Pay 560 at the end of each year, or
2. Pay K at the end of year 1 and 800 at the end of year 2.
Find K
A) 329.42 B) 331.66 C) 334.82 D) 337.57 E) 341.65
2. A company has liabilities of 2000 payable in 1 year and 5000 payable in 3
years. The investments available to the company are the following zero-
coupon bonds:
Maturity
(years)
1
3
Effective
Annual Rate
6.5%
7.5%
Par
1000
1000
Determine the cost for matching liabilities exactly.
A) 5903 B) 5935 C) 5952 D) 5970 E) 5988
3. A woman has a fixed rate mortgage on her home. Her payments are level
and made at the end of the month. The principal repaid in the 20th payment
is 3 times the principal repaid in the 5th payment. Find the rate of interest on
this mortgage.
A) 6.8% B) 7.0% C) 7.2% D) 7.4% E) 7.6%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE2-2 Practice Exam 2 - Exam FM / Exam 2
4. You are given the following n-year forward rates:
Year
0
1
2
3
Forward Rate
2.9%
3.7%
4.4%
5.2%
Find s4.
A) 3.92% B) 4.05% C) 4.17% D) 4.31% E 4.46%
5. A man buys a 20-year annuity-immediate for 10,000. He receives annual
payments of 910. He invests these payments in a fund that earns 7.5%
annually. What is his annual yield on this investment?
A) 6.5% B) 6.7 C) 6.9% D) 7.1% E) 7.3%
6. An investment pays 2000 at the end of year one and 4000 at the end of year
three. It is purchased to yield 7.2% annual effective rate. What is the
Macaulay duration for this investment?
A) 2.270 B) 2.301 C) 2.334 D) 2.358 E) 2.515
7. A woman buys two 5-year 1000 par bonds. The first has 7.5% semiannual
coupons and is priced to yield 8% convertible semiannually. The second has
6% semiannual coupons and is priced to yield 7% convertible semiannually.
The coupon payments from the two bonds are deposited in a fund that pays
6.8% convertible semiannually.
What is her annual effective yield for this combined investment?
A) 7.3% B) 7.5% C) 7.7% D) 7.9% E) 8.1%
8. The spot rate for year k is given by the equation
sk = 0.08 + 0.003k - 0.0015k2.
Find the three-year forward rate implied by this yield curve.
A) 4.36% B) 4.41% C) 4.58% D) 4.65% E) 4.74%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 2 - Exam FM / Exam 2
Page PE2- 3
9. A 10-year annuity-due pays 50 quarterly for the first 5 years and 100
quarterly for the last 5 years. The annuity earns at a nominal rate of 6%
convertible quarterly. What is the present value of this annuity?
A) 1978 B) 2034 C) 2077 D) 2119 E) 2165
10. A 20-year annuity-immediate pays 100 a year for the first 10 years. Starting
with the 11th payment, each payment is increased by 6% over the previous
one. The annuity earns at an annual effective rate of 7%.
Find the present value of this annuity.
A) 1150 B) 1185 C) 1235 D) 1262 E) 1288
11. A special 3-year 1000 par bond has 8% annual coupons and has an effective
annual interest rate of 7%. Find the Macaulay duration of this bond.
A) 2.5 B) 2.6 C) 2.7 D) 2.8 E) 2.9
12. A new company expects the dividends on its common stock to be 1 the first
year and increase by 1 each year until it reaches 10. Thereafter it expects
the dividend to grow by 3% each year. Assume an annual interest rate of
5%.
Calculate the price of this stock using the dividend discount model.
A) 344 B) 351 C) 356 D) 365 E) 372
13. A company has a loan of 100,000 to be repaid with 30 annual end of year
level payments. The principal and the interest in the 21st payment are the
same. Find the principal repaid in the 10th payment.
A) 1862 B) 1871 C) 1884 D) 1901 E) 1913
14. A man deposits money into a fund. For the first four years the fund
accumulates at a nominal interest rate of 6% convertible quarterly. For the
next six years the fund accumulates at a nominal discount 8% convertible
semiannually.
For the 10 year period what is the equivalent force of interest?
A) 0.0719 B) 0.0728 C) 0.0731 D) 0.0737 E) 0.0742
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE2-4
Practice Exam 2 - Exam FM / Exam 2
IS. A 20-year annuity-immediate has annual payments. The first payment is
1000. Subsequent payments decrease by 100 each year until they reach 100.
The remaining payments stay at 100. The annual effective interest rate is
6.5%. Find the present value of this annuity.
A) 4708 B) 4765 C) 4815 D) 4853 E) 4894
16. A man buys a house for 100,000. He finances it for 30 years with level
monthly payments made at the end of each month at a fixed interest rate of
7.5% convertible monthly. After 10 years he refinances the outstanding
balance principal for 15 years at 6% convertible monthly.
Calculate his new monthly payments.
A) 702.45 B) 717.68 C) 732.43 D) 750.65 E) 762.38
17. A woman is asked to invest 20,000 in a project. She is promised returns of
5,000 in one year, 6,000 in two years, 7,000 in three years and 10,000 in four
years. Find the IRR for this investment.
A) 12.71% B) 12.84% C) 12.96% D) 13.11% E) 13.23%
18. Consider the following yield curve:
Year
1
2
3
4
Spot Rate
2.0%
2.5%
3.0%
4.0%
A 4-year 1000 par bond has an annual coupon rate of 3.5%. Use the yield
curve to find the price of this bond.
A) 980 B) 984 C) 989 D) 994 E0 999
19. A man buys a 10-year 1000 par bond with 7% semiannual coupons. The bond
is priced to yield 6.5% convertible semiannually. The coupon payments are
invested in a fund that earns 6% convertible semiannually.
His wife makes annual end of year payments of K into a fund that earns
6.5% annually. At the end of 10 years their accumulated funds are the same.
Find K.
A) 126.28 B) 131.45 C) 139.25 D) 143.80 E) 151.38
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 2 - Exam FM / Exam 2 Page PE2- 5
20. For an unknown interest rate i, the following payments have the same
present value:
1. 675 at the end of two years.
2. 200 at the end of one year and 500 at the end of three years.
Find the value of i. (Assume i < 100%)
A) 9.0% B) 9.2% C) 9.4% D) 9.6% E) 9.8%
21. The S&R index currently has a price of 1100. The price of a three month
1120-strike put is 71.32. The annual interest rate is 3.5% compounded
continuously. What is the profit on this put in three months if the spot price
then is 1080?
A) -84.35 B) -31.95 C) 0 D) 30.95 E) 83.52
22. Your home has a value of 340,000. Your annual insurance premium is 6,000
and your deductible is 25,000. If you look at your insurance as a put option,
what is the strike price?
A) 315,000 B) 295,000 C) 280,000 D) 275,000 E) 270,000
23. An insurance company sells single premium deferred annuity contracts
with return linked to a stock index, the time-t value of one unit of which is
denoted by S(t). The contracts offer a minimum guarantee return rate of
g=2.0%. At time 0, a single premium of amount n is paid by the
policyholder, and n xy% is deducted by the insurance company.
In one year the insurance company will pay the policyholder
n x (1 - y%) x Max[S(T)/S(0), (1 + g%)]., where ) S(0) =100
You are given the following information:
i) Dividends are incorporated in the stock index. That is, the stock index is
constructed with all stock dividends reinvested,
ii) The price of a one-year European put option, with strike price of $102, on
the stock index is $15.80.
Determine y%, so that the insurance company does not make or lose money
on this contract.
A) 13.2% B) 13.35% C) 13.5% D) 13.64% E) 13.80%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE2-6
Practice Exam 2 - Exam FM / Exam 2
24. Investor C buys the S&R index at time 0 for 1300 and buys a 1300-strike put
with T = .25 for a price of 71.85. If the interest rate is r=.035, what is his
minimum profit (loss)?
A) -82.33 B) -63.015 C) -57.64
D) -83.91 E) There is no minimum
25. Near market closing time on a given day, the European call and put prices
for a stock are available as follows:
Strike Price
40
50
55-
Call Price
11
6
3
Put Price
3
8
11
The options have expiration time T = .5. The continuously compounded
annual interest rate is r = .04.
Mary constructs the following portfolio: Long two call options with strike
price 40; short six call options with strike price 50; lend $2; and long some
calls with strike price 55. The $2 she lends is obtained from the sale and
purchase of the options.
What is her profit at T = .5 if the price of the stock is 52 at that time?
A) 2 B) 5.02 C) 4 D) 6.08 E) 14.04
26. Investor F sells a 1300-strike S&R put for 71.85 and a 1300-strike S&R call
for 83.18. The interest rate is r = .035 and T = .25. What is his maximum
profit?
A) 71.85 B) 83.18 C) 155.03
D) 156.39 E) There is no maximum
27. A stock has current price S0 = 40. The annual continuous interest rate is
r = .03 and the continuous dividend yield is S = .01. You observe a one year
prepaid forward price of 39.60. Which of the following is true?
A) No arbitrage is possible.
B) You can create an arbitrage by buying one prepaid forward and selling
one share of the stock short
C) You can create an arbitrage by selling the prepaid forward and buying
one share of the stock.
D) You can create an arbitrage by buying the prepaid forward and selling
e"01 shares of the stock short
E) You can create an arbitrage by selling the prepaid forward and buying
e~m shares of the stock
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 2 - Exam FM / Exam 2
Page PE2- 7
28. The S&R index has a spot price of S0 = 1300. The continuous interest rate is
r = .03 and the continuous dividend yield is 5 = 0 The one year forward
price is 1339.59. You enter into a forward sale contract and buy the index.
Which of the following positions is this equivalent to:
A) A short sale of the index.
B) Purchase of a one year zero-coupon bond with r = .03
C) A reverse cash and carry hedge.
D) A cash and carry arbitrage
E) None of these.
In Problems 29-30, use the following table of quarterly oil forward prices and
zero-coupon bond prices.
Quarter
Oil Forward Price
Zero-coupon bond price
1
20.9
.984
2
21.2
.969
3
20.8
.953
4
20.7
.935;
29. Suppose you enter a three quarter oil swap. What payment per barrel will be
made to you in the second quarter if the spot rate for the second quarter is
21.25?
A) .28 B) .22 C) .18 D) .12 E) .08
30. What is the guaranteed quarterly rate on a four quarter interest rate swap?
A) .0118 B) .0137 C) .0158 D) .0169 E) .0195
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE2-8
Practice Exam 2 - Exam FM / Exam 2
Solutions
1. We first need to find i. We can use the BA II Plus and set
N = 2, PMT = 560, PV = -1000 and FV =0. CPT I/Y = 7.9
To find K set
1000 = K/1.079 + 800/1.0792. K = 337.57
Answer D
2. The company must invest the present value of 2000 in 1 year at 6.5% plus
the present value of 5000 in 3 years at 7.5%
The cost is
2000/1.065 + 5000/1.0753 = 1877.93 + 4024.80 = 5902.73
Answer A
3. If P is the annual payment then the principal repaid in the 20th payment is
pvn-2o+i ^he p^Qipai repaid in the 5th payment is Pvn"5+1.
Dividing these we get v"15 = (1 + i)15 = 3. Then i = 3ms - 1 = 0.076.
Answer E
4. (1 + s4)4 = (1 + io,i)Q + ii,2)(l + i2,3)(l + i3,4)
= (1.029)(1.037)(1.044)(1.052) = 1.17195
s4 =1.171951/4-1 = 0.0405
Answer B
5. To get the accumulated amount of fund using the BA II Plus, set
N = 20, I/Y = 7.5, PV = 0, PMT = -910. CPT FV = 39,407.26
The annual yield rate is r = (39,407.26/10,000)1/20 - 1 = 0.071.
Answer D
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Practice Exam 2 - Exam FM / Exam 2
Page PE2- 9
6. The present values of these investments are
2000/1.072 = 1865.67 and 4000/1.0723 = 3246.95.
The total is 5112.62. The weights for the Macaulay duration are
Wi = 1865.67/5112.62 = 0.3649 and w2 = 3246.95/5112.62 = 0.6351.
D = (1)(0.3649) + (3)(0.6351) = 2.270
Answer A
7. Using the BA II Plus to get the price of the first bond, set
N = 10,1/Y = 4, PMT = 37.5, FV = 1000. CPT PV = -979.72.
To get the price of the second bond, set
N = 10,1/Y = 3.5, PMT = 30, FV = 1000. CPT PV = -958.42
The total price of the bonds is 1938.14.
To get the accumulation of the deposited coupon payments set
N = 10,1/Y = 3.4, PMT= -67.5, PV =0. CPT FV = 788.22.
Accumulation plus redemption values is 2788.22.
(1 + r)5 = 2788.22/1938.14 = 1.4386
r = 1.43861/5-1 = 0.075
Answer B
8. We need to find i3,4.
1 + i3.4 = (1 + s4)4/(l + s3)3
s3= 0.08 + 0.003(3) - 0.0015(9) = 0.0755
s4= 0.08 + 0.003(4) - 0.0015(16) = 0.068
i3f4 = (1.068)4/(1.0755)3 - 1 = 0.0458
Answer C
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE2-10
Practice Exam 2 - Exam FM / Exam 2
9. This annuity can be viewed as the difference between a 10-year annuity-due
with payments of 100 and a 5-year annuity-due with payments of 50.
To get the present values of these annuities using the BA II Plus, first set
the mode to BGN. For the 10-year annuity, set
N = 40,1/Y = 1.5, PMT = -100, FV = 0. CPT PV = 3036.46
For the 5-year annuity set
N = 20,1/Y =1.5, PMT = -50, FV = 0. CPT PV = 871.31
Present value of difference is 3036.46 - 871.31 = 2165.15.
Answer E
10. The present value of the annuity is lOOa^ +
106
1.07
ii
, 1.06 Tl.06
1.07
1.07
lOOajoi = 702.36
106
1.0711
, 1.06 Tl.06
1 + + ...+
1.07
1.07
=
f 106 )[
a.07nJ~
H
i-
1.06N
1.07,
'1.06
,1.07
10 ~i
))
= 482.94
The present value is 702.36 + 482.94 = 1185.30
Answer B
11. The Macaulay duration is
D = [80v + 2(80)v2 +3(1080)v3]/(80v + 80v2 + 1080v3)
v = 1/1.07 = .93458
D = 2859.32/1026.24 = 2.786
Answer D
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Exam FM / Exam 2 - Financial Mathematics

Practice Exam 2 - Exam FM / Exam 2
PagePE2-ll
12. The dividends for the first 10 years form an increasing arithmetic sequence.
The present value of these dividends is
_ (a, - 10v'°) _ [8.X078 -10(0.6139)] _
v m i 0.05
The dividends thereafter form a constant growth perpetuity. The present
value of these dividends at time t = 10 years is
P = D/(i - r) = 10(1.03)/(0.05-0.03) = 515
This is deferred for 10 years so the stock price is
39.376 + 515/1.0510 = 355.54
Answer C
13. If P is the annual payment, the amount of principal repaid in the 21st is
Pu3o-2i+i _ Pvio = p/2 Hence v10 = V2.
So (1 + i)10 = 2. The i = 2mo - 1 = .0718.
Using the BAII Plus to get the payment, set
N = 30,1/Y = 7.18, PV = -100,000, FV = 0. CPT PMT = 8,204.84
The principal repaid in the 10th payment is
8,204.84v3010+1 = 8,204.84(1.0718)-21 = 1,912.85
Answer E
14. If D is the amount deposited into the fund, the accumulation at the end of
ten years is D(1.015)16/(0.96)12 = D(2.0711).
To get the force of interest, set e10<?= 2.0711.
Then 5= (l/10)ln(2.0711) = 0.0728
Answer B
15. The present value of this annuity is 100(00)^ + 100v10 am.
Then am = 7.189 and v10 = 0.5327.
(Da)mJ10-amK 43.246
v m 0.065
Present value of annuity is 4,324.60 + 100(7.189X0.5327) = 4,707.56.
Answer A
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Page PE2-12
Practice Exam 2 - Exam FM / Exam 2
16. To compute the payment with the BA II Plus set
N = 360, I/Y = 0.625, PV = 100,00, FV = 0. CPT PMT = -699.215.
To get outstanding principal after 10 years, reset N = 240. Then
CPT PV = 86,794.987. To get new payment, reset N = 180 and I/Y = 0.5.
CPT PMT = - 732.425.
Answer C
17. To find the IRR put the BA II Plus in CF mode. Then enter the following
cash flows: C0 = -20,000, Ci = 5,000, C2 = 6,000, C3 = 7,000 and C4 = 10,000.
Then IRR CPT = 13.23
Answer E
18. The price of the bond is
P = 35/1.02 + 35/1.0252 + 35/1.033 + 1035/1.044 = 984.38.
Answer B
19. The man's accumulation (using I/Y = 3.0) is 35s2oi +1000 = 1940.46.
The wife's accumulation (using I/Y = 6.5) is
Ksm=K (13.4944)
Therefore K= 1,940.46/13.4944 = 143.797
Answer D
20. Equating the present values of the two payments we get
675v2 = 200v + 500v3. Dividing by v we get the following quadratic equation:
500v2 - 675v + 200 = 0.
Using the quadratic formula we get 2 positive values for v, 0.911 and
0.439. The only meaningful root is v = 0.911, or i = .098.
Answer E
21. The put profit is
max(0,1120-ST)-71.32e035(25)=max(0,1120-1080)-71.95 = -31.95
Answer B
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Practice Exam 2 - Exam FM / Exam 2
Page PE2-13
22. Let VT be the value of the house at time T. The payoff has value
max(0,340,000-25,000-Vr) = max(0,315,000-VT)
This is the payoff of a put with K = 315,000.
Answer A
23. Using g = .02, T = 1,S0 = 100, the total payoff is
= ioV1"y)[Sl+max(102"Sl,0)]
The expression in square brackets is the payoff of a single share of the
index and a put, while the two lead terms give the number of units of this
combination the company needs to buy to pay off the single premium
deferred annuity. The company wants to use the premium n to buy the
shares and the options needed. The cost of those shares and options today is
^(l-y^So* put cost] = -^-(l-y)115.80 = 1.158^(1 -y)
To break even this cost must equal the premium collected.
1.158;r(l-y) = ;r->y = .1364
The required percentage is 13.64%
Answer D
24. Buying the index and buying a put with strike 1300 creates a floor. The floor
has the same profit function as a long call with strike 1300.
The minimum profit on the floor is the (negative) loss of the future value of
the call premium when the call expires unexercised. By parity, the value of
the call is 83.18. The minimum profit is -83.18e035(25) = -83.91
Alternatively, we could write the profit for stock prices less than 1300 as the
put strike payoff less the future value cost of the put premium and
repayment of a loan of 1300 to buy the stock
1300- 71.85e035(25) - 1300e035(25) = -83.91
Answer D
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Page PE2-14
Practice Exam 2 - Exam FM / Exam 2
25. For Mary's portfolio the number of long calls at K = 55 is not given.
However you can quickly figure out what it is.
The arbitrage lends $2, so in order to have 0 outlay at the beginning there
must be $2 of excess cash obtained from the sale and purchase of calls. If
there are n long calls at K = 55 we have the following proceeds from
options.
Strike
Position
Proceeds
40
Long 2
-22
50
Short 6
+36
55
Long n
-3n
Since total proceeds are 2 to lend, we have
-22 + 36 - 3n = 2 -> n = 4
Mary has no out-of-pocket cost at time 0. She earns $2 and invests it at the
continuous rate r = .04. Her profit at time .5 is the future value of the
invested $2 + the sum of the payoffs of the options in the portfolio.
2e02+2(52-40)-6 (52-50)+ 4(0) = 14.04
Answer E
26. This is a written straddle. It assumes its maximum profit value at the strike
price of 1300, where both sold options expire worthless and the writer
retains the future value of the two premiums.
(71.85 + 83.18) e035(25) =156.39
Answer D
27. The correct forward price is S0e"^ = 40e"01 = 39.60. Thus the market price is
correct and there is no arbitrage.
Answer A
28. Your position is - LONG FORWARD + STOCK. This is equivalent to BOND,
or purchase of a zero coupon bond at the interest rate r = .03 The forward
price is the correct theoretical price.
Answer B
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Practice Exam 2 - Exam FM / Exam 2
Page PE2-15
29. The swap price is
n
gP(0.*Q/o(*t) 20.9 (.984)+ 21.2 (.969)+ 20.8 (.953)
lP(0,t,) -984+ .969+ .953 =
The spot price in the second quarter is 21.25, and the payment is 21.25 -
20.97 = .28
Answer A
30. The guaranteed interest rate is the four year par coupon bond rate.
1-P(0,4)
c =
P(0,l) + P(0,2) + P(0,3) + P(0,4)
1-.935
.984+ .969+ .953+ .935
= .0169
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 3 - Exam FM / Exam 2
PagePE3- 1
Exam FM
Questions
1. A man has two annuities-immediate with the same interest rate and the
same level payments. The first is a 30-year annuity and the second one is a
15-year deferred 15-year annuity. The present value of the first is 4 times
the present value of the second. Find the interest rate.
A) 7.3% B) 7.4% C) 7.5% D) 7.6% E) 7.7%
2. A 10-year 1000 par bond has 6% semi-annual coupons. The bond is sold at a
premium of 35. What is the bond's nominal annual yield convertible
semiannually?
A) 5.48% B) 5.54% C) 5.62% D) 5.71% E) 5.79%
3. A company has liabilities of 2000 and 5000 due at the end of years one and
three respectively. The investments available to the company are two zero
coupon bonds. The first is a one-year 1000 par value bond with an annual
effective rate of 5.6%. The second is a three-year 1000 par bond. If the cost
of exactly matching liabilities is 6068.36, what in the annual effective yield
on the second bond?
A) 5.2% B) 5.5% C) 5.8% D) 6.0% E) 6.2%
4. An investment pays 2000 at the end of year one, 4000 at the end of year 3
and 6000 at the end of year 5. It was purchased to yield an annual rate of
6.2%. Find the Macaulay duration of this investment.
A) 3.28 B) 3.44 C) 3.49 D) 3.53 E) 3.56
5. A 10-year 1000 par bond with 6.5% semi-annual coupons is priced to yield at
an annual rate of j convertible semi-annually. The amount of premium
amortized in period 7 is 2.346, and the amount amortized in period 12 is
2.706. Findj.
A) 5.8% B) 6.0% C) 6.2% D) 6.4% E) 6.6%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE3-2
Practice Exam 3 - Exam FM / Exam 2
6. Money is deposited in a bank. For the first 4 years interest accumulates at
annual nominal rate of 6% convertible monthly. For the next 6 years it
accumulates at a force of interest of 5%. For the 10-year period what is the
equivalent nominal discount rate convertible quarterly?
A) 4.9% B) 5.2% C) 5.4% D) 5.7% E) 5.9%
7. A man planning to work for the next 30 years sets up a retirement account
by making monthly end of the month payments in to a fund. The first
payment is 100 and each subsequent payment is 1 more that the previous
one. The fund earns at a nominal rate of 7.2% convertible monthly. At the
end of the 30 years he plans to make end of month withdrawals of 2000 per
month. If interest rates stay the same, how many payments will he expect to
receive?
A) 292 B) 302 C) 312 D) 322 E) 332
8. For a given yield curve the implied forward rates are i0,i = 0.030 and ii,2 =
0.032. The spot rate i3 = 0.04. Find i2,3.
A) 0.0542 B) 0.0547 C) 0.0553 D) 0.0561 E) 0.0582
9. Consider the following account summary:
Balance
Date Before Activity Deposits Withdrawals
January 1 10,000
April 1 10,500 2000
September 1 12,800 2600
December 31 X
If the time weighted yield is 6.466%, what is the dollar weighted yield?
A) 6.58% B) 6.62% C) 6.65% D) 6.71% E) 6.74%
10. A man buys a home for 200,000 and takes out a 30-year mortgage with
monthly payments. The interest rate is 5.4% convertible monthly. At the end
of 15 years he decides to add 500 a month to each subsequent payment.
Assuming there are no penalties, how many more payments, including the
final partial payment, are there?
A) 102 B) 108 C) 111 D) 115 E) 120
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 3 - Exam FM / Exam 2
Page PE3- 3
11. A woman buys a 10-year 1000 par bond with 7.0% semi-annual coupons. The
coupon payments are deposited into an account that pays 6.6% convertible
semi-annually. After the 10th deposit the bank drops its rate to 5.8%
convertible semi-annually. At the end of the 10 years period what is her
annual yield for this investment?
A) 6.5% B) 6.7% C) 6.9% D) 7.1% E) 7.3%
12. A man has a 30-year loan with level annual end of year payments. The
principal repaid in the 10th payment is 408.12, and the principal in the 20th
payment is 766.10. What is the principal repaid in the 15th payment?
A) 540.33 B) 544.02 C) 548.65 D) 552.25 E) 559.16
13. Tom has a 10-year increasing annuity-immediate that pays 100 for the first
year and increases by 100 each year thereafter. Jerry has a 10-year
decreasing annuity-immediate that pays X the first year and decreases by
X/10 each year thereafter. Each has an annual interest rate of 6.5%, and they
have the same present value. Find X.
A) 821 B) 828 C) 835 D) 842 E) 849
14. Sally and Linus each make annual end of year deposits into a savings
accounts that have the same annual interest rate. Sally's annual deposits are
100. Linus deposits 100 per year for the first 10 years and 200 per year
thereafter. At the end of 20 years Linus has accumulated 4/3 the amount that
Sally has. What is their common, nonzero, interest rate?
A) 6.0% B) 6.5% C) 6.9% D) 7.2% E) 7.5%
15. You are given the following yield curve:
Year Spot Rate
1 4.5%
2 4.0%
3 3.8%
4 3.6%
A 3-year 1000 par bond has a 5% annual coupon rate. Use the yield
curve to find the price of the bond.
A) 1033 B) 1038 C) 1042 D) 1046 E) 1051
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Page PE3-4
Practice Exam 3 - Exam FM / Exam 2
16. A 3-year 1000 par bond with 5.8% annual coupons is priced to yield 6.4%.
What is the Macaulay duration for the bond?
A) 2.795 B) 2.801 C) 2.837 D) 2.862 E) 2.890
17. A man invests 1000 at the beginning of each year into a fund that pays an
annual interest of 5.6%. The annual interest payments are deposited into a
fund that that pays 6.2% annually. What is his total accumulation at the end
of 10 years?
A) 13,261 B) 13,585 C) 13,730 D) 14,020 E) 14,318
18. A perpetuity immediate pays 100 a year for the first 10 years. Starting with
year 11, each payment is 3% more than the previous one. The annual yield is
4.5%. Find the present value of this perpetuity.
A) 5213 B) 5324 C) 5375 D) 5431 E) 5486
19. Lucy deposits 1000 into an account and makes an additional deposit of 2000
two years later. The account accumulates at a constant force of interest. At
the end of 4 years the accumulation is 3431.75. Find the force of interest.
A) 0.035 B) 0.040 C) 0.045 D) 0.050 E) 0.055
20. A man buys a house using a thirty year mortgage loan for 300,000. The loan
has an interest rate of 6% convertible monthly. He also owns a fifteen year
zero-coupon bond which will make a payment of 100,000 to him on the same
day as he makes the last payment of the fifteenth year of the mortgage. He
plans to increase his monthly payment over the first fifteen years so that at
the end of year fifteen he can use the 100,000 from the bond to retire the
loan. What should his new monthly payment for the first fifteen years be?
A) 1798.65 B) 1995.83 C) 2187.71 D) 2297.81 E) 2798.65
21. The current price of a stock that pays no dividends is 40. The continuously
compounded risk free rate is 4%. Investor A buys a six month 41-strike put
for 3.48. Investor B enters into a six month short forward contract to sell
that stock for the forward price 40.81. At what stock price do the two
investors have the same profit in six months?
A) 40.81 B) 41 C) 44.29 D) 44.36
E) They do not have the same profit at any stock price.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 3 - Exam FM / Exam 2
Page PE3- 5
22. An investor buys a 30-strike put and a 30-strike call on a stock. Both options
have the same expiration date. Which of the following is the most likely
reason for taking this position?
A) To profit from an expected increase in the stock price.
B) To profit from an expected decrease in the stock price.
C) To profit from high volatility in the stock price.
D) To profit from low volatility in the stock.
E) To create a synthetic forward sale.
23. You buy a 35-strike put and write a 45-strike call on a stock. The options
have the same expiration date. Which of the following can be the graph of
your profit?
A)
D)
E) None of these
24. The current price of a stock is 40. The price of a 35-strike call is 6.13 and the
price of a 45 strike call is 0.97. Consider buying n 35-strike calls and selling
m 45-strike calls. What ratio n/m gives you a zero premium for this
position?
A) .158
B) .172
C) .567
D) 5.814 E) 6.320
©ACTEX 2009
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Exam FM / Exam 2 - Financial Mathematics

Page PE3-6
Practice Exam 3 - Exam FM / Exam 2
25. You write a 35-strike put and a 45-strike call on a stock. The options both
expire in three months. The price of the put is 0.44 and the price of the call
is 0.97. The continuous risk free rate is 4%. What is your maximum profit?
A) 0.53 B) 0.535 C) 1.41 D) 1.424
E) There is no maximum
26. Which of the following are true?
I. Future and forward prices at expiration for otherwise identical contracts
must be the same, since futures are standardized forwards.
II. When the interest rate is positively correlated with the futures price, the
futures price will exceed the forward price for an otherwise identical
contract.
III.All forward and futures contracts will require a maintenance margin
account which is marked to market on a regular basis.
A) I only B) II only C) III only D) I and III E) II and III
27. The price of an S&P 500 Index futures contract is 1520. An investor enters a
short forward position. When the position is closed the futures price is 1540.
If there is no settlement requirement, what is the dollar gain or loss?
A) $20 gain B) $20 loss C) $5000 gain
D) $5000 loss E) None of these
28. A stock has current price S0 = 35. The annual continuous interest rate is
r = .04 and the continuous dividend yield is 8 = .02 . You observe a one year
prepaid forward price of 34.20. Which of the following is true?
A) No arbitrage is possible.
B) You can create an arbitrage by buying one prepaid forward and selling
one share of the stock short
C) You can create an arbitrage by selling the prepaid forward and buying
one share of the stock.
D) You can create an arbitrage by buying the prepaid forward and selling
e"02 shares of the stock short
E) You can create an arbitrage by selling the prepaid forward and buying
e"02 shares of the stock.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 3 - Exam FM / Exam 2 Page PE3- 7
29. The zero-coupon bond prices for the next 3 quarters are
Quarter
Zero-coupon bond price
1
.985
2
.971
3
.954
The guaranteed rate on a four quarter interest rate swap is 1.74%. Find the
zero coupon bond rate for the fourth quarter.
A) .929 B).933 C) .935 D) .937 E) .939
30. Zero coupon bond yields and oil forward prices for the next three years are
Year
Oil Forward Price
Zero-coupon bond yield
1
60
5%
2
62
6%
3
64
7%
What is the level swap payment for a three year oil price swap?
A) 61.90 B) 62.13 C) 62.27 D) 62.38 E) 62.43
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE3-8 Practice Exam 3 - Exam FM / Exam 2
Solutions
1. We may assume the payments are 1. Relating the present values we get
(1 - v30)/i = 4vls(l - v15)/i.
Dividing by (1 - v15) yields
1 + v15 = 4v15 ^ (1 + i)15 = 3 => i = 0.076.
Answer D
2. The price of this bond is 1035. To get the yield using the BA II Plus
calculator set N = 20, PV = -1035, PMT = 30 and FV = 1000. Then CPT I/Y =
2.770. Yield is 5.54%.
Answer B
3. We have 2000/1.056 + 5000/(1 + i)3 = 6068.36. Hence
(1 + i)3 = 5000/4174.42 = 1.1978 ^ 1 + i = 1.062 =* i = .062
Answer E
4. The present values of the payments are
2000/1.062 + 4000/1.0623 + 6000/1.0625
= 1883.24 + 3339.54 + 4441.49 = 9664.27
The weights for the Macaulay duration are
wi = 1883.24/9664.27 = 0.1949
w2 = 3339.54/9664.27 = 0.3456
w3 = 4441.49/9664.27 = 0.4596
D = 0.1949(1) + 0.3456(3) + 0.4596(5) = 3.5297
Answer D
5. The amount of premium amortized in period k is
1000(0.0325 -j/2)v20fc+1 where v = 1/(1 + j/2).
For period 7 we have
2.346 = 1000(0.0325 - j/2)(l + j/2)14.
For period 12 we have
2.706 = 1000(0.0325-j/2)(l +j/2)9
So (l+j/2)5 = 2.706/2.346 = 1.1535 => j/2 = 0.029
Thus; = 0.058
Answer A
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 3 - Exam FM / Exam 2
Page PE3- 9
6. The accumulation factor for the 10 year period is
(1.005)48e005(6) = 1.7150.
Then (l-d(4)/4)-40 = 1.7150
And 1 - d(4)/4 = 0.9866 => d(4) = 0.054
Answer C
7. The total accumulation is given by the future value version of the P-Q
present value formula (2.53) in module 2 of the study guide.
A =100s3-^+(s3-^-360)/i
= 100(1,269.225) + (1,269.225 - 360)/0.006
= 126,922.50 + 151,537.50 = 278,460
For withdrawals using the BA II Plus calculator set I/Y = 0.6, PV = 278,460,
PMT = -2000 and FV = 0. Then CPT N = 301.59.
Answer B
8. (1 + s3)3 = (1 + io,i)(l + iu)(l + 12.3)
(1 + i2,3) = (1.04)3/[(1.03)(1.032)] = 1.0582
Answer E
9. Using the time weighted yield to get X we have
(10,500/10,000)(12,800/12,500)(X/10,200) = 1.06466
X = 10,100
The amount of interest earned is found from
10,000 + 2000 - 2600 + J = 10,100 => J = 700
To get the dollar weighted yield set
j = 700/[10,000 + (3/4)2000 -(1/3)2600] = 0.0658
Answer A
10. First find the monthly payments using the calculator. Set N = 360, I/Y = 0.45,
PV = -200,000 and FV =0. Then CPT PMT = 1123.06.
Reset N = 180, then CPT PV = -138,344.43
Reset PMT = 1623.06 and CPT N = 107.75
Answer B
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE3-10
Practice Exam 3 - Exam FM / Exam 2
11. The accumulation of deposits is
A = 35 s^ o.o33(1.029)10 +35 s^ 0.029
= (406.82)(1.3309) + 399.39 = 940.83
Total return is 1940.83
Annual yield is (1940.83/1000)1'10 - 1 = 0.069
Answer C
12. The amount of principal repaid in the fcth period is
PMTv3o-k+i For the 10th period, PMTv21 = 408.12
For the 20th period, PMTv11 = 766.10
v10 = 408.12/766.10 = 0.53272
Principal repaid in the 15th period is
PMTv16 = (PMTvn)v5 = 766.10(0.53272)1/2 = 559.16
Answer E
13. The present value of Tom's annuity is
lOOda)^ = lOOKfiioi- 10v10)/i] =3582.84
The present value of Jerry's annuity is
(X/lOXDa) m = (X/10)[(n-amW] = 4.325X
X = 3582.84/4.325 = 828.40
Answer B
14. Sally's accumulation is A = 100[(1 + i)20 - l]/i
Linus's accumulation is
B = 100[(1 + i)20 - l]/i + 100[(1+ i)10 - l]/i.
B = (4/3) A, or 4A = 3B. Let x = (1 + i)10. Then
4jc2 - 4 = 3jc2 - 3 + 3x - 3 or x2 -3x + 2 = 0.
The roots are x = 1, 2 (x = 1 yields i = 0.)
Hence (1 + i)10 = 2 => i = 0.072
Answer D
15. The price of the bond is
P = 50/1.045 + 50/1.0402 + 1050/1.0383 = 1032.93
Answer A
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 3 - Exam FM / Exam 2
Page PE3-11
16. To get the price of the bond using the calculator,
set N = 3,1/Y = 6.4, PMT = 58 and FV = 1000. Then
CPT PV = -984.08. The price is 984.08
The Macaulay duration is
D = [(58/1.064) + (58)(2)/1.0642 + (1058)(3)/1.0643]/984.08
= 2.837
Answer C
17. The amount of interest for year k is 1000k(0.056) = 56k. The
accumulation of these payments plus interest is
56(18)^= 3730.48. Total accumulation is 13,730.48.
Answer C
18. This can be visualized as a 10-year annuity immediate plus a
10-year deferred geometrically increasing perpetuity.
The present value is
100a m + [103/(0.045 - 0.03)](1.045)10 = 791.27 + 6866.67(0.6439) = 5212.72
Answer A
19.The accumulation is A = lOOOe45 + 2000e25 = 3431.75.
Let x = e25. Then we have
1000*2 + 2000* - 3431.75 = 0
The positive root of this equation is x = e25 = 1.1052.
5 = 0.05
Answer D
20. The normal monthly payment would be 1798.65. The man wants to make a
larger payment that would leave a balance of 100,000 at the end of fifteen
years so that he can pay the 100,000 from the bond and retire the loan. Use
the calculator.
300,000 PV
.5 I/Y
180 N
100,000 ±_ FV
CPT PMT
The payment is 2187.71.
Answer C
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE3-12
Practice Exam 3 - Exam FM / Exam 2
21. Let S be the stock price in six months. The profit function of the forward is
40.81 - S. The profit function of the put is
Max (41 - S, 0) - 3.48e04( 5) = Max (41 - S, 0) - 3.55
[37.45 -S, S<41
1-3.55 S>41
If you think graphically, it is clear that the intersection occurs when S > 41.
3.00 -
1.00
£ -1.00 -
Q_
-3.00 -
-5.00 -
.7 nn -
\.
^\
•>, ^s.
37 39 41 43
Stock Price
45
i orwara
Put |
47
To find the intersection point we solve the equation
40.81 - S = -3.55 -> S = 44.36
Answer D
22. This position is a straddle, which is designed to capture returns from high
volatility.
Answer C
23. This is a collar.
Answer A
24. We need 6.13n = .97m -> — = .158.
m
Answer A
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 3 - Exam FM / Exam 2
Page PE3-13
25. This is a written strangle. The maximum value occurs for stock prices
between 35 and 45 where the options expire worthless and you pocket the
total call premiums.
(.44 + .97)e04(-25)= 1.424
Answer D
26. I) is false. Final prices can differ since margin is required for futures but
not necessarily for forwards.
II) is true. It is taken from page 147 of Derivatives Markets.
III) is false, as we see from I) above.
Answer B
27. The notional value of the contract is 250 times the index contract price.
Thus the short loses 20(250) = 5000.
Answer D
28. The correct prepaid forward price is S0e~ST = 35e"02 = 34.31. Thus the
forward price of 34.20 is too low. You can buy the forward for 34.20 and sell
short a tailed position in the stock for a price of S0e~ST = 35e"02 = 34.31. This
gives a profit of .11 at time 0. In one year the prepaid forward will deliver a
share of stock which can be used to cover the short sale.
Answer D
29. The guaranteed interest rate is the four year par coupon bond rate.
1-P(0,4)
c =
P(0,l) + P(0,2) + P(0,3) + P(0,4)
It follows that
1-P(0,4)
.985 + .971 + .954 + P(0,4)
Answer B
= .0174->P (0,4) = .933
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE3-14
Practice Exam 3 - Exam FM / Exam 2
30. We will use the general formula
£p(o,to/0(tO
p=^—„
The required zero-coupon bond prices are
p(0'1)=n>5=-952' p(a2)=r^=-890' p(°'3)=ii^=-816
The swap payment is
60 (-952)+ 62 (,890) + 64 (-816) _
.952+ .890+ .816
Answer A
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 4 - Exam FM / Exam 2
PagePE4- 1
Exam FM
Questions
1. If 5 = 0.08, find i(6) + d(4).
A) 0.1591 B) 0.1597 C) 0.1608 D) 0.1615 E) 0.1621
2. A man has children aged 15,18 and 20. He purchases annuities for each one
that pay 5000 a year beginning now and continuing as long as the recipient is
under 30. The annual interest rate is 5.5%. What is the total cost of these
annuities?
A) 131,000 B) 135,000 C) 138,000 D) 142,000 E) 147,000
3. A company has liabilities of 5,000 and 2,000 due at the end of years 2 and 4
respectively. It purchases zero coupon bonds maturing in 2 and 4 years, both
earning the same interest rate. The cost of matching liabilities exactly is
6000. What is the common interest rate for these bonds?
A) 5.4% B) 5.6% C) 5.8% D) 6.0% E) 6.2%
4. A man buys a house for with a 30-year 6.4% monthly payment mortgage for
150,00. After 12 years he refinances the house at a new rate of 5.8% and a
new term of 10 years. What are his new monthly payments?
A) 1322 B) 1330 C) 1337 D) 1342 E) 1349
5. A woman buys a 20-year 1000 par bond with 6% semiannual coupons. The
bond is priced to yield 5.6% convertible semiannually. The coupon payments
are deposited into a fund that earns 5% convertible semiannually for the
first 10 years and 5.4% convertible semiannually for the last 10 years. What
is her annual yield on this investment?
A) 5.2% B) 5.4% C) 5.6% D) 5.8% E) 6.0%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE4-2
Practice Exam 4 - Exam FM / Exam 2
6. Given the spot rate of s2 = 0.046 and the forward rates i2,3= 0.037 and i3,4 =
0.039, find s4.
A) 0.036 B) 0.038 C) 0.040 D) 0.042 (D) 0.044
7. A man purchases a 30-year annuity immediate that makes annual payments.
The first 10 payments are 200, the next 10 are 400 and the last 10 are 300.
The annuity earns 6.5% annually. What is the present value of this annuity?
A) 3582 B) 3617 C) 3675 D) 3713 E) 3753
8. A man has 30-year 6.6% home mortgage with monthly end of month
payments of 766.39. What is the first period in which the principal repaid is
over 500?
A) 269 B) 274 C) 278 D) 281 E) 284
9. An annual corporate bond is priced to yield 6.7% annually and has a price of
1023.68. Its Macaulay duration is D = 8.2135. Estimate the change in price if
rates decrease by 0.10%.
A) 7.880 B) 7.893 C) 8.010 D) 8.018 E) 8.023
10. A woman has a 20-year annuity immediate with annual payments and an
annual interest rate of 6.3%. The annuity pays 1000 the first year.
Subsequent payments decrease by 100 per year until they reach 100. The
remaining payments stay at 100 per year. Find the present value of this
annuity.
A) 4695 B) 4723 C) 4749 D) 4801 E) 4862
11. A man borrows 50,000 for 10 years at 7.6% annual interest. For the first 3
years he makes payments of only 3000. What will his payments need to be
for the final 7 years?
A) 9,876 B) 9,915 C) 9,963 D) 10,020 E) 10,088
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 4 - Exam FM / Exam 2
Page PE4- 3
12. An investor has a portfolio consisting 20,000 worth of a 2-year bond with a
modified duration of 1.92, 35,000 worth of a 3-year bond with a modified
duration of 2.84 and 45,000 worth of a 5-year bond with a modified duration
of 4.79. Find the modified duration for the entire portfolio.
A) 3.49 B) 3.53 C) 3.57 D) 3.61 E) 3.65
13. A woman invests 12,000 in a project. She is promised returns of 4,000 in 2
years, 6,000 in 3 years and 8,000 in 4 years. Find the IRR for this investment.
A) 13.05% B) 13.27% C) 13.44% D) 13.59% E) 13.71%
14. A three-year $1000 par value bond with 4.5% annual coupons is priced using
the spot rates implied by the forward rates i0,i = 0.051, ii,2 = 0.047 and i2,3 =
0.043. Find the price of the bond.
A) 964 B) 974 C) 984 D) 994 E) 1004
15. A woman deposits 1000 into a savings account. For the first 5 years the
money accumulates with a force of interest of 5 = 0.04. For the nest 3 years
the money accumulates at a nominal discount rate of 0.06 convertible
semiannually. At the end of 10 years the money has earned at an annual
effective rate of 5.2%. What was the nominal annual interest rate
convertible quarterly for the last 2 years?
A) 5.5% B) 5.7% C) 5.9% D) 6.1% E) 6.3%
16. You begin the year with 8000 in an account. You make deposits of 2000 on
March 1 and 1000 on November 1. You withdraw 500 on July 1. Your dollar-
weighted yield for the year is 8.87%. How much interest did you earn?
A) 850 B) 861 C) 869 D) 873 E) 882
17. A 20-year annuity immediate pays 100 the first year and increases by 100 a
year through year 10. Starting in year 11 each yearly payment is 5% greater
than the previous payment. The annuity earns 6.8% annually. What is the
present value of this annuity?
A) 8175 B) 8239 C) 8290 D) 8344 E) 8395
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page PE4-4
Practice Exam 4 - Exam FM / Exam 2
18. A 3-year 1000 par bond with 4.8% annual coupons is priced to yield 5.5%.
What is the Macaulay duration of the bond?
A) 2.836 B) 2.844 C) 2.851 D) 2.863 E) 2.875
19. A man borrows 65,000 for 20 years and makes the annual interest payments
to the lender. He makes annual contributions to a sinking fund to raise
money to pay off the principal. He makes payments to the sinking fund of X
for the first 10 years and 2X for the last 10 years. The fund earns 6.5%. Find
X.
A) 1232 B) 1237 C) 1242 D) 1247 E) 1252
20. You are given the yield curve sk = 0.068 + 0.002k - 0.001/c2. Find the 3 year
forward rate implied by this yield curve.
A) 3.9% B) 4.1% C) 4.3% D) 4.5% E) 4.7%
21. The current price of a stock that pays no dividends is 40. The continuously
compounded risk free rate is 4%. Investor A buys a six month 41-strike put
for 3.48. Investor B enters into a six month short forward contract to sell
that stock for the forward price 40.81. At what stock price do the two
investors have the same payoff in six months?
A) 40.81 B) 41 C) 44.29 D) 44.36
E) They do not have the same profit at any stock price.
22. An investor buys a 30-strike call and a sells a 35-strike call on a stock. Both
options have the same expiration date.
Which of the following is the most likely reason for taking this position?
A) To profit from an expected increase in the stock price.
B) To profit from an expected decrease in the stock price.
C) To profit from high volatility in the stock price.
D) To profit from low volatility in the stock.
E) To create a synthetic forward sale.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 4 - Exam FM / Exam 2
Page PE4- 5
23. You buy a 35-strike put and a 45-strike call on a stock. The options have the
same expiration date. Which of the following can be the graph of your
profit?
A)
C)
D)
E) None of these
24. The current price of a stock is 40. The price of a 35-strike three month call
is 6.13 and the price of a 35 strike three month put is 0.44. The continuous
risk-free rate is 4%. What is the price of a forward contract to buy the stock
in three months for 35?
A) 5.69
B) 5.75
C) 6.77
D) 6.84
E)0
25. The Wiresguys Company manufactures wire, for which it must buy copper.
Two pounds of copper will produce one unit of wire, which sells for the
price of the two pounds of copper plus $8. The fixed cost of the unit of wire
is $4 and the variable cost is $3. The current cost of copper is 1.10 per
pound. Which of the following might be advisable for Wiresguys?
A) Hedge by buying a .90 strike call for copper.
B) Hedge by designing a paylater option strategy for copper.
C) Hedge with a collar for copper.
D) It is not necessary to hedge the cost of copper
E) None of the above.
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page PE4-6
Practice Exam 4 - Exam FM / Exam 2
26. Which of the following are true for forward price arbitrages when
transaction costs and differences between borrowing and lending rates are
taken into account?
I) The no-arbitrage region becomes wider when transaction costs
increase.
II) The no-arbitrage region becomes narrower if the borrowing rate
increases and the lending rate decreases.
III) It is likely that the no-arbitrage region will be different for different
arbitragers.
A) I only B) II only C) III only D) I and III E) II and III
27. A stock has current price S0 = 40. The annual continuous interest rate and
dividend yield are r = .025 and 8 = .01. If the expiration time for a forward
contract is T = .5, what is the difference between the forward price and the
prepaid forward price?
A) 0.10 B)0.20 C)0.30 D) 0.40 E) 0.50
28. The S&R index has a spot price of S0 = 1300. The continuous interest rate is
. r = .05 and the continuous dividend yield is 8 - 0 You observe a six month
forward price of 1340. What arbitrage profit can be made in 6 months?
A)0 B)5.23 Q7.09 D) 9.80 E) 10.17
29. Zero coupon bond yields and oil forward prices for the next two years are
Year
Oil Forward Price
Zero-coupon bond yield
1
60
5%
2
62
6%
A dealer provides a two year fixed price oil swap to a client and hedges it
with forward contracts. Which of the following best describes the dealer's
position after hedging?
A) Between year 1 and 2 he will be borrowing at a rate of 5.5%.
B) Between year 1 and 2 he will be lending at a rate of 5.5%.
C) Between year 1 and 2 he will be borrowing at a rate of 7%.
D) Between year 1 and 2 he will be lending at a rate of 7%.
E) None of the above.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 4 - Exam FM / Exam 2
Page PE4- 7
30. Zero coupon bond yields for the next three years are
Year
Zero-coupon bond yield
1
5%
2
6%
3
7%
What is the level swap rate for a three year interest swap?
A) 5.8% B) 6.1% C) 6.7% D) 6.8% E) 6.9%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE4-8
Practice Exam 4 - Exam FM / Exam 2
Solutions
1. The relations are e5 = (1 + i{6)l6f = (1 - d(4)/4)^
So i(6) = (e5/6 - 1)16 = 0.0805 and d(4) = (1 - e-5/4)/4 = 0.0792.
Thus i(6) + d(4) = 0.1597
Answer B
2. The children will receive 15,12 and 10 payments respectively. The annuities
are annuities due. The sum of the present values is
5000(6^ +d^ + am) = 5000(10.590 + 9.093 + 7.952) = 138,175.
Answer C
3. We have 5000/(1 + i)2 + 2000/(1 + i)4 = 6000.
Letting x = (1 + i)2, we have 5000/x + 2000/x2 = 6000.
This reduces to 6x2 - 5x - 2 = 0.
The positive root is x = (1 + i)2 = 1.1287 => i = 0.062
Answer E
4. To find the payment using the calculator set N =360,1/Y = 6.4/12,
PV = -150,000 and FV = 0. Then CPT PMT = 938.26.
To find the balance after 12 years, reset N = 144 and CPT FV = 120,160.54.
Now reset I/Y = 5.8/12, N= 120, PV= -120,160.54 and FV = 0. Then CPT PMT =
1321.99.
Answer A
5. To find the price of the bond with the calculator set N = 40, I/Y = 2.8, PMT =
30 and FV = 1000. Then CPT PV = -1047.76.
The accumulation of coupon payments plus interest is
A = 30[s^ o.o25 (1.027)20 + S2010.027] = 2087.66.
Total accumulation is 3087.66.
Annual yield = (3087.66/1047.76)1/20 - 1 = 0.056
Answer C
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Practice Exam 4 - Exam FM / Exam 2
Page PE4- 9
6. The basic formula is (1 + sn)n = (1 + sn-i)nl(l + in-i,n).
(1 + s3)3 = (1.046)2(1.037) = 1.1346
(1 + s4)4 = 1.1346(1.039) = 1.788
Then s4 = 0.042
Answer D
7. The present value of this annuity is
A =300a^ +100 a^ -200aioi
= 300(13.059) + 100(11.019) - 200(7.189) = 3581.80
Answer A
8. The amount of principal repaid in period k is PMTv360"^1.
Set 766.39v361-* = 500.
Then (1 + i)361fc = 1.0055361fc = 766.39/500 = 1.5328.
361 - k = ln(1.5328)/ln(1.0055) = 77.867 => k = 283.135
The first period in which principal repaid is over 500 is the 284th.
Answer E
9. AP = -(D)P(i)(Ai)/(l + i)
= -(8.2135)(1023.68)(-0.001)/1.067 = 7.880
Answer A
10. The present value of this annuity is
A =100(Da)^ +100v10a^
= 100(10 - am)/i + 100(1.063)10a^
a^= 7.2566. Hence A = 4748.52
Answer C
11. We first need to find the principal balance due after the third payment.
Using the calculator set N= 3,1/Y = 7.6, PV = 50,000 and PMT = -3000. Then
CPT FV = -52,587.02. The new balance is 52,587.02. To get the new payments
set N = 7,1/Y = 7.6, PV = -52,587.02 and FV = 0. Then CPT PMT = 9962.76
Answer C
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page PE4-10
Practice Exam 4 - Exam FM / Exam 2
12. The weights for the individual bonds are wi = 20,000/100,000 = 0.2,
w2 = 35,000/100,000 = 0.35 and w3 = 45,000/100/000 = 0.45.
The duration of the portfolio is 0.2(1.92) + 0.35(2.84) + 0.45(4.79) = 3.534
Answer B
13. Using the CF worksheet on the calculator set CF0 = -12,000, C01 = 0, C02 =
4,000, C03 = 6,000 and C04 = 8,000. Then key IRR CPT. The yield is 13.59%.
Answer D
14. We need to find (1 + s„)n for n = 1, 2 and 3.
(1 + si) = 1 + io.i = 1.051
(1 + s2)2 = (1 + si)(l + iu) = (1.0S1)(1.047) = 1.1004
(1 + s3)3 = (1+ s2)2(l + i2,3) = (1.1004X1.043) = 1.1477
P = 45/1.051 + 45/1.1004 + 1045/1.1477 = 994.23
Answer D
15. The accumulation is A = e004(5)(l - 0.06/2)6(l + i(4)/4)8 = (1.052)10.
(1 + i(4)/4)8 = 1.6602(e02)(0.97)6 = 1.1322
Then i(4> = 0.063
Answer E
16. The amount interest can be found using the formula
I/[8,000 + 2000(5/6) + 1000(1/6) - 500(1/2)] = j = 0.0887
I = 0.0887(9583.33) = 850.04
Answer A
17. The present value of the annuity is
PV = 100(Ia) m + 1050v:0
1-11±«
1 + i
'(*-*)
where i = 0.068 and g = 0.05
(la) ^=(a^-10v10)/i = 35.170
PV = 100(35.17) + 1050(0.5179X8.684) = 8239.4
Answer B
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 4 - Exam FM / Exam 2
PagePE4-ll
18. The Macaulay duration is
D = [48/1.055 + 2(48)/1.0552 + 3(1048)/1.0553]/(Bond Price)
To find the bond price set N = 3,1/Y= 5.5, PMT = 48 and FV = 1000. Then CPT
PV = -981.11. So bond price is 981.11.
D = 2809.22/981.11 = 2.863
Answer D
19. The accumulation in the sinking fund is
Xs^ + Xs^ = X(38.825 + 13.494) = 52.319X
Hence X = 65,000/52.319 = 1242.38
Answer C
20. The 3-year forward rate is i3>4 = (s4)4/(s3)3
s4= 0.068 + 0.002(4) - 0.001(16) = 0.060
s3 = 0.068 + 0.002(3) - 0.001(9) = 0.065
i3,4 = 1.060V1.0653 = 1.045
Answer D
21. Let S be the stock price in six months. The payoff function of the forward is
40.81 - S. The payoff function of the put is
,„ /„, n^ f41-S, S<41
Max(41-S,0) = ^
v ; [0 S>41
The put payoff is greater than the forward payoff for all values of S.
Answer E
22. This position is a bull spread, which is designed to capture returns from an
increase in stock price while costing less than the purchase of the call only.
Answer A
23. This is a strangle.
Answer B
24. You create a synthetic forward purchase for 35 by buying the 35-strike call
and selling the 35-strike put. The cost is 6.13 - 0.44 = 5.69.
Answer A
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page PE4-12
Practice Exam 4 - Exam FM / Exam 2
25. The profit of Wiresguys does not depend on the cost of copper. If C is the
cost of copper, we have:
Revenue per unit of wire = 2C + 8
Cost per unit of wire = 2C + 4 + 3
Profit per unit of wire = 1
Attempts to hedge the cost of copper are not needed. (This is based on
problem 4.13 in the text.)
Answer D
26. The no-arbitrage region has bounds
F-=(S0b-2/c)er'T and F+ = (S0a +2fc)er"T.
The changes described in I) and II) decrease F" and increase F+. Thus I) is
true and II) is false.
Ill) is true, since different arbitragers can have cost and rate differences
due to discounts, special agreements with banks or the ability to process
transactions in house.
Answer D
27. The forward price and prepaid forward price are
Fo.o.5 =40e('025-01)'5 =40.30 , F0Po.5 = 40e("01)"5 =39.80
The difference is 40.30 - 39.80 = 0.50.
Answer E
28. The forward price should be 1300e05( 5) = 1332.91. Thus you can create an
arbitrage by entering a forward sale contract at the price of 1320, and
borrowing 1300 to buy the stock today. In six months you will deliver the
share of stock and receive the forward price of 1340. The loan repayment
due is 1300e05(5) =1332.91.
Thus there is a profit of 1340 - 1332.91 = 7.09
Answer C
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Practice Exam 4 - Exam FM / Exam 2
Page PE4-13
29. The fixed swap payment x is given by
60 62 ( 1 1 1 *no^
- + t = x + =- -> x = 60.966
1.05 1.062 11.05 1.06
The dealer's payments after hedging are
Year
Payment
1
60.966 -60 = .966
2
60.966-62 = -1.034
Thus his position is equivalent to borrowing .966 and paying back 1.034 in
„. . . 1.034 i n_
one year. His rate is 1 = .07.
.966
Answer C
Note: the text pointed out in its similar example for an upward sloping yield
curve that the dealer was borrowing at the implied forward rate. The answer
of 7% is the implied forward rate here.
30. The guaranteed interest rate is the three year par coupon bond rate.
1-P(0,3)
c =
P(0,l) + P(0,2) + P(0,3)
The required zero-coupon bond prices are
P(0,1) = —!- = .952, p (0,2) = -i—= .890, P(0,3) = —^ = .816
v ; 1.05 v ; 1.062 v ; 1.073
The swap rate is
1-.816
.952+ .890+ .816
Answer E
= .069
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Practice Exam 5 - Exam FM / Exam 2
PagePE5- 1
Exam FM
Questions
1. A man wishes to accumulate 100,000 by making monthly end of month
contributions for 30 years into an account that earns 5.4% interest
convertible monthly. After 10 years the interest rate increases to 6.6%
convertible monthly. What should his new contributions be if he still wishes
to accumulate 100,000?
A) 60.70 B) 63.85 C) 68.50 D) 71.25 E) 74.65
2. Charlie deposits 5000 into an account that earns an annual rate of i. Lucy
buys a 25-year annuity-immediate for 5000. The annuity has annual
payments and earns 7% annually. She deposits her annual payments into a
fund that pays 6.5% annually. After 25 years Charlie and Lucy have
accumulated the same amount. Find i.
A) 6.7% B) 6.9% C) 7.1% D) 7.3% E) 7.5%
3. An investor buys a 10-year 1000 par bond that has 7.5% semiannual coupons
and is priced to yield 6.8% convertible semiannually. The bond is called at
the end of 6 years with a redemption value of X. The yield to the investor is
still 6.8% convertible semiannually. Find X.
A) 1009 B) 1014 C) 1019 D) 1024 (E) 1029
4. A man borrows 10,000 to be paid back in 30 years with level end of year
payments at an annual interest rate of i. The sum of principal repayments in
years 5 and 10 is equal to the principal repaid in year 15. Find i.
A) 8.9% B) 9.2% C) 9.5% D) 9.8% E) 10.1%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page PE5-2
Practice Exam 5 - Exam FM / Exam 2
S. A man has two 20-year annuities-immediate. Each has a present value of
1000. The first has annual payments and earns 5.8% annually. The second
earns 5.4% convertible semiannually with semiannual payments. All
payments are deposited into a fund that pays an annual effective rate of 6%.
What is his accumulation at the end of 20 years?
A) 6170 B) 6190 C) 6210 D) 6230 E) 6250
6. A woman has 10,000 in an account on January 1. She makes withdrawals of
400 on April 1 and 600 on November 1. Her dollar-weighted return for the
year is 12.77%. What is her balance on December 31?
A) 10,123 B) 10,226 C) 10,317 D) 10,437 E) 10,501
7. Given s2 = 0.053, s4 = 0.0575 and i2>3 = 0.058, find i3>4.
A) 0.066 B) 0.067 C) 0.068 D) 0.069 E) 0.070
8. A 3-year 1000 par bond has 4.5% annual coupons. The forward rates implied
by the yield curve are i0>i = 0.033, ili2 = 0.038 and i2>3 = 0.042. Find the price of
the bond using the spot rates from the yield curve.
A) 1011 B) 1016 C) 1021 D) 1026 E) 1031
9. Given d(4) = 0.064, find 5 + i(6).
A) 0.123 B) 0.125 C) 0.127 D) 0.129 E) 0.131
10. A company has liabilities of 3000, 5000 and 2000 due at the end of years 1, 2
and 3 respectively. It can purchase zero-coupon bonds to match its
liabilities. Each bond has a par value of 1000. The first ones mature in one
year with a rate of 5%, the second ones in two years with a rate of i, and the
third ones in three years with a rate of 6%. The cost of matching its
liabilities is 9028.64. Find i.
A) 5.1% B) 5.3% C) 5.5% D) 5.7% E) 5.9%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 5 - Exam FM / Exam 2
Page PES- 3
11. A woman buys a 30-year annuity-immediate with monthly payments that
earns 5.4% convertible monthly. The present value of the annuity is 10,000.
At the end of 15 years the interest rate is increased to 6.3% convertible
monthly. What will her new monthly payments be?
A) 57.50 B) 58.00 C) 58.50 D) 59.00 E) 59.50
12. Jack deposits 1000 into an account on 01/01/03. Jill deposits 500 into an
account on 01/01/04, and another 600 into the account on 01/01/05. On 01/01/07
the accounts have the same amount in them. The accounts earned the same
annual interest. What was the interest rate?
A) 6.2% B) 6.4% C) 6.6% D) 6.8% E) 7.0%
13. An investment pays 2000 at the end of year 1, 2500 at the end of year 2 and X
at the end of year 3. The investment earns 8% annually. The present value
of the investment is 6773.60. What is the Macaulay duration of the
investment?
A) 2.137 B) 2.175 C) 2.204 D) 2.229 E) 2.253
14. A man wants to accumulate 250,000 in 25 years by making monthly end of
month payments into a fund that earns 6.3% convertible monthly. His first
payment is 100 and each subsequent payment is increased by X over the
previous one. What must X be to achieve his goal?
A) 2.04 B) 2.09 C) 2.14 D) 2.19 E) 2.24
IS. A 10-year 1000 par bond with 6% semiannual coupons is purchased to yield
5.6% convertible semiannually. How much of the premium is amortized in
the seventh period?
A) 1.33 B) 1.36 C) 1.39 D) 1.42 E) 1.45
16. A 10-year 1000 par bond with 6% annual coupons is priced to yield 5.5%.
Find the Macaulay duration of this bond.
A) 7.847 B) 7.898 C) 7.937 D) 7.962 E) 7.995
17. Money in an account earns 7% simple interest per year. What is the
effective rate of interest for the time interval [3,4]?
A) 5.61% B) 5.66% C) 5.71% D) 5.75% E) 5.79%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page PE5-4
Practice Exam 5 - Exam FM / Exam 2
18. A man wants to retire in 25 years. He sets up an account by making monthly
end of month payments of X. The account earns 6% convertible monthly.
When he retires he wants to be able to make annual end of year withdrawals
for 25 years. He wants the first to be 10,000 and each subsequent one to be
3% more than the previous one. What should X be if interest rates stay the
same?
A) 236 B) 239 C) 242 D) 245 E) 248
19. A man has a 30-year loan with level end of year payments. The principal
repaid in year 5 is 159.68 and in year 10 it is 213.73. What is the payment?
A) 706 B) 711 C) 716 D) 721 E) 726
20. A man receives an inheritance of X which he uses to buy a 30-year annuity
immediate. The annuity will have monthly payments of 100 for the first 10
years, 300 for the next ten years and 1000 for the final 10 years. The annuity
earns 5.7% convertible monthly. Find X.
A) 53,750 B) 53,925 C) 54,175 D) 54,350 E) 54,525
21. The current price of a stock that pays no dividends is 40. The continuously
compounded risk free rate is 4%. A six month short forward contract to sell
the stock at the forward price 42 is offered to you.
A) You should be paid 1.214 for entering the contract
B) You should pay 1.214 for entering the contract
C) You should be paid 1.166 for entering the contract
D) You should pay 1.166 for entering the contract
E) No payment is needed.
22. Which of the following can have a net premium of 0? All options below are
for the same stock and have the same expiration date.
A) Buy a 30 strike put and write a higher strike call.
B) Buy a call and a put with the same strike price.
C) Buy an out of the money put and in the money call.
D) Buy the stock for S0 and sell a call with K = S0
E) None of these
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Practice Exam 5 - Exam FM / Exam 2
Page PES- 5
23. You enter a position based on options on the same stock. It has the following
profit function graph.
A
Which of the following could have that graph?
A) Buy a 35-strike call and sell a 40-strike put.
B) Sell a 35-strike put and buy a 40-strike call.
C) Sell a 40-strike put and a 40-strike call, and buy a 35-strike put and a
45-strike call.
D) Buy a 40-strike put and a 40-strike call, and sell a 35-strike put and a
45-strike call.
E) None of these
24. The current price of a stock is 40. The price of a 35-strike three month call
is 6.13 and the price of a 35 strike three month put is 0.44. The continuous
risk-free rate is 4%. What payment should be made for you to enter into a
short forward contract to sell the stock for 35 in 3 months?
A) You should be paid 5.69
B) You should pay 5.69
C) You should be paid 5.75
D) You should pay 5.75
E) No payment is needed on a forward contract.
25. Which of the following are reasons that a firm might decide not to hedge?
1. The option prices and other associated costs appear to be too high.
2. The necessary accounting and financial management skills require
expertise that the company does not have.
3. The firm fears that a hedging operation will reduce its debt capacity.
A) 2 B)l,2 C)l,3 D)2,3 E) 1,2,3
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE5-6
Practice Exam 5 - Exam FM / Exam 2
26. Which of the following are true of futures and forwards contracts?
I) Both forward and futures contracts can be settled on any day up to
expiration.
II) The credit risk is always the same for futures and forwards.
III) Futures contract trading is never halted, so you can trade them on the
exchanges at any time.
A) I only B) II only C) III only D) All E) None
27. A stock has current price S0 = 30. The annual continuous dividend rate is
8 = .02. If the expiration time for a forward contract is T = .5 and the
correct forward price is 30.15 , what is the continuous interest rate r?
A) 0.01 B) 0.016 C) 0.020 D) 0.025 E) 0.03
28. The S&R index has a spot price of S0 = 1300. The continuous interest rate is
r = .025 and the continuous dividend yield is 8 = 0 The one year forward
price is 1332.91. You enter into a forward sale contract and buy the index.
Which of the following positions is this equivalent to:
A) A short sale of the index.
B) Purchase of a one year zero-coupon bond with r = .025
C) A reverse cash and carry hedge.
D) A cash and carry arbitrage
E) None of these.
29. Zero coupon bond yields and oil forward prices for the next two years are
Year
Oil Forward Price
Zero-coupon bond yield
1
60
5%
2
62
6%
What is the market value of a two year oil swap contract based on these
forward prices if the forward prices for the next two years go up to 61 and
64 respectively on the same day but the yield curve does not change?
A) 0 B) 1.483 C) 1.557 D) 2.073 E) 2.733
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Practice Exam 5 - Exam FM / Exam 2
Page PES- 7
30. Zero coupon bond yields for the next three years are
Year
Zero-coupon bond yield
1
5.2%
2
?
3
7.1%
The level swap rate for a three year interest swap is 7% What is the zero
coupon yield for two years?
A) 5.9% B) 6.0% C) 6.1% D) 6.2% E) 6.3%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page PE5-8 Practice Exam 5 - Exam FM / Exam 2
Solutions
1. To get the original payments set N = 360,1/Y = 0.45, FV = 100,000 and PV = 0.
Then CPT PMT = -111.53.
To get the accumulation after 10 years reset N = 120 and
CPT FV = 17,694.47.
Now set N = 240,1/Y = 0.55, PV = -17,694.47 and FV = 100,000.
Then CPT PMT = -68.50
Answer C
2. To get Lucy's payments set N = 25,1/Y = 7, PV = -5000 and FV = 0. Then CPT
PMT = 429.05. Her accumulation in the fund is 429.05s^= 25,265.91. To find
i set 5000(1 + i)25 = 25,265.91. Then i = 0.067.
Answer A
3. To find the price of the bond set N = 20,1/Y = 3.4, PMT = 37.5 and FV = 1000.
Then CPT PV = -1050.20. To find X reset N = 12. Then CPT FV = 1024.16.
Answer D
4. The principal repaid in year k is PMTv3"*1. So we have
PMTv26 + PMTv21 = PMTv16, or v10 + vs = 1. If we let x = v5 we get x2 + x = 1.
So x = 0.6180, and i = (0.6180)1'5 - 1 = 0.101.
Answer E
5. The annual payments from the first annuity are 85.77
The semiannual payments from the second annuity are 41.19.
For each year the accumulation of deposits plus interest for the year is
85.77 + 41.19U.06)1'2 + 41.19 = 169.37.
The total accumulation is 169.37s Mo6 = 6,230.38.
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Practice Exam 5 - Exam FM / Exam 2
Page PE5- 9
6. The denominator for the dollar-weighted interest rate is
10,000 - 400(3/4) - 600(1/6) = 9600.
The interest amount I is
9600(0.1277) = 1225.92.
The balance on December 31 is
10,000 - 1000 + 1225.92 = 10,225.92.
Answer B
7. (1 + s3)3 = (1 + s2)2(l + i2,3) = (1.053)2(1.058) = 1.1731
1 + i3,4 = (1 + s4)4/(l + S3)3 = 1.05754/1.1731 = 1.066
Answer A
8. The bond price is P = 45/(1 + sO +45/(1 + s2)2 + 1045/(1 + s3)3.
1 + Si = 1 + io.i = 1.033. (1 + s2)2 = (1 + Si)(l + ii,2) = 1.0723
(1 + S3)3 = (1 + s2)2(l + i2,3) = (1.0723X1.042) = 1.1173
P = 45/1.033 + 45/1.0723 + 1045/1.1173 = 1020.82.
Answer C
9. The fundamental relations are e8 = (1 + i<-6)/6)6 = (1 - d(4)/4)^.
(1 - d^M)"4 = 0.984"4 = 1.0666. Hence 6 = ln(1.0666) = 0.0645.
Then i(6) = (1.06661'6 - 1)(6) = 0.0648.
Thus 8 + i(6) = 0.0645 + 0.0648 = 0.1293
Answer D
10. To match liabilities we need
3000/1.05 + 5000/(1 + if + 2000/1.063 = 9028.64
5000/(1 + i)2 = 4492.26 => 1 + i = 1.055
Answer C
11. To get the original payments set N = 360,1/Y = 0.45, PV = 10,000 and FV = 0.
The CPT PMT = 56.15. The reset N = 180 and CPT FV = 6,917.22 gives the
balance after 15 years. Then reset I/Y = 0.525. PV = -6917.77 and FV = 0.
Then CPT PMT = 59.50.
Answer E
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page PE5-10
Practice Exam 5 - Exam FM / Exam 2
12. Let r be the rate earned. The total in Jack's account is 1000(1 + r)4. The
amount in JilPs account is 500(1 + r)3 + 600(1 + r)2.
Equating these values and setting 1 + r = x, the equation reduces to
10x2 - 5x - 6 = 0.
The positive root is x = 1 + i = 1.064.
Hence r = 0.064.
Answer B
13. The present values of the first two payments are 2000/1.08 = 1,851.85 and
2500/1.082 = 2,143.35. The corresponding weights are
wi = 1,851.85/6.773.6 = 0.2734 and w2 = 2,143.35/6,773.6 = 0.3164.
Then w3 = 1 - 0.2734 - 0.3164 = 0.4102. The Macaulay duration is
D = 0.2734(1) + 0.3164(2) + 0.4102(3) = 2.1368
Answer A
14. The accumulation can be written as
A = 100s 3ooi+X(s 3001 - 300)/i , i = 0.525%
250,000= 100(725.88) + X(725.88 - 300)/0.00525
X = 177,412/81,120 = 2.187
Answer D
15. The amount of the premium amortized in the fcth period is
(r- i)(1000)v20fc+1 = 1000(0.002)(1.028)(21fc)
For the 7th period the amount is 2(1.028)14 = 1.359
Answer B
16. The price of the bond is P = 1037.69.
The Macaulay duration is [60(Ia) m + 10(1000)v10]/P for I =5.5%.
(la) ^=38.143, so D = (2288.60 + 5854.30)/1037.69 = 7.847
Answer A
17. The effective interest rate U is [a(4) - a(3)]/a(3).
So U = (1.28 - 1.21)/1.21 = 0.0579
Answer E
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Practice Exam 5 - Exam FM / Exam 2
PagePE5-ll
18. For retirement years the annual effective interest rate will be
g = (1.005)12 - 1 = 0.06168. The present value of withdrawals is
10,000[1 - (1.03/1.06168)25]/0.03168 = 167,642.
To find the monthly deposits set N = 300,1/Y = 0.5, PV = 0 and FV = 167,642.
Then CPT PMT = -241.91
Answer C
19. The principal repaid in the repaid year k is PMTv30fc+1.
For year 5, PMTv26 = 159.68. For year 10, PMTv21 = 213.73.
Then v5 = (1 + i)s = 213.73/159.68 = 1.3385 => 1 + i = 1.060.
Then PMT = 159.68(1.06)26 = 726.45
Answer E
20. This can be viewed as the sum of 3 annuities-immediate. The first is a 30-
year annuity with monthly payments of 100, the second a 10-year deferred
20-year annuity with monthly payments of 200, and the third a 20-year
deferred 10-year annuity with monthly payments of 700. The present value
of this annuity is
PV= 100a 3^ + 200v120a^+ 700v240aI2oi
= 100(172.295) + 200(0.5663X143.014) + 700(0.3207)(91.308)
= 53,925
Answer B
21. The forward price should be 40e04(5) = 40.81 Thus the futures price is priced
too high by 42-40.81 = 1.19.
Since the short position will pay an extra 1.19 in six months, you must pay
the present value of that amount today. You pay 1.19e"02 = 1.166.
Answer D
22. A) is a collar, which can have a 0-cost.
Answer A
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page PE5-12
Practice Exam 5 - Exam FM / Exam 2
23. This is the graph of a butterfly spread in which you write a straddle and buy
a strangle.
Answer C
24. You create a synthetic forward sale for 35 by selling the 35-strike call and
buying the 35-strike put. The net to you is 6.13 - 0.44 = 5.69.
Answer A
25.1) and 2) are given in the text as reasons not to hedge on page 106. However
hedging is regarded as a tool to protect and increase debt capacity, so that 3)
is not a valid reason to avoid hedging.
Answer B
26. The answers are based on text quoted below from the text on page 142.
I) False. "Whereas forward contracts are settled at expiration, futures
contracts are settled daily."
II) False. "Because of daily settlement, the nature of the credit risk is
different with the futures contract. In fact, futures contracts are
structured so as to minimize the effects of credit risk.
III) False. "A price limit is a move in the futures price that triggers a
temporary halt in trading."
Answer E)
27.
F0 T = S0e{T~s)T -* 30.15 = 30e(r-02)-s
In
30.15^
30 ,
Answer E
= .5r-.01->r = .03
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Practice Exam 5 - Exam FM / Exam 2
Page PE5-13
28. Apply the basic identity
STOCK = LONG FORWARD + ZERO COUPON BOND.
Your position is - LONG FORWARD + STOCK. This is equivalent to BOND,
or purchase of a zero coupon bond at the interest rate r = .025 The forward
price is the correct theoretical price.
Answer B
29. The level swap payments before and after the change are given by
60 62 ( 1 1 ^ ,no,.
1.05 1.062 7 11.05 1.062
61 64 ( 1 1
■ + . , -o = Xnfter . _ _ +
1.05 'l.062""fier{l.0S 1.062
\
Xafter = 62.449
J
The holder of the original swap can sell a swap under the new forward
prices. Then in each of the next two years he will have payments of Spot -
60.966 and 62.449-Spot, and get a net payment of 1.483. The market value of
the swap is the present value
M83+M83 = 2733
1.05 1.062
Answer E
30. The guaranteed interest rate is the three year par coupon bond rate.
1-P(0,3)
C P(0,l) + P(0,2) + P(0,3)
The known zero-coupon bond prices are
P(0,1) = —— = .951, P(0,3) = —^- = .814
1 ; 1.052 K ' 1.0713
Thus .07 = 1"/81f ► P (0,2) = .892 -► r (0,2) = 5.9%.
.951+ P (0,2)+ .814 K ' v ;
Answer A
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Practice Exam 6 - Exam FM / Exam 2
Page PE6- 1
Exam FM
Questions
1. An 8 year par value bond with semiannual coupons at 6% convertible
semiannual has a price of 1050. The bond can be called at par value of X on
any coupon date starting at the end of year 6. The price guarantees that Sue
will receive a yield of at least 5% convertible semiannually. Calculate X.
A) 986 B)721 C) 999 D) 944 E) 1,276
2. A stock currently is priced at 50. It does not pay dividends. The risk-free
rate is r - .02. You sell short one share of the stock for three months and
enter into a three month long forward purchase contract. Which of the
following is equivalent to this position?
A) Borrowing 50.25 at the risk free rate.
B) Borrowing 50 at the risk-free rate
C) Lending 50.25 at the risk free rate.
D) Lending 50 at the risk-free rate
E) None of these
3. Terry purchases an annuity with payments made at the beginning of each
month for 36 payments. The monthly payments are a constant amount of 15
for the first 24 payments, however the 25th payment is 20, the 26th payment is
25, the 27th payment is 30, and this arithmetic sequence continues until the
36th payment. The nominal interest rate is 6% convertible monthly. What is
the present value of this annuity?
A) 823.1 B) 764.0 C) 829.1 D) 827.5 E) 871.6
4. The price of a stock is currently selling for 39.35. The next dividend
payable one year from today is expected to be 1.00. Suppose the price
included a forecasted future growth rate of 6% for the dividends. What is
the annual effective interest rate, i?
A) 2.54% B) 3.15% C) 3.46% D) 6.00% E) 8.54%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE6-2
Practice Exam 6 - Exam FM / Exam 2
5. Paul pays $100,000 today for a 4-year investment that returns cash flows of
$60,000 and the end of each of years 3 and 4. Suppose, at 15%, the net
present value of Paul's cash flows is equal to the net present value of Kelly's
cash flows, where Kelly makes an investment of X one year from today that
returns cash flows of $60,000 at the end of each of years 4 and 5. Calculate
X.
A) 94,316 B) 98,503 C) 105,380 D) 103,937 E) 90,379
6. An appliance store offers to sell a television for $5000. Suppose the current
market loan rate is a nominal rate of 10% convertible monthly. As an
inducement, the dealer offers 100% financing at an effective annual interest
rate of 6%. The loan is to be repaid in equal installments at the end of each
month for a 3 year period.
If the dealer himself is paying monthly payments on the market loan, but
finances his customer with the inducement loan, what is the final cost to the
dealer, in terms of total paid, for the inducement?
A) 311 B)420 C)175 D) 332 E) 308
7. A fund earned investment income of 8,000 during 2004. The beginning and
ending balances of the fund were 95,000 and 120,000 respectively. A deposit
was made at time K during the year. No other deposits or withdrawals were
made. The fund earned 7.5235% in 2004 using the dollar-weighted method.
Determine K.
A) Feb 1 B) Mar 1 C) May 1 D) July 1 E) October 1
8. Andy purchases a 16 year annuity immediate paying 100 the first year and
increasing by 4% each year thereafter. Rick purchases a 16 year annuity
immediate paying X the first year and decreasing by 2% each year
thereafter. At an effective annual rate of 5%, both annuities have the same
present value. Calculate X.
A) 148.7 B) 145.2 C) 124.5 D) 123.2 E) 120.0
9. Katie purchases a 15 year par value bond with 5% semiannual coupons at a
price of 2345. The bond can be called at par value X on any coupon date
starting at the end of year 10. The price guarantees that Katie will receive
a nominal semiannual yield of at least 4%. Mark purchases a 15 year par
value bond identical to Katie's except it is not callable. Assuming the same
yield, what is the price of Mark's bond?
A) 2,168 B) 2,170 C) 2,405 D) 2,300 E) 2,411
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 6 - Exam FM / Exam 2 Page PE6- 3
10. A trader is dealing in three month S&R index options. He writes a straddle
by selling a 1000 strike call and a 1000 strike put, and buys a strangle by
buying a 975 strike put and a 1025 strike call. Which of the following could
be the graph of his profit function?
A)
i
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE6-4
Practice Exam 6 - Exam FM / Exam 2
11. Suppose the amount in a fund one and a half years from today is 100. Find
the present value of the fund if the nominal rate of discount is 5%
convertible quarterly.
A) 86.8 B)96.4 C) 92.7 D) 92.9 E) 92.2
12. An annuity due pays an initial benefit of 1 per year, with the benefit
increasing by 10.25% every four years. The annuity is payable for 40 annual
payments. Using an annual effective rate of 2%, calculate the future value
of this annuity.
A) 42 B)69 C)83 D) 59 E) 93
13. A stock has current price 50. It pays no dividends. The risk-free rate is
r = .025. You observe an actual six month forward price of 50.68. Which of
the following describes a possible arbitrage of the forward price?
A) You can arbitrage this price by selling the forward at 50.68, buying the
stock at 50 and borrowing 50 for 6 months at the risk-free rate.
B) You can arbitrage this price by selling the forward at 50.68, buying the
stock at 50 and borrowing 49.48 for six months at the risk-free rate.
C) You can arbitrage this price by selling the stock forward at 50.68, selling
the stock short at 50 and borrowing 50 for six months at the risk-free
rate.
D) You can arbitrage this price by buying the stock at 50 and lending 49.38
for six months at the risk-free rate.
E) You can arbitrage this price by selling the stock short at 50 and lending
50 for six months at the risk-free rate.
14. A stock currently is priced at 85. The continuous dividend rate is S = .02.
The risk-free rate is r = .04. A call and a put with the same strike price and
T=.5 have premiums 4.91 and 4.56 respectively. Find the strike price.
A)84.51 B) 84.95 C) 85 D) 85.50 E) 85.93
15. A 20 year 5,000 bond that pays 4% annual coupons matures at par. It is
purchased to yield 5% annual for the first 12 years and 6% annual
thereafter. Calculate the amount for accumulation of discount for the 8th
coupon.
A)-15 B)+25 C)-9 D)-58 E)-160
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 6 - Exam FM / Exam 2
Page PE6- 5
16. Todd borrows X for nine years at an annual effective interest rate of 8%, to
be paid with equal payments at the end of each year. The outstanding
balance immediately after the fifth payment is 4,506.74. Calculate the
principal repaid in the first payment.
A) 551 B)565 C) 681 D) 574 E) 384
17. Suppose a yield curve for spot rates is given by the following equation:
st = 0.08 -0.001t + 0.002t2
What would be the effective annual forward interest rate for a loan
originating at time t=4, with a term of 3 years?
A) 0.3603 B) 0.0569 C) 0.0033 D) 0.2606 E) 0.1805
18. Ken purchases a $200,000 home. Mortgage payments are to be made
monthly for 30 years with the first payment to be made one month from
now. The annual effective rate of interest is 5%. Starting with the 100th
payment, each monthly payment is increased by $400 in order to repay the
mortgage more quickly.
Calculate the total amount of interest paid during the duration of the loan.
A) 136,216 B) 136,215 C) 135,648 D) 136,558 E) 136,159
19. SeventiesCo sells gold chains. Each chain sells at a price equal to the cost of
gold used to make the chain plus $20. The fixed cost per chain is $10.
Forward contracts and put and call options on gold are available. What
should SeventiesCo do to control risk ?
A) Enter into long forward contracts to purchase gold at the forward price.
B) Buy calls on gold to assure that gold can be obtained at a set price.
C) Create a straddle using purchased calls and puts at the same strike to
combat volatility
D) Hedge with s zero-cost collar
E) None of these
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE6-6
Practice Exam 6 - Exam FM / Exam 2
20. A company has two traders. Trader A buys the stock of MegaFirm at its
current price S0 and buys a call with strike price K. Trader B sells short the
stock of MegaFirm at its current price S0 and buys a put with strike price K.
Which of the following graphs describes the combined position of the two
traders?
A) t
C)
D)
E) None of these
21. An annuity immediate has 32 initial quarterly payments of 20 followed by a
perpetuity of quarterly payments of 25 starting in the 9th year. Find the
present value at a nominal rate of 16% convertible quarterly.
A) 510 B)165 C)814 D) 536 E) 506
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 6 - Exam FM / Exam 2
Page PE6- 7
22. A buys the S&R index and a K-strike put. B lends 1014.80 and buys a It-
strike call, r = .04 and T = .25 for the put, the call and the loan. The index
does not pay dividends. A and B have the same payoff function. Find K.
A)1000 B)1012 C)1018 D)1020 E) 1025
23. Brent would like to accumulate $100,000 at the end of 17 years to pay college
expenses for his daughter. If the effective annual rate is 6% and Brent will
be making monthly payments, how much does he need to deposit each month
if his first payment is today and he makes a total of 204 payments?
A) 286 B)288 C) 283 D) 282 E) 285
24. Below is a 4 year yield curve with one missing entry.
Years to maturity
Zero Coupon Bond Yield
1
3.0%
2
4.0%
3
4
5%
The theoretically correct yield for a 4 year fixed interest rate swap is 4.94%.
Find the range for the missing spot rate in the table above.
A) 4.0%-4.15%
B) 4.16%-4.3%
C) 4.31% - 4.45%
D) 4.46% - 4.6%
E) 4.61% - 4.75%
25. Suppose Chris takes out a loan of amount X and makes annual payments of
2000 at the end of each year for 15 years. The total amount of interest paid
towards the loan is 6,124. Calculate the interest paid in the first payment.
A) 408 B)60 C)716 D) 672 E) 464
26. You are given an annuity-immediate paying 10 annually for twenty years.
After the twenty years, the payments decrease by one per year until it
reaches a payment of 1. The payments of one continue forever. The annual
effective rate of interest is 6%. Calculate the present value of this annuity.
A) 129 B)133 C)132 D) 131 E) 134
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE6-8
Practice Exam 6 - Exam FM / Exam 2
27. The non-dividend paying S&R index is currently at 1350. The risk-free rate
is r = .04. You are offered a six-month long forward on the index with a
forward price for purchase in six months quoted at 1410. Which of the
following applies if you enter into this forward contract?
A) You should be paid 32.73.
B) You should pay 32.73.
C) You should be paid 32.08.
D) You should pay 32.08
E) You do not pay or receive anything.
28. What is the modified duration of a five year 2000 par value bond with 8%
annual coupons and an effective rate of interest equal to 7%?
A) 4.327 B) 4.004 C) 3.550 D) 3.802 E) 3.287
29. The present value of a perpetuity of 6,000 paid at the end of each year plus
the present value of a perpetuity of 8,000 paid at the end of every 4 years is
equal to the present value of an annuity of X paid at the end of each year for
30 years. Interest is 6% convertible quarterly. Calculate X.
A) 9,479 B) 9,400 C) 9,475 D) 9,410 E) 9,264
30. The following are the prices of $100 zero-coupon bonds redeemable at par
Term to Maturity Price
1 96.23
2 94.12
3 89.23
4 84.59
5 82.48
Determine the forward rate for year 4.
A) 2.55% B) 5.20% C) 5.49% D) 12.10% E) 13.76%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 6 - Exam FM / Exam 2
Page PE6- 9
Solutions
1. Since this is a callable bond that is also a premium bond, we want to price it
at the term that is the worst case situation for the investor. Therefore, it
should be priced at the earliest call date. So, n = 12.
1050 = X(0.03)aI2|0.02S + Xv12
X = *°^ = 998.7741
(0.03)anio.o25 + v12
Answer C
2. The correct forward price is 50e02( 25) = 50.25. Thus you have a correctly
priced forward contract. You also have a short position in the stock.
Thus the word equation STOCK = FORWARD + BOND
applies in the form -STOCK + FORWARD = -BOND
The left hand side above describes your combination of a short position in
the stock and a long forward contract. The right side describes the sale of a
three month zero coupon bond, which is a borrowing.
In your actual position you will receive 50 in cash, as a borrower would. If
you invest this cash at the risk-free rate, in three months it will grow to
50e02( 25) = 50.25. You will then pay 50.25 for the forward purchase of the
stock and deliver it against the short, just as a borrower would pay 50.25 to
pay off a loan of 50.
Answer B
3. You could approach this problem many different ways. One way is to think
of a 36 month annuity due with payments of 15 and then on top of that a
deferred geometrically Increasing annuity with a constant difference of 5,
starting with the 25th payment.
i = °^ = 0.005
12
PV = 15daa +vM5(/d)m =495.5305698 + 333.6115032 = 829.1421
Answer C
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE6-10
Practice Exam 6 - Exam FM / Exam 2
4. To price a stock using the dividend growth model:
P = , where g is the constant percentage growth rate.
(i-g)
39.35 = -> 1 = 0.085413
(i-0.06)
Answer E
5. First, calculate the net present value of Paul's cash flows. There is a cash
flow of -100,000 at time 0, and cash flows of 60,000 at time 3 and 4. So,
NPV = -100,000 + ^0 + 6010^ = _26,243.83132
1.153 1.154
Now set this net present value equal to Kelly's cash flows, which consists of
a value of -X at time 1, and cash flows of 60,000 at time 4 and 5.
-X 60,000 60,000
1.15+ 1.154 + 1.15s
-26,243.83132 = ^ + ^^ + ^^ -> X = 103,936.5747
Answer D
6. First, calculate the payments towards both loans, then compare the total out
of pocket costs for both loans and subtract the difference.
For the market loan you can use TVM on the BAII Plus or the logic below to
get the payment and total payments:
5000 = (Pmt)ami0/12
Pmt = 161.34
Total payment = 161.34*36=5,808.24
For the inducement loan you can use TVM on the BAII Plus or the logic
below to get the payment and total payments:
5000 = (Pmt)a^006/12
Pmt = 152.11
Total payment = 152.11*36=5,475.96
The cost to the dealer for the inducement, in terms of total paid, is
5,808.24 - 5,475.96 = 332.28
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 6 - Exam FM / Exam 2
PagePE6-ll
7. You know the beginning and ending balances of 95,000 and 120,000. You also
know that the investment income was 8,000. Therefore, you know that the
deposit must have been 17,000. The dollar-weighted method implies that
8000 =0.075235
95000 +17000(1-K)
This leads to K=0.3333, so the deposit is made after 4 months. The best
answer is May 1.
Answer C
8. First, calculate the present value of Andy's annuity.
PV = lOOvoos + 100(1.04)v£o5 +100(1.042)Vo3.os + ...100(1.0415)vJ6os
= 100vo.oS[l + 1.04vo.o5 + 1.042v02os +... + 1.0415vJ50S)
= lOOvoosC1,"1,'0^6^] = 1419.6571
1 - 1.04V0.05
Then, set this present value equal to Rick's annuity.
1419.6571 = Xv00S +X(0.98)v2.os +X(0.982)v030S + ...X(0.9815)vJ60S
1419.6571 = Xvoostl + 0.98vo.os + 0.982 v2.0S +... + 0.9815 vj50s)
X = 148.67
Answer A
Since the coupon rate is higher than the yield rate, this is a premium bond
and therefore should be priced at the earliest possible call date, or after 10
years. The price of the bond equation would be set up as follows to solve for
the par value:
2345 = X(0.025)a^a02 + Xv02002
2345 = X[0.025a^0.02+Vo20o2]
2345 = X(1.081757)
X = 2,167.7693
For Mark's bond, you would price this par value at the full term of 15 years.
P = 2167.77(0.025)a3«^02 + 2167.77v030o2
P = 2410.52
Answer E
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page PE6-12
Practice Exam 6 - Exam FM / Exam 2
10. This is a butterfly spread.
Answer B
11. We are given that d(4) = 0.05. We need to solve for an interest rate. You
could solve for the quarterly rate or the semiannual rate. I will solve for
i(2).
(l+i^)2=(l_£^5)-4
2 4
i(2) = 0.05095337286, and the effective semiannual rate = 0.02547668643
Therefore, the present value is PV = lOOvf =92.73
Answer C
12.1 will solve first for the present value and then find the future value. You
will end up needed to use a geometric series to solve this present value
calculation.
PV = a4-lo.o2 + v4(1.1025)a^02 + v8(1.1025)2d^02 + ■.. + v36(1.1025)9d^0
PV = <^o.o2[l + v4(1.1025) + v8(1.1025)2 + • • ■ + v36(1.1025)9]
PV^aj0,[1-(l',4(11025))"l = 42.2448
41002 l-v4(1.1025)
FV = 42.2448(1.02)40 = 93.2782
.02
Answer E
13. The correct forward price is S0e{r~s)T = 50e025( 5) = 50.629. Thus the actual
price of 50.68 is too high and there is an arbitrage. You can sell the stock
forward at the higher price of 50.68 and offset this with a synthetic forward
with the correct price of 50.629.
The synthetic forward is constructed using the relation
FORWARD = STOCK - BOND.
This implies that you will buy the stock and sell a zero coupon bond that
gives you 50 today to pay for that purchase.
Answer A
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 6 - Exam FM / Exam 2
Page PE6-13
14. Use parity.
C-P = S0e-5T-Ke-rT
4.91-4.56 = 85e-02(5)-Ke-04(5)
K = 85.50
Answer D
15. We need to price the bond first. Instead of pricing the bond at time zero, I
would price the bond at time 7 and then use amortization techniques to
calculate the principal portion for the next coupon.
To price at time 7, we need the present value of the remaining coupons and
also the present value of the redemption value.
P = 200^05 + Vo5 os 200^.06 + v05o5Vo8o6 5000 = 4296.9726
This is the book value of the bond immediately before the 8th coupon. The
interest portion of the 8th coupon will be: 4296.9726*0.05=214.8486. Since
each coupon is 200, the principal portion of the 8th coupon will be:
200-214.8486=-14.8486.
Answer A
16. First, we know that the outstanding balance after the fifth payment is
4506.74. We could use this to find the payment amount by treating it as a
four year loan amount.
4506.74 = Pmt(a4io.o8)
Pmt = 1360.6786
Now that we know the payment amount, we could solve for the initial loan
amount by solving for a present value:
py = 1360.68(a9,0.08)
PV = 8,500.02
Now we can solve for the interest portion of the first payment:
h= 8500.02* 0.08 = 680.00
If the interest portion of the payment is 680, the principal portion must be
680.68.
Note: the solutions for Pmt and PV above can be done entirely on the BA II
Plus to save time.
Answer C
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE6-14
Practice Exam 6 - Exam FM / Exam 2
17. We need to calculate (1 + s4>7)3 = ,„ ,4
(l + s4)
First, we need s7 and s4
From the curve: s7 = 0.08 - 0.001 * 7 + 0.002 * 7 A 2 = 0.171
s4 = 0.08 - 0.001 * 4 + 0.002 * 4 A 2 = 0.108
From the first equation,
(1 + S4 7)3 = (1±£t)1 = 1^71^ = 2.0032688
' (l + s4)4 1.1084
(l + s4|7) = 1.2606071
Answer D
18. First, calculate the monthly payment amount for a 30 year 200,000 loan at
5% annual effective interest using the BA II Plus or the math below.
i = (1.05)1/12-l = .00407
200,000 ^a^
K = 1060.11
Then, we need to know the outstanding balance after the 99th payment:
OB99 =200,000(1 + 0" -1060.11s99ii =170,162.81
Now, we add the extra $400 to the payment amount and see when the loan
would be paid off.
170,162.81 = 1460.11a^
n = 158.3880
So, there would be 158 MORE payments of 1460.11 and one last partial
payment. To figure out the amount of the last partial payment, we need the
OB at time after the 158th payment.
OB15S = 170162.81(1 + i)158 - 1460.11s158ii = 564.98
The last payment includes this outstanding balance plus interest. So, the
last payment is: 564.98(1 + i) = 567.28
So, there are 99 payments of 1060.11,158 payments of 1460.11, and a final
payment of 567.27. Therefore, the out of pocket contribution is 336,215.55,
and the interest paid towards the loan is 136,215.55
This question can also be done entirely on the BA II Plus.
Answer A
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 6 - Exam FM / Exam 2
Page PE6-15
19. Seventies Co has no risk since they include the price of gold in the price
they charge. They do not need to do anything. (This is based on problem 4.12
in the text).
Answer E
20. The purchase of the stock by trader A is offset by the sale by trader B. The
resulting position is the purchase of a put and a call at the same strike price,
which is a straddle.
Answer B
21. Think of this as a 32 payment annuity with a payment amount of 20, followed
by deferred perpetuity with a payment amount of 25.
PV = 20a32io.o4 + — v32 = 357.4710 +178.1612 = 535.6322.
0.04
Answer D
22. We use the relation
Payoff [Index + Put with strike K]
= Payoff [Call with strike K + Zero-coupon bond for K]
The first payoff is for A's position and the second is for B's position. Thus
B's lending is a zero-coupon bond for K and 1014.80 = Ke"04(2S) -> K = 1025.
Answer E
23. Solve for the payment amount of an annuity due using the BA II Plus or the
math below.
i = (1.06)1/12-l = .00487
100,000 = Pmt(s"204t)
Pmt = 286.1561
Answer A
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE6-16
Practice Exam 6 - Exam FM / Exam 2
24. The theoretically correct yield for such a swap is
R_ 1-P(0,4)
P(0,l) + P(0,2) + P(0,3) + P(0,4)
Thus we have
l-d.05)-4
.0494 =
1.03"1 + 1.04"2 +(l + r(0,3)r3 +1.05
-4
This gives
(1 + r (0,3))"3 = .872534 -> r (0,3) » 4.65%
Answer E
Note: Answers will vary according to the degree of precision used. That is
why the choices were given in ranges.
25. First, let's calculate the loan amount. You know that the total interest is
6,124 and that 15 payments of 2000 were made. So, the loan amount would
be the difference of 15*2000 and 6,124
15 * 2000 -X = 6,124
X = 23,876
Now, solve for the interest rate. I would use my calculator, setting PV to
23,876, Pmt to -2000, n to 15, FV to 0, and solving for I/Y. You should get
I/Y=2.99992482%
The interest owed on the first payment would be :
23,876*0.0299992482=716.2621.
Answer C
26. This consists of a 20 year annuity immediate paying 10, a deferred nine year
decreasing annuity due, and then a deferred perpetuity.
PV = 10a^0M + v2\Da)^06 + *>29(^)
PV = 114.6992 + 11.4240 + 3.0759 = 129.1991
Answer A
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 6 - Exam FM / Exam 2
Page PE6-17
27. The correct forward price is 1350e04/2 = 1377.27. The contract requires you
to buy in six months at the higher price of 1410, which is too high by
1410 -1377.27 = 32.73. You should be paid the present value of 32.73 which
is 32.73e"04/2 = 32.08
Answer C
28. Use the Macaulay duration formula for a bond, and then transform the
Macaulay duration to the modified duration. First, price the bond
P = 160(a5io.o7) + 2000v5 = 2082.0039
D = 160(Ia)5.o,o7+5(2000)v5 = 4 327254
DM = ^= 4.327254 =
1 + i 1.07
Answer B
29. The present value of the two perpetuities is
_6000_ + _8000_
1.0154-1 1.01516-1
We can use this as the present value of the 30 year annuity with i = 1.0154 -1
127,519.2892 = Pmt(a30U)
Pmt = 9,399.70
Answer B
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE6-18
Practice Exam 6 - Exam FM / Exam 2
30. The four year forward rate would be UtS.
(1 + Ss)5
(1 + 14.5) =
(1 + S4)4
To solve for the spot rates:
84.59 = ^^
(I + S4)4
(I + S4)4 =1.1822
and
100
82.48 = -
(l + Ss)S
(l + ss)s =1.2124
Then,
(1 + i ,_d + s5)5_ 1.2124
(1+k5)-a^4y-n822-1-0255
Answer A
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 7 - Exam FM / Exam 2
PagePE7- 1
Exam FM
Questions
1. For call and put options on a stock with price S0 and strike price K, you are
given information on the difference between C-Pcall and put prices. For
T = .5, C-P = 2.99.For T = .25, C-P = 2.50. You are give r = .04 and S = 0.
Find K.
A) 48.52 B) 49.74 C) 50.82 D) 51.70 E) 52.95
2. A 30 year 10,000 bond pays 3% annual coupons and matures at par. It is
purchased to yield 5% for the first 15 years and 7% thereafter. Calculate
the price of the bond.
A) 5,848 B) 6,172 C) 5,637 D) 6,418 E) 4,862
3. George borrows X for 20 years at a nominal rate of 12% convertible
monthly, to be repaid with equal payments at the end of each month. The
outstanding balance immediately after the 10th payment is 297,000. How
much total interest will George pay for this loan?
A) 793,243 B) 658,660 C) 300,175 D) 487,854 E) 493,069
4. Suppose a total of 30 semiannual payments of amount 5 are made starting
exactly six years from today. Assuming an annual effective rate of 6%,
what is the future value at a time 30 years from today? Assume that after
the payments are complete, the investment is left in the same account
earning interest.
A) 708 B) 411 C) 243 D) 399 E) 450
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE7-2
Practice Exam 7 - Exam FM / Exam 2
5. The table below gives the prices of puts and calls on a stock at two different
strike prices.
Strike Call Put
35.00 6.13 0.44
40.00 2.78 1.99
The current price of the stock is 40. A trader buys a 40-strike call, sells a 40-
strike put, sells a 45-strike call and buys a 45-strike put. All options are 6-
month European. What is his maximum profit?
A)0 B).90 Q1.90 D)2.90 E) 3.90
6. Andy deposits X into an account that earns 10% annual effective interest for
3 years and then a nominal interest rate of 5% convertible semiannual for
the 3 years after that. If, after the 6 years, his future value is 200,000, how
much interest did he earn during the 3rd year?
A) 15,678 B) 18,750 C) 129,571 D) 24,200 E) 56,130
7. An association had an initial balance of 200 on Jan 1 and also had deposits of
25 on March 31st, June 30th, and Sep 30th. The association had a withdrawal
of 30 on Feb 28th, a withdrawal of 60 on June 30th, and ended with a balance
of 250 on Dec 31st. Calculate their dollar weighted rate of return.
A) 32.10% B) 42.99% C) 35.62% D) 19.18% E) 23.34%
8. A two year oil swap will enable you to assure a level price of 20.80 per
barrel for each of the next two years. If you had instead decided to enter
into two separate forward agreements and invest the present value of the
two forward prices today to assure the purchase, you would have had to
invest $38. The one year spot rate is 6% and the two year spot rate is i.
Find i.
A) 6.21% B) 6.32% C) 6.39% D) 6.44% E) 6.50%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 7 - Exam FM / Exam 2
Page PE7- 3
9. At time t=0, Mark puts a one-time deposit of 1,000 into a fund crediting
interest at an annual effective rate of i.
At time t=2, Lewis puts a one-time deposit of 1,000 into a different fund
crediting interest at a force St = .
3 + t
At time t=18, the amounts in each fund will be equal. Calculate i.
A) 2.9% B) 5.3% C) 8.3% D) 9.4% E) 10.5%
10. Money accumulates in a fund at an effect annual interest rate of i during the
first 6 years and at an annual interest rate of 3i thereafter. A deposit of 1 is
made into the fund at time zero. It accumulates to 1.84 at the end of 11
years and 2.83 at the end of 16 years. What is the value of the deposit at the
end of 9 years?
A) 3.43 B)2.17 C) 1.55 D) 1.22 E) 1.65
11. An annuity due has 40 quarterly payments of $50 followed immediately by a
perpetuity with quarterly payments of X. Find X, if the present value is
2000, at an annual effective rate of 16%.
A) 194 B)151 C)157 D) 179 E) 167
12. A six-month European put on the S&R index has price of 74.20. The strike
price is equal to the six month forward price. You are given the continuous
interest rate r = .04. The current value of the index is 1000. Find the value S
of the index for which the put profit equals the six month short forward
profit at expiration.
A) 944.50 B) 946.00 C) 1020.20 D) 1094.40 E) 1095.90
13. A n-year 1000 par value bond with 8% annual coupons has an annual
effective yield of i, i>0. The book value of the bond at the end of year 3 is
1099.84 and the book value at the end of year 5 is 1087.27. What is the
effective yield interest rate?
A) 6.7% B) 5.9% C) 7.3% D) 6.2% E) 5.5%
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE7-4
Practice Exam 7 - Exam FM / Exam 2
14. At time zero, Sal makes a deposit of 300 into an account earning nominal
annual interest rate of 3% compounded monthly. Also, Rick makes a deposit
of 250 into another account earning an annual effective interest rate of i.
During the 2nd year, both accounts earn the same amount of interest. Solve
for the amount of interest that Rick will make during the 6th year.
A) 15.9 B)16.7 Q10.8 D) 11.2 E) 14.5
15. The current price of a stock is 40. Harry sets up a position in six month
calls. He buys two 35-strike calls, sells six 40-strike calls and buys four 45
strike calls. Which of the following could be his profit graph?
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 7 - Exam FM / Exam 2
Page PE7- 5
16. The time weighted return for the fund with the transactions in the table
below is 12%. What is the dollar weighted rate of return?
Date
1/1/2005
6/1/2005
10/1/2005
12/31/2005
Value before
transaction
980
1010
1055
1060
Transaction
Deposit
30
Transaction
Withdrawal
X
A) 12.25% B) 1.53% C) 12.00% D) 11.78% E) 5.12%
17 A dealer enters into a long forward agreement to buy 1000 shares of a stock
in three months for $51 per share. The stock is currently priced at 50 and
pays no dividend. Which of the following strategies would hedge his risk so
that his total profit would be zero in all cases?
A) Buy 1000 shares of the stock today with $50,000 cash borrowed on a
three month loan at the risk free rate.
B) Buy 1000 shares of the stock today and invest an amount of $50,000 at the
risk free rate for 3 months.
C) Sell 1000 shares of the stock short and invest an amount of $50,000 at the
risk free rate for 3 months.
D) Sell 1000 shares of the stock short and borrow an amount of $50,000 at
the risk free rate for 3 months.
E) No hedging is necessary.
18. Suppose that Julia finances a $315,000 mortgage for 25 years at a nominal
rate of 6.5% convertible monthly. Julia will be making monthly payments
with her first payment due one month from receiving the loan. Suppose
Julia adds $125 to each financed payment to pay of the loan faster. What is
the dollar amount of Julia's last payment?
A) 918 B)936 C) 2,252 D) 1,254 E) 923
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE7-6
Practice Exam 7 - Exam FM / Exam 2
19. Suppose a company expects liabilities of 100,000 in one year, 200,000 in two
years, 300,000 in three years, and 400,000 in four years. Also, suppose that
they want to fund those liabilities by an exact match of investments in the
following zero coupon bonds and annual coupon bonds. How many bond
A's should they buy, assuming that fractional bonds can be purchased?
Bond
A
B
C
D
Yield Rate -
Annual
4.5%
5%
5.5%
6%
Coupon Rate -
Annual
Zero Coupon
6%
6%
6%
Par
Value
1000
1000
1000
1000
Maturity
Term
1
2
3
4
A) 40.7 B)45.6 C) 52.5 D) 89.6 E) 100
20. At the same time, Dan and Darci deposit money into two different funds.
Dan deposits 200 and Darci deposits 80. Both accounts earn the same rate of
interest. The amount of interest earned in Dan's account during the 10th
year is the same as the amount of interest earned in Darci's account during
the 20th year. Determine the amount of interest earned in Dan's account
during the 13th year.
A) 23.1 B)57.6 C) 49.1 D) 63.2 E) 52.6
21. How much should you pay today for an annuity with 30 payments where the
initial payment of 500 is three years from today and each subsequent annual
payment is 6% greater than the previous payment? Let the annual effective
interest rate equal 8%.
A) 8,969 B) 11,589 C) 9,426 D) 9,200 E) 9,731
22. The price of a stock is currently 40. A trader buys a 40-strike put and sells a
45-strike put with the same maturity. Which of the following best describes
the trader's most likely expectation for the price of the stock?
A) It will go down.
B) It will go up.
C) It will have high volatility.
D) It will have low volatility.
E) The price is theoretically incorrect and an arbitrage is possible.
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 7 - Exam FM / Exam 2
Page PE7- 7
23. You are buying a perpetuity with annual payments as follows:
i) Payments of X at the end of the first year and every three years
thereafter
ii) Payments of X+l at the end of the second year and every three years
thereafter
iii) Payments of X+2 at the end of the third year and every three years
thereafter.
The interest rate is 5% convertible semiannually. If the present value is
38.86, calculate X.
A) 0.98 B)1.00 Q1.20 D) 1.23 E) 1.25
24. Annual payments of 500 are made at the beginning of each year for 30 years
to a annuity earning an annual effective rate of 7%. The interest is
immediately reinvested into another fund earning 4.5% annual effective
interest.
At the end of the 30 ears, what is the accumulated value of the 30 payments
and the reinvested interest?
A) 36,325 B) 47,230 C) 26,300 D) 30,504 E) 41,252
25. The S&R index, which does not pay a dividend, is currently priced at 1000.
The 6 month forward price is 1015.11. A 1000-strike 6-month call on the
index is priced at 63.71. Find the price of a 1000-strike 6-month put.
A) 14.88 B) 15.00 C) 29.88 D) 36.75 E) 48.82
26. The current price of a stock is 50. A trader creates a synthetic three-month
forward position for 1000 shares by buying $49,502.5 worth of the stock with
an amount of $49,502.50 borrowed at the risk-free rate of r = .04.
Find the continuous dividend yield 5 for the stock.
A) .01 B).02 Q.03 D).04 E) .05
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE7-8
Practice Exam 7 - Exam FM / Exam 2
27. A 1,000 loan is repaid with equal payments at the end of each quarter for 10
years. The principal portion of the 13th payment is 1.5 times the principal
portion of the 5th payment. Calculate the quarterly payment towards the
loan.
A) 60 B)26 C)57 D) 69 E) 131
28. Michael is the Chief Financial Officer for ABC company. He needs to
determine the amount of interest that will be paid at the end of the 5th year
of the bank loan for tax purposes. The loan is for 7 years, for $1,000,000,
with annual payments at the end of each year, at 10% effective interest rate.
Which of the following is closest to the number that he needs?
A) 35,650 B) 51,080 C) 65,000 D) 89,450 E) 170,000
29. Use the following table to represent spot rates. Calculate the total future
value at time 5 of a payment of $3000 made today and a payment of $3000
made at time 3. Assume that the payment at time 3 will be invested at
today's forward rates.
Term (years)
1
2
3
4
5
Annual yield
6.00%
6.10%
6.40%
6.80%
7.50%
A) 7,684 B) 7,411 C) 7,882 D) 7,566 E) 8,568
30. Amanda receives a 10 annual payment increasing annuity paying 30 at the
end of the first year and increasing by 5 each year thereafter. Kevin
receives a 10 annual payment decreasing annuity that pays X at the end of
the first year and decreases by 2 each year thereafter. At an annual interest
rate of 4%, both annuities have the same present value. Calculate X.
A) 61.60 B) 42.53 C) 28.60 D) 59.24 E) 47.99
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 7 - Exam FM / Exam 2
Page PE7- 9
Solutions
1. We use the parity relation C-P = S0e ST -Ke rT. Then we have the two
equations
2.99 = S0-e-04(5)K
2.50 = S0-e-04(25)K
Subtracting the second from the first, we obtain
.49 = (e"01 - e"02) K -> K = 49.74
Answer B
2. The easiest way to solve this problem is to find the present value of the
coupons plus the present value of the redemption value.
P = 300a^005 + vj505300a^007 + vJ505Vo5o7(10,000) = 6171.64
Answer B
3. First, calculate the payment amount by using the information given
regarding the outstanding balance and using the retrospective method.
OBt=Ka^
297,000 = Ka23o^0.oi
Using the calculator, K = 3305.1809
Next, calculate the original loan amount with a payment of 3305.1809 and
240 monthly payments with a monthly effective rate of 1%. You can use the
calculator to get the original loan amount, X, of 300,174.6003.
Next, to calculate the total interest paid, calculate the difference between
the out of pocket contribution towards the loan and the loan amount. The
contribution is 240 times 3305.1809, or 793,243.416. This gives us a total
interest paid amount of 793,243.416 - 300,174.6003 = 493,068.8157.
Answer E
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE7-10
Practice Exam 7 - Exam FM / Exam 2
4. This is a deferred annuity due. We will think of the time in terms of
semiannual periods. It will take 15 years to make the semiannual payments.
The last payment will be made at time 20.5. However, the future value
calculation of an annuity due will calculate the value at time 21. Then, the
payments will be left in the fund for an additional 9 years.
i = (1.06)1/2 -1 = 0.0295630141
FV = 5sm (1 + i)18 =410.85
Answer B
5. This is a box spread, which is equivalent to a zero-coupon bond at the risk-
free rate. The profit on such a bond is 0.
Answer A
6. This is a lump sum deposit that earns different rates. First, calculate X, the
lump sum deposit, by setting up a future value calculation.
200,000 = X(l + 0.10)3[(1 + Mil)2]3
X = 129,571.2796
To calculate the amount of interest earned during the third year we need the
difference between the future value at time 2 and time 3. Or, you could
multiply the future value at time 2 by 10%.
FV3-FV2 =129,571.2976(1 + 0.10)3-129,571.27961 + 0.10)2 =15,678.1270
Answer A
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 7 - Exam FM / Exam 2 Page PE7-11
7. After organizing the information, we have the following transactions.
1 Date
1/1
2/28
3/31
6/30
9/30
12/31
Value before
transaction
200
250
Transaction
Deposit
25
25
25
Transaction
Withdrawal
30
60
Now, using the dollar-weighted equation:
.= 250 - 200 - [-30 + 25 - 35 + 25]
1 "200-30(1-2/12)+ 25(1-3/12)-35(1-6/12)+ 25(1-9/12)
1 = 0.3561644
Answer C
8. The present value of the level payments equals the present value of the two
separate forward agreement payments. (See page 248 of the second edition
of Derivatives Markets). Thus
20.80 20.80 OQ
+ T = 38
1.06 (l + i)2
This gives us 1 + i = 1.063874.
Answer C
9. The accumulation function for Lewis would be:
a(t) = e'>rIdt
_ 6ln(3+t)|28 _ 6ln21-ln5 _ 4^
Now, set that amount to Donald's accumulation function and solve for i:
(l + i)18 =4.2
1 = 0.08299128
Answer C
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE7-12 Practice Exam 7 - Exam FM / Exam 2
10. Let's set up two equations for the two future values that are given.
(l + i)6d + 3i)5=1.84 and (1 +i)6(l + 3i)10 = 2.83
Solve both equations for (1 + i)6 and set them equal to each other.
1.84 _ 2.83
(l + 3i)5 ~(l + 3i)10
(l + 3i)10=2.83
(l + 3i)s ~1.84
i = 0.02997258091
Now, solve for the value in nine years:
(l + i)6(l + 3i)3 =1.55
Answer C
11. This is a 40 payment annuity due and a deferred perpetuity due. Set up a
present value calculation with the perpetuity payment as X.
i = (1 + .16)1/4 -1 = 0.03780198565
2000 = 50(2^+v40Xa^
2000 = 1061.5174 + v40X(—)
i
X = 150.8017
Answer B
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 7 - Exam FM / Exam 2
Page PE7-13
12. The short forward price is lOOOe04( 5) = 1020.20.
The future value of the put premium is 74.20e04( 5) = 75.70. The put profit at
index price S is Max(1020.20-S,0)-75.70
The short forward profit at index price S is 1020.20 - S
The short forward profit function is decreasing, and when S=1020.20, the
short forward profit of 0 is greater than the put profit of -75.70. Thus the put
and forward profits are equal for some S > 1020.20, and we solve the
equation 1020.20 - S = -75.70 -> S = 1095.90
Answer E
13. Some concepts that will simplify this problem is knowing that the principal
of one coupon is just a factor of (1 + i)m different from the principal of
another coupon.
Pr ink+m =Prink(l + i)m
Also, using amortization techniques, the principal of a coupon can be found
by taking the previous book value, multiplying it by i to get the interest
portion, and then subtracting the interest portion from the coupon amount.
The principal portion of the 4th coupon is 80-1099.84i
The principal portion of the 6th coupon is 80-1087.27i
80 - 1087.27i = (80 - 1099.84i)(l + if
80 - 1087.27i = (80 - 1099.84i)(l + 2i + i2)
1099.84i3 + 2119.68i2 - 147.43i = 0
i(1099.84i2 + 2119.68i -147.43) = 0
Using the quadratic formula, i = 0.06720917418
Answer A
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE7-14
Practice Exam 7 - Exam FM / Exam 2
14. First, we need to solve for i. By finding the future value of both accounts
after one year and multiplying that amount by the respective annual
effective interest rates of both accounts.
300[(1 + ^)12][(1 + ^)12 -1] = 250(1 + i)(0
300[(1 + ^)12][(1 + ^)12 -1] = 250(i + i2)
0 = 250i2+250i-9.4023
Using the quadratic formula: i= 0.03629208
Now, to figure the amount of interest that Rick will make during the 6th
year: 250(1 + i)5i = 10.8433
Answer C
15. The payoff from this position is 0 on the interval (0,35), increasing on the
interval (35,40), decreasing on the interval (40,45) and constant for S > 45.
Since the expense is a constant, the profit graph will have the same shape as
the payoff graph. Thus the only possibility is B.
Answer B
16. First, use the given time weighted return to solve for X. Then, once you
have X, solve for the time weighted return.
(1 + 12) 101° 105S 106°
980 1010 + 30 1055-X
X = 65.5308
Now, the dollar weighted equation:
1060 - 980 - [30 - 65.5308]
l~ 980+ 30(1-5/12)-65.5308(1-9/12)
= 0.1177543195
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 7 - Exam FM / Exam 2
Page PE7-15
17. A long forward can be hedged by a forward sale. The word equation
-Forward = -STOCK + Bond
shows that a forward sale can be created by selling the stock short and
buying a zero-coupon bond (investing the cash at the risk-free rate).
Answer C
18. First, calculate the payment amount without the extra money added.
ao65=00054166667
12
315,000 = Pmt{a^m)
Pmt = 2,126.90
Now, Julia is adding $125 to each payment, so Julia is making a payment of
$2,251.90. This will pay off the loan faster, so we need to solve for the new
n.
315,000 = 2,251.90(a^)
n = 262.4094
That means Julia will make 262 full payments and one partial payment. To
calculate the amount of her last partial payment, we need to know the
Outstanding Balance after the 262nd payment using the retrospective
method.
OB262 = 315,000(1 + i)262 - 2,251.902558s262t
OB262= 918.33
Her last payment will consist of this balance and interest charged.
= 918.3311(1 + 0 = 923.3054
This problem could have been done entirely on a finance calculator.
Answer E
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE7-16
Practice Exam 7 - Exam FM / Exam 2
19. To approach this problem, work from the longer term bonds to the shorter
term bonds.
For bond D, you will receive 1000+60 per bond at maturity. If you need
400,000 in four years, then you will need 400>000 = 377.3584 of bond D. If
1060
you have 377.3584 D bonds, then at time 3, you will get
377.3584*60=22,641.504 in coupons from the D bonds. Your need at time 3
has decreased to 300,000-22,641.504 = 277,358.496.
For bond C, you will receive 1000+60 per bond at maturity. If you need
277,358.496 in three years, then you will need 277>358-496 = 261.6590 of
1060
bond C. If you have 377.3584 D bonds and 261.6590 C bonds, then at time 2,
you will get 377.3584*60 + 261.6590*60=38,341.044 in coupons. Your need at
time 2 has decreased to 200,000-38,341.044 = 161,658.956.
For bond B, you will receive 1000+60 per bond at maturity. If you need
161,658.956 in two years, then you will need 161>658-956 = 152.5084 of bond
1060
B. If you have 377.3584 D bonds, 261.6590 C bonds, and 152.5084 B bonds
then at time 1, you will get 377.3584*60 + 261.6590*60 +
152.5084*60=47,491.548 in coupons. Your need at time 1 has decreased to
100,000-47,491.548 = 52,508.452
For bond A, you will receive 1000 per bond at maturity. If you need
52 508 452
52,508.452 in one year, then you will need —- = 52.5085 of bond A.
1000
Answer C
20. The tenth year is from time 9 to time 10. The amount of interest that Dan
earns during the tenth year is then 200(1 + i)10 - 200(1 + i)9.
The twentieth year is from time 19 to time 20. The amount of interest that
Darci earns during the twentieth year is then 80(1 + i)20 - 80(1 + i)19.
Set these two equations equal to each other to solve for i.
200(1 + i)10 - 200(1 + i)9 = 80(1 + i)20 - 80(1 + i)19
200(1 + i)9[(l + i) -1] = 80(1 + i)19[d + 0-1]
200(1 + i)9= 80(1 + i)19
2.5 = (l + i)10
i = 0.09595822639
The amount of interest earned for Dan during the 13th year is
200(1 + i)ui = 57.6289
Answer B
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 7 - Exam FM / Exam 2
Page PE7-17
21. The first payment is a deferred payment and the remaining payments are
geometric increases of that initial payment. A present value set up would
look like the following.
PV = 500v3 +500(1.06)v4 +500(1.06)2v5 +.... + 500(1.06)29v32
Use the formula for a geometric series, to get the following.
PV = 500v3(l + 1.06v +1.062 v2 +... +1.0629 v29)
= 500v31~(1-06y)3° = 9199.8278
l-(1.06v)
Answer D
22. The trader's position is a bull spread, which is typically used to profit from a
price increase.
Answer B
23. This perpetuity would look like the following series of payments:
X, X+l, X+2, X, X+l, X+2, etc.
Break this perpetuity into three perpetuities due. One perpetuity due that is
deferred one year and has payments of X every three years, a second that
is deferred two years and has payments of X+l every three years, and a
third that is deferred three years and has payments of X+2 every three
years. We need a three year effective interest rate:
i = (1 + —)6 -1 = 0.15969342
d = —= 0.137703134
1 + i
The present values of the three perpetuities are:
mr -A. nTr 9 A + 1 nTr -j -A. 4- JL
add
where j is equal to the one year effective interest rate:
j = (l + ®^ly _i = 0.050625
The sum of these three present values should be equal to 38.86
PV1+PV2+PV3 =38.86
VjX + v2(X +1) + v3(X + 2) = 38.86d
2.72006X = 2.7206 -> X = 1.0002
Answer B
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Page PE7-18
Practice Exam 7 - Exam FM / Exam 2
24. The accumulated value of the payments without interest is 300*500=150,000
The accumulated value of the interest is an annuity with geometrically
increasing payments. The first payment is 35, the second is 70, and the last
payment would be 1050.
This is a geometrically increasing annuity with 30 payments at 4.5%
effective interest.
FV = 300 • 500 + 35(Js)^0045 = 41,251.857
Answer E
25. We do not know the interest rate r, but we can find it using the forward
price.
1015.11 = S0e{r-S)T = lOOOe5r -> r = .03
Then we can use the parity equation to find the put price P.
C-P = S0e-ST-Ke-rT
63.71 -P = 1000- lOOOe"015
P = 48.82
Answer E
26. When you create a synthetic forward for a single share you buy e~5T units of
the stock for a price of S0e~JT . Thus we have 49.5025 = 50e"J( 25) =.04.
Answer D
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
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Practice Exam 7 - Exam FM / Exam 2
Page PE7-19
27. First, we need to solve for i. We know that the principal portion of the 13th
payment is 1.5 times the principal portion of the 5th payment. The principal
in the tth payment is:
PRt = Kvn~t+1
If there are 10 years with quarterly payments, then n equals 40. Now, set
the principal portion of the 13th payment equal to 1.5 times the principal
portion of the 5th payment 59.87.
Kv40-13+1 = l45Kv40-5+l
v28=1.5v36
1.5
(1 + i)8 =1.5
i = 0.0519895
Now, that you know i, find the payment required to pay a loan of 1000, with
40 payments and an interest rate i.
1000 = Ka4oi00519895
. K = 59.8742
Answer A
28. First, figure the payment required for a 1,000,000 loan for 7 years with
payments made at the end of each year at 10% effective interest.
1,000,000 = Ka7]010
K = 205,405.4997
If you want to know the amount of interest paid in the 5th payment, you need
to know the outstanding balance after the 4th payment:
OB4 =205,405.4997a^010 =510,813.0759
The amount of interest due in the 5th payment will be 10% of the outstanding
balance after the 4th payment, or 51,081.30759
This problem could have been done entirely on a finance calculator.
Answer B
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page PE7-20
Practice Exam 7 - Exam FM / Exam 2
29. The trick to this problem is recognizing that you need to calculate a forward
rate for the second payment. You will need i3,5. Then, it is a future value
calculation.
(l + i3,5)2 =
2 (l + ss)5 (L075)5
= 1.191838574
(l + s3)3 (1.064)3
FV = 3000(1 + s5)5 + 3000(1 + Uf5)2
FV = 3000(1.075)5 + 3000(1.191838574)) = 7882.4037
Answer C
30. For both Amanda and Kevin, we can use the formula for a present value of
an annuity with payments of P, P+Q, P+2Q, ..., P+(n-l)Q. The formula is:
Pa;A+Q(^Ll^L)
Substituting in for Amanda, her present value is:
30a^nn. + 5f aioio.o4 ~ 10vl° 1 = 412.7336
0.04
^1010.04
J
Substituting in for Kevin, his present value is:
****-> N^)
Set the present value of Kevin's equal to 412.7336, and solve for X.
X = 59.2408
Answer D
^ISIcm-IOv
10 A
0.04
= 412.7336
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Index
Page Ind-1
Page
Accreting Swap
Accrued Interest
Accumulation Functions
Accumulation Functions, Continuous Interest
American Option
Amortization
Amortization of Discount
Amortization of Premium
Amortization Table
Amortization with Arithmetic Payments
Amortization with Geometric Payments
Amortization with Level Payments
Amortization with Monthly Payments
Amortization with Variable Payments
Amortizing Swap
Amount for Accumulation of Discount
Annuities
Annuities with Arithmetic Progression
Annuities with Geometric Progression
Annuities with Level Payments
Annuities with Varying Payments
Annuity Due
Annuity Immediate
Arbitrage Pricing
Assets
At the Money Option
M14-11
M4-12
Ml-4
Ml-15
M9-9
M3-1
M4-6
M4-6
M3-2
M3-10
M3-9
M3-6
M3-11
M3-4
M14-11
M4-7
M2-1
M2-19
M2-20
M2-7
M2-15
M2-8
M2-3
M12-3
M7-1
M9-17
B
Bear Spread
Bermudan Option
Bid-ask spread
Bond Price
Bonds
Borrowing Projects
Box Spread
Bull Spread
M10-11
M9-9
M8-3
M4-4
M4-1
M5-6
MlO-12
MlO-10
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics

Page Ind-2
Index
C
Call Option M9-9
Call Provisions M4-9
Callable Bond M4-9
Cap Strategy M10-3
Capitalization of Interest M3-19
Cash and Carry Hedge M12-11
Cash Flows M5-1
Cash settlement M9-2
Certificate of Deposit M7-22
Change in Price M7-13
Clearing House M12-18
Collar M10-13
Collar, Hedging with M10-14
Compound Interest Ml-1
Constant Force of Interest Ml-15
Continuous Annuities M2-11
Conversion of Nominal Interest rate to Discount Rate Ml-13
Convertible Interest Ml-8
Convexity M7-12
Cost of Carry M12:17
Coupon Rate M4-1
CoveredCall M10-4 "
CoveredPut MlO-5
D
Dealer as Swap Counterparty M14-4
Decreasing Annuities M2-18
Deferred Annuities M2-24
Derivative Security M8-1
Discount Bond M4-2
Discount Rate Ml-10
Dividends M7-20
Dollar Weighted Rate M5-7
Duration M7-4
Duration Matching M7-18
Duration of Portfolio M7-14
E
Effective Interest Rate Ml-6
Equation of Value Ml-19
Equity Linked CD M9-19
Equity Linked Note, Marshall & Isley M10-21
Eurodollar M13-6
Eurodollar Future M13-6
European Option M9-9
M4-1
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Index Page Ind- 3
iodic j-age
F
Face Value
Flat Price
Flat Yield Curve
Floor Strategy
Force of Interest
Forward and Futures Prices Compared
Forward Contract
Forward Contract on Stock, Pricing
Forward Interest Rate
Forward Premium
Forward Rate
FRA (Forward Rate Agreement)
Fully Immunized
Future Value
Future Value of Annuities
Futures Contracts
G
Geometric Series
Greater Convexity for Assets
H
Hedging
Hedging with a Collar
Hedging with a Synthetic Stock
Hedging, Buyer
Hedging, Producer-Seller
Hedging, Reasons for
I
Immunization
Implied Forward Rate
Implied Forward Rate
In the Money Option
Increasing Annuities
Inflation
Installment Loan
Insurance, Options s
Interest Rate Risk
Interest Rate Swap
Internal Rate of Return
Inverted Yield Curve
Investment Year Method
L
Law of One Price
Lease Rate
M4-11
M6-3
M10-2
Ml-15
M12-22
M9-1
M12-7
M13-2
M12-8
M6-5
M13-4
M7-19
Ml-2
M2-4
M12-18
M2-2
M7-18
M8-2
Mll-9
M12-10
Mll-5
MlT-2
Mll-7
M7-16
M6-5
M13-3
M9-17
M2-16
M2-32
M3-14
M9-18
mm"
M14-7
M5-1
M6-3
M5-11
M6-4
M12-17
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Ind-4
Index
ILiUUQl^HII^HHIHHHHHHHiHIHIHHHHHHIIHHH^KjuISS
Liabilities M7-1
Liability Management M7-1
Long forward M9-2
Long Position in Stock M8-3
M
Macaulay Duration M7-4
Macaulay Duration of Coupon Bond M7-8
Makeham's Formula M4-5
Margin M12-19
Mark to Market M12-19
Market Price M4-12
Matching Assets and Liabilities M7-1
Modified Duration M7-6
Modified Internal Rate of Return M5-5
Money Market Funds M7-22
Mortgage-Backed Securities M7-22
Mutual Funds M7-21
N
Negative Amortization M3-19
Negative Amortization of Discount M4-8
Net Interest M3-16
Net Present Value M5-13
New Money Rate M5-12
Nominal Discount Rate Ml-12
Nominal Interest Rate Ml-7
O
Open Outcry M12-18
Out of the Money Option M9-17
P
Par Value M4-1
Parallel Shift in Yield Curve M7-15
Parity, Put-Call M10-6
Paylater Strategy Ml 1-11
Payoff for Forward M9-3
Periodic Interest Rate Ml-8
Perpetuities M2-6
Portfolio Method M5-11
Premium M9-9
Premium Bond M4-2
Premium-Discount Formula for Bonds M4-5
Prepaid Forward Price M12-3
Present Value Ml-2
Present Value Matching M7-18
Price Function, P(i) M7-10
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Index
Page Ind- 5
Price of Stock M7-20
Price-Plus Accrued M4-11
Pricing Bonds between Payment Dates M4-11
Profit for forward M9-3
Prospective Method M3-7
Put Option M9-14
Q
Quanto Index Contracts M12-24
Quasi Arbitrage M12-15
R
Redemption Value M4-1
Reinvestment Problems M2-31
Relating Interest Rate and Force of Interest Ml-i8
Retrospective Method M3-8
Risk-Free Rates M6-1
S
S&P 500 Futures Contract M12-19
Settlement Date M4-11
Short Forward M9-2
Short Sale of Stock M8-3
Simple Interest Ml-1
Sinking Fund M3-15
Sinking Fund Balance M3-17
Sinking Fund Deposit M3-16
Spot price M9-1
Spot Rate M6-2
Spot Rate M13-2
Spread M10-9
Stock index M9-1
Stocks M7-20
Straddle M10-16
Strangle M10-18
Swap, Oil M14-2
Swap Curve M14-10
Swap Payment M14-3
SwapRateR JM14:8
Swap rate, general formula M14-13
Swap, Market Value M14-5
Synthetic Forward M10-6
Synthetic Stock M12-9
T
Taylor Series M7-10
Term Structure of Interest Rates M6-1
Time Value of Money Ml-1
©ACTEX 2009 Exam FM / Exam 2 - Financial Mathematics
Hassett, Ratliff, Garcia, & Steeby

Page Ind-6
Index
Time Weighted Rate
Timelines
Treasury STRIP bond
True Price
M5-7
M2-2
M6-2
M4-12
u
Uniqueness of Internal Rate of Return
Unit Annuity
M5-5
M2-1
Variable Annuities
Volatility
M2-26
M7-6
w
Weighted Average
Written Call Option
Written Put Option
M7-4
M9-12
M9-16
Y
Yield Curve
M6-2
Zero Cost Collar
Zero Coupon Bond
Zero Coupon Bond Profit
Zero-coupon Bond Price
Zero-coupon Bond Yield
M10-15
M6-1
M9-7
M13-3
M13-3
©ACTEX 2009
Hassett, Ratliff, Garcia, & Steeby
Exam FM / Exam 2 - Financial Mathematics