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Exam FM Study Manual

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Fall 2019 Edition | Volume IJohn B. Dinius, FSA | Matthew J. Hassett, Ph.D.

Michael I. Ratliff, Ph.D., ASA | Toni Coombs GarciaAmy C. Steeby, MBA, MEd

ACTEX

Exam FM Study Manual ACTEX

ACTEX Learning | Learn Today. Lead Tomorrow.

Fall 2019 Edition | Volume IJohn B. Dinius, FSA | Matthew J. Hassett, Ph.D.

Michael I. Ratliff, Ph.D., ASA | Toni Coombs GarciaAmy C. Steeby, MBA, MEd

Copyright © 2019 SRBooks, Inc.

ISBN: 978-1-63588-811-9

Printed in the United States of America.

No portion of this ACTEX Study Manual may bereproduced or transmitted in any part or by any means

without the permission of the publisher.

Actuarial & Financial Risk Resource Materials

Since 1972

Learn Today. Lead Tomorrow. ACTEX Learning

ACTEX Learning Exam FM – Financial Mathematics Dinius, Hassett, Ratliff, Garcia, & Steeby

ACTEX first published a study manual for the Society of Actuaries’ Exam FM (“Financial Mathematics”) in 2004. That manual was prepared by lead author Matthew Hassett, assisted by Michael Ratliff, Toni Coombs Garcia, and Amy Steeby. The manual has been regularly updated and expanded to keep pace with changes in the syllabus, including the 2017 and 2018 changes in the SOA’s Exam FM syllabus. This latest edition of the ACTEX Study Manual for Exam FM, edited by lead author John Dinius, has been enhanced with additional problem-solving techniques and more practice exams, including over 100 new problems with solutions. In total, there are more than 1,000 examples, exercises, and problems here to help you prepare for Exam FM. This manual has 9 modules that are arranged in 3 groups of 3 modules each. The first 3 modules present basic concepts (the time value of money, annuities, and loan repayment). The next 3 modules apply these concepts to a range of topics (bonds, yield rates, and the term structure of interest rates). The final 3 modules examine more complex real-world concepts (asset-liability management, the factors that influence market interest rates, and interest rate swaps). After each group of three modules there is a “midterm exam,” providing an opportunity to check your progress. At the end of the manual are 14 full-length practice exams of 35 problems each. All of the problems in these exams are original and are not available anywhere else. They are intended to provide realistic exam-taking experience to complete your preparation for Exam FM. This manual is designed to address all of the topics on the SOA’s Exam FM syllabus. However, you might consider obtaining one of the official textbooks for SOA Exam FM and using that text in combination with this manual and the SOA study notes. The following pages provide recommendations on how to prepare for actuarial exams and suggestions for using this manual most effectively.

A note about Errors: If you find a possible error in this manual, please let us know. Use the “Feedback” link on the left side of the ACTEX homepage (www.actexmadriver.com) and describe the issue. We will review and respond to all comments. Any confirmed errata will be posted on the ACTEX website under the “Errata” link.

Preface


How to Learn Actuarial Mathematics and Pass Exams On the next page you will find a list of study tips for learning the material in the Exam FM syllabus and passing Exam FM. But first it is important to state the basic learning philosophy that we are using in this guide: You must master the basics before you proceed to the more difficult problems.

Think about your basic calculus course. There were some very challenging applications in which you used derivatives to solve hard max-min problems.

It is important to learn how to solve these hard problems, but if you did not have the basic skills of taking derivatives and manipulating algebraic expressions, you could not do the more advanced problems. Thus every calculus book has you practice derivative skills before presenting the tougher sections on applied problems.

You should approach interest theory the same way. The first 2 or 3 modules give you the basic tools you will need to solve the problems in the later modules. Learn these concepts and methods (and the related formulas) very well, as you will need them in each of the remaining modules.

This guide is designed to progress from simpler problems to harder ones.

In each module we start with the basic concepts and simple examples, and then progress to more difficult material so that you will be prepared to attack actual exam problems by the end of the module.

The same philosophy is used in our practice exams at the end of this manual. The first few practice exams have simpler problems, and the problems become more difficult as you progress through the practice exams.

A good strategy when taking an exam is to answer all of the easier problems before you tackle the harder ones.

An exam is scored in percentage terms, and a multiple choice exam like Exam FM will have a mix of problems at different difficulty levels.

If an exam has ten problems and three are very hard, getting the right answers to only the three hard problems and missing the others gets you a score of 30%. This could happen if the hardest problems are the first ones on the exam and you attempt them first and never get to the easy problems.

A useful exam strategy is to go through the exam and quickly solve all the more basic problems before spending extra time on the hard ones. Strive to answer all of the easy problems correctly.

On passing exams


Study Tips 1) Develop a schedule so that you will complete your studying in time for the

exam. Divide your schedule into time for each module, plus time at the end to review and to solve practice problems. Your schedule will depend on how much time you have before the exam, but a reasonable approach might be to complete one module per week.

2) If possible, join a study group of your peers who are studying for Exam FM.

3) For each module:

a) Read the module in the FM manual (and the associated SOA study note, if any).

b) As you read through the examples in the text, make sure that you can correctly compute the answers.

c) Summarize each concept you learn in the manual’s margins or in a notebook.

d) Understand the main idea of each concept and be able to summarize it in your own words. Imagine that you are trying to teach someone else this concept.

e) While reading, create flash cards for the formulas to facilitate memorization.

f) Learn the calculator skills thoroughly and know all of the calculator’s functions.

g) Review the corresponding chapter in one of the SOA’s recommended texts (if you are using a text to supplement this manual).

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 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 write 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.

j) Do the Supplemental Exercises and review your solutions. 4) After learning the material in each module, it is a good idea to go back to

previous modules and do a quick half-hour or 1-hour review, so that information isn't forgotten.

5) Go back and redo the sample exam problems that you marked as Medium or Hard when you worked through them the first time.


6) At the end of Modules 3, 6, and 9, we have included practice exams that are like midterms. Taking these tests will help you consolidate your knowledge.

7) After learning the material in all of the modules and taking the midterms, go to the practice exams. a) The first 6 practice exams are relatively straightforward to enable you

to review the basics of each topic. b) The next 5 practice exams introduce more difficult questions in order to

replicate the actual exam experience. c) The last 3 practice exams include especially challenging problems to test

your understanding of the material and your ability to apply solution techniques.

Please keep in mind that the actual exam questions are confidential, and there is no guarantee that the questions you encounter on Exam FM will look exactly like the problems in this manual.

Table of Contents Page TOC-


1

Topic Page

Volume I

Introduction Intro-1 Module 1 Interest Rates and the Time Value of Money Section 1.1 Time Value of Money M1-1 Section 1.2 Present Value and Future Value M1-3 Section 1.3 Functions of Investment Growth M1-6 Section 1.4 Effective Rate of Interest M1-10 Section 1.5 Nominal Rate of Interest M1-13 Section 1.6 Rate of Discount M1-17 Section 1.7 Present Value Factor M1-20 Section 1.8 Nominal Rate of Discount M1-21 Section 1.9 Continuous Compounding and the Force of Interest M1-24 Section 1.10 Quoted Rates for Treasury Bills M1-29 Section 1.11 Relating Discount, Force of Interest, and Interest Rate M1-35 Section 1.12 Solving for PV, FV, n, and i M1-37 Section 1.13 The Rule of 72 M1-46 Section 1.14 Formula Sheet M1-48 Section 1.15 Basic Review Problems M1-49 Section 1.16 Basic Review Problem Solutions M1-50 Section 1.17 Sample Exam Problems M1-55 Section 1.18 Sample Exam Problem Solutions M1-60 Section 1.19 Supplemental Exercises M1-69 Section 1.20 Supplemental Exercise Solutions M1-70 Module 2 Annuities Section 2.1 Introduction to Annuities M2-1 Section 2.2 Annuity-Immediate Calculations M2-3 Section 2.3 Perpetuities M2-7 Section 2.4 Annuity-Due Calculations M2-8 Section 2.5 Continuously Payable Annuities M2-11 Section 2.6 Basic Annuity Problems for Calculator Practice M2-14 Section 2.7 Annuities with Varying Payments M2-19 Section 2.8 Increasing Annuities with Terms in Arithmetic Progression M2-20 Section 2.9 Decreasing Annuities with Terms in Arithmetic Progression M2-23 Section 2.10 A Single Formula for Annuities with

Terms in Arithmetic Progression M2-25

Section 2.11 Annuities with Terms in Geometric Progression M2-27 Section 2.12 Equations of Value and Loan Payments M2-34 Section 2.13 Deferred Annuities M2-36

CCoonntteennttss

Page TOC-2 Table of Contents


Topic Page Section 2.14 Annuities with More Complex Payment Patterns M2-38 Section 2.15 Annuities with Payments More Frequent than Annual M2-42 Section 2.16 Annuities with Payments Less Frequent than Annual M2-47Section 2.17 Payment Periods that Don’t Match

the Interest Conversion Period M2-51

Section 2.18 Continuously Payable Annuities with Continuously Varying Payments

M2-58

Section 2.19 Reinvestment Problems M2-63 Section 2.20 Inflation M2-65 Section 2.21 Formula Sheet M2-67 Section 2.22 Basic Review Problems M2-71 Section 2.23 Basic Review Problem Solutions M2-73 Section 2.24 Sample Exam Problems M2-80 Section 2.25 Sample Exam Problem Solutions M2-87 Section 2.26 Supplemental Exercises M2-102 Section 2.27 Supplemental Exercise Solutions M2-104

Module 3 Loan Repayment Section 3.1 The Amortization Method of Loan Repayment M3-1 Section 3.2 Calculating the Loan Balance M3-4 Section 3.3 Loans with Varying Payments M3-10 Section 3.4 Formulas for Level-Payment Loan Amortization M3-13 Section 3.5 Monthly-Payment Loans M3-15 Section 3.6 An Example with Level Payment of Principal M3-19 Section 3.7 Capitalization of Interest and Negative Amortization M3-20 Section 3.8 Formula Sheet M3-22 Section 3.9 Basic Review Problems M3-23 Section 3.10 Basic Review Problem Solutions M3-24 Section 3.11 Sample Exam Problems M3-27 Section 3.12 Sample Exam Problem Solutions M3-29 Section 3.13 Supplemental Exercises M3-32 Section 3.14 Supplemental Exercise Solutions M3-33

Midterm 1 Interest Rates, Annuities, and Loans MT1-1

Module 4 Bonds Section 4.1 Introduction to Bonds M4-1 Section 4.2 Amortization of Premium or Discount M4-6 Section 4.3 Callable Bonds M4-10 Section 4.4 Pricing Bonds Between Payment Dates M4-14 Section 4.5 Formula Sheet M4-18 Section 4.6 Basic Review Problems M4-19 Section 4.7 Basic Review Problem Solutions M4-20 Section 4.8 Sample Exam Problems M4-21 Section 4.9 Sample Exam Problem Solutions M4-25 Section 4.10 Supplemental Exercises M4-32 Section 4.11 Supplemental Exercise Solutions M4-33

Table of Contents Page TOC-


3

Topic Page Module 5 Yield Rate of an Investment Section 5.1 Internal Rate of Return (IRR)

M5-1

Section 5.2 Time-Weighted and Dollar-Weighted Rates of Return M5-8 Section 5.3 Net Present Value M5-13 Section 5.4 Formula Sheet M5-15 Section 5.5 Basic Review Problems M5-16 Section 5.6 Basic Review Problem Solutions M5-17 Section 5.7 Sample Exam Problems M5-19 Section 5.8 Sample Exam Problem Solutions M5-22 Section 5.9 Supplemental Exercises M5-27 Section 5.10 Supplemental Exercise Solutions M5-29 Module 6 The Term Structure of Interest Rates Section 6.1 Spot Rates and the Yield Curve M6-1 Section 6.2 Forward Rates M6-9 Section 6.3 Formula Sheet M6-13 Section 6.4 Basic Review Problems M6-14 Section 6.5 Basic Review Problem Solutions M6-15 Section 6.6 Sample Exam Problems M6-18 Section 6.7 Sample Exam Problem Solutions M6-20 Section 6.8 Supplemental Exercises M6-22 Section 6.9 Supplemental Exercise Solutions M6-23 Midterm 2 Bonds, Yield Rate, and The Term Structure of Interest Rates MT2-1 Module 7 Asset-Liability Management Section 7.1 Basic Asset-Liability Management:

Matching Asset & Liability Cash Flows M7-1 Section 7.2 Duration M7-4 Section 7.3 Modified Duration M7-7 Section 7.4 Helpful Formulas for Duration Calculations M7-11 Section 7.5 Using Duration to Approximate Change in Price M7-13 Section 7.6 Approximations Using Duration and Convexity M7-24 Section 7.7 The Duration of a Portfolio M7-29 Section 7.8 Immunization M7-32 Section 7.9 Stocks and Other Investments M7-38 Section 7.10 Formula Sheet M7-43 Section 7.11 Basic Review Problems M7-46 Section 7.12 Basic Review Problem Solutions M7-48 Section 7.13 Sample Exam Problems M7-52 Section 7.14 Sample Exam Problem Solutions M7-57 Section 7.15 Supplemental Exercises M7-64 Section 7.16 Supplemental Exercise Solutions M7-66

Page TOC-4 Table of Contents


Topic Page

Module 8 Determinants of Interest Rates Section 8.1 Background M8-1 Section 8.2 Components of the Interest Rate M8-5 Section 8.3 Retail Savings and Lending Interest Rates M8-18 Section 8.4 Bond Yields M8-22 Section 8.5 The Role of Central Banks M8-34 Section 8.6 Formula Sheet M8-37 Section 8.7 Basic Review Problems M8-38 Section 8.8 Basic Review Problem Solutions M8-40 Section 8.9 Sample Exam Problems M8-43 Section 8.10 Sample Exam Problem Solutions M8-45 Section 8.11 Supplemental Exercises M8-47 Section 8.12 Supplemental Exercise Solutions M8-50

Index Index-1

Volume II

Module 9 Interest Rate Swaps Section 9.1 Introduction to Derivative Securities M9-1 Section 9.2 Variable-Rate Loans M9-2 Section 9.3 Example of an Interest Rate Swap M9-4 Section 9.4 Interest Rate Swap Terminology M9-8 Section 9.5 Calculating the Swap Rate M9-13 Section 9.6 Simplified Formulas for the Swap Rate M9-20 Section 9.7 Market Value of a Swap M9-25 Section 9.8 Formula Sheet M9-30 Section 9.9 Basic Review Problems M9-32 Section 9.10 Basic Review Problem Solutions M9-33 Section 9.11 Sample Exam Problems M9-37 Section 9.12 Sample Exam Problem Solutions M9-42 Section 9.13 Supplemental Exercises M9-48 Section 9.14 Supplemental Exercise Solutions M9-50

Midterm 3 Asset-Liability Mgmt., Determinants of Int. Rates, and Interest Rate Swaps MT3-1

Practice Exams PE-1

Appendix The Texas Instruments BA II Plus Calculator Section Appx-1 Calculator Settings Appx-1 Section Appx-2 Calculator Features Appx-3 Section Appx-3 Time Value of Money (TVM) Functions Appx-5 Section Appx-4 The Cash Flow Worksheet Appx-9 Section Appx-5 The Bond Worksheet Appx-11Section Appx-6 The Interest Conversion (ICONV) Worksheet Appx-13

Index Index-1

Introduction Page Intro - 1


As you begin your preparation for the Society of Actuaries’ Exam FM, you should be aware that studying Financial Mathematics (or “interest theory,” as I like to call it) is not a matter of learning mathematics. Instead, financial mathematics involves applying mathematics to situations that involve financial transactions. This will require you to learn a new language, the language of the financial world, and then to apply your existing math skills to solve problems that are presented in this new language. It is important that you spend adequate time to fully understand the meanings of all the terms that will be introduced in this manual. Nearly all of the problems on Exam FM will be word problems (rather than just formulas), and it is very difficult to solve these problems unless you understand the language that is being used. In this manual, we assume that you have a solid working knowledge of differential and integral calculus and some familiarity with probability. We also assume that you have an excellent knowledge of algebraic methods. Depending on what mathematics courses you have taken (and how recently), you may need to review these topics in order to understand some of the material and work the problems in this manual. Throughout this manual, a large number of the examples and practice problems are solved using the Texas Instruments BA II Plus calculator, which is the financial calculator approved for use on Exam FM. It is essential for you to have a BA II Plus calculator in order to understand the solutions presented here, and also to solve the problems on the actual exam. This calculator is available in a standard model, and also as the “BA II Plus Professional.” The Professional model, which is somewhat more expensive than the standard model, is a bit easier to work with, which could be important when taking a timed exam. At the end of Volume II is an appendix with information about the BA II Plus. This appendix is included to help you learn the calculator’s functions and adjust its settings so that you will be able to solve problems more quickly. Very importantly, the appendix explains that your calculator will be “reset” by the exam staff when you check in to take Exam FM, and provides instructions for returning the calculator to the settings you prefer. Reading this material will help you avoid having calculator difficulties on the day of your exam.

IInnttrroodduuccttiioonn

Page Intro - 2 Introduction


Over the years, most actuarial students have found that the best way to prepare for Exam FM is to work a very large number of problems (hundreds and hundreds of problems). There are many examples, exercises, problems, and practice exams included in this manual. Many more problems can be found on the Society of Actuaries website (www.soa.org) or by searching the Web. You should plan to spend a significant proportion of your study time working problems and reviewing the solutions that are provided in this manual and on the websites. Financial mathematics is an integral part of an actuary’s skill set, and you can expect to apply interest theory regularly throughout your career. Mastering the topics covered in this manual will provide you a valuable tool for understanding financial and economic matters both on and off the job. Best of luck to you in learning Financial Mathematics and passing Exam FM! John Dinius June 2019

Module 1 – Interest Rates and the Time Value of Money Page M1 –


1

II nn tt ee rr ee ss tt RR aa tt ee ss aa nn dd tt hh ee TT ii mm ee VV aa ll uu ee oo ff MM oo nn ee yy

Section 1.1 Time Value of Money Interest theory deals with the time value of money. For example, a dollar invested today at 6% interest per year will be worth $1.06 one year from today. Because a dollar invested today can provide more than one dollar a year from now, it follows that receiving a dollar today has a greater value than receiving a dollar one year from now. In other words, money has a “time value,” and in order to assess the value of a payment, we need to know not only the amount of the payment, but also when the payment occurs. That is the underlying principle of interest theory. In the example of the investment at 6% interest, the dollar that is invested today is called the principal, and the $0.06 increase in value is called interest. What happens to the investment after the first year depends on whether it is earning compound interest or simple interest. We illustrate this with an example based on an investment of 100 that earns 6% interest for two years.

a) Compound interest: Interest is earned during each year on the total amount in the account at the beginning of that year. The amounts in the account at the end of Year 1 and Year 2 are: Year 1: 100 0.06 100 100 1.06 106 Year 2: 2106 0.06 106 106 1.06 100 1.06 112.36 Interest is “compounded” at the end of Year 1. That is, the interest earned during Year 1 is “converted” to principal at the end of Year 1 and it becomes part of the principal that earns interest during Year 2.

Module 11

Page M1 – 2 Module 1 – Interest Rates and the Time Value of Money


b) Simple interest: In each year interest is earned on only the original principal of 100. The amounts in the account at the end of Year 1 and Year 2 are: Year 1: 100 0.06 100 100 1.06 106 Year 2: 106 0.06 100 100 1 2 0.06 112

Because the interest earned during the first year is not converted to principal (not “compounded”) at the end of the first year, it does not earn interest during the second year. The principal is 100 in both years, and the amount of interest earned in each year is 6.

Simple interest is generally used only for shorter-term investments (usually less than one year). Compound interest is the most widely used method of computing interest, especially for multi-period investments. Because it is so widely used, we will begin our study of interest theory with compound interest. Note: In this manual, amounts of money will generally be given without an indication of what currency is being used. You may want to think of these amounts as U.S. or Canadian dollars ($100, in the case of the above example), or you may just treat them as amounts of money with no specific denomination.



3

Section 1.2 Present Value and Future Value The value of an investment today (time 0) is its present value (PV); its value n periods from today is called its future value (FV) as of time n. More broadly, if we know the value of an investment as of a particular date and we want to find its value as of an earlier date, we are calculating a present value as of the earlier date. And if we want to find the value as of a later date, then we are calculating a future value (or an accumulated value) as of that later date. If funds are invested at a compound interest rate of i per period for n periods, the basic relationships are: (1.1) Example (1.2)

Let n = 10 and i = 0.06.

a) If PV = 1,000, then 101,000 1.06 1,790.85FV

b) If FV = 1,000, then 10

1,000558.39

1.06PV

Calculation a) demonstrates that if we invest 1,000 today at 6% interest, in 10 years it will have accumulated to a future value of 1,790.85. Calculation b) shows that if we need 1,000 ten years from now, we can accumulate that amount by investing 558.39 now at 6% interest.

Exercise (1.3)

Using an interest rate of 5% compounded annually, find a) the present value (today) of 20,000 payable in 15 years, and b) the future value 6 years from today of 5,000 deposited today.

Answers: a) 9,620.34 b) 6,700.48

1n

FV PV i (1 )n

FVPV

i



Calculator Note

The BA II Plus calculator has 5 “Time Value of Money” keys:

N Number of periods

I/Y Interest rate per period (usually per year)

PV Present Value

PMT Periodic Payment

FV Future Value

These keys are used for performing calculations in the Time Value of Money (TVM) Worksheet. In this module we will not look at any problems that involve periodic payments. The PMT key will be used beginning in Module 2. Using the other four keys, we can 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 entries from prior problems. Note that the legend “CLR TVM” appears above the FV key on the BA II Plus calculator. To clear the TVM registers use the keystrokes 2ND CLR TVM . This sets all 5 of the TVM values to 0. Before we do a calculation, we must choose a sign convention that applies to the values we enter into the calculator (as well as the answer we calculate). In this manual, we will use the following convention:

Money that you receive is positive; money that you pay out is negative.

Thus, if you put 1,000 into an account now, you should enter 1,000 into the calculator for PV to indicate that this amount is being paid out. (You can make an entry negative by pressing the +|- key.)

In Example (1.4), we will rework Example (1.2) using the TVM worksheet.



5

Example (1.4)

To find the future value of 1,000 in 10 years at 6% compound interest with the BA II Plus, use the following keystrokes:

10 N 6 I/Y 1,000 +|- PV CPT FV

You will see in the display: FV = 1,790.85 The answer is positive, since this is money that you will receive. Note that the interest rate is 6% 0.06 , but it is entered into the calculator as 6, not 0.06. The calculator treats your entry as a percentage; that is, it divides the amount entered by 100 when it performs the calculation. To find the present value at 6% compound interest of 1,000 to be received 10 years in the future, use the following keystrokes:

10 N 6 I/Y 1000 FV CPT PV

You will see in the display: PV = −558.39 The answer is negative; this is money that you must put into the account now in order to receive 1,000 at the end of 10 years. Note: You may have noticed that there is an asterisk (*) above the value −558.39 in your calculator’s display. The asterisk indicates that −558.39 is a computed value, not a number that was entered. As long as the inputs (the values for N, I/Y, PMT, and FV) are not changed, the asterisk will continue to appear over the calculated value of −558.39, even if you perform other calculations and then press RCL PV. But if one of the inputs (such as N or FV) is changed, then when you press RCL PV, the −558.39 will appear without the asterisk, indicating that it is not a computed value based on the current entries in the TVM variables.

Exercise (1.5)

Rework Exercise (1.3) using the calculator’s TVM worksheet.

Note: Unlike other worksheets in the BA II Plus calculator, you do not need to press a special key to activate the TVM worksheet. You can enter a value in one of the TVM registers at any time, even when another worksheet is active. However, you cannot compute (CPT) an answer in the TVM worksheet while another worksheet is active. (Exit the other worksheet by pressing 2ND QUIT or CE|C ; then compute the answer.)



Section 1.3 Functions of Investment Growth An investor might wish to plot the growth of an investment over time. Two functions are commonly used:

The accumulation function, a(t), is the value at time t of an initial investment of 1 made at time 0. The amount function, A(t), is the value at time t of an initial investment of A(0) made at time 0.

For compound interest at a constant annual rate i, these functions are: (1.6)

For compound interest at a rate of i = 60% per year, the values of a(t) at the end of each of the first 4 years are as shown in the following table:

t 0 1 2 3 4 a(t) 1 1.60 2.56 4.096 6.5536

Note: An extremely high interest rate (60%) is used here so that the following graphs will clearly show the exponential (and non-linear) form of the accumulation function. What is the value of this a(t) function when t is not an integer? For example, what is the value of a(t) for t = 1.5? There can be instances where interest is considered to be earned only at the end of each year. In that case, the value of the accumulation function at t = 1.5 would be the same as at t = 1, that is, it would be 1.60. At the end of the second year, all of the interest for the period from t = 1 to t = 2 would be credited, and the accumulated value would increase instantaneously from 1.60 to 2.56.

Compound interest:

( ) (1 )ta t i ( ) (0) 1t

A t A i



7

Thus, if interest is considered to be earned only at the end of each year, the graph of a(t) is a step function:

t Real-world contracts that involve interest should specify how interest for partial periods will be calculated. For our purposes in studying interest theory, we will generally assume that interest is earned continuously. When interest is

earned continuously, the formulas ( ) (1 )ta t i and ( ) (0) 1t

A t A i are valid

for all values of t, not just integer values, and the accumulation and amount functions are continuous functions (not step functions). Unless an exam question specifies that interest is credited only at the end of each year (or the end of each month or each quarter, etc.), you should assume that interest is earned continuously.

In the current example, the value of the accumulation function at time 1.5 is: 1.5(1.5) (1.60) 2.0239a

If interest is earned continuously, the graph of a(t) is a smooth, continuous function:

t

a(t)

a(t)



For simple interest at a constant annual rate i, the accumulation and amount functions are: (1.7)

For simple interest at a rate i = 60% per year, the values of a(t) at the end of each of the first 4 years are:

t 0 1 2 3 4 a(t) 1 1.60 2.20 2.80 3.40

Again, a(t) can be a step function if interest is considered to be earned only at the end of each year. However, as with compound interest, we will generally treat simple interest as being earned continuously, so that a(t) is a continuous function. The following graph includes plots for both the step function and the continuous function.

t

Simple interest: ( ) 1a t i t ( ) (0) 1A t A i t

a(t) earned continuously

earned at end of each year



9

The following graph compares the growth of 2 investments. In each case, an amount of 1 is invested at time 0. Each investment earns interest continuously. One earns simple interest at a 60% annual rate; the other earns compound interest at a 60% annual rate.

t

Both investments have a value of 1.60 at the end of 1 year. After the first year, the investment earning compound interest grows much faster, as it earns “interest on interest.” The simple-interest investment earns interest on only the original principal of 1, so its rate of growth (its slope) is constant at 0.60 per year. Note, however, that the investment at simple interest has a larger value between time 0 and time 1 than the investment earning compound interest,

since 1 1t

t i i for values of t between 0 and 1.

The simple-interest investment is growing faster (has a steeper slope) than the compound-interest investment at the beginning of the first year. But the compound-interest investment grows faster and faster as the year progresses (because it earns interest on a larger and larger principal). At the end of the first year, the compound-interest investment has the same value as the simple-interest investment. At all times after the first year, the compound interest investment has a larger value than the simple interest investment.

a(t) Compound

Interest

Simple Interest



Section 1.4 Effective Rate of Interest We will now use the amount function to define the effective rate of interest for any specified time period. For the one-year time period , 1t t , the beginning

and ending amounts are ( )A t and ( 1)A t . The amount of interest earned over

the interval is 1A t A t . The effective rate of interest for the one-year

period from t to t+1 is: (1.8)

Note: The notation 1 2,t ti will be used in this module to represent an effective

interest rate for the period 1 2,t t . This is not standard actuarial notation, and it

will not be used in the other modules of this manual (or on Exam FM). Example (1.9)

Let the interest rate be 6% and the time interval be [1,2].

For compound interest:

1,2

2 1 1.1236 1.060.06

1 1.06

a ai

a

For simple interest:

1,2

2 1 1.12 1.060.0566

1 1.06

a ai

a

Exercise (1.10)

Let the interest rate be 6% and the time interval be [2,3]. Find 2,3i for a) compound interest at 6%, and b) simple interest at 6%. Answers: a) 0.06 b) 0.0536

Note that over multi-year periods a compound interest rate of 6% per year gives a constant effective rate of 6% for each one-year period, while a simple interest rate of 6% leads to declining effective rates over time. This is because the investment at compound interest always earns 6% on the entire beginning-of-year balance, but the investment at simple interest earns 6% on only the original principal.

t tt t

it

, 1interest earned between and +1

value of investment at time

A t A t a t a t

A t a t

1 1



11

The above discussion involves effective rates over a 1-year period of time. These are called annual effective rates. An annual effective rate is the most common way to express a rate of compound interest, but we can also calculate an effective rate for a shorter or longer time period (e.g., a quarterly effective rate, or a 2-year effective rate). In each case, the effective rate equals the amount of interest earned during the period (e.g., a 3-month period or a 2-year period), divided by the value of the investment at the beginning of that period. Example (1.11)

An investment earns a 6% annual interest rate. We will calculate the quarterly (3-month) effective rates for the periods [0.25,0.50] and [1.25,1.50]. We will do each of these calculations based on a 6% rate of compound interest, and also for a 6% rate of simple interest.

At 6% compound interest

For the period [0.25,0.50]:

0.5 0.25

[0.25,0.50] 0.25

0.50 0.25 1.06 1.060.01467

0.25 1.06

a ai

a


1.50 1.25

[1.25,1.50] 1.25

1.50 1.25 1.06 1.060.01467

1.25 1.06

a ai

a

At 6% simple interest


[0.25,0.50]

1 0.5 0.06 1 0.25 0.060.50 0.250.01478

0.25 1 0.25 0.06

a ai

a


[1.25,1.50]

1 1.50 0.06 1 1.25 0.061.50 1.250.01395

1.25 1 1.25 0.06

a ai

a

During the period [0.25,0.50], 6% simple interest generated a higher quarterly effective rate than 6% compound interest. As we noted previously, early in the first year, simple interest produces faster growth than compound interest at the same numerical interest rate (but compound interest catches up at the end of the first year). During the period [1.25,1.50], of course, compound interest produces a higher effective rate than simple interest. The quarterly effective rate for compound interest during this period is 0.01467, the same as it was for the period [0.25,0.50]. But the quarterly effective rate for simple interest has decreased from 0.01478 to 0.01395.



Exercise (1.12)

An investment earns a 6% annual interest rate. Calculate the quarterly (3-month) effective rates for the periods [0.50,0.75] and [1.50,1.75]:

a) at 6% compound interest, and b) at 6% simple interest

Are the rates calculated in a) (at compound interest) higher or lower than the rates calculated in b) (at simple interest)?

Answer: a) 0.01467 for each period b) 0.01456 and 0.01376; Compound interest produces a higher effective rate in each period.



13

Section 1.5 Nominal Rate of Interest In many instances where payments (such as loan payments) are made more frequently than once a year (e.g., monthly, quarterly, or semi-annually), the interest rate is expressed as a nominal annual rate. A nominal annual rate of interest is equal to the effective interest rate per period multiplied by the number of periods per year. For example, if an investment is earning interest at a 2% quarterly effective rate, you could multiply 2% by 4 and refer to this as a “nominal annual rate of 8%, convertible quarterly” (or compounded quarterly). This gives us a simple way of referring to the interest rate on an annual scale, but 8% is not the rate actually earned each year. In this example, the investment earns more than 8%. One dollar accumulates to (1.02)4 = 1.0824 in one year, so a nominal annual rate of 8% convertible quarterly is equivalent to an annual effective rate of 8.24%. Many students find this confusing, so we will go over it again for clarification: 1. The effective rate per period is your starting point. Example: 2% per quarter (a 2% “quarterly effective rate”) 2. Calculate the nominal annual rate.

Nominal Rate = (effective rate/period) × (number of periods per year) Example: 2% × 4 = 8% (an 8% “nominal annual rate convertible quarterly”)

3. The annual effective rate is the annual rate that the investment

actually earns with compounding of interest. Example: End-of-year accumulated value is 41.02 1.0824 . The annual effective rate is 8.24%.

A nominal rate is an artificial rate that provides a way of talking about a periodic rate (such as a quarterly or monthly effective rate) in familiar annual terms. The annual effective rate is not artificial. It is the rate that is actually earned in a year. Similarly, a quarterly effective rate is the rate actually earned in a quarter. It is important to understand that interest calculations are always done using effective rates (whether annual, quarterly, monthly, etc.). Nominal rates are not used in calculations; a nominal rate must first be converted to an effective rate, and then calculations are performed using the effective rate.



Exercise (1.13)

Suppose that interest is earned at a rate of 1% per month, compounded monthly (i.e., a 1% monthly effective rate). a) What is the nominal annual rate? b) What is the annual effective rate?

Answers: a) 12% convertible monthly b) 12.6825%

In the general case of m conversion periods per year, we denote the nominal

annual rate by i(m). The effective interest rate per period is ( )mim

, and the annual

effective rate is: (1.14) This has the important consequence that: (1.15) You will often see the statement that interest is “convertible” or “compounded” m times per year. This means that the interest earned during each period (of length 1 / m years) is “compounded” (converted to principal) at the end of that period and earns interest during the following period. Example (1.16)

Suppose interest is convertible monthly and the nominal rate is (12) 0.09i . Then the annual effective rate is:

12120.09

1 1 1.0075 1 0.093812

(or 9.38%)

This process can easily be reversed to find the nominal rate if we are given the effective rate.

( )

1 1mmi

im

( )

1 1mmi

im



15

Example (1.17)

Interest is convertible semi-annually and results in an annual effective rate of 10.25%. Find the nominal annual rate convertible semi-annually.

Solution.

m = 2, so we need to find 2i . By (1.15):

2(2)

1 1.10252

i

(2)

1 1.1025 1.052

i

(2) 2 1.05 1 10%i

Thus the nominal annual rate is 10% convertible semi-annually, and the semi-annual effective rate is 5%.

Note that you can derive a formula that solves for mi given i and m. It is:

1

1 1m mi m i

It is not necessary to memorize this formula. Formula (1.15) is intuitive and easy to remember, and we can always substitute the given values of i and m

into (1.15) to solve for mi . This is the approach we used in Example (1.17).



Calculator Note

The BA II Plus calculator has an Interest Conversion Worksheet that can be used to convert between nominal and effective interest rates. The legend above the 2 key is ICONV, which stands for “interest conversion.” You can activate this worksheet by using the keystrokes 2ND ICONV. The worksheet has three variables:

NOM for nominal annual rate EFF for annual effective rate C/Y for number of conversion periods per year

You can scroll among these variables using the and keys on the top row of your calculator. In Example (1.16) we found the effective rate corresponding to a nominal annual rate of 9% convertible 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 press the ENTER key. Then scroll to the line for C/Y, key in 12, and press the ENTER key. Then scroll to the line for EFF and use the CPT key to compute the effective rate. The rate displayed is EFF = 9.38 (to two decimal places). This means 9.38%, so the rate is 0.0938. In Example (1.17) we found the nominal rate corresponding to an effective rate of 10.25% convertible semi-annually. 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 press the ENTER key. Then scroll to the line for C/Y, key in 2, and press the ENTER key. Then scroll to the line for NOM and press the CPT key to compute the effective rate. The rate displayed is NOM = 10 (that is, 10%, or 0.10). The ICONV worksheet can be used to calculate EFF or NOM, but not C/Y (the number of conversion periods per year). To exit the ICONV worksheet, press either CE|C or 2ND QUIT. These keys allow you to exit any of the BAII Plus worksheets. (Note: Sometimes you will have to press CE|C more than once to exit a worksheet.)

Exercise (1.18)

a) Given i(12) = 6%, find the annual effective rate i. b) Given an annual effective rate of i = 5%, find i(12).

Answers: a) 6.168% b) 4.889%



17

Section 1.6 Rate of Discount

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 will receive a payment of 1,060 at the end of the year.

b) Target a given sum at the end of the year, and “discount” that amount to determine how much to invest. Suppose that you want to have 1,000 at the end of the year. The present value of 1,000 at 6% interest is

1,000 / 1.06 943.40 . In b), where you invested 943.40 and received 1,000, the difference of 56.60 is referred to as the amount of discount, and 56.60 / 1,000 0.0566 is the rate of discount. This rate equals the amount of interest that will be earned during the year (56.60), divided by the end-of-year value (1,000). (As we already know, dividing the amount of interest earned by the beginning-of-year value produces the annual effective rate of interest: 56.60 / 943.40 0.06 .) Although 56.60 is described as the amount of discount, it is also the amount of interest earned during the year. If we view 56.60 as an amount subtracted from the ending balance of 1,000 to find the beginning balance, we call it discount. If we view 56.60 as an amount added to the beginning balance of 993.40 to produce the ending balance, we call it interest. The rate of discount is used extensively in interest theory and actuarial mathematics. We will derive an expression for the rate of discount in terms of i. If you wish to invest at an annual effective rate i and obtain a future value of 1

one year later, the present value you need to invest is: 1

1PV

i

. The amount of

interest earned during the year is 1

1 11

PVi

. The annual effective rate of

discount, d, is defined as the amount of interest earned during the year (which

is 1

11 i

), divided by the ending balance (which is 1), so we have:

(1.19)

11

1d

i



From this definition of the rate of discount, we can use algebra to develop the key relationship: (1.20)

The annual effective rate of discount, d, can also be expressed in terms of the amount function, A t( ) , or the accumulation function, a t( ) . In each case, d is equal to the amount of interest earned during a one-year period, divided by the value of the investment at the end of that year. (1.21) Note: We can also define the effective rate of discount for periods other than one year. For example, the monthly effective rate of discount for a particular month equals the amount of interest earned during that month, divided by the balance at the end of the month. When t = 0 and A(0) = 1, Equation (1.21) is identical to Formula (1.20), because the amount of interest earned is i and the end-of-year balance is (1 )i , so d equals i divided by (1 )i . The annual effective interest rate, of course, is the amount of interest earned ( i ), divided by the beginning-of-year balance (1). Example (1.22)

For i = 0.06, d = 0.061.06

= 0.0566

Exercise (1.23)

Given 0.10i , find d. Answer: 0.0909

From Formula (1.20), we can derive an expression for i in terms of d:

11 1

i dd d di i d i d i

i d

(1.24)

1i

di

interest earned between and +1value of investment at time 1

t td

t

1 1

1 1

A t A t a t a t

A t a t

1

d

id



19

Example (1.25)

For d = 0.06, 0.06

0.06381 1 0.06

d

id

Exercise (1.26)

Given 0.10d , find i. Answer: 0.1111

You should be aware that the word “discount” is used in various ways besides those described in this section. It is important to make sure you understand how the word is being used each time it appears. For example, the verb form, “to discount,” typically means “to calculate a present value.” The phrase “discount at a rate of 5%” likely means “calculate a present value using a 5% rate of interest” (not a 5% rate of discount). Even the phrase “discount rate” frequently refers to a rate of interest at which discounting (present valuing) is to be done. Always read each exam question carefully to make sure you know whether it is referring to an interest rate or a rate of discount.



Section 1.7 Present Value Factor Another variable that is used in actuarial interest problems is the present value factor, v:

(1.27) The variable v equals the value at the beginning of the year of 1 unit payable at the end of the year. From the definition of d in (1.19), we see that: (1.28)

And from Formula (1.20), we can derive the following important relation: (1.29) The difference i − d simplifies nicely:

i − d = i − (1 )i

i =

2

(1 )i

i = id

(1.30)

The preceding relationships are often useful in solving exam problems.

Example (1.31)

Given 0.07d , find v and i.

Solution.

1 0.93v d iv1 1

1 1.07530.93

, so 0.0753i or 7.53%

Exercise (1.32)

Given 0.05d , find v and i. Answers: v = 0.95 i = 0.0526

Note that we can now write:

1n

n

FVPV v FV

i

The use of v as a present value factor is common in actuarial texts and is essential for actuarial exams. Many other financial professions do not use v.

1

1v

i

1d v and 1v d

d i v

i d i d



21

Section 1.8 Nominal Rate of Discount A rate of discount can be quoted as a nominal annual rate. For example, if the quarterly effective rate of discount is 2%, we can say that the nominal annual rate of discount is 8% convertible quarterly. The annual effective rate of discount would not be 8%, as we shall see below.

The nominal rate of discount convertible m-thly is denoted by md . For example, a nominal rate of discount convertible quarterly would be written as (4)d . It is related to the present value factor v and the annual effective rate of discount d by the equation: (1.33) Example (1.34)

Find the annual effective rate of discount that is equivalent to a nominal annual rate of discount of 8% convertible quarterly.

Solution.

4

40.081 1 0.98 0.9224

4d

1 0.9224 0.0776d

Example (1.35)

Find the nominal annual rate of discount convertible semi-annually that corresponds to an annual effective rate of discount of 6%.

Solution.

22

1 1 1 0.06 0.942

dd

2

1 0.94 0.9695362

d

2 2 1 0.969536 0.060928d

mmdv d

m1 1



Calculator Note

In the ICONV worksheet, if you enter either EFF or NOM as a negative number, the BA II Plus will interpret the negative number as a rate of discount and solve for the corresponding (NOM or EFF) rate of discount, which will also be displayed as a negative number. In Example (1.34) we found the annual effective rate of discount that is equivalent to a nominal rate of discount of 8% convertible quarterly. To do this using the BA II Plus calculator, open the ICONV worksheet and scroll to the line for NOM. Key in 8 +|- and press the ENTER key. Then scroll to the line for C/Y, key in 4, and press the ENTER key. Then scroll to the line for EFF and use the CPT key to compute the effective rate. The calculator displays EFF = –7.76 (to two decimal places). That value (7.76%) is the same rate of discount that was calculated in Example (1.34). You can clear the computed values in the ICONV worksheet by keying in 2ND CLR WORK (2nd function of the CE/C key). The value of C/Y will remain unchanged until you enter a new value, but NOM and EFF will be set to 0.

Exercise (1.36)

Find: a) the annual effective rate of discount that is equivalent to a nominal

rate of discount of 7.5% convertible every 4 months (m=3), and b) the nominal annual rate of discount convertible monthly that is

equivalent to an annual effective rate of discount of 6%

Answers: a) 7.31% b) 6.17%

Some problems may require conversion of a nominal interest rate convertible m times per year to an equivalent nominal rate of discount convertible p times per year. The equation for this type of problem is:

1 1

m pm pi dm p

The left-hand side of this equation is equivalent to 1 i , and the right-hand side

is equivalent to 1

1 iv

. Notice that the exponent in the right-hand expression

is –p, not p.



23

Example (1.37)

Find the rate of discount convertible semi-annually that is equivalent to a nominal rate of interest of 8% convertible monthly.

Solution.

212 20.081 1.0830 1

12 2d

12

1.083 1.0407 12

d

2 12 1 1.0407 0.0782d

Example (1.38)

Find the rate of discount convertible quarterly that is equivalent to a nominal rate of interest of 6% convertible monthly.

Answer: 5.94%

As was previously mentioned for nominal rates of interest, you should never perform calculations using nominal rates of discount. Always convert a nominal rate to an effective rate and use the effective rate of discount in your calculations.



Section 1.9 Continuous Compounding and the Force of Interest We have already noted that unless an exam problem states otherwise, compound interest should be regarded as being earned continuously throughout the year, so that the accumulation function, a(t), is a continuous exponential function of t. In this section, we will consider the case where interest is compounded continuously. That is, interest is converted to principal as soon as it is earned, and immediately begins earning interest. As we will see, this is equivalent to earning a nominal annual rate of interest i(m) where m is infinite. If an investment earns compound interest at an annual effective rate of 8%, its value increases annually by a factor of 1.08. However, if it earns interest at a nominal annual rate of 8% convertible semi-annually, then it is earning a semi-annual effective rate of 4%, and its value increases annually by a factor of

21.0816 1.04 . This is equivalent to an annual effective interest rate of 8.16%.

The following table shows the equivalent annual effective rates earned by money invested at a nominal annual rate of 8%, but with different conversion periods that range from semi-annual to monthly to hourly. For a nominal annual rate of 8% with m conversion periods per year, the equivalent annual effective

rate is:

0.081 1

m

m

Nominal Rate: 8%

Conversion Period m Ann. Eff. Rate Semi-annual 2 8.16000%

Quarterly 4 8.24322% Monthly 12 8.29995%

Daily 365 8.32776% Hourly 8,760 8.32867%

Note that a higher frequency of conversion (a larger value of m) produces a higher equivalent annual effective interest rate. Now consider the situation mentioned in the first paragraph above, where interest is convertible to principal continuously as it is earned. In that case,

m is infinite, and the annual effective interest rate is

0.08lim 1 1

m

m m. This is

equal to 0.08 – 1e , which is approximately 0.0832871, so the equivalent annual effective interest rate is 8.32871%. Note that the result of compounding interest every hour (as shown in the above table) matches the result of continuous compounding to 5 significant digits.



25

When interest is compounded continuously, we call the interest rate a “force of interest.” The force of interest is the rate per year at which the investment is earning interest, expressed as a percentage of the current value of the investment. In the preceding example, the force of interest was 8%, so if the investment has a value of 100 at time 0, then it is earning interest at a rate of 8.00 per year at time 0. But as soon as interest is earned, that interest is added to the principal, and the increased principal earns interest at a rate of 8% per year. (For example, after 1 day, the principal equals 100.02 and is earning interest at a rate of 8.0016 per year. On the last day of the year, the principal has grown to 108.305 and is earning interest at a rate of 8.6644 per year.)

Based on a force of interest of 0.08, at the end of one year, 8.3287 of interest will have been earned on a beginning-of-year investment of 100. By definition, the annual effective interest rate for the year is 8.3287%. You might want to think of “8.3287 per year” as the average rate at which the investment’s value increased during the year. At the beginning of the year, interest was accumulating at a slower rate (8.00 per year); at the end of the year, it accumulated at a faster rate (8.6644 per year). But on average the investment earned interest at a rate equal to 8.3287% of the beginning-of-year principal. Since 8.3287% is the annual effective interest rate for this investment, we see that the annual effective interest rate is really an average rate at which interest is accumulating during the year, expressed as a percentage of the beginning-of-year balance. By contrast, the force of interest is the instantaneous rate at which interest is being earned, expressed as a percentage of the current balance at any given moment.

The Greek letter delta ( ) is used to represent the force of interest. For a constant force of interest , the equivalent annual effective rate, i, can be calculated using the following formula: (1.39)

Note: This relationship between i and is derived using calculus on page M1-28.

Remember, continuous compounding is not a different type of compound interest. The accumulation and amount functions are still exponential. The force of interest, , is simply a different way to describe the rate at which an investment is increasing with compound interest. We are able to use the above formula to translate rates expressed as a force of interest to the equivalent annual effective rate (or vice-versa), just as we are able to translate a nominal rate into the equivalent annual effective rate. For a constant force of interest and the equivalent annual effective interest rate i, we have the following relationships:

(1.40) (1.41)

lim 1 1 1m

mi e

m

1 i e ln 1 i

(1 )n ni e 1nn nv i e



The preceding discussion assumes a constant force of interest. That is, we have assumed that is a constant. However, there are also situations where the force of interest varies over time. In the case of a varying force of interest, is a function of t and is expressed as ( )t or t , the instantaneous force of interest at time t, which is defined as: (1.42) We can analyze this formula as follows: The accumulation function, a(t), is the value of an investment at time t (based on an investment of 1 at time 0). Its derivative, a'(t), is the rate at which that investment is earning interest,

expressed as a rate per year. The ratio of these two values, ( )( )

a ta t

, is the rate

(per year) at which the investment is earning interest at time t, expressed as a percentage of a(t). This is the definition of a force of interest. And because it is measured at time t and can change as a function of time, ( )t or t is called the instantaneous force of interest at time t. For the accumulation function with a constant force of interest (that is, when ta t e ), definition (1.42) yields:

/t t

t t

de dt et

e e

Thus t is equal to for all values of t, so definition (1.42) is valid for a

constant force of interest, as well as for a varying force of interest. The following example involves a varying force of interest: Example (1.43)

Let 2

1a t t .

(Note: This is an unrealistic accumulation function, but it is easy to analyze.)

Then

2

2 2

1 / 2 1'( ) 2( ) 11 1

d t dt ta tt

a t tt t

( )( )

( )a t

ta t



27

There is another useful relationship that enables us to create an expression for a(t) if only t is given:

(1.44)

Thus: 0 0( ) ln ln ln 0 ln ln 1 ln[ ( )]

kkt dt a t a k a a k a k

This implies that:

(1.45) This formula for the accumulation function is important for solving problems that involve a varying force of interest. Example (1.46)

Given 2

( )1

tt

, find an expression for a(t).

Solution. To find ( )a t , we first need to integrate t :

0 0 0

2( ) 2 ln( 1) 2 ln( 1)

1

tt tu du du u t

u

Then we can write:

2 22 ln( 1) ln( 1)( ) 1t ta t e e t

Note: If your calculus is rusty, you may need to review calculus before doing these problems.

Exercise (1.47)

Given 6

( )2 1

tt

, find a(t).

Answer: 33 ln(2 1)( ) 2 1

ta t e t

In the above examples, we have developed expressions for the accumulation function, ( )a t , when the force of interest varies continuously. This produces the t-year accumulation factor for an investment made at time 0. However, sometimes we need an accumulation factor for a different period, such as the period from time t1 to time t2.

( )ln[ ( )]

( )d a t

a t tdt a t

0( ) ln ( )

tu du a te e a t



The required factor in this case is equal to 2 1( ) / ( )a t a t , so we could calculate it by finding 1( )a t and 2( )a t , and then performing a division. However, a more direct method involves integrating ( )t for the period from t1 to t2:

2

1( )2

1

( )( )

u dutta t

ea t

This technique is applied in Example (1.82) on page M1-45. Deriving the Relationship Between and i:

In this section, we have used the relationship ln 1 i . The following

analysis derives that relationship from the definition of . Consider the accumulation function, a(t), for an investment of 1 made at time 0 that earns interest at a rate of per year compounded continuously, i.e., is the force of interest. At any time t, the value of this investment is a(t) and interest is accumulating at a rate of a t . That is, a(t) is growing

continuously at a rate of a t per year, leading to the following equation:

( ) / ( )da t dt a t

We can solve for 1a as follows:

( )( )

da t

dta t

1 1

0 0

( )( )

t t

da tdt

a t

1 1

00ln ( )a t t

ln (1) ln (0) (1 0) a a

ln (1) ln 1 a

ln (1) 0 a

(1) a e

Since (1) 1a i , we have:

1 1 ln(1 )i e i e i

These are the key relationships between a constant force of interest and the equivalent annual effective interest rate i.

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