Actex_2fm-act-18fsmp_sample_5-24-18.pdf

ACTEX

SOA Exam FM Study Manual

With StudyPlus +

StudyPlus + gives you digital access* to: • Flashcards & Formula Sheet • Actuarial Exam & Career Strategy Guides • Technical Skill eLearning Tools • Samples of Supplemental Texts & Study Tools • And more! *See inside for keycode access and login instructions

Fall 2018 Edition | Volume I John B. Dinius, FSA | Matthew J. Hassett, Ph.D. Michael I. Ratliff, Ph.D., ASA | Toni Coombs Garcia Amy C. Steeby, MBA, MEd

ACTEX Learning | Learn Today. Lead Tomorrow.

ACTEX

SOA Exam FM Study Manual Fall 2018 Edition John B. Dinius, FSA | Matthew J. Hassett, Ph.D. Michael I. Ratliff, Ph.D., ASA | Toni Coombs Garcia Amy C. Steeby, MBA, MEd

ACTEX Learning New Hartford, Connecticut

ACTEX Learning Learn Today. Lead Tomorrow.

Actuarial & Financial Risk Resource Materials

Since 1972

Copyright © 2018 SRBooks, Inc. ISBN: 978-1-63588-438-8 Printed in the United States of America.

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.

Your Opinion is Important to Us ACTEX is eager to provide you with helpful study material to assist you in gaining the necessary knowledge to become a successful actuary. In turn we would like your help in evaluating our manuals so we can help you meet that end. We invite you to provide us with a critique of this manual by sending this form to us at your convenience. We appreciate your time and value your input.

Publication:

ACTEX FM Study Manual, Fall 2018 Edition I found Actex by: (Check one) A Professor

School/Internship Program

Employer

Friend

Facebook/Twitter

In preparing for my exam I found this manual: (Check one) Good

Very Good

Satisfactory

Unsatisfactory

I found the following helpful:

I found the following problems: (Please be specific as to area, i.e., section, specific item, and/or page number.)

To improve this manual I would:

Name: Address: Phone:

E-mail: (Please provide this information in case clarification is needed.) Send to: Stephen Camilli ACTEX Learning P.O. Box 715 New Hartford, CT 06057

Or visit our website at www.ActexMadRiver.com to complete the survey on-line. Click on the “Send Us Feedback” link to access the online version. You can also e-mail your comments to [email protected].

ACTEX

Exam Prep Online Courses!

Your Path To Exam Success Full time students are eligible for a 50% discount on ACTEX Exam Preparation courses.

Learn from the experts and feel confident on exam day • Comprehensive instructor support, video lectures, and suggested exercises/solutions, supplemental resources • Responsive, student-focused instructors with engaging style and helpful support make preparing for your exam efficient and enjoyable • Complete at your own pace • Weekly timed practice tests with video solutions • Live discussion forums and real-time chat sessions • Emphasizes problem-solving techniques • Comparable to a one-semester, undergraduate-level course • One-on-one instructor support Student Testimonials “I took the P exam today and passed! The course has been extremely effective and I wanted to thank you for all your help!” “The instructor was very knowledgeable about all subjects covered. He was also quick to respond which led to less time waiting for an answer before moving forward.”

“The video lectures were extremely helpful as were the office hours, but the problems and solutions for the book were the most helpful aspects.” “Rich Owens was a great instructor. He was very knowledgeable, helpful, available, and any other positive adjective you could think of.”

Visit www.ActexMadRiver.com for specific course information and required materials.

NOT FOR PRINT: ADD KEYCODE TEMPLATE PAGE HERE

NOT FOR PRINT: ADD KEYCODE TEMPLATE PAGE HERE

Preface 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 for Exam FM and to increase the number of sample problems and practice exams. This latest edition of the ACTEX Study Manual for Exam FM, edited by lead author John Dinius, has been extensively revised and edited to reflect the 2017 and 2018 changes in the SOA’s Exam FM syllabus, and an Appendix has been added with detailed information about the BA II Plus calculator that the student will use for this exam. The 2017 syllabus for Exam FM eliminated most of the material on Financial Derivatives and added sections on “The Determination of Interest Rates” and “Interest Rate Swaps.” The SOA issued new study notes on these two topics, as well as a study note on “Using Duration and Convexity to Approximate Present Value.” This material is covered in Modules 7, 8, and 9 of this manual. The 2018 syllabus, which is effective with the October 2018 administration of the exam, has eliminated reference to “sinking funds,” so sinking funds are no longer covered here. (They were previously included in Module 3.) 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 the student an opportunity to check his/her progress. Also included at the end of the manual are 11 practice exams of 35 problems each. These are intended to provide realistic exam-taking experience to complete the student’s preparation for Exam FM. This manual is designed to address all of the topics on the SOA’s Exam FM syllabus. However, we suggest that you obtain one of the official textbooks for SOA Exam FM and use that text in combination with this manual. The following pages provide recommendations on how to prepare for actuarial exams and suggestions on how to use 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 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.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

On passing exams 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 doing algebraic simplification, 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 is actually a possibility if the very hard problems are the first ones on the exam and you try to solve them first. 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.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

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. 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. 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 have 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. You may want to attempt them in a non-timed environment to evaluate your skills and understanding. b) The final 5 practice exams introduce more difficult questions in order to replicate the actual exam experience. You should take each of these in a timed environment to give yourself experience with exam conditions. 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. ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Table of Contents

Page TOC- 1

C on t e n t s Topic

Page

Volume I Introduction

Intro-1

Module 1

Interest Rates and the Time Value of Money

Section 1.1 Section 1.2 Section 1.3 Section 1.4 Section 1.5 Section 1.6 Section 1.7 Section 1.8 Section 1.9 Section 1.10 Section 1.11 Section 1.12 Section 1.13 Section 1.14 Section 1.15 Section 1.16 Section 1.17 Section 1.18 Section 1.19 Section 1.20

Time Value of Money Present Value and Future Value Functions of Investment Growth Effective Rate of Interest Nominal Rate of Interest Rate of Discount Present Value Factor Nominal Rate of Discount Continuous Compounding and the Force of Interest Quoted Rates for Treasury Bills Relating Discount, Force of Interest, and Interest Rate Solving for PV, FV, n, and i The Rule of 72 Formula Sheet Basic Review Problems Basic Review Problem Solutions Sample Exam Problems Sample Exam Problem Solutions Supplemental Exercises Supplemental Exercise Solutions

Module 2

Annuities

Section 2.1 Section 2.2 Section 2.3 Section 2.4 Section 2.5 Section 2.6 Section 2.7 Section 2.8 Section 2.9 Section 2.10 Section 2.11

Introduction to Annuities Annuity-Immediate Calculations Perpetuities Annuities with Level Payments Other Than 1 Annuity-Due Calculations Continuously Payable Annuities Basic Annuity Problems for Calculator Practice Annuities with Varying Payments Increasing Annuities with Terms in Arithmetic Progression Decreasing Annuities with Terms in Arithmetic Progression A Single Formula for Annuities with Terms in Arithmetic Progression Annuities with Terms in Geometric Progression Equations of Value and Loan Payments

Section 2.12 Section 2.13

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

M1-1 M1-3 M1-6 M1-10 M1-13 M1-17 M1-20 M1-21 M1-24 M1-29 M1-35 M1-37 M1-46 M1-48 M1-49 M1-50 M1-55 M1-60 M1-69 M1-70

M2-1 M2-3 M2-7 M2-8 M2-9 M2-11 M2-14 M2-19 M2-21 M2-24 M2-26 M2-28 M2-35

Exam FM – Financial Mathematics

Page TOC-2

Section 2.14 Section 2.15 Section 2.16 Section 2.17 Section 2.18 Section 2.19 Section 2.20 Section 2.21 Section 2.22 Section 2.23 Section 2.24 Section 2.25 Section 2.26 Section 2.27

Table of Contents

Deferred Annuities Annuities with More Complex Payment Patterns Annuities with Payments More Frequent than Annual Payment Periods that Don’t Match the Interest Conversion Period Continuously Payable Annuities with Continuously Varying Payments Reinvestment Problems Inflation Formula Sheet Basic Review Problems Basic Review Problem Solutions Sample Exam Problems Sample Exam Problem Solutions Supplemental Exercises Supplemental Exercise Solutions

Module 3

Loan Repayment

Section 3.1 Section 3.2 Section 3.3 Section 3.4 Section 3.5 Section 3.6 Section 3.7 Section 3.8 Section 3.9 Section 3.10 Section 3.11 Section 3.12 Section 3.13 Section 3.14

The Amortization Method of Loan Repayment Calculating the Loan Balance Loans with Varying Payments Formulas for Level-Payment Loan Amortization Monthly-Payment Loans An Example with Level Payment of Principal Capitalization of Interest and Negative Amortization Formula Sheet Basic Review Problems Basic Review Problem Solutions Sample Exam Problems Sample Exam Problem Solutions Supplemental Exercises Supplemental Exercise Solutions

M2-37 M2-39 M2-43 M2-48 M2-53 M2-58 M2-60 M2-62 M2-65 M2-67 M2-74 M2-82 M2-98 M2-100

M3-1 M3-4 M3-10 M3-12 M3-14 M3-18 M3-19 M3-21 M3-22 M3-23 M3-26 M3-28 M3-32 M3-33

Midterm 1 Interest Rates, Annuities, and Loans

Module 4

Bonds

Section 4.1 Section 4.2 Section 4.3 Section 4.4 Section 4.5 Section 4.6 Section 4.7 Section 4.8 Section 4.9 Section 4.10 Section 4.11

Introduction to Bonds Amortization of Premium or Discount Callable Bonds Pricing Bonds Between Payment Dates Formula Sheet Basic Review Problems Basic Review Problem Solutions Sample Exam Problems Sample Exam Problem Solutions Supplemental Exercises Supplemental Exercise Solutions

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

MT1-1

M4-1 M4-6 M4-10 M4-14 M4-18 M4-19 M4-20 M4-21 M4-25 M4-32 M4-33

Exam FM – Financial Mathematics

Table of Contents

Page TOC- 3

Module 5

Yield Rate of an Investment

Section 5.1 Section 5.2 Section 5.3 Section 5.4 Section 5.5 Section 5.6 Section 5.7 Section 5.8 Section 5.9 Section 5.10

Internal Rate of Return (IRR) Time-Weighted and Dollar-Weighted Rates of Return Net Present Value Formula Sheet Basic Review Problems Basic Review Problem Solutions Sample Exam Problems Sample Exam Problem Solutions Supplemental Exercises Supplemental Exercise Solutions

Module 6

The Term Structure of Interest Rates

Section 6.1 Section 6.2 Section 6.3 Section 6.4 Section 6.5 Section 6.6 Section 6.7 Section 6.8 Section 6.9

Spot Rates and the Yield Curve Forward Rates Formula Sheet Basic Review Problems Basic Review Problem Solutions Sample Exam Problems Sample Exam Problem Solutions Supplemental Exercises Supplemental Exercise Solutions

M5-1 M5-8 M5-13 M5-15 M5-16 M5-17 M5-19 M5-23 M5-28 M5-30

M6-1 M6-9 M6-13 M6-14 M6-15 M6-17 M6-19 M6-21 M6-22

Midterm 2 Bonds, Yield Rate, and The Term Structure of Interest Rates

Module 7

Asset-Liability Management

Section 7.1

Basic Asset-Liability Management: Matching Asset & Liability Cash Flows Duration Modified Duration Helpful Formulas for Duration Calculations Using Duration to Approximate Change in Price Approximations Using Duration and Convexity The Duration of a Portfolio Immunization Stocks and Other Investments Formula Sheet Basic Review Problems Basic Review Problem Solutions Sample Exam Problems Sample Exam Problem Solutions Supplemental Exercises Supplemental Exercise Solutions

Section 7.2 Section 7.3 Section 7.4 Section 7.5 Section 7.6 Section 7.7 Section 7.8 Section 7.9 Section 7.10 Section 7.11 Section 7.12 Section 7.13 Section 7.14 Section 7.15 Section 7.16

Index

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

MT2-1

M7-1 M7-4 M7-7 M7-11 M7-13 M7-24 M7-29 M7-32 M7-38 M7-42 M7-45 M7-47 M7-50 M7-55 M7-62 M7-63 Index-1

Exam FM – Financial Mathematics

Page TOC-4

Table of Contents

Volume II Module 8

Determinants of Interest Rates

Section 8.1 Section 8.2 Section 8.3 Section 8.4 Section 8.5 Section 8.6 Section 8.7 Section 8.8 Section 8.9 Section 8.10 Section 8.11 Section 8.12

Background Components of the Interest Rate Retail Savings and Lending Interest Rates Bond Yields The Role of Central Banks Formula Sheet Basic Review Problems Basic Review Problem Solutions Sample Exam Problems Sample Exam Problem Solutions Supplemental Exercises Supplemental Exercise Solutions

Module 9

Interest Rate Swaps

Section 9.1 Section 9.2 Section 9.3 Section 9.4 Section 9.5 Section 9.6 Section 9.7 Section 9.8 Section 9.9 Section 9.10 Section 9.11 Section 9.12 Section 9.13 Section 9.14

Introduction to Derivative Securities Variable-Rate Loans Example of an Interest Rate Swap Interest Rate Swap Terminology Calculating the Swap Rate Simplified Formulas for the Swap Rate Market Value of a Swap Formula Sheet Basic Review Problems Basic Review Problem Solutions Sample Exam Problems Sample Exam Problem Solutions Supplemental Exercises Supplemental Exercise Solutions

M8-1 M8-5 M8-18 M8-22 M8-34 M8-37 M8-38 M8-40 M8-43 M8-45 M8-47 M8-50

M9-1 M9-2 M9-4 M9-8 M9-13 M9-20 M9-25 M9-30 M9-32 M9-33 M9-37 M9-42 M9-48 M9-50

Midterm 3 Asset-Liability Management, Determinants of Interest Rates, and Interest Rate Swaps

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

MT3-1

Exam FM – Financial Mathematics

Table of Contents

Page TOC- 5

Practice Exams About the Practice Exams Practice Exam 1 Practice Exam 2 Practice Exam 3 Practice Exam 4 Practice Exam 5 Practice Exam 6 Practice Exam 7 Practice Exam 8 Practice Exam 9 Practice Exam 10 Practice Exam 11

PE0-1 PE1-1 PE2-1 PE3-1 PE4-1 PE5-1 PE6-1 PE7-1 PE8-1 PE9-1 PE10-1 PE11-1

Appendix

The Texas Instruments BA II Plus Calculator

Section App-1 Section App-2 Section App-3 Section App-4 Section App-5 Section App-6

Calculator Settings Other Calculator Features The Time Value of Money (TVM) Functions The Cash Flow Worksheet The Bond Worksheet The Interest Conversion (ICONV) Worksheet

Index

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Appx-1 Appx-3 Appx-5 Appx-9 Appx-11 Appx-13 Index-1

Exam FM – Financial Mathematics

Introduction

Page Intro - 1

Introduction 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 the 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. An appendix at the end of Volume II has information about the BA II Plus to help you learn its 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 it shows you how to return the calculator to the settings you prefer. Reading this material will help you avoid having calculator difficulties on the day of your exam.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

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. A strong understanding of 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 May 2018

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 1

Module

1

Interest Rates and the 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 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: 106  0.06 106   106 1.06   100 1.06   112.36 2

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.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

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 (“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.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 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)

FV  PV 1  i 

n

PV 

FV (1  i) n

Example (1.2) Let n = 10 and i = 0.06. a) If PV = 1,000, then FV  1,000 1.06 

b) If FV = 1,000, then PV =

10

 1,790.85

1, 000 = 558.39 (1.06) 10

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. Answer: a) 9,620.34 b) 6,700.48

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 4

Module 1 – Interest Rates and the Time Value of Money

Calculator Note The BA II Plus calculator has 5 “Time Value of Money” keys: N I/Y PV PMT FV

Number of periods Interest rate per period (usually per year) Present value Periodic payment 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.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 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 −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 functions.

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.)

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 6

Module 1 – Interest Rates and the Time Value of Money

 

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)

Compound interest:

a(t)  (1  i)t

A(t)  A(0) 1  i 

t

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 a(t)

0 1

1 1.60

2 2.56

3 4.096

4 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.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 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: a(t)

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 t t earned continuously, the formulas a(t)  (1  i) and A (t)  A(0) 1  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:

a(1.5)  (1.60)1.5  2.0239 If interest is earned continuously, the graph of a(t) is a smooth, continuous function: a(t)

t ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 8

Module 1 – Interest Rates and the Time Value of Money

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

Simple interest:

a(t)  (1  i t)

A(t)  A(0) 1  i t 

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 a(t)

0 1

1 1.60

2 2.20

3 2.80

4 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.

a(t)

earned continuously

earned at end of each year

t

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 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. a(t) Compound Interest

Simple Interest

t Both investments have a value of 1.60 at the end of 1 year. After the first year, the one 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 t 1  t  i  1  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 simpleinterest investment. At all times after the first year, the compound interest investment has a larger value than the simple interest investment.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 10

Module 1 – Interest Rates and the Time Value of Money

 

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 t, t  1 , the beginning and ending amounts are A(t) and A(t  1) . The amount of interest earned over the interval is A  t  1  A  t  . The effective rate of interest for the one-year period from t to t+1 is: (1.8)

it ,t 1  

interest earned between t and t+1 value of investment at time t A  t  1  A  t  A t 



a  t  1  a  t  a t

Note: The notation it1 ,t2  will be used in this module to represent an effective interest rate for the period t1 , t2  . 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:

i1,2 

a  2   a  1 a  1



1.1236  1.06  0.06 1.06



1.12  1.06  0.0566 1.06

For simple interest:

i1,2 

a  2   a  1 a  1

Exercise (1.10)

Let the interest rate be 6% and the time interval be [2,3]. Find i2,3 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 beginningof-year balance, but the investment at simple interest earns 6% on only the original principal. ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 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]:

i[0.25,0.50] 

a  0.50   a  0.25  a  0.25 



1.06 0.5  1.06 0.25  0.01467 1.06 0.25



1.061.50  1.061.25  0.01467 1.061.25

For the period [1.25,1.50]:

i[1.25,1.50] 

a 1.50   a 1.25  a 1.25 

At 6% simple interest For the period [0.25,0.50]: i[0.25,0.50] 

1   0.5  0.06   1   0.25  0.06    0.01478 1   0.25  0.06

a  0.50   a  0.25  a  0.25 

For the period [1.25,1.50]: i[1.25,1.50] 

a 1.50   a 1.25  a 1.25 

1  1.50  0.06   1  1.25  0.06    0.01395 1  1.25  0.06

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.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 12

Module 1 – Interest Rates and the Time Value of Money

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.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 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 (1.02)4 = 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.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 14

Module 1 – Interest Rates and the Time Value of Money

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? Answer: a) 12% convertible monthly b) 12.6825%

In the general case of m conversion periods per year, we denote the nominal i (m) annual rate by i(m). The effective interest rate per period is , and the annual m effective rate is: (1.14)

m

 i (m)  i  1   1 m  

This has the important consequence that: (1.15)

 i (m)  1  i  1   m  

m

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 i (12)  0.09 . Then the annual effective rate is: 12

0.09   12  1  12   1  1.0075  1  0.0938  

(or 9.38%)

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

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 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. 2 m = 2, so we need to find i   . By (1.15):

2

 i (2)   1    1.1025 2    i (2)   1    1.1025  1.05 2  

i(2)  2(1.05  1)  10% Thus the nominal annual rate is 10% convertible semi-annually, and the semi-annual effective rate is 5%. m Note that you can derive a formula that solves for i   given i and m. It is:

i

m

1    m 1  i  m  1  

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 m into (1.15) to solve for i   . This is the approach we used in Example (1.17).

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 16

Module 1 – Interest Rates and the Time Value of Money

Calculator Note The BA II Plus calculator has an Interest Conversion Worksheet that can be used to solve these problems. The legend above the 2 key is ICONV, which stands for “interest conversion.” You can activate the 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 use 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) b)

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

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

a) 6.168%

b) 4.889%

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 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 1 . The amount one year later, the present value you need to invest is: PV  (1  i) 1 . The annual effective of interest earned during the year is 1  PV  1  (1  i) rate of discount, d, is defined as the amount of interest earned during the year 1 ), divided by the ending balance (which is 1), so we have: (which is 1  (1  i)

(1.19)

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

d 1

1 (1  i)

Exam FM – Financial Mathematics

Page M1 – 18

Module 1 – Interest Rates and the Time Value of Money

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

d

(1.20)

i (1  i)

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)

interest earned between t and t+1 value of investment at time t  1 A  t  1  A  t  a  t  1  a  t    A  t  1 a  t  1

d

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 =

0.06 = 0.0566 1.06

Exercise (1.23)

Given i  0.10 , find d. Answer:

0.0909

From Formula (1.20), we can derive an expression for i in terms of d: i d d  d  di  i  d  i 1  d   i  (1  i) 1d (1.24)

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

i

d 1 d

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 19

Example (1.25)

For d = 0.06, i 

d 0.06   0.0638 1  d 1  0.06

Exercise (1.26)

Given d  0.10 , 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.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 20

Module 1 – Interest Rates and the Time Value of Money

 

Section 1.7 Present Value Factor Another variable that is used in actuarial interest problems is the present value factor, v: (1.27)

v

1 1 i

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)

d 1 v

v 1 d

and

And from formula (1.20), we can derive the following important relation: (1.29)

d  iv

The difference i − d simplifies nicely: i i−d=i− = (1  i) (1.30)

i2 = id (1  i)

id  id

The preceding relationships are often useful in solving exam problems. Example (1.31)

Given d  0.07 , find v and i. Solution.

v  1  d  0.93

1i 

1 1   1.0753 , so i  0.0753 or 7.53% v 0.93

Exercise (1.32)

Given d  0.05 , find v and i. Answers:

v = 0.95

i = 0.0526

Note that we can now write:

PV 

FV

1  i 

n

 v n  FV

The use of v as the present value factor is common in actuarial texts and is essential for actuarial exams. Many other financial professions do not use v. ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 21

 

Section 1.8 Nominal Rate of Discount A rate of discount can also 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 d   . For example, a nominal rate of discount convertible quarterly would be written as d (4) . It is related to the present value factor v and the annual effective rate of discount d by the equation: m

(1.33)

m  d   v  1  d  1    m  

m

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

0.08  4    0.98   0.9224 1  d  1   4  

d  1  0.9224  0.0776

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. 2

2  d   1     1  d  1  0.06  0.94 2   2  d   1  2 

   0.94  0.969536 

d    2  1  0.969536   0.060928 2

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 22

Module 1 – Interest Rates and the Time Value of Money

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). For example, in (1.34) we found the annual effective rate of discount corresponding to a nominal rate of discount of 8% convertible quarterly. To do this on 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 problem is: m

m p   i   d   1     1  m  p  

  

p

In this equation the left-hand side is equivalent to 1  i and the right-hand side 1 is equivalent to  1  i . Notice that the exponent in the right-hand expression v is –p, not p.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 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.

0.08    1  12   

12

2  d    1.0830   1    2  

2  d   1.083  1.0407   1    2  



2

1



d    2  1  1.0407 1  0.0782 2

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 for your calculations.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 24

Module 1 – Interest Rates and the Time Value of Money

 

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 the interest rate 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 semiannual effective rate of 4%, and its value increases annually by a factor of 1.0816  1.04 2 . 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 m

0.08    1. rate is:  1  m   Nominal Rate: 8% Conversion Period Ann. Eff. Rate m 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 m

0.08    1 . This is is infinite, and the annual effective interest rate is lim  1  m m   equal to e 0.08 – 1 , 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. ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 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-ofyear 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)

m

   i  lim  1    1  e  1 m m  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  i  e

(1.41)

(1  i) n  en

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

  ln 1  i  v n  1  i 

n

 e  n

Exam FM – Financial Mathematics

Page M1 – 26

Module 1 – Interest Rates and the Time Value of Money

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 that case,  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)

 (t) 

a '(t) a(t)

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, a '(t) , is the rate expressed as a rate per year. The ratio of these two values, a(t) (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 a  t   e t ), definition (1.42) yields:

 t  

de t / dt  e t  t   e t 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 a  t    t  1 . 2

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

d  t  1  / dt 2  t  1 2    Then   t    . 2 2  t  1 t  1  t  1 2

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 27

There is another useful relationship that enables us to create an expression for a(t) if only   t  is given: (1.44)

Thus:

d a '(t) ln[a(t)]    t  dt a(t)



k

0

 (t)dt  ln  a  t    ln a  k    ln a  0    ln a  k    ln 1  ln[a(k)] k

0

This implies that: (1.45)

t

e 0

 (u ) du

e

ln a t  

 a(t)

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

Given  (t) =

2 , find an expression for a(t). (t  1)

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

t

0

0

  (u)du  

t 2 du  2 ln(u  1)  2 ln(t  1) 0 (u  1)

Then we can write:

a(t)  e2ln(t1)  (eln(t1) )2  (t  1)2 Note: If your calculus is rusty, you may need to review calculus before doing these problems.

Exercise (1.47)

Given  ( t ) =

6 , find a(t). (2t  1) Answer: a(t)  e

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

3 ln(2 t  1)

  2t  1 

3

Exam FM – Financial Mathematics

Page M1 – 28

Module 1 – Interest Rates and the Time Value of Money

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 a 1 as follows:

da(t)    dt a(t) 1

1

da(t) t0 a(t)  t0   dt ln a(t)

1 0

1

 t 0

ln a(1)  ln a(0)    (1  0) ln a(1)  ln 1  

ln a(1)  0   a(1)  e

Since a(1)  1  i , we have:

1  i  e i  e  1   ln(1  i) These are the key relationships between a constant force of interest  and the equivalent annual effective interest rate i.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 29

 

Section 1.10 Quoted Rates for Treasury Bills In this section, we will consider a real-world example: the methods used to quote rates for United States Treasury bills and Government of Canada Treasury bills. The rates quoted for these securities are neither effective rates nor nominal rates. For Exam FM, you need to know how these rates are calculated and be able to convert them to annual effective rates. This material is included in the SOA study note on Determinants of Interest Rates.

United States Treasury Bills United States Treasury bills (T-bills) are short-term securities issued by the United States Treasury. An investor who purchases a Treasury bill is lending money to the United States government for a term of 4 weeks, 13 weeks, 26 weeks, or 52 weeks. The investor will receive the face amount of the T-bill on its maturity date. For example, an investor might pay 960 for a 52-week T-bill with a face amount of 1,000. At the end of 52 weeks, the U.S. Treasury will pay the maturity value of 1,000, and the investor will have earned a return of: 1,000  1  4.167% 960 This is the effective interest rate for the 52-week (364-day) term of the T-bill. Technically, the annual effective rate is a bit higher than this, since 4.167% was earned in slightly less than one year. Quoted rates for U.S. T-bills are neither effective interest rates nor nominal rates. Instead, they are expressed as “bank discount yield,” which is calculated by the following formula: (1.48)

For a U.S. Treasury bill:

Quoted Rate 

360 Amount of Interest  Days to Maturity Maturity Value

This formula produces a rate that is not consistent with any of the standard interest functions that we have studied. The next two examples demonstrate how this formula can be used to calculate the quoted rate for a U.S. T-bill if its price is known, or to calculate its price from the quoted rate.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 30

Module 1 – Interest Rates and the Time Value of Money

Example (1.49)

A newly-issued U.S. Treasury bill will mature in 182 days for its face amount of 1,000. It is priced at 985. What is the quoted rate, and what is the annual effective interest rate that is earned? Solution. Since the bond is priced at 985 and will mature for 1,000, the amount of interest is 15. We therefore have all the values needed to find the quoted rate using Formula (1.48):

Quoted Rate 

360 15   0.02967 182 1,000

The quoted rate is 2.967%. In 182 days, the value of the T-bill increases by a factor of:

1,000  1.015228 985 The equivalent annual effective interest rate is: 1.015228 365/182  1  3.077% We can also compute the equivalent annual effective rate of discount:

d

i 0.03077   2.985% 1  i 1.03077

Rates quoted for U.S. Treasury bills are similar to, but not the same as, nominal Amount of Interest rates of discount. The last term in Formula (1.48), , is the Maturity Value definition of an effective rate of discount for the term of the T-bill (interest earned, divided by ending balance). However, the other part of the formula, 360 , has a numerator that assumes 30-day months and a Days to Maturity denominator that uses the actual number of days until maturity. Because of this inconsistency, the quoted rate for a Treasury bill is smaller than the nominal rate of discount that is actually being earned. The next example and the comments that follow it demonstrate this relationship.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 31

Example (1.50)

A newly-issued U.S. Treasury bill matures in 182 days for its face amount of 1,000. Its rate is quoted as 4%. What is its current price, and what is its annual effective interest rate does it earn? Solution. From Formula (1.48), we can develop the following formula for the amount of interest that will be earned:

Quoted Rate 

Days to Maturity  Maturity Value  Amount of Interest 360

Using the values given for this T-bill, we have:

0.04 

182  1,000  20.22 360

The amount of interest is 20.22. The current price of this T-bill is: 1,000  20.22  979.78 Since the price increases from 979.78 to 1,000 in 182 days, the annual effective interest rate earned by this T-bill is:

 1,000   979.78   

365/182

 1  4.182%

The 4% quoted rate in Example (1.50) is similar to a nominal rate of discount

d   , 2

but it is slightly different because it is calculated using the actual

number of days to maturity and the assumption of a 360-day year. To analyze how this affects the calculation, we will compare the calculations for this Treasury bill to the corresponding calculations for a 6-month zero-coupon bond with a 1,000 maturity value and a yield that is described as “a nominal rate of discount of 4%, convertible semi-annually.” d   0.04   0.02 . 2 2 2

The 6-month effective rate of discount for the bond is: This corresponds to 0.04 

182  0.02022 , the 182-day effective rate of 360

discount for the 4% T-bill.

The interest earned during the bond’s 6-month term is: 0.02  1,000  20 . This corresponds to 0.02022  1, 000  20.22 for the T-bill. The bond’s price is 1,000  20  980 . This corresponds to 1,000  20.22  979.78 for the T-bill.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 32

Module 1 – Interest Rates and the Time Value of Money

The 6-month effective interest rate for the bond is 1,000 / 980  1  2.0408% . This corresponds to 1,000 / 979.78  1  2.0637% for the T-bill, which is a 182-day effective rate. The annual effective interest rate earned by the bond is 1.020408 2  1  4.123% . This corresponds to 1.020637 365/182  1  4.182% for the T-bill. Conclusion: The annual effective rate for a 6-month T-bill with a quoted rate of 4% is slightly higher than the annual effective rate for a 6-month zero-coupon bond earning a nominal rate of discount of 4% convertible semi-annually. Government of Canada Treasury Bills The Canadian government also issues Treasury bills. Rates for Government of Canada T-bills are quoted differently from U.S. T-bills. Basically, they are similar to nominal interest rates, rather than nominal rates of discount, and they assume a 365-day year rather than 360, as the following formula indicates. (1.51)

For a Government of Canada Treasury bill:

Quoted Rate 

365 Amount of Interest  Days to Maturity Current Price

Example (1.52)

A newly-issued Government of Canada Treasury bill matures in 182 days for its face amount of 1,000. It is priced at 985. What is the quoted rate, and what is the annual effective yield? Solution. The amount of interest the bond will earn is 15. Applying Formula (1.51), we have: 365 15 Quoted Rate    0.03054 182 985

The quoted rate is 3.054%. In 182 days, the value of the T-bill increases by a factor of:

1,000 / 985  1.015228 Its annual effective yield is the equivalent annual effective interest rate: 1.015228 365/182  1  3.077% The price, term, yield, and maturity value for the Government of Canada T-bill of Example (1.52) are the same as for the U.S. T-bill of Example (1.49). But because the quotation convention is different, the Canadian security is quoted at 3.054% and the U.S. security is quoted at 2.967%. Neither of these quotes matches the annual effective yield (3.077% for both securities), because the quoted values are not effective rates. They are similar to (but not identical to) nominal rates: a nominal rate of discount in the case of U.S. T-bills, and a nominal rate of interest in the case of Government of Canada T-bills. ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 33

Example (1.53)

A newly-issued Government of Canada Treasury bill matures in 182 days for its face amount of 1,000. Its rate is quoted as 4%. What is its current price, and what is its annual effective yield? Solution. From Formula (1.51), we can develop the following formula for the amount of interest that will be earned:

Days to Maturity  Current Price  Amount of Interest 365 182   Current Price  Amount of Interest 365

Quoted Rate  0.04

We don’t know the Current Price for this T-bill, but we do know that:

Current Price  Amount of Interest  Maturity Value  1,000 This gives us 2 equations in 2 unknowns. Using P for Price and I for Interest, we have: 182 P  I 365 P  I  1,000

0.04 

We can solve for P and I as follows: I  1,000  P 182  P  1, 000  P 0.04  365 182   P   1+0.04   1,000 365   P  980.44

I  1,000  P  19.56

To find the yield, we note that the price increases from 980.44 to 1,000 in 182 days, so the annual effective yield is:

 1,000   980.44   

365/182

 1  4.0401%

Note: The 4% quoted rate is similar to a 4% nominal rate of interest convertible semi-annually, which is equivalent to an annual effective rate of 4.04% (almost identical to the 4.0401% rate calculated above). Since Government of Canada T-bill quotes are based on a 365-day year, the yield of a T-bill quoted at 4% is nearly identical to the yield based on a 4% nominal rate of interest.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 34

Module 1 – Interest Rates and the Time Value of Money

The term, maturity value, and quoted rate for the Government of Canada T-bill in Example (1.53) are the same as for the U.S. T-bill of Example (1.50). But because the quotation conventions are not the same, the Canadian security has a price of 980.44 and a yield of 4.0401%, while the U.S. security has a price of 979.78 and a yield of 4.182%. For Exam FM, it is important to understand the special formulas used to quote rates for U.S. and Canadian T-bills. When performing calculations for T-bills, it is usually best to convert quoted rates to the equivalent annual effective rates.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 35

 

Section 1.11 Relating Discount, Force of Interest, and Interest Rate A very important relationship is: (1.54)

d  d

m

   i

m

(given that i  0 and m  1 )

i

This relationship is useful when checking whether a calculated value is reasonable, or determining which answer choices are plausible. It is not hard to see that d    i , since for i > 0:

i  ln(1  i)  i 1 i For a concrete example, let i  0.05 and m  4 . Then: 0.05   ln 1.05   0.0488 d  0.0476 1.05 4 4 i    4  1.05 1/ 4  1   0.049089 d    4  1  1.05 1/ 4   0.048494 m It should be noted that, just as  is the limit of i   as m becomes infinite,  is

also the limit of d   as m becomes infinite. That is, the force of discount and the force of interest are equal. For a given rate of compound growth, the annual effective rate of interest, i, is a larger number than the annual effective rate of m

m discount, d. The corresponding nominal rates of interest, i   , decrease as m m increases, and the nominal rates of discount, d   , increase as m increases. At m m the limit, when m becomes infinite, i   and d   are both equal to  :

(1.55)

i



 d





   It is reasonable that i   and d   should be equal. Just as i   is the instantaneous rate

of increase in an investment of 1 as t increases, d    is the instantaneous rate of decrease in an investment of 1 as t decreases. So long as the accumulation function a  t  and its first derivative, a '  t  are continuous functions, these two values must be equal.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 36

Module 1 – Interest Rates and the Time Value of Money

As an example of equivalent rates of interest that are expressed in different ways, consider the U.S. T-bill of Example (1.49) and the Government of Canada T-bill of Example (1.52). In both cases, the T-bill is priced at 985 and matures for 1,000 at the end of 182 days. Therefore, each of the two securities is accumulating interest at the same rate. However, the quoted rates for the U.S. and Canadian securities are not the same, and neither rate matches the annual effective rate. Those three different rates are shown in the following table, along with the continuously compounded yield (force of interest) for that same T-bill. The four measures are arranged in order of increasing size. U.S. T-bill Quote 2.967%

<

Force of Interest 3.031%

Gov. of Canada Annual T-bill Quote Effective Rate < 3.054% < 3.077%

Formula (1.54) provides an indication of why these different measures are related in this way. As mentioned earlier, the U.S. quotation method for T-bills m is similar to a nominal rate of discount ( d   ), so we would expect it to have a smaller value than the equivalent force of interest. By contrast, the Canadian

quotation method is essentially a nominal rate of interest ( i   ), so it is larger than the force of interest. The equivalent annual effective interest rate has the largest value. m

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 37

Section 1.12 Solving for PV, FV, n, and i Time value of money calculations can be done easily with the BAII Plus. This section will demonstrate how to solve for PV, FV, n, and i. In most cases, we will first use algebra to solve the problem, in order to establish the logic, and then show how the calculator’s Time Value of Money [TVM] keys can be used to save time. Example (1.56)

You want to have 120,000 in a college fund in 18 years. How much should you deposit now into an account earning a 6% annual effective rate of interest? Solution. You need a future value (FV) of 120,000 in 18 years. To find the present value (PV) that you should deposit, we have:

PV  120,000  v18 

120,000  42,041.25 1.06 18

For the BA II Plus, the keystrokes are: 18 N

6 I/Y

120000 FV

CPT PV

The equation used in the above example, PV  120, 000 v 18 , is called an equation of value. It is an equation expressing that the value a lender pays to a borrower equals the value that the borrower pays to the lender. In this case, the lender is you, and you are lending (or investing) the present value of 120,000 by depositing that amount into an account. The borrower is the bank (or other financial institution) that accepts the deposit and agrees to pay 6% interest. The value of what you deposit into the account at time 0 is unknown, so we call it PV. The future value that the bank pays you is 120,000, but since that amount is paid at time 18, the value at time 0 of what the bank pays you is 120,000 v 18 . In each of the following problems we will develop the appropriate equation of value in order to solve for the unknown value. An important concept that is always present in an equation of value is that of valuation date. In the above example, we chose time 0 as the valuation date, and determined the value of the payments as of time 0. (PV is an amount paid at time 0, so its value at time 0 is PV; 120,000 is an amount paid at time 18, so its value at time 0 is 120,000 v 18 .) An equation of value is not valid unless the value of each payment is determined as of the same date as every other payment. It is therefore critical to choose a valuation date and make sure that each term in the equation of value represents the value as of that date. As we will see, a wise choice of valuation date can make a problem much easier to solve. ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 38

Module 1 – Interest Rates and the Time Value of Money

Exercise (1.57)

How much should you deposit into the fund described in Example (1.56) if you want to have 100,000 in 16 years? Answer: 39,364.63

Example (1.58) You deposit 1,000 into an account earning interest at an annual effective rate of 5.75%. How much will you have in 5 years? Solution. The equation of value is: 5 FV  1,000 1.0575   1, 322.52

For the BA II Plus, the keystrokes are: 5 N

5.75 I/Y

1000 +|- PV

CPT FV

Note that in this case the valuation date is time 5. Both FV and 1,000 1.0575 

5

represent values determined as of time 5. We could have chosen time 0 as the valuation date instead, in which case the equation of value would be:

FV  1.0575 

5

 1,000 . Both the left and right sides of this equation of value

represent values as of time 0. Exercise (1.59)

In Example (1.58), how much will be in the account at the end of 10 years? Answer: 1,749.06

Example (1.60) You deposit 1,000 into an account earning a force of interest of 0.06. How long will it take to double your money? Solution. Doubling your money gives FV = 2,000. The equation of value is:

2,000  1,000e0.06t It follows that: 2  e 0.06t t

ln(2)  0.06t

ln  2 

 11.55 0.06 Note: Since the BA II Plus calculator’s TVM functions do not use force of interest, there is no TVM method for solving this problem.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 39

In Example (1.60), the valuation date was unknown. It was time t. We know that 2,000 is an amount that will be paid at time t, and 1,000e.06t is the value at time t of your deposit of 1,000 made at time 0. Once we solve for t, of course, we know that the valuation date was time 11.55. Exercise (1.61)

For the account in Example (1.60) how long would it take to triple your money? Answer: 18.31 years

Now let’s look at a variation on the preceding example that requires careful thinking: Example (1.62) You deposit 1,000 into an account earning an annual effective rate of 6%, but with interest payable only at the end of each year. If the account value is withdrawn before the end of a year, no interest is payable for that year. After how many years will the account balance be at least 2,000? Solution. If interest were being earned continuously, 2,000 would be reached after exactly 11.8957 years: 1, 000  1.06 t  2, 000 1.06 t  2 t

ln 2  11.8957 ln 1.06

However, because interest is not considered to be earned until the end of the year, you will not have 2,000 after 11.8957 years, but you will have more than 2,000 at the end of the 12th year. The answer is therefore 12 years.

Exercise (1.63) You deposit 1,000 into an account earning an annual effective rate of 5%, but with interest payable only at the end of each year. After how many years will the account balance be at least 2,000? Answer: 15

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 40

Module 1 – Interest Rates and the Time Value of Money

Example (1.64) You make an investment where you pay 1,000 now and get 1,500 back in 5 years. What interest rate will you earn? Solution. The equation of value is: 5 1, 000 1  i   1, 500

Thus:

1  i 

5

 1.5

1  i  1.5 1/ 5  1.08447 i  8.447%

For the BA II Plus the keystrokes are 5N 1000 +|- PV 1500 FV

CPT

I/Y

Exercise (1.65)

You make an investment where you pay 1,000 now and will get 2,000 back in 12 years. What annual effective interest rate will you earn? Answer: 5.9463%

The problems can be made more complex, as you will see when you move to the Sample 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 (1.66)

You deposit 1,000 into an account earning 5.75% convertible semi-annually. How much will you have in 5 years? Solution.

Now we have an effective interest rate of

0.0575  0.02875 per 2

semi-annual period for 2  5  1 0 periods. (Note that we cannot use the nominal rate of 5.75% for our calculations. We must convert it to an effective rate.) The equation of value is: FV  1,000 1.02875 

10

 1, 327.70

The BA II Plus keystrokes are: 10 N

2.875 I/Y

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

1000 +|- PV

CPT FV

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 41

Calculator Note The BA II Plus calculator has an option that allows you to choose 2 interest conversion periods per year (or 4 conversion periods, or any other number). P/Y is the 2nd function of the I/Y key; it allows you to set both the number of payments per year (P/Y) and (after pressing the DOWN arrow key) the number of interest conversion periods per year (C/Y). We advise against using this feature, since it can be confusing. Also, and very importantly, it is easy to forget to reset P/Y and C/Y after finishing a calculation, and that can lead to errors on later problems. All of the calculator solutions in this manual will be based on setting your calculator for P/Y=1 and C/Y=1. The I/Y key stands for “Interest per Year.” However, you should think of it as the effective “Interest Rate per Period.” For example, if we invest 1,000 for 5 years at a nominal interest rate of 6% convertible quarterly, we can analyze it as an effective interest rate of 1.5% per period for 20 periods (where a period is a quarter-year). Set N=20, I/Y=1.5, PV=–1,000, and CPT FV=1,346.86.

Exercise (1.67)

You deposit 1,000 into an account earning 6.5% convertible quarterly. How much will you have after 4.5 years? Answer: 1,336.63

Example (1.68)

You make an investment where you pay 1,000 now and get 1,500 back in 5 years. What nominal rate of interest convertible quarterly did you earn? Solution. In 5 years, there are 20 quarterly periods. We will first calculate the effective interest rate per period and then convert it to a nominal annual rate convertible quarterly.  i (4)  Let j be the quarterly effective interest rate  j  . 4  

Then the equation of value (with a valuation date of time 5) is: 20 1,000  1  j   1, 500 1

 1, 500  20 j Solving for j, we have:   1  0.02048  1,000  i (4)  4  j  8.192% The nominal rate convertible quarterly is:

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 42

Module 1 – Interest Rates and the Time Value of Money

To find the quarterly effective interest rate using the BA II Plus, the keystrokes are: 20 N

1000 +|- PV

1500 FV

CPT I/Y

The resulting quarterly effective rate is 2.048%. The nominal rate is 4  2.048%  8.192% convertible quarterly. Exercise (1.69)

An investment of 1,000 accumulates to 2,000 in 12 years. What nominal rate of interest convertible semi-annually did the investment earn? Answer: 5.860%

Other problems may have more than one future payment, or may have an unknown payment amount at some point. We see this in the next two examples. Example (1.70)

How much should you deposit now in a bank account earning a 5% annual effective rate to be able to withdraw 1,000 in 2 years and 2,000 in 4 years? Solution. The equation of value (with a valuation date of time 0) is:

PV  1,000v 2  2,000v 4 

1,000 2,000   2, 552.43 1.05 2 1.05 4

Exercise (1.71)

An amount of 3,000 is deposited into an account at time 0. The account earns a 6% annual effective rate. At the end of 2 years, 500 is withdrawn from the account. What is the account balance at the end of 3 years? Answer: 3,043.05

Example (1.72) You deposit 1,000 into an account now and an amount X in one year. The account earns an annual effective rate of 6%. What value of X will result in an account balance of 2,000 at the end of two years? Solution. This problem could be solved in multiple steps using the TVM functions. However, it is much easier to write and solve the equation of value. Using time 2 for our valuation date, we have:

1,000 1.06  X 1.06   2,000 → X  826.79 2

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 43

Exercise (1.73)

You deposit an amount X into an account at time 0 and 2X into the same account at time 3. The account balance at time 5 is 5,000. If the account has earned a 4% annual effective rate, what is the value of X? Answer: 1,479.35

Another type of problem that requires more thought is one in which the interest rate changes over time. Example (1.74) You deposit 5,000 into an account that earns interest at an annual effective rate of 5% during the first two years, and 7% in all subsequent years. What is the account balance at the end of 5 years? Solution.

FV  5, 000(1.05) 2 1.07   6, 753.05 3

You could also do the problem on the BA II Plus in two steps. Amount in 2 years: 2 N 5 I/Y (Answer: 5,512.50)

5000 +|- PV

CPT FV

Amount in 5 years: First press +|- and PV . (Since 5,512.50, the accumulated value at time 2, is already in your calculator’s display and is the amount you “deposit” for the last 3 years, we make it negative and enter it as PV.) Then press:

3 N

7 I/Y

CPT FV

Answer: 6,753.05

Exercise (1.75)

You deposit 2,000 into an account that earns an annual effective rate of 4% during the first 2.5 years, and 6% in all subsequent years. What is the account balance at the end of 4 years? Answer: 2,407.53

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 44

Module 1 – Interest Rates and the Time Value of Money

Example (1.76) What constant annual effective interest rate would have produced the same 5-year accumulation as in Example (1.74)? Solution. The BA II Plus can be used to solve this quickly. We accumulated FV = 6,753.05 in 5 years from an initial investment of 5,000. Solve for the interest rate as follows:

5N

5,000 +|- PV

6,753.05 FV

CPT I/Y

Answer: 6.20% To solve mathematically, denote the unknown interest rate by i . Then:

1  i 

5

 1.05 2  1.07   1.3506

1  i  1.3506 

3

1/ 5

 1.0620

i  6.20%

Exercise (1.77)

What constant annual effective interest rate would have produced the same 4-year accumulation as in Exercise (1.75)? Answer: 4.75%

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 45

Example (1.78) You deposit 5,000 at t 0 into an account that earns interest at an annual effective rate of 5% for two years, and earns continuously 2 in all compounded interest at a varying force of interest  (t) = (t  1) subsequent years. What is the account balance at the end of 5 years? Solution. At the end of two years the account contains 5,512.50. [See Example (1.74), above.] The force of interest must be applied from time 2 through time 5. The accumulated value in 5 years is:

  2 dt  5, 512.50   e 2 t 1    5

Note the limits on the integral. A common mistake is to integrate from 0 to 3. That produces a 3-year accumulation factor, but it is based on the growth rates  (t)  for the period t = 0 to t = 3, rather than the correct period (t = 2 to t = 5).

We now calculate: 5 2 5 2 t  1  dt  2 ln  t  1 2  2 ln  6   ln  3 5

e

2

2 t 1dt

e

2ln  6   2ln  3 



62 4 32

The final answer is:

  2 dt  5, 512.50  e 2 t 1   5, 512.50  4   22,050   5

Exercise (1.79)

You deposit 4,000 at t 0 into an account that earns continuously 1 compounded interest at a varying force of interest   t   . 4t  4 What is the account balance at the end of 4 years? Answer: 5,981.40

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 46

Module 1 – Interest Rates and the Time Value of Money

 

Section 1.13 The Rule of 72 The Rule of 72 is a simple method for estimating how long it will take an investment to double in value. (1.80)

Rule of 72:

If an investment earns an annual effective interest rate of x%, the investment will double in value in 72 approximately years. x (Similarly, if the investment earns an effective rate 72 periods.) of x% per period, it will double in about x

As an example, an investment at an annual effective rate of 8% will double in value in approximately 9 years  72 / 8  9 . (Actually, the 9-year accumulation factor is 1.9990.) The Rule of 72 is most accurate for interest rates close to 8%, but it produces values within 2% of the exact answer for interest rates from 2% to 14%. At interest rates lower than 8%, the resulting accumulation factor will be somewhat more than 2 (e.g., 1.0172  2.047 ), because the Rule of 72 slightly overstates the doubling period. At interest rates greater than 8%, the resulting accumulation factor will be somewhat less than 2 (e.g., 1.18 4  1.939 ), because the Rule of 72 slightly understates the doubling period. This rule can be very useful for making quick estimates or for checking whether your answer to a problem is reasonable. To see how this works, consider Example (1.56) in Section 1.12, which asks for the amount you would need to invest at 6% in order to have 120,000 after 18 years. We now know that an investment earning an annual effective rate of 6% will double in value in approximately 12 years. That means that in 18 years it will double approximately one-and-a-half times (since 18 / 12  1.5 ). So in 18 years the amount invested will increase by a factor of 21.5  2 2 . If you recall that

2  1.414 , then you know that the investment will grow by a factor of about 2.828. In other words, it will not quite triple in value, so you will need to invest a bit more than 40,000 in order to have 120,000 in 18 years. If you had tried to solve the problem and your answer was less than 40,000, or significantly more than 40,000, you would know that you had made an error, since the answer has to be a little more than 40,000. The value we calculated (42,041.45) appears to be reasonable. ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Module 1 – Interest Rates and the Time Value of Money

Page M1 – 47

Next, consider Example (1.60), where we calculated how long it would take an investment to double at a force of interest of 0.06. If it were an annual effective interest rate of 6%, the answer would be very close to 12 years. Since a force of interest of 0.06 represents a slightly larger growth rate, we can expect the answer to be a little less than 12. Our answer of 11.5525 for that example appears reasonable, whereas an answer of 11 would be too small, and anything over 12 would be too large. Exercise (1.65) provides a very obvious case where the Rule of 72 can be applied. It describes an investment that doubles in 12 years (from 1,000 to 2,000) and asks what annual effective interest rate was earned. Just as we would divide 72 by the interest rate to find the number of years for the investment to double, we can divide 72 by the number of years to get the interest rate (as a percent). Since 72 / 12  6 , the answer is about 6% (or a little less than 6%, since the Rule of 72 overstates the doubling time for interest rates less than 8%). Based on that reasoning, our calculated answer of 5.9463% is reasonable. Exercise (1.69) is very similar to Exercise (1.65) (again, an investment doubles in 12 years), but we are asked to find the nominal rate convertible semiannually. There are 24 semi-annual periods in 12 years, so our estimate of the semi-annual effective rate is 3% (since 72 / 24  3 ). This corresponds to a nominal rate of 6% convertible semi-annually, so it appears that our calculated answer of 5.86% is reasonable.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics

Page M1 – 48

Module 1 – Interest Rates and the Time Value of Money

 

Section 1.14 Formula Sheet a(t): accumulated value at time t of an initial investment of 1 made at time 0 A(t): accumulated value at time t of an initial investment of A(0) made at time 0 For simple interest at a constant rate i:

A(t)  A(0)  1  t  i

a(t)  1  t  i

For compound interest at a constant rate i: t a(t)  (1  i)t A(t)  A(0)  1  i  FV  PV 1  i 

n

PV =

FV (1  i) n

v

1 1 i

d 1 v

v 1 d

d=

i 1 i

d  iv

i

e  1  i

a(t)  e t

  ln 1  i  (1  i) n  en

d 1 d

i  d  id

vn  (1  i) n  en

For a variable force of interest:

 (t) 

a(t) d  ln  a  t   a(t) dt 

t

a(t)  e 0

 ( u ) du

Equivalent Rates:  i (m )  1  i  1   m  

m  d   1  d  1    m  

m

m

m

m p   i   d   1 1        m  p   

p

Note the negative exponent, -p.

 



  lim i  m  lim d  m m 

m 



d  d

m

   i

m

i

(given that i>0 and m>1)

Quoted Rates for Treasury bills: U.S. T-bills:

Quoted Rate 

360 Amount of Interest  Days to Maturity Maturity Value

Canadian T-bills:

Quoted Rate 

365 Amount of Interest  Days to Maturity Current Price

Rule of 72:

At an effective interest rate of x% per period, an investment will double in about 72 / x periods.

ACTEX Learning Dinius, Hassett, Ratliff, Garcia, & Steeby

Exam FM – Financial Mathematics