Mathematics Of Investment & Credit

PREFACEWhile teaching an intermediate level university course in mathematics ofinvestment over a number of years, I found an increasing need for atextbook that provided a thorough and modern treatment of the subject,while incorporating theory and applications. This book is an attempt (as a7th edition, it must be a seventh attempt) to satisfy that need. It is based, toa large extent, on notes that I have developed while teaching and my use ofa number of textbooks for the course. The university course for which thisbook was written has also been intended to help students prepare for themathematics of investment topic that is covered on one of the professionalexaminations of the Society of Actuaries and the Casualty ActuarialSociety. A number of the examples and exercises in this book are taken oradapted from questions on past SOA/CAS examinations.As in many areas of mathematics, the subject of mathematics ofinvestment has aspects that do not become outdated over time, but ratherbecome the foundation upon which new developments are based. Thetraditional topics of compound interest and dated cashflow valuations, andtheir applications, are developed in the first five chapters of the book. Inaddition, in Chapters 6 to 10, a number of topics are introduced whichhave become of increasing importance in modern financial practice overthe past number of years. The past three decades or so have seen a greatincrease in the use of derivative securities, particularly financial options.The subjects covered in Chapters 6 to 9, such as the term structure ofinterest rates, interest rate swaps and forward contracts, form thefoundation for the mathematical models used to describe and valuederivative securities, which are introduced in Chapter 10. This 7th editionexpands on and updates the 6th edition’s coverage of measures of durationand convexity, interest rate swaps and topics in fixed income securities.The purpose of the methods developed in this book is to facilitate financialvaluations. This book emphasizes a direct calculation approach, assumingthat the reader has access to a financial calculator with standard financialfunctions.xi

xii PREFACEThe mathematical background required for the book is a course in calculusat the freshman level. Chapters 8 and 10 cover a couple of topics thatinvolve the notion of probability, but mostly at an elementary level. A verybasic understanding of probability concepts should be sufficientbackground for those topics.The topics in the first five Chapters of this book are arranged in an orderthat is similar to traditional approaches to the subject, with Chapter 1introducing the various measures of interest rates, Chapter 2 developingmethods for valuing a series of payments, Chapter 3 consideringamortization of loans, Chapter 4 covering bond valuation, and Chapter 5introducing the various methods of measuring the rate of return earned byan investment.The content of this book is probably more than can reasonably be coveredin a one-semester course at an introductory or even intermediate level, butit might be possible for the FM Exam material to be covered in a onesemester course. At the University of Toronto, the contents of this booksare covered in two consecutive one-semester courses at the Sophomorelevel.I would like to acknowledge the support of the Actuarial Education andResearch Foundation, which provided support for the early stages ofdevelopment of this book. I would also like to thank those who providedso much help and insight in the earlier and current editions of this book:John Mereu, Michael Gabon, Steve Linney, Walter Lowrie, SrinivasaRamanujam, Peter Ryall, David Promislow, Robert Marcus, Sandi LynnScherer, Marlene Lundbeck, Richard London, David Scollnick and SamCox. I would like to thank Robert Alps particularly for providing me withsome the insights on duration that have been included in the expandedcoverage of that topic in this edition.I would like to acknowledge ACTEX Learning for their great support forthis book over the years and particularly for their editorial and technicalsupport.Finally, I am grateful to have had the continuous support of my wife, SueFoster, throughout the development of each edition of this book.Samuel A. Broverman, ASA, Ph.D. University of Toronto, October 2017

CHAPTER 1INTEREST RATE MEASUREMENT“Money makes the world go round, the world go round, the world go round.”– Fred Ebb, lyricist for the 1966 Broadway musical “Cabaret”1.0 INTRODUCTIONAlmost everyone, at one time or another, will be a saver, borrower, or investor, and will have access to insurance, pension plans, or other financialbenefits and liabilities. There is a wide variety of financial transactionsin which individuals, corporations, or governments can become involved.The range of available investments is continually expanding, accompaniedby an increase in the complexity of many of these investments.Financial transactions involve numerical calculations, and, dependingon their complexity, may require detailed mathematical formulations. It istherefore important to establish fundamental principles upon which thesecalculations and formulations are based. The objective of this book is tosystematically develop insights and mathematical techniques which leadto these fundamental principles upon which financial transactions can bemodeled and analyzed.The initial step in the analysis of a financial transaction is to translate a verbal description of the transaction into a mathematical model. Unfortunately,in practice, a transaction may be described in language that is vague andwhich may result in disagreements regarding its interpretation. The need forprecision in the mathematical model of a financial transaction requires thatthere be a correspondingly precise and unambiguous understanding of theverbal description before the translation to the model is made. To this end,terminology and notation, much of which is in standard use in financial andactuarial practice, will be introduced.A component that is common to virtually all financial transactions is interest, the “time value of money.” Most people are aware that interestrates play a central role in their own personal financial situations as wellas in the economy as a whole. Many governments and private enterprises1

2 CHAPTER 1employ economists and analysts who make forecasts regarding the levelof interest rates.The U.S. Federal Reserve Board sets the “federal funds discount rate,” atarget rate at which banks can borrow and invest funds with one another.This rate affects the more general cost of borrowing and also has an effect on the stock and bond markets. Bonds and stocks will be consideredin more detail later in the book. For now, it is not unreasonable to acceptthe hypothesis that higher interest rates tend to reduce the value of otherinvestments, if for no other reason than that the increased attraction ofinvesting at a higher rate of interest makes another investment earning alower rate relatively less attractive.Irrational ExuberanceAfter the close of trading on North American financial markets onThursday, December 5, 1996, U.S. Federal Reserve Board chairmanAlan Greenspan delivered a lecture at The American Enterprise Institute for Public Policy Research.In that speech, Mr. Greenspan commented on the possible negativeconsequences of “irrational exuberance” in the financial markets.The speech was widely interpreted by investment traders as indicatingthat stocks in the U.S. market were overvalued, and that the FederalReserve Board might increase U.S. interest rates, which might affectinterest rates worldwide.Although U.S. markets had already closed, those in the Far East werejust opening for trading on December 6, 1996. Japan’s main stockmarket index dropped 3.2%, the Hong Kong stock market droppedalmost 3%. As the opening of trading in the various world marketsmoved westward throughout the day, market drops continued to occur.The German market fell 4% and the London market fell 2%. When theNew York Stock Exchange opened at 9:30 AM EST on Friday, December 6, 1996, it dropped about 2% in the first 30 minutes of trading,although the market did recover later in the /speeches/1996/19961205.htm

INTEREST RATE MEASUREMENT 3The variety of interest rates and the investments and transactions to whichthey relate is extensive. Figure 1.1 provides a snapshot of a variety of current and historic interest rates as of April 17, 2017 and is an illustration ofjust a few of the types of interest rates that arise in practice. Libor refers tothe London Interbank Overnight Rate, which is an international ratecharged by one bank to another for very short term loans denominated inU.S. dollars. The prime rate is the interest rate that banks charge their mostcreditworthy customers. An ARM is an Adjustable Rate Mortgage, whichis a mortgage loan whose interest rate is periodically reset based on changesin market interest rates.KEY RATES (in %)Fed Reserve Target Rate3-Month LiborPrime RateAAA Average 20-YearCorporate Bond YieldsHigh Yield l20162.863.582.84April20075.896.175.92MORTGAGE RATES provided by Bankrate.com15-Year Mortgage30-Year Mortgage1-Year ARMU.S. ount Rate*.207281.03639*This term will be defined in Section 1.5FIGURE 1.1To analyze financial transactions, a clear understanding of the concept ofinterest is required. Interest can be defined in a variety of contexts, andmost people have at least a vague notion of what it is. In the most commoncontext, interest refers to the consideration or rent paid by a borrower ofmoney to a lender for the use of the money over a period of time.

4 CHAPTER 1This chapter provides a detailed development of the mechanics of interestrates: how they are measured and applied to amounts of principal over timeto calculate amounts of interest. A standard measure of interest rates willbe defined and two commonly used growth patterns for investment, simple and compound interest, will be described. Various alternative standardmeasures of interest, such as nominal annual rate of interest, rate of discount, and force of interest, are discussed. A general way in which a financial transaction is modeled in mathematical form will be presented usingthe notions of accumulated value, present value, and equation of value.1.1 INTEREST ACCUMULATION ANDEFFECTIVE RATES OF INTERESTAn interest rate is most typically quoted as an annual percentage. Ifinterest is credited or charged annually, the quoted annual rate, indecimal or fraction form, is multiplied by the amount invested or loanedto calculate the amount of interest that accrues over a one-year period. Itis generally understood that as interest is credited or paid, it is reinvested.This reinvesting of interest leads to the process of compounding interest.The following example illustrates this process.EXAMPLE 1.1(Compound interest calculation)The current rate of interest quoted by a bank on its savings account is 9% perannum (per year), with interest credited annually. Smith opens an accountwith a deposit of 1000. Assuming that there are no transactions on the account other than the annual crediting of interest, determine the account balance just after interest is credited at the end of 3 years.SOLUTIONAfter one year the interest credited will be 1000 .09 90, resulting in abalance (with interest) of 1000 1000 .09 1000(1.09) 1090. It isstandard practice that this balance is reinvested and earns interest in thesecond year, producing an interest amount of 1090 .09 98.10, and atotal balance of1090 1090 .09 1090(1.09) 1000(1.09) 2 1188.10

INTEREST RATE MEASUREMENT 5at the end of the second year. The balance at the end of the third year will be1188.10 1188.10 .09 (1188.10)(1.09) 1000(1.09)3 1295.03.The following time diagram illustrates this process.0 1000DepositTotal12 1000 .09 90 1090 .09 98.10InterestInterest1000 90 1090 1000 1.091090 98.10 1188.10 1090 1.09 1000(1.09) 23 1188.10 .09 106.93Interest1188.10 106.93 1295.03 1188.10 1.09 1000(1.09)3 FIGURE 1.2It can be seen from Example 1.1 that with an interest rate of i per annumand interest credited annually, an initial deposit of C will earn interest ofCi for the following year. The accumulated value or future value at theend of the year will be C Ci C (1 i ). If this amount is reinvested andleft on deposit for another year, the interest earned in the second yearwill be C (1 i )i, so that the accumulated balance is C (1 i ) C (1 i )i C (1 i ) 2 at the end of the second year. The account will continue togrow by a factor of 1 i per year, resulting in a balance of C (1 i ) n atthe end of n years. This is the pattern of accumulation that results fromcompounding, or reinvesting, the interest as it is credited.01 CiCDepositInterestTotal C Ci C (1 i )2 C (1 i )iInterestn–1 C (1 i )n 2 iInterest C (1 i ) n 2C (1 i ) C (1 i ) n 2 i C (1 i )in 1 C (1 i ) 2 C (1 i )FIGURE 1.3n C (1 i )n 1iInterest C (1 i ) n 1 C (1 i ) n 1 i C (1 i ) n

6 CHAPTER 1In Example 1.1, if Smith were to observe the accumulating balance in theaccount by looking at regular bank statements, Smith would see only oneentry of interest credited each year. If Smith made the initial deposit onJanuary 1, 2017 then Smith would have interest added to his account onDecember 31 of 2017 and every December 31 after that for as long as theaccount remained open.The rate of interest may change from one year to the next. If the interestrate is i1 in the first year, i2 in the second year, and so on, then after nyears an initial amount C will accumulate to C (1 i1 )(1 i2 ) (1 in ),where the growth factor for year t is 1 it and the interest rate for year tis it . Note that “year t” starts at time t 1 and ends at time t.EXAMPLE 1.2(Average annual rate of return)The excerpts below are taken from the 2016 year-end report of NationalBank Global Equity Fund, a fund managed by a Canadian mutual fundcompany. The excerpts below focus on the performance of the fund andthe Dow Jones Industrial Average during the five year period ending December 31, 2016. The Dow Jones Industrial Average is a price-weightedaverage of stocks traded on major American stock exchanges.Annual Rate of ReturnNB Global EquityDow Jones Ind. 201333.57%26.50%201214.14%7.26%Average Annual ReturnNB Global EquityDow Jones Ind. Avg.1 yr.%0.00%13.42%2 yr.%8.66%5.30%3 yr.%10.21%6.04%5 yr.%15.34%10.10%FIGURE 1.4For the five year period ending December 31, 2016, the total compoundgrowth in the Global Equity Fund can be found by compounding the annual rates of return for the 5 years.(1 0.00)(1 .1808)(1 .1338)(1 .3357)(1 .1414) 2.0411This would be the value on December 31, 2016 of an investment of 1made into the fund on January 1, 2012.This five year growth can be described by means of an average annual re-

INTEREST RATE MEASUREMENT 7turn per year for the five-year period. In practice the phrase “average annual return” refers to an annual compound rate of interest for the period ofyears being considered. The average annual return would be i, where(1 i )5 2.0411 . Solving for i results in a value of i .1534. This is theaverage annual return for the five year period ending December 31, 2016.For the Dow Jones average, an investment of 1 made January 1, 2012would have a value on December 31, 2016 of(1 .1342)(1 .0223)(1 .0752)(1 .2650)(1 .0726) 1.6178Solving for i in the equation (1 i )5 1.6178 results in i .1010, or a 5year average annual return of 10.10%.The Global Equity Fund is described on the National Bank website asfollows: “The fund’s investment objective is to achieve long-term capitalgrowth. It builds a diversified portfolio of common and preferred shareslisted on recognized stock exchanges.”The Dow Jones Industrial Average is a stock index of 30 large publiclyowned U.S. companies. 1.1.1 EFFECTIVE RATES OF INTERESTIn practice, interest may be credited or charged more frequently than onceper year. Many bank accounts pay interest monthly and credit cards generally charge interest monthly on unpaid balances. If a deposit is allowed toaccumulate in an account over time, the algebraic form of the accumulationwill be similar to the one given earlier for annual interest. At interest rate jper compounding period, an initial deposit of amount C will accumulate toC (1 j ) n after n compounding periods. (It is typical to use i to denote anannual rate of interest, and in this text j will often be used to denote an interest rate for a period of time other than a year.)At an interest rate of .75% per month on a bank account, with interest credited monthly, the growth factor for a one-year period at this rate would be(1.0075)12 1.0938. The account earns 9.38% over the full year and9.38% is called the annual effective rate of interest earned on the account.

8 CHAPTER 1Definition 1.1 – Annual Effective Rate of InterestThe annual effective rate of interest earned by an investment during aone-year period is the percentage change in the value of the investmentfrom the beginning to the end of the year, without regard to the investment behavior at intermediate points in the year.In Example 1.2, the annual effective rates of return for the fund and theindex are given for years 2012 through 2016. Comparisons of the performance of two or more investments are often done by comparing the respective annual effective interest rates earned by the investments over aparticular year. The mutual fund earned an annual effective rate of interestof 13.38% for 2014, but the Dow index earned 7.52%. For the 5-year period from January 1, 2012 to December 31, 2016, the mutual fund earned anaverage annual effective rate of interest of 15.34%, but the Dow index average annual effective rate was 10.10%.Equivalent Rates of InterestIf the monthly compounding at .75% described earlier continued foranother year, the accumulated or future value after two years would beC (1.0075) 24 C (1.0938) 2 . We see that over an integral number of years amonth-by-month accumulation at a monthly rate of .75% is equivalent toannual compounding at an annual rate of 9.38%; the word “equivalent” isused in the sense that they result in the same accumulated value.Definition 1.2 - Equivalent Rates of InterestTwo rates of interest are said to be equivalent if they result in the sameaccumulated values at each point in time.1.1.2 COMPOUND INTERESTWhen compound interest is in effect, and deposits and withdrawals areoccurring in an account, the resulting balance at some future point intime can be determined by accumulating all individual transactions tothat future time point.

INTEREST RATE MEASUREMENT 9EXAMPLE 1.3(Compound interest calculation)Smith deposits 1000 into an account on January 1, 2011. The account credits interest at an annual e

1.6.1 Continuous Investment Growth 41 1.6.2 Investment Growth Based on the Force of Interest 43 1.6.3 Constant Force of Interest 46 1.7 Inflation and the “Real” Rate of Interest 47 1.8 Factors Affecting Interest Rates 51 1.8.1 Government Policy 52 1.8.2 Risk Premium 54 1.8.3 Theories of the Term Structure of Interest Rates 56File Size: 452KBPage Count: 46Explore furtherMathematical Economics Practice Problems and Solutions .www.rationalargumentator.comInvestment Banking Book PDF - Valuation, Financial .corporatefinanceinstitute.comBasics of Finance pdf - cs for Finance: An Introduction to Financial .fac.ksu.edu.saFundamentals of Finance - The Basics Global Finance Sch globalfinanceschool.comRecommended to you based on what's popular Feedback