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<ul><li><p>A Basic Course in the Theory of Interest and DerivativesMarkets:</p><p>A Preparation for the Actuarial Exam FM/2</p><p>Marcel B. FinanArkansas Tech University</p><p>cAll Rights ReservedPreliminary Draft</p><p>Last updated</p><p>November 30, 2012</p></li><li><p>2In memory of my parents</p><p>August 1, 2008January 7, 2009</p></li><li><p>Preface</p><p>This manuscript is designed for an introductory course in the theory of interest and annuity. Thismanuscript is suitablefor a junior level course in the mathematics of finance.A calculator, such as TI BA II Plus, either the solar or battery version, will be useful in solvingmany of the problems in this book. A recommended resource link for the use of this calculator canbe found at</p><p>http://www.scribd.com/doc/517593/TI-BA-II-PLUS-MANUAL.</p><p>The recommended approach for using this book is to read each section, work on the embeddedexamples, and then try the problems. Answer keys are provided so that you check your numericalanswers against the correct ones.Problems taken from previous exams will be indicated by the symbol .This manuscript can be used for personal use or class use, but not for commercial purposes. If youfind any errors, I would appreciate hearing from you: mfinan@atu.eduThis project has been supported by a research grant from Arkansas Tech University.</p><p>Marcel B. FinanRussellville, ArkansasMarch 2009</p><p>3</p></li><li><p>4 PREFACE</p></li><li><p>Contents</p><p>Preface 3</p><p>The Basics of Interest Theory 91 The Meaning of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Accumulation and Amount Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Effective Interest Rate (EIR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Linear Accumulation Functions: Simple Interest . . . . . . . . . . . . . . . . . . . . . . 285 Date Conventions Under Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . 356 Exponential Accumulation Functions: Compound Interest . . . . . . . . . . . . . . . . 417 Present Value and Discount Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 Interest in Advance: Effective Rate of Discount . . . . . . . . . . . . . . . . . . . . . . 569 Nominal Rates of Interest and Discount . . . . . . . . . . . . . . . . . . . . . . . . . . 6610 Force of Interest: Continuous Compounding . . . . . . . . . . . . . . . . . . . . . . . 7811 Time Varying Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9312 Equations of Value and Time Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 10013 Solving for the Unknown Interest Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 10714 Solving for Unknown Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116</p><p>The Basics of Annuity Theory 14315 Present and Accumulated Values of an Annuity-Immediate . . . . . . . . . . . . . . . 14416 Annuity in Advance: Annuity Due . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15717 Annuity Values on Any Date: Deferred Annuity . . . . . . . . . . . . . . . . . . . . . 16818 Annuities with Infinite Payments: Perpetuities . . . . . . . . . . . . . . . . . . . . . . 17619 Solving for the Unknown Number of Payments of an Annuity . . . . . . . . . . . . . . 18420 Solving for the Unknown Rate of Interest of an Annuity . . . . . . . . . . . . . . . . . 19221 Varying Interest of an Annuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20222 Annuities Payable at a Different Frequency than Interest is Convertible . . . . . . . . 20623 Analysis of Annuities Payable Less Frequently than Interest is Convertible . . . . . . . 210</p><p>5</p></li><li><p>6 CONTENTS</p><p>24 Analysis of Annuities Payable More Frequently than Interest is Convertible . . . . . . 218</p><p>25 Continuous Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228</p><p>26 Varying Annuity-Immediate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234</p><p>27 Varying Annuity-Due . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249</p><p>28 Varying Annuities with Payments at a Different Frequency than Interest is Convertible 257</p><p>29 Continuous Varying Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268</p><p>Rate of Return of an Investment 275</p><p>30 Discounted Cash Flow Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276</p><p>31 Uniqueness of IRR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285</p><p>32 Interest Reinvested at a Different Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 292</p><p>33 Interest Measurement of a Fund: Dollar-Weighted Interest Rate . . . . . . . . . . . . 302</p><p>34 Interest Measurement of a Fund: Time-Weighted Rate of Interest . . . . . . . . . . . . 311</p><p>35 Allocating Investment Income: Portfolio and Investment Year Methods . . . . . . . . 320</p><p>36 Yield Rates in Capital Budgeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328</p><p>Loan Repayment Methods 333</p><p>37 Finding the Loan Balance Using Prospective and Retrospective Methods. . . . . . . . 334</p><p>38 Amortization Schedules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342</p><p>39 Sinking Fund Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353</p><p>40 Loans Payable at a Different Frequency than Interest is Convertible . . . . . . . . . . 365</p><p>41 Amortization with Varying Series of Payments . . . . . . . . . . . . . . . . . . . . . . 371</p><p>Bonds and Related Topics 379</p><p>42 Types of Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380</p><p>43 The Various Pricing Formulas of a Bond . . . . . . . . . . . . . . . . . . . . . . . . . 384</p><p>44 Amortization of Premium or Discount . . . . . . . . . . . . . . . . . . . . . . . . . . . 396</p><p>45 Valuation of Bonds Between Coupons Payment Dates . . . . . . . . . . . . . . . . . . 405</p><p>46 Approximation Methods of Bonds Yield Rates . . . . . . . . . . . . . . . . . . . . . . 413</p><p>47 Callable Bonds and Serial Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420</p><p>Stocks and Money Market Instruments 427</p><p>48 Preferred and Common Stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428</p><p>49 Buying Stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433</p><p>50 Short Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438</p><p>51 Money Market Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445</p></li><li><p>CONTENTS 7</p><p>Measures of Interest Rate Sensitivity 45352 The Effect of Inflation on Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 45453 The Term Structure of Interest Rates and Yield Curves . . . . . . . . . . . . . . . . . 45954 Macaulay and Modified Durations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46855 Redington Immunization and Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . 47956 Full Immunization and Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486</p><p>An Introduction to the Mathematics of Financial Derivatives 49357 Financial Derivatives and Related Issues . . . . . . . . . . . . . . . . . . . . . . . . . 49458 Derivatives Markets and Risk Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . 50059 Forward and Futures Contracts: Payoff and Profit Diagrams . . . . . . . . . . . . . . 50460 Call Options: Payoff and Profit Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 51461 Put Options: Payoff and Profit Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 52362 Stock Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53363 Options Strategies: Floors and Caps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54064 Covered Writings: Covered Calls and Covered Puts . . . . . . . . . . . . . . . . . . . 54765 Synthetic Forward and Put-Call Parity . . . . . . . . . . . . . . . . . . . . . . . . . . 55366 Spread Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56067 Collars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56868 Volatility Speculation: Straddles, Strangles, and Butterfly Spreads . . . . . . . . . . . 57469 Equity Linked CDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58570 Prepaid Forward Contracts On Stock . . . . . . . . . . . . . . . . . . . . . . . . . . . 59171 Forward Contracts on Stock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59872 Futures Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61073 Understanding the Economy of Swaps: A Simple Commodity Swap . . . . . . . . . . 61874 Interest Rate Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62975 Risk Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638</p><p>BIBLIOGRAPHY 645</p></li><li><p>8 CONTENTS</p></li><li><p>The Basics of Interest Theory</p><p>A component that is common to all financial transactions is the investment of money at interest.When a bank lends money to you, it charges rent for the money. When you lend money to a bank(also known as making a deposit in a savings account), the bank pays rent to you for the money.In either case, the rent is called interest.In Sections 1 through 14, we present the basic theory concerning the study of interest. Our goalhere is to give a mathematical background for this area, and to develop the basic formulas whichwill be needed in the rest of the book.</p><p>9</p></li><li><p>10 THE BASICS OF INTEREST THEORY</p><p>1 The Meaning of Interest</p><p>To analyze financial transactions, a clear understanding of the concept of interest is required. Inter-est can be defined in a variety of contexts, such as the ones found in dictionaries and encyclopedias.In the most common context, interest is an amount charged to a borrower for the use of thelenders money over a period of time. For example, if you have borrowed $100 and you promisedto pay back $105 after one year then the lender in this case is making a profit of $5, which is thefee for borrowing his money. Looking at this from the lenders perspective, the money the lender isinvesting is changing value with time due to the interest being added. For that reason, interest issometimes referred to as the time value of money.Interest problems generally involve four quantities: principal(s), investment period length(s), inter-est rate(s), amount value(s).The money invested in financial transactions will be referred to as the principal, denoted by P. Theamount it has grown to will be called the amount value and will be denoted by A. The differenceI = A P is the amount of interest earned during the period of investment. Interest expressedas a percent of the principal will be referred to as an interest rate.Interest takes into account the risk of default (risk that the borrower cant pay back the loan). Therisk of default can be reduced if the borrowers promise to release an asset of theirs in the event oftheir default (the asset is called collateral).The unit in which time of investment is measured is called the measurement period. The mostcommon measurement period is one year but may be longer or shorter (could be days, months,years, decades, etc.).</p><p>Example 1.1Which of the following may fit the definition of interest?(a) The amount I owe on my credit card.(b) The amount of credit remaining on my credit card.(c) The cost of borrowing money for some period of time.(d) A fee charged on the money youve earned by the Federal government.</p><p>Solution.The answer is (c)</p><p>Example 1.2Let A(t) denote the amount value of an investment at time t years.(a) Write an expression giving the amount of interest earned from time t to time t + s in terms ofA only.(b) Use (a) to find the annual interest rate, i.e., the interest rate from time t years to time t + 1years.</p></li><li><p>1 THE MEANING OF INTEREST 11</p><p>Solution.(a) The interest earned during the time t years and t+ s years is</p><p>A(t+ s) A(t).</p><p>(b) The annual interest rate isA(t+ 1) A(t)</p><p>A(t)</p><p>Example 1.3You deposit $1,000 into a savings account. One year later, the account has accumulated to $1,050.(a) What is the principal in this investment?(b) What is the interest earned?(c) What is the annual interest rate?</p><p>Solution.(a) The principal is $1,000.(b) The interest earned is $1,050 - $1,000 = $50.(c) The annual interest rate is 50</p><p>1000= 5%</p><p>Interest rates are most often computed on an annual basis, but they can be determined for non-annual time periods as well. For example, a bank offers you for your deposits an annual interestrate of 10% compounded semi-annually. What this means is that if you deposit $1000 now, thenafter six months, the bank will pay you 5% 1000 = $50 so that your account balance is $1050.Six months later, your balance will be 5% 1050 + 1050 = $1102.50. So in a period of one year youhave earned $102.50. The annual interest rate is then 10.25% which is higher than the quoted 10%that pays interest semi-annually.In the next several sections, various quantitative measures of interest are analyzed. Also, the mostbasic principles involved in the measurement of interest are discussed.</p></li><li><p>12 THE BASICS OF INTEREST THEORY</p><p>Practice Problems</p><p>Problem 1.1You invest $3,200 in a savings account on January 1, 2004. On December 31, 2004, the account hasaccumulated to $3,294.08. What is the annual interest rate?</p><p>Problem 1.2You borrow $12,000 from a bank. The loan is to be repaid in full in one years time with a paymentdue of $12,780.(a) What is the interest amount paid on the loan?(b) What is the annual interest rate?</p><p>Problem 1.3The current interest rate quoted by a bank on its savings accounts is 9% per year. You openan account with a deposit of $1,000. Assuming there are no transactions on the account such asdepositing or withdrawing during one full year, what will be the amount value in the account atthe end of the year?</p><p>Problem 1.4The simplest example of interest is a loan agreement two children might make:I will lend you adollar, but every day you keep it, you owe me one more penny. Write down a formula expressingthe amount value after t days.</p><p>Problem 1.5When interest is calculated on the original principal only it is called simple interest. Accumulatedinterest from prior periods is not used in calculations for the following periods. In this case, theamount value A, the principal P, the period of investment t, and the annual interest rate i arerelated by the formula A = P (1 + it). At what rate will $500 accumulate to $615 in 2.5 years?</p><p>Problem 1.6Using the formula of the previous problem, in how many years will 500 accumulate to 630 if theannual interest rate is 7.8%?</p><p>Problem 1.7Compounding is the process of adding accumulated interest back to the principal, so that interestis earned on interest from that moment on. In this case, we have the formula A = P (1 + i)t and wecall i a yearly compound interest. You can think of compound interest as a series of back-to-back simple interest contracts. The interest earned in each period is added t...</p></li></ul>