a basic course in the theory of interest and derivatives markets

A Basic Course in the Theory of Interest and Derivatives Markets: A Preparation for the Actuarial Exam FM/2 Marcel B. Finan Arkansas Tech University c All Rights Reserved Preliminary Draft Last updated October 6, 2014 2 In memory of my parents August 1, 2008 January 7, 2009 Preface This manuscript is designed for an introductory course in the theory of in- terest and annuity. This manuscript 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 solving many of the problems in this book. A recommended resource link for the use of this calculator can be found at http://www.scribd.com/doc/517593/TI-BA-II-PLUS-MANUAL. The recommended approach for using this book is to read each section, work on the embedded examples, and then try the problems. Answer keys are provided so that you check your numerical answers 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 com- mercial purposes. If you find any errors, I would appreciate hearing from you: mfinan@atu.edu This project has been supported by a research grant from Arkansas Tech University. Marcel B. Finan Russellville, Arkansas March 2009 3 4 PREFACE Contents Preface 3 The Basics of Interest Theory 9 1 The Meaning of Interest . . . . . . . . . . . . . . . . . . . . . . . 10 2 Accumulation and Amount Functions . . . . . . . . . . . . . . . . 15 3 Effective Interest Rate (EIR) . . . . . . . . . . . . . . . . . . . . 25 4 Linear Accumulation Functions: Simple Interest . . . . . . . . . . 32 5 Date Conventions Under Simple Interest . . . . . . . . . . . . . . 40 6 Exponential Accumulation Functions: Compound Interest . . . . 46 7 Present Value and Discount Functions . . . . . . . . . . . . . . . 56 8 Interest in Advance: Effective Rate of Discount . . . . . . . . . . 63 9 Nominal Rates of Interest and Discount . . . . . . . . . . . . . . 75 10 Force of Interest: Continuous Compounding . . . . . . . . . . . 88 11 Time Varying Interest Rates . . . . . . . . . . . . . . . . . . . . 104 12 Equations of Value and Time Diagrams . . . . . . . . . . . . . . 111 13 Solving for the Unknown Interest Rate . . . . . . . . . . . . . . 118 14 Solving for Unknown Time . . . . . . . . . . . . . . . . . . . . . 127 The Basics of Annuity Theory 155 15 Present and Accumulated Values of an Annuity-Immediate . . . 156 16 Annuity in Advance: Annuity Due . . . . . . . . . . . . . . . . . 170 17 Annuity Values on Any Date: Deferred Annuity . . . . . . . . . 181 18 Annuities with Infinite Payments: Perpetuities . . . . . . . . . . 191 19 Solving for the Unknown Number of Payments of an Annuity . . 199 20 Solving for the Unknown Rate of Interest of an Annuity . . . . . 209 21 Varying Interest of an Annuity . . . . . . . . . . . . . . . . . . . 219 22 Annuities Payable at a Different Frequency than Interest is Con- vertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 5 6 CONTENTS 23 Analysis of Annuities Payable Less Frequently than Interest is Convertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 24 Analysis of Annuities Payable More Frequently than Interest is Convertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 25 Continuous Annuities . . . . . . . . . . . . . . . . . . . . . . . . 249 26 Varying Annuity-Immediate . . . . . . . . . . . . . . . . . . . . 255 27 Varying Annuity-Due . . . . . . . . . . . . . . . . . . . . . . . . 272 28 Varying Annuities with Payments at a Different Frequency than Interest is Convertible . . . . . . . . . . . . . . . . . . . . . . 281 29 Continuous Varying Annuities . . . . . . . . . . . . . . . . . . . 294 Rate of Return of an Investment 301 30 Discounted Cash Flow Technique . . . . . . . . . . . . . . . . . 302 31 Uniqueness of IRR . . . . . . . . . . . . . . . . . . . . . . . . . 313 32 Interest Reinvested at a Different Rate . . . . . . . . . . . . . . 320 33 Interest Measurement of a Fund: Dollar-Weighted Interest Rate 331 34 Interest Measurement of a Fund: Time-Weighted Rate of Interest 341 35 Allocating Investment Income: Portfolio and Investment Year Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 36 Yield Rates in Capital Budgeting . . . . . . . . . . . . . . . . . 360 Loan Repayment Methods 365 37 Finding the Loan Balance Using Prospective and Retrospective Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 38 Amortization Schedules . . . . . . . . . . . . . . . . . . . . . . . 374 39 Sinking Fund Method . . . . . . . . . . . . . . . . . . . . . . . . 387 40 Loans Payable at a Different Frequency than Interest is Convertible401 41 Amortization with Varying Series of Payments . . . . . . . . . . 407 Bonds and Related Topics 417 42 Types of Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 43 The Various Pricing Formulas of a Bond . . . . . . . . . . . . . 424 44 Amortization of Premium or Discount . . . . . . . . . . . . . . . 437 45 Valuation of Bonds Between Coupons Payment Dates . . . . . . 447 46 Approximation Methods of Bonds’ Yield Rates . . . . . . . . . . 456 47 Callable Bonds and Serial Bonds . . . . . . . . . . . . . . . . . . 464 CONTENTS 7 Stocks and Money Market Instruments 473 48 Preferred and Common Stocks . . . . . . . . . . . . . . . . . . . 475 49 Buying Stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 50 Short Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 51 Money Market Instruments . . . . . . . . . . . . . . . . . . . . . 493 Measures of Interest Rate Sensitivity 501 52 The Effect of Inflation on Interest Rates . . . . . . . . . . . . . . 502 53 The Term Structure of Interest Rates and Yield Curves . . . . . 507 54 Macaulay and Modified Durations . . . . . . . . . . . . . . . . . 517 55 Redington Immunization and Convexity . . . . . . . . . . . . . . 528 56 Full Immunization and Dedication . . . . . . . . . . . . . . . . . 536 An Introduction to the Mathematics of Financial Derivatives 545 57 Financial Derivatives and Related Issues . . . . . . . . . . . . . 546 58 Derivatives Markets and Risk Sharing . . . . . . . . . . . . . . . 552 59 Forward and Futures Contracts: Payoff and Profit Diagrams . . 556 60 Call Options: Payoff and Profit Diagrams . . . . . . . . . . . . . 568 61 Put Options: Payoff and Profit Diagrams . . . . . . . . . . . . . 578 62 Stock Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 63 Options Strategies: Floors and Caps . . . . . . . . . . . . . . . . 597 64 Covered Writings: Covered Calls and Covered Puts . . . . . . . 605 65 Synthetic Forward and Put-Call Parity . . . . . . . . . . . . . . 611 66 Spread Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . 618 67 Collars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 68 Volatility Speculation: Straddles, Strangles, and Butterfly Spreads634 69 Equity Linked CDs . . . . . . . . . . . . . . . . . . . . . . . . . 645 70 Prepaid Forward Contracts On Stock . . . . . . . . . . . . . . . 652 71 Forward Contracts on Stock . . . . . . . . . . . . . . . . . . . . 659 72 Futures Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . 673 73 Understanding the Economy of Swaps: A Simple Commodity Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 74 Interest Rate Swaps . . . . . . . . . . . . . . . . . . . . . . . . . 693 75 Risk Management . . . . . . . . . . . . . . . . . . . . . . . . . . 703 Answer Key 711 BIBLIOGRAPHY 745 8 CONTENTS The Basics of Interest Theory 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 goal here is to give a mathematical background for this area, and to develop the basic formulas which will be needed in the rest of the book. 9 10 THE BASICS OF INTEREST THEORY 1 The Meaning of Interest To analyze financial transactions, a clear understanding of the concept of interest is required. Interest can be defined in a variety of contexts, such as the ones found in dictionaries and encyclopedias. In the most common con- text, interest is an amount charged to a borrower for the use of the lender’s money over a period of time. For example, if you have borrowed $100 and you promised to pay back $105 after one year then the lender in this case is making a profit of $5, which is the fee for borrowing his money. Looking at this from the lender’s perspective, the money the lender is investing is changing value with time due to the interest being added. For that reason, interest is sometimes referred to as the time value of money. Interest problems generally involve four quantities: principal(s), investment period length(s), interest rate(s), amount value(s). The money invested in financial transactions will be referred to as the prin- cipal, denoted by P. The amount it has grown to will be called the amount value and will be denoted by A. The difference I = A − P is the amount of interest earned during the period of investment. Interest expressed as 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 can’t pay back the loan). The risk of default can be reduced if the borrowers promise to release an asset of theirs in the event of their default (the asset is called collateral). The unit in which time of investment is measured is called the measure- ment period. The most common measurement period is one year but may be longer or shorter (could be days, months, years, decades, etc.). Example 1.1 Which 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 you’ve earned by the Federal government. Solution. The answer is (c) Example 1.2 Let A(t) denote the amount value of an investment at time t years. [...]... A( n − 1) 26 THE BASICS OF INTEREST THEORY where In = A( n) − A( n − 1) Note that In represents the amount of growth of the function A( t) in the nth period whereas in is the rate of growth (based on the amount in the fund at the beginning of the period) Thus, the effective rate of interest in is the ratio of the amount of interest earned during the period to the amount of principal invested at the beginning... between the amount value at the end of the period and the amount value at the beginning of the period It should be noted that In involves the effect of interest over an interval of time, whereas A( n) is an amount at a specific point in time In general, the amount of interest earned on an original investment of $k between time s and t > s is I[s,t] = A( t) − A( s) = k (a( t) − a( s)) Example 2.5 Consider the amount... have earned $102.50 in interest 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 most basic principles involved in the measurement of interest are discussed 12 THE BASICS OF INTEREST THEORY Practice Problems Problem 1.1 You invest $3,200 in a savings... (a) What is the balance in the account at the end of year (b) What is the interest earned over the year period? (c) What is the effective interest rate? 2 ACCUMULATION AND AMOUNT FUNCTIONS 15 2 Accumulation and Amount Functions Imagine a fund growing at interest It would be very convenient to have a function representing the accumulated value, i.e., principal plus interest, of an invested principal at... $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? Solution (a) The principal is $1,000 (b) The interest earned is $1,050 - $1,000 = $50 50 (c) The annual interest rate is 1,000 = 5% Interest rates are most often computed on an annual basis, but they can... down a formula expressing the amount value after t days Problem 1.5 When interest is calculated on the original principal ONLY it is called simple interest Accumulated interest from prior periods is not used in calculations for the following periods In this case, the amount value A, the principal P, the period of investment t, and the annual interest rate i are related by the formula A = P (1 + it) At... formula A = P (1 + i)t and we call i a annual 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 to the principal of the previous period to become the principal for the next period You borrow $10,000 for three years at 5% annual interest compounded annually What is the amount value at the end of three years?... annual interest rate changes from one year to the next Let i1 be the interest rate for the first year, i2 the interest rate for the second year,· · · , in the interest rate for the nth year What will be the amount value of an investment of P at the end of the nth year? Problem 1.13 Discounting is the process of finding the present value of an amount of cash at some future date By the present value we mean... example Example 4.3 Consider the following investments by John and Peter John deposits $100 into a savings account paying 6% simple interest for 2 years Peter deposits $100 now with the same bank and at the same simple interest rate At the end of the year, he withdraws his balance and closes his account He then reinvests the total money in a new savings account offering the same rate Who has the greater...1 THE MEANING OF INTEREST 11 (a) Write an expression giving the amount of interest earned from time t to time t + s in terms of A only (b) Use (a) to find the annual interest rate, i.e., the interest rate from time t years to time t + 1 years Solution (a) The interest earned during the time t years and t + s years is A( t + s) − A( t) (b) The annual interest rate is A( t + 1) − A( t) A( t) Example 1.3 . the book. 9 10 THE BASICS OF INTEREST THEORY 1 The Meaning of Interest To analyze financial transactions, a clear understanding of the concept of interest is required. Interest can be defined in a variety of. A Basic Course in the Theory of Interest and Derivatives Markets: A Preparation for the Actuarial Exam FM/2 Marcel B. Finan Arkansas Tech University c All Rights Reserved Preliminary Draft Last. sections, various quantitative measures of interest are analyzed. Also, the most basic principles involved in the measurement of interest are discussed. 12 THE BASICS OF INTEREST THEORY Practice Problems Problem