Mathematical Interest Theory (Mathematical Association of America Textbooks)

Now available in Third Edition: TEXT/57Mathematical Interest Theory gives an introduction of how investments grow over time. This is done in a mathematically precise manner. The emphasis is on practical applications that give the reader a concrete understanding of why the various relationships should be true. Among the modern financial topics introduced are: arbitrage, options, futures, and swaps. The content of the book, along with an understanding of probability, will provide a solid foundation for readers embarking on actuarial careers. On the other hand, Mathematical Interest Theory is written for anyone who has a strong high-school algebra background and is interested in being an informed borrower or investor. The content is suitable for a mid-level or upper-level undergraduate course or a beginning graduate course.Mathematical Interest Theory includes more than 240 carefully worked examples. There are over 430 problems, and numerical answers are included in an appendix. A companion student solution manual (see TEXT/15) has detailed solutions to the odd-numbered problems. Most of the examples involve computation, and detailed instruction is provided on how the Texas Instruments BA II Plus and BA II Plus Professional calculators can be used to efficiently solve the problems. This is important for readers wishing to pass the SOA/CAS joint financial mathematics exam FM/2. However, this part of the book can be skipped without disturbing the flow of the exposition. interest theory cover 1 copyright page 3 title page 4 Contents 10 Preface 14 To students 14 Examples 15 Problems 15 Special Features 15 Coverage 16 Second edition 17 Financial transactions 17 Acknowledgments 17 Contacting the authors 18 0 An introduction to the Texas Instruments BA II Plus 20 0.1 CHOOSING A CALCULATOR 20 0.2 FONT CONVENTION 20 0.3 BA II PLUS BASICS 21 0.4 PROBLEMS, CHAPTER 0 26 1 The growth of money 28 1.1 INTRODUCTION 28 1.2 WHAT IS INTEREST ? 29 1.3 ACCUMULATION AND AMOUNT FUNCTIONS 30 1.4 SIMPLE INTEREST / LINEAR ACCUMULATION FUNCTIONS 34 1.5 COMPOUND INTEREST (THE USUAL CASE!) 38 1.6 INTEREST IN ADVANCE / THE EFFECTIVE DISCOUNT RATE 43 1.7 DISCOUNT FUNCTIONS / THE TIME VALUE OF MONEY 47 1.8 SIMPLE DISCOUNT 56 1.9 COMPOUND DISCOUNT 58 1.10 NOMINAL RATES OF INTEREST AND DISCOUNT 62 1.11 A FRIENDLY COMPETITION (CONSTANT FORCE OF INTEREST) 70 1.12 FORCE OF INTEREST 73 1.13 NOTE FOR THOSE WHO SKIPPED SECTIONS (1.11) AND (1.12) 76 1.14 INFLATION 77 1.15 PROBLEMS, CHAPTER 1 80 1.3) Accumulation and amount functions 81 (1.4) Simple interest 81 (1.5) Compound interest 82 (1.6) Effective discount rates/ Interest in advance 83 (1.7) Discount functions/ The time value of money 84 (1.8) Simple discount 84 (1.9) Compound discount 85 (1.10) Nominal rates of interest and discount 85 (1.11) A friendly competition (Constant force of interest) 86 (1.12) Force of interest 87 (1.13) Note for those who skipped Section (1.11) and (1.12) 88 (1.14) Inflation 88 Chapter 1 review problems 89 2 Equations of value and yield rates 92 2.1 INTRODUCTION 92 2.2 EQUATIONS OF VALUE FOR INVESTMENTS INVOLVING A SINGLE DEPOSIT MADE UNDER COMPOUND INTEREST 93 2.3 EQUATIONS OF VALUE FOR INVESTMENTS WITH MULTIPLE CONTRIBUTIONS 95 2.4 INVESTMENT RETURN 103 2.5 REINVESTMENT CONSIDERATIONS 109 2.6 APPROXIMATE DOLLAR-WEIGHTED YIELD RATES 110 2.7 FUND PERFORMANCE 116 2.8 PROBLEMS, CHAPTER 2 119 (2.0) Chapter 2 writing problems 119 (2.2) Equations of value for investments involving a single deposit made under compound interest 119 2.3) Equations of value for investments with multiple contributions 120 (2.4) Investment return 122 (2.5) Reinvestment considerations 124 (2.6) Approximate dollar-weighted yield rates 124 (2.7) Fund performance 125 Chapter 2 review problems 125 3 Annuities (annuities certain) 128 3.1 INTRODUCTION 128 3.2 ANNUITIES - IMMEDIATE 130 3.3 ANNUITIES -DUE 143 3.4 PERPETUITIES 147 3.5 DEFERRED ANNUITIES AND VALUES ON ANY DATE 149 3.6 OUTSTANDING LOAN BALANCES 153 3.7 NONLEVEL ANNUITIES 159 3.8 ANNUITIES WITH PAYMENTS IN GEOMETRIC PROGRESSION 161 3.9 ANNUITIES WITH PAYMENTS IN ARITHMETIC PROGRESSION 164 3.10 YIELD RATE EXAMPLES INVOLVING ANNUITIES 170 3.11 ANNUITY SYMBOLS FOR NONINTEGRAL TERMS 175 3.12 ANNUITIES GOVERNED BY GENERAL ACCUMULATION FUNCTIONS 178 3.13 THE INVESTMENT YEAR METHOD 182 3.14 PROBLEMS, CHAPTER 3 185 (3.0) Chapter 3 writing problems 185 (3.2) Annuities-immediate 185 (3.3) Annuities-due 187 (3.4) Perpetuities 188 3.5) Deferred annuities and annuity values on any date 189 (3.6) Outstanding loan balances 190 (3.7) Nonlevel annuities 191 (3.8) Annuities with payments in geometric progressions 192 (3.9) Annuities with payments in arithmetic progressions 193 (3.10) Yield rate examples involving annuities 194 (3.11) Annuity-symbols for nonintegral terms 195 (3.12) Annuities governed by general accumulation functions 196 (3.13) The investment year Method 196 Chapter 3 review problems 197 4 Annuities with different payment and conversion periods 198 4.1 INTRODUCTION 198 4.2 LEVEL ANNUITIES WITH PAYMENTS LESS FREQUENT THAN EACH INTEREST PERIOD 199 4.3 LEVEL ANNUITIES WITH PAYMENTS MORE FREQUENT THAN EACH INTEREST PERIOD 203 4.4 ANNUITIES WITH PAYMENTS LESS FREQUENT THAN EACH INTEREST PERIOD AND PAYMENTS IN ARITHMETIC PROGRESSION 209 4.5 ANNUITIES WITH PAYMENTS MORE FREQUENT THAN EACH INTEREST PERIOD AND PAYMENTS IN ARITHMETIC PROGRESSION 211 4.6 CONTINUOUSLY PAYING ANNUITIES 216 4.7 A YIELD RATE EXAMPLE 220 4.8 PROBLEMS, CHAPTER 4 222 (4.0) Chapter 4 writing problems 222 (4.2) Level annuities with payments less frequent than each interest period 223 (4.3) Level annuities with payments more frequent than each interest period 224 (4.4) Annuities with payments less frequent than each interest period and payments in arithmetic progression 225 (4.5) Annuitieswith payments more frequent than each interest period and payments in arithmetic progression 225 (4.6) Continuously paying annuities 227 (4.7) A yield rate example 227 Chapter 4 review problems 228 5 Loan repayment 230 5.1 INTRODUCTION 230 5.2 AMORTIZED LOANS AND AMORTIZATION SCHEDULES 230 5.3 THE SINKING FUND METHOD 239 5.4 LOANS WITH OTHER REPAYMENT PATTERNS 245 5.5 YIELD RATE EXAMPLES AND REPLACEMENT OF CAPITAL 247 5.6 PROBLEMS, CHAPTER 5 255 (5.0) Chapter 5 writing problems 255 (5.2) Amortized loans and amortization schedules 255 (5.3) The sinking fund method 257 (5.4) Loans with other repayment patterns 258 (5.5) Yield rate examples and replacement of capital 259 Chapter 5 review problems 260 6 Bonds 262 6.1 INTRODUCTION 262 6.2 BOND ALPHABET SOUP AND THE BASIC PRICE FORMULA 263 6.3 THE PREMIUM-DISCOUNT FORMULA 269 6.4 OTHER PRICING FORMULAS FOR BONDS 271 6.5 BOND AMORTIZATION SCHEDULES 272 6.6 VALUING A BOND AFTER ITS DATE OF ISSUE 282 6.7 SELLING A BOND AFTER ITS DATE OF ISSUE 288 6.8 YIELD RATE EXAMPLES 297 6.9 CALLABLE BONDS 300 6.10 FLOATING-RATE BONDS 305 6.11 THE BA II PLUS CALCULATOR BOND WORKSHEET 306 6.12 PROBLEMS, CHAPTER 6 312 (6.0) Chapter 6 writing problems 312 (6.2) Bond alphabet soup and the basic price formula 312 (6.3) The premium-discount formula 313 (6.4) Other pricing formulas for bonds 313 (6.5) Bond amortization schedules 314 (6.6) Valuing a bond after its date of issue 314 (6.7) Selling a bond after its date of issue 315 (6.8) Yield rate examples 316 (6.9) Callable bonds 317 (6.10) Floating-rate bonds 317 (6.11) The BAII Plus calculator Bond worksheet 318 Chapter 6 review problems 318 7 Stocks and financial markets 320 7.1 COMMON AND PREFERRED STOCK 320 7.2 BROKERAGE ACCOUNTS 324 7.3 GOING LONG: BUYING STOCK WITH BORROWED MONEY 330 7.4 SELLING SHORT: SELLING BORROWED STOCKS 332 7.5 PROBLEMS, CHAPTER 7 337 (7.0) Chapter 7 writing problems 337 (7.1) Common and preferred stock 337 (7.2) Brokerage accounts 337 (7.3) Going long: buying stock with borrowed money 338 (7.4) Selling short: selling borrowed stocks 338 Chapter 7 review problems 339 8 Arbitrage, the term structure of interest rates, and derivatives 342 8.1 INTRODUCTION 342 8.2 ARBITRAGE 343 8.3 THE TERM STRUCTURE OF INTEREST RATES 346 8.4 FORWARD CONTRACTS 356 8.5 COMMODITY FUTURES HELD UNTIL DELIVERY 358 8.6 OFFSETTING POSITIONS AND LIQUIDITY OF FUTURES CONTRACTS 364 8.7 PRICE DISCOVERY AND MORE KINDS OF FUTURES 368 8.8 OPTIONS 370 8.9 USING REPLICATING PORTFOLIOS TO PRICE OPTIONS 376 8.10 USING WEIGHTED AVERAGES TO PRICE OPTIONS 387 8.11 SWAPS 394 8.12 PROBLEMS, CHAPTER 8 400 (8.0) Chapter 8 writing problems 400 (8.2) Arbitrage 401 (8.3) The term structure of interest rates 401 (8.4) Forward contracts 402 (8.5) Commodity futures held until delivery 403 (8.6) Offsetting positions and liquidity of futures contracts 403 (8.7) Price discovery and more kinds of futures 404 (8.8) Options 405 (8.9) Using replicating portfolios to price options 406 (8.10) Using weighted averages to price options: risk-neutral probabilities 407 (8.11) Swaps 408 Chapter 8 review problems 409 9 Interest rate sensitivity 412 9.1 OVERVIEW 412 9.2 DURATION 416 9.3 CONVEXITY 428 9.4 IMMUNIZATION 436 9.5 OTHER TYPES OF DURATION 443 9.6 PROBLEMS, CHAPTER 9 447 (9.0) Chapter 9 writing problems 447 (9.1) Overview 447 (9.2) Macaulay and modified duration 447 (9.3) Convexity 448 (9.4) Immunization 449 (9.5) Other types of duration 450 Chapter 9 review problems 451 Appendix A Some useful formulas 452 Appendix B Answers to end of chapter problems 458 PROBLEMS, CHAPTER 0 458 PROBLEMS, CHAPTER 1 459 PROBLEMS, CHAPTER 2 462 PROBLEMS, CHAPTER 3 464 PROBLEMS, CHAPTER 4 468 PROBLEMS, CHAPTER 5 470 PROBLEMS, CHAPTER 6 471 PROBLEMS, CHAPTER 7 475 PROBLEMS, CHAPTER 8 476 PROBLEMS, CHAPTER 9 479 Bibliography 482 Index 484 About the Authors 494 actuary,studies.,interest,theory actuary studies. interest theory Mathematical Interest Theory gives an introduction of how investments grow over time. This is done in a mathematically precise manner. The emphasis is on practical applications that give the reader a concrete understanding of why the various relationships should be true. Among the modern financial topics introduced are: arbitrage, options, futures, and swaps. The content of the book, along with an understanding of probability, will provide a solid foundation for readers embarking on actuarial careers. On the other hand, Mathematical Interest Theory is written for anyone who has a strong high-school algebra background and is interested in being an informed borrower or investor. The content is suitable for a mid-level or upper-level undergraduate course or a beginning graduate course. Mathematical Interest Theory includes more than 240 carefully worked examples. There are over 430 problems, and numerical answers are included in an appendix. A companion student solution manual has detailed solutions to the odd-numbered problems. Most of the examples involve computation, and detailed instruction is provided on how the Texas Instruments BA II Plus and BA II Plus Professional calculators can be used to efficiently solve the problems. This is important for readers wishing to pass the SOA/CAS joint financial mathematics exam FM/2. However, this part of the book can be skipped without disturbing the flow of the exposition. Mathematical Interest Theory gives an introduction to how investments vary over time, and this book provides a solid foundation for readers embarking on actuarial careers.. This is done in a mathematically precise manner, but the emphasis is on practical applications and giving the reader a concrete understanding as to why the various relationships should be true. Modern financial topics including arbitrage, options, futures, and swaps are introduced. Along with an understanding of probability,this book provides a solid foundation for readers embarking on actuarial careers. It also includes detailed instruction on how to use the Texas Instruments BA II Plus and BA II Plus Professional calculators. This text is among the recommended reading options for the Society of Actuaries/Casualty Actuarial Society FM/2 exam.