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1 April 1991 Stephen G. Kellison Stressed throughout the entire book. Discounted cash flow analysis and capital budgeting as decision tools have been given more emphasis. Finally, the second edition contains a more thorough treatment of non-unique yield rates including various ways to resolve the difficulties they create. The material on amortization of loans has been significantly expanded. A section on real estate mortgages has been added, including an illustrative 30-year amortization schedule. New types of mortgage loans are defined and analyzed, such as the adjustable rate mortgage. The discussion of truth in lending requirements is more complete than in the first edition. Finally, new material has been added on amortization with step-rate amounts of principal and on continuous amortization. The second edition contains several other extensions to the material in the first edition. A sounder theoretical foundation for compound interest and simple interest is presented at the beginning of the book. A section has been added exploring the pitfalls of attempting to value annuities at simple interest. The approach used to value annuities payable at a different frequency than interest is convertible has been modified. More emphasis is placed on annuity calculations involving varying interest and on annuities with payments varying in geometric progression. The material on bonds has been expanded to include the peculiar algorithm widely used in practice to value bonds between coupons payment dates, which is a mixture of simple and compound interest. Finally, illustrations have been added on such "real-world" practices as penalties for early withdrawal and prepayment penalties. The computational approaches in the book have been modernized to reflect the widespread availability of pocket calculators with exponential and logarithmic functions. In a few instances, results are obtained which require a calculator with built-in financial functions, but these have been kept to a minimum. Compound interest tables are still included and have been extended to include higher rates of interest. Iteration techniques are used in solving for unknown rates of interest throughout the book. The pedagogical approach of the second edition is similar to the first edition. Verbal inte retations of key results are emphasized throughout. Illustrative examples are provided at the end of most sections. The number of exercises has been expanded substantially to 480. Each exercise is intended to illustrate a jomewhat different point, so that the number of repetitious exercises has been icept to a minimum. Many of the exercises involve more realistic rates of merest than in the first edition. Answers to the exercises are provided at the ?ack of the book. A working knowledge of basic calculus is required, since the continuous nature of interest is recognized throughout the book. A background in numerical analysis will be helpful in connection with the iteration techniques, although the reader without such a background can still follow the development. The new sections on stochastic approaches to interest assume a knowledge of basic statistics. The book is designed to be appropriate for both classroom use with an instructor and for self-study by those learning the subject without the aid of an instructor. The amount of material probably exceeds that which can be covered thoroughly in a one-semester university course. The second edition contains a total of 10 appendices. These appendices cover topics which may be of value to certain readers, but are not of sufficiently broad interest to warrant inclusion in the chapters. An extensive bibliography has been added at the end of the book. Also, a glossary of notation has been added indicating the section in which each new symbol has been introduced. The author is indebted to a number of correspondents who have written over the years in connection with first edition. As a result of this correspondence, a number of improvements have been incorporated into the second edition. The author also wishes to express his appreciation to the Society of Actuaries for using both editions as the syllabus reference on the actuarial examination covering the mathematics of compound interest. A committee of the Society of Actuaries revised a working draft of the second edition and as a result a number of very helpful suggestions were received which resulted in a significant improvement in the final manuscript. The author wishes to give special recognition and appreciation to Tabitha Miller who spent uncountable hours in typing the manuscript for the book. Since final copy was produced by word processing rather than by typesetting, this was a very painstaking and often frustrating process. The author will forever be in her debt for the many months of arduous effort in producing this book. Finally, the author wishes to recognize the patience and understanding of his wife Toni for her unfailing support. Without her dedication and support this book could not have been completed. She bore a very special burden, since much of the work on the book was completed during a period in which we adopted two lovely children.
Contents The measurement of interest........................1 1.1 Introduction, 1 1.2 The accumulation and amount functions, 1 1.3 The effective rate of interest, 4 1.4 Simple interest, 5 1.5 Compound interest, 7 1.6 Present value, 10 1.7 The effective rate of discount, 12 1.8 Nominal rates of interest and discount, 17 1.9 Forces of interest and discount, 22 1.10 Varying interest, 28 1.11 Summary of results, 29 Solution of problems in interest..................... 35 2.1 Introduction, 35 2.2 Obtaining numerical results, 35 2.3 Determining time periods, 38 2.4 The basic problem, 40 2.5 Equations of value, 42 2.6 Unknown time, 44 2.7 Unknown rate of interest, 48 2.8 Practical examples, 51 Basic annuities................................ 58 3.1 Introduction, 58 3.2 Annuity-immediate, 59 3.3 Annuity-due, 62 3.4 Annuity values on any date, 65 3.5 Pe etuities, 69 3.6 Nonstandard terms and interest rates, 71
xii Contents 3.7 Unknown time, 72 3.8 Unknown rate of interest, 75 3.9 Varying interest, 80 3.10 Annuities not involving compound interest, 82 More general annuities .......................... 95 4.1 Introduction, 95 4.2 Annuities payable at a different frequency than interest is convertible, 95 4.3 Further analysis of annuities payable less frequently than interest is convertible, 97 4.4 Further analysis of annuities payable more frequently than interest is convertible, 102 4.5 Continuous annuities, 107 4.6 Basic varying annuities, 109 4.7 More general varying annuities, 117 4.8 Continuous varying annuities, 119 4.9 Summary of results, 121 Yield rates ................................. 128 5.1 Introduction, 128 5.2 Discounted cash flow analysis, 129 5.3 Uniqueness of the yield rate, 133 5.4 Reinvestment rates, 136 5.5 Interest measurement of a fund, 140 5.6 Time-weighted rates of interest, 145 5.7 Portfolio methods and investment year methods, 149 5.8 Capital budgeting, 152 5.9 More general borrowing/lending models, 156 Amortization schedules and sinking funds ............. 166 6.1 Introduction, 166 6.2 Finding the outstanding loan balance, 167 6.3 Amortization schedules, 169 6.4 Sinking funds, 175 6.5 Differing payment periods and interest conversion periods, 181 6.6 Varying series of payments, 185 6.7 Amortization with continuous payments, 189 6.8 Step-rate amounts of principal, 192 Bonds and other securities............... 7.1 Introduction, 204 7.2 Types of securities, 204 7.3 Price of a bond, 208 7.4 Premium and discount, 213 7.5 Valuation between coupon payment dates, 220 7.6 Determination of yield rates, 225 7.7 Callable bonds, 230 7.8 Serial bonds, 232 7.9 Some generalizations, 234 7.10 Other securities, 236 7.11 Valuation of securities, 238 Contents xiii ..... 204 Practical applications ........... 8.1 Introduction, 248 8.2 Truth in lending, 248 8.3 Real estate mortgages, 255 8.4 Approximate methods, 260 8.5 Depreciation methods, 270 8.6 Capitalized cost, 277 8.7 Short sales, 279 8.8 Modern financial instruments, 282 More advanced financial analysis........... 9.1 Introduction, 294 9.2 An economic rationale for interest, 295 9.3 Determinants of the level of interest rates, 296 9.4 Recognition of inflation, 298 9.5 Reflecting risk and uncertainty, 302 9.6 Yield curves, 309 9.7 Interest rate assumptions, 313 9.8 Duration, 315 9.9 Immunization, 320 9.10 Matching assets and liabilities, 326 10 Stochastic approaches to interest . . . 10.1 Introduction, 336 10.2 Independent rates of interest, 337 10.3 Dependent rates of interest, 345
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