**AMS/MAA Textbooks**

Volume: 14;
2009;
475 pp;
Hardcover

Print ISBN: 978-0-88385-754-0

Product Code: TEXT/14

List Price: $92.00

AMS Member Price: $69.00

MAA Member Price: $69.00

**Electronic ISBN: 978-1-61444-600-2
Product Code: TEXT/14.E**

List Price: $92.00

AMS Member Price: $69.00

MAA Member Price: $69.00

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# Mathematical Interest Theory: Second Edition

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*Leslie Jane Federer Vaaler; James W. Daniel*

MAA Press: An Imprint of the American Mathematical Society

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 (see TEXT/15) has detailed solutions to the odd-numbered
problems. Most of the examples involve computation, and detailed
instruction is provided on how to 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.

Most questions from this textbook are available in
WebAssign. WebAssign is a leading provider of online
instructional tools for both faculty and students.

#### Reviews & Endorsements

This is an excellent book on interest theory; one of the four books recognized by the Society of Actuaries and the Casualty Actuarial Society as a basis of study for the interest theory component of their joint Financial Mathematics (FM) exam. What I particularly like about 'Mathematical Interest Theory' is that many problems are intrinsically multi-stepped requiring use of several core functions. By providing a multitude of superior problems, the authors are able to familiarize the student, not only with core actuarial function, but also develop their skills in studying the interaction between these functions and real-world problems.

-- Russell Jay Hendel, UMAP Journal

Students pursuing an actuarial career as well as those seeking a mathematically based finances course stand to benefit from this informative, up-to-date, and above all, skillfully written treatise. Instructors and students of interest theory owe Daniel and Vaaler a debt of gratitude for their fine efforts.

-- Susan Staples, Texas Christian University

We use the Vaaler and Daniel text as our primary learning resource for actuarial science majors as they prepare for the SOA FM/CAS2 financial mathematics exam. Each concept is followed by several illustrative and detailed examples that help students master the big ideas in interest theory.

-- Mark Maxwell, Director of Actuarial Science, Robert Morris University

# Table of Contents

## Mathematical Interest Theory: Second Edition

Table of Contents pages: 1 2

- cover cover11
- copyright page ii3
- title page iii4
- Contents ix10
- Preface xiii14
- 0 An introduction to the Texas Instruments BA II Plus 120
- 1 The growth of money 928
- 1.1 INTRODUCTION 928
- 1.2 WHAT IS INTEREST ? 1029
- 1.3 ACCUMULATION AND AMOUNT FUNCTIONS 1130
- 1.4 SIMPLE INTEREST / LINEAR ACCUMULATION FUNCTIONS 1534
- 1.5 COMPOUND INTEREST (THE USUAL CASE!) 1938
- 1.6 INTEREST IN ADVANCE / THE EFFECTIVE DISCOUNT RATE 2443
- 1.7 DISCOUNT FUNCTIONS / THE TIME VALUE OF MONEY 2847
- 1.8 SIMPLE DISCOUNT 3756
- 1.9 COMPOUND DISCOUNT 3958
- 1.10 NOMINAL RATES OF INTEREST AND DISCOUNT 4362
- 1.11 A FRIENDLY COMPETITION (CONSTANT FORCE OF INTEREST) 5170
- 1.12 FORCE OF INTEREST 5473
- 1.13 NOTE FOR THOSE WHO SKIPPED SECTIONS (1.11) AND (1.12) 5776
- 1.14 INFLATION 5877
- 1.15 PROBLEMS, CHAPTER 1 6180
- 1.3) Accumulation and amount functions 6281
- (1.4) Simple interest 6281
- (1.5) Compound interest 6382
- (1.6) Effective discount rates/ Interest in advance 6483
- (1.7) Discount functions/ The time value of money 6584
- (1.8) Simple discount 6584
- (1.9) Compound discount 6685
- (1.10) Nominal rates of interest and discount 6685
- (1.11) A friendly competition (Constant force of interest) 6786
- (1.12) Force of interest 6887
- (1.13) Note for those who skipped Section (1.11) and (1.12) 6988
- (1.14) Inflation 6988
- Chapter 1 review problems 7089

- 2 Equations of value and yield rates 7392
- 2.1 INTRODUCTION 7392
- 2.2 EQUATIONS OF VALUE FOR INVESTMENTS INVOLVING A SINGLE DEPOSIT MADE UNDER COMPOUND INTEREST 7493
- 2.3 EQUATIONS OF VALUE FOR INVESTMENTS WITH MULTIPLE CONTRIBUTIONS 7695
- 2.4 INVESTMENT RETURN 84103
- 2.5 REINVESTMENT CONSIDERATIONS 90109
- 2.6 APPROXIMATE DOLLAR-WEIGHTED YIELD RATES 91110
- 2.7 FUND PERFORMANCE 97116
- 2.8 PROBLEMS, CHAPTER 2 100119
- (2.0) Chapter 2 writing problems 100119
- (2.2) Equations of value for investments involving a single deposit made under compound interest 100119
- 2.3) Equations of value for investments with multiple contributions 101120
- (2.4) Investment return 103122
- (2.5) Reinvestment considerations 105124
- (2.6) Approximate dollar-weighted yield rates 105124
- (2.7) Fund performance 106125
- Chapter 2 review problems 106125

- 3 Annuities (annuities certain) 109128
- 3.1 INTRODUCTION 109128
- 3.2 ANNUITIES - IMMEDIATE 111130
- 3.3 ANNUITIES -DUE 124143
- 3.4 PERPETUITIES 128147
- 3.5 DEFERRED ANNUITIES AND VALUES ON ANY DATE 130149
- 3.6 OUTSTANDING LOAN BALANCES 134153
- 3.7 NONLEVEL ANNUITIES 140159
- 3.8 ANNUITIES WITH PAYMENTS IN GEOMETRIC PROGRESSION 142161
- 3.9 ANNUITIES WITH PAYMENTS IN ARITHMETIC PROGRESSION 145164
- 3.10 YIELD RATE EXAMPLES INVOLVING ANNUITIES 151170
- 3.11 ANNUITY SYMBOLS FOR NONINTEGRAL TERMS 156175
- 3.12 ANNUITIES GOVERNED BY GENERAL ACCUMULATION FUNCTIONS 159178
- 3.13 THE INVESTMENT YEAR METHOD 163182
- 3.14 PROBLEMS, CHAPTER 3 166185
- (3.0) Chapter 3 writing problems 166185
- (3.2) Annuities-immediate 166185
- (3.3) Annuities-due 168187
- (3.4) Perpetuities 169188
- 3.5) Deferred annuities and annuity values on any date 170189
- (3.6) Outstanding loan balances 171190
- (3.7) Nonlevel annuities 172191
- (3.8) Annuities with payments in geometric progressions 173192
- (3.9) Annuities with payments in arithmetic progressions 174193
- (3.10) Yield rate examples involving annuities 175194
- (3.11) Annuity-symbols for nonintegral terms 176195
- (3.12) Annuities governed by general accumulation functions 177196
- (3.13) The investment year Method 177196
- Chapter 3 review problems 178197

- 4 Annuities with different payment and conversion periods 179198
- 4.1 INTRODUCTION 179198
- 4.2 LEVEL ANNUITIES WITH PAYMENTS LESS FREQUENT THAN EACH INTEREST PERIOD 180199
- 4.3 LEVEL ANNUITIES WITH PAYMENTS MORE FREQUENT THAN EACH INTEREST PERIOD 184203
- 4.4 ANNUITIES WITH PAYMENTS LESS FREQUENT THAN EACH INTEREST PERIOD AND PAYMENTS IN ARITHMETIC PROGRESSION 190209
- 4.5 ANNUITIES WITH PAYMENTS MORE FREQUENT THAN EACH INTEREST PERIOD AND PAYMENTS IN ARITHMETIC PROGRESSION 192211
- 4.6 CONTINUOUSLY PAYING ANNUITIES 197216
- 4.7 A YIELD RATE EXAMPLE 201220
- 4.8 PROBLEMS, CHAPTER 4 203222
- (4.0) Chapter 4 writing problems 203222
- (4.2) Level annuities with payments less frequent than each interest period 204223
- (4.3) Level annuities with payments more frequent than each interest period 205224
- (4.4) Annuities with payments less frequent than each interest period and payments in arithmetic progression 206225
- (4.5) Annuitieswith payments more frequent than each interest period and payments in arithmetic progression 206225
- (4.6) Continuously paying annuities 208227
- (4.7) A yield rate example 208227
- Chapter 4 review problems 209228

- 5 Loan repayment 211230
- 6 Bonds 243262
- 6.1 INTRODUCTION 243262
- 6.2 BOND ALPHABET SOUP AND THE BASIC PRICE FORMULA 244263
- 6.3 THE PREMIUM-DISCOUNT FORMULA 250269
- 6.4 OTHER PRICING FORMULAS FOR BONDS 252271
- 6.5 BOND AMORTIZATION SCHEDULES 253272
- 6.6 VALUING A BOND AFTER ITS DATE OF ISSUE 263282
- 6.7 SELLING A BOND AFTER ITS DATE OF ISSUE 269288
- 6.8 YIELD RATE EXAMPLES 278297
- 6.9 CALLABLE BONDS 281300
- 6.10 FLOATING-RATE BONDS 286305
- 6.11 THE BA II PLUS CALCULATOR BOND WORKSHEET 287306
- 6.12 PROBLEMS, CHAPTER 6 293312
- (6.0) Chapter 6 writing problems 293312
- (6.2) Bond alphabet soup and the basic price formula 293312
- (6.3) The premium-discount formula 294313
- (6.4) Other pricing formulas for bonds 294313
- (6.5) Bond amortization schedules 295314
- (6.6) Valuing a bond after its date of issue 295314
- (6.7) Selling a bond after its date of issue 296315
- (6.8) Yield rate examples 297316
- (6.9) Callable bonds 298317
- (6.10) Floating-rate bonds 298317
- (6.11) The BAII Plus calculator Bond worksheet 299318
- Chapter 6 review problems 299318

- 7 Stocks and financial markets 301320
- 8 Arbitrage, the term structure of interest rates, and derivatives 323342
- 8.1 INTRODUCTION 323342
- 8.2 ARBITRAGE 324343
- 8.3 THE TERM STRUCTURE OF INTEREST RATES 327346
- 8.4 FORWARD CONTRACTS 337356
- 8.5 COMMODITY FUTURES HELD UNTIL DELIVERY 339358
- 8.6 OFFSETTING POSITIONS AND LIQUIDITY OF FUTURES CONTRACTS 345364
- 8.7 PRICE DISCOVERY AND MORE KINDS OF FUTURES 349368
- 8.8 OPTIONS 351370
- 8.9 USING REPLICATING PORTFOLIOS TO PRICE OPTIONS 357376
- 8.10 USING WEIGHTED AVERAGES TO PRICE OPTIONS 368387
- 8.11 SWAPS 375394
- 8.12 PROBLEMS, CHAPTER 8 381400
- (8.0) Chapter 8 writing problems 381400
- (8.2) Arbitrage 382401
- (8.3) The term structure of interest rates 382401
- (8.4) Forward contracts 383402
- (8.5) Commodity futures held until delivery 384403
- (8.6) Offsetting positions and liquidity of futures contracts 384403
- (8.7) Price discovery and more kinds of futures 385404
- (8.8) Options 386405
- (8.9) Using replicating portfolios to price options 387406
- (8.10) Using weighted averages to price options: risk-neutral probabilities 388407
- (8.11) Swaps 389408
- Chapter 8 review problems 390409

- 9 Interest rate sensitivity 393412

Table of Contents pages: 1 2