# Mathematical Finance

## Books

An introduction to the mathematical skills needed to understand finance and make better financial decisions

Mathematical Finance enables readers to develop the mathematical skills needed to better understand and solve financial problems that arise in business, from small entrepreneurial operations to large corporations, and to also make better personal financial decisions. Despite the availability of automated tools to perform financial calculations, the author demonstrates that a basic grasp of the underlying mathematical formulas and tables is essential to truly understand finance.

The book begins with an introduction to the most fundamental mathematical concepts, including numbers, exponents, and logarithms; mathematical progressions; and statistical measures. Next, the author explores the mathematics of the time value of money through a discussion of simple interest, bank discount, compound interest, and annuities. Subsequent chapters explore the mathematical aspects of various financial scenarios, including:

• Mortgage debt, leasing, and credit and loans

• Capital budgeting, depreciation, and depletion

• Break-even analysis and leverage

• Investing, with coverage of stocks, bonds, mutual funds, options, cost of capital, and ratio analysis

• Return and risk, along with a discussion of the Capital Asset Pricing Model (CAPM)

• Life annuities as well as life, property, and casualty insurance

Throughout the book, numerous examples and exercises present realistic financial scenarios that aid readers in applying their newfound mathematical skills to devise solutions. The author does not promote the use of financial calculators and computers, but rather guides readers through problem solving using formulas and tables with little emphasis on derivations and proofs.

Extensively class-tested to ensure an easy-to-follow presentation, Mathematical Finance is an excellent book for courses in business, economics, and mathematics of finance at the upper-undergraduate and graduate levels. The book is also appropriate for consumers and entrepreneurs who need to build their mathematical skills in order to better understand financial problems and make better financial choices.

Preface xv

UNIT I MATHEMATICAL INTRODUCTION 1

1 Numbers, Exponents, and Logarithms 3

1.1. Numbers, 3

1.2. Fractions, 3

1.3. Decimals, 5

1.4. Repetends, 6

1.5. Percentages, 7

1.6. Base Amount, Percentage Rate, and Percentage Amount, 8

1.7. Ratios, 9

1.8. Proportions, 10

1.9. Aliquots, 10

1.10. Exponents, 11

1.11. Laws of Exponents, 11

1.12. Exponential Function, 12

1.13. Natural Exponential Function, 13

1.14. Laws of Natural Exponents, 14

1.15. Scientific Notation, 15

1.16. Logarithms, 15

1.17. Laws of Logarithms, 16

1.18. Characteristic, Mantissa, and Antilogarithm, 16

1.19. Logarithmic Function, 18

2 Mathematical Progressions 20

2.1. Arithmetic Progression, 20

2.2. Geometric Progression, 23

2.3. Recursive Progression, 26

2.4. Infinite Geometric Progression, 28

2.5. Growth and Decay Curves, 29

2.6. Growth and Decay Functions with a Natural Logarithmic Base, 34

3 Statistical Measures 35

3.1. Basic Combinatorial Rules and Concepts, 35

3.2. Permutation, 37

3.3. Combination, 40

3.4. Probability, 41

3.5. Mathematical Expectation and Expected Value, 44

3.6. Variance, 46

3.7. Standard Deviation, 48

3.8. Covariance, 49

3.9. Correlation, 50

3.10. Normal Distribution, 52

Unit I Summary 54

List of Formulas 55

Exercises for Unit I 60

UNIT II MATHEMATICS OF THE TIME VALUE OF MONEY 63

Introduction 65

1 Simple Interest 67

1.1. Total Interest, 67

1.2. Rate of Interest, 67

1.3. Term of Maturity, 68

1.4. Current Value, 68

1.5. Future Value, 69

1.6. Finding n and r When the Current and Future Values are Both Known, 69

1.7. Simple Discount, 70

1.8. Calculating the Term in Days, 72

1.9. Ordinary Interest and Exact Interest, 73

1.10. Obtaining Ordinary Interest and Exact Interest in Terms of Each Other, 73

1.11. Focal Date and Equation of Value, 75

1.12. Equivalent Time: Finding an Average due Date, 78

1.13. Partial Payments, 80

1.14. Finding the Simple Interest Rate by the Dollar-Weighted Method, 81

2 Bank Discount 83

2.1. Finding FV Using the Discount Formula, 84

2.2. Finding the Discount Term and the Discount Rate, 84

2.3. Difference between a Simple Discount and a Bank Discount, 85

2.4. Comparing the Discount Rate to the Interest Rate, 87

2.5. Discounting a Promissory Note, 88

2.6. Discounting a Treasury Bill, 90

3 Compound Interest 93

3.1. The Compounding Formula, 94

3.2. Finding the Current Value, 97

3.3. Discount Factor, 98

3.4. Finding the Rate of Compound Interest, 100

3.5. Finding the Compounding Term, 100

3.6. The Rule of 72 and Other Rules, 101

3.7. Effective Interest Rate, 102

3.8. Types of Compounding, 104

3.9. Continuous Compounding, 105

3.10. Equations of Value for a Compound Interest, 106

3.11. Equated Time for a Compound Interest, 108

4 Annuities 110

4.1. Types of Annuities, 110

4.2. Future Value of an Ordinary Annuity, 111

4.3. Current Value of an Ordinary Annuity, 114

4.4. Finding the Payment of an Ordinary Annuity, 116

4.5. Finding the Term of an Ordinary Annuity, 118

4.6. Finding the Interest Rate of an Ordinary Annuity, 120

4.7. Annuity Due: Future and Current Values, 121

4.8. Finding the Payment of an Annuity Due, 123

4.9. Finding the Term of an Annuity Due, 124

4.10. Deferred Annuity, 126

4.11. Future and Current Values of a Deferred Annuity, 127

4.12. Perpetuities, 128

Unit II Summary 130

List of Formulas 132

Exercises for Unit II 138

UNIT III MATHEMATICS OF DEBT AND LEASING 145

1 Credit and Loans 147

1.1. Types of Debt, 147

1.2. Dynamics of Interest–Principal Proportions, 148

1.3. Premature Payoff, 152

1.4. Assessing Interest and Structuring Payments, 154

1.5. Cost of Credit, 158

1.6. Finance Charge and Average Daily Balance, 160

1.7. Credit Limit vs. Debt Limit, 162

2 Mortgage Debt 164

2.1. Analysis of Amortization, 164

2.2. Effects of Interest Rate, Term, and Down Payment on the Monthly Payment, 170

2.4. Mortgage Points and the Effective Rate, 176

2.5. Assuming a Mortgage Loan, 176

2.6. Prepayment Penalty on a Mortgage Loan, 177

2.7. Refinancing a Mortgage Loan, 178

2.8. Wraparound and Balloon Payment Loans, 180

2.9. Sinking Funds, 182

2.10. Comparing Amortization to Sinking Fund Methods, 187

3 Leasing 189

3.1. For the Lessee, 189

3.2. For the Lessor, 196

Unit III Summary 198

List of Formulas 199

Exercises for Unit III 202

UNIT IV MATHEMATICS OF CAPITAL BUDGETING AND DEPRECIATION 205

1 Capital Budgeting 207

1.1. Net Present Value, 207

1.2. Internal Rate of Return, 210

1.3. Profitability Index, 212

1.4. Capitalization and Capitalized Cost, 213

1.5. Other Capital Budgeting Methods, 216

2 Depreciation and Depletion 219

2.1. The Straight-Line Method, 220

2.2. The Fixed-Proportion Method, 223

2.3. The Sum-of-Digits Method, 226

2.4. The Amortization Method, 229

2.5. The Sinking Fund Method, 231

2.6. Composite Rate and Composite Life, 233

2.7. Depletion, 235

Unit IV Summary 239

List of Formulas 240

Exercises for Unit IV 243

UNIT V MATHEMATICS OF THE BREAK-EVEN POINT AND LEVERAGE 247

1 Break-Even Analysis 249

1.1. Deriving BEQ and BER, 249

1.2. BEQ and BER Variables, 251

1.3. Cash Break-Even Technique, 254

1.4. The Break-even Point and the Target Profit, 256

1.5. Algebraic Approach to the Break-Even Point, 257

1.6. The Break-Even Point When Borrowing, 261

1.7. Dual Break-Even Points, 264

1.8. Other Applications of the Break-Even Point, 267

1.9. BEQ and BER Sensitivity to their Variables, 272

1.10. Uses and Limitations of Break-Even Analysis, 272

2 Leverage 274

2.1. Operating Leverage, 274

2.2. Operating Leverage, Fixed Cost, and Business Risk, 277

2.3. Financial Leverage, 278

2.4. Total or Combined Leverage, 284

Unit V Summary 287

List of Formulas 289

Exercises for Unit V 291

UNIT VI MATHEMATICS OF INVESTMENT 295

1 Stocks 297

1.1. Buying and Selling Stocks, 298

1.2. Common Stock Valuation, 300

1.3. Cost of New Issues of Common Stock, 306

1.4. Stock Value with Two-Stage Dividend Growth, 307

1.5. Cost of Stock through the CAPM, 307

1.6. Other Methods for Common Stock Valuation, 308

1.7. Valuation of Preferred Stock, 309

1.8. Cost of Preferred Stock, 310

2 Bonds 311

2.1. Bond Valuation, 311

2.2. Premium and Discount Prices, 315

2.4. Discount Accumulation, 319

2.5. Bond Purchase Price Between Interest Days, 321

2.6. Estimating the Yield Rate, 324

2.7. Duration, 328

3 Mutual Funds 330

3.1. Fund Evaluation, 331

3.3. Performance Measures, 332

3.4. The Effect of Systematic Risk (β), 338

3.5. Dollar-Cost Averaging, 340

4 Options 341

4.1. Dynamics of Making Profits with Options, 343

4.2. Intrinsic Value of Calls and Puts, 344

4.3. Time Value of Calls and Puts, 347

4.4. The Delta Ratio, 348

4.5. Determinants of Option Value, 350

4.6. Option Valuation, 351

4.7. Combined Intrinsic Values of Options, 353

5 Cost of Capital and Ratio Analysis 357

5.1. Before- and After-Tax Cost of Capital, 357

5.2. Weighted-Average Cost of Capital, 358

5.3. Ratio Analysis, 359

5.4. The DuPont Model, 374

5.5. A Final Word about Ratios, 376

Unit VI Summary 377

List of Formulas 379

Exercises for Unit VI 384

UNIT VII MATHEMATICS OF RETURN AND RISK 387

1 Measuring Return and Risk 389

1.1. Expected Rate of Return, 389

1.2. Measuring the Risk, 390

1.3. Risk Aversion and Risk Premium, 394

1.4. Return and Risk at the Portfolio Level, 394

1.5. Markowitz’s Two-Asset Portfolio, 405

1.6. Lending and Borrowing at a Risk-Free Rate of Return, 408

1.7. Types of Risk, 409

2 The Capital Asset Pricing Model (CAPM) 411

2.1. The Financial Beta (β), 411

2.2. The CAPM Equation, 414

2.3. The Security Market Line, 416

2.4. SML Swing by Risk Aversion, 418

Unit VII Summary 422

List of Formulas 423

Exercises for Unit VII 425

UNIT VIII MATHEMATICS OF INSURANCE 429

1 Life Annuities 431

1.1. Mortality Table, 431

1.2. Commutation Terms, 436

1.3. Pure Endowment, 438

1.4. Types of Life Annuities, 439

2 Life Insurance 448

2.1. Whole Life Insurance Policy, 448

2.2. Annual Premium: Whole Life Basis, 449

2.3. Annual Premium: m-Payment Basis, 450

2.4. Deferred Whole Life Policy, 451

2.5. Deferred Annual Premium: Whole Life Basis, 452

2.6. Deferred Annual Premium: m-Payment Basis, 453

2.7. Term Life Insurance Policy, 454

2.8. Endowment Insurance Policy, 456

2.9. Annual Premium for the Endowment Policy, 457

2.10. Less than Annual Premiums, 458

2.12. Reserve and Terminal Reserve Funds, 461

2.13. Benefits of the Terminal Reserve, 465

2.14. How Much Life Insurance Should You Buy?, 465

3 Property and Casualty Insurance 470

3.1. Deductibles and Co-Insurance, 472

3.2. Health Care Insurance, 473

3.3. Policy Limit, 476

Unit VIII Summary 477

List of Formulas 478

Exercises for Unit VIII 482

References 485

Appendix 487

Index 515

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