Quantitative Methods for Finance and Investments

PREFACE......Page 12
ACKNOWLEDGMENTS......Page 14
1.1 THE IMPORTANCE OF MATHEMATICS IN FINANCE......Page 15
1.2 MATHEMATICAL AND COMPUTER MODELING IN FINANCE......Page 16
1.3 MONEY, SECURITIES, AND MARKETS......Page 17
1.4 TIME VALUE, RISK, ARBITRAGE, AND PRICING......Page 19
1.5 THE ORGANIZATION OF THIS BOOK......Page 20
2.2 VARIABLES, EQUATIONS, AND INEQUALITIES......Page 21
2.3 EXPONENTS......Page 22
APPLICATION 2.1: INTEREST AND FUTURE VALUE (A more complete presentation of this material is provided in chapter 4)......Page 23
2.4 THE ORDER OF ARITHMETIC OPERATIONS AND THE RULES OF ALGEBRA......Page 24
2.5 THE NUMBER e......Page 25
2.6 LOGARITHMS (Background reading: sections 2.2 and 2.5)......Page 26
APPLICATION 2.3: THE TIME NEEDED TO DOUBLE YOUR MONEY (You may also wish to read related material in chapter 4)......Page 27
2.8 SUMMATIONS (Background reading: sections 2.3 and 2.7)......Page 28
APPLICATION 2.4: MEAN VALUES (You may also wish to read related material in section 5.1)......Page 29
2.9 DOUBLE SUMMATIONS (Background reading: section 2.8)......Page 30
APPLICATION 2.5: GEOMETRIC MEANS (Background reading: application 2.4)......Page 31
APPLICATION 2.6: THE TERM STRUCTURE OF INTEREST RATES (Background reading: application 2.5)......Page 32
APPLICATION 2.7: DERIVING THE NUMBER e (Background reading: section 2.5)......Page 33
2.12 PERMUTATIONS AND COMBINATIONS (Background reading: section 2.11)......Page 34
EXERCISES......Page 35
APPENDIX 2.A AN INTRODUCTION TO THE EXCEL™ SPREADSHEET......Page 37
3.1 ALGEBRAIC MANIPULATIONS (Background reading: section 2.4)......Page 39
APPLICATION 3.1: PURCHASE POWER PARITY......Page 41
APPLICATION 3.2: FINDING BREAK-EVEN PRODUCTION LEVELS......Page 42
3.2 THE QUADRATIC FORMULA (Background reading: section 3.1)......Page 43
APPLICATION 3.4: FINDING BREAK-EVEN PRODUCTION LEVELS (Background reading: application 3.2)......Page 44
APPLICATION 3.5: FINDING THE PERFECTLY HEDGED PORTFOLIO......Page 45
3.3 SOLVING SYSTEMS OF EQUATIONS THAT CONTAIN MULTIPLE VARIABLES (Background reading: section 3.1)......Page 46
APPLICATION 3.7: EXTERNAL FINANCING NEEDS (Background reading: section 3.3)......Page 49
3.4 GEOMETRIC EXPANSIONS (Background reading: section 3.1)......Page 52
APPLICATION 3.8: MONEY MULTIPLIERS......Page 54
3.5 FUNCTIONS AND GRAPHS (Background reading: section 2.3)......Page 55
APPLICATION 3.9: UTILITY OF WEALTH......Page 57
EXERCISES......Page 58
APPENDIX 3.A SOLVING SYSTEMS OF EQUATIONS ON A SPREADSHEET (Background reading: section 3.3 and appendix 2.A)......Page 62
4.2 SIMPLE INTEREST (Background reading: sections 2.4, 2.7, and 4.1)......Page 65
4.3 COMPOUND INTEREST (Background reading: sections 2.7, 3.1, and 4.2)......Page 66
4.4 FRACTIONAL PERIOD COMPOUNDING OF INTEREST......Page 67
APPLICATION 4.1: APY AND BANK ACCOUNT COMPARISONS......Page 69
4.5 CONTINUOUS COMPOUNDING OF INTEREST (Background reading: sections 2.5 and 4.4)......Page 70
4.6 ANNUITY FUTURE VALUES (Background reading: sections 2.8, 3.4, and 4.3)......Page 71
APPLICATION 4.2: PLANNING FOR RETIREMENT (Background reading: sections 2.5, 3.1, and 4.5)......Page 73
4.7 DISCOUNTING AND PRESENT VALUE (Background reading: section 4.3)......Page 74
4.8 THE PRESENT VALUE OF A SERIES OF CASH FLOWS (Background reading: sections 2.8 and 4.7)......Page 75
4.9 ANNUITY PRESENT VALUES (Background reading: sections 3.4, 4.6, and 4.8)......Page 76
Deriving the present-value annuity formula......Page 77
APPLICATION 4.4: VALUING A BOND......Page 78
4.10 AMORTIZATION (Background reading: section 4.9)......Page 79
APPLICATION 4.5: DETERMINING THE MORTGAGE PAYMENT......Page 80
4.11 PERPETUITY MODELS (Background reading: section 4.9)......Page 81
4.12 SINGLE-STAGE GROWTH MODELS (Background reading: sections 4.9 and 4.11)......Page 82
APPLICATION 4.6: STOCK VALUATION MODELS......Page 84
4.13 MULTIPLE-STAGE GROWTH MODELS* (Background reading: section 4.12)......Page 86
EXERCISES......Page 87
APPENDIX 4.A TIME VALUE SPREADSHEET APPLICATIONS......Page 91
5.1 RETURN ON INVESTMENT (Background reading: section 2.8 and application 2.4)......Page 93
APPLICATION 5.1: FUND PERFORMANCE (Background reading: sections 2.8 and 5.1 and application 2.4)......Page 95
5.2 GEOMETRIC MEAN RETURN ON INVESTMENT (Background reading: sections 2.10 and 5.1 and application 2.5)......Page 96
APPLICATION 5.2: FUND PERFORMANCE, PART II (Background reading: section 5.2 and application 5.1)......Page 97
5.3 INTERNAL RATE OF RETURN (Background reading: sections 4.7, 4.8, and 4.9)......Page 98
5.4 BOND YIELDS......Page 101
5.6 EXPECTED RETURN (Background reading: sections 2.8 and 5.5)......Page 102
5.7 VARIANCE AND STANDARD DEVIATION......Page 103
5.8 HISTORICAL VARIANCE AND STANDARD DEVIATION......Page 105
5.9 COVARIANCE......Page 107
5.10 THE COEFFICIENT OF CORRELATION AND THE COEFFICIENT OF DETERMINATION......Page 108
EXERCISES......Page 109
APPENDIX 5.A RETURN AND RISK SPREADSHEET APPLICATIONS......Page 113
6.2 PORTFOLIO RETURN (Background reading: sections 2.8, 5.5, and 6.1)......Page 117
6.3 PORTFOLIO VARIANCE (Background reading: sections 2.9, 5.7–5.10, and 6.2)......Page 118
6.4 DIVERSIFICATION AND EFFICIENCY......Page 120
6.5 THE MARKET PORTFOLIO AND BETA......Page 124
6.6 DERIVING THE PORTFOLIO VARIANCE EXPRESSION (Background reading: section 6.3)......Page 125
EXERCISES......Page 127
7.1 AN INTRODUCTION TO MATRICES......Page 129
APPLICATION 7.1: PORTFOLIO MATHEMATICS (Background reading: sections 2.9, 6.2, 6.3, and 7.1)......Page 130
7.2 MATRIX ARITHMETIC (Background reading: sections 2.9 and 7.1)......Page 131
APPLICATION 7.2: PORTFOLIO MATHEMATICS, PART II (Background reading: application 7.1 and section 7.2)......Page 134
APPLICATION 7.3: PUT–CALL PARITY (Background reading: section 7.2)......Page 135
7.3 INVERTING MATRICES (Background reading: sections 3.3 and 7.2)......Page 137
7.4 SOLVING SYSTEMS OF LINEAR EQUATIONS (Background reading: sections 1.3, 3.3, and 7.3)......Page 139
APPLICATION 7.4: EXTERNAL FUNDING REQUIREMENTS (Background reading: section 7.5 and application 3.7)......Page 140
APPLICATION 7.5: COUPON BONDS AND DERIVING YIELD CURVES (Background reading: sections 4.8, 4.9, and 7.4 and applications 2.6 and 4.4)......Page 141
APPLICATION 7.6: ARBITRAGE WITH RISKLESS BONDS (Background reading: application 7.5)......Page 144
APPLICATION 7.7: FIXED INCOME PORTFOLIO DEDICATION (Background reading: application 7.6)......Page 145
APPLICATION 7.8: BINOMIAL OPTION PRICING (Background reading: section 7.4 and application 7.3)......Page 146
7.5 SPANNING THE STATE SPACE (Background reading: sections 7.3 and 7.4)......Page 147
APPLICATION 7.9: USING OPTIONS TO SPAN THE STATE SPACE (Background reading: section 7.5 and application 7.8)......Page 150
EXERCISES......Page 151
APPENDIX 7.A MATRIX MATHEMATICS ON A SPREADSHEET (Background reading: sections 7.3, 7.4, and 7.5 and appendix 3.A)......Page 156
8.1 FUNCTIONS AND LIMITS (Background reading: sections 2.2 and 3.5)......Page 159
APPLICATION 8.1: THE NATURAL LOG (Background reading: sections 2.5, 2.11, 4.4, 4.5, and 8.1 and application 2.7)......Page 160
8.2 SLOPES, DERIVATIVES, MAXIMA, AND MINIMA (Background reading: section 8.1)......Page 161
8.3 DERIVATIVES OF POLYNOMIALS (Background reading: section 8.2)......Page 163
APPLICATION 8.2: MARGINAL UTILITY (Background reading: section 8.3 and application 3.9)......Page 165
APPLICATION 8.3: DURATION AND IMMUNIZATION (Background reading: sections 4.8, 4.9, 5.4, and 8.3 and application 4.4)......Page 167
APPLICATION 8.4: PORTFOLIO RISK AND DIVERSIFICATION (Background reading: sections 6.1, 6.4, 6.5 and 8.2)......Page 170
8.4 PARTIAL AND TOTAL DERIVATIVES (Background reading: section 8.3)......Page 171
8.5 THE CHAIN RULE, PRODUCT RULE, AND QUOTIENT RULE (Background reading: sections 8.3 and 8.4)......Page 172
APPLICATION 8.5: PLOTTING THE CAPITAL MARKET LINE (Background reading: sections 6.4, 7.4, and 8.5)......Page 173
8.6 LOGARITHMIC AND EXPONENTIAL FUNCTIONS (Background reading: sections 2.5, 2.6, 8.1, and 8.2)......Page 179
8.7 TAYLOR SERIES EXPANSIONS (Background reading: sections 2.11 and 8.3)......Page 180
APPLICATION 8.6: CONVEXITY AND IMMUNIZATION (Background reading: section 8.7 and application 8.4)......Page 181
8.8 THE METHOD OF LAGRANGE MULTIPLIERS (Background reading: sections 7.4 and 8.4)......Page 182
APPLICATION 8.7: OPTIMAL PORTFOLIO SELECTION (Background reading: sections 6.3, 6.4, and 8.7)......Page 184
EXERCISES......Page 186
APPENDIX 8.A DERIVATIVES OF POLYNOMIALS (Background reading: sections 2.12 and 8.3)......Page 190
APPENDIX 8.C PORTFOLIO RISK MINIMIZATION ON A SPREADSHEET (Background reading: section 8.8, application 8.8, and appendix 7.C)......Page 192
9.1 ANTIDIFFERENTIATION AND THE INDEFINITE INTEGRAL (Background reading: section 8.3)......Page 194
9.2 RIEMANN SUMS (Background reading: section 9.1)......Page 195
9.3 DEFINITE INTEGRALS AND AREAS (Background reading: sections 9.1 and 9.2)......Page 199
APPLICATION 9.1: CUMULATIVE DENSITIES (Background reading: sections 2.4 and 9.3)......Page 200
APPLICATION 9.2: EXPECTED VALUE AND VARIANCE (Background reading: sections 5.6, 5.7, and 9.2 and application 9.1)......Page 202
APPLICATION 9.3: VALUING CONTINUOUS DIVIDEND PAYMENTS (Background reading: sections 4.5, 4.8, 9.2, and 9.3)......Page 203
9.4 DIFFERENTIAL EQUATIONS......Page 205
APPLICATION 9.5: SECURITY RETURNS IN CONTINUOUS TIME (Background reading: section 9.4 and application 9.3)......Page 207
APPLICATION 9.6: ANNUITIES AND GROWING ANNUITIES (Background reading: section 9.4 and application 9.5)......Page 208
EXERCISES......Page 209
APPENDIX 9.A RULES FOR FINDING INTEGRALS......Page 212
APPENDIX 9.B RIEMANN SUMS ON A SPREADSHEET (Background reading: section 9.2)......Page 213
10.1 AN INTRODUCTION TO STOCK OPTIONS......Page 217
10.2 BINOMIAL OPTION PRICING: ONE TIME PERIOD (Background reading: section 10.1 and application 7.8)......Page 219
10.3 BINOMIAL OPTION PRICING: MULTIPLE TIME PERIODS (Background reading: section 10.2)......Page 221
10.4 THE BLACK–SCHOLES OPTION PRICING MODEL (Background reading: section 10.3)......Page 224
10.5 PUTS AND VALUATION (Background reading: sections 10.1 and 10.4 and application 7.9)......Page 226
10.6 BLACK–SCHOLES MODEL SENSITIVITIES (Background reading: sections 8.4, 8.5, and 10.4)......Page 227
10.7 ESTIMATING IMPLIED VOLATILITIES (Background reading: sections 8.6 and 10.6)......Page 229
EXERCISES......Page 233
REFERENCES......Page 236
A P P E N D I X A SOLUTIONS TO EXERCISES......Page 238
A P P E N D I X B THE Z-TABLE......Page 280
A P P E N D I X C NOTATION......Page 281
A P P E N D I X D GLOSSARY......Page 284
INDEX......Page 288