Investment Mathematics

Investment Mathematics......Page 3
Contents......Page 7
Preface......Page 15
Acknowledgements......Page 17
Part I Security Analysis......Page 19
1.2 Accumulated values......Page 21
1.3 Effective and nominal rates of interest......Page 23
1.4 The accumulated value of an annuity-certain......Page 25
1.5 Present values......Page 26
1.6 The present value of an annuity-certain......Page 28
1.8 Net present value......Page 33
1.9 Internal rate of return......Page 34
1.10 Discounted payback period......Page 35
1.12 Sensitivity analysis......Page 37
Annex 1.1 Exponents......Page 38
Annex 1.2 Geometric series......Page 39
2.2 Types of bond......Page 43
2.3 Accrued interest......Page 44
2.5 Interest yield......Page 46
2.7 Gross redemption yield......Page 47
2.8 Net redemption yield......Page 50
2.10 Volatility......Page 51
2.12 The relationship between duration and volatility......Page 53
2.14 Yield curves......Page 54
2.15 The expectations theory......Page 55
2.16 The liquidity preference theory......Page 56
2.20 Spot and forward interest rates......Page 57
2.21 Spot rates and redemption yields......Page 58
2.22 Strips......Page 59
2.23 Corporate bonds......Page 60
3.2 Discounted dividend model......Page 61
3.3 Investment ratios......Page 64
3.4 Scrip issues and stock splits......Page 65
3.5 Rights issues......Page 67
3.6 Market efficiency......Page 69
3.7 Real estate......Page 71
3.8 Yield gaps......Page 75
4.2 The calculation of real returns given a constant rate of inflation......Page 77
4.3 Valuation of a series of cash flows given a constant rate of inflation......Page 78
4.4 The relationship between real and nominal yields......Page 80
4.6 Real returns from equity investments......Page 81
4.7 Estimation of equity values for a given real rate of return......Page 85
4.8 Calculating real returns with varying rates of inflation......Page 86
5.2 Characteristics of index-linked bonds......Page 91
5.4 Index-linked bonds: a more general approach......Page 93
5.5 The effect of indexation lags......Page 97
5.6 A further generalisation of the model......Page 98
5.7 Holding period returns......Page 100
5.9 The real yield gap......Page 102
5.10.1 Index-linked and conventional bonds: basic relationships......Page 104
5.10.3 Solving the problem of internal consistency: break-even inflation rates......Page 106
5.10.5 Forward and spot inflation expectations......Page 108
6.2 Exchange rates......Page 111
6.3 Exchanges rates, inflation rates and interest rates......Page 112
6.4 Covered interest arbitrage......Page 113
6.5 The operation of speculators......Page 114
6.7 The international Fisher effect......Page 116
6.8 Interactions between exchange rates, interest rates and inflation......Page 117
6.9 International bond investment......Page 120
6.11 Foreign currency hedging......Page 122
7.2 Forward and futures contracts......Page 125
7.2.1 Pricing of forwards and futures......Page 126
7.2.2 Forward pricing on a security paying no income......Page 127
7.2.3 Forward pricing on a security paying a known cash income......Page 128
7.2.5 Stock index futures......Page 130
7.2.6 Basis relationships......Page 131
7.2.7 Bond futures......Page 132
7.3.1 Comparative advantage argument for swaps market......Page 134
7.3.2 Pricing interest rate swap contracts......Page 135
7.3.3 Using swaps in risk management......Page 136
7.4 Option contracts......Page 137
7.4.1 Payoff diagrams for options......Page 138
7.4.2 Intrinsic value and time value......Page 139
7.4.3 Factors affecting option prices......Page 140
Part II Statistics for Investment......Page 143
8.2 Data sources......Page 145
8.3 Sampling and data types......Page 146
8.4.1 Frequency tables......Page 147
8.4.3 Bar charts......Page 149
8.4.4 Histograms......Page 150
8.4.5 Stem and leaf plots......Page 153
8.4.6 Pie charts......Page 154
8.4.7 Time series graphs......Page 158
8.4.9 Scatter diagrams......Page 159
8.4.10 The misrepresentation of data......Page 161
8.5.1 Arithmetic mean......Page 163
8.5.4 Link between the mean, median and mode......Page 165
8.5.5 Weighted average......Page 166
8.5.7 Range......Page 167
8.5.9 Mean deviation (from the mean)......Page 168
8.5.12 Coefficient of variation......Page 169
9.2.1 Relative frequency definition of probability......Page 171
9.2.4 Mutually exclusive events......Page 172
9.2.6 Independent events......Page 173
9.2.8 Bayes’ theorem......Page 174
9.3 Probability distributions......Page 176
9.3.1 Cumulative distribution function (c.d.f.)......Page 177
9.3.3 Expected values of probability distributions......Page 178
9.3.4 Properties of the expected value......Page 179
9.3.6 Variance......Page 180
9.3.9 Probability density function (p.d.f.)......Page 181
9.4 The binomial distribution......Page 183
9.5 The normal distribution......Page 184
9.5.1 The standard normal distribution......Page 185
9.6 The normal approximation to the binomial......Page 187
9.7 The lognormal distribution......Page 189
9.8 The concept of probability applied to investment returns......Page 190
9.9 Some useful probability results......Page 191
9.10 Accumulation of investments using a stochastic approach: one time period......Page 193
9.11 Accumulation of single investments with independent rates of return......Page 195
9.12 The accumulation of annual investments with independent rates of return......Page 197
Annex 9.1 Properties of the expected value......Page 203
Annex 9.2 Properties of the variance......Page 204
10.2 Unbiased estimators......Page 205
10.3 Confidence interval for the mean......Page 206
10.5 Small samples......Page 209
10.6 Confidence interval for a proportion......Page 211
10.7 Classical hypothesis testing......Page 212
10.9 Power......Page 214
10.10 Operating characteristic......Page 215
10.11 Hypothesis test for a proportion......Page 216
10.12 Some problems with classical hypothesis testing......Page 217
10.13 An alternative to classical hypothesis testing: the use of p-values......Page 218
10.14 Statistical and practical significance......Page 219
Annex 10.1 Standard error of the sample mean......Page 220
11.3 Measuring linear association......Page 221
11.4 Pearson’s product moment correlation coefficient......Page 223
11.6 Spearman’s rank correlation coefficient......Page 225
11.7 Pearson’s versus Spearman’s......Page 226
11.8 Non-linear association......Page 227
11.9 Outliers......Page 228
11.10 Significance test for r......Page 229
11.12 Simple linear regression......Page 231
11.13 The least-squares regression line......Page 232
11.14 The least-squares regression line of x on y......Page 235
11.15 Prediction intervals for the conditional mean......Page 238
11.16 The coefficient of determination......Page 240
11.17 Residuals......Page 242
11.19 A warning......Page 244
Part III Applications......Page 245
12.2 Expected return and risk for a portfolio of two investments......Page 247
12.3 Expected return and risk for a portfolio of many investments......Page 252
12.4 The efficient frontier......Page 253
12.5 Indifference curves and the optimum portfolio......Page 254
12.7 The Market Model......Page 255
12.9 Portfolio selection models incorporating liabilities......Page 258
12.10 Modern portfolio theory and international diversification......Page 261
12.11 The Capital Asset Pricing Model......Page 263
12.12 International CAPM......Page 272
12.13 Arbitrage Pricing Theory......Page 275
12.14 Downside measures of risk......Page 280
12.15 Markowitz semi-variance......Page 282
12.16 Mean semi-variance efficient frontiers......Page 283
Annex 12.1 Using Excel to calculate efficient frontiers......Page 284
13.2 Equity indices......Page 289
13.3 Bond indices......Page 297
13.4 Ex-dividend adjustment......Page 298
13.5 Calculating total return indices within a calendar year......Page 299
13.6 Net and gross indices......Page 300
13.7.1 US real estate indices......Page 301
14.2 Money-weighted rate of return......Page 303
14.3 Time-weighted rate of return......Page 305
14.4 Linked internal rate of return......Page 309
14.5 Notional funds......Page 310
14.6 Consideration of risk......Page 312
14.7 Information ratios......Page 316
14.8 Survivorship bias......Page 317
14.9 Transitions......Page 319
15.2 Volatility......Page 321
15.3 Duration......Page 322
15.4 The relationship between volatility and duration......Page 323
15.5 Factors affecting volatility and duration......Page 326
15.6 Convexity......Page 327
15.7 Non-government bonds......Page 332
15.8 Some applications of the concepts of volatility and duration......Page 333
15.9 The theory of immunisation......Page 335
15.10 Some practical issues with immunisation and matching......Page 338
16.2 Stock options......Page 341
16.3 The riskless hedge......Page 342
16.4 Risk neutrality......Page 343
16.5 A more general binomial model......Page 347
16.6 The value of p......Page 348
16.7 Estimating the parameters u, u and n......Page 349
16.8 The Black–Scholes model......Page 351
16.9 Call options......Page 352
16.10 Computational considerations......Page 356
16.11 Put options......Page 357
16.13 Estimation of volatility from historical data......Page 360
16.14 Implied volatility......Page 361
16.15 Put/call parity......Page 362
16.16 Adjustments for known dividends......Page 365
16.17 Put/call parity with known dividends......Page 367
16.18 American-style options......Page 368
16.19 Option trading strategies......Page 369
16.21 Bond options......Page 375
16.23 Currency options......Page 376
Annex 16.1 The heuristic derivation of the Black–Scholes model......Page 377
17.1 Introduction......Page 383
17.2 Persistence in economic series......Page 385
17.3 Autocorrelation......Page 389
17.4 The random walk model......Page 392
17.5 Autoregressive models......Page 394
17.6 ARIMA models......Page 398
17.7 ARCH models......Page 399
17.8 Asset-liability modelling......Page 402
17.9 The Wilkie model......Page 403
17.11 Interest rate modelling......Page 406
17.12 Value at risk......Page 409
Compound Interest Tables......Page 417
Student’s t Distribution: Critical Points......Page 426
Areas in the Right-hand Tail of the Normal Distribution......Page 427
Index......Page 429