Investment Mathematics provides an introductory analysis of investments from a quantitative viewpoint, drawing together many of the tools and techniques required by investment professionals. Using these techniques, the authors provide simple analyses of a number of securities including fixed interest bonds, equities, index-linked bonds, foreign currency and derivatives. The book concludes with coverage of other applications, including modern portfolio theory, portfolio performance measurement and stochastic investment models.

Table of Contents:
Preface xiii Acknowledgements xv Part I Security Analysis 1 1 Compound Interest 3 1.1 Introduction 3 1.2 Accumulated values 3 1.3 Effective and nominal rates of interest 5 1.4 The accumulated value of an annuity-certain 7 1.5 Present values 8 1.6 The present value of an annuity-certain 10 1.7 Investment project analysis 15 1.8 Net present value 15 1.9 Internal rate of return 16 1.10 Discounted payback period 17 1.11 Analysis of decision criteria 19 1.12 Sensitivity analysis 19 Annex 1.1 Exponents 20 Annex 1.2 Geometric series 21 2 Fixed-interest Bonds 25 2.1 Introduction 25 2.2 Types of bond 25 2.3 Accrued interest 26 2.4 Present value of payments 28 2.5 Interest yield 28 2.6 Simple yield to maturity 29 2.7 Gross redemption yield 29 2.8 Net redemption yield 32 2.9 Holding period return 33 2.10 Volatility 33 2.11 Duration 35 2.12 The relationship between duration and volatility 35 2.13 Convexity 36 2.14 Yield curves 36 2.15 The expectations theory 37 2.16 The liquidity preference theory 38 2.17 The market segmentation theory 39 2.18 Inflation risk premium 39 2.19 Par yield curves 39 2.20 Spot and forward interest rates 39 2.21 Spot rates and redemption yields 40 2.22 Strips 41 2.23 Corporate bonds 42 3 Equities and Real Estate 43 3.1 Introduction 43 3.2 Discounted dividend model 43 3.3 Investment ratios 46 3.4 Scrip issues and stock splits 47 3.5 Rights issues 49 3.6 Market efficiency 51 3.7 Real estate 53 3.8 Yield gaps 57 4 Real Returns 59 4.1 Introduction 59 4.2 The calculation of real returns given a constant rate of inflation 59 4.3 Valuation of a series of cash flows given a constant rate of inflation 60 4.4 The relationship between real and nominal yields 62 4.5 Estimation of the rate of inflation 63 4.6 Real returns from equity investments 63 4.7 Estimation of equity values for a given real rate of return 67 4.8 Calculating real returns with varying rates of inflation 68 5 Index-linked Bonds 73 5.1 Introduction 73 5.2 Characteristics of index-linked bonds 73 5.3 Index-linked bonds: simple case 75 5.4 Index-linked bonds: a more general approach 75 5.5 The effect of indexation lags 79 5.6 A further generalisation of the model 80 5.7 Holding period returns 82 5.8 Accrued interest 84 5.9 The real yield gap 84 5.10 Estimating market expectations of inflation 86 5.10.1 Index-linked and conventional bonds: basic relationships 86 5.10.2 Problems with the simple approach to estimating inflation expectations 88 5.10.3 Solving the problem of internal consistency: break-even inflation rates 88 5.10.4 Solving the problem of differing durations 90 5.10.5 Forward and spot inflation expectations 90 6 Foreign Currency Investments 93 6.1 Introduction 93 6.2 Exchange rates 93 6.3 Exchanges rates, inflation rates and interest rates 94 6.4 Covered interest arbitrage 95 6.5 The operation of speculators 96 6.6 Purchasing power parity theory 98 6.7 The international Fisher effect 98 6.8 Interactions between exchange rates, interest rates and inflation 99 6.9 International bond investment 102 6.10 International equity investment 104 6.11 Foreign currency hedging 104 7 Derivative Securities 107 7.1 Introduction 107 7.2 Forward and futures contracts 107 7.2.1 Pricing of forwards and futures 108 7.2.2 Forward pricing on a security paying no income 109 7.2.3 Forward pricing on a security paying a known cash income 110 7.2.4 Forward pricing on assets requiring storage 112 7.2.5 Stock index futures 112 7.2.6 Basis relationships 113 7.2.7 Bond futures 114 7.3 Swap contracts 116 7.3.1 Comparative advantage argument for swaps market 116 7.3.2 Pricing interest rate swap contracts 117 7.3.3 Using swaps in risk management 118 7.4 Option contracts 119 7.4.1 Payoff diagrams for options 120 7.4.2 Intrinsic value and time value 121 7.4.3 Factors affecting option prices 122 Part II Statistics for Investment 125 8 Describing Investment Data 127 8.1 Introduction 127 8.2 Data sources 127 8.3 Sampling and data types 128 8.4 Data presentation 129 8.4.1 Frequency tables 129 8.4.2 Cumulative frequency tables 131 8.4.3 Bar charts 131 8.4.4 Histograms 132 8.4.5 Stem and leaf plots 135 8.4.6 Pie charts 136 8.4.7 Time series graphs 140 8.4.8 Cumulative frequency graphs 141 8.4.9 Scatter diagrams 141 8.4.10 The misrepresentation of data 143 8.5 Descriptive statistics 145 8.5.1 Arithmetic mean 145 8.5.2 Median 147 8.5.3 Mode 147 8.5.4 Link between the mean, median and mode 147 8.5.5 Weighted average 148 8.5.6 Geometric mean 149 8.5.7 Range 149 8.5.8 Inter-quartile range 150 8.5.9 Mean deviation (from the mean) 150 8.5.10 Sample variance 151 8.5.11 Sample standard deviation 151 8.5.12 Coefficient of variation 151 9 Modelling Investment Returns 153 9.1 Introduction 153 9.2 Probability 153 9.2.1 Relative frequency definition of probability 153 9.2.2 Subjective probability 154 9.2.3 The addition rule 154 9.2.4 Mutually exclusive events 154 9.2.5 Conditional probability 155 9.2.6 Independent events 155 9.2.7 Complementary events 156 9.2.8 Bayes’ theorem 156 9.3 Probability distributions 158 9.3.1 Cumulative distribution function (c.d.f.) 159 9.3.2 The mean and variance of probability distributions 160 9.3.3 Expected values of probability distributions 160 9.3.4 Properties of the expected value 161 9.3.5 The general linear transformation 162 9.3.6 Variance 162 9.3.7 Covariance 163 9.3.8 Moments of random variables 163 9.3.9 Probability density function (p.d.f.) 163 9.4 The binomial distribution 165 9.5 The normal distribution 166 9.5.1 The standard normal distribution 167 9.6 The normal approximation to the binomial 169 9.6.1 Binomial proportions 171 9.7 The lognormal distribution 171 9.8 The concept of probability applied to investment returns 172 9.9 Some useful probability results 173 9.10 Accumulation of investments using a stochastic approach: one time period 175 9.11 Accumulation of single investments with independent rates of return 177 9.12 The accumulation of annual investments with independent rates of return 179 Annex 9.1 Properties of the expected value 185 Annex 9.2 Properties of the variance 186 10 Estimating Parameters and Hypothesis Testing 187 10.1 Introduction 187 10.2 Unbiased estimators 187 10.3 Confidence interval for the mean 188 10.4 Levels of confidence 191 10.5 Small samples 191 10.6 Confidence interval for a proportion 193 10.7 Classical hypothesis testing 194 10.8 Type I and Type II errors 196 10.9 Power 196 10.10 Operating characteristic 197 10.11 Hypothesis test for a proportion 198 10.12 Some problems with classical hypothesis testing 199 10.13 An alternative to classical hypothesis testing: the use of p-values 200 10.14 Statistical and practical significance 201 Annex 10.1 Standard error of the sample mean 202 11 Measuring and Testing Comovements in Returns 203 11.1 Introduction 203 11.2 Correlation 203 11.3 Measuring linear association 203 11.4 Pearson’s product moment correlation coefficient 205 11.5 Covariance and the population correlation coefficient 207 11.6 Spearman’s rank correlation coefficient 207 11.7 Pearson’s versus Spearman’s 208 11.8 Non-linear association 209 11.9 Outliers 210 11.10 Significance test for r 211 11.11 Significance test for Spearman’s rank correlation coefficient 213 11.12 Simple linear regression 213 11.13 The least-squares regression line 214 11.14 The Least-squares Regression Line of X on Y 217 11.15 Prediction intervals for the conditional mean 220 11.16 The coefficient of determination 222 11.17 Residuals 224 11.18 Multiple regression 226 11.19 A warning 226 Part III Applications 227 12 Modern Portfolio Theory and Asset Pricing 229 12.1 Introduction 229 12.2 Expected return and risk for a portfolio of two investments 229 12.3 Expected return and risk for a portfolio of many investments 234 12.4 The efficient frontier 235 12.5 Indifference curves and the optimum portfolio 236 12.6 Practical application of the Markowitz model 237 12.7 The Market Model 237 12.8 Estimation of expected returns and risks 240 12.9 Portfolio selection models incorporating liabilities 240 12.10 Modern portfolio theory and international diversification 243 12.11 The Capital Asset Pricing Model 245 12.12 International CAPM 254 12.13 Arbitrage Pricing Theory 257 12.14 Downside measures of risk 262 12.15 Markowitz semi-variance 264 12.16 Mean semi-variance efficient frontiers 265 Annex 12.1 Using Excel to calculate efficient frontiers 266 13 Market Indices 271 13.1 Introduction 271 13.2 Equity indices 271 13.3 Bond indices 279 13.4 Ex-dividend adjustment 280 13.5 Calculating total return indices within a calendar year 281 13.6 Net and gross indices 282 13.7 Commercial real estate indices 283 13.7.1 US real estate indices 283 14 Portfolio Performance Measurement 285 14.1 Introduction 285 14.2 Money-weighted rate of return 285 14.3 Time-weighted rate of return 287 14.4 Linked internal rate of return 291 14.5 Notional funds 292 14.6 Consideration of risk 294 14.7 Information ratios 298 14.8 Survivorship bias 299 14.9 Transitions 301 15 Bond Analysis 303 15.1 Introduction 303 15.2 Volatility 303 15.3 Duration 304 15.4 The relationship between volatility and duration 305 15.5 Factors affecting volatility and duration 308 15.6 Convexity 309 15.7 Non-government bonds 314 15.8 Some applications of the concepts of volatility and duration 315 15.9 The theory of immunisation 317 15.10 Some practical issues with immunisation and matching 320 16 Option Pricing Models 323 16.1 Introduction 323 16.2 Stock options 323 16.3 The riskless hedge 324 16.4 Risk neutrality 325 16.5 A more general binomial model 329 16.6 The value of p 330 16.7 Estimating the parameters u, and n 331 16.8 The Black–Scholes model 333 16.9 Call options 334 16.10 Computational considerations 338 16.11 Put options 339 16.12 Volatility 342 16.13 Estimation of volatility from historical data 342 16.14 Implied volatility 343 16.15 Put=call parity 344 16.16 Adjustments for known dividends 347 16.17 Put=call parity with known dividends 349 16.18 American-style options 350 16.19 Option trading strategies 351 16.20 Stock index options 357 16.21 Bond options 357 16.22 Futures options 358 16.23 Currency options 358 16.24 Exotic options 359 Annex 16.1 The heuristic derivation of the Black–Scholes model 359 17 Stochastic Investment Models 365 17.1 Introduction 365 17.2 Persistence in economic series 367 17.3 Autocorrelation 371 17.4 The random walk model 374 17.5 Autoregressive models 376 17.6 ARIMA models 380 17.7 ARCH models 381 17.8 Asset-liability modelling 384 17.9 The Wilkie model 385 17.10 A note on calibration 388 17.11 Interest rate modelling 388 17.12 Value at risk 391 Compound Interest Tables 399 Student’s t Distribution: Critical Points 408 Areas in the Right-hand Tail of the Normal Distribution 409 Index 411