Investment Mathematics

ANDREW ADAMS is Senior Lecturer in Finance and Director of the Centre for Financial Markets Research at the University of Edinburgh. He has studied financial markets for over thirty years, as a practitioner in the City of London and as an academic. His research interests focus mainly on investment trust pricing and risk. PHILIP BOOTH is Professor of Insurance and Risk Management at the Sir John Cass Business School, City of London and Editorial and Programme Director at the Institute of Economic Affairs. He is a former special adviser at the Bank of England and previously held the Chair in Real Estate Finance and Investment at the Sir John Cass Business School. He has a long experience of teaching and researching in the fields of investment and social insurance and is author or co-author of a number of books and papers in these fields. Philip Booth is a Fellow of the Institute of Actuaries and of the Royal Statistical Society. DAVID BOWIE is a Partner and head of quantitative analysis in the Investment Practice of Hymans Robertson Consultants & Actuaries. His focus is on the development and application of asset/liability modelling and the use of capital market theory in providing investment advice to pension funds and other institutional investors. DELLA FREETH is Reader in Education for Health Care Practice at St Bartholomew School of Nursing and Midwifery, City University, where she conducts quantitative and qualitative research.

• Preface xiiiAcknowledgements xvPart I Security Analysis 11 Compound Interest 31.1 Introduction 31.2 Accumulated values 31.3 Effective and nominal rates of interest 51.4 The accumulated value of an annuity-certain 71.5 Present values 81.6 The present value of an annuity-certain 101.7 Investment project analysis 151.8 Net present value 151.9 Internal rate of return 161.10 Discounted payback period 171.11 Analysis of decision criteria 191.12 Sensitivity analysis 19Annex 1.1 Exponents 20Annex 1.2 Geometric series 212 Fixed-interest Bonds 252.1 Introduction 252.2 Types of bond 252.3 Accrued interest 262.4 Present value of payments 282.5 Interest yield 282.6 Simple yield to maturity 292.7 Gross redemption yield 292.8 Net redemption yield 322.9 Holding period return 332.10 Volatility 332.11 Duration 352.12 The relationship between duration and volatility 352.13 Convexity 362.14 Yield curves 362.15 The expectations theory 372.16 The liquidity preference theory 382.17 The market segmentation theory 392.18 Inflation risk premium 392.19 Par yield curves 392.20 Spot and forward interest rates 392.21 Spot rates and redemption yields 402.22 Strips 412.23 Corporate bonds 423 Equities and Real Estate 433.1 Introduction 433.2 Discounted dividend model 433.3 Investment ratios 463.4 Scrip issues and stock splits 473.5 Rights issues 493.6 Market efficiency 513.7 Real estate 533.8 Yield gaps 574 Real Returns 594.1 Introduction 594.2 The calculation of real returns given a constant rate of inflation 594.3 Valuation of a series of cash flows given a constant rate of inflation 604.4 The relationship between real and nominal yields 624.5 Estimation of the rate of inflation 634.6 Real returns from equity investments 634.7 Estimation of equity values for a given real rate of return 674.8 Calculating real returns with varying rates of inflation 685 Index-linked Bonds 735.1 Introduction 735.2 Characteristics of index-linked bonds 735.3 Index-linked bonds: simple case 755.4 Index-linked bonds: a more general approach 755.5 The effect of indexation lags 795.6 A further generalisation of the model 805.7 Holding period returns 825.8 Accrued interest 845.9 The real yield gap 845.10 Estimating market expectations of inflation 865.10.1 Index-linked and conventional bonds: basic relationships 865.10.2 Problems with the simple approach to estimating inflation expectations 885.10.3 Solving the problem of internal consistency: break-even inflation rates 885.10.4 Solving the problem of differing durations 905.10.5 Forward and spot inflation expectations 906 Foreign Currency Investments 936.1 Introduction 936.2 Exchange rates 936.3 Exchanges rates, inflation rates and interest rates 946.4 Covered interest arbitrage 956.5 The operation of speculators 966.6 Purchasing power parity theory 986.7 The international Fisher effect 986.8 Interactions between exchange rates, interest rates and inflation 996.9 International bond investment 1026.10 International equity investment 1046.11 Foreign currency hedging 1047 Derivative Securities 1077.1 Introduction 1077.2 Forward and futures contracts 1077.2.1 Pricing of forwards and futures 1087.2.2 Forward pricing on a security paying no income 1097.2.3 Forward pricing on a security paying a known cash income 1107.2.4 Forward pricing on assets requiring storage 1127.2.5 Stock index futures 1127.2.6 Basis relationships 1137.2.7 Bond futures 1147.3 Swap contracts 1167.3.1 Comparative advantage argument for swaps market 1167.3.2 Pricing interest rate swap contracts 1177.3.3 Using swaps in risk management 1187.4 Option contracts 1197.4.1 Payoff diagrams for options 1207.4.2 Intrinsic value and time value 1217.4.3 Factors affecting option prices 122Part II Statistics for Investment 1258 Describing Investment Data 1278.1 Introduction 1278.2 Data sources 1278.3 Sampling and data types 1288.4 Data presentation 1298.4.1 Frequency tables 1298.4.2 Cumulative frequency tables 1318.4.3 Bar charts 1318.4.4 Histograms 1328.4.5 Stem and leaf plots 1358.4.6 Pie charts 1368.4.7 Time series graphs 1408.4.8 Cumulative frequency graphs 1418.4.9 Scatter diagrams 1418.4.10 The misrepresentation of data 1438.5 Descriptive statistics 1458.5.1 Arithmetic mean 1458.5.2 Median 1478.5.3 Mode 1478.5.4 Link between the mean, median and mode 1478.5.5 Weighted average 1488.5.6 Geometric mean 1498.5.7 Range 1498.5.8 Inter-quartile range 1508.5.9 Mean deviation (from the mean) 1508.5.10 Sample variance 1518.5.11 Sample standard deviation 1518.5.12 Coefficient of variation 1519 Modelling Investment Returns 1539.1 Introduction 1539.2 Probability 1539.2.1 Relative frequency definition of probability 1539.2.2 Subjective probability 1549.2.3 The addition rule 1549.2.4 Mutually exclusive events 1549.2.5 Conditional probability 1559.2.6 Independent events 1559.2.7 Complementary events 1569.2.8 Bayes’ theorem 1569.3 Probability distributions 1589.3.1 Cumulative distribution function (c.d.f.) 1599.3.2 The mean and variance of probability distributions 1609.3.3 Expected values of probability distributions 1609.3.4 Properties of the expected value 1619.3.5 The general linear transformation 1629.3.6 Variance 1629.3.7 Covariance 1639.3.8 Moments of random variables 1639.3.9 Probability density function (p.d.f.) 1639.4 The binomial distribution 1659.5 The normal distribution 1669.5.1 The standard normal distribution 1679.6 The normal approximation to the binomial 1699.6.1 Binomial proportions 1719.7 The lognormal distribution 1719.8 The concept of probability applied to investment returns 1729.9 Some useful probability results 1739.10 Accumulation of investments using a stochastic approach: one time period 1759.11 Accumulation of single investments with independent rates of return 1779.12 The accumulation of annual investments with independent rates of return 179Annex 9.1 Properties of the expected value 185Annex 9.2 Properties of the variance 18610 Estimating Parameters and Hypothesis Testing 18710.1 Introduction 18710.2 Unbiased estimators 18710.3 Confidence interval for the mean 18810.4 Levels of confidence 19110.5 Small samples 19110.6 Confidence interval for a proportion 19310.7 Classical hypothesis testing 19410.8 Type I and Type II errors 19610.9 Power 19610.10 Operating characteristic 19710.11 Hypothesis test for a proportion 19810.12 Some problems with classical hypothesis testing 19910.13 An alternative to classical hypothesis testing: the use of p-values 20010.14 Statistical and practical significance 201Annex 10.1 Standard error of the sample mean 20211 Measuring and Testing Comovements in Returns 20311.1 Introduction 20311.2 Correlation 20311.3 Measuring linear association 20311.4 Pearson’s product moment correlation coefficient 20511.5 Covariance and the population correlation coefficient 20711.6 Spearman’s rank correlation coefficient 20711.7 Pearson’s versus Spearman’s 20811.8 Non-linear association 20911.9 Outliers 21011.10 Significance test for r 21111.11 Significance test for Spearman’s rank correlation coefficient 21311.12 Simple linear regression 21311.13 The least-squares regression line 21411.14 The Least-squares Regression Line of X on Y 21711.15 Prediction intervals for the conditional mean 22011.16 The coefficient of determination 22211.17 Residuals 22411.18 Multiple regression 22611.19 A warning 226Part III Applications 22712 Modern Portfolio Theory and Asset Pricing 22912.1 Introduction 22912.2 Expected return and risk for a portfolio of two investments 22912.3 Expected return and risk for a portfolio of many investments 23412.4 The efficient frontier 23512.5 Indifference curves and the optimum portfolio 23612.6 Practical application of the Markowitz model 23712.7 The Market Model 23712.8 Estimation of expected returns and risks 24012.9 Portfolio selection models incorporating liabilities 24012.10 Modern portfolio theory and international diversification 24312.11 The Capital Asset Pricing Model 24512.12 International CAPM 25412.13 Arbitrage Pricing Theory 25712.14 Downside measures of risk 26212.15 Markowitz semi-variance 26412.16 Mean semi-variance efficient frontiers 265Annex 12.1 Using Excel to calculate efficient frontiers 26613 Market Indices 27113.1 Introduction 27113.2 Equity indices 27113.3 Bond indices 27913.4 Ex-dividend adjustment 28013.5 Calculating total return indices within a calendar year 28113.6 Net and gross indices 28213.7 Commercial real estate indices 28313.7.1 US real estate indices 28314 Portfolio Performance Measurement 28514.1 Introduction 28514.2 Money-weighted rate of return 28514.3 Time-weighted rate of return 28714.4 Linked internal rate of return 29114.5 Notional funds 29214.6 Consideration of risk 29414.7 Information ratios 29814.8 Survivorship bias 29914.9 Transitions 30115 Bond Analysis 30315.1 Introduction 30315.2 Volatility 30315.3 Duration 30415.4 The relationship between volatility and duration 30515.5 Factors affecting volatility and duration 30815.6 Convexity 30915.7 Non-government bonds 31415.8 Some applications of the concepts of volatility and duration 31515.9 The theory of immunisation 31715.10 Some practical issues with immunisation and matching 32016 Option Pricing Models 32316.1 Introduction 32316.2 Stock options 32316.3 The riskless hedge 32416.4 Risk neutrality 32516.5 A more general binomial model 32916.6 The value of p 33016.7 Estimating the parameters u, and n 33116.8 The Black–Scholes model 33316.9 Call options 33416.10 Computational considerations 33816.11 Put options 33916.12 Volatility 34216.13 Estimation of volatility from historical data 34216.14 Implied volatility 34316.15 Put=call parity 34416.16 Adjustments for known dividends 34716.17 Put=call parity with known dividends 34916.18 American-style options 35016.19 Option trading strategies 35116.20 Stock index options 35716.21 Bond options 35716.22 Futures options 35816.23 Currency options 35816.24 Exotic options 359Annex 16.1 The heuristic derivation of the Black–Scholes model 35917 Stochastic Investment Models 36517.1 Introduction 36517.2 Persistence in economic series 36717.3 Autocorrelation 37117.4 The random walk model 37417.5 Autoregressive models 37617.6 ARIMA models 38017.7 ARCH models 38117.8 Asset-liability modelling 38417.9 The Wilkie model 38517.10 A note on calibration 38817.11 Interest rate modelling 38817.12 Value at risk 391Compound Interest Tables 399Student’s t Distribution: Critical Points 408Areas in the Right-hand Tail of the Normal Distribution 409Index 411