Essential Mathematics for Market Risk Management

Preface xiii 1 Introduction 1 1.1 Basic Challenges in Risk Management 1 1.2 Value at Risk 3 1.3 Further Challenges in Risk Management 6 2 Applied Linear Algebra for Risk Managers 11 2.1 Vectors and Matrices 11 2.2 Matrix Algebra in Practice 17 2.3 Eigenvectors and Eigenvalues 21 2.4 Positive Definite Matrices 24 3 Probability Theory for Risk Managers 27 3.1 Univariate Theory 27 3.1.1 Random variables 27 3.1.2 Expectation 31 3.1.3 Variance 32 3.2 Multivariate Theory 33 3.2.1 The joint distribution function 33 3.2.2 The joint and marginal density functions 34 3.2.3 The notion of independence 34 3.2.4 The notion of conditional dependence 35 3.2.5 Covariance and correlation 35 3.2.6 The mean vector and covariance matrix 37 3.2.7 Linear combinations of random variables 38 3.3 The Normal Distribution 39 4 Optimization Tools 43 4.1 Background Calculus 43 4.1.1 Single-variable functions 43 4.1.2 Multivariable functions 44 4.2 Optimizing Functions 47 4.2.1 Unconstrained quadratic functions 48 4.2.2 Constrained quadratic functions 50 4.3 Over-determined Linear Systems 52 4.4 Linear Regression 54 5 Portfolio Theory I 63 5.1 Measuring Returns 63 5.1.1 A comparison of the standard and log returns 64 5.2 Setting Up the Optimal Portfolio Problem 67 5.3 Solving the Optimal Portfolio Problem 70 6 Portfolio Theory II 77 6.1 The Two-Fund Investment Service 77 6.2 A Mathematical Investigation of the Optimal Frontier 78 6.2.1 The minimum variance portfolio 78 6.2.2 Covariance of frontier portfolios 78 6.2.3 Correlation with the minimum variance portfolio 79 6.2.4 The zero-covariance portfolio 79 6.3 A Geometrical Investigation of the Optimal Frontier 80 6.3.1 Equation of a tangent to an efficient portfolio 80 6.3.2 Locating the zero-covariance portfolio 82 6.4 A Further Investigation of Covariance 83 6.5 The Optimal Portfolio Problem Revisited 86 7 The Capital Asset Pricing Model (CAPM) 91 7.1 Connecting the Portfolio Frontiers 91 7.2 The Tangent Portfolio 94 7.2.1 The market s supply of risky assets 94 7.3 The CAPM 95 7.4 Applications of CAPM 96 7.4.1 Decomposing risk 97 8 Risk Factor Modelling 101 8.1 General Factor Modelling 101 8.2 Theoretical Properties of the Factor Model 102 8.3 Models Based on Principal Component Analysis (PCA) 105 8.3.1 PCA in two dimensions 106 8.3.2 PCA in higher dimensions 112 9 The Value at Risk Concept 117 9.1 A Framework for Value at Risk 117 9.1.1 A motivating example 120 9.1.2 Defining value at risk 121 9.2 Investigating Value at Risk 122 9.2.1 The suitability of value at risk to capital allocation 124 9.3 Tail Value at Risk 126 9.4 Spectral Risk Measures 127 10 Value at Risk under a Normal Distribution 131 10.1 Calculation of Value at Risk 131 10.2 Calculation of Marginal Value at Risk 132 10.3 Calculation of Tail Value at Risk 134 10.4 Sub-additivity of Normal Value at Risk 135 11 Advanced Probability Theory for Risk Managers 137 11.1 Moments of a Random Variable 137 11.2 The Characteristic Function 140 11.2.1 Dealing with the sum of several random variables 142 11.2.2 Dealing with a scaling of a random variable 143 11.2.3 Normally distributed random variables 143 11.3 The Central Limit Theorem 145 11.4 The Moment-Generating Function 147 11.5 The Log-normal Distribution 148 12 A Survey of Useful Distribution Functions 151 12.1 The Gamma Distribution 151 12.2 The Chi-Squared Distribution 154 12.3 The Non-central Chi-Squared Distribution 157 12.4 The F-Distribution 161 12.5 The t-Distribution 164 13 A Crash Course on Financial Derivatives 169 13.1 The Black Scholes Pricing Formula 169 13.1.1 A model for asset returns 170 13.1.2 A second-order approximation 172 13.1.3 The Black Scholes formula 174 13.2 Risk-Neutral Pricing 176 13.3 A Sensitivity Analysis 179 13.3.1 Asset price sensitivity: The delta and gamma measures 179 13.3.2 Time decay sensitivity: The theta measure 182 13.3.3 The remaining sensitivity measures 183 14 Non-linear Value at Risk 185 14.1 Linear Value at Risk Revisited 185 14.2 Approximations for Non-linear Portfolios 186 14.2.1 Delta approximation for the portfolio 188 14.2.2 Gamma approximation for the portfolio 189 14.3 Value at Risk for Derivative Portfolios 190 14.3.1 Multi-factor delta approximation 190 14.3.2 Single-factor gamma approximation 191 14.3.3 Multi-factor gamma approximation 192 15 Time Series Analysis 197 15.1 Stationary Processes 197 15.1.1 Purely random processes 198 15.1.2 White noise processes 198 15.1.3 Random walk processes 199 15.2 Moving Average Processes 199 15.3 Auto-regressive Processes 201 15.4 Auto-regressive Moving Average Processes 203 16 Maximum Likelihood Estimation 207 16.1 Sample Mean and Variance 209 16.2 On the Accuracy of Statistical Estimators 211 16.2.1 Sample mean example 211 16.2.2 Sample variance example 212 16.3 The Appeal of the Maximum Likelihood Method 215 17 The Delta Method for Statistical Estimates 217 17.1 Theoretical Framework 217 17.2 Sample Variance 219 17.3 Sample Skewness and Kurtosis 221 17.3.1 Analysis of skewness 222 17.3.2 Analysis of kurtosis 223 18 Hypothesis Testing 227 18.1 The Testing Framework 227 18.1.1 The null and alternative hypotheses 227 18.1.2 Hypotheses: simple vs compound 228 18.1.3 The acceptance and rejection regions 228 18.1.4 Potential errors 229 18.1.5 Controlling the testing errors/defining the acceptance region 229 18.2 Testing Simple Hypotheses 230 18.2.1 Testing the mean when the variance is known 231 18.3 The Test Statistic 233 18.3.1 Example: Testing the mean when the variance is unknown 234 18.3.2 The p-value of a test statistic 236 18.4 Testing Compound Hypotheses 237 19 Statistical Properties of Financial Losses 241 19.1 Analysis of Sample Statistics 244 19.2 The Empirical Density and Q Q Plots 247 19.3 The Auto-correlation Function 247 19.4 The Volatility Plot 252 19.5 The Stylized Facts 253 20 Modelling Volatility 255 20.1 The RiskMetrics Model 256 20.2 ARCH Models 258 20.2.1 The ARCH(1) volatility model 260 20.3 GARCH Models 264 20.3.1 The GARCH(1, 1) volatility model 265 20.3.2 The RiskMetrics model revisited 268 20.3.3 Summary 269 20.4 Exponential GARCH 269 21 Extreme Value Theory 271 21.1 The Mathematics of Extreme Events 271 21.1.1 A naive attempt 273 21.1.2 Example 1: Exponentially distributed losses 273 21.1.3 Example 2: Normally distributed losses 274 21.1.4 Example 3: Pareto distributed losses 275 21.1.5 Example 4: Uniformly distributed losses 275 21.1.6 Example 5: Cauchy distributed losses 276 21.1.7 The extreme value theorem 277 21.2 Domains of Attraction 278 21.2.1 The Fr'echet domain of attraction 280 21.3 Extreme Value at Risk 283 21.4 Practical Issues 286 21.4.1 Parameter estimation 286 21.4.2 The choice of threshold 287 22 Simulation Models 291 22.1 Estimating the Quantile of a Distribution 291 22.1.1 Asymptotic behaviour 293 22.2 Historical Simulation 296 22.3 Monte Carlo Simulation 299 22.3.1 The Choleski algorithm 300 22.3.2 Generating random numbers 302 23 Alternative Approaches to VaR 309 23.1 The t-Distributed Assumption 309 23.2 Corrections to the Normal Assumption 313 24 Backtesting 319 24.1 Quantifying the Performance of VaR 319 24.2 Testing the Proportion of VaR Exceptions 320 24.3 Testing the Independence of VaR Exceptions 323 References 327 Index 331