About the authorDR SIMON HUBBERT is a lecturer in Mathematics and Mathematical Finance at Birkbeck College, University of London, where he is currently the programme director for the graduate diploma in Financial Engineering. He has taught masters level courses on Risk Management and Financial Mathematics for many years and also has valuable experience in the financial industry having engaged in consultation work with IBM global business services and as a risk analyst for the debt management office, a branch of HM Treasury.

• Preface xiii 1 Introduction 11.1 Basic Challenges in Risk Management 11.2 Value at Risk 31.3 Further Challenges in Risk Management 62 Applied Linear Algebra for Risk Managers 112.1 Vectors and Matrices 112.2 Matrix Algebra in Practice 172.3 Eigenvectors and Eigenvalues 212.4 Positive Definite Matrices 243 Probability Theory for Risk Managers 273.1 Univariate Theory 273.1.1 Random variables 273.1.2 Expectation 313.1.3 Variance 323.2 Multivariate Theory 333.2.1 The joint distribution function 333.2.2 The joint and marginal density functions 343.2.3 The notion of independence 343.2.4 The notion of conditional dependence 353.2.5 Covariance and correlation 353.2.6 The mean vector and covariance matrix 373.2.7 Linear combinations of random variables 383.3 The Normal Distribution 394 Optimization Tools 434.1 Background Calculus 434.1.1 Single-variable functions 434.1.2 Multivariable functions 444.2 Optimizing Functions 474.2.1 Unconstrained quadratic functions 484.2.2 Constrained quadratic functions 504.3 Over-determined Linear Systems 524.4 Linear Regression 545 Portfolio Theory I 635.1 Measuring Returns 635.1.1 A comparison of the standard and log returns 645.2 Setting Up the Optimal Portfolio Problem 675.3 Solving the Optimal Portfolio Problem 706 Portfolio Theory II 776.1 The Two-Fund Investment Service 776.2 A Mathematical Investigation of the Optimal Frontier 786.2.1 The minimum variance portfolio 786.2.2 Covariance of frontier portfolios 786.2.3 Correlation with the minimum variance portfolio 796.2.4 The zero-covariance portfolio 796.3 A Geometrical Investigation of the Optimal Frontier 806.3.1 Equation of a tangent to an efficient portfolio 806.3.2 Locating the zero-covariance portfolio 826.4 A Further Investigation of Covariance 836.5 The Optimal Portfolio Problem Revisited 867 The Capital Asset Pricing Model (CAPM) 917.1 Connecting the Portfolio Frontiers 917.2 The Tangent Portfolio 947.2.1 The market’s supply of risky assets 947.3 The CAPM 957.4 Applications of CAPM 967.4.1 Decomposing risk 978 Risk Factor Modelling 1018.1 General Factor Modelling 1018.2 Theoretical Properties of the Factor Model 1028.3 Models Based on Principal Component Analysis (PCA) 1058.3.1 PCA in two dimensions 1068.3.2 PCA in higher dimensions 1129 The Value at Risk Concept 1179.1 A Framework for Value at Risk 1179.1.1 A motivating example 1209.1.2 Defining value at risk 1219.2 Investigating Value at Risk 1229.2.1 The suitability of value at risk to capital allocation 1249.3 Tail Value at Risk 1269.4 Spectral Risk Measures 12710 Value at Risk under a Normal Distribution 13110.1 Calculation of Value at Risk 13110.2 Calculation of Marginal Value at Risk 13210.3 Calculation of Tail Value at Risk 13410.4 Sub-additivity of Normal Value at Risk 13511 Advanced Probability Theory for Risk Managers 13711.1 Moments of a Random Variable 13711.2 The Characteristic Function 14011.2.1 Dealing with the sum of several random variables 14211.2.2 Dealing with a scaling of a random variable 14311.2.3 Normally distributed random variables 14311.3 The Central Limit Theorem 14511.4 The Moment-Generating Function 14711.5 The Log-normal Distribution 14812 A Survey of Useful Distribution Functions 15112.1 The Gamma Distribution 15112.2 The Chi-Squared Distribution 15412.3 The Non-central Chi-Squared Distribution 15712.4 The F-Distribution 16112.5 The t-Distribution 16413 A Crash Course on Financial Derivatives 16913.1 The Black–Scholes Pricing Formula 16913.1.1 A model for asset returns 17013.1.2 A second-order approximation 17213.1.3 The Black–Scholes formula 17413.2 Risk-Neutral Pricing 17613.3 A Sensitivity Analysis 17913.3.1 Asset price sensitivity: The delta and gamma measures 17913.3.2 Time decay sensitivity: The theta measure 18213.3.3 The remaining sensitivity measures 18314 Non-linear Value at Risk 18514.1 Linear Value at Risk Revisited 18514.2 Approximations for Non-linear Portfolios 18614.2.1 Delta approximation for the portfolio 18814.2.2 Gamma approximation for the portfolio 18914.3 Value at Risk for Derivative Portfolios 19014.3.1 Multi-factor delta approximation 19014.3.2 Single-factor gamma approximation 19114.3.3 Multi-factor gamma approximation 19215 Time Series Analysis 19715.1 Stationary Processes 19715.1.1 Purely random processes 19815.1.2 White noise processes 19815.1.3 Random walk processes 19915.2 Moving Average Processes 19915.3 Auto-regressive Processes 20115.4 Auto-regressive Moving Average Processes 20316 Maximum Likelihood Estimation 20716.1 Sample Mean and Variance 20916.2 On the Accuracy of Statistical Estimators 21116.2.1 Sample mean example 21116.2.2 Sample variance example 21216.3 The Appeal of the Maximum Likelihood Method 21517 The Delta Method for Statistical Estimates 21717.1 Theoretical Framework 21717.2 Sample Variance 21917.3 Sample Skewness and Kurtosis 22117.3.1 Analysis of skewness 22217.3.2 Analysis of kurtosis 22318 Hypothesis Testing 22718.1 The Testing Framework 22718.1.1 The null and alternative hypotheses 22718.1.2 Hypotheses: simple vs compound 22818.1.3 The acceptance and rejection regions 22818.1.4 Potential errors 22918.1.5 Controlling the testing errors/defining the acceptance region 22918.2 Testing Simple Hypotheses 23018.2.1 Testing the mean when the variance is known 23118.3 The Test Statistic 23318.3.1 Example: Testing the mean when the variance is unknown 23418.3.2 The p-value of a test statistic 23618.4 Testing Compound Hypotheses 23719 Statistical Properties of Financial Losses 24119.1 Analysis of Sample Statistics 24419.2 The Empirical Density and Q–Q Plots 24719.3 The Auto-correlation Function 24719.4 The Volatility Plot 25219.5 The Stylized Facts 25320 Modelling Volatility 25520.1 The RiskMetrics Model 25620.2 ARCH Models 25820.2.1 The ARCH(1) volatility model 26020.3 GARCH Models 26420.3.1 The GARCH(1, 1) volatility model 26520.3.2 The RiskMetrics model revisited 26820.3.3 Summary 26920.4 Exponential GARCH 26921 Extreme Value Theory 27121.1 The Mathematics of Extreme Events 27121.1.1 A naive attempt 27321.1.2 Example 1: Exponentially distributed losses 27321.1.3 Example 2: Normally distributed losses 27421.1.4 Example 3: Pareto distributed losses 27521.1.5 Example 4: Uniformly distributed losses 27521.1.6 Example 5: Cauchy distributed losses 27621.1.7 The extreme value theorem 27721.2 Domains of Attraction 27821.2.1 The Fr´echet domain of attraction 28021.3 Extreme Value at Risk 28321.4 Practical Issues 28621.4.1 Parameter estimation 28621.4.2 The choice of threshold 28722 Simulation Models 29122.1 Estimating the Quantile of a Distribution 29122.1.1 Asymptotic behaviour 29322.2 Historical Simulation 29622.3 Monte Carlo Simulation 29922.3.1 The Choleski algorithm 30022.3.2 Generating random numbers 30223 Alternative Approaches to VaR 30923.1 The t-Distributed Assumption 30923.2 Corrections to the Normal Assumption 31324 Backtesting 31924.1 Quantifying the Performance of VaR 31924.2 Testing the Proportion of VaR Exceptions 32024.3 Testing the Independence of VaR Exceptions 323References 327Index 331

About the authorDR SIMON HUBBERT is a lecturer in Mathematics and Mathematical Finance at Birkbeck College, University of London, where he is currently the programme director for the graduate diploma in Financial Engineering. He has taught masters level courses on Risk Management and Financial Mathematics for many years and also has valuable experience in the financial industry having engaged in consultation work with IBM global business services and as a risk analyst for the debt management office, a branch of HM Treasury.

• Preface xiii 1 Introduction 11.1 Basic Challenges in Risk Management 11.2 Value at Risk 31.3 Further Challenges in Risk Management 62 Applied Linear Algebra for Risk Managers 112.1 Vectors and Matrices 112.2 Matrix Algebra in Practice 172.3 Eigenvectors and Eigenvalues 212.4 Positive Definite Matrices 243 Probability Theory for Risk Managers 273.1 Univariate Theory 273.1.1 Random variables 273.1.2 Expectation 313.1.3 Variance 323.2 Multivariate Theory 333.2.1 The joint distribution function 333.2.2 The joint and marginal density functions 343.2.3 The notion of independence 343.2.4 The notion of conditional dependence 353.2.5 Covariance and correlation 353.2.6 The mean vector and covariance matrix 373.2.7 Linear combinations of random variables 383.3 The Normal Distribution 394 Optimization Tools 434.1 Background Calculus 434.1.1 Single-variable functions 434.1.2 Multivariable functions 444.2 Optimizing Functions 474.2.1 Unconstrained quadratic functions 484.2.2 Constrained quadratic functions 504.3 Over-determined Linear Systems 524.4 Linear Regression 545 Portfolio Theory I 635.1 Measuring Returns 635.1.1 A comparison of the standard and log returns 645.2 Setting Up the Optimal Portfolio Problem 675.3 Solving the Optimal Portfolio Problem 706 Portfolio Theory II 776.1 The Two-Fund Investment Service 776.2 A Mathematical Investigation of the Optimal Frontier 786.2.1 The minimum variance portfolio 786.2.2 Covariance of frontier portfolios 786.2.3 Correlation with the minimum variance portfolio 796.2.4 The zero-covariance portfolio 796.3 A Geometrical Investigation of the Optimal Frontier 806.3.1 Equation of a tangent to an efficient portfolio 806.3.2 Locating the zero-covariance portfolio 826.4 A Further Investigation of Covariance 836.5 The Optimal Portfolio Problem Revisited 867 The Capital Asset Pricing Model (CAPM) 917.1 Connecting the Portfolio Frontiers 917.2 The Tangent Portfolio 947.2.1 The market’s supply of risky assets 947.3 The CAPM 957.4 Applications of CAPM 967.4.1 Decomposing risk 978 Risk Factor Modelling 1018.1 General Factor Modelling 1018.2 Theoretical Properties of the Factor Model 1028.3 Models Based on Principal Component Analysis (PCA) 1058.3.1 PCA in two dimensions 1068.3.2 PCA in higher dimensions 1129 The Value at Risk Concept 1179.1 A Framework for Value at Risk 1179.1.1 A motivating example 1209.1.2 Defining value at risk 1219.2 Investigating Value at Risk 1229.2.1 The suitability of value at risk to capital allocation 1249.3 Tail Value at Risk 1269.4 Spectral Risk Measures 12710 Value at Risk under a Normal Distribution 13110.1 Calculation of Value at Risk 13110.2 Calculation of Marginal Value at Risk 13210.3 Calculation of Tail Value at Risk 13410.4 Sub-additivity of Normal Value at Risk 13511 Advanced Probability Theory for Risk Managers 13711.1 Moments of a Random Variable 13711.2 The Characteristic Function 14011.2.1 Dealing with the sum of several random variables 14211.2.2 Dealing with a scaling of a random variable 14311.2.3 Normally distributed random variables 14311.3 The Central Limit Theorem 14511.4 The Moment-Generating Function 14711.5 The Log-normal Distribution 14812 A Survey of Useful Distribution Functions 15112.1 The Gamma Distribution 15112.2 The Chi-Squared Distribution 15412.3 The Non-central Chi-Squared Distribution 15712.4 The F-Distribution 16112.5 The t-Distribution 16413 A Crash Course on Financial Derivatives 16913.1 The Black–Scholes Pricing Formula 16913.1.1 A model for asset returns 17013.1.2 A second-order approximation 17213.1.3 The Black–Scholes formula 17413.2 Risk-Neutral Pricing 17613.3 A Sensitivity Analysis 17913.3.1 Asset price sensitivity: The delta and gamma measures 17913.3.2 Time decay sensitivity: The theta measure 18213.3.3 The remaining sensitivity measures 18314 Non-linear Value at Risk 18514.1 Linear Value at Risk Revisited 18514.2 Approximations for Non-linear Portfolios 18614.2.1 Delta approximation for the portfolio 18814.2.2 Gamma approximation for the portfolio 18914.3 Value at Risk for Derivative Portfolios 19014.3.1 Multi-factor delta approximation 19014.3.2 Single-factor gamma approximation 19114.3.3 Multi-factor gamma approximation 19215 Time Series Analysis 19715.1 Stationary Processes 19715.1.1 Purely random processes 19815.1.2 White noise processes 19815.1.3 Random walk processes 19915.2 Moving Average Processes 19915.3 Auto-regressive Processes 20115.4 Auto-regressive Moving Average Processes 20316 Maximum Likelihood Estimation 20716.1 Sample Mean and Variance 20916.2 On the Accuracy of Statistical Estimators 21116.2.1 Sample mean example 21116.2.2 Sample variance example 21216.3 The Appeal of the Maximum Likelihood Method 21517 The Delta Method for Statistical Estimates 21717.1 Theoretical Framework 21717.2 Sample Variance 21917.3 Sample Skewness and Kurtosis 22117.3.1 Analysis of skewness 22217.3.2 Analysis of kurtosis 22318 Hypothesis Testing 22718.1 The Testing Framework 22718.1.1 The null and alternative hypotheses 22718.1.2 Hypotheses: simple vs compound 22818.1.3 The acceptance and rejection regions 22818.1.4 Potential errors 22918.1.5 Controlling the testing errors/defining the acceptance region 22918.2 Testing Simple Hypotheses 23018.2.1 Testing the mean when the variance is known 23118.3 The Test Statistic 23318.3.1 Example: Testing the mean when the variance is unknown 23418.3.2 The p-value of a test statistic 23618.4 Testing Compound Hypotheses 23719 Statistical Properties of Financial Losses 24119.1 Analysis of Sample Statistics 24419.2 The Empirical Density and Q–Q Plots 24719.3 The Auto-correlation Function 24719.4 The Volatility Plot 25219.5 The Stylized Facts 25320 Modelling Volatility 25520.1 The RiskMetrics Model 25620.2 ARCH Models 25820.2.1 The ARCH(1) volatility model 26020.3 GARCH Models 26420.3.1 The GARCH(1, 1) volatility model 26520.3.2 The RiskMetrics model revisited 26820.3.3 Summary 26920.4 Exponential GARCH 26921 Extreme Value Theory 27121.1 The Mathematics of Extreme Events 27121.1.1 A naive attempt 27321.1.2 Example 1: Exponentially distributed losses 27321.1.3 Example 2: Normally distributed losses 27421.1.4 Example 3: Pareto distributed losses 27521.1.5 Example 4: Uniformly distributed losses 27521.1.6 Example 5: Cauchy distributed losses 27621.1.7 The extreme value theorem 27721.2 Domains of Attraction 27821.2.1 The Fr´echet domain of attraction 28021.3 Extreme Value at Risk 28321.4 Practical Issues 28621.4.1 Parameter estimation 28621.4.2 The choice of threshold 28722 Simulation Models 29122.1 Estimating the Quantile of a Distribution 29122.1.1 Asymptotic behaviour 29322.2 Historical Simulation 29622.3 Monte Carlo Simulation 29922.3.1 The Choleski algorithm 30022.3.2 Generating random numbers 30223 Alternative Approaches to VaR 30923.1 The t-Distributed Assumption 30923.2 Corrections to the Normal Assumption 31324 Backtesting 31924.1 Quantifying the Performance of VaR 31924.2 Testing the Proportion of VaR Exceptions 32024.3 Testing the Independence of VaR Exceptions 323References 327Index 331

About the authorDR SIMON HUBBERT is a lecturer in Mathematics and Mathematical Finance at Birkbeck College, University of London, where he is currently the programme director for the graduate diploma in Financial Engineering. He has taught masters level courses on Risk Management and Financial Mathematics for many years and also has valuable experience in the financial industry having engaged in consultation work with IBM global business services and as a risk analyst for the debt management office, a branch of HM Treasury.

• Preface xiii 1 Introduction 11.1 Basic Challenges in Risk Management 11.2 Value at Risk 31.3 Further Challenges in Risk Management 62 Applied Linear Algebra for Risk Managers 112.1 Vectors and Matrices 112.2 Matrix Algebra in Practice 172.3 Eigenvectors and Eigenvalues 212.4 Positive Definite Matrices 243 Probability Theory for Risk Managers 273.1 Univariate Theory 273.1.1 Random variables 273.1.2 Expectation 313.1.3 Variance 323.2 Multivariate Theory 333.2.1 The joint distribution function 333.2.2 The joint and marginal density functions 343.2.3 The notion of independence 343.2.4 The notion of conditional dependence 353.2.5 Covariance and correlation 353.2.6 The mean vector and covariance matrix 373.2.7 Linear combinations of random variables 383.3 The Normal Distribution 394 Optimization Tools 434.1 Background Calculus 434.1.1 Single-variable functions 434.1.2 Multivariable functions 444.2 Optimizing Functions 474.2.1 Unconstrained quadratic functions 484.2.2 Constrained quadratic functions 504.3 Over-determined Linear Systems 524.4 Linear Regression 545 Portfolio Theory I 635.1 Measuring Returns 635.1.1 A comparison of the standard and log returns 645.2 Setting Up the Optimal Portfolio Problem 675.3 Solving the Optimal Portfolio Problem 706 Portfolio Theory II 776.1 The Two-Fund Investment Service 776.2 A Mathematical Investigation of the Optimal Frontier 786.2.1 The minimum variance portfolio 786.2.2 Covariance of frontier portfolios 786.2.3 Correlation with the minimum variance portfolio 796.2.4 The zero-covariance portfolio 796.3 A Geometrical Investigation of the Optimal Frontier 806.3.1 Equation of a tangent to an efficient portfolio 806.3.2 Locating the zero-covariance portfolio 826.4 A Further Investigation of Covariance 836.5 The Optimal Portfolio Problem Revisited 867 The Capital Asset Pricing Model (CAPM) 917.1 Connecting the Portfolio Frontiers 917.2 The Tangent Portfolio 947.2.1 The market’s supply of risky assets 947.3 The CAPM 957.4 Applications of CAPM 967.4.1 Decomposing risk 978 Risk Factor Modelling 1018.1 General Factor Modelling 1018.2 Theoretical Properties of the Factor Model 1028.3 Models Based on Principal Component Analysis (PCA) 1058.3.1 PCA in two dimensions 1068.3.2 PCA in higher dimensions 1129 The Value at Risk Concept 1179.1 A Framework for Value at Risk 1179.1.1 A motivating example 1209.1.2 Defining value at risk 1219.2 Investigating Value at Risk 1229.2.1 The suitability of value at risk to capital allocation 1249.3 Tail Value at Risk 1269.4 Spectral Risk Measures 12710 Value at Risk under a Normal Distribution 13110.1 Calculation of Value at Risk 13110.2 Calculation of Marginal Value at Risk 13210.3 Calculation of Tail Value at Risk 13410.4 Sub-additivity of Normal Value at Risk 13511 Advanced Probability Theory for Risk Managers 13711.1 Moments of a Random Variable 13711.2 The Characteristic Function 14011.2.1 Dealing with the sum of several random variables 14211.2.2 Dealing with a scaling of a random variable 14311.2.3 Normally distributed random variables 14311.3 The Central Limit Theorem 14511.4 The Moment-Generating Function 14711.5 The Log-normal Distribution 14812 A Survey of Useful Distribution Functions 15112.1 The Gamma Distribution 15112.2 The Chi-Squared Distribution 15412.3 The Non-central Chi-Squared Distribution 15712.4 The F-Distribution 16112.5 The t-Distribution 16413 A Crash Course on Financial Derivatives 16913.1 The Black–Scholes Pricing Formula 16913.1.1 A model for asset returns 17013.1.2 A second-order approximation 17213.1.3 The Black–Scholes formula 17413.2 Risk-Neutral Pricing 17613.3 A Sensitivity Analysis 17913.3.1 Asset price sensitivity: The delta and gamma measures 17913.3.2 Time decay sensitivity: The theta measure 18213.3.3 The remaining sensitivity measures 18314 Non-linear Value at Risk 18514.1 Linear Value at Risk Revisited 18514.2 Approximations for Non-linear Portfolios 18614.2.1 Delta approximation for the portfolio 18814.2.2 Gamma approximation for the portfolio 18914.3 Value at Risk for Derivative Portfolios 19014.3.1 Multi-factor delta approximation 19014.3.2 Single-factor gamma approximation 19114.3.3 Multi-factor gamma approximation 19215 Time Series Analysis 19715.1 Stationary Processes 19715.1.1 Purely random processes 19815.1.2 White noise processes 19815.1.3 Random walk processes 19915.2 Moving Average Processes 19915.3 Auto-regressive Processes 20115.4 Auto-regressive Moving Average Processes 20316 Maximum Likelihood Estimation 20716.1 Sample Mean and Variance 20916.2 On the Accuracy of Statistical Estimators 21116.2.1 Sample mean example 21116.2.2 Sample variance example 21216.3 The Appeal of the Maximum Likelihood Method 21517 The Delta Method for Statistical Estimates 21717.1 Theoretical Framework 21717.2 Sample Variance 21917.3 Sample Skewness and Kurtosis 22117.3.1 Analysis of skewness 22217.3.2 Analysis of kurtosis 22318 Hypothesis Testing 22718.1 The Testing Framework 22718.1.1 The null and alternative hypotheses 22718.1.2 Hypotheses: simple vs compound 22818.1.3 The acceptance and rejection regions 22818.1.4 Potential errors 22918.1.5 Controlling the testing errors/defining the acceptance region 22918.2 Testing Simple Hypotheses 23018.2.1 Testing the mean when the variance is known 23118.3 The Test Statistic 23318.3.1 Example: Testing the mean when the variance is unknown 23418.3.2 The p-value of a test statistic 23618.4 Testing Compound Hypotheses 23719 Statistical Properties of Financial Losses 24119.1 Analysis of Sample Statistics 24419.2 The Empirical Density and Q–Q Plots 24719.3 The Auto-correlation Function 24719.4 The Volatility Plot 25219.5 The Stylized Facts 25320 Modelling Volatility 25520.1 The RiskMetrics Model 25620.2 ARCH Models 25820.2.1 The ARCH(1) volatility model 26020.3 GARCH Models 26420.3.1 The GARCH(1, 1) volatility model 26520.3.2 The RiskMetrics model revisited 26820.3.3 Summary 26920.4 Exponential GARCH 26921 Extreme Value Theory 27121.1 The Mathematics of Extreme Events 27121.1.1 A naive attempt 27321.1.2 Example 1: Exponentially distributed losses 27321.1.3 Example 2: Normally distributed losses 27421.1.4 Example 3: Pareto distributed losses 27521.1.5 Example 4: Uniformly distributed losses 27521.1.6 Example 5: Cauchy distributed losses 27621.1.7 The extreme value theorem 27721.2 Domains of Attraction 27821.2.1 The Fr´echet domain of attraction 28021.3 Extreme Value at Risk 28321.4 Practical Issues 28621.4.1 Parameter estimation 28621.4.2 The choice of threshold 28722 Simulation Models 29122.1 Estimating the Quantile of a Distribution 29122.1.1 Asymptotic behaviour 29322.2 Historical Simulation 29622.3 Monte Carlo Simulation 29922.3.1 The Choleski algorithm 30022.3.2 Generating random numbers 30223 Alternative Approaches to VaR 30923.1 The t-Distributed Assumption 30923.2 Corrections to the Normal Assumption 31324 Backtesting 31924.1 Quantifying the Performance of VaR 31924.2 Testing the Proportion of VaR Exceptions 32024.3 Testing the Independence of VaR Exceptions 323References 327Index 331