Everything you need to know in order to manage risk effectively within your organization You cannot afford to ignore the explosion in mathematical finance in your quest to remain competitive. This exciting branch of mathematics has very direct practical implications: when a new model is tested and implemented it can have an immediate impact on the financial environment. With risk management top of the agenda for many organizations, this book is essential reading for getting to grips with the mathematical story behind the subject of financial risk management. It will take you on a journey--from the early ideas of risk quantification up to today's sophisticated models and approaches to business risk management. To help you investigate the most up-to-date, pioneering developments in modern risk management, the book presents statistical theories and shows you how to put statistical tools into action to investigate areas such as the design of mathematical models for financial volatility or calculating the value at risk for an investment portfolio.* Respected academic author Simon Hubbert is the youngest director of a financial engineering program in the U.K. He brings his industry experience to his practical approach to risk analysis* Captures the essential mathematical tools needed to explore many common risk management problems* Website with model simulations and source code enables you to put models of risk management into practice* Plunges into the world of high-risk finance and examines the crucial relationship between the risk and the potential reward of holding a portfolio of risky financial assets This book is your one-stop-shop for effective 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 N
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