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InhaltsangabePreface 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 Singlevariable 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 Overdetermined 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 TwoFund 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 Subadditivity 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 Lognormal Distribution 148 12 A Survey of Useful Distribution Functions 151 12.1 The Gamma Distribution 151 12.2 The ChiSquared Distribution 154 12.3 The Noncentral ChiSquared Distribution 157 12.4 The FDistribution 161 12.5 The tDistribution 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 RiskNeutral Pricing 176

In finance the universally held view is that the more risk we take the more reward we stand to gain but, just as importantly, the greater the chance of loss. The role of the financial risk manager is to be aware of the presence of risk, to understand how it can damage a potential investment and, most of all, be able to reduce the exposure to it in order to avert a potential disaster. Essential Mathematics for Market Risk Management provides readers with the mathematical tools for managing and controlling the major sources of risk in the financial markets. Unlike most books on investment risk management which tend to be either panoptic in their coverage or narrowly focused on advanced mathematical procedures, this book offers a thorough understanding of the basic mathematical concepts and procedures required to satisfy the two key criteria of financial risk management: to ensure a healthy return on investment for a tolerable amount of risk, and to insulate a portfolio against catastrophic market events. To this end, Dr Simon Hubbert, has drawn from his previous industrial experience to develop a format which clearly and methodically * Traces the evolution of quantitative risk management - from Markowitz's landmark solution to the portfolio problem in the 1950s, to the emergence of Value at Risk (VaR) in the mid 1990s and its subsequent impact. * Provides the basic mathematical tools needed to understand and solve common risk management problems, including applied linear algebra, probability theory and mathematical optimization. * Introduces and explains the statistical theory, tools and techniques behind cutting-edge research into financial risk management taking place in professional and academic institutions globally. * Explores a range of advanced topics in quantitative risk management, including derivative pricing, non-linear Value at Risk, volatility modelling and extreme value theory. By focusing on the key issues a typical financial risk manager faces on both a daily and long-term basis - from monitoring portfolio performance to modelling the volatility of specific assets - this book is essential reading for finance professionals and students who recognize the need to be conversant in modern quantitative methods for financial risk management.