Market Risk Analysis Volume I

Transcription

1 Market Risk Analysis Volume I Quantitative Methods in Finance Carol Alexander John Wiley & Sons, Ltd

2 List of Figures List of Tables List of Examples Foreword Preface to Volume I xiii xvi xvii xix xxiii 1.1 Basic Calculus for Finance Introduction Functions and Graphs, Equations and Roots Linear and Quadratic Functions Continuous and Differentiable Real-Valued Functions Inverse Functions The Exponential Function The Natural Logarithm Differentiation and Integration Definitions Rules for Differentiation Monotonic, Concave and Convex Functions Stationary Points and Optimization Integration Analysis of Financial Returns Discrete and Continuous Time Notation Portfolio Holdings and Portfolio Weights Profit and Loss Percentage and Log Returns Geometric Brownian Motion Discrete and Continuous Compounding in Discrete Time Period Log Returns in Discrete Time Return on a Linear Portfolio Sources of Returns Functions of Several Variables Partial Derivatives: Function of Two Variables Partial Derivatives: Function of Several Variables 27

3 Stationary Points Optimization Total Derivatives Taylor Expansion Definition and Examples Risk Factors and their Sensitivities Some Financial Applications of Taylor Expansion Multivariate Taylor Expansion Summary and Conclusions Essential Linear Algebra for Finance Introduction Matrix Algebra and its Mathematical Applications Basic Terminology Laws of Matrix Algebra Singular Matrices Determinants Matrix Inversion Solution of Simultaneous Linear Equations Quadratic Forms Definite Matrices Eigenvectors and Eigenvalues Matrices as Linear Transformations Formal Definitions The Characteristic Equation Eigenvalues and Eigenvectors of a 2 x 2 Correlation Matrix Properties of Eigenvalues and Eigenvectors Using Excel to Find Eigenvalues and Eigenvectors Eigenvalue Test for Definiteness Applications to Linear Portfolios A Covariance and Correlation Matrices Portfolio Risk and Return in Matrix Notation Positive Definiteness of Covariance and Correlation Matrices Eigenvalues and Eigenvectors of Covariance and Correlation Matrices Matrix Decomposition Spectral Decomposition of a Symmetric Matrix Similarity Transforms Cholesky Decomposition LU Decomposition Principal Component Analysis Definition of Principal Components Principal Component Representation Case Study: PCA of European Equity Indices Summary and Conclusions 70

4 1.3 Probability and Statistics Introduction Basic Concepts Classical versus Bayesian Approaches Laws of Probability Density and Distribution Functions Samples and Histograms Expected Value and Sample Mean Variance Skewness and Kurtosis Quantiles, Quartiles and Percentiles Univariate Distributions Binomial Distribution Poisson and Exponential Distributions Uniform Distribution Normal Distribution Lognormal Distribution Normal Mixture Distributions Student t Distributions Sampling Distributions Generalized Extreme Value Distributions Generalized Pareto Distribution Stable Distributions Kernels Multivariate Distributions Bivariate Distributions Independent Random Variables Covariance Correlation Multivariate Continuous Distributions Multivariate Normal Distributions Bivariate Normal Mixture Distributions Multivariate Student t Distributions Introduction to Statistical Inference Quantiles, Critical Values and Confidence Intervals Central Limit Theorem Confidence Intervals Based on Student t Distribution Confidence Intervals for Variance Hypothesis Tests Tests on Means Tests on Variances Non-Parametric Tests on Distributions Maximum Likelihood Estimation The Likelihood Function Finding the Maximum Likelihood Estimates Standard Errors on Mean and Variance Estimates 133

5 Stochastic Summary Processes in Discrete and Continuous Time Stationary and Integrated Processes in Discrete Time Mean Reverting Processes and Random Walks in Continuous Time Stochastic Models for Asset Prices and Returns Jumps and the Poisson Process and Conclusions [.4 Introduction to Linear Regression Introduction Simple Linear Regression Properties Simple Linear Model Ordinary Least Squares Properties of the Error Process ANOVA and Goodness of Fit Hypothesis Tests on Coefficients Reporting the Estimated Regression Model Excel Estimation of the Simple Linear Model of OLS Estimators Estimates and Estimators Unbiasedness and Efficiency Gauss-Markov Theorem Consistency and Normality of OLS Estimators Testing for Normality Multivariate Linear Regression Simple Linear Model and OLS in Matrix Notation General Linear Model Case Study: A Multiple Regression Multiple Regression in Excel Hypothesis Testing in Multiple Regression Testing Multiple Restrictions Confidence Intervals Multicollinearity Case Study: Determinants of Credit Spreads Orthogonal Regression Autocorrelation and Heteroscedasticity Causes of Autocorrelation and Heteroscedasticity Consequences of Autocorrelation and Heteroscedasticity Testing for Autocorrelation Testing for Heteroscedasticity Generalized Least Squares Applications of Linear Regression in Finance Testing a Theory Analysing Empirical Market Behaviour Optimal Portfolio Allocation

6 Regression-Based Hedge Ratios Trading on Regression Models Summary and Conclusions Numerical Methods in Finance Introduction Iteration Method of Bisection Newton-Raphson Iteration Gradient Methods Interpolation and Extrapolation Linear and Bilinear Interpolation Polynomial Interpolation: Application to Currency Options Cubic Splines: Application to Yield Curves Optimization Least Squares Problems Likelihood Methods The EM Algorithm Case Study: Applying the EM Algorithm to Normal Mixture Densities Finite Difference Approximations First and Second Order Finite Differences Finite Difference Approximations for the Greeks Finite Difference Solutions to Partial Differential Equations Binomial Lattices Constructing the Lattice Arbitrage Free Pricing and Risk Neutral Valuation Pricing European Options Lognormal Asset Price Distributions Pricing American Options Monte Carlo Simulation Random Numbers Simulations from an Empirical or a Given Distribution Case Study: Generating Time Series of Lognormal Asset Prices Simulations on a System of Two Correlated Normal Returns Multivariate Normal and Student t Distributed Simulations Summary and Conclusions Introduction to Portfolio Theory Introduction Utility Theory Properties of Utility Functions Risk Preference How to Determine the Risk Tolerance of an Investor A Coefficients of Risk Aversion 231

7 Some Standard Utility Functions Mean-Variance Criterion Extension of the Mean-Variance Criterion to Higher Moments Portfolio Allocation Portfolio Diversification Minimum Variance Portfolios The Markowitz Problem Minimum Variance Portfolios with Many Constraints Efficient Frontier Optimal Allocations Theory of Asset Pricing Capital Market Line Capital Asset Pricing Model Security Market Line Testing the CAPM Extensions to CAPM Risk Adjusted Performance Measures CAPM RAPMs Making Decisions Using the Sharpe Ratio Adjusting the Sharpe Ratio for Autocorrelation Adjusting the Sharpe Ratio for Higher Moments Generalized Sharpe Ratio Kappa Indices, Omega and Sortino Ratio Summary and Conclusions 266 References 269 Statistical Tables 273 Index 279