Market Risk Analysis Volume II. Practical Financial Econometrics

Transcription

1 Market Risk Analysis Volume II Practical Financial Econometrics Carol Alexander John Wiley & Sons, Ltd

2 List of Figures List of Tables List of Examples Foreword Preface to Volume II xiii xvii xx xxii xxvi H.1 Factor Models 1 II. 1.1 Introduction 1 n.1.2 Single Factor Models 2 H Single Index Model 2 II. 1.2.T Estimating Portfolio Characteristics using OLS 4 II Estimating Portfolio Risk using EWMA 6 II Relationship between Beta, Correlation and Relative Volatility 8 II Risk Decomposition in a Single Factor Model 10 II.1.3 Multi-Factor Models 11 II Multi-factor Models of Asset or Portfolio Returns Style Attribution Analysis General Formulation of Multi-factor Model 16 II Multi-factor Models of International Portfolios.17 II. 1.4 Case Study: Estimation of Fundamental Factor Models 21 II Estimating Systematic Risk for a Portfolio of US Stocks 22 II Multicollinearity: A Problem with Fundamental Factor Models 23 II Estimating Fundamental Factor Models by Orthogonal Regression 25 II. 1.5 Analysis of Barra Model. 27 II Risk Indices, Descriptors and Fundamental Betas 28 II Model Specification and Risk Decomposition 30 II. 1.6 Tracking Error and Active Risk 31 II Ex Post versus Ex Ante Measurement of Risk and Return 32 II Definition of Active Returns 32 II Definition of Active Weights. 33 II Ex Post Tracking Error 33

3 II Ex Post Mean-Adjusted Tracking Error 36 II Ex Ante Tracking Error 39 II Ex Ante Mean-Adjusted Tracking Error 40 II Clarification of the Definition of Active Risk 42 II.L7 Summary and Conclusions Principal Component Analysis Introduction Review of Principal Component Analysis Definition of Principal Components Principal Component Representation Frequently Asked Questions Case Study: PCA of UK Government Yield Curves Properties of UK Interest Rates Volatility and Correlation of UK Spot Rates PCA on UK Spot Rates Correlation Matrix Principal Component Representation PCA on UK Short Spot Rates Covariance Matrix Term Structure Factor Models Interest Rate Sensitive Portfolios Factor Models for Currency Forward Positions Factor Models for Commodity Futures Portfolios Application to Portfolio Immunization Application to Asset-Liability Management Application to Portfolio Risk Measurement Multiple Curve Factor Models Equity PCA Factor Models Model Structure Specific Risks and Dimension Reduction Case Study: PCA Factor Model for DJIA Portfolios Summary and Conclusions Classical Models of Volatility and Correlation Introduction Variance and Volatility Volatility and the Square-Root-of-Time Rule Constant Volatility Assumption Volatility when Returns are Autocorrelated Remarks about Volatility Covariance and Correlation 94 II Definition of Covariance and Correlation 94 II Correlation Pitfalls Covariance Matrices Scaling Covariance Matrices Equally Weighted Averages Unconditional. Variance and Volatility Unconditional Covariance and Correlation Forecasting with Equally Weighted Averages 103

4 II.3.5 Precision of Equally Weighted Estimates Confidence Intervals for Variance and Volatility Standard Error of Variance Estimator :5.3 Standard Error of Volatility Estimator 107 II Standard Error of Correlation Estimator Case Study: Volatility arid Correlation of US Treasuries Choosing the Data Our Data. Ill Effect of Sample Period How to Calculate Changes in Interest Rates Equally Weighted Moving Averages Effect of Volatility Clusters Pitfalls of the Equally Weighted Moving Average Method Three Ways to Forecast Long Term Volatility Exponentially Weighted Moving Averages Statistical Methodology Interpretation of Lambda Properties of EWMA Estimators Forecasting with EWMA Standard Errors for EWMA Forecasts RiskMetrics Methodology Orthogonal EWMA versus RiskMetrics EWMA Summary and Conclusions 129 II.4 Introduction to GARCH Models Introduction The Symmetric Normal GARCH Model Model Specification Parameter Estimation Volatility Estimates 141 IL4.2.4 GARCH Volatility Forecasts Imposing Long Term Volatility Comparison of GARCH and EWMA Volatility Models Asymmetric GARCH Models A-GARCH GJR-GARCH Exponential GARCH Analytic E-GARCH Volatility Term Structure Forecasts Volatility Feedback ' Non-Normal GARCH Models Student t GARCH Models. ' ' Case Study: Comparison of GARCH Models for the FTSE Normal Mixture GARCH Models Markov Switching GARCH 163 H.4.5 GARCH Covariance Matrices Estimation of Multivariate GARCH Models Constant and Dynamic Conditional Correlation GARCH Factor GARCH 169

5 Orthogonal GARCH Model Specification Case Study: A Comparison of RiskMetrics and O-GARCH 173 H Splicing Methods for Constructing Large Covariance Matrices Monte Carlo Simulation with GARCH Models Simulation with Volatility Clustering Simulation with Volatility Clustering Regimes Simulation with Correlation Clustering Applications of GARCH Models Option Pricing with GARCH Diffusions Pricing Path-Dependent European Options Value-at-Risk Measurement 192 H Estimation of Time Varying Sensitivities 193 II Portfolio Optimization 195 H.4.9 Summary and Conclusions 197 II.5 Time Series Models and Cointegration 201 H.5.1 Introduction Stationary Processes Time Series Models Inversion and the Lag Operator Response to Shocks Estimation Prediction Multivariate Models for Stationary Processes Stochastic Trends Random Walks and Efficient Markets Integrated Processes and Stochastic Trends 213 H Deterministic Trends Unit Root Tests 215 H Unit Roots in Asset Prices Unit Roots in Interest Rates, Credit Spreads and Implied Volatility Reconciliation of Time Series and Continuous Time Models Unit Roots in Commodity Prices Long Term Equilibrium Cointegration and Correlation Compared Common Stochastic Trends Formal Definition of Cointegration 228 / ' II Evidence of Cointegration in Financial Markets Estimation and Testing in Cointegrated Systems Application to Benchmark Tracking Case Study: Cointegration Index Tracking in the Dow Jones Index, Modelling Short Term Dynamics, 243 H Error Correction Models ' 243

6 Granger Causality Case Study: Pairs Trading Volatility Index Futures 247 II.5.6 Summary and Conclusions ' Introduction to Copulas Introduction Concordance Metrics ' ' Concordance Rank Correlations Copulas and Associated Theoretical Concepts Simulation of a Single Random Variable Definition of a Copula Conditional Copula Distributions and their Quantile Curves Tail Dependence Bounds for Dependence Examples of Copulas Normal or Gaussian Copulas Student t Copulas ' Normal Mixture Copulas Archimedean Copulas Conditional Copula Distributions and Quantile Curves : Normal or Gaussian Copulas Student t Copulas "Normal Mixture Copulas.,, , Archimedean Copulas Examples Calibrating Copulas Correspondence between Copulas and Rank Correlations Maximum Likelihood Estimation How to Choose the Best Copula Simulation with Copulas, Using Conditional Copulas for Simulation Simulation from Elliptical Copulas H Simulation with Normal and Student t Copulas Simulation from Archimedean Copulas Market Risk Applications Value-at-Risk Estimation Aggregation and Portfolio Diversification Using Copulas for Portfolio Optimization Summary and Conclusions Advanced Econometric Models : Introduction Quantile Regression Review of Standard Regression What is Quantile Regression? Parameter Estimation in Quantile Regression 305

7 Inference on Linear Quantile Regressions Using Copulas for Non-linear Quantile Regression 307 II.7.3 Case Studies on Quantile Regression 309 H Case Study 1: Quantile Regression of Vftse on FTSE 100 Index II Case Study 2: Hedging with Copula Quantile Regression 314 n.7.4 Other Non-Linear Regression Models Non-linear Least Squares Discrete Choice Models Markov Switching Models Testing for Structural Breaks Model Specification Financial Applications and Software Modelling Ultra High Frequency Data Data Sources and Filtering Modelling the Time between Trades Forecasting Volatility Summary and Conclusions 337 II.8 Forecasting and Model Evaluation 341 H.8.1 Introduction Returns Models Goodness of Fit Forecasting Simulating Critical Values for Test Statistics Specification Tests for Regime Switching Models Volatility Models 350 n Goodness of Fit of GARCH Models 351 n Forecasting with GARCH Volatility Models 352 n Moving Average Models Forecasting the Tails of a Distribution 356 II Confidence Intervals for Quantiles 356 n Coverage Tests Application of Coverage Tests to GARCH Models Forecasting Conditional Correlations Operational Evaluation General Backtesting Algorithm Alpha Models Portfolio Optimization Hedging with Futures Value-at-Risk Measurement Trading Implied Volatility Trading Realized Volatility Pricing and Hedging Options 373 n.8.6 Summary and Conclusions 375 References 377 Index 387