Market Risk Analysis Volume II -- Contents -- List of Figures -- List of Tables -- List of Examples -- Foreword -- Preface to Volume II -- II.1 Factor Models -- II.1.1 Introduction -- II.1.2 Single Factor Models -- II.1.2.1 Single Index Model -- II.1.2.2 Estimating Portfolio Characteristics using OLS -- II.1.2.3 Estimating Portfolio Risk using EWMA -- II.1.2.4 Relationship between Beta, Correlation and Relative Volatility -- II.1.2.5 Risk Decomposition in a Single Factor Model -- II.1.3 Multi-Factor Models -- II.1.3.1 Multi-factor Models of Asset or Portfolio Returns -- II.1.3.2 Style Attribution Analysis -- II.1.3.3 General Formulation of Multi-factor Model -- II.1.3.4 Multi-factor Models of International Portfolios -- II.1.4 Case Study: Estimation of Fundamental Factor Models -- II.1.4.1 Estimating Systematic Risk for a Portfolio of US Stocks -- II.1.4.2 Multicollinearity: A Problem with Fundamental Factor Models -- II.1.4.3 Estimating Fundamental Factor Models by Orthogonal Regression -- II.1.5 Analysis of Barra Model -- II.1.5.1 Risk Indices, Descriptors and Fundamental Betas -- II.1.5.2 Model Specification and Risk Decomposition -- II.1.6 Tracking Error and Active Risk -- II.1.6.1 Ex Post versus Ex Ante Measurement of Risk and Return -- II.1.6.2 Definition of Active Returns -- II.1.6.3 Definition of Active Weights -- II.1.6.4 Ex Post Tracking Error -- II.1.6.5 Ex Post Mean-Adjusted Tracking Error -- II.1.6.6 Ex Ante Tracking Error -- II.1.6.7 Ex Ante Mean-Adjusted Tracking Error -- II.1.6.8 Clarification of the Definition of Active Risk -- II.1.7 Summary and Conclusions -- II.2 Principal Component Analysis -- II.2.1 Introduction -- II.2.2 Review of Principal Component Analysis -- II.2.2.1 Definition of Principal Components -- II.2.2.2 Principal Component Representation -- II.2.2.3 Frequently Asked Questions.

II.2.3 Case Study: PCA of UK Government Yield Curves -- II.2.3.1 Properties of UK Interest Rates -- II.2.3.2 Volatility and Correlation of UK Spot Rates -- II.2.3.3 PCA on UK Spot Rates Correlation Matrix -- II.2.3.4 Principal Component Representation -- II.2.3.5 PCA on UK Short Spot Rates Covariance Matrix -- II.2.4 Term Structure Factor Models -- II.2.4.1 Interest Rate Sensitive Portfolios -- II.2.4.2 Factor Models for Currency Forward Positions -- II.2.4.3 Factor Models for Commodity Futures Portfolios -- II.2.4.4 Application to Portfolio Immunization -- II.2.4.5 Application to Asset-Liability Management -- II.2.4.6 Application to Portfolio Risk Measurement -- II.2.4.7 Multiple Curve Factor Models -- II.2.5 Equity PCA Factor Models -- II.2.5.1 Model Structure -- II.2.5.2 Specific Risks and Dimension Reduction -- II.2.5.3 Case Study: PCA Factor Model for DJIA Portfolios -- II.2.6 Summary and Conclusions -- II.3 Classical Models of Volatility and Correlation -- II.3.1 Introduction -- II.3.2 Variance and Volatility -- II.3.2.1 Volatility and the Square-Root-of-Time Rule -- II.3.2.2 Constant Volatility Assumption -- II.3.2.3 Volatility when Returns are Autocorrelated -- II.3.2.4 Remarks about Volatility -- II.3.3 Covariance and Correlation -- II.3.3.1 Definition of Covariance and Correlation -- II.3.3.2 Correlation Pitfalls -- II.3.3.3 Covariance Matrices -- II.3.3.4 Scaling Covariance Matrices -- II.3.4 Equally Weighted Averages -- II.3.4.1 Unconditional Variance and Volatility -- II.3.4.2 Unconditional Covariance and Correlation -- II.3.4.3 Forecasting with Equally Weighted Averages -- II.3.5 Precision of Equally Weighted Estimates -- II.3.5.1 Confidence Intervals for Variance and Volatility -- II.3.5.2 Standard Error of Variance Estimator -- II.3.5.3 Standard Error of Volatility Estimator -- II.3.5.4 Standard Error of Correlation Estimator.

II.3.6 Case Study: Volatility and Correlation of US Treasuries -- II.3.6.1 Choosing the Data -- II.3.6.2 Our Data -- II.3.6.3 Effect of Sample Period -- II.3.6.4 How to Calculate Changes in Interest Rates -- II.3.7 Equally Weighted Moving Averages -- II.3.7.1 Effect of Volatility Clusters -- II.3.7.2 Pitfalls of the Equally Weighted Moving Average Method -- II.3.7.3 Three Ways to Forecast Long Term Volatility -- II.3.8 Exponentially Weighted Moving Averages -- II.3.8.1 Statistical Methodology -- II.3.8.2 Interpretation of Lambda -- II.3.8.3 Properties of EWMA Estimators -- II.3.8.4 Forecasting with EWMA -- II.3.8.5 Standard Errors for EWMA Forecasts -- II.3.8.6 RiskMetricsTM Methodology -- II.3.8.7 Orthogonal EWMA versus RiskMetrics EWMA -- II.3.9 Summary and Conclusions -- II.4 Introduction to GARCH Models -- II.4.1 Introduction -- II.4.2 The Symmetric Normal GARCH Model -- II.4.2.1 Model Specification -- II.4.2.2 Parameter Estimation -- II.4.2.3 Volatility Estimates -- II.4.2.4 GARCH Volatility Forecasts -- II.4.2.5 Imposing Long Term Volatility -- II.4.2.6 Comparison of GARCH and EWMA Volatility Models -- II.4.3 Asymmetric GARCH Models -- II.4.3.1 A-GARCH -- II.4.3.2 GJR-GARCH -- II.4.3.3 Exponential GARCH -- II.4.3.4 Analytic E-GARCH Volatility Term Structure Forecasts -- II.4.3.5 Volatility Feedback -- II.4.4 Non-Normal GARCH Models -- II.4.4.1 Student t GARCH Models -- II.4.4.2 Case Study: Comparison of GARCH Models for the FTSE 100 -- II.4.4.3 Normal Mixture GARCH Models -- II.4.4.4 Markov Switching GARCH -- II.4.5 GARCH Covariance Matrices -- II.4.5.1 Estimation of Multivariate GARCH Models -- II.4.5.2 Constant and Dynamic Conditional Correlation GARCH -- II.4.5.3 Factor GARCH -- II.4.6 Orthogonal GARCH -- II.4.6.1 Model Specification -- II.4.6.2 Case Study: A Comparison of RiskMetrics and O-GARCH.

II.4.6.3 Splicing Methods for Constructing Large Covariance Matrices -- II.4.7 Monte Carlo Simulation with GARCH Models -- II.4.7.1 Simulation with Volatility Clustering -- II.4.7.2 Simulation with Volatility Clustering Regimes -- II.4.7.3 Simulation with Correlation Clustering -- II.4.8 Applications of GARCH Models -- II.4.8.1 Option Pricing with GARCH Diffusions -- II.4.8.2 Pricing Path-Dependent European Options -- II.4.8.3 Value-at-Risk Measurement -- II.4.8.4 Estimation of Time Varying Sensitivities -- II.4.8.5 Portfolio Optimization -- II.4.9 Summary and Conclusions -- II.5 Time Series Models and Cointegration -- II.5.1 Introduction -- II.5.2 Stationary Processes -- II.5.2.1 Time Series Models -- II.5.2.2 Inversion and the Lag Operator -- II.5.2.3 Response to Shocks -- II.5.2.4 Estimation -- II.5.2.5 Prediction -- II.5.2.6 Multivariate Models for Stationary Processes -- II.5.3 Stochastic Trends -- II.5.3.1 Random Walks and Efficient Markets -- II.5.3.2 Integrated Processes and Stochastic Trends -- II.5.3.3 Deterministic Trends -- II.5.3.4 Unit Root Tests -- II.5.3.5 Unit Roots in Asset Prices -- II.5.3.6 Unit Roots in Interest Rates, Credit Spreads and Implied Volatility -- II.5.3.7 Reconciliation of Time Series and Continuous Time Models -- II.5.3.8 Unit Roots in Commodity Prices -- II.5.4 Long Term Equilibrium -- II.5.4.1 Cointegration and Correlation Compared -- II.5.4.2 Common Stochastic Trends -- II.5.4.3 Formal Definition of Cointegration -- II.5.4.4 Evidence of Cointegration in Financial Markets -- II.5.4.5 Estimation and Testing in Cointegrated Systems -- II.5.4.6 Application to Benchmark Tracking -- II.5.4.7 Case Study: Cointegration Index Tracking in the Dow Jones Index -- II.5.5 Modelling Short Term Dynamics -- II.5.5.1 Error Correction Models -- II.5.5.2 Granger Causality.

II.5.5.3 Case Study: Pairs Trading Volatility Index Futures -- II.5.6 Summary and Conclusions -- II.6 Introduction to Copulas -- II.6.1 Introduction -- II.6.2 Concordance Metrics -- II.6.2.1 Concordance -- II.6.2.2 Rank Correlations -- II.6.3 Copulas and Associated Theoretical Concepts -- II.6.3.1 Simulation of a Single Random Variable -- II.6.3.2 Definition of a Copula -- II.6.3.3 Conditional Copula Distributions and their Quantile Curves -- II.6.3.4 Tail Dependence -- II.6.3.5 Bounds for Dependence -- II.6.4 Examples of Copulas -- II.6.4.1 Normal or Gaussian Copulas -- II.6.4.2 Student t Copulas -- II.6.4.3 Normal Mixture Copulas -- II.6.4.4 Archimedean Copulas -- II.6.5 Conditional Copula Distributions and Quantile Curves -- II.6.5.1 Normal or Gaussian Copulas -- II.6.5.2 Student t Copulas -- II.6.5.3 Normal Mixture Copulas -- II.6.5.4 Archimedean Copulas -- II.6.5.5 Examples -- II.6.6 Calibrating Copulas -- II.6.6.1 Correspondence between Copulas and Rank Correlations -- II.6.6.2 Maximum Likelihood Estimation -- II.6.6.3 How to Choose the Best Copula -- II.6.7 Simulation with Copulas -- II.6.7.1 Using Conditional Copulas for Simulation -- II.6.7.2 Simulation from Elliptical Copulas -- II.6.7.3 Simulation with Normal and Student t Copulas -- II.6.7.4 Simulation from Archimedean Copulas -- II.6.8 Market Risk Applications -- II.6.8.1 Value-at-Risk Estimation -- II.6.8.2 Aggregation and Portfolio Diversification -- II.6.8.3 Using Copulas for Portfolio Optimization -- II.6.9 Summary and Conclusions -- II.7 Advanced Econometric Models -- II.7.1 Introduction -- II.7.2 Quantile Regression -- II.7.2.1 Review of Standard Regression -- II.7.2.2 What is Quantile Regression? -- II.7.2.3 Parameter Estimation in Quantile Regression -- II.7.2.4 Inference on Linear Quantile Regressions -- II.7.2.5 Using Copulas for Non-linear Quantile Regression.

II.7.3 Case Studies on Quantile Regression.