Market Risk Analysis, Practical Financial Econometrics

Carol Alexander is a Professor of Risk Management at the ICMA Centre, University of Reading, and Chair of the Academic Advisory Council of the Professional Risk Manager’s International Association (PRMIA). She is the author of Market Models: A Guide to Financial Data Analysis(John Wiley & Sons Ltd, 2001) and has been editor and contributor of a very large number of books in finance and mathematics, including the multi-volume Professional Risk Manager's Handbook(McGraw-Hill, 2008 and PRMIA Publications). Carol has published nearly 100 academic journal articles, book chapters and books, the majority of which focus on financial risk management and mathematical finance. Professor Alexander is one of the world's leading authorities on market risk analysis. For further details, see www.icmacentre.rdg.ac.uk/alexander.

List of Figures xiii

List of Tables xvii

List of Examples xx

Foreword xxii

Preface to Volume II xxvi

II. 1 Factor Models 1

II.1. 1 Introduction 1

II.1. 2 Single Factor Models 2

II.1.2. 1 Single Index Model 2

II.1.2. 2 Estimating Portfolio Characteristics using OLS 4

II.1.2. 3 Estimating Portfolio Risk using EWMA 6

II.1.2. 4 Relationship between Beta, Correlation and Relative Volatility 8

II.1.2. 5 Risk Decomposition in a Single Factor Model 10

II.1. 3 Multi-Factor Models 11

II.1.3. 1 Multi-factor Models of Asset or Portfolio Returns 11

II.1.3. 2 Style Attribution Analysis 13

II.1.3. 3 General Formulation of Multi-factor Model 16

II.1.3. 4 Multi-factor Models of International Portfolios 18

II.1. 4 Case Study: Estimation of Fundamental Factor Models 21

II.1.4. 1 Estimating Systematic Risk for a Portfolio of US Stocks 22

II.1.4. 2 Multicollinearity: A Problem with Fundamental Factor Models 23

II.1.4. 3 Estimating Fundamental Factor Models by Orthogonal Regression 25

II.1. 5 Analysis of Barra Model 27

II.1.5. 1 Risk Indices, Descriptors and Fundamental Betas 28

II.1.5. 2 Model Specification and Risk Decomposition 30

II.1. 6 Tracking Error and Active Risk 31

II.1.6. 1 Ex Post versus Ex Ante Measurement of Risk and Return 32

II.1.6. 2 Definition of Active Returns 32

II.1.6. 3 Definition of Active Weights 33

II.1.6. 4 Ex Post Tracking Error 33

II.1.6. 5 Ex Post Mean-Adjusted Tracking Error 36

II.1.6. 6 Ex Ante Tracking Error 39

II.1.6. 7 Ex Ante Mean-Adjusted Tracking Error 40

II.1.6. 8 Clarification of the Definition of Active Risk 42

II.1. 7 Summary and Conclusions 44

II. 2 Principal Component Analysis 47

II.2. 1 Introduction 47

II.2. 2 Review of Principal Component Analysis 48

II.2.2. 1 Definition of Principal Components 49

II 2 Principal Component Representation 49

II.2.2. 3 Frequently Asked Questions 50

II.2. 3 Case Study: PCA of UK Government Yield Curves 53

II.2.3. 1 Properties of UK Interest Rates 53

II.2.3. 2 Volatility and Correlation of UK Spot Rates 55

II.2.3. 3 PCA on UK Spot Rates Correlation Matrix 56

II.2.3. 4 Principal Component Representation 58

II.2.3. 5 PCA on UK Short Spot Rates Covariance Matrix 60

II.2. 4 Term Structure Factor Models 61

II.2.4. 1 Interest Rate Sensitive Portfolios 62

II.2.4. 2 Factor Models for Currency Forward Positions 66

II.2.4. 3 Factor Models for Commodity Futures Portfolios 70

II.2.4. 4 Application to Portfolio Immunization 71

II.2.4. 5 Application to Asset–Liability Management 72

II.2.4. 6 Application to Portfolio Risk Measurement 73

II.2.4. 7 Multiple Curve Factor Models 76

II.2. 5 Equity PCA Factor Models 80

II.2.5. 1 Model Structure 80

II.2.5. 2 Specific Risks and Dimension Reduction 81

II.2.5. 3 Case Study: PCA Factor Model for DJIA Portfolios 82

II.2. 6 Summary and Conclusions 86

II. 3 Classical Models of Volatility and Correlation 89

II.3. 1 Introduction 89

II.3. 2 Variance and Volatility 90

II.3.2. 1 Volatility and the Square-Root-of-Time Rule 90

II.3.3. 2 Constant Volatility Assumption 92

II.3.2. 3 Volatility when Returns are Autocorrelated 92

II.3.2. 4 Remarks about Volatility 93

II.3. 3 Covariance and Correlation 94

II.3.3. 1 Definition of Covariance and Correlation 94

II.3.3. 2 Correlation Pitfalls 95

II 3 Covariance Matrices 96

II.3.3. 4 Scaling Covariance Matrices 97

II.3. 4 Equally Weighted Averages 98

II.3.4. 1 Unconditional Variance and Volatility 99

II.3.4. 2 Unconditional Covariance and Correlation 102

II.3.4. 3 Forecasting with Equally Weighted Averages 103

II.3. 5 Precision of Equally Weighted Estimates 104

II.3.5. 1 Confidence Intervals for Variance and Volatility 104

II.3.5. 2 Standard Error of Variance Estimator 106

II.3.5. 3 Standard Error of Volatility Estimator 107

II.3.5. 4 Standard Error of Correlation Estimator 109

II.3. 6 Case Study: Volatility and Correlation of US Treasuries 109

II.3.6. 1 Choosing the Data 110

II.3.6. 2 Our Data 111

II.3.6. 3 Effect of Sample Period 112

II.3.6. 4 How to Calculate Changes in Interest Rates 113

II.3. 7 Equally Weighted Moving Averages 115

II.3.7. 1 Effect of Volatility Clusters 115

II.3.7. 2 Pitfalls of the Equally Weighted Moving Average Method 117

II.3.7. 3 Three Ways to Forecast Long Term Volatility 118

II.3. 8 Exponentially Weighted Moving Averages 120

II.3.8. 1 Statistical Methodology 120

II.3.8. 2 Interpretation of Lambda 121

II.3.8. 3 Properties of EWMA Estimators 122

II.3.8. 4 Forecasting with EWMA 123

II.3.8. 5 Standard Errors for EWMA Forecasts 124

II.3.8. 6 RiskMetrics TM Methodology 126

II.3.8. 7 Orthogonal EWMA versus RiskMetrics EWMA 128

II.3. 9 Summary and Conclusions 129

II. 4 Introduction to GARCH Models 131

II.4. 1 Introduction 131

II.4. 2 The Symmetric Normal GARCH Model 135

II.4.2. 1 Model Specification 135

II.4.2. 2 Parameter Estimation 137

II.4.2. 3 Volatility Estimates 141

II.4.2. 4 GARCH Volatility Forecasts 142

II.4.2. 5 Imposing Long Term Volatility 144

II.4.2. 6 Comparison of GARCH and EWMA Volatility Models 147

II.4. 3 Asymmetric GARCH Models 147

II.4.3. 1 A-garch 148

II.4.3. 2 Gjr-garch 150

II.4.3. 3 Exponential GARCH 151

II.4.3. 4 Analytic E-GARCH Volatility Term Structure Forecasts 154

II.4.3. 5 Volatility Feedback 156

II.4. 4 Non-Normal GARCH Models 157

II.4.4. 1 Student t GARCH Models 157

II.4.4. 2 Case Study: Comparison of GARCH Models for the Ftse 100 159

II.4.4. 3 Normal Mixture GARCH Models 161

II 4 Markov Switching GARCH 163

II.4. 5 GARCH Covariance Matrices 164

II.4.5. 1 Estimation of Multivariate GARCH Models 165

II.4.5. 2 Constant and Dynamic Conditional Correlation GARCH 166

II.4.5. 3 Factor GARCH 169

II.4. 6 Orthogonal GARCH 171

II.4.6. 1 Model Specification 171

II.4.6. 2 Case Study: A Comparison of RiskMetrics and O-GARCH 173

II.4.6. 3 Splicing Methods for Constructing Large Covariance Matrices 179

II.4. 7 Monte Carlo Simulation with GARCH Models 180

II.4.7. 1 Simulation with Volatility Clustering 180

II.4.7. 2 Simulation with Volatility Clustering Regimes 183

II.4.7. 3 Simulation with Correlation Clustering 185

II.4. 8 Applications of GARCH Models 188

II.4.8. 1 Option Pricing with GARCH Diffusions 188

II.4.8. 2 Pricing Path-Dependent European Options 189

II.4.8. 3 Value-at-Risk Measurement 192

II.4.8. 4 Estimation of Time Varying Sensitivities 193

II.4.8. 5 Portfolio Optimization 195

II.4. 9 Summary and Conclusions 197

II. 5 Time Series Models and Cointegration 201

II.5. 1 Introduction 201

II.5. 2 Stationary Processes 202

II.5.2. 1 Time Series Models 203

II.5.2. 2 Inversion and the Lag Operator 206

II.5.2. 3 Response to Shocks 206

II.5.2. 4 Estimation 208

II.5.2. 5 Prediction 210

II.5.2. 6 Multivariate Models for Stationary Processes 211

II.5. 3 Stochastic Trends 212

II.5.3. 1 Random Walks and Efficient Markets 212

II.5.3. 2 Integrated Processes and Stochastic Trends 213

II.5.3. 3 Deterministic Trends 214

II.5.3. 4 Unit Root Tests 215

II.5.3. 5 Unit Roots in Asset Prices 218

II.5.3. 6 Unit Roots in Interest Rates, Credit Spreads and Implied Volatility 220

II.5.3. 7 Reconciliation of Time Series and Continuous Time Models 223

II.5.3. 8 Unit Roots in Commodity Prices 224

II.5. 4 Long Term Equilibrium 225

II.5.4. 1 Cointegration and Correlation Compared 225

II.5.4. 2 Common Stochastic Trends 227

II.5.4. 3 Formal Definition of Cointegration 228

II.5.4. 4 Evidence of Cointegration in Financial Markets 229

II.5.4. 5 Estimation and Testing in Cointegrated Systems 231

II.5.4. 6 Application to Benchmark Tracking 239

II.5.4. 7 Case Study: Cointegration Index Tracking in the Dow Jones Index 240

II.5.5 Modelling Short Term Dynamics 243

II.5.5.1 Error Correction Models 243

II.5.5. 2 Granger Causality 246

II.5.5. 3 Case Study: Pairs Trading Volatility Index Futures 247

II.5. 6 Summary and Conclusions 250

II. 6 Introduction to Copulas 253

II.6. 1 Introduction 253

II.6. 2 Concordance Metrics 255

II.6.2. 1 Concordance 255

II.6.2. 2 Rank Correlations 256

II.6. 3 Copulas and Associated Theoretical Concepts 258

II.6.3. 1 Simulation of a Single Random Variable 258

II.6.3. 2 Definition of a Copula 259

II.6.3. 3 Conditional Copula Distributions and their Quantile Curves 263

II.6.3. 4 Tail Dependence 264

II.6.3. 5 Bounds for Dependence 265

II.6. 4 Examples of Copulas 266

II.6.4. 1 Normal or Gaussian Copulas 266

II.6.4. 2 Student t Copulas 268

II.6.4. 3 Normal Mixture Copulas 269

II.6.4. 4 Archimedean Copulas 271

II.6. 5 Conditional Copula Distributions and Quantile Curves 273

II.6.5. 1 Normal or Gaussian Copulas 273

II.6.5. 2 Student t Copulas 274

II.6.5. 3 Normal Mixture Copulas 275

II.6.5. 4 Archimedean Copulas 275

II.6.5. 5 Examples 276

II.6. 6 Calibrating Copulas 279

II.6.6. 1 Correspondence between Copulas and Rank Correlations 280

II.6.6. 2 Maximum Likelihood Estimation 281

II.6.6. 3 How to Choose the Best Copula 283

II.6. 7 Simulation with Copulas 285

II.6.7. 1 Using Conditional Copulas for Simulation 285

II.6.7. 2 Simulation from Elliptical Copulas 286

II.6.7. 3 Simulation with Normal and Student t Copulas 287

II.6.7. 4 Simulation from Archimedean Copulas 290

II.6. 8 Market Risk Applications 290

II.6.8. 1 Value-at-Risk Estimation 291

II.6.8. 2 Aggregation and Portfolio Diversification 292

II.6.8. 3 Using Copulas for Portfolio Optimization 295

II.6. 9 Summary and Conclusions 298

II. 7 Advanced Econometric Models 301

II.7. 1 Introduction 301

II.7. 2 Quantile Regression 303

II.7.2. 1 Review of Standard Regression 304

II.7.2. 2 What is Quantile Regression? 305

II.7.2. 3 Parameter Estimation in Quantile Regression 305

II.7.2. 4 Inference in Linear Quantile Regression 307

II.7.2. 5 Using Copulas for Non-linear Quantile Regression 307

II.7. 3 Case Studies on Quantile Regression 309

II.7.3. 1 Case Study 1: Quantile Regression of Vftse on FTSE 100 Index 309

II.7.3. 2 Case Study 2: Hedging with Copula Quantile Regression 314

II.7. 4 Other Non-Linear Regression Models 319

II.7.4. 1 Non-linear Least Squares 319

II.7.4. 2 Discrete Choice Models 321

II.7. 5 Markov Switching Models 325

II.7.5. 1 Testing for Structural Breaks 325

II.7.5. 2 Model Specification 327

II.7.5. 3 Financial Applications and Software 329

II.7. 6 Modelling Ultra High Frequency Data 330

II.7.6. 1 Data Sources and Filtering 330

II.7.6. 2 Modelling the Time between Trades 332

II.7.6. 3 Forecasting Volatility 334

II.7. 7 Summary and Conclusions 337

II. 8 Forecasting and Model Evaluation 341

II.8. 1 Introduction 341

II.8. 2 Returns Models 342

II.8.2. 1 Goodness of Fit 343

II.8.2. 2 Forecasting 347

II.8.2. 3 Simulating Critical Values for Test Statistics 348

II.8.2. 4 Specification Tests for Regime Switching Models 350

II.8. 3 Volatility Models 350

II.8.3. 1 Goodness of Fit of GARCH Models 351

II.8.3. 2 Forecasting with GARCH Volatility Models 352

II.8.3. 3 Moving Average Models 354

II.8. 4 Forecasting the Tails of a Distribution 356

II.8.4. 1 Confidence Intervals for Quantiles 356

II.8.4. 2 Coverage Tests 357

II.8.4. 3 Application of Coverage Tests to GARCH Models 360

II.8.4. 4 Forecasting Conditional Correlations 361

II.8. 5 Operational Evaluation 363

II.8.5. 1 General Backtesting Algorithm 363

II.8.5. 2 Alpha Models 365

II.8.5. 3 Portfolio Optimization 366

II.8.5. 4 Hedging with Futures 366

II.8.5. 5 Value-at-Risk Measurement 367

II.8.5. 6 Trading Implied Volatility 370

II.8.5. 7 Trading Realized Volatility 372

II.8.5. 8 Pricing and Hedging Options 373

II.8. 6 Summary and Conclusions 375

References 377

Index 387