6.2 Application to Term Structures* 147

6.2.1 The Trend, Tilt and Convexity Components of a

Single Yield Curve 147

6.2.2 Modelling Multiple Yield Curves with PCA 149

6.2.3 Term Structures of Futures Prices 153

6.3 Modelling Volatility Smiles and Skews 154

6.3.1 PCA of Deviations from ATM Volatility 157

6.3.2 The Dynamics of Fixed Strike Volatilities in

Different Market Regimes 159

6.3.3 Parameterization of the Volatility Surface and

Quantification of da/dS 167

6.3.4 Summary 170

6.4 Overcoming Data Problems Using PCA 171

6.4.1 Multicollinearity 172

6.4.2 Missing Data 175

Chapter 7: Covariance Matrices 179

7.1 Applications of Covariance Matrices in Risk Management 180

7.1.1 The Variance of a Linear Portfolio 180

7.1.2 Simulating Correlated Risk Factor Movements

in Derivatives Portfolios 182

7.1.3 The Need for Positive Semi-definite Covariance Matrices* 183

7.1.4 Stress Testing Portfolios Using the Covariance Matrix* 184

7.2 Applications of Covariance Matrices in Investment Analysis 186

7.2.1 Minimum Variance Portfolios 187

7.2.2 The Relationship between Risk and Return 189

7.2.3 Capital Allocation and Risk-Adjusted Performance

Measures 193

7.2.4 Modelling Attitudes to Risk 194

7.2.5 Efficient Portfolios in Practice 198

7.3 The RiskMetrics Data 201

7.4 Orthogonal Methods for Generating Covariance Matrices 204

7.4.1 Using PCA to Construct Covariance Matrices 205

7.4.2 Orthogonal EWMA 206

7.4.3 Orthogonal GARCH 210

7.4.4 Splicing Methods for Obtaining Large Covariance Matrices 221

7.4.5 Summary 227

Chapter 8: Risk Measurement in Factor Models 229

8.1 Decomposing Risk in Factor Models 230

8.1.1 The Capital Asset Pricing Model 230

8.1.2 Multi-factor Fundamental Models 233

8.1.3 Statistical Factor Models 235

8.2 Classical Risk Measurement Techniques* 236

8.2.1 The Different Perspectives of Risk Managers and

Asset Managers 236

8.2.2 Methods Relevant for Constant Parameter Assumptions 237

8.2.3 Methods Relevant for Time-Varying Parameter

Assumptions 238

8.2.4 Index Stripping 238

8.3 Bayesian Methods for Estimating Factor Sensitivities 239

8.3.1 Bayes Rule 240

8.3.2 Bayesian Estimation of Factor Models 242

8.3.3 Confidence in Beliefs and the Effect on Bayesian Estimates 245

8.4 Remarks on Factor Model Specification Procedures 246

Chapter 9: Value-at-Risk 249

9.1 Controlling the Risk in Financial Markets 250

9.1.1 The 1988 Basel Accord and the 1996 Amendment 251

9.1.2 Internal Models for Calculating Market Risk Capital Requirements 252

9.1.3 Basel 2 Proposals 255

9.2 Advantages and Limitations of Value-at-Risk 255

9.2.1 Comparison with Traditional Risk Measures 256

9.2.2 VaR-Based Trading Limits 257

9.2.3 Alternatives to VaR 257

9.3 Covariance VaR Models* 260

9.3.1 Basic Assumptions 260

9.3.2 Simple Cash Portfolios 261

9.3.3 Covariance VaR with Factor Models 262

9.3.4 Covariance VaR with Cash-Flow Maps 263

9.3.5 Aggregation 266

9.3.6 Advantages and Limitations 266

9.4 Simulation VaR Models* 267

9.4.1 Historical Simulation 268

9.4.2 Monte Carlo Simulation 270

9.4.3 Delta-Gamma Approximations 273

9.5 Model Validation 275

9.5.1 Backtesting Methodology and Regulatory

Classification 275

9.5.2 Sensitivity Analysis and Model Comparison 277

9.6 Scenario Analysis and Stress Testing* 278

9.6.1 Scenario Analysis 279

9.6.2 Probabilistic Scenario Analysis 280

9.6.3 Stress-Testing Portfolios 281

Chapter 10: Modelling Non-normal Returns 285

10.1 Testing for Non-normality in Returns Distributions* 286

10.1.1 Skewness and Excess Kurtosis 286

10.1.2 QQ Plots 288

10.2 Non-normal Distributions 290

10.2.1 Extreme Value Distributions 290

10.2.2 Hyperbolic Distributions 296

10.2.3 Normal Mixture Distributions* 297

10.3 Applications of Normal-Mixture Distributions* 301

10.3.1 Covariance VaR Measures 302

10.3.2 Term Structure Forecasts of Excess Kurtosis 303

10.3.3 Applications of Normal Mixtures to Option

Pricing and Hedging 305

Part III: Statistical Models for Financial Markets

Chapter 11: Time Series Models 315

11.1 Basic Properties of Time Series 316

11.1.1 Time Series Operators 316

11.1.2 Stationary Processes and Mean-Reversion 317

11.1.3 Integrated Processes and Random Walks 320

11.1.4 Detrending Financial Time Series Data 322

11.1.5 Unit Root Tests* 324

11.1.6 Testing for the Trend in Financial Markets 328

11.2 Univariate Time Series Models 329

11.2.1 AR Models 329

11.2.2 MA Models 331

11.2.3 ARMA Models 332

11.3 Model Identification* 333

11.3.1 Correlograms 333

11.3.2 Autocorrelation Tests 335

11.3.3 Testing Down 337

11.3.4 Forecasting with ARMA Models 338

11.4 Multivariate Time Series 340

11.4.1 Vector Autoregressions 340

11.4.2 Testing for Joint Covariance Stationarity 341

11.4.3 Granger Causality 344

Chapter 12: Cointegration 347

12.1 Introducing Cointegration 348

12.1.1 Cointegration and Correlation 349

12.1.2 Common Trends and Long-Run Equilibria 350

12.2 Testing for Cointegration* 353

12.2.1 The Engle-Granger Methodology 354

12.2.2 The Johansen Methodology 357

12.3 Error Correction and Causality 361

12.4 Cointegration in Financial Markets 366

12.4.1 Foreign Exchange 366

12.4.2 Spot and Futures 367

12.4.3 Commodities 367

12.4.4 Spread Options 367

12.4.5 Term Structures 368

12.4.6 Market Integration 368

12.5 Applications of Cointegration to Investment Analysis 369 12.5.1 Selection and Allocation 370