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2 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
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