# Market Risk Analysis Volume IV. Value-at-Risk Models

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1 Market Risk Analysis Volume IV Value-at-Risk Models Carol Alexander John Wiley & Sons, Ltd

2 List of Figures List of Tables List of Examples Foreword Preface to Volume IV xiii xvi xxi xxv xxix IV.l Value at Risk and Other Risk Metrics 1 IV. 1.1 Introduction 1 IV, 1.2 An Overview of Market Risk Assessment 4 IV Risk Measurement in Banks 4 IV Risk Measurement in Portfolio Management 6 IV Risk Measurement in Large Corporations 7 IV. 1.3 Downside and Quantile Risk Metrics 9 IV Semi-Standard Deviation and Second Order Lower Partial Moment 9 IV Other Lower Partial Moments 10 IV Quantile Risk Metrics 11 IV. 1.4 Defining Value at Risk 13 IV Confidence Level and Risk Horizon 13 IV Discounted P&L 15 IV Mathematical Definition of VaR 15 IV. 1.5 Foundations of Value-at-Risk Measurement 17 IV Normal Linear VaR Formula: Portfolio Level 18 IV Static Portfolios 20 IV Scaling VaR 21 IV Discounting and the Expected Return 23 IV. 1.6 Risk Factor Value at Risk 25 IV Motivation 26 IV Normal Linear Equity VaR 27 IV1.6.3 Normal Linear Interest Rate VaR 29

3 IV. 1.7 Decomposition of Value at Risk 30 IV Systematic and Specific VaR 31 IV Stand-alone VaR 31 I V.I. 7.3 Marginal and Incremental VaR 32 IV. 1.8 Risk Metrics Associated with Value at Risk 33 IV Benchmark VaR 34 IV Conditional VaR: Expected Tail Loss and Expected Shortfall 35 IV Coherent Risk Metrics 38 IV.1.9 Introduction to Value-at-Risk Models 41 IV Normal Linear VaR 41 IV Historical Simulation 42 IV Monte Carlo Simulation 44 IV Case Study: VaR of the S&P 500 Index 45 IV Summary and Conclusions 47 IV.2 Parametric Linear VaR Models 53 IV.2.1 Introduction 53 IV.2.2 Foundations of Normal Linear Value at Risk 56 IV Understanding the Normal Linear VaR Formula 56 IV Analytic Formula for Normal VaR when Returns are Autocorrelated 60 IV Systematic Normal Linear VaR 63 IV Stand-Alone Normal Linear VaR 64 IV Marginal and Incremental Normal Linear VaR 66 IV.2.3 Normal Linear Value at Risk for Cash-How Maps 67 IV Normal Linear Interest Rate VaR 67 IV Calculating PV01 68 IV Approximating Marginal and Incremental VaR 70 IV Disaggregating Normal Linear Interest Rate VaR 72 IV Normal Linear Credit Spread VaR 75 IV.2.4 Case Study: PC Value at Risk of a UK Fixed Income Portfolio 79 IV Calculating the Volatility and VaR of the Portfolio 80 IV Combining Cash-Flow Mapping with PCA 81 IV Advantages of Using PC Factors for Interest Rate VaR 85 IV.2.5 Normal Linear Value at Risk for Stock Portfolios 85 ~ IV Cash Positions on a Few Stocks 86 IV Systematic and Specific VaR for Domestic Stock Portfolios 87 IV Empirical Estimation of Specific VaR 90 IV EWMA Estimates of Specific VaR 91 IV.2.6 Systematic Value-at-Risk Decomposition for Stock Portfolios 93 IV Portfolios Exposed to One Foreign Currency 93 IV Portfolios Exposed to Several Foreign Currencies 97 IV Interest Rate VaR of Equity Portfolios 100 IV Hedging the Risks of International Equity Portfolios 101 IV.2.7 Case Study: Normal Linear Value at Risk for Commodity Futures 103

4 ix IV.2.8 Student t Distributed Linear Value at Risk 106 IV Effect of Leptokurtosis and Skewness on VaR 106 IV Student t Linear VaR Formula 107 IV Empirical Examples of Student t Linear VaR 109 IV.2.9 Linear Value at Risk with Mixture Distributions 111 IV Mixture Distributions 111 IV Mixture Linear VaR Formula 113 IV Mixture Parameter Estimation 114 IV Examples of Mixture Linear VaR 115 IV Normal Mixture Risk Factor VaR 119 IV.2.10 Exponential Weighting with Parametric Linear Value at Risk 121 IV Exponentially Weighted Moving Averages 121 IV EWMA VaR at the Portfolio Level 124 IV RiskMetrics VaR Methodology 126 IV.2.11 Expected Tail Loss (Conditional VaR) 128 IV ETL in the Normal Linear VaR Model 129 IV ETL in the Student t Linear VaR Model 130 IV ETL in the Normal Mixture Linear VaR Model 132 IV ETL under a Mixture of Student t Distributions 133 IV.2.12 Case Study: Credit Spread Parametric Linear Value at Risk and ETL 135 IV The itraxx Europe Index 135 IV VaR Estimates 137 IV.2.13 Summary and Conclusions 138 IV.3 Historical Simulation 141 IV.3.1 Introduction 141 IV.3.2 Properties of Historical Value at Risk 144 IV Definition of Historical VaR 144 IV Sample Size and Data Frequency 145 IV Power Law Scale Exponents 146 IV Case Study: Scale'Exponents for Major Risk Factors 147 IV Scaling Historical VaR for Linear Portfolios 150 IV Errors from Square-Root Scaling of Historical VaR 151 IV Overlapping Data and Multi-Step Historical Simulation 151 IV.3.3 Improving the Accuracy of Historical Value at Risk 152 IV Case Study: Equally Weighted Historical and Linear VaR 153 IV Exponential Weighting of Return Distributions 156 IV Volatility Adjustment 158 IV Filtered Historical Simulation 163 IV.3.4 Precision of Historical Value at Risk at Extreme Quantiles 165 IV Kernel Fitting 165 IV Extreme Value Distributions 167 IV Cornish-Fisher Approximation 170 IV Johnson Distributions 172 IV.3.5 Historical Value at Risk for Linear Portfolios 175 IV Historical VaR for Cash Flows 176 IV Total, Systematic and Specific VaR of a Stock Portfolio 179

5 IV Equity and Forex VaR of an International Stock Portfolio 185 IV Interest Rate and Forex VaR of an International Bond Position 190 IV Case Study: Historical VaR for a Crack Spread Trader 192 IV.3.6 Estimating Expected Tail Loss in the Historical Value-at-Risk Model 195 IV Parametric Historical ETL 195 IV Empirical Results on Historical ETL 195 IV Disaggregation of Historical ETL 197 IV.3.7 Summary and Conclusions 198 IV.4 Monte Carlo VaR 201 IV.4.1 Introduction 201 IV.4.2 Basic Concepts 203 IV Pseudo-Random Number Generation 203 IV Low Discrepancy Sequences 204 IV Variance Reduction 206 IV Sampling from Univariate Distributions 211 IV Sampling from Multivariate Distributions 213 IV Introduction to Monte Carlo VaR 213 IV.4.3 Modelling Dynamic Properties in Risk Factor Returns 215 IV Multi-Step Monte Carlo 215 IV Volatility Clustering and Mean Reversion 218 IV Regime Switching Models 223 IV.4.4 Modelling Risk Factor Dependence 225 IV Multivariate Distributions for i.i.d. Returns 226 IV Principal Component Analysis 230 IV Behavioural Models 232 IV Case Study: Modelling the Price - Volatility Relationship 232 IV.4.5 Monte Carlo Value at Risk for Linear Portfolios 233 IV Algorithms for VaR and ETL 235 IV Cash-Flow Portfolios: Copula VaR and PC VaR 236 IV Equity Portfolios: 'Crash' Scenario VaR 239 IV Currency Portfolios: VaR with Volatility Clustering 241 IV.4.6 Summary and Conclusions 244 IV.5 Value at Risk for Option Portfolios 247 IV.5.1 Introduction 247 IV.5.2 Risk Characteristics of Option Portfolios 250 IV Gamma Effects 250 IV Delta and Vega Effects 252 IV Theta and Rho Effects 253 IV Static and Dynamic VaR Estimates 254 IV.5.3 Analytic Value-at-Risk Approximations 257 IV Delta Approximation and Delta-Normal VaR 257 IV P&L Distributions for Option Portfolios 259 IV Delta-Gamma VaR 260

6 IV.5.4 Historical Value at Risk for Option Portfolios 262 IV VaR and ETL with Exact Revaluation 263 IV Dynamically Hedged Option Portfolios 272 IV Greeks Approximation 273 IV Historical VaR for Path-Dependent Options 278 IV Case Study: Historical VaR for an Energy Options Trading Book 280 IV.5.5 Monte Carlo Value at Risk for Option Portfolios 282 IV Monte Carlo VaR and ETL with Exact Revaluation 283 IV Risk Factor Models for Simulating Options VaR 287 IV Capturing Non-normality and Non-linearity 287 IV Capturing Gamma, Vega and Theta Effects 290 IV Path Dependency 292 IV Option Portfolios with a Single Underlying 296 IV Option Portfolios with Several Underlyings 299 IV Case Study: Monte Carlo VaR for an Energy Options Trading Book 302 IV.5.6 Summary and Conclusions 307 IV.6 Risk Model Risk 311 IV.6.1 Introduction 311 IV.6.2 Sources of Risk Model Risk 313 IV Risk Factor Mapping 314 IV Risk Factor or Asset Returns Model 319 IV VaR Resolution Method 322 IV Scaling 323 IV.6.3 Estimation Risk 324 IV Distribution of VaR Estimators in Parametric Linear Models 324 IV Distribution of VaR Estimators in Simulation Models 328 IV.6.4 Model Validation. 332 IV Backtesting Methodology 332 IV Guidelines for Backtesting from Banking Regulators 335 IV Coverage Tests 337 IV Backtests Based on Regression 340 IV Backtesting ETL Forecasts 344 IV Bias Statistics for Normal Linear VaR 345 IV Distribution Forecasts 348 IV Some Backtesting Results 351 IV.6.5 Summary and Conclusions 353 IV.7 Scenario Analysis and Stress Testing 357 IV.7.1 Introduction 357 IV.7.2 Scenarios on Financial Risk Factors 359 IV Broad Categorization of Scenarios 360 IV Historical Scenarios 361 IV Hypothetical Scenarios 362 IV Distribution Scenario Design 366

7 xii Contents IV.7.3 Scenario Value at Risk and Expected Tail Loss 367 IV Normal Distribution Scenarios 367 IV Compound Distribution Scenario VaR 371 IV Bayesian VaR 375 IV.7.4 Introduction to Stress Testing 378 IV Regulatory Guidelines 379 IV Systemic Risk 381 IV Stress Tests Based on Worst Case Loss 381 IV.7.5 A Coherent Framework for Stress Testing 384 IV VaR Based on Stressed Covariance Matrices 385 IV Generating Hypothetical Covariance Matrices 388 IV Stress Tests Based on Principal Component Analysis 390 IV Modelling Liquidity Risk 392 IV Incorporating Volatility Clustering 397 IV.7.6 Summary and Conclusions 398 IV.8 Capital Allocation 401 IV.8.1 Introduction 401 IV.8.2 Minimum Market Risk Capital Requirements for Banks 403 IV Basel Accords 404 IV Banking and Trading Book Accounting 405 IV Regulatory Framework for Market Risk 406 IV Internal Models 408 IV Standardized Rules 411 IV Incremental Risk Charge 412 IV.8.3 Economic Capital Allocation 416 IV Measurement of Economic Capital 416 IV Banking Applications of Economic Capital 421 IV Aggregation Risk 422 IV Risk Adjusted Performance Measures 424 IV Optimal Allocation of Economic Capital 430 IV.8.4 Summary and Conclusions 433 References 437 Index 441

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