KEVIN DOWD is Professor of Financial Risk Management at Nottingham University Business School and a member of the School's Centre for Research in Risk and Insurance Studies.

• Preface xiAcknowledgements xix1 The Risk Measurement Revolution 11.1 Contributory Factors 11.1.1 A Volatile Environment 11.1.2 Growth in Trading Activity 21.1.3 Advances in Information Technology 21.2 Risk Measurement Before VaR 31.2.1 Gap Analysis 31.2.2 Duration Analysis 41.2.3 Scenario Analysis 41.2.4 Portfolio Theory 51.2.5 Derivatives Risk Measures 61.3 Value at Risk 71.3.1 The Origin and Development of VaR 71.3.2 Attractions of VaR 101.3.3 Criticisms of VaR 111.4 Recommended Reading 122 Measures of Financial Risk 132.1 The Mean–Variance Framework for Measuring Financial Risk 132.1.1 The Normality Assumption 132.1.2 Limitations of the Normality Assumption 152.1.3 Traditional Approaches to Financial Risk Measurement 182.1.3.1 Portfolio Theory 182.1.3.2 Duration Approaches to Fixed-income Risk Measurement 182.2 Value at Risk 192.2.1 VaR Basics 192.2.2 Choice of VaR Parameters 242.2.3 Limitations of VaR as a Risk Measure 252.2.3.1 VaR Uninformative of Tail Losses 252.2.3.2 VaR Can Create Perverse Incentive Structures 262.2.3.3 VaR Can Discourage Diversification 272.2.3.4 VaR Not Sub-additive 272.3 Expected Tail Loss 282.3.1 Coherent Risk Measures 282.3.2 The Expected Tail Loss 292.4 Conclusions 332.5 Recommended Reading 333 Basic Issues in Measuring Market Risk 353.1 Data 353.1.1 Profit/Loss Data 353.1.2 Loss/Profit Data 353.1.3 Arithmetic Returns Data 363.1.4 Geometric Returns Data 363.2 Estimating Historical Simulation VaR 363.3 Estimating Parametric VaR 373.3.1 Estimating VaR with Normally Distributed Profits/Losses 383.3.2 Estimating VaR with Normally Distributed Arithmetic Returns 393.3.3 Estimating Lognormal VaR 403.4 Estimating Expected Tail Loss 423.5 Summary 44Appendix: Mapping Positions to Risk Factors 45A3.1 Selecting Core Instruments or Factors 46A3.1.1 Selecting Core Instruments 46A3.1.2 Selecting Core Factors 47A3.2 Mapping Positions and VaR Estimation 47A3.2.1 The Basic Building Blocks 47A3.2.1.1 Basic FX Positions 47A3.2.1.2 Basic Equity Positions 48A3.2.1.3 Zero-coupon Bonds 50A3.2.1.4 Basic Forward/Futures 51A3.2.2 More Complex Positions 52A3.3 Recommended Reading 534 Non-parametric VaR and ETL 554.1 Compiling Historical Simulation Data 554.2 Estimation of Historical Simulation VaR and ETL 564.2.1 Basic Historical Simulation 564.2.2 Estimating Curves and Surfaces for VaR and ETL 574.3 Estimating Confidence Intervals for Historical Simulation VaR and ETL 584.3.1 A Quantile Standard Error Approach to the Estimation of Confidence Intervals for HS VaR and ETL 584.3.2 An Order Statistics Approach to the Estimation of Confidence Intervals for HS VaR and ETL 584.3.3 A Bootstrap Approach to the Estimation of Confidence Intervals for HS VaR and ETL 594.4 Weighted Historical Simulation 614.4.1 Age-weighted Historical Simulation 624.4.2 Volatility-weighted Historical Simulation 634.4.3 Filtered Historical Simulation 644.5 Advantages and Disadvantages of Historical Simulation 664.5.1 Advantages 664.5.2 Disadvantages 674.5.2.1 Total Dependence on the Data Set 674.5.2.2 Problems of Data Period Length 684.6 Principal Components Approaches to VaR and ETL Estimation 684.7 Conclusions 694.8 Recommended Reading 705 Parametric VaR and ETL 715.1 Normal VaR and ETL 725.1.1 General Features 725.1.2 Disadvantages of Normality 765.2 The Student t-distribution 775.3 The Lognormal Distribution 785.4 Extreme Value Distributions 815.4.1 The Generalised Extreme Value Distribution 815.4.2 The Peaks Over Threshold (Generalised Pareto) Approach 825.5 The Multivariate Normal Variance–Covariance Approach 845.6 Conclusions 865.7 Recommended Reading 87Appendix: Delta–Gamma and Related Approximations 88A5.1 Delta–normal Approaches 88A5.2 Delta–Gamma Approaches 90A5.2.1 The Delta–Gamma Approximation 90A5.2.2 The Delta–Gamma Normal Approach 90A5.2.3 Wilson’s Delta–Gamma Approach 91A5.2.4 Other Delta–Gamma Approaches 93A5.3 Conclusions 94A5.4 Recommended Reading 956 Simulation Approaches to VaR and ETL Estimation 976.1 Options VaR and ETL 976.1.1 Preliminary Considerations 976.1.2 An Example: Estimating the VaR and ETL of an American Put 986.1.3 Refining MCS Estimation of Options VaR and ETL 996.2 Estimating VaR by Simulating Principal Components 996.2.1 Basic Principal Components Simulation 996.2.2 Scenario Simulation 1006.3 Fixed-income VaR and ETL 1026.3.1 General Considerations 1026.3.1.1 Stochastic Processes for Interest Rates 1026.3.1.2 The Term Structure of Interest Rates 1036.3.2 A General Approach to Fixed-income VaR and ETL 1036.4 Estimating VaR and ETL under a Dynamic Portfolio Strategy 1056.5 Estimating Credit-related Risks with Simulation Methods 1076.6 Estimating Insurance Risks with Simulation Methods 1096.7 Estimating Pensions Risks with Simulation Methods 1106.7.1 Estimating Risks of Defined-benefit Pension Plans 1116.7.2 Estimating Risks of Defined-contribution Pension Plans 1136.8 Conclusions 1156.9 Recommended Reading 1157 Incremental and Component Risks 1177.1 Incremental VaR 1177.1.1 Interpreting Incremental VaR 1177.1.2 Estimating IVaR by Brute Force: The ‘Before and After’ Approach 1187.1.3 Estimating IVaR Using Marginal VaRs 1197.1.3.1 Garman’s ‘delVaR’ Approach 1197.1.3.2 Potential Drawbacks of the delVaR Approach 1227.2 Component VaR 1227.2.1 Properties of Component VaR 1227.2.2 Uses of Component VaR 1247.2.2.1 ‘Drill-down’ Capability 1247.2.2.2 Reporting Component VaRs 1257.3 Conclusions 1267.4 Recommended Reading 1268 Estimating Liquidity Risks 1278.1 Liquidity and Liquidity Risks 1278.2 Estimating Liquidity-adjusted VaR and ETL 1288.2.1 A Transactions Cost Approach 1288.2.2 The Exogenous Spread Approach 1318.2.3 The Market Price Response Approach 1328.2.4 Derivatives Pricing Approaches 1328.2.5 The Liquidity Discount Approach 1338.2.6 A Summary and Comparison of Alternative Approaches 1348.3 Estimating Liquidity at Risk (LaR) 1358.4 Estimating Liquidity in Crises 1378.5 Recommended Reading 1399 Backtesting Market Risk Models 1419.1 Preliminary Data Issues 1419.1.1 Obtaining Data 1419.2 Statistical Backtests Based on the Frequency of Tail Losses 1439.2.1 The Basic Frequency-of-tail-losses (or Kupiec) Test 1439.2.2 The Time-to-first-tail-loss Test 1459.2.3 A Tail-loss Confidence-interval Test 1469.2.4 The Conditional Backtesting (Christoffersen) Approach 1479.3 Statistical Backtests Based on the Sizes of Tail Losses 1479.3.1 The Basic Sizes-of-tail-losses Test 1479.3.2 The Crnkovic–Drachman Backtest Procedure 1499.3.3 The Berkowitz Approach 1519.4 Forecast Evaluation Approaches to Backtesting 1539.4.1 Basic Ideas 1539.4.2 The Frequency-of-tail-losses (Lopez I) Approach 1549.4.3 The Size-adjusted Frequency (Lopez II) Approach 1549.4.4 The Blanco–Ihle Approach 1559.4.5 An Alternative Sizes-of-tail-losses Approach 1559.5 Other Methods of Comparing Models 1569.6 Assessing the Accuracy of Backtest Results 1569.7 Backtesting with Alternative Confidence Levels, Positions and Data 1579.7.1 Backtesting with Alternative Confidence Levels 1589.7.2 Backtesting with Alternative Positions 1599.7.3 Backtesting with Alternative Data 1599.8 Summary 1599.9 Recommended Reading 16010 Stress Testing 16110.1 Benefits and Difficulties of Stress Testing 16310.1.1 Benefits of Stress Testing 16310.1.2 Difficulties with Stress Tests 16510.2 Scenario Analysis 16710.2.1 Choosing Scenarios 16710.2.1.1 Stylised Scenarios 16710.2.1.2 Actual Historical Events 16810.2.1.3 Hypothetical One-off Events 17010.2.2 Evaluating the Effects of Scenarios 17010.3 Mechanical Stress Testing 17210.3.1 Factor Push Analysis 17210.3.2 Maximum Loss Optimisation 17410.4 Conclusions 17510.5 Recommended Reading 17511 Model Risk 17711.1 Models and Model Risk 17711.1.1 Models 17711.1.2 Model Risk 17811.2 Sources of Model Risk 17911.2.1 Incorrect Model Specification 17911.2.2 Incorrect Model Application 18111.2.3 Implementation Risk 18111.2.4 Other Sources of Model Risk 18211.2.4.1 Incorrect Calibration 18211.2.4.2 Programming Problems 18211.2.4.3 Data Problems 18311.3 Combating Model Risk 18311.3.1 Combating Model Risk: Some Guidelines for Risk Practitioners 18411.3.2 Combating Model Risk: Some Guidelines for Managers 18411.3.3 Institutional Methods to Combat Model Risk 18611.3.3.1 Procedures to Vet, Check and Review Models 18611.3.3.2 Independent Risk Oversight 18711.4 Conclusions 18811.5 Recommended Reading 188Toolkit 189Bibliography 261Author Index 271Subject Index 275Software Index 283

KEVIN DOWD is Professor of Financial Risk Management at Nottingham University Business School and a member of the School's Centre for Research in Risk and Insurance Studies.

• Preface xiAcknowledgements xix1 The Risk Measurement Revolution 11.1 Contributory Factors 11.1.1 A Volatile Environment 11.1.2 Growth in Trading Activity 21.1.3 Advances in Information Technology 21.2 Risk Measurement Before VaR 31.2.1 Gap Analysis 31.2.2 Duration Analysis 41.2.3 Scenario Analysis 41.2.4 Portfolio Theory 51.2.5 Derivatives Risk Measures 61.3 Value at Risk 71.3.1 The Origin and Development of VaR 71.3.2 Attractions of VaR 101.3.3 Criticisms of VaR 111.4 Recommended Reading 122 Measures of Financial Risk 132.1 The Mean–Variance Framework for Measuring Financial Risk 132.1.1 The Normality Assumption 132.1.2 Limitations of the Normality Assumption 152.1.3 Traditional Approaches to Financial Risk Measurement 182.1.3.1 Portfolio Theory 182.1.3.2 Duration Approaches to Fixed-income Risk Measurement 182.2 Value at Risk 192.2.1 VaR Basics 192.2.2 Choice of VaR Parameters 242.2.3 Limitations of VaR as a Risk Measure 252.2.3.1 VaR Uninformative of Tail Losses 252.2.3.2 VaR Can Create Perverse Incentive Structures 262.2.3.3 VaR Can Discourage Diversification 272.2.3.4 VaR Not Sub-additive 272.3 Expected Tail Loss 282.3.1 Coherent Risk Measures 282.3.2 The Expected Tail Loss 292.4 Conclusions 332.5 Recommended Reading 333 Basic Issues in Measuring Market Risk 353.1 Data 353.1.1 Profit/Loss Data 353.1.2 Loss/Profit Data 353.1.3 Arithmetic Returns Data 363.1.4 Geometric Returns Data 363.2 Estimating Historical Simulation VaR 363.3 Estimating Parametric VaR 373.3.1 Estimating VaR with Normally Distributed Profits/Losses 383.3.2 Estimating VaR with Normally Distributed Arithmetic Returns 393.3.3 Estimating Lognormal VaR 403.4 Estimating Expected Tail Loss 423.5 Summary 44Appendix: Mapping Positions to Risk Factors 45A3.1 Selecting Core Instruments or Factors 46A3.1.1 Selecting Core Instruments 46A3.1.2 Selecting Core Factors 47A3.2 Mapping Positions and VaR Estimation 47A3.2.1 The Basic Building Blocks 47A3.2.1.1 Basic FX Positions 47A3.2.1.2 Basic Equity Positions 48A3.2.1.3 Zero-coupon Bonds 50A3.2.1.4 Basic Forward/Futures 51A3.2.2 More Complex Positions 52A3.3 Recommended Reading 534 Non-parametric VaR and ETL 554.1 Compiling Historical Simulation Data 554.2 Estimation of Historical Simulation VaR and ETL 564.2.1 Basic Historical Simulation 564.2.2 Estimating Curves and Surfaces for VaR and ETL 574.3 Estimating Confidence Intervals for Historical Simulation VaR and ETL 584.3.1 A Quantile Standard Error Approach to the Estimation of Confidence Intervals for HS VaR and ETL 584.3.2 An Order Statistics Approach to the Estimation of Confidence Intervals for HS VaR and ETL 584.3.3 A Bootstrap Approach to the Estimation of Confidence Intervals for HS VaR and ETL 594.4 Weighted Historical Simulation 614.4.1 Age-weighted Historical Simulation 624.4.2 Volatility-weighted Historical Simulation 634.4.3 Filtered Historical Simulation 644.5 Advantages and Disadvantages of Historical Simulation 664.5.1 Advantages 664.5.2 Disadvantages 674.5.2.1 Total Dependence on the Data Set 674.5.2.2 Problems of Data Period Length 684.6 Principal Components Approaches to VaR and ETL Estimation 684.7 Conclusions 694.8 Recommended Reading 705 Parametric VaR and ETL 715.1 Normal VaR and ETL 725.1.1 General Features 725.1.2 Disadvantages of Normality 765.2 The Student t-distribution 775.3 The Lognormal Distribution 785.4 Extreme Value Distributions 815.4.1 The Generalised Extreme Value Distribution 815.4.2 The Peaks Over Threshold (Generalised Pareto) Approach 825.5 The Multivariate Normal Variance–Covariance Approach 845.6 Conclusions 865.7 Recommended Reading 87Appendix: Delta–Gamma and Related Approximations 88A5.1 Delta–normal Approaches 88A5.2 Delta–Gamma Approaches 90A5.2.1 The Delta–Gamma Approximation 90A5.2.2 The Delta–Gamma Normal Approach 90A5.2.3 Wilson’s Delta–Gamma Approach 91A5.2.4 Other Delta–Gamma Approaches 93A5.3 Conclusions 94A5.4 Recommended Reading 956 Simulation Approaches to VaR and ETL Estimation 976.1 Options VaR and ETL 976.1.1 Preliminary Considerations 976.1.2 An Example: Estimating the VaR and ETL of an American Put 986.1.3 Refining MCS Estimation of Options VaR and ETL 996.2 Estimating VaR by Simulating Principal Components 996.2.1 Basic Principal Components Simulation 996.2.2 Scenario Simulation 1006.3 Fixed-income VaR and ETL 1026.3.1 General Considerations 1026.3.1.1 Stochastic Processes for Interest Rates 1026.3.1.2 The Term Structure of Interest Rates 1036.3.2 A General Approach to Fixed-income VaR and ETL 1036.4 Estimating VaR and ETL under a Dynamic Portfolio Strategy 1056.5 Estimating Credit-related Risks with Simulation Methods 1076.6 Estimating Insurance Risks with Simulation Methods 1096.7 Estimating Pensions Risks with Simulation Methods 1106.7.1 Estimating Risks of Defined-benefit Pension Plans 1116.7.2 Estimating Risks of Defined-contribution Pension Plans 1136.8 Conclusions 1156.9 Recommended Reading 1157 Incremental and Component Risks 1177.1 Incremental VaR 1177.1.1 Interpreting Incremental VaR 1177.1.2 Estimating IVaR by Brute Force: The ‘Before and After’ Approach 1187.1.3 Estimating IVaR Using Marginal VaRs 1197.1.3.1 Garman’s ‘delVaR’ Approach 1197.1.3.2 Potential Drawbacks of the delVaR Approach 1227.2 Component VaR 1227.2.1 Properties of Component VaR 1227.2.2 Uses of Component VaR 1247.2.2.1 ‘Drill-down’ Capability 1247.2.2.2 Reporting Component VaRs 1257.3 Conclusions 1267.4 Recommended Reading 1268 Estimating Liquidity Risks 1278.1 Liquidity and Liquidity Risks 1278.2 Estimating Liquidity-adjusted VaR and ETL 1288.2.1 A Transactions Cost Approach 1288.2.2 The Exogenous Spread Approach 1318.2.3 The Market Price Response Approach 1328.2.4 Derivatives Pricing Approaches 1328.2.5 The Liquidity Discount Approach 1338.2.6 A Summary and Comparison of Alternative Approaches 1348.3 Estimating Liquidity at Risk (LaR) 1358.4 Estimating Liquidity in Crises 1378.5 Recommended Reading 1399 Backtesting Market Risk Models 1419.1 Preliminary Data Issues 1419.1.1 Obtaining Data 1419.2 Statistical Backtests Based on the Frequency of Tail Losses 1439.2.1 The Basic Frequency-of-tail-losses (or Kupiec) Test 1439.2.2 The Time-to-first-tail-loss Test 1459.2.3 A Tail-loss Confidence-interval Test 1469.2.4 The Conditional Backtesting (Christoffersen) Approach 1479.3 Statistical Backtests Based on the Sizes of Tail Losses 1479.3.1 The Basic Sizes-of-tail-losses Test 1479.3.2 The Crnkovic–Drachman Backtest Procedure 1499.3.3 The Berkowitz Approach 1519.4 Forecast Evaluation Approaches to Backtesting 1539.4.1 Basic Ideas 1539.4.2 The Frequency-of-tail-losses (Lopez I) Approach 1549.4.3 The Size-adjusted Frequency (Lopez II) Approach 1549.4.4 The Blanco–Ihle Approach 1559.4.5 An Alternative Sizes-of-tail-losses Approach 1559.5 Other Methods of Comparing Models 1569.6 Assessing the Accuracy of Backtest Results 1569.7 Backtesting with Alternative Confidence Levels, Positions and Data 1579.7.1 Backtesting with Alternative Confidence Levels 1589.7.2 Backtesting with Alternative Positions 1599.7.3 Backtesting with Alternative Data 1599.8 Summary 1599.9 Recommended Reading 16010 Stress Testing 16110.1 Benefits and Difficulties of Stress Testing 16310.1.1 Benefits of Stress Testing 16310.1.2 Difficulties with Stress Tests 16510.2 Scenario Analysis 16710.2.1 Choosing Scenarios 16710.2.1.1 Stylised Scenarios 16710.2.1.2 Actual Historical Events 16810.2.1.3 Hypothetical One-off Events 17010.2.2 Evaluating the Effects of Scenarios 17010.3 Mechanical Stress Testing 17210.3.1 Factor Push Analysis 17210.3.2 Maximum Loss Optimisation 17410.4 Conclusions 17510.5 Recommended Reading 17511 Model Risk 17711.1 Models and Model Risk 17711.1.1 Models 17711.1.2 Model Risk 17811.2 Sources of Model Risk 17911.2.1 Incorrect Model Specification 17911.2.2 Incorrect Model Application 18111.2.3 Implementation Risk 18111.2.4 Other Sources of Model Risk 18211.2.4.1 Incorrect Calibration 18211.2.4.2 Programming Problems 18211.2.4.3 Data Problems 18311.3 Combating Model Risk 18311.3.1 Combating Model Risk: Some Guidelines for Risk Practitioners 18411.3.2 Combating Model Risk: Some Guidelines for Managers 18411.3.3 Institutional Methods to Combat Model Risk 18611.3.3.1 Procedures to Vet, Check and Review Models 18611.3.3.2 Independent Risk Oversight 18711.4 Conclusions 18811.5 Recommended Reading 188Toolkit 189Bibliography 261Author Index 271Subject Index 275Software Index 283

KEVIN DOWD is Professor of Financial Risk Management at Nottingham University Business School and a member of the School's Centre for Research in Risk and Insurance Studies.

• Preface xiAcknowledgements xix1 The Risk Measurement Revolution 11.1 Contributory Factors 11.1.1 A Volatile Environment 11.1.2 Growth in Trading Activity 21.1.3 Advances in Information Technology 21.2 Risk Measurement Before VaR 31.2.1 Gap Analysis 31.2.2 Duration Analysis 41.2.3 Scenario Analysis 41.2.4 Portfolio Theory 51.2.5 Derivatives Risk Measures 61.3 Value at Risk 71.3.1 The Origin and Development of VaR 71.3.2 Attractions of VaR 101.3.3 Criticisms of VaR 111.4 Recommended Reading 122 Measures of Financial Risk 132.1 The Mean–Variance Framework for Measuring Financial Risk 132.1.1 The Normality Assumption 132.1.2 Limitations of the Normality Assumption 152.1.3 Traditional Approaches to Financial Risk Measurement 182.1.3.1 Portfolio Theory 182.1.3.2 Duration Approaches to Fixed-income Risk Measurement 182.2 Value at Risk 192.2.1 VaR Basics 192.2.2 Choice of VaR Parameters 242.2.3 Limitations of VaR as a Risk Measure 252.2.3.1 VaR Uninformative of Tail Losses 252.2.3.2 VaR Can Create Perverse Incentive Structures 262.2.3.3 VaR Can Discourage Diversification 272.2.3.4 VaR Not Sub-additive 272.3 Expected Tail Loss 282.3.1 Coherent Risk Measures 282.3.2 The Expected Tail Loss 292.4 Conclusions 332.5 Recommended Reading 333 Basic Issues in Measuring Market Risk 353.1 Data 353.1.1 Profit/Loss Data 353.1.2 Loss/Profit Data 353.1.3 Arithmetic Returns Data 363.1.4 Geometric Returns Data 363.2 Estimating Historical Simulation VaR 363.3 Estimating Parametric VaR 373.3.1 Estimating VaR with Normally Distributed Profits/Losses 383.3.2 Estimating VaR with Normally Distributed Arithmetic Returns 393.3.3 Estimating Lognormal VaR 403.4 Estimating Expected Tail Loss 423.5 Summary 44Appendix: Mapping Positions to Risk Factors 45A3.1 Selecting Core Instruments or Factors 46A3.1.1 Selecting Core Instruments 46A3.1.2 Selecting Core Factors 47A3.2 Mapping Positions and VaR Estimation 47A3.2.1 The Basic Building Blocks 47A3.2.1.1 Basic FX Positions 47A3.2.1.2 Basic Equity Positions 48A3.2.1.3 Zero-coupon Bonds 50A3.2.1.4 Basic Forward/Futures 51A3.2.2 More Complex Positions 52A3.3 Recommended Reading 534 Non-parametric VaR and ETL 554.1 Compiling Historical Simulation Data 554.2 Estimation of Historical Simulation VaR and ETL 564.2.1 Basic Historical Simulation 564.2.2 Estimating Curves and Surfaces for VaR and ETL 574.3 Estimating Confidence Intervals for Historical Simulation VaR and ETL 584.3.1 A Quantile Standard Error Approach to the Estimation of Confidence Intervals for HS VaR and ETL 584.3.2 An Order Statistics Approach to the Estimation of Confidence Intervals for HS VaR and ETL 584.3.3 A Bootstrap Approach to the Estimation of Confidence Intervals for HS VaR and ETL 594.4 Weighted Historical Simulation 614.4.1 Age-weighted Historical Simulation 624.4.2 Volatility-weighted Historical Simulation 634.4.3 Filtered Historical Simulation 644.5 Advantages and Disadvantages of Historical Simulation 664.5.1 Advantages 664.5.2 Disadvantages 674.5.2.1 Total Dependence on the Data Set 674.5.2.2 Problems of Data Period Length 684.6 Principal Components Approaches to VaR and ETL Estimation 684.7 Conclusions 694.8 Recommended Reading 705 Parametric VaR and ETL 715.1 Normal VaR and ETL 725.1.1 General Features 725.1.2 Disadvantages of Normality 765.2 The Student t-distribution 775.3 The Lognormal Distribution 785.4 Extreme Value Distributions 815.4.1 The Generalised Extreme Value Distribution 815.4.2 The Peaks Over Threshold (Generalised Pareto) Approach 825.5 The Multivariate Normal Variance–Covariance Approach 845.6 Conclusions 865.7 Recommended Reading 87Appendix: Delta–Gamma and Related Approximations 88A5.1 Delta–normal Approaches 88A5.2 Delta–Gamma Approaches 90A5.2.1 The Delta–Gamma Approximation 90A5.2.2 The Delta–Gamma Normal Approach 90A5.2.3 Wilson’s Delta–Gamma Approach 91A5.2.4 Other Delta–Gamma Approaches 93A5.3 Conclusions 94A5.4 Recommended Reading 956 Simulation Approaches to VaR and ETL Estimation 976.1 Options VaR and ETL 976.1.1 Preliminary Considerations 976.1.2 An Example: Estimating the VaR and ETL of an American Put 986.1.3 Refining MCS Estimation of Options VaR and ETL 996.2 Estimating VaR by Simulating Principal Components 996.2.1 Basic Principal Components Simulation 996.2.2 Scenario Simulation 1006.3 Fixed-income VaR and ETL 1026.3.1 General Considerations 1026.3.1.1 Stochastic Processes for Interest Rates 1026.3.1.2 The Term Structure of Interest Rates 1036.3.2 A General Approach to Fixed-income VaR and ETL 1036.4 Estimating VaR and ETL under a Dynamic Portfolio Strategy 1056.5 Estimating Credit-related Risks with Simulation Methods 1076.6 Estimating Insurance Risks with Simulation Methods 1096.7 Estimating Pensions Risks with Simulation Methods 1106.7.1 Estimating Risks of Defined-benefit Pension Plans 1116.7.2 Estimating Risks of Defined-contribution Pension Plans 1136.8 Conclusions 1156.9 Recommended Reading 1157 Incremental and Component Risks 1177.1 Incremental VaR 1177.1.1 Interpreting Incremental VaR 1177.1.2 Estimating IVaR by Brute Force: The ‘Before and After’ Approach 1187.1.3 Estimating IVaR Using Marginal VaRs 1197.1.3.1 Garman’s ‘delVaR’ Approach 1197.1.3.2 Potential Drawbacks of the delVaR Approach 1227.2 Component VaR 1227.2.1 Properties of Component VaR 1227.2.2 Uses of Component VaR 1247.2.2.1 ‘Drill-down’ Capability 1247.2.2.2 Reporting Component VaRs 1257.3 Conclusions 1267.4 Recommended Reading 1268 Estimating Liquidity Risks 1278.1 Liquidity and Liquidity Risks 1278.2 Estimating Liquidity-adjusted VaR and ETL 1288.2.1 A Transactions Cost Approach 1288.2.2 The Exogenous Spread Approach 1318.2.3 The Market Price Response Approach 1328.2.4 Derivatives Pricing Approaches 1328.2.5 The Liquidity Discount Approach 1338.2.6 A Summary and Comparison of Alternative Approaches 1348.3 Estimating Liquidity at Risk (LaR) 1358.4 Estimating Liquidity in Crises 1378.5 Recommended Reading 1399 Backtesting Market Risk Models 1419.1 Preliminary Data Issues 1419.1.1 Obtaining Data 1419.2 Statistical Backtests Based on the Frequency of Tail Losses 1439.2.1 The Basic Frequency-of-tail-losses (or Kupiec) Test 1439.2.2 The Time-to-first-tail-loss Test 1459.2.3 A Tail-loss Confidence-interval Test 1469.2.4 The Conditional Backtesting (Christoffersen) Approach 1479.3 Statistical Backtests Based on the Sizes of Tail Losses 1479.3.1 The Basic Sizes-of-tail-losses Test 1479.3.2 The Crnkovic–Drachman Backtest Procedure 1499.3.3 The Berkowitz Approach 1519.4 Forecast Evaluation Approaches to Backtesting 1539.4.1 Basic Ideas 1539.4.2 The Frequency-of-tail-losses (Lopez I) Approach 1549.4.3 The Size-adjusted Frequency (Lopez II) Approach 1549.4.4 The Blanco–Ihle Approach 1559.4.5 An Alternative Sizes-of-tail-losses Approach 1559.5 Other Methods of Comparing Models 1569.6 Assessing the Accuracy of Backtest Results 1569.7 Backtesting with Alternative Confidence Levels, Positions and Data 1579.7.1 Backtesting with Alternative Confidence Levels 1589.7.2 Backtesting with Alternative Positions 1599.7.3 Backtesting with Alternative Data 1599.8 Summary 1599.9 Recommended Reading 16010 Stress Testing 16110.1 Benefits and Difficulties of Stress Testing 16310.1.1 Benefits of Stress Testing 16310.1.2 Difficulties with Stress Tests 16510.2 Scenario Analysis 16710.2.1 Choosing Scenarios 16710.2.1.1 Stylised Scenarios 16710.2.1.2 Actual Historical Events 16810.2.1.3 Hypothetical One-off Events 17010.2.2 Evaluating the Effects of Scenarios 17010.3 Mechanical Stress Testing 17210.3.1 Factor Push Analysis 17210.3.2 Maximum Loss Optimisation 17410.4 Conclusions 17510.5 Recommended Reading 17511 Model Risk 17711.1 Models and Model Risk 17711.1.1 Models 17711.1.2 Model Risk 17811.2 Sources of Model Risk 17911.2.1 Incorrect Model Specification 17911.2.2 Incorrect Model Application 18111.2.3 Implementation Risk 18111.2.4 Other Sources of Model Risk 18211.2.4.1 Incorrect Calibration 18211.2.4.2 Programming Problems 18211.2.4.3 Data Problems 18311.3 Combating Model Risk 18311.3.1 Combating Model Risk: Some Guidelines for Risk Practitioners 18411.3.2 Combating Model Risk: Some Guidelines for Managers 18411.3.3 Institutional Methods to Combat Model Risk 18611.3.3.1 Procedures to Vet, Check and Review Models 18611.3.3.2 Independent Risk Oversight 18711.4 Conclusions 18811.5 Recommended Reading 188Toolkit 189Bibliography 261Author Index 271Subject Index 275Software Index 283