An Introduction to Market Risk Measurement

Preface xi

Acknowledgements xix

1 The Risk Measurement Revolution 1

1.1 Contributory Factors 1

1.1.1 A Volatile Environment 1

1.1.2 Growth in Trading Activity 2

1.1.3 Advances in Information Technology 2

1.2 Risk Measurement Before VaR 3

1.2.1 Gap Analysis 3

1.2.2 Duration Analysis 4

1.2.3 Scenario Analysis 4

1.2.4 Portfolio Theory 5

1.2.5 Derivatives Risk Measures 6

1.3 Value at Risk 7

1.3.1 The Origin and Development of VaR 7

1.3.2 Attractions of VaR 10

1.3.3 Criticisms of VaR 11

1.4 Recommended Reading 12

2 Measures of Financial Risk 13

2.1 The Mean–Variance Framework for Measuring Financial Risk 13

2.1.1 The Normality Assumption 13

2.1.2 Limitations of the Normality Assumption 15

2.1.3 Traditional Approaches to Financial Risk Measurement 18

2.1.3.1 Portfolio Theory 18

2.1.3.2 Duration Approaches to Fixed-income Risk Measurement 18

2.2 Value at Risk 19

2.2.1 VaR Basics 19

2.2.2 Choice of VaR Parameters 24

2.2.3 Limitations of VaR as a Risk Measure 25

2.2.3.1 VaR Uninformative of Tail Losses 25

2.2.3.2 VaR Can Create Perverse Incentive Structures 26

2.2.3.3 VaR Can Discourage Diversification 27

2.2.3.4 VaR Not Sub-additive 27

2.3 Expected Tail Loss 28

2.3.1 Coherent Risk Measures 28

2.3.2 The Expected Tail Loss 29

2.4 Conclusions 33

2.5 Recommended Reading 33

3 Basic Issues in Measuring Market Risk 35

3.1 Data 35

3.1.1 Profit/Loss Data 35

3.1.2 Loss/Profit Data 35

3.1.3 Arithmetic Returns Data 36

3.1.4 Geometric Returns Data 36

3.2 Estimating Historical Simulation VaR 36

3.3 Estimating Parametric VaR 37

3.3.1 Estimating VaR with Normally Distributed Profits/Losses 38

3.3.2 Estimating VaR with Normally Distributed Arithmetic Returns 39

3.3.3 Estimating Lognormal VaR 40

3.4 Estimating Expected Tail Loss 42

3.5 Summary 44

Appendix: Mapping Positions to Risk Factors 45

A3.1 Selecting Core Instruments or Factors 46

A3.1.1 Selecting Core Instruments 46

A3.1.2 Selecting Core Factors 47

A3.2 Mapping Positions and VaR Estimation 47

A3.2.1 The Basic Building Blocks 47

A3.2.1.1 Basic FX Positions 47

A3.2.1.2 Basic Equity Positions 48

A3.2.1.3 Zero-coupon Bonds 50

A3.2.1.4 Basic Forward/Futures 51

A3.2.2 More Complex Positions 52

A3.3 Recommended Reading 53

4 Non-parametric VaR and ETL 55

4.1 Compiling Historical Simulation Data 55

4.2 Estimation of Historical Simulation VaR and ETL 56

4.2.1 Basic Historical Simulation 56

4.2.2 Estimating Curves and Surfaces for VaR and ETL 57

4.3 Estimating Confidence Intervals for Historical Simulation VaR and ETL 58

4.3.1 A Quantile Standard Error Approach to the Estimation of Confidence Intervals for HS VaR and ETL 58

4.3.2 An Order Statistics Approach to the Estimation of Confidence Intervals for HS VaR and ETL 58

4.3.3 A Bootstrap Approach to the Estimation of Confidence Intervals for HS VaR and ETL 59

4.4 Weighted Historical Simulation 61

4.4.1 Age-weighted Historical Simulation 62

4.4.2 Volatility-weighted Historical Simulation 63

4.4.3 Filtered Historical Simulation 64

4.5 Advantages and Disadvantages of Historical Simulation 66

4.5.1 Advantages 66

4.5.2 Disadvantages 67

4.5.2.1 Total Dependence on the Data Set 67

4.5.2.2 Problems of Data Period Length 68

4.6 Principal Components Approaches to VaR and ETL Estimation 68

4.7 Conclusions 69

4.8 Recommended Reading 70

5 Parametric VaR and ETL 71

5.1 Normal VaR and ETL 72

5.1.1 General Features 72

5.1.2 Disadvantages of Normality 76

5.2 The Student t-distribution 77

5.3 The Lognormal Distribution 78

5.4 Extreme Value Distributions 81

5.4.1 The Generalised Extreme Value Distribution 81

5.4.2 The Peaks Over Threshold (Generalised Pareto) Approach 82

5.5 The Multivariate Normal Variance–Covariance Approach 84

5.6 Conclusions 86

5.7 Recommended Reading 87

Appendix: Delta–Gamma and Related Approximations 88

A5.1 Delta–normal Approaches 88

A5.2 Delta–Gamma Approaches 90

A5.2.1 The Delta–Gamma Approximation 90

A5.2.2 The Delta–Gamma Normal Approach 90

A5.2.3 Wilson’s Delta–Gamma Approach 91

A5.2.4 Other Delta–Gamma Approaches 93

A5.3 Conclusions 94

A5.4 Recommended Reading 95

6 Simulation Approaches to VaR and ETL Estimation 97

6.1 Options VaR and ETL 97

6.1.1 Preliminary Considerations 97

6.1.2 An Example: Estimating the VaR and ETL of an American Put 98

6.1.3 Refining MCS Estimation of Options VaR and ETL 99

6.2 Estimating VaR by Simulating Principal Components 99

6.2.1 Basic Principal Components Simulation 99

6.2.2 Scenario Simulation 100

6.3 Fixed-income VaR and ETL 102

6.3.1 General Considerations 102

6.3.1.1 Stochastic Processes for Interest Rates 102

6.3.1.2 The Term Structure of Interest Rates 103

6.3.2 A General Approach to Fixed-income VaR and ETL 103

6.4 Estimating VaR and ETL under a Dynamic Portfolio Strategy 105

6.5 Estimating Credit-related Risks with Simulation Methods 107

6.6 Estimating Insurance Risks with Simulation Methods 109

6.7 Estimating Pensions Risks with Simulation Methods 110

6.7.1 Estimating Risks of Defined-benefit Pension Plans 111

6.7.2 Estimating Risks of Defined-contribution Pension Plans 113

6.8 Conclusions 115

6.9 Recommended Reading 115

7 Incremental and Component Risks 117

7.1 Incremental VaR 117

7.1.1 Interpreting Incremental VaR 117

7.1.2 Estimating IVaR by Brute Force: The ‘Before and After’ Approach 118

7.1.3 Estimating IVaR Using Marginal VaRs 119

7.1.3.1 Garman’s ‘delVaR’ Approach 119

7.1.3.2 Potential Drawbacks of the delVaR Approach 122

7.2 Component VaR 122

7.2.1 Properties of Component VaR 122

7.2.2 Uses of Component VaR 124

7.2.2.1 ‘Drill-down’ Capability 124

7.2.2.2 Reporting Component VaRs 125

7.3 Conclusions 126

7.4 Recommended Reading 126

8 Estimating Liquidity Risks 127

8.1 Liquidity and Liquidity Risks 127

8.2 Estimating Liquidity-adjusted VaR and ETL 128

8.2.1 A Transactions Cost Approach 128

8.2.2 The Exogenous Spread Approach 131

8.2.3 The Market Price Response Approach 132

8.2.4 Derivatives Pricing Approaches 132

8.2.5 The Liquidity Discount Approach 133

8.2.6 A Summary and Comparison of Alternative Approaches 134

8.3 Estimating Liquidity at Risk (LaR) 135

8.4 Estimating Liquidity in Crises 137

8.5 Recommended Reading 139

9 Backtesting Market Risk Models 141

9.1 Preliminary Data Issues 141

9.1.1 Obtaining Data 141

9.2 Statistical Backtests Based on the Frequency of Tail Losses 143

9.2.1 The Basic Frequency-of-tail-losses (or Kupiec) Test 143

9.2.2 The Time-to-first-tail-loss Test 145

9.2.3 A Tail-loss Confidence-interval Test 146

9.2.4 The Conditional Backtesting (Christoffersen) Approach 147

9.3 Statistical Backtests Based on the Sizes of Tail Losses 147

9.3.1 The Basic Sizes-of-tail-losses Test 147

9.3.2 The Crnkovic–Drachman Backtest Procedure 149

9.3.3 The Berkowitz Approach 151

9.4 Forecast Evaluation Approaches to Backtesting 153

9.4.1 Basic Ideas 153

9.4.2 The Frequency-of-tail-losses (Lopez I) Approach 154

9.4.3 The Size-adjusted Frequency (Lopez II) Approach 154

9.4.4 The Blanco–Ihle Approach 155

9.4.5 An Alternative Sizes-of-tail-losses Approach 155

9.5 Other Methods of Comparing Models 156

9.6 Assessing the Accuracy of Backtest Results 156

9.7 Backtesting with Alternative Confidence Levels, Positions and Data 157

9.7.1 Backtesting with Alternative Confidence Levels 158

9.7.2 Backtesting with Alternative Positions 159

9.7.3 Backtesting with Alternative Data 159

9.8 Summary 159

9.9 Recommended Reading 160

10 Stress Testing 161

10.1 Benefits and Difficulties of Stress Testing 163

10.1.1 Benefits of Stress Testing 163

10.1.2 Difficulties with Stress Tests 165

10.2 Scenario Analysis 167

10.2.1 Choosing Scenarios 167

10.2.1.1 Stylised Scenarios 167

10.2.1.2 Actual Historical Events 168

10.2.1.3 Hypothetical One-off Events 170

10.2.2 Evaluating the Effects of Scenarios 170

10.3 Mechanical Stress Testing 172

10.3.1 Factor Push Analysis 172

10.3.2 Maximum Loss Optimisation 174

10.4 Conclusions 175

10.5 Recommended Reading 175

11 Model Risk 177

11.1 Models and Model Risk 177

11.1.1 Models 177

11.1.2 Model Risk 178

11.2 Sources of Model Risk 179

11.2.1 Incorrect Model Specification 179

11.2.2 Incorrect Model Application 181

11.2.3 Implementation Risk 181

11.2.4 Other Sources of Model Risk 182

11.2.4.1 Incorrect Calibration 182

11.2.4.2 Programming Problems 182

11.2.4.3 Data Problems 183

11.3 Combating Model Risk 183

11.3.1 Combating Model Risk: Some Guidelines for Risk Practitioners 184

11.3.2 Combating Model Risk: Some Guidelines for Managers 184

11.3.3 Institutional Methods to Combat Model Risk 186

11.3.3.1 Procedures to Vet, Check and Review Models 186

11.3.3.2 Independent Risk Oversight 187

11.4 Conclusions 188

11.5 Recommended Reading 188

Toolkit 189

Bibliography 261

Author Index 271

Subject Index 275

Software Index 283