Measuring Market Risk

Measuring Market Risk......Page 4
Contents......Page 8
Preface......Page 14
Acknowledgements......Page 24
1.1.1 A Volatile Environment......Page 26
1.1.2 Growth in Trading Activity......Page 27
1.2.1 Gap Analysis......Page 28
1.2.2 Duration Analysis......Page 29
1.2.4 Portfolio Theory......Page 30
1.2.5 Derivatives Risk Measures......Page 32
1.3.1 The Origin and Development of VaR......Page 33
1.3.2 Attractions of VaR......Page 35
1.3.3 Criticisms of VaR......Page 36
1.4 Recommended Reading......Page 38
2.1.1 The Normality Assumption......Page 40
2.1.2 Limitations of the Normality Assumption......Page 42
2.1.3.1 Portfolio Theory......Page 45
2.2.1 VaR Basics......Page 46
2.2.2 Choice of VaR Parameters......Page 52
2.2.3.2 VaR Can Create Perverse Incentive Structures......Page 53
2.2.3.3 VaR Can Discourage Diversification......Page 54
2.2.3.4 VaR Not Sub-additive......Page 55
2.3.1 Coherent Risk Measures......Page 56
2.3.2 The Expected Tail Loss......Page 57
2.5 Recommended Reading......Page 61
3.1.1 Profit/Loss Data......Page 62
3.1.4 Geometric Returns Data......Page 63
3.2 Estimating Historical Simulation VaR......Page 64
3.3.1 Estimating VaR with Normally Distributed Profits/Losses......Page 65
3.3.3 Estimating Lognormal VaR......Page 67
3.4 Estimating Expected Tail Loss......Page 68
3.5 Summary......Page 71
Appendix: Mapping Positions to Risk Factors......Page 72
A3.1.1 Selecting Core Instruments......Page 73
A3.2 Mapping Positions and VaR Estimation......Page 74
A3.2.1.2 Basic Equity Positions......Page 75
A3.2.1.3 Zero-coupon Bonds......Page 77
A3.2.1.4 Basic Forwards/Futures......Page 79
A3.2.2 More Complex Positions......Page 80
A3.3 Recommended Reading......Page 81
4.1 Compiling Historical Simulation Data......Page 82
4.2.1 Basic Historical Simulation......Page 83
4.2.2 Historical Simulation Using Non-parametric Density Estimation......Page 84
4.2.3 Estimating Curves and Surfaces for VaR and ETL......Page 86
4.3.2 An Order Statistics Approach to the Estimation of Confidence Intervals for HS VaR and ETL......Page 87
4.3.3 A Bootstrap Approach to the Estimation of Confidence Intervals for HS VaR and ETL......Page 88
4.4 Weighted Historical Simulation......Page 90
4.4.1 Age-weighted Historical Simulation......Page 91
4.4.2 Volatility-weighted Historical Simulation......Page 92
4.4.3 Filtered Historical Simulation......Page 94
4.5.2.1 Total Dependence on the Data Set......Page 97
4.5.2.2 Problems of Data Period Length......Page 98
4.7 Conclusions......Page 99
4.8 Recommended Reading......Page 100
5 Parametric VaR and ETL......Page 102
5.1.1 General Features......Page 103
5.2 The Student t-distribution......Page 107
5.3 The Lognormal Distribution......Page 110
5.4 Extreme Value Distributions......Page 113
5.4.1 The Generalised Extreme Value Distribution......Page 114
5.4.2 The Peaks Over Threshold (Generalised Pareto) Approach......Page 115
5.5.2 Elliptical and Hyperbolic Approaches......Page 117
5.5.3 Normal Mixture Approaches......Page 118
5.5.4 The Cornish–Fisher Approximation......Page 120
5.6 The Multivariate Normal Variance–Covariance Approach......Page 121
5.7.1 Elliptical Variance–Covariance Approaches......Page 124
5.7.2 The Hull–White Transformation-to-normality Approach......Page 125
5.8 Handling Multivariate Return Distributions With Copulas......Page 126
5.9 Conclusions......Page 127
5.10 Recommended Reading......Page 129
A5.1 Delta–Normal Approaches......Page 130
A5.2.2 The Delta–Gamma Normal Approach......Page 132
A5.2.3 Wilson’s Delta–Gamma Approach......Page 133
A5.2.4 Other Delta–Gamma Approaches......Page 135
A5.3 Conclusions......Page 136
A5.4 Recommended Reading......Page 137
A5.5 When and How Can We Solve for Options VaR......Page 138
A5.6.1 A General Framework for Measuring Options Risks......Page 140
A5.6.2 A Worked Example: Measuring the VaR of a European Call Option......Page 141
A5.6.3 VaR/ETL Approaches and Greek Approaches to Options Risk......Page 146
A5.7 Recommended Reading......Page 147
6.1.1 Preliminary Considerations......Page 148
6.1.2 An Example: Estimating the VaR and ETL of an American Put......Page 149
6.1.3 Refining MCS Estimation of Options VaR and ETL......Page 150
6.2.1 Basic Principal Components Simulation......Page 151
6.2.2 Scenario Simulation......Page 152
6.3.1 General Considerations......Page 153
6.3.1.2 The Term Structure of Interest Rates......Page 154
6.3.2 A General Approach to Fixed-income VaR and ETL......Page 155
6.4 Estimating VaR and ETL under a Dynamic Portfolio Strategy......Page 156
6.5 Estimating Credit-related Risks with Simulation Methods......Page 159
6.7 Estimating Pensions Risks with Simulation Methods......Page 161
6.7.1 Estimating Risks of Defined-benefit Pension Plans......Page 163
6.7.2 Estimating Risks of Defined-contribution Pension Plans......Page 165
6.8 Conclusions......Page 166
6.9 Recommended Reading......Page 167
7.1.1 Introduction to Binomial Tree Methods......Page 168
7.1.2 A Worked Example: Estimating the VaR and ETL of an American Put with a Binomial Tree......Page 170
7.1.3 Other Considerations......Page 173
7.2 Trinomial Tree Methods......Page 174
7.4 Recommended Reading......Page 176
8.1.1 Interpreting Incremental VaR......Page 178
8.1.2 Estimating IVaR by Brute Force: The ‘Before and After’ Approach......Page 179
8.1.3.1 Garman’s ‘delVaR’ Approach......Page 180
8.1.3.2 Potential Drawbacks of the delVaR Approach......Page 183
8.2.1 Properties of Component VaR......Page 184
8.2.2.2 Reporting Component VaRs......Page 186
8.4 Recommended Reading......Page 188
9.1 Liquidity and Liquidity Risks......Page 190
9.2.1 A Transactions Cost Approach......Page 191
9.2.2 The Exogenous Spread Approach......Page 194
9.2.4 Derivatives Pricing Approaches......Page 195
9.2.5 The Liquidity Discount Approach......Page 196
9.2.6 A Summary and Comparison of Alternative Approaches......Page 197
9.3 Estimating Liquidity at Risk......Page 198
9.4 Estimating Liquidity in Crises......Page 201
9.5 Recommended Reading......Page 202
10.1.1 Obtaining Data......Page 204
10.2.1 The Basic Frequency-of-tail-losses (or Kupiec) Test......Page 206
10.2.2 The Time-to-first-tail-loss Test......Page 208
10.2.3 A Tail-loss Confidence-interval Test......Page 209
10.3.1 The Basic Sizes-of-tail-losses Test......Page 210
10.3.2 The Crnkovic–Drachman Backtest Procedure......Page 213
10.3.3 The Berkowitz Approach......Page 215
10.4.1 Basic Ideas......Page 216
10.4.2 The Frequency-of-tail-losses (Lopez I) Approach......Page 217
10.4.4 The Blanco–Ihle Approach......Page 218
10.5 Comparing Alternative Models......Page 219
10.6 Assessing the Accuracy of Backtest Results......Page 220
10.7 Backtesting With Alternative Confidence Levels, Positions and Data......Page 221
10.7.2 Backtesting with Alternative Positions......Page 222
10.8 Summary......Page 223
10.9 Recommended Reading......Page 224
11 Stress Testing......Page 226
11.1.1 Benefits of Stress Testing......Page 228
11.1.2 Difficulties with Stress Tests......Page 230
11.2 Scenario Analysis......Page 232
11.2.1.2 Actual Historical Events......Page 233
11.2.1.3 Hypothetical One-off Events......Page 235
11.2.2 Evaluating the Effects of Scenarios......Page 236
11.3 Mechanical Stress Testing......Page 237
11.3.1 Factor Push Analysis......Page 238
11.3.2 Maximum Loss Optimisation......Page 239
11.4 Conclusions......Page 240
11.5 Recommended Reading......Page 241
12.1.1 Models......Page 242
12.1.2 Model Risk......Page 243
12.2.1 Incorrect Model Specification......Page 244
12.2.2 Incorrect Model Application......Page 246
12.2.4.1 Incorrect Calibration......Page 247
12.2.4.3 Data Problems......Page 248
12.3.1 Combating Model Risk: Some Guidelines for Risk Practitioners......Page 249
12.3.2 Combating Model Risk: Some Guidelines for Managers......Page 250
12.3.3.1 Procedures to Vet, Check and Review Models......Page 251
12.3.3.2 Independent Risk Oversight......Page 252
12.4 Conclusions......Page 253
12.5 Recommended Reading......Page 254
Toolkit......Page 256
Bibliography......Page 366
Author Index......Page 380
Subject Index......Page 384
Software Index......Page 394