Measuring Market Risk

Fully revised and restructured, Measuring Market Risk, Second Edition includes a new chapter on options risk management, as well as substantial new information on parametric risk, non-parametric measurements and liquidity risks, more practical information to help with specific calculations, and new examples including Q&A

Preface to the Second Edition xiii
Acknowledgements xix
1 The Rise of Value at Risk 1
1. 1 The emergence of financial risk management 2
1. 2 Market risk measurement 4
1. 3 Risk measurement before VaR 5
1. 3. 1 Gap analysis 5
1. 3. 2 Duration analysis 5
1. 3. 3 Scenario analysis 6
1. 3. 4 Portfolio theory 7
1. 3. 5 Derivatives risk measures 8
1. 4 Value at risk 9
1. 4. 1 The origin and development of VaR 9
1. 4. 2 Attractions of VaR 11
1. 4. 3 Criticisms of VaR 13
Appendix: Types of Market Risk 15
2 Measures of Financial Risk 19
2. 1 The mean-variance framework for measuring financial risk 20
2. 2 Value at risk 27
2. 2. 1 Basics of VaR 27
2. 2. 2 Determination of the VaR parameters 29
2. 2. 3 Limitations of VaR as a risk measure 31
2. 3 Coherent risk measures 32
2. 3. 1 The coherence axioms and their implications 32
2. 3. 2 The expected shortfall 35
2. 3. 3 Spectral risk measures 37
2. 3. 4 Scenarios as coherent risk measures 42
2. 4 Conclusions 44
Appendix 1: Probability Functions 45
Appendix 2: Regulatory Uses of VaR 52
3 Estimating Market Risk Measures: An Introduction and Overview 53
3. 1 Data 53
3. 1. 1 Profit/loss data 53
3. 1. 2 Loss/profit data 54
3. 1. 3 Arithmetic return data 54
3. 1. 4 Geometric return data 54
3. 2 Estimating historical simulation VaR 56
3. 3 Estimating parametric VaR 57
3. 3. 1 Estimating VaR with normally distributed profits/losses 57
3. 3. 2 Estimating VaR with normally distributed arithmetic returns 59
3. 3. 3 Estimating lognormal VaR 61
3. 4 Estimating coherent risk measures 64
3. 4. 1 Estimating expected shortfall 64
3. 4. 2 Estimating coherent risk measures 64
3. 5 Estimating the standard errors of risk measure estimators 69
3. 5. 1 Standard errors of quantile estimators 69
3. 5. 2 Standard errors in estimators of coherent risk measures 72
3. 6 The core issues: an overview 73
Appendix 1: Preliminary Data Analysis 75
Appendix 2: Numerical Integration Methods 80
4 Non-parametric Approaches 83
4. 1 Compiling historical simulation data 84
4. 2 Estimation of historical simulation VaR and ES 84
4. 2. 1 Basic historical simulation 84
4. 2. 2 Bootstrapped historical simulation 85
4. 2. 3 Historical simulation using non-parametric density estimation 86
4. 2. 4 Estimating curves and surfaces for VAR and ES 88
4. 3 Estimating confidence intervals for historical simulation VaR and ES 89
4. 3. 1 An order-statistics approach to the estimation of confidence intervals for HS VaR and ES 89
4. 3. 2 A bootstrap approach to the estimation of confidence intervals for HS VaR and ES 90
4. 4 Weighted historical simulation 92
4. 4. 1 Age-weighted historical simulation 93
4. 4. 2 Volatility-weighted historical simulation 94
4. 4. 3 Correlation-weighted historical simulation 95
4. 4. 4 Filtered historical simulation 96
4. 5 Advantages and disadvantages of non-parametric methods 99
4. 5. 1 Advantages 99
4. 5. 2 Disadvantages 100
4. 6 Conclusions 101
Appendix 1: Estimating Risk Measures with Order Statistics 102
Appendix 2: The Bootstrap 105
Appendix 3: Non-parametric Density Estimation 111
Appendix 4: Principal Components Analysis and Factor Analysis 118
5 Forecasting Volatilities, Covariances and Correlations 127
5. 1 Forecasting volatilities 127
5. 1. 1 Defining volatility 127
5. 1. 2 Historical volatility forecasts 128
5. 1. 3 Exponentially weighted moving average volatility 129
5. 1. 4 GARCH models 131
5. 1. 5 Implied volatilities 136
5. 2 Forecasting covariances and correlations 137
5. 2. 1 Defining covariances and correlations 137
5. 2. 2 Historical covariances and correlations 138
5. 2. 3 Exponentially weighted moving average covariances 140
5. 2. 4 GARCH covariances 140
5. 2. 5 Implied covariances and correlations 141
5. 2. 6 Some pitfalls with correlation estimation 141
5. 3 Forecasting covariance matrices 142
5. 3. 1 Positive definiteness and positive semi-definiteness 142
5. 3. 2 Historical variance-covariance estimation 142
5. 3. 3 Multivariate EWMA 142
5. 3. 4 Multivariate GARCH 142
5. 3. 5 Computational problems with covariance and correlation matrices 143
Appendix: Modelling Dependence: Correlations and Copulas 145
6 Parametric Approaches (I) 151
6. 1 Conditional vs unconditional distributions 152
6. 2 Normal VaR and ES 154
6. 3 The t-distribution 159
6. 4 The lognormal distribution 161
6. 5 Miscellaneous parametric approaches 165
6. 5. 1 Lé vy approaches 165
6. 5. 2 Elliptical and hyperbolic approaches 167
6. 5. 3 Normal mixture approaches 167
6. 5. 4 Jump diffusion 168
6. 5. 5 Stochastic volatility approaches 169
6. 5. 6 The Cornish-Fisher approximation 171
6. 6 The multivariate normal variance-covariance approach 173
6. 7 Non-normal variance-covariance approaches 176
6. 7. 1 Multivariate t-distributions 176
6. 7. 2 Multivariate elliptical distributions 177
6. 7. 3 The Hull-White transformation-into-normality approach 177
6. 8 Handling multivariate return distributions with copulas 178
6. 8. 1 Motivation 178
6. 8. 2 Estimating VaR with copulas 179
6. 9 Conclusions 182
Appendix: Forecasting Longer-term Risk Measures 184
7 Parametric Approaches (II): Extreme Value 189
7. 1 Generalised extreme-value theory 190
7. 1. 1 Theory 190
7. 1. 2 A short-cut EV method 194
7. 1. 3 Estimation of EV parameters 195
7. 2 The peaks-over-threshold approach: the generalised Pareto distribution 201
7. 2. 1 Theory 201
7. 2. 2 Estimation 203
7. 2. 3 GEV vs POT 204
7. 3 Refinements to EV approaches 204
7. 3. 1 Conditional EV 204
7. 3. 2 Dealing with dependent (or non-iid) data 205
7. 3. 3 Multivariate EVT 206
7. 4 Conclusions 206
8 Monte Carlo Simulation Methods 209
8. 1 Uses of Monte carlo simulation 210
8. 2 Monte Carlo simulation with a single risk factor 213
8. 3 Monte Carlo simulation with multiple risk factors 215
8. 4 Variance-reduction methods 217
8. 4. 1 Antithetic variables 218
8. 4. 2 Control variates 218
8. 4. 3 Importance sampling 219
8. 4. 4 Stratified sampling 220
8. 4. 5 Moment matching 223
8. 5 Advantages and disadvantages of Monte Carlo simulation 225
8. 5. 1 Advantages 225
8. 5. 2 Disadvantages 225
8. 6 Conclusions 225
9 Applications of Stochastic Risk Measurement Methods 227
9. 1 Selecting stochastic processes 227
9. 2 Dealing with multivariate stochastic processes 230
9. 2. 1 Principal components simulation 230
9. 2. 2 Scenario simulation 232
9. 3 Dynamic risks 234
9. 4 Fixed-income risks 236
9. 4. 1 Distinctive features of fixed-income problems 237
9. 4. 2 Estimating fixed-income risk measures 237
9. 5 Credit-related risks 238
9. 6 Insurance risks 240
9. 6. 1 General insurance risks 241
9. 6. 2 Life insurance risks 242
9. 7 Measuring pensions risks 244
9. 7. 1 Estimating risks of defined-benefit pension plans 245
9. 7. 2 Estimating risks of defined-contribution pension plans 246
9. 8 Conclusions 248
10 Estimating Options Risk Measures 249
10. 1 Analytical and algorithmic solutions for options VaR 249
10. 2 Simulation approaches 253
10. 3 Delta-gamma and related approaches 256
10. 3. 1 Delta-normal approaches 257
10. 3. 2 Delta-gamma approaches 258
10. 4 Conclusions 264
11 Incremental and Component Risks 265
11. 1 Incremental VaR 265
11. 1. 1 Interpreting Incremental VaR 265
11. 1. 2 Estimating IVaR by brute force: the 'before and after' approach 266
11. 1. 3 Estimating IVaR using analytical solutions 267
11. 2 Component VaR 271
11. 2. 1 Properties of component VaR 271
11. 2. 2 Uses of component VaR 274
11. 3 Decomposition of coherent risk measures 277
12 Mapping Positions to Risk Factors 279
12. 1 Selecting core instruments 280
12. 2 Mapping positions and VaR estimation 281
12. 2. 1 Basic building blocks 281
12. 2. 2 More complex positions 287
13 Stress Testing 291
13. 1 Benefits and difficulties of stress testing 293
13. 1. 1 Benefits of stress testing 293
13. 1. 2 Difficulties with stress tests 295
13. 2 Scenario analysis 297
13. 2. 1 Choosing scenarios 297
13. 2. 2 Evaluating the effects of scenarios 300
13. 3 Mechanical stress testing 303
13. 3. 1 Factor push analysis 303
13. 3. 2 Maximum loss optimisation 305
13. 3. 3 CrashMetrics 305
13. 4 Conclusions 306
14 Estimating Liquidity Risks 309
14. 1 Liquidity and liquidity risks 309
14. 2 Estimating liquidity-adjusted VaR 310
14. 2. 1 The constant spread approach 311
14. 2. 2 The exogenous spread approach 312
14. 2. 3 Endogenous-price approaches 314
14. 2. 4 The liquidity discount approach 315
14. 3 Estimating liquidity at risk (LaR) 316
14. 4 Estimating liquidity in crises 319
15 Backtesting Market Risk Models 321
15. 1 Preliminary data issues 321
15. 2 Backtests based on frequency tests 323
15. 2. 1 The basic frequency backtest 324
15. 2. 2 The conditional testing (Christoffersen) backtest 329
15. 3 Backtests based on tests of distribution equality 331
15. 3. 1 Tests based on the Rosenblatt transformation 331
15. 3. 2 Tests using the Berkowitz transformation 333
15. 3. 3 Overlapping forecast periods 335
15. 4 Comparing alternative models 336
15. 5 Backtesting with alternative positions and data 339
15. 5. 1 Backtesting with alternative positions 340
15. 5. 2 Backtesting with alternative data 340
15. 6 Assessing the precision of backtest results 340
15. 7 Summary and conclusions 342
Appendix: Testing Whether Two Distributions are Different 343
16 Model Risk 351
16. 1 Models and model risk 351
16. 2 Sources of model risk 353
16. 2. 1 Incorrect model specification 353
16. 2. 2 Incorrect model application 354
16. 2. 3 Implementation risk 354
16. 2. 4 Other sources of model risk 355
16. 3 Quantifying model risk 357
16. 4 Managing model risk 359
16. 4. 1 Managing model risk: some guidelines for risk practitioners 359
16. 4. 2 Managing model risk: some guidelines for senior managers 360
16. 4. 3 Institutional methods to manage model risk 361
16. 5 Conclusions 363
Bibliography 365
Index 379