1 Introduction to Value at Risk (VaR) 1
1.1 Economics underlying VaR measurement 2
1.1.1 What is VaR? 4
1.1.2 Calculating VaR 6
1.1.3 The assumptions behind VaR calculations 8
1.1.4 Inputs into VaR calculations 10
1.2 Diversification and VaR 13
1.2.1 Factors affecting portfolio diversification 16
1.2.2 Decomposing volatility into systematic and
idiosyncratic risk 17
1.2.3 Diversification: Words of caution – the
case of long-term capital management
(LTCM) 18
2 Quantifying Volatility in VaR Models 21
2.1 The Stochastic Behavior of Returns 22
2.1.1 Revisiting the assumptions 22
2.1.2 The distribution of interest rate changes 23
2.1.3 Fat tails 25
2.1.4 Explaining fat tails 26
2.1.5 Effects of volatility changes 29
2.1.6 Can (conditional) normality be salvaged? 31
2.1.7 Normality cannot be salvaged 34
x UNDERSTANDING MARKET, CREDIT, AND OPERATIONAL RISK
2.2 VaR Estimation Approaches 35
2.2.1 Cyclical volatility 36
2.2.2 Historical standard deviation 36
2.2.3 Implementation considerations 38
2.2.4 Exponential smoothing – RiskMetrics™
volatility 40
2.2.4.1 The optimal smoother lambda 43
2.2.4.2 Adaptive volatility estimation 44
2.2.4.3 The empirical performance of
RiskMetrics™ 45
2.2.4.4 GARCH 45
2.2.5 Nonparametric volatility forecasting 48
2.2.5.1 Historical simulation 48
2.2.5.2 Multivariate density estimation 51
2.2.6 A comparison of methods 54
2.2.7 The hybrid approach 56
2.3 Return Aggregation and VaR 59
2.4 Implied Volatility as a Predictor of
Future Volatility 62
2.5 Long Horizon Volatility and VaR 66
2.6 Mean Reversion and Long Horizon Volatility 69
2.7 Correlation Measurement 71
2.8 Summary 74
Appendix 2.1 Backtesting Methodology
and Results 74
3 Putting VaR to Work 82
3.1 The VaR of Derivatives – Preliminaries 82
3.1.1 Linear derivatives 83
3.1.2 Nonlinear derivatives 86
3.1.3 Approximating the VaR of derivatives 86
3.1.4 Fixed income securities with embedded
optionality 93
3.1.5 “Delta normal” vs. full-revaluation 95
3.2 Structured Monte Carlo, Stress Testing, and
Scenario Analysis 97
3.2.1 Motivation 97
3.2.2 Structured Monte Carlo 98
3.2.3 Scenario analysis 101
3.2.3.1 Correlation breakdown 101
3.2.3.2 Generating reasonable stress 103
CONTENTS xi
3.2.3.3 Stress testing in practice 104
3.2.3.4 Stress testing and historical
simulation 106
3.2.3.5 Asset concentration 107
3.3 Worst Case Scenario (WCS) 110
3.3.1 WCS vs. VaR 110
3.3.2 A comparison of VaR to WCS 111
3.3.3 Extensions 112
3.4 Summary 113
Appendix 3.1 Duration 114
4 Extending the VaR Approach to
Non-tradable Loans 119
4.1 Traditional Approaches to Credit Risk
Measurement 120
4.1.1 Expert systems 121
4.1.2 Rating systems 122
4.1.3 Credit scoring models 124
4.2 Theoretical Underpinnings: Two Approaches 128
4.2.1 Options-theoretic structural models of credit
risk measurement 128
4.2.2 Reduced form or intensity-based models
of credit risk measurement 132
4.2.3 Proprietary VaR models of credit risk
measurement 138
4.3 CreditMetrics 138
4.3.1 The distribution of an individual loan’s value 138
4.3.2 The value distribution for a portfolio of loans 143
4.3.2.1 Calculating the correlation between
equity returns and industry indices
for each borrower in the loan
portfolio 144
4.3.2.2 Calculating the correlation between
borrower equity returns 144
4.3.2.3 Solving for joint migration
probabilities 145
4.3.2.4 Valuing each loan across the entire
credit migration spectrum 147
4.3.2.5 Calculating the mean and standard
deviation of the normal portfolio
value distribution 149
xii UNDERSTANDING MARKET, CREDIT, AND OPERATIONAL RISK
4.4 Algorithmics’ Mark-to-Future 151
4.5 Summary 153
Appendix 4.1 CreditMetrics: Calculating Credit VaR
Using the Actual Distribution 155
5 Extending the VaR Approach to
Operational Risks 158
5.1 Top-Down Approaches to Operational Risk
Measurement 161
5.1.1 Top-down vs. bottom-up models 162
5.1.2 Data requirements 163
5.1.3 Top-down models 165
5.1.3.1 Multi-factor models 165
5.1.3.2 Income-based models 166
5.1.3.3 Expense-based models 167
5.1.3.4 Operating leverage models 167
5.1.3.5 Scenario analysis 167
5.1.3.6 Risk profiling models 168
5.2 Bottom-Up Approaches to Operational Risk
Measurement 170
5.2.1 Process approaches 170
5.2.1.1 Causal networks or scorecards 170
5.2.1.2 Connectivity models 173
5.2.1.3 Reliability models 175
5.2.2 Actuarial approaches 176
5.2.2.1 Empirical loss distributions 176
5.2.2.2 Parametric loss distributions 176
5.2.2.3 Extreme value theory 179
5.2.3 Proprietary operational risk models 182
5.3 Hedging Operational Risk 185
5.3.1 Insurance 186
5.3.2 Self-insurance 188
5.3.3 Hedging using derivatives 190
5.3.3.1 Catastrophe options 191
5.3.3.2 Cat bonds 193
5.3.4 Limitations to operational risk hedging 195
5.4 Summary 196
Appendix 5.1 Copula Functions 196
6 Applying VaR to Regulatory Models 200
6.1 BIS Regulatory Models of Market Risk 203
CONTENTS xiii
6.1.1 The standardized framework for market risk 203
6.1.1.1 Measuring interest rate risk 203
6.1.1.2 Measuring foreign exchange
rate risk 204
6.1.1.3 Measuring equity price risk 205
6.1.2 Internal models of market risk 205
6.2 BIS Regulatory Models of Credit Risk 206
6.2.1 The Standardized Model for credit risk 207
6.2.2 The Internal Ratings-Based Models for
credit risk 209
6.2.2.1 The Foundation IRB Approach 210
6.2.2.2 The Advanced IRB Approach 214
6.2.3 BIS regulatory models of off-balance sheet
credit risk 215
6.2.4 Assessment of the BIS regulatory models
of credit risk 218
6.3 BIS Regulatory Models of Operational Risk 221
6.3.1 The Basic Indicator Approach 223
6.3.2 The Standardized Approach 224
6.3.3 The Advanced Measurement Approach 225
6.3.3.1 The internal measurement approach 227
6.3.3.2 The loss distribution approach 230
6.3.3.3 The scorecard approach 230
6.4 Summary 231
7 VaR: Outstanding Research 233
7.1 Data Availability 233
7.2 Model Integration 234
7.3 Dynamic Modeling 235