An Introduction to Value-at-Risk
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Preface xvii
Preface to the first edition xxi
About the author xxiii
1 INTRODUCTION TO RISK 1
Defining risk 2
The elements of risk: characterising risk 3
Forms of market risk 4
Other risks 5
Risk estimation 6
Risk management 7
The risk management function 7
Managing risk 9
Quantitative measurement of risk-reward 9
Standard deviation 10
Sharpe Ratio 10
Van Ratio 11
2 VOLATILITY AND CORRELATION 13
Statistical concepts 14
Arithmetic mean 14
Probability distributions 16
Confidence intervals 18
Volatility 20
The normal distribution and VaR 26
Correlation 28
3 VALUE-AT-RISK 29
What is VaR? 30
Definition 30
Methodology 32
Centralised database 32
Correlation assumptions 33
Correlation method 33
Historical simulation method 34
Monte Carlo simulation method 35
Validity of the volatility-correlation VaR estimate 35
How to calculate VaR 35
Historical method 36
Simulation method 37
Variance-covariance, analytic or parametric method 37
Mapping 44
Confidence intervals 47
Comparison between methods 48
Choosing between methods 48
Comparison with the historical approach 53
Comparing VaR calculation for different methodologies 54
Summary 56
4 VALUE-AT-RISK FOR FIXED INTEREST INSTRUMENTS 59
Fixed income products 60
Bond valuation 60
Duration 62
Modified duration 64
Convexity 64
Interest rate products 65
Forward rate agreements 65
Fixed income portfolio 68
Applying VaR for a FRA 70
VaR for an interest rate swap 72
Applying VaR for a bond futures contract 76
Calculation illustration 76
The historical method 79
Simulation methodology 80
Volatility over time 81
Application 81
Bloomberg screens 82
5 OPTIONS: RISK AND VALUE-AT-RISK 85
Option valuation using the Black-Scholes model 86
Option pricing 86
Volatility 88
The Greeks 89
Delta 90
Gamma 90
Vega 91
Other Greeks 92
Risk measurement 92
Spot ladder 93
Maturity ladder 93
Across-time ladder 93
Jump risk 93
Applying VaR for Options 94
6 MONTE CARLO SIMULATION AND VALUE-AT-RISK 99
Introduction: Monte Carlo simulation 100
Option value under Monte Carlo 100
Monte Carlo distribution 103
Monte Carlo simulation and VaR 104
7 REGULATORY ISSUES AND STRESS-TESTING 107
Capital adequacy 108
Model compliance 108
CAD II 109
Specific risk 111
Back-testing 112
Stress-testing 112
Simulating stress 113
Stress-testing in practice 114
Issues in stress-testing 115
The crash and Basel III 116
Stressed VaR 116
8 CREDIT RISK AND CREDIT VALUE-AT-RISK 119
Types of credit risk 120
Credit spread risk 120
Credit default risk 121
Credit ratings 121
Credit ratings 121
Ratings changes over time 123
Corporate recovery rates 125
Credit derivatives 126
Measuring risk for a CDS contract 128
Modelling credit risk 129
Time horizon 131
Data inputs 131
CreditMetrics 131
Methodology 132
Time horizon 133
Calculating the credit VaR 134
CreditRiskþ 137
Applications of credit VaR 142
Prioritising risk-reducing actions 142
Standard credit limit setting 143
Concentration limits 144
Integrating the credit risk and market risk functions 144
9 A REVIEW OF VALUE-AT-RISK 147
VaR in Crisis 149
Weaknesses Revealed 151
Market risk 151
Credit risk 153
Portfolio effects 155
New Regulation and Development 158
Procyclicality: stressed VaR (SVaR) 158
Default and migration risks: incremental risk charge (IRC) 159
Liquidity risks: differing liquidity horizons 161
Counterparty risks: CVA VaR 162
Fat tail risk: over-buffering 164
New framework for trading book 164
Beyond the Current Paradigm 166
Exercises 171
Appendix: Taylor's Expansion 179
Abbreviations 183
Selected bibliography 185
Index 187
