An Introduction to Value-at-Risk, 5th Edition

Foreword xv

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