An Introduction to Value-at-Risk

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 CreditRisk? 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