Preface xiii
1 What is risk management? 1
1.1 Introduction 2
1.2 Identifying and documenting risk 5
1.3 Fallacies and traps in risk management 7
1.4 Why safety is different 9
1.5 The Basel framework 11
1.6 Hold or hedge? 12
1.7 Learning from a disaster 13
1.7.1 What went wrong? 15
Notes 17
References 18
Exercises 19
2 The structure of risk 22
2.1 Introduction to probability and risk 23
2.2 The structure of risk 25
2.2.1 Intersection and union risk 25
2.2.2 Maximum of random variables 28
2.3 Portfolios and diversification 30
2.3.1 Adding random variables 30
2.3.2 Portfolios with minimum variance 33
2.3.3 Optimal portfolio theory 37
2.3.4 When risk follows a normal distribution 38
2.4 The impact of correlation 40
2.4.1 Using covariance in combining random variables 41
2.4.2 Minimum variance portfolio with covariance 43
2.4.3 The maximum of variables that are positively correlated 44
2.4.4 Multivariate normal 46
2.5 Using copulas to model multivariate distributions 49
2.5.1 *Details on copula modeling 52
Notes 58
References 59
Exercises 60
3 Measuring risk 63
3.1 How can we measure risk? 64
3.2 Value at risk 67
3.3 Combining and comparing risks 73
3.4 VaR in practice 76
3.5 Criticisms of VaR 79
3.6 Beyond value at risk 82
3.6.1 *More details on expected shortfall 86
Notes 88
References 88
Exercises 89
4 Understanding the tails 92
4.1 Heavy-tailed distributions 93
4.1.1 Defining the tail index 93
4.1.2 Estimating the tail index 95
4.1.3 *More details on the tail index 98
4.2 Limiting distributions for the maximum 100
4.2.1 *More details on maximum distributions and Fisher-Tippett 106
4.3 Excess distributions 109
4.3.1 *More details on threshold exceedances 114
4.4 Estimation using extreme value theory 115
4.4.1 Step 1. Choose a threshold u 116
4.4.2 Step 2. Estimate the parameters ¿ and ß 118
4.4.3 Step 3. Estimate the risk measures of interest 119
Notes 121
References 122
Exercises 123
5 Making decisions under uncertainty 125
5.1 Decisions, states and outcomes 126
5.1.1 Decisions 126
5.1.2 States 127
5.1.3 Outcomes 127
5.1.4 Probabilities 128
5.1.5 Values 129
5.2 Expected Utility Theory 130
5.2.1 Maximizing expected profit 130
5.2.2 Expected utility 132
5.2.3 No alternative to Expected Utility Theory 135
5.2.4 *A sketch proof of the theorem 139
5.2.5 What shape is the utility function? 142
5.2.6 *Expected utility when probabilities are subjective 145
5.3 Stochastic dominance and risk profiles 148
5.3.1 *More details on stochastic dominance 152
5.4 Risk decisions for managers 156
5.4.1 Managers and shareholders 156
5.4.2 A single company-wide view of risk 158
5.4.3 Risk of insolvency 158
Notes 160
References 161
Exercises 162
6 Understanding risk behavior 164
6.1 Why decision theory fails 165
6.1.1 The meaning of utility 165
6.1.2 Bounded rationality 167
6.1.3 Inconsistent choices under uncertainty 168
6.1.4 Problems from scaling utility functions 171
6.2 Prospect Theory 172
6.2.1 Foundations for behavioral decision theory 173
6.2.2 Decision weights and subjective values 175
6.3 Cumulative Prospect Theory 180
6.3.1 *More details on Prospect Theory 183
6.3.2 Applying Prospect Theory 185
6.3.3 Why Prospect Theory does not always predict well 187
6.4 Decisions with ambiguity 189
6.5 How managers treat risk 191
Notes 194
References 194
Exercises 195
7 Stochastic optimization 198
7.1 Introduction to stochastic optimization 199
7.1.1 A review of optimization 199
7.1.2 Two-stage recourse problems 203
7.1.3 Ordering with stochastic demand 208
7.2 Choosing scenarios 212
7.2.1 How to carry out Monte Carlo simulation 213
7.2.2 Alternatives to Monte Carlo 217
7.3 Multistage stochastic optimization 218
7.3.1 Non-anticipatory constraints 220
7.4 Value at risk constraints 224
Notes 228
References 228
Exercises 229
8 Robust optimization 232
8.1 True uncertainty: Beyond probabilities 233
8.2 Avoiding disaster when there is uncertainty 234
8.2.1 *More details on constraint reformulation 240
8.2.2 Budget of uncertainty 243
8.2.3 *More details on budgets of uncertainty 247
8.3 Robust optimization and the minimax approach 250
8.3.1 *Distributionally robust optimization 254
Notes 261
References 262
Exercises 263
9 Real options 265
9.1 Introduction to real options 266
9.2 Calculating values with real options 267
9.2.1 *Deriving the formula for the surplus with a normal distribution 272
9.3 Combining real options and net present value 273
9.4 The connection with financial options 278
9.5 Using Monte Carlo simulation to value real options 282
9.6 Some potential problems with the use of real options 285
Notes 287
References 287
Exercises 288
10 Credit risk 291
10.1 Introduction to credit risk 292
10.2 Using credit scores for credit risk 294
10.2.1 A Markov chain analysis of defaults 296
10.3 Consumer credit 301
10.3.1 Probability, odds and log odds 302
10.4 Logistic regression 308
10.4.1 *More details on logistic regression 313
10.4.2 Building a scorecard 315
10.4.3 Other scoring applications 317
Notes 317
References 318
Exercises 319
Appendix A Tutorial on probability theory 323
A. 1 Random events 323
A. 2 Bayes' rule and independence 326
A. 3 Random variables 327
A. 4 Means and variances 329
A. 5 Combinations of random variables 332
A. 6 The normal distribution and the Central Limit Theorem 336
Appendix B Answers to even-numbered exercises 340
Index 361
