Contents
Preface xiii
Acknowledgments xv
About the Authors xvii
CHAPTER 1
Concepts of Probability 1
1.1 Introduction 1
1.2 Basic Concepts 2
1.3 Discrete Probability Distributions 2
1.4 Continuous Probability Distributions 5
1.5 Statistical Moments and Quantiles 13
1.6 Joint Probability Distributions 17
1.7 Probabilistic Inequalities 30
1.8 Summary 32
CHAPTER 2
Optimization 35
2.1 Introduction 35
2.2 Unconstrained Optimization 36
2.3 Constrained Optimization 48
2.4 Summary 58
CHAPTER 3
Probability Metrics 61
3.1 Introduction 61
3.2 Measuring Distances: The Discrete Case 62
3.3 Primary, Simple, and Compound Metrics 72
3.4 Summary 90
3.5 Technical Appendix 90
CHAPTER 4
Ideal Probability Metrics 103
4.1 Introduction 103
4.2 The Classical Central Limit Theorem 105
4.3 The Generalized Central Limit Theorem 120
4.4 Construction of Ideal Probability Metrics 124
4.5 Summary 131
4.6 Technical Appendix 131
CHAPTER 5
Choice under Uncertainty 139
5.1 Introduction 139
5.2 Expected Utility Theory 141
5.3 Stochastic Dominance 147
5.4 Probability Metrics and Stochastic Dominance 157
5.5 Summary 161
5.6 Technical Appendix 161
CHAPTER 6
Risk and Uncertainty 171
6.1 Introduction 171
6.2 Measures of Dispersion 174
6.3 Probability Metrics and Dispersion Measures 180
6.4 Measures of Risk 181
6.5 Risk Measures and Dispersion Measures 198
6.6 Risk Measures and Stochastic Orders 199
6.7 Summary 200
6.8 Technical Appendix 201
CHAPTER 7
Average Value-at-Risk 207
7.1 Introduction 207
7.2 Average Value-at-Risk 208
7.3 AVaR Estimation from a Sample 214
7.4 Computing Portfolio AVaR in Practice 216
7.5 Backtesting of AVaR 220
Contents xi
7.6 Spectral Risk Measures 222
7.7 Risk Measures and Probability Metrics 224
7.8 Summary 227
7.9 Technical Appendix 227
CHAPTER 8
Optimal Portfolios 245
8.1 Introduction 245
8.2 Mean-Variance Analysis 247
8.2.4 Adding a Risk-Free Asset 256
8.3 Mean-Risk Analysis 258
8.4 Summary 274
8.5 Technical Appendix 274
CHAPTER 9
Benchmark Tracking Problems 287
9.1 Introduction 287
9.2 The Tracking Error Problem 288
9.3 Relation to Probability Metrics 292
9.4 Examples of r.d. Metrics 296
9.5 Numerical Example 300
9.6 Summary 304
9.7 Technical Appendix 304
CHAPTER 10
Performance Measures 317
10.1 Introduction 317
10.2 Reward-to-Risk Ratios 318
10.3 Reward-to-Variability Ratios 333
10.4 Summary 343
10.5 Technical Appendix 343
Index 361