Stochastic Dominance and Applications to Finance, Risk and Economics Book

Preface xi

1 Utility in Decision Theory 1

1.1 Choice under certainty 1

1.2 Basic probability background 5

1.2.1 Probability measures and distributions 5

1.2.2 Integration 17

1.2.3 Notes on topological spaces 31

1.3 Choice under uncertainty 34

1.4 Utilities and risk attitudes 48

1.4.1 Qualitative representations of risk attitudes 48

1.4.2 Notes on convex functions 53

1.4.3 Jensen's inequality 54

1.5 Exercises 55

2 Foundations of Stochastic Dominance 59

2.1 Some preliminary mathematics 59

2.1.1 Approximation of utility functions 60

2.1.2 A fundamental lemma 65

2.2 Deriving representations of preferences 66

2-2.1 Representation for risk neutral individuals 66

2.2.2 Representation for risk averse individuals 68

2.2.3 Representation for risk seeking individuals 69

2.2.4 A subclass of risk averse individuals 70

2.3 Stochastic dominance 76

2.3.1 First-order stochastic dominance 77

2.3.2 Second-order stochastic dominance 81

2.3.3 Third-order stochastic dominance 83

2.4 Exercises 85

3 Issues in Stochastic Dominance 89

3.1 A closer look at the mean-variance rule 89

3.2 Multivariate stochastic dominance 93

3.3 Stochastic dominance via quantile functions 96

3.3.1 First-order stochastic dominance 99

3.3.2 Second-order stochastic dominance 99

3.3.3 Stochastic dominance rule for risk seekers 101

3.4 Exercises 104

4 Financial Risk Measures 107

4.1 The problem of risk modeling 107

4.2 Some popular risk measures 110

4.2.1 Variance 110

4.2.2 Value-at-risk Ill

4.2.3 Tail value-at-risk 114

4.3 Desirable properties of risk measures 119

4.4 Exercises 123

5 Choquet Integrals as Risk Measures 129

5.1 Extended theory ofmeasures 129

5.2 Capacities 130

5.3 The Choquet integral 135

5.4 Basic properties of the Choquet integral 142

5.5 Comonotonicity 149

5.6 Notes on copulas 154

5.7 A characterization theorem 161

5.8 A class of coherent risk measures 164

5.9 Consistency with stochastic dominance 172

5.10 Exercises 176

6 Foundational Statistics for Stochastic Dominance 179

6.1 From theory to applications 179

6.2 Structure of statistical inference 182

6.3 Generalities on statistical estimation 189

6.4 Nonparametric estimation 196

6.4.1 The Glivenko-Cantelli theorem 196

6.4.2 Estimating probability density functions 201

6.4.3 Method of excess mass 205

6.4.4 Nonparametric regression 207

6.4.5 Risk estimation 209

6.5 Basics of hypothesis testing 213

6.5.1 The Neyman-Pearson lemma 215

6.5.2 Consistent tests 217

6.5.3 The Kolmogorov-Smirnov statistic 218

6.5.4 Two-sample KS tests 221

6.5.5 Chi-squared testing 222

6.6 Exercises 226

7 Models and Data in Econometrics 233

7.1 Justifications of models 233

7.1.1 The logit model 234

7.1.2 Stochastic volatility 238

7.1.3 Financial risk models 244

7.2 Coarse data 248

7.2.1 Indirect observations in auctions 248

7.2.2 Game theory 250

7.2.3 Measurement-error data in linear models 252

7.2.4 Censored data 253

7.2.5 Missing data 258

7.3 Modeling dependence structure 264

7.3.1 Copulas 264

7.3.2 Copulas for sampling designs 268

7.3.3 Estimation of copulas 270

7.4 Some additional statistical tools 271

7.4.1 Bayesian statistics 271

7.4.2 From linear regression to filtering models 271

7.4.3 Recursive estimation 275

7.4.4 What is a filter? 276

7.4.5 The Kalman filter 278

7.5 Exercises 281

8 Applications to Finance 285

8.1 Diversification 285

8.1.1 Convex stochastic dominance 285

8.1.2 Diversification for risk averters and risk seekers 291

8.2 Diversification on convex combinations 293

8.3 Prospect and Markowitz SD 300

8.3.1 Illustration 304

8.4 Market rationality and efficiency 305

8.4.1 Applications of SD to calendar anomalies 308

8.4.2 Data 309

8.4.3 Results 309

8.5 SD and rationality of momentum effect 318

8.5.1 Evidence on profitability of momentum strategies 319

8.5.2 Data and methodology 321

8.5.3 Profitability of momentum strategy 321

8.5.4 Results of stochastic dominance tests 325

8.5.5 Robustness checks 328

8.6 Exercises 331

9 Applications to Risk Management 333

9.1 Measures of profit/loss for risk analysis 333

9.1.1 SD criterion for decision-making in risk analysis 335

9.1.2 MV criterion for decision-making in risk analysis 337

9.2 REITs and stocks and fixed-income assets 341

9.2.1 Data and methodology 343

9.2.2 Empirical findings 344

9.2.3 Discussion 350

9.3 Evaluating hedge funds performance 351

9.3.1 Data and methodology 354

9.3.2 Discussion 363

9.4 Evaluating iShare performance 364

9.4.1 Data and methodology 365

9.4.2 Results 367

9.4.3 Discussion 374

9.5 Exercises 374

10 Applications to Economics 381

10.1 Indifference curves/location-scale family 381

10.1.1 Portfolio and expected utility 381

10.1.2 A dilemma in using the mean-variance criterion 386

10.2 LS family for n random seed sources 387

10.2.1 Location-scale expected utility 389

10.2.2 Indifference curves 393

10.2.3 Expected versus non-expected LS utility functions 394

10.2.4 Dominance relationships over the LS family 398

10.3 Elasticity of risk aversion and trade 400

10.3.1 International trade and uncertainty 400

10.3.2 LS parameter condition and elasticity 401

10.3.3 Risk and mean effects on international trade 402

10.4 Income inequality 403

10.5 Exercises 406

Appendix Stochastic Dominance Tests 409

A.l CAPM statistics 409

A.2 Testing equality of multiple Sharpe ratios 410

A.3 Hypothesis testing 413

A.4 Davidson-Duclos (DD) test 414

A.4.1 Stochastic dominance tests for risk averters 414

A.4.2 Stochastic dominance tests for risk seekers 416

A.5 Barrett and Donald (BD) test 417

A.5.1 Stochastic dominance tests for risk averters 417

A.5.2 Stochastic dominance tests for risk seekers 418

A.6 Linton, Maasoumi and Whang test 419

A.6.1 Stochastic dominance tests for risk averters 419

A.6.2 Stochastic dominance tests for risk seekers 419

A.7 Stochastic dominance tests for MSD and PSD 420

Bibliography 425

Index 439