Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures

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

1.3.1 Bernoulli Distribution 3

1.3.2 Binomial Distribution 3

1.3.3 Poisson Distribution 4

1.4 Continuous Probability Distributions 5

1.4.1 Probability Distribution Function, Probability Density Function, and Cumulative Distribution Function 5

1.4.2 The Normal Distribution 8

1.4.3 Exponential Distribution 10

1.4.4 Student’s t-distribution 11

1.4.5 Extreme Value Distribution 12

1.4.6 Generalized Extreme Value Distribution 12

1.5 Statistical Moments and Quantiles 13

1.5.1 Location 13

1.5.2 Dispersion 13

1.5.3 Asymmetry 13

1.5.4 Concentration in Tails 14

1.5.5 Statistical Moments 14

1.5.6 Quantiles 16

1.5.7 Sample Moments 16

1.6 Joint Probability Distributions 17

1.6.1 Conditional Probability 18

1.6.2 Definition of Joint Probability Distributions 19

1.6.3 Marginal Distributions 19

1.6.4 Dependence of Random Variables 20

1.6.5 Covariance and Correlation 20

1.6.6 Multivariate Normal Distribution 21

1.6.7 Elliptical Distributions 23

1.6.8 Copula Functions 25

1.7 Probabilistic Inequalities 30

1.7.1 Chebyshev’s Inequality 30

1.7.2 Fr´echet-Hoeffding Inequality 31

1.8 Summary 32

CHAPTER 2 Optimization 35

2.1 Introduction 35

2.2 Unconstrained Optimization 36

2.2.1 Minima and Maxima of a Differentiable Function 37

2.2.2 Convex Functions 40

2.2.3 Quasiconvex Functions 46

2.3 Constrained Optimization 48

2.3.1 Lagrange Multipliers 49

2.3.2 Convex Programming 52

2.3.3 Linear Programming 55

2.3.4 Quadratic Programming 57

2.4 Summary 58

CHAPTER 3 Probability Metrics 61

3.1 Introduction 61

3.2 Measuring Distances: The Discrete Case 62

3.2.1 Sets of Characteristics 63

3.2.2 Distribution Functions 64

3.2.3 Joint Distribution 68

3.3 Primary, Simple, and Compound Metrics 72

3.3.1 Axiomatic Construction 73

3.3.2 Primary Metrics 74

3.3.3 Simple Metrics 75

3.3.4 Compound Metrics 84

3.3.5 Minimal and Maximal Metrics 86

3.4 Summary 90

3.5 Technical Appendix 90

3.5.1 Remarks on the Axiomatic Construction of Probability Metrics 91

3.5.2 Examples of Probability Distances 94

3.5.3 Minimal and Maximal Distances 99

CHAPTER 4 Ideal Probability Metrics 103

4.1 Introduction 103

4.2 The Classical Central Limit Theorem 105

4.2.1 The Binomial Approximation to the Normal Distribution 105

4.2.2 The General Case 112

4.2.3 Estimating the Distance from the Limit Distribution 118

4.3 The Generalized Central Limit Theorem 120

4.3.1 Stable Distributions 120

4.3.2 Modeling Financial Assets with Stable Distributions 122

4.4 Construction of Ideal Probability Metrics 124

4.4.1 Definition 125

4.4.2 Examples 126

4.5 Summary 131

4.6 Technical Appendix 131

4.6.1 The CLT Conditions 131

4.6.2 Remarks on Ideal Metrics 133

CHAPTER 5 Choice under Uncertainty 139

5.1 Introduction 139

5.2 Expected Utility Theory 141

5.2.1 St. Petersburg Paradox 141

5.2.2 The von Neumann–Morgenstern Expected Utility Theory 143

5.2.3 Types of Utility Functions 145

5.3 Stochastic Dominance 147

5.3.1 First-Order Stochastic Dominance 148

5.3.2 Second-Order Stochastic Dominance 149

5.3.3 Rothschild-Stiglitz Stochastic Dominance 150

5.3.4 Third-Order Stochastic Dominance 152

5.3.5 Efficient Sets and the Portfolio Choice Problem 154

5.3.6 Return versus Payoff 154

5.4 Probability Metrics and Stochastic Dominance 157

5.5 Summary 161

5.6 Technical Appendix 161

5.6.1 The Axioms of Choice 161

5.6.2 Stochastic Dominance Relations of Order n 163

5.6.3 Return versus Payoff and Stochastic Dominance 164

5.6.4 Other Stochastic Dominance Relations 166

CHAPTER 6 Risk and Uncertainty 171

6.1 Introduction 171

6.2 Measures of Dispersion 174

6.2.1 Standard Deviation 174

6.2.2 Mean Absolute Deviation 176

6.2.3 Semistandard Deviation 177

6.2.4 Axiomatic Description 178

6.2.5 Deviation Measures 179

6.3 Probability Metrics and Dispersion Measures 180

6.4 Measures of Risk 181

6.4.1 Value-at-Risk 182

6.4.2 Computing Portfolio VaR in Practice 186

6.4.3 Backtesting of VaR 192

6.4.4 Coherent Risk Measures 194

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

6.8.1 Convex Risk Measures 201

6.8.2 Probability Metrics and Deviation Measures 202

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

7.4.1 The Multivariate Normal Assumption 216

7.4.2 The Historical Method 217

7.4.3 The Hybrid Method 217

7.4.4 The Monte Carlo Method 218

7.5 Backtesting of AVaR 220

7.6 Spectral Risk Measures 222

7.7 Risk Measures and Probability Metrics 224

7.8 Summary 227

7.9 Technical Appendix 227

7.9.1 Characteristics of Conditional Loss Distributions 228

7.9.2 Higher-Order AVaR 230

7.9.3 The Minimization Formula for AVaR 232

7.9.4 AVaR for Stable Distributions 235

7.9.5 ETL versus AVaR 236

7.9.6 Remarks on Spectral Risk Measures 241

CHAPTER 8 Optimal Portfolios 245

8.1 Introduction 245

8.2 Mean-Variance Analysis 247

8.2.1 Mean-Variance Optimization Problems 247

8.2.2 The Mean-Variance Efficient Frontier 251

8.2.3 Mean-Variance Analysis and SSD 254

8.2.4 Adding a Risk-Free Asset 256

8.3 Mean-Risk Analysis 258

8.3.1 Mean-Risk Optimization Problems 259

8.3.2 The Mean-Risk Efficient Frontier 262

8.3.3 Mean-Risk Analysis and SSD 266

8.3.4 Risk versus Dispersion Measures 267

8.4 Summary 274

8.5 Technical Appendix 274

8.5.1 Types of Constraints 274

8.5.2 Quadratic Approximations to Utility Functions 276

8.5.3 Solving Mean-Variance Problems in Practice 278

8.5.4 Solving Mean-Risk Problems in Practice 279

8.5.5 Reward-Risk Analysis 281

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

9.7.1 Deviation Measures and r.d. Metrics 305

9.7.2 Remarks on the Axioms 305

9.7.3 Minimal r.d. Metrics 307

9.7.4 Limit Cases of L∗p(X, Y) and Θ∗p(X, Y) 310

9.7.5 Computing r.d. Metrics in Practice 311

CHAPTER 10 Performance Measures 317

10.1 Introduction 317

10.2 Reward-to-Risk Ratios 318

10.2.1 RR Ratios and the Efficient Portfolios 320

10.2.2 Limitations in the Application of Reward-to-Risk Ratios 324

10.2.3 The STARR 325

10.2.4 The Sortino Ratio 329

10.2.5 The Sortino-Satchell Ratio 330

10.2.6 A One-Sided Variability Ratio 331

10.2.7 The Rachev Ratio 332

10.3 Reward-to-Variability Ratios 333

10.3.1 RV Ratios and the Efficient Portfolios 335

10.3.2 The Sharpe Ratio 337

10.3.3 The Capital Market Line and the Sharpe Ratio 340

10.4 Summary 343

10.5 Technical Appendix 343

10.5.1 Extensions of STARR 343

10.5.2 Quasiconcave Performance Measures 345

10.5.3 The Capital Market Line and Quasiconcave Ratios 353

10.5.4 Nonquasiconcave Performance Measures 356

10.5.5 Probability Metrics and Performance Measures 357

Index 361