Svetlozar T. Rachev, PhD, Doctor of Science, is Chair-Professor at the University of Karlsruhe in the School of Economics and Business Engineering; Professor Emeritus at the University of California, Santa Barbara; and Chief-Scientist of FinAnalytica Inc.Stoyan V. Stoyanov, PhD, is the Chief Financial Researcher at FinAnalytica Inc.Frank J. Fabozzi, PhD, CFA, is Professor in the Practice of Finance and Becton Fellow at Yale University's School of Management and the Editor of the Journal of Portfolio Management.

• Preface xiiiAcknowledgments xvAbout the Authors xviiChapter 1 Concepts of Probability 11.1 Introduction 11.2 Basic Concepts 21.3 Discrete Probability Distributions 21.3.1 Bernoulli Distribution 31.3.2 Binomial Distribution 31.3.3 Poisson Distribution 41.4 Continuous Probability Distributions 51.4.1 Probability Distribution Function, Probability Density Function, and Cumulative Distribution Function 51.4.2 The Normal Distribution 81.4.3 Exponential Distribution 101.4.4 Student’s t-distribution 111.4.5 Extreme Value Distribution 121.4.6 Generalized Extreme Value Distribution 121.5 Statistical Moments and Quantiles 131.5.1 Location 131.5.2 Dispersion 131.5.3 Asymmetry 131.5.4 Concentration in Tails 141.5.5 Statistical Moments 141.5.6 Quantiles 161.5.7 Sample Moments 161.6 Joint Probability Distributions 171.6.1 Conditional Probability 181.6.2 Definition of Joint Probability Distributions 191.6.3 Marginal Distributions 191.6.4 Dependence of Random Variables 201.6.5 Covariance and Correlation 201.6.6 Multivariate Normal Distribution 211.6.7 Elliptical Distributions 231.6.8 Copula Functions 251.7 Probabilistic Inequalities 301.7.1 Chebyshev’s Inequality 301.7.2 Fréchet-Hoeffding Inequality 311.8 Summary 32Chapter 2 Optimization 352.1 Introduction 352.2 Unconstrained Optimization 362.2.1 Minima and Maxima of a Differentiable Function 372.2.2 Convex Functions 402.2.3 Quasiconvex Functions 462.3 Constrained Optimization 482.3.1 Lagrange Multipliers 492.3.2 Convex Programming 522.3.3 Linear Programming 552.3.4 Quadratic Programming 572.4 Summary 58Chapter 3 Probability Metrics 613.1 Introduction 613.2 Measuring Distances: The Discrete Case 623.2.1 Sets of Characteristics 633.2.2 Distribution Functions 643.2.3 Joint Distribution 683.3 Primary, Simple, and Compound Metrics 723.3.1 Axiomatic Construction 733.3.2 Primary Metrics 743.3.3 Simple Metrics 753.3.4 Compound Metrics 843.3.5 Minimal and Maximal Metrics 863.4 Summary 903.5 Technical Appendix 903.5.1 Remarks on the Axiomatic Construction of Probability Metrics 913.5.2 Examples of Probability Distances 943.5.3 Minimal and Maximal Distances 99Chapter 4 Ideal Probability Metrics 1034.1 Introduction 1034.2 The Classical Central Limit Theorem 1054.2.1 The Binomial Approximation to the Normal Distribution 1054.2.2 The General Case 1124.2.3 Estimating the Distance from the Limit Distribution 1184.3 The Generalized Central Limit Theorem 1204.3.1 Stable Distributions 1204.3.2 Modeling Financial Assets with Stable Distributions 1224.4 Construction of Ideal Probability Metrics 1244.4.1 Definition 1254.4.2 Examples 1264.5 Summary 1314.6 Technical Appendix 1314.6.1 The CLT Conditions 1314.6.2 Remarks on Ideal Metrics 133Chapter 5 Choice under Uncertainty 1395.1 Introduction 1395.2 Expected Utility Theory 1415.2.1 St. Petersburg Paradox 1415.2.2 The von Neumann–Morgenstern Expected Utility Theory 1435.2.3 Types of Utility Functions 1455.3 Stochastic Dominance 1475.3.1 First-Order Stochastic Dominance 1485.3.2 Second-Order Stochastic Dominance 1495.3.3 Rothschild-Stiglitz Stochastic Dominance 1505.3.4 Third-Order Stochastic Dominance 1525.3.5 Efficient Sets and the Portfolio Choice Problem 1545.3.6 Return versus Payoff 1545.4 Probability Metrics and Stochastic Dominance 1575.5 Summary 1615.6 Technical Appendix 1615.6.1 The Axioms of Choice 1615.6.2 Stochastic Dominance Relations of Order n 1635.6.3 Return versus Payoff and Stochastic Dominance 1645.6.4 Other Stochastic Dominance Relations 166Chapter 6 Risk and Uncertainty 1716.1 Introduction 1716.2 Measures of Dispersion 1746.2.1 Standard Deviation 1746.2.2 Mean Absolute Deviation 1766.2.3 Semistandard Deviation 1776.2.4 Axiomatic Description 1786.2.5 Deviation Measures 1796.3 Probability Metrics and Dispersion Measures 1806.4 Measures of Risk 1816.4.1 Value-at-Risk 1826.4.2 Computing Portfolio VaR in Practice 1866.4.3 Backtesting of VaR 1926.4.4 Coherent Risk Measures 1946.5 Risk Measures and Dispersion Measures 1986.6 Risk Measures and Stochastic Orders 1996.7 Summary 2006.8 Technical Appendix 2016.8.1 Convex Risk Measures 2016.8.2 Probability Metrics and Deviation Measures 202Chapter 7 Average Value-at-Risk 2077.1 Introduction 2077.2 Average Value-at-Risk 2087.3 AVaR Estimation from a Sample 2147.4 Computing Portfolio AVaR in Practice 2167.4.1 The Multivariate Normal Assumption 2167.4.2 The Historical Method 2177.4.3 The Hybrid Method 2177.4.4 The Monte Carlo Method 2187.5 Backtesting of AVaR 2207.6 Spectral Risk Measures 2227.7 Risk Measures and Probability Metrics 2247.8 Summary 2277.9 Technical Appendix 2277.9.1 Characteristics of Conditional Loss Distributions 2287.9.2 Higher-Order AVaR 2307.9.3 The Minimization Formula for AVaR 2327.9.4 AVaR for Stable Distributions 2357.9.5 ETL versus AVaR 2367.9.6 Remarks on Spectral Risk Measures 241Chapter 8 Optimal Portfolios 2458.1 Introduction 2458.2 Mean-Variance Analysis 2478.2.1 Mean-Variance Optimization Problems 2478.2.2 The Mean-Variance Efficient Frontier 2518.2.3 Mean-Variance Analysis and SSD 2548.2.4 Adding a Risk-Free Asset 2568.3 Mean-Risk Analysis 2588.3.1 Mean-Risk Optimization Problems 2598.3.2 The Mean-Risk Efficient Frontier 2628.3.3 Mean-Risk Analysis and SSD 2668.3.4 Risk versus Dispersion Measures 2678.4 Summary 2748.5 Technical Appendix 2748.5.1 Types of Constraints 2748.5.2 Quadratic Approximations to Utility Functions 2768.5.3 Solving Mean-Variance Problems in Practice 2788.5.4 Solving Mean-Risk Problems in Practice 2798.5.5 Reward-Risk Analysis 281Chapter 9 Benchmark Tracking Problems 2879.1 Introduction 2879.2 The Tracking Error Problem 2889.3 Relation to Probability Metrics 2929.4 Examples of r.d. Metrics 2969.5 Numerical Example 3009.6 Summary 3049.7 Technical Appendix 3049.7.1 Deviation Measures and r.d. Metrics 3059.7.2 Remarks on the Axioms 3059.7.3 Minimal r.d. Metrics 3079.7.4 Limit Cases of L∗p(X, Y) and Θ∗p(X, Y) 3109.7.5 Computing r.d. Metrics in Practice 311Chapter 10 Performance Measures 31710.1 Introduction 31710.2 Reward-to-Risk Ratios 31810.2.1 RR Ratios and the Efficient Portfolios 32010.2.2 Limitations in the Application of Reward-to-Risk Ratios 32410.2.3 The STARR 32510.2.4 The Sortino Ratio 32910.2.5 The Sortino-Satchell Ratio 33010.2.6 A One-Sided Variability Ratio 33110.2.7 The Rachev Ratio 33210.3 Reward-to-Variability Ratios 33310.3.1 RV Ratios and the Efficient Portfolios 33510.3.2 The Sharpe Ratio 33710.3.3 The Capital Market Line and the Sharpe Ratio 34010.4 Summary 34310.5 Technical Appendix 34310.5.1 Extensions of STARR 34310.5.2 Quasiconcave Performance Measures 34510.5.3 The Capital Market Line and Quasiconcave Ratios 35310.5.4 Nonquasiconcave Performance Measures 35610.5.5 Probability Metrics and Performance Measures 357Index 361

Svetlozar T. Rachev, PhD, Doctor of Science, is Chair-Professor at the University of Karlsruhe in the School of Economics and Business Engineering; Professor Emeritus at the University of California, Santa Barbara; and Chief-Scientist of FinAnalytica Inc.Stoyan V. Stoyanov, PhD, is the Chief Financial Researcher at FinAnalytica Inc.Frank J. Fabozzi, PhD, CFA, is Professor in the Practice of Finance and Becton Fellow at Yale University's School of Management and the Editor of the Journal of Portfolio Management.

• Preface xiiiAcknowledgments xvAbout the Authors xviiChapter 1 Concepts of Probability 11.1 Introduction 11.2 Basic Concepts 21.3 Discrete Probability Distributions 21.3.1 Bernoulli Distribution 31.3.2 Binomial Distribution 31.3.3 Poisson Distribution 41.4 Continuous Probability Distributions 51.4.1 Probability Distribution Function, Probability Density Function, and Cumulative Distribution Function 51.4.2 The Normal Distribution 81.4.3 Exponential Distribution 101.4.4 Student’s t-distribution 111.4.5 Extreme Value Distribution 121.4.6 Generalized Extreme Value Distribution 121.5 Statistical Moments and Quantiles 131.5.1 Location 131.5.2 Dispersion 131.5.3 Asymmetry 131.5.4 Concentration in Tails 141.5.5 Statistical Moments 141.5.6 Quantiles 161.5.7 Sample Moments 161.6 Joint Probability Distributions 171.6.1 Conditional Probability 181.6.2 Definition of Joint Probability Distributions 191.6.3 Marginal Distributions 191.6.4 Dependence of Random Variables 201.6.5 Covariance and Correlation 201.6.6 Multivariate Normal Distribution 211.6.7 Elliptical Distributions 231.6.8 Copula Functions 251.7 Probabilistic Inequalities 301.7.1 Chebyshev’s Inequality 301.7.2 Fréchet-Hoeffding Inequality 311.8 Summary 32Chapter 2 Optimization 352.1 Introduction 352.2 Unconstrained Optimization 362.2.1 Minima and Maxima of a Differentiable Function 372.2.2 Convex Functions 402.2.3 Quasiconvex Functions 462.3 Constrained Optimization 482.3.1 Lagrange Multipliers 492.3.2 Convex Programming 522.3.3 Linear Programming 552.3.4 Quadratic Programming 572.4 Summary 58Chapter 3 Probability Metrics 613.1 Introduction 613.2 Measuring Distances: The Discrete Case 623.2.1 Sets of Characteristics 633.2.2 Distribution Functions 643.2.3 Joint Distribution 683.3 Primary, Simple, and Compound Metrics 723.3.1 Axiomatic Construction 733.3.2 Primary Metrics 743.3.3 Simple Metrics 753.3.4 Compound Metrics 843.3.5 Minimal and Maximal Metrics 863.4 Summary 903.5 Technical Appendix 903.5.1 Remarks on the Axiomatic Construction of Probability Metrics 913.5.2 Examples of Probability Distances 943.5.3 Minimal and Maximal Distances 99Chapter 4 Ideal Probability Metrics 1034.1 Introduction 1034.2 The Classical Central Limit Theorem 1054.2.1 The Binomial Approximation to the Normal Distribution 1054.2.2 The General Case 1124.2.3 Estimating the Distance from the Limit Distribution 1184.3 The Generalized Central Limit Theorem 1204.3.1 Stable Distributions 1204.3.2 Modeling Financial Assets with Stable Distributions 1224.4 Construction of Ideal Probability Metrics 1244.4.1 Definition 1254.4.2 Examples 1264.5 Summary 1314.6 Technical Appendix 1314.6.1 The CLT Conditions 1314.6.2 Remarks on Ideal Metrics 133Chapter 5 Choice under Uncertainty 1395.1 Introduction 1395.2 Expected Utility Theory 1415.2.1 St. Petersburg Paradox 1415.2.2 The von Neumann–Morgenstern Expected Utility Theory 1435.2.3 Types of Utility Functions 1455.3 Stochastic Dominance 1475.3.1 First-Order Stochastic Dominance 1485.3.2 Second-Order Stochastic Dominance 1495.3.3 Rothschild-Stiglitz Stochastic Dominance 1505.3.4 Third-Order Stochastic Dominance 1525.3.5 Efficient Sets and the Portfolio Choice Problem 1545.3.6 Return versus Payoff 1545.4 Probability Metrics and Stochastic Dominance 1575.5 Summary 1615.6 Technical Appendix 1615.6.1 The Axioms of Choice 1615.6.2 Stochastic Dominance Relations of Order n 1635.6.3 Return versus Payoff and Stochastic Dominance 1645.6.4 Other Stochastic Dominance Relations 166Chapter 6 Risk and Uncertainty 1716.1 Introduction 1716.2 Measures of Dispersion 1746.2.1 Standard Deviation 1746.2.2 Mean Absolute Deviation 1766.2.3 Semistandard Deviation 1776.2.4 Axiomatic Description 1786.2.5 Deviation Measures 1796.3 Probability Metrics and Dispersion Measures 1806.4 Measures of Risk 1816.4.1 Value-at-Risk 1826.4.2 Computing Portfolio VaR in Practice 1866.4.3 Backtesting of VaR 1926.4.4 Coherent Risk Measures 1946.5 Risk Measures and Dispersion Measures 1986.6 Risk Measures and Stochastic Orders 1996.7 Summary 2006.8 Technical Appendix 2016.8.1 Convex Risk Measures 2016.8.2 Probability Metrics and Deviation Measures 202Chapter 7 Average Value-at-Risk 2077.1 Introduction 2077.2 Average Value-at-Risk 2087.3 AVaR Estimation from a Sample 2147.4 Computing Portfolio AVaR in Practice 2167.4.1 The Multivariate Normal Assumption 2167.4.2 The Historical Method 2177.4.3 The Hybrid Method 2177.4.4 The Monte Carlo Method 2187.5 Backtesting of AVaR 2207.6 Spectral Risk Measures 2227.7 Risk Measures and Probability Metrics 2247.8 Summary 2277.9 Technical Appendix 2277.9.1 Characteristics of Conditional Loss Distributions 2287.9.2 Higher-Order AVaR 2307.9.3 The Minimization Formula for AVaR 2327.9.4 AVaR for Stable Distributions 2357.9.5 ETL versus AVaR 2367.9.6 Remarks on Spectral Risk Measures 241Chapter 8 Optimal Portfolios 2458.1 Introduction 2458.2 Mean-Variance Analysis 2478.2.1 Mean-Variance Optimization Problems 2478.2.2 The Mean-Variance Efficient Frontier 2518.2.3 Mean-Variance Analysis and SSD 2548.2.4 Adding a Risk-Free Asset 2568.3 Mean-Risk Analysis 2588.3.1 Mean-Risk Optimization Problems 2598.3.2 The Mean-Risk Efficient Frontier 2628.3.3 Mean-Risk Analysis and SSD 2668.3.4 Risk versus Dispersion Measures 2678.4 Summary 2748.5 Technical Appendix 2748.5.1 Types of Constraints 2748.5.2 Quadratic Approximations to Utility Functions 2768.5.3 Solving Mean-Variance Problems in Practice 2788.5.4 Solving Mean-Risk Problems in Practice 2798.5.5 Reward-Risk Analysis 281Chapter 9 Benchmark Tracking Problems 2879.1 Introduction 2879.2 The Tracking Error Problem 2889.3 Relation to Probability Metrics 2929.4 Examples of r.d. Metrics 2969.5 Numerical Example 3009.6 Summary 3049.7 Technical Appendix 3049.7.1 Deviation Measures and r.d. Metrics 3059.7.2 Remarks on the Axioms 3059.7.3 Minimal r.d. Metrics 3079.7.4 Limit Cases of L∗p(X, Y) and Θ∗p(X, Y) 3109.7.5 Computing r.d. Metrics in Practice 311Chapter 10 Performance Measures 31710.1 Introduction 31710.2 Reward-to-Risk Ratios 31810.2.1 RR Ratios and the Efficient Portfolios 32010.2.2 Limitations in the Application of Reward-to-Risk Ratios 32410.2.3 The STARR 32510.2.4 The Sortino Ratio 32910.2.5 The Sortino-Satchell Ratio 33010.2.6 A One-Sided Variability Ratio 33110.2.7 The Rachev Ratio 33210.3 Reward-to-Variability Ratios 33310.3.1 RV Ratios and the Efficient Portfolios 33510.3.2 The Sharpe Ratio 33710.3.3 The Capital Market Line and the Sharpe Ratio 34010.4 Summary 34310.5 Technical Appendix 34310.5.1 Extensions of STARR 34310.5.2 Quasiconcave Performance Measures 34510.5.3 The Capital Market Line and Quasiconcave Ratios 35310.5.4 Nonquasiconcave Performance Measures 35610.5.5 Probability Metrics and Performance Measures 357Index 361