Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ide

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This groundbreaking book extends traditional approaches of risk measurement and portfolio optimization by combining distributional models with risk or performance measures into one framework.

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This groundbreaking book extends traditional approaches of risk measurement and portfolio optimization by combining distributional models with risk or performance measures into one framework. Throughout these pages, the expert authors explain the fundamentals of probability metrics, outline new approaches to portfolio optimization, and discuss a variety of essential risk measures. Using numerous examples, they illustrate a range of applications to optimal portfolio choice and risk theory, as well as applications to the area of computational finance that may be useful to financial engineers.

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Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization The finance industry is seeing increased interest in new risk measures and techniques for portfolio optimization when parameters of the model are uncertain. This groundbreaking book extends traditional approaches of risk measurement and portfolio optimization by combining distributional models with risk or performance measures into one framework. Throughout these pages, the expert authors explain the fundamentals of probability metrics, outline new approaches to portfolio optimization, and discuss a variety of essential risk measures. Using numerous examples, they illustrate a range of applications to optimal portfolio choice and risk theory, as well as applications to the area of computational finance that may be useful to financial engineers. They also clearly show how stochastic models, risk assessment, and optimization are essential to mastering risk, uncertainty, and performance measurement. Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization provides quantitative portfolio managers (including hedge fund managers), financial engineers, consultants, and academic researchers with answers to the key question of which risk measure is best for any given problem.

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Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization The finance industry is seeing increased interest in new risk measures and techniques for portfolio optimization when parameters of the model are uncertain. This groundbreaking book extends traditional approaches of risk measurement and portfolio optimization by combining distributional models with risk or performance measures into one framework. Throughout these pages, the expert authors explain the fundamentals of probability metrics, outline new approaches to portfolio optimization, and discuss a variety of essential risk measures. Using numerous examples, they illustrate a range of applications to optimal portfolio choice and risk theory, as well as applications to the area of computational finance that may be useful to financial engineers. They also clearly show how stochastic models, risk assessment, and optimization are essential to mastering risk, uncertainty, and performance measurement. Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization provides quantitative portfolio managers (including hedge fund managers), financial engineers, consultants, and academic researchers with answers to the key question of which risk measure is best for any given problem.

Author Biography

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.

Table of 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 1.3.1 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 Fréchet-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

Long Description

An introduction to stochastic models for risk evaluation and portfolio selection enhanced by insights from the field of probability metrics and optimization theory This book extends traditional approaches of risk measurement and portfolio optimization by combining distributional models with risk or performance measures into one framework. They introduce the reader to the rudiments of probability metrics, outline new approaches to portfolio optimization, and discuss a variety of essential risk measures. Svetlozar T. Rachev, PhD, DrSci (Karlsruhe, Germany) is currently Chair-Professor in Statistics, Econometrics, and Mathematical Finance at the University of Karlsruhe. Stoyan V. Stoyanov, PhD (Sofia, Bulgaria) is the Chief Financial Researcher at FinAnlytica Inc. Frank J. Fabozzi, PhD, CFA (New Hope, PA) is Professor in the Practice of Finance at Yale University's School of Management.

Details ISBN047005316X Author Frank J. Fabozzi Language English ISBN-10 047005316X ISBN-13 9780470053164 Media Book Format Hardcover Illustrations Yes Year 2008 Subtitle The Ideal Risk, Uncertainty, and Performance Measures Short Title ADVD STOCHASTIC MODELS RISK AS Residence Santa Barbara Affiliation Univ. of California Univ. of California, Santa Barbara Univ. of Califo Edition 1st DOI 10.1604/9780470053164 Series Number 149 UK Release Date 2008-04-11 AU Release Date 2008-04-11 NZ Release Date 2008-04-11 US Release Date 2008-04-11 Pages 400 Publisher John Wiley & Sons Inc Series Frank J. Fabozzi Series Publication Date 2008-04-11 Imprint John Wiley & Sons Inc Place of Publication New York Country of Publication United States DEWEY 332.015195 Audience Professional & Vocational We've got this

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