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

ISBN: 978-0-470-05316-4
382 pages
February 2008
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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Preface.

Acknowledgments.

CHAPTER 1: Concepts of Probability.

1.1 Introduction.

1.2 Basic Concepts.

1.3 Discrete Probability Distributions.

1.3.1 Bernoulli Distribution.

1.3.2 Binomial Distribution.

1.3.3 Poisson Distribution.

1.4 Continuous Probability Distributions.

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

1.4.2 The Normal Distribution.

1.4.3 Exponential Distribution.

1.4.4 Student’s t-distribution.

1.4.5 Extreme Value Distribution.

1.4.6 Generalized Extreme Value Distribution.

1.5 Statistical Moments and Quantiles.

1.5.1 Location.

1.5.2 Dispersion.

1.5.3 Asymmetry.

1.5.4 Concentration in Tails.

1.5.5 Statistical Moments.

1.5.6 Quantiles.

1.5.7 Sample Moments.

1.6 Joint Probability Distributions.

1.6.1 Conditional Probability.

1.6.2 Definition of Joint Probability Distributions.

1.6.3 Marginal Distributions.

1.6.4 Dependence of Random Variables.

1.6.5 Covariance and Correlation.

1.6.6 Multivariate Normal Distribution.

1.6.7 Elliptical Distributions.

1.6.8 Copula Functions.

1.7 Probabilistic Inequalities.

1.7.1 Chebyshev’s Inequality.

1.7.2 Fr´echet-Hoeffding Inequality.

1.8 Summary.

CHAPTER 2: Optimization.

2.1 Introduction.

2.2 Unconstrained Optimization.

2.2.1 Minima and Maxima of a Differentiable Function.

2.2.2 Convex Functions.

2.2.3 Quasiconvex Functions.

2.3 Constrained Optimization.

2.3.1 Lagrange Multipliers.

2.3.2 Convex Programming.

2.3.3 Linear Programming.

2.4 Summary.

CHAPTER 3: Probability Metrics.

3.1 Introduction.

3.2 Measuring Distances: The Discrete Case.

3.2.1 Sets of Characteristics.

3.2.2 Distribution Functions.

3.2.3 Joint Distribution.

3.3 Primary, Simple, and Compound Metrics.

3.3.1 Axiomatic Construction.

3.3.2 Primary Metrics.

3.3.3 Simple Metrics.

3.3.4 Compound Metrics.

3.3.5 Minimal and Maximal Metrics.

3.4 Summary.

3.5 Technical Appendix.

3.5.1 Remarks on the Axiomatic Construction of Probability Metrics.

3.5.2 Examples of Probability Distances.

3.5.3 Minimal and Maximal Distances.

CHAPTER 4: Ideal Probability Metrics.

4.1 Introduction.

4.2 The Classical Central Limit Theorem.

4.2.1 The Binomial Approximation to the Normal Distribution.

4.2.2 The General Case.

4.2.3 Estimating the Distance from the Limit Distribution.

4.3 The Generalized Central Limit Theorem.

4.3.1 Stable Distributions.

4.3.2 Modeling Financial Assets with Stable Distributions.

4.4 Construction of Ideal Probability Metrics.

4.4.1 Definition.

4.4.2 Examples.

4.5 Summary.

4.6 Technical Appendix.

4.6.1 The CLT Conditions.

4.6.2 Remarks on Ideal Metrics.

CHAPTER 5: Choice under Uncertainty.

5.1 Introduction.

5.2 Expected Utility Theory.

5.2.2 The von Neumann–Morgenstern Expected Utility Theory.

5.2.3 Types of Utility Functions.

5.3 Stochastic Dominance.

5.3.1 First-Order Stochastic Dominance.

5.3.2 Second-Order Stochastic Dominance.

5.3.3 Rothschild-Stiglitz Stochastic Dominance.

5.3.4 Third-Order Stochastic Dominance.

5.3.5 Efficient Sets and the Portfolio Choice Problem.

5.3.6 Return versus Payoff.

5.4 Probability Metrics and Stochastic Dominance.

5.5 Summary.

5.6 Technical Appendix.

5.6.1 The Axioms of Choice.

5.6.2 Stochastic Dominance Relations of Order n.

5.6.3 Return versus Payoff and Stochastic Dominance.

5.6.4 Other Stochastic Dominance Relations.

CHAPTER 6: Risk and Uncertainty.

6.1 Introduction.

6.2 Measures of Dispersion.

6.2.1 Standard Deviation.

6.2.2 Mean Absolute Deviation.

6.2.3 Semistandard Deviation.

6.2.4 Axiomatic Description.

6.2.5 Deviation Measures.

6.3 Probability Metrics and Dispersion Measures.

6.4 Measures of Risk.

6.4.1 Value-at-Risk.

6.4.2 Computing Portfolio VaR in Practice.

6.4.3 Backtesting of VaR.

6.4.4 Coherent Risk Measures.

6.5 Risk Measures and Dispersion Measures.

6.6 Risk Measures and Stochastic Orders.

6.7 Summary.

6.8 Technical Appendix.

6.8.1 Convex Risk Measures.

6.8.2 Probability Metrics and Deviation Measures.

CHAPTER 7: Average Value-at-Risk.

7.1 Introduction.

7.2 Average Value-at-Risk.

7.3 AVaR Estimation from a Sample.

7.4 Computing Portfolio AVaR in Practice.

7.4.1 The Multivariate Normal Assumption.

7.4.2 The Historical Method.

7.4.3 The Hybrid Method 217

7.4.4 The Monte Carlo Method.

7.5 Backtesting of AVaR.

7.6 Spectral Risk Measures.

7.7 Risk Measures and Probability Metrics.

7.8 Summary.

7.9 Technical Appendix.

7.9.1 Characteristics of Conditional Loss Distributions.

7.9.2 Higher-Order AVaR.

7.9.3 The Minimization Formula for AVaR.

7.9.4 AVaR for Stable Distributions.

7.9.5 ETL versus AVaR.

7.9.6 Remarks on Spectral Risk Measures.

CHAPTER 8: Optimal Portfolios.

8.1 Introduction.

8.2 Mean-Variance Analysis.

8.2.1 Mean-Variance Optimization Problems.

8.2.2 The Mean-Variance Efficient Frontier.

8.2.3 Mean-Variance Analysis and SSD.

8.3 Mean-Risk Analysis.

8.3.1 Mean-Risk Optimization Problems.

8.3.2 The Mean-Risk Efficient Frontier.

8.3.3 Mean-Risk Analysis and SSD.

8.3.4 Risk versus Dispersion Measures.

8.4 Summary.

8.5 Technical Appendix.

8.5.1 Types of Constraints.

8.5.2 Quadratic Approximations to Utility Functions.

8.5.3 Solving Mean-Variance Problems in Practice.

8.5.4 Solving Mean-Risk Problems in Practice.

8.5.5 Reward-Risk Analysis.

CHAPTER 9: Benchmark Tracking Problems.

9.1 Introduction.

9.2 The Tracking Error Problem.

9.3 Relation to Probability Metrics.

9.4 Examples of r.d. Metrics.

9.5 Numerical Example.

9.6 Summary.

9.7 Technical Appendix.

9.7.1 Deviation Measures and r.d. Metrics.

9.7.2 Remarks on the Axioms.

9.7.3 Minimal r.d. Metrics.

CHAPTER 10: Performance Measures.

10.1 Introduction.

10.2 Reward-to-Risk Ratios.

10.2.1 RR Ratios and the Efficient Portfolios.

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

10.2.3 The STARR.

10.2.4 The Sortino Ratio.

10.2.5 The Sortino-Satchell Ratio.

10.2.6 A One-Sided Variability Ratio.

10.2.7 The Rachev Ratio.

10.3 Reward-to-Variability Ratios.

10.3.1 RV Ratios and the Efficient Portfolios.

10.3.2 The Sharpe Ratio.

10.3.3 The Capital Market Line and the Sharpe Ratio.

10.4 Summary.

10.5 Technical Appendix.

10.5.1 Extensions of STARR.

10.5.2 Quasiconcave Performance Measures.

10.5.3 The Capital Market Line and Quasiconcave Ratios.

10.5.4 Nonquasiconcave Performance Measures.

10.5.5 Probability Metrics and Performance Measures.

Index.

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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.

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