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


S ince the 1990s, significant progress has been made indeveloping the concept of a risk measure from both a theoreticaland a practical viewpoint. This notion has evolved into amaterially different form from the original idea behind traditionalmean-variance analysis. As a consequence, the distinction betweenrisk and uncertainty, which translates into a distinction between arisk measure and a dispersion measure, offers a new way of lookingat the problem of optimal portfolio ...

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S ince the 1990s, significant progress has been made indeveloping the concept of a risk measure from both a theoreticaland a practical viewpoint. This notion has evolved into amaterially different form from the original idea behind traditionalmean-variance analysis. As a consequence, the distinction betweenrisk and uncertainty, which translates into a distinction between arisk measure and a dispersion measure, offers a new way of lookingat the problem of optimal portfolio selection.

In Advanced Stochastic Models, Risk Assessment, and PortfolioOptimization, the authors assert that the ideas behind the conceptof probability metrics can be borrowed and applied in the field ofasset management in order to construct an ideal risk measure whichwould be "ideal" for a given optimal portfolio selection problem.They provide a basic introduction to the theory of probabilitymetrics and the problem of optimal portfolio selection consideredin the general context of risk and reward measures.

Generally, the theory of probability metrics studies the problemof measuring distances between random quantities. There are nolimitations in the theory of probability metrics concerning thenature of the random quantities, which makes its methodsfundamental and appealing. Actually, it is more appropriate torefer to the random quantities as random elements: they can berandom variables, random vectors, random functions, or randomelements of general spaces. In the context of financialapplications, we can study the distance between two random stocksprices, or between vectors of financial variables buildingportfolios, or between entire yield curves that are much morecomplicated objects. The methods of the theory remain the same, nomatter the nature of the random elements.

Using numerous illustrative examples, this book shows howprobability metrics can be applied to a range of areas in finance,including: stochastic dominance orders, the construction of riskand dispersion measures, problems involving average value-at-riskand spectral risk measures in particular, reward-risk analysis,generalizing mean-variance analysis, benchmark tracking, and theconstruction of performance measures. For each chapter where moretechnical knowledge is necessary, an appendix is included.

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Product Details

  • ISBN-13: 9780470053164
  • Publisher: Wiley
  • Publication date: 2/25/2008
  • Series: Frank J. Fabozzi Series , #149
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 382
  • Product dimensions: 6.36 (w) x 9.37 (h) x 1.41 (d)

Meet the Author

Svetlozar T. Rachev, PhD, Doctor of Science, isChair-Professor at the University of Karlsruhe in the School ofEconomics and Business Engineering; Professor Emeritus at theUniversity of California, Santa Barbara; and Chief-Scientist ofFinAnalytica Inc.

Stoyan V. Stoyanov, PhD, is the Chief FinancialResearcher at FinAnalytica Inc.

Frank J. Fabozzi, PhD, CFA, is Professor in the Practiceof Finance and Becton Fellow at Yale University's School ofManagement and the Editor of the Journal of PortfolioManagement.

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Table of Contents



About the Authors.

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 DensityFunction, 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 Frèchet-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.

2.3.4 Quadratic 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 ProbabilityMetrics.

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.

5.2.1 St. Petersburg Paradox.

5.2.2 The von Neumann–Morgenstern Expected UtilityTheory.

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.

8.2.4 Adding a Risk-Free Asset.

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-RiskRatios.

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.


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