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

by Svetlozar T. Rachev (Univ. of California, Santa Barbara); Stoyan V. Stoyanov; Frank J. Fabozzi (School of Management, Yale Univ.)

• ISBN13: 9780470253601
• ISBN10: 0470253606
• Format: eBook
• Copyright: 2008-05-01
• Publisher: Wiley

Summary

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.

Table of Contents

Preface

Acknowledgments

About the Authors

Concepts of Probability

Introduction

Basic Concepts

Discrete Probability Distributions

Bernoulli Distribution

Binomial Distribution

Poisson Distribution

Continuous Probability Distributions

Probability Distribution Function, Probability Density Function, and Cumulative Distribution Function

The Normal Distribution

Exponential Distribution

StudentÆs t-distribution

Extreme Value Distribution

Generalized Extreme Value Distribution

Statistical Moments and Quantiles

Location

Dispersion

Asymmetry

Concentration in Tails

Statistical Moments

Quantiles

Sample Moments

Joint Probability Distributions

Conditional Probability

Definition of Joint Probability Distributions

Marginal Distributions

Dependence of Random Variables

Covariance and Correlation

Multivariate Normal Distribution

Elliptical Distributions

Copula Functions

Probabilistic Inequalities

ChebyshevÆs Inequality

Fr´echet-Hoeffding Inequality

Summary

Optimization

Introduction

Unconstrained Optimization

Minima and Maxima of a Differentiable Function

Convex Functions

Quasiconvex Functions

Constrained Optimization

Lagrange Multipliers

Convex Programming

Linear Programming

Quadratic Programming

Summary

Probability Metrics

Introduction

Measuring Distances: The Discrete Case

Sets of Characteristics

Distribution Functions

Joint Distribution

Primary, Simple, and Compound Metrics

Axiomatic Construction

Primary Metrics

Simple Metrics

Compound Metrics

Minimal and Maximal Metrics

Summary

Technical Appendix

Remarks on the Axiomatic Construction of Probability Metrics

Examples of Probability Distances

Minimal and Maximal Distances

Ideal Probability Metrics

Introduction

The Classical Central Limit Theorem

The Binomial Approximation to the Normal Distribution

The General Case

Estimating the Distance from the Limit Distribution

The Generalized Central Limit Theorem

Stable Distributions

Modeling Financial Assets with Stable Distributions

Construction of Ideal Probability Metrics

Definition

Examples

Summary

Technical Appendix

The CLT Conditions

Remarks on Ideal Metrics

Choice under Uncertainty

Introduction

Expected Utility Theory

St. Petersburg Paradox

The von NeumannûMorgenstern Expected Utility Theory

Types of Utility Functions

Stochastic Dominance

First-Order Stochastic Dominance

Second-Order Stochastic Dominance

Rothschild-Stiglitz Stochastic Dominance

Third-Order Stochastic Dominance

Efficient Sets and the Portfolio Choice Problem

Return versus Payoff

Probability Metrics and Stochastic Dominance

Summary

Technical Appendix

The Axioms of Choice

Stochastic Dominance Relations of Order n

Return versus Payoff and Stochastic Dominance

Other Stochastic Dominance Relations

Risk and Uncertainty

Introduction

Measures of Dispersion

Standard Deviation

Mean Absolute Deviation

Semistandard Deviation

Axiomatic Description

Deviation Measures

Probability Metrics and Dispersion Measures

Measures of Risk

Value-at-Risk

Computing Portfolio VaR in Practice

Backtesting of VaR

Coherent Risk Measures

Risk Measures and Dispersion Measures

Risk Measures and Stochastic Orders

Summary

Technical Appendix

Convex Risk Measures

Probability Metrics and Deviation Measures

Average Value-at-Risk

Introduction

Average Value-at-Risk

AVaR Estimation from a Sample

Computing Portfolio AVaR in Practice

The Multivariate Normal Assumption

The Historical Method

The Hybrid Method 217

The Monte Carlo Method

Backtesting of AVaR

Spectral Risk Measures

Risk Measures and Probability Metrics

Summary

Technical Appendix

Characteristics of Conditional Loss Distributions

Higher-Order AVaR

The Minimization Formula for AVaR

AVaR for Stable Distributions

ETL versus AVaR

Remarks on Spectral Risk Measures

Optimal Portfolios

Introduction

Mean-Variance Analysis

Mean-Variance Optimization Problems

The Mean-Variance Efficient Frontier

Mean-Variance Analysis and SSD

Adding a Risk-Free Asset

Mean-Risk Analysis

Mean-Risk Optimization Problems

The Mean-Risk Efficient Frontier

Mean-Risk Analysis and SSD

Risk versus Dispersion Measures

Summary

Technical Appendix

Types of Constraints

Quadratic Approximations to Utility Functions

Solving Mean-Variance Problems in Practice

Solving Mean-Risk Problems in Practice

Reward-Risk Analysis

Benchmark Tracking Problems

Introduction

The Tracking Error Problem

Relation to Probability Metrics

Examples of r.d. Metrics

Numerical Example

Summary

Technical Appendix

Deviation Measures and r.d. Metrics

Remarks on the Axioms

Minimal r.d. Metrics

Performance Measures

Introduction

Reward-to-Risk Ratios

RR Ratios and the Efficient Portfolios

Limitations in the Application of Reward-to-Risk Ratios

The STARR

The Sortino Ratio

The Sortino-Satchell Ratio

A One-Sided Variability Ratio

The Rachev Ratio

Reward-to-Variability Ratios

RV Ratios and the Efficient Portfolios

The Sharpe Ratio

The Capital Market Line and the Sharpe Ratio

Summary

Technical Appendix

Extensions of STARR

Quasiconcave Performance Measures

The Capital Market Line and Quasiconcave Ratios

Nonquasiconcave Performance Measures

Probability Metrics and Performance Measures

Index

Table of Contents provided by Publisher. All Rights Reserved.