A Probability Metrics Approach to Financial Risk Measures E-Book

Svetlozar T. Rachev

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 441

Veröffentlichungsjahr: 2011

Beschreibung

A Probability Metrics Approach to Financial Risk Measures relates the field of probability metrics and risk measures to one another and applies them to finance for the first time.

• Helps to answer the question: which risk measure is best for a given problem?
• Finds new relations between existing classes of risk measures
• Describes applications in finance and extends them where possible
• Presents the theory of probability metrics in a more accessible form which would be appropriate for non-specialists in the field
• Applications include optimal portfolio choice, risk theory, and numerical methods in finance
• Topics requiring more mathematical rigor and detail are included in technical appendices to chapters

Leseprobe

Contents

Cover

Title Page

Copyright

Dedication

Preface

About the Authors

Chapter 1: Introduction

1.1 Probability Metrics

1.2 Applications in Finance

References

Chapter 2: Probability Distances and Metrics

2.1 Introduction

2.2 Some Examples of Probability Metrics

2.3 Distance and Semidistance Spaces

2.4 Definitions of Probability Distances and Metrics

2.5 Summary

2.6 Technical Appendix

References

Chapter 3: Choice under Uncertainty

3.1 Introduction

3.2 Expected Utility Theory

3.3 Stochastic Dominance

3.4 Probability Metrics and Stochastic Dominance

3.5 Cumulative Prospect Theory

3.6 Summary

3.7 Technical Appendix

References

Chapter 4: A Classification of Probability Distances

4.1 Introduction

4.2 Primary Distances and Primary Metrics

4.3 Simple Distances and Metrics

4.4 Compound Distances and Moment Functions

4.5 Ideal Probability Metrics

4.6 Summary

4.7 Technical Appendix

References

Chapter 5: Risk and Uncertainty

5.1 Introduction

5.2 Measures of Dispersion

5.3 Probability Metrics and Dispersion Measures

5.4 Measures of Risk

5.5 Risk Measures and Dispersion Measures

5.6 Risk Measures and Stochastic Orders

5.7 Summary

5.8 Technical Appendix

References

Chapter 6: Average Value-at-Risk

6.1 Introduction

6.2 Average Value-at-Risk

6.3 AVaR Estimation from a Sample

6.4 Computing Portfolio AVaR in Practice

6.5 Back-Testing of AVaR

6.6 Spectral Risk Measures

6.7 Risk Measures and Probability Metrics

6.8 Risk Measures Based on Distortion Functionals

6.9 Summary

6.10 Technical Appendix

References

Chapter 7: Computing AVaR through Monte Carlo

7.1 Introduction

7.2 An Illustration of Monte Carlo Variability

7.3 Asymptotic Distribution, Classical Conditions

7.4 Rate of Convergence to the Normal Distribution

7.5 Asymptotic Distribution, Heavy-tailed Returns

7.6 Rate of Convergence, Heavy-tailed Returns

7.7 On the Choice of a Distributional Model

7.8 Summary

7.9 Technical Appendix

References

Chapter 8: Stochastic Dominance Revisited

8.1 Introduction

8.2 Metrization of Preference Relations

8.3 The Hausdorff Metric Structure

8.4 Examples

8.5 Utility-type Representations

8.6 Almost Stochastic Orders and Degree of Violation

8.7 Summary

8.8 Technical Appendix

References

Index

This edition first published 2011

© 2011 Svetlozar T. Rachev, Stoyan V. Stoyanov and Frank J. Fabozzi

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Library of Congress Cataloging-in-Publication Data

Rachev, S. T. (Svetlozar Todorov)

A probability metrics approach to financial risk measures / Svetlozar T. Rachev, Stoyan V. Stoyanov, Frank J. Fabozzi, CFA.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-4051-8369-7 (hardback)

1. Financial risk management. 2. Probabilities. I. Stoyanov, Stoyan V. II. Fabozzi, Frank J. III. Title.

HD61.R33 2010

332.01′5192–dc22

2010040519

A catalogue record for this book is available from the British Library.

STR

To my grandchildren Iliana, Zoya, and Zari

SVS

To my parents Veselin and Evgeniya Kolevi and my brother Pavel Stoyanov

FJF

To my wife Donna and my children Francesco, Patricia, and Karly

Preface

The theory of probability metrics is a branch of probability theory. It finds application in different theoretical and applied fields such as probability theory, queuing theory, insurance risk theory, and finance. The theory of probability metrics looks for answers to the following basic question: How can one measure the difference between random quantities? In finance, for example, we assume a stochastic model for asset return distributions and, in order to estimate the risk of a portfolio of assets, we sample from the fitted distribution. Then, we use the generated simulations to calculate portfolio risk. In this context, there are two issues arising on two different levels. First, the assumed stochastic model should be “close” to the empirical data. In this sense, we say that we need a realistic model in the first place. Second, since the risk estimate is essentially computed from random scenarios, we have to be aware of the variability of the estimator and how it depends on the assumed asset return distributions.

Although based on universal principles and ideas, the field of probability metrics is very specialized. Most of the literature is highly technical and is accessible mostly to specialists in probability theory. As far as applications are concerned, apart from our book Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: Ideal Risk, Uncertainty, and Performance Measures (John Wiley & Sons, 2008), we are unaware of other literature describing applications in finance.

This book has two goals. The first goal is to describe applications in finance and extend them where possible. The second goal is to present the theory of probability metrics in a more accessible form which would be appropriate for non-specialists in the field. Topics requiring more mathematical rigor and detail are included in technical appendices to chapters.

The book is organized in the following way. Chapter 1 provides a conceptual description of the method of probability metrics and reviews direct and indirect applications in the field of finance. Chapter 2 provides an introduction to the theory of probability metrics. The classical theory describing investor choice under uncertainty is provided in Chapter 3. Chapter 4 discusses the classification of probability distances to primary, simple, and compound types. The information in Chapter 2 is a prerequisite. Chapters 5, 6, and 7 are devoted to risk and uncertainty measures and discuss in detail AVaR and the Monte Carlo method for AVaR estimation. Chapter 6 is a prerequisite to Chapter 10. Finally, Chapter 8 considers the problem of quantifying stochastic dominance relations and takes advantage of the terms introduced in Chapter 3.

Svetlozar T. Rachev

Stoyan V. Stoyanov

Frank J. Fabozzi

About the Authors

Svetlozar (Zari) T. Rachev completed his Ph.D. degree in 1979 from Moscow State (Lomonosov) University, and his Doctor of Science Degree in 1986 from Steklov Mathematical Institute in Moscow. Currently he is Chair-Professor in Statistics, Econometrics and Mathematical Finance at the University of Karlsruhe in the School of Economics and Business Engineering. He is also Professor Emeritus at the University of California, Santa Barbara in the Department of Statistics and Applied Probability. He has published seven monographs, eight handbooks and special-edited volumes, and over 300 research articles. His recently coauthored books published by Wiley in mathematical finance and financial econometrics include Fat-Tailed and Skewed Asset Return Distributions: Implications for Risk Management, Portfolio selection, and Option Pricing (2005), Operational Risk: A Guide to Basel II Capital Requirements, Models, and Analysis (2007), Financial Econometrics: From Basics to Advanced Modeling Techniques (2007), and Bayesian Methods in Finance (2008). Professor Rachev is cofounder of Bravo Risk Management Group, specializing in financial risk-management software. Bravo Group was recently acquired by FinAnalytica, for which he currently serves as Chief-Scientist.

Stoyan V. Stoyanov is a Professor of Finance at EDHEC Business School and Scientific Director for EDHEC-Risk Institute in Asia. Prior to joining EDHEC, he was the Head of Quantitative Research at FinAnalytica, specializing in financial risk management software. He completed his Ph.D. degree with honors in 2005 from the School of Economics and Business Engineering (Chair of Statistics, Econometrics and Mathematical Finance) at the University of Karlsruhe and is author and co-author of numerous papers. His research interests include probability theory, heavy-tailed modeling in the field of finance, and optimal portfolio theory. His articles have recently appeared in Economics Letters, Journal of Banking and Finance, Applied Mathematical Finance, Applied Financial Economics, and International Journal of Theoretical and Applied Finance. He is a co-author of the mathematical finance book Advanced Stochastic Models, Risk Assessment and Portfolio Optimization: The Ideal Risk, Uncertainty and Performance Measures (2008) published by Wiley.

Frank J. Fabozzi is Professor in the Practice of Finance in the School of Management at Yale University. Prior to joining the Yale faculty, he was a Visiting Professor of Finance in the Sloan School of Management at MIT. Professor Fabozzi is a Fellow of the International Center for Finance at Yale University and on the Advisory Council for the Department of Operations Research and Financial Engineering at Princeton University. He is the editor of the Journal of Portfolio Management. His recently co-authored books published by Wiley in mathematical finance and financial econometrics include The Mathematics of Financial Modeling and Investment Management (2004), Financial Modeling of the Equity Market: From CAPM to Cointegration (2006), Robust Portfolio Optimization and Management (2007), Financial Econometrics: From Basics to Advanced Modeling Techniques (2007), and Bayesian Methods in Finance (2008). He earned a doctorate in economics from the City University of New York in 1972. In 2002 Professor Fabozzi was inducted into the Fixed Income Analysts Society's Hall of Fame and he is the 2007 recipient of the C. Stewart Sheppard Award given by the CFA Institute. He earned the designation of Chartered Financial Analyst and Certified Public Accountant.

Chapter 1

Introduction

In this chapter, we provide a conceptual description of the method of probability metrics and discuss direct and indirect applications in the field of finance, which are described in more detail throughout the book.

1.1 Probability Metrics

The development of the theory of probability metrics started with the investigation of problems related to limit theorems in probability theory. Limit theorems occupy a very important place in probability theory, statistics, and all their applications. A well-known example is the celebrated central limit theorem (CLT) but there are many other limit theorems, such as the generalized CLT, the max-stable CLT, functional limit theorems, etc. In general, the applicability of the limit theorems stems from the fact that the limit law can be regarded as an approximation to the stochastic model under consideration and, therefore, can be accepted as an approximate substitute. The central question arising is how large an error we make by adopting the approximate model and this question can be investigated by studying the distance between the limit law and the stochastic model. It turns out that this distance is not influenced by the particular problem. Rather, it can be studied by a theory based on some universal principles.

Generally, the theory of probability metrics studies the problem of measuring distances between random quantities. On one hand, it provides the fundamental principles for building probability metrics – the means of measuring such distances. On the other, it studies the relationships between various classes of probability metrics. Another realm of study concerns problems which require a particular metric while the basic results can be obtained in terms of other metrics. In such cases, the metrics relationship is of primary importance.

Certainly, the problem of measuring distances is not limited to random quantities only. In its basic form, it originated in different fields of mathematics. Nevertheless, the theory of probability metrics was developed due to the need of metrics with specific properties. Their choice is very often dictated by the stochastic model under consideration and to a large extent determines the success of the investigation. Rachev (1991) provides more details on the methods of the theory of probability metrics and its numerous applications in both theoretical and more practical problems.

1.2 Applications in Finance

There are no limitations in the theory of probability metrics concerning the nature of the random quantities. This makes its methods fundamental and appealing. Actually, in the general case, it is more appropriate to refer to the random quantities as random elements. They can be random variables, random vectors, random functions or random elements in general spaces. For instance, in the context of financial applications, we can study the distance between two random stocks prices, or between vectors of financial variables that are used to construct portfolios, or between yield curves which are much more complicated objects. The methods of the theory remain the same, irrespective of the nature of the random elements. This represents the most direct application of the theory of probability metrics in finance: that is, it provides a method for measuring how different two random elements are. We explain the axiomatic construction of probability metrics and provide financial interpretations in Chapter 2.

Financial economics, like any other science relying on statistical methods, considers statistical information about the objects it studies on several levels. In some theories in the area of finance, conclusions are drawn only on the basis of certain characteristics of the corresponding distributions. For example, an investor would oftentimes use a risk-reward ratio to rank investment opportunities. Essentially, this reduces to computing the measure of reward (e.g., the expected return) and the measure of risk (e.g., value-at-risk, conditional value-at-risk, standard deviation). Both the measure of reward and the measure of risk represent two characteristics of the corresponding distributions. In effect, the final decision is made on the basis of these two characteristics which, from the investor's perspective, aggregate the information available in the distribution functions.

The theory describing investor choice under uncertainty, the fundamentals of which we discuss in Chapter 3, uses a different approach. Various criteria were developed for first-, second-, and higher-order stochastic dominance based on the distributions themselves. As a consequence, investment opportunities are compared directly through their distribution functions, which is a superior approach from the standpoint of the utilized information.

As another example, consider the problem of building a diversified portfolio. The investor would be interested not only in the marginal distribution characteristics (i.e., the characteristics of the assets on a stand-alone basis), but also in how the assets depend on each another. This requires an additional piece of information which cannot be recovered from the distribution functions of the asset returns. The notion of stochastic dependence can be described by considering the joint behavior of assets returns.

The theory of probability metrics offers a systematic approach towards such a hierarchy of ways to utilize statistical information. It distinguishes between primary, simple, and compound types of distances which are defined on the space of characteristics, the space of distribution functions, and the space of joint distributions, respectively. Therefore, depending on the particular problem, one can choose the appropriate distance type and this represents another direct application of the theory of probability metrics in the field of finance. This classification of probability distances is explained in Chapter 4.

Besides direct applications, there are also a number of indirect ones. For instance, one of the most important problems in risk estimation is formulating a realistic hypothesis for the asset return distributions. This is largely an empirical question because no arguments exist that can be used to derive a model from some general principles. Therefore, we have to hypothesize a model that best describes a number of empirically confirmed phenomena about asset returns: (1) volatility clustering, (2) autoregressive behavior, (3) short- and long-range dependence, and (4) fat-tailed behavior of the building blocks of the time-series model which varies depending on the frequency (e.g., intra-day, daily, monthly). The theory of probability metrics can be used to suggest a solution to (4). The fact that the degree of heavy-tailedness varies with the frequency may be related to the process of aggregation of higher-frequency returns to obtain lower frequency returns. Generally, the residuals from higher-frequency return models tend to have heavier tails and this observation together with a result known as a pre-limit theorem can be used to derive a suggestion for the overall shape of the return distribution. Furthermore, the probability distance used in the pre-limit theorem indicates that the derived shape is most relevant for the body of the distribution. As a result, through the theory of probability metrics we can obtain an approach to construct reasonable models for asset return distributions. We discuss in more detail limit and pre-limit theorems in Chapter 7.

Another central topic in finance is quantification of risk and uncertainty. The two notions are related but are not synonymous. Functionals quantifying risk are called risk measures and functionals quantifying uncertainty are called deviation measures or dispersion measures. Axiomatic constructions are suggested in the literature for all of them. It turns out that the axioms defining measures of uncertainty can be linked to the axioms defining probability distances, however, with one important modification. The axiom of symmetry, which every distance function should satisfy, appears unnecessarily restrictive. Therefore, we can derive the class of deviation measures from the axiomatic construction of asymmetric probability distances which are also called probability quasi-distances. The topic is discussed in detail in Chapter 5.

As far as risk measures are concerned, we consider in detail advantages and disadvantages of value-at-risk, average value-at-risk (AVaR), and spectral risk measures in Chapter 5 and Chapter 6. Since Monte Carlo-based techniques are quite common among practitioners, we discuss in Chapter 7 Monte Carlo-based estimation of AVaR and the problem of stochastic stability in particular. The discussion is practical, based on simulation studies, and is inspired by the classical application of the theory of probability metrics in estimating the stochastic stability of probabilistic models. We apply the CLT and the Generalized CLT to derive the asymptotic distribution of the AVaR estimator under different distributional hypotheses and we discuss approaches to improve its stochastic stability.

We mentioned that adopting stochastic dominance rules for prospect selection rather than rules based on certain characteristics leads to a more efficient use of the information contained in the corresponding distribution functions. Stochastic dominance rules, however, are of the type “X dominates ” or “X does not dominate ”: that is, the conclusion is qualitative. As a consequence, computational problems are hard to solve in this setting. A way to overcome this difficulty is to transform the nature of the relationship from qualitative to quantitative. We describe how this can be achieved in Chapter 8, which is the last chapter in the book. Our approach is fundamental and is based on asymmetric probability semidistances, which are also called probability quasi-semidistances.

The link with probability metrics theory allows a classification of stochastic dominance relations in general. They can be primary, simple, or compound but also, depending on the underlying structure, they may or may not be generated by classes of investors, which is a typical characterization in the classical theory of choice under uncertainty. This is also a topic discussed in Chapter 8.

References

Rachev, S. T. (1991), Probability Metrics and the Stability of Stochastic Models, Wiley, New York.