Robust Equity Portfolio Management E-Book

Woo Chang Kim

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Frank J. Fabozzi Series
• Sprache: Englisch

Seitenzahl: 320

Veröffentlichungsjahr: 2015

Beschreibung

A comprehensive portfolio optimization guide, with provided MATLAB code Robust Equity Portfolio Management + Website offers the most comprehensive coverage available in this burgeoning field. Beginning with the fundamentals before moving into advanced techniques, this book provides useful coverage for both beginners and advanced readers. MATLAB code is provided to allow readers of all levels to begin implementing robust models immediately, with detailed explanations and applications in the equity market included to help you grasp the real-world use of each technique. The discussion includes the most up-to-date thinking and cutting-edge methods, including a much-needed alternative to the traditional Markowitz mean-variance model. Unparalleled in depth and breadth, this book is an invaluable reference for all risk managers, portfolio managers, and analysts. Portfolio construction models originating from the standard Markowitz mean-variance model have a high input sensitivity that threatens optimization, spawning a flurry of research into new analytic techniques. This book covers the latest developments along with the basics, to give you a truly comprehensive understanding backed by a robust, practical skill set. * Get up to speed on the latest developments in portfolio optimization * Implement robust models using provided MATLAB code * Learn advanced optimization methods with equity portfolio applications * Understand the formulations, performances, and properties of robust portfolios The Markowitz mean-variance model remains the standard framework for portfolio optimization, but the interest in--and need for--an alternative is rapidly increasing. Resolving the sensitivity issue and dramatically reducing portfolio risk is a major focus of today's portfolio manager. Robust Equity Portfolio Management + Website provides a viable alternative framework, and the hard skills to implement any optimization method.

Leseprobe

Table of Contents

The Frank J. Fabozzi Series

Title Page

Copyright

Dedication

Preface

Chapter 1: Introduction

1.1 Overview of the Chapters

1.2 Use of MATLAB

Notes

Chapter 2: Mean-Variance Portfolio Selection

2.1 Return of Portfolios

2.2 Risk of Portfolios

2.3 Diversification

2.4 Mean-Variance Analysis

2.5 Factor Models

2.6 Example

Key Points

Notes

Chapter 3: Shortcomings of Mean-Variance Analysis

3.1 Limitations on the Use of Variance

3.2 Difficulty in Estimating the Inputs

3.3 Sensitivity of Mean-Variance Portfolios

3.4 Improvements on Mean-Variance Analysis

Key Points

Notes

Chapter 4: Robust Approaches for Portfolio Selection

4.1 Robustness

4.2 Robust statistics

4.3 Shrinkage Estimation

4.4 Monte Carlo Simulation

4.5 Constraining Portfolio Weights

4.6 Bayesian Approach

4.7 Stochastic Programming

4.8 Additional Approaches

Key Points

Notes

Chapter 5: Robust Optimization

5.1 Worst-Case Decision Making

5.2 Convex Optimization

5.3 Robust Counterparts

5.4 Interior Point Methods

Key Points

Notes

Chapter 6: Robust Portfolio Construction

6.1 Some Preliminaries

6.2 Mean-Variance Portfolios

6.3 Constructing Robust Portfolios

6.4 Robust Portfolios with Box Uncertainty

6.5 Robust Portfolios with Ellipsoidal Uncertainty

6.6 Closing Remarks

Key Points

Notes

Chapter 7: Controlling Third and Fourth Moments of Portfolio Returns via Robust Mean-Variance Approach

7.1 Controlling Higher Moments of Portfolio Return

7.2 Why Robust Formulation Controls Higher Moments

7.3 Empirical Tests

Key Points

Notes

Chapter 8: Higher Factor Exposures of Robust Equity Portfolios

8.1 Importance of Portfolio Factor Exposure

8.2 Fundamental Factor Models in the Equity Market

8.3 Factor Dependency of Robust Portfolios: Theoretical Arguments

8.4 Factor Dependency of Robust Portfolios: Empirical Findings

8.5 Factor Movements and Robust Portfolios

8.6 Robust Formulations That Control Factor Exposure

Key Points

Notes

Chapter 9: Composition of Robust Portfolios

9.1 Overview of Analyses

9.2 Composition Based on Investment Styles

9.3 Composition Based on Additional Factors

9.4 Composition Based on Stock Betas

9.5 Robust Portfolio Construction Based on Stock Beta Attributes

Key Points

Notes

Chapter 10: Robust Portfolio Performance

10.1 Portfolio Performance Measures

10.2 Historical Performance of Robust Portfolios

10.3 Measuring Robustness

Key Points

Notes

Chapter 11: Robust Optimization Software

11.1 YALMIP

11.2 ROME (Robust Optimization Made Easy)

11.3 AIMMS

Key Points

Notes

About the Authors

About the Companion Website

Index

End User License Agreement

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Guide

Cover

Table of Contents

Begin Reading

List of Illustrations

Chapter 2: Mean-Variance Portfolio Selection

Exhibit 2.1 Daily stock price of Twitter and Tesla Motors from January to June 2014

Exhibit 2.2 Portfolio with 50-50 allocation in Twitter and Tesla Motors

Exhibit 2.3 Risk of the 50-50 portfolio for three correlation levels

Exhibit 2.4 Portfolio return and risk for various proportions

Exhibit 2.5 Minimum-variance set and the efficient frontier

Exhibit 2.6 Mean-variance efficient portfolios

Chapter 3: Shortcomings of Mean-Variance Analysis

Exhibit 3.1 Histogram of S&P 500 returns from 1995 to 2013 and estimated normal distribution

Exhibit 3.2 Histogram of weekly returns from 1995 to 2013 and estimated normal distribution

Exhibit 3.3 Two efficient frontiers from different sets of data

Exhibit 3.4 Estimated, actual, and true frontiers for investing in 10 stocks

Exhibit 3.5 Expected return of 10 stocks

Exhibit 3.6 Portfolio weights given to each stock when expected return of a single stock is shifted (horizontal-axis: index of stock with change in expected return; 0: no change)

Exhibit 3.7 Histograms of portfolio returns with the same VaR at 5% level

Chapter 4: Robust Approaches for Portfolio Selection

Exhibit 4.1 Daily returns of a stock for 20 trading days

Exhibit 4.2 Comparison between portfolio selection from estimators and simulations

Exhibit 4.3 Weights of mean-variance portfolios with various conditions on weights

Chapter 5: Robust Optimization

Exhibit 5.1 Probability of loss for a normal distribution

Exhibit 5.2 Uncertainty Sets That Form Tractable Robust Counterparts

Chapter 7: Controlling Third and Fourth Moments of Portfolio Returns via Robust Mean-Variance Approach

Exhibit 7.1 PDFs for skew normal distribution (

α

= 5, 0, and –5)

Exhibit 7.2 Sample mean and sample variance from three skew normal distributions Panel A. From positively skewed distribution Panel B. From symmetric distribution Panel C. From negatively skewed distribution

Exhibit 7.3 Sample paths of log-prices (dark lines) and realized volatilities (grey lines) generated from skew normal distributions (Panel A: skewness = 0.995; Panel B: skewness = -0.995) Panel A. Sample log-price and realized volatility paths from positively skew normal distribution Panel B. Sample log-price and realized volatility paths from negatively skew normal distribution

Exhibit 7.4 Dow-Jones Industrial Average Index (DJIA) log-price with implied and realized volatilities from 1997 to 2005 Panel A. Log-price (dark line) and implied volatility (grey line) of DJIA Panel B. Log-price (dark line) and 60-day realized volatility (grey line) of DJIA

Exhibit 7.5 Sample means and sample variances of skew normal random variables

Exhibit 7.6 Summary statistics of 10 industry portfolios with market portfolio based on daily returns

Exhibit 7.7 Test results for base case (, , and )

Exhibit 7.8 Test results when

β

is varied Panel A. Robust portfolio dominates mean-variance optimal portfolio in both third and fourth central moments. Panel B. Robust portfolio dominates mean-variance optimal portfolio in third central moment. Panel C. Robust portfolio dominates mean-variance optimal portfolio in fourth central moment. Panel D. Robust portfolio is dominated by mean-variance portfolio in both third and fourth central moments.

Exhibit 7.9 Test results when

I

is varied Panel A. Robust portfolio dominates mean-variance optimal portfolio in both third and fourth central moments. Panel B. Robust portfolio dominates mean-variance optimal portfolio in third central moment. Panel C. Robust portfolio dominates mean-variance optimal portfolio in fourth central moment. Panel D. Robust portfolio is dominated by mean-variance portfolio in both third and fourth central moments.

Exhibit 7.10 Test results when

J

is varied Panel A. Robust portfolio dominates mean-variance optimal portfolio in both third and fourth central moments. Panel B. Robust portfolio dominates mean-variance optimal portfolio in third central moment. Panel C. Robust portfolio dominates mean-variance optimal portfolio in fourth central moment. Panel D. Robust portfolio is dominated by mean-variance portfolio in both third and fourth central moments.

Chapter 8: Higher Factor Exposures of Robust Equity Portfolios

Exhibit 8.1 Monthly returns of U.S. large-cap stocks and five industries

Exhibit 8.2 Monthly returns of a portfolio and the Fama-French three factors

Exhibit 8.3 Factors included in the Northfield U.S. Fundamental Equity Risk Model and MSCI Barra U.S. Total Market Equity Models

Exhibit 8.4 Framework of the empirical analysis

Exhibit 8.5

R

2

values for mean-variance and robust portfolios

Exhibit 8.6

R

2

values for robust portfolios with various confidence levels

Exhibit 8.7

R

2

values for mean-variance and robust portfolios (box uncertainty)

Exhibit 8.8

R

2

values for robust portfolios (box uncertainty) with various confidence levels

Exhibit 8.9

R

2

values for mean-variance and robust portfolios (1-year estimation period)

Exhibit 8.10

R

2

values for robust portfolios (ellipsoid uncertainty) with various confidence levels for a 1-year estimation period

Exhibit 8.11

R

2

values for robust portfolios (box uncertainty) with various confidence levels for a 1-year estimation period

Exhibit 8.12

R

2

values for mean-variance and robust portfolios (five factors and three-year estimation)

Exhibit 8.13

R

2

values for robust portfolios (ellipsoid uncertainty) with various confidence levels (five factors and three-year estimation)

Exhibit 8.14

R

2

values for robust portfolios (box uncertainty) with various confidence levels (five factors and three-year estimation)

Exhibit 8.15

R

2

values for mean-variance and robust portfolios (five factors and two-year estimation)

Exhibit 8.16

R

2

values for robust portfolios (ellipsoid uncertainty) with various confidence levels (five factors and two-year estimation)

Exhibit 8.17

R

2

values for robust portfolios (box uncertainty) with various confidence levels (five factors and two-year estimation)

Exhibit 8.18 Graphical representation of constraint based on portfolio variance

Chapter 9: Composition of Robust Portfolios

Exhibit 9.1 Average allocation in 100 funds (

λ

= 0.01)

Exhibit 9.2 Average allocation in 100 funds (

λ

= 0.05)

Exhibit 9.3 Details on average weights allocated in 100 funds

Exhibit 9.4 Allocation in small-cap growth, small-cap value, large-cap growth, and large-cap value

Exhibit 9.5 Allocation in small-cap growth, small-cap value, large-cap growth, and large-cap value (

λ

= 0.01)

Exhibit 9.6 Correlation among the 10 momentum funds

Exhibit 9.7 Descriptive statistics of the 10 momentum funds

Exhibit 9.8 Average allocation in 10 momentum funds (

λ

= 0.01)

Exhibit 9.9 Average allocation in 10 momentum funds (

λ

= 0.1)

Exhibit 9.10 List of 49 industries

Exhibit 9.11 Details on average weights allocated in 49 industry funds

Exhibit 9.12 Average allocations in 100 funds sorted by beta (primary axis: weight, secondary axis: beta)

Exhibit 9.13 Allocation and beta of 100 funds

Exhibit 9.14 Allocation and beta of 49 industry funds

Exhibit 9.15 Rule-based robust portfolio algorithm

Exhibit 9.16 Investment performance with various estimation periods from 1973 to 2011

Exhibit 9.17 Log wealth of portfolios from 1976 to 2011 (wealth starting from one)

Chapter 10: Robust Portfolio Performance

Exhibit 10.1 Stock price of Twitter, Inc. (TWTR) during December 2013 (positive drawdown shown as black bars)

Exhibit 10.2 Description of conventional portfolios

Exhibit 10.3 Description of robust portfolio strategies

Exhibit 10.4 Wealth of investment in major stock market indices (in logarithm)

Exhibit 10.5 Summary of monthly performance from 1981 to 2013 with three-month estimation

Exhibit 10.6 Summary of monthly performance from 1981 to 2013 with six-month estimation

Exhibit 10.7 Summary of monthly performance from 1981 to 2013 with 12-month estimation

Exhibit 10.8 Wealth from investment from 1981 to 2013 (in logarithm)

Exhibit 10.9 The 49 industry classifications

Exhibit 10.10 Summary of weekly performance from 2007 to 2012 with six-month estimation

Exhibit 10.11 Summary of weekly performance from 2007 to 2012 with 12-month estimation

Exhibit 10.12 Wealth from investment from 2007 to 2012 (in logarithm)

Chapter 11: Robust Optimization Software

Exhibit 11.1 Comparison of optimal robust allocations for box uncertainty

The Frank J. Fabozzi Series

Fixed Income Securities, Second Edition by Frank J. Fabozzi

Focus on Value: A Corporate and Investor Guide to Wealth Creation

by James L. Grant and James A. Abate

Handbook of Global Fixed Income Calculations

by Dragomir Krgin

Managing a Corporate Bond Portfolio

by Leland E. Crabbe and Frank J. Fabozzi

Real Options and Option-Embedded Securities

by William T. Moore

Capital Budgeting: Theory and Practice

by Pamela P. Peterson and Frank J. Fabozzi

The Exchange-Traded Funds Manual

by Gary L. Gastineau

Professional Perspectives on Fixed Income Portfolio Management, Volume 3

edited by Frank J. Fabozzi

Investing in Emerging Fixed Income Markets

edited by Frank J. Fabozzi and Efstathia Pilarinu

Handbook of Alternative Assets

by Mark J. P. Anson

The Global Money Markets

by Frank J. Fabozzi, Steven V. Mann, and Moorad Choudhry

The Handbook of Financial Instruments

edited by Frank J. Fabozzi

Interest Rate, Term Structure, and Valuation Modeling

edited by Frank J. Fabozzi

Investment Performance Measurement

by Bruce J. Feibel

The Handbook of Equity Style Management

edited by T. Daniel Coggin and Frank J. Fabozzi

The Theory and Practice of Investment Management

edited by Frank J. Fabozzi and Harry M. Markowitz

Foundations of Economic Value Added, Second Edition

by James L. Grant

Financial Management and Analysis, Second Edition

by Frank J. Fabozzi and Pamela P. Peterson

Measuring and Controlling Interest Rate and Credit Risk, Second Edition

by Frank J. Fabozzi, Steven V. Mann, and Moorad Choudhry

Professional Perspectives on Fixed Income Portfolio Management, Volume 4

edited by Frank J. Fabozzi

The Handbook of European Fixed Income Securities

edited by Frank J. Fabozzi and Moorad Choudhry

The Handbook of European Structured Financial Products

edited by Frank J. Fabozzi and Moorad Choudhry

The Mathematics of Financial Modeling and Investment Management

by Sergio M. Focardi and Frank J. Fabozzi

Short Selling: Strategies, Risks, and Rewards

edited by Frank J. Fabozzi

The Real Estate Investment Handbook

by G. Timothy Haight and Daniel Singer

Market Neutral Strategies

edited by Bruce I. Jacobs and Kenneth N. Levy

Securities Finance: Securities Lending and Repurchase Agreements

edited by Frank J. Fabozzi and Steven V. Mann

Fat-Tailed and Skewed Asset Return Distributions

by Svetlozar T. Rachev, Christian Menn, and Frank J. Fabozzi

Financial Modeling of the Equity Market: From CAPM to Cointegration

by Frank J. Fabozzi, Sergio M. Focardi, and Petter N. Kolm

Advanced Bond Portfolio Management: Best Practices in Modeling and Strategies

edited by Frank J. Fabozzi, Lionel Martellini, and Philippe Priaulet

Analysis of Financial Statements, Second Edition

by Pamela P. Peterson and Frank J. Fabozzi

Collateralized Debt Obligations: Structures and Analysis, Second Edition

by Douglas J. Lucas, Laurie S. Goodman, and Frank J. Fabozzi

Handbook of Alternative Assets, Second Edition

by Mark J. P. Anson

Introduction to Structured Finance

by Frank J. Fabozzi, Henry A. Davis, and Moorad Choudhry

Financial Econometrics

by Svetlozar T. Rachev, Stefan Mittnik, Frank J. Fabozzi, Sergio M. Focardi, and Teo Jasic

Developments in Collateralized Debt Obligations: New Products and Insights

by Douglas J. Lucas, Laurie S. Goodman, Frank J. Fabozzi, and Rebecca J. Manning

Robust Portfolio Optimization and Management

by Frank J. Fabozzi, Peter N. Kolm, Dessislava A. Pachamanova, and Sergio M. Focardi

Advanced Stochastic Models, Risk Assessment, and Portfolio Optimizations

by Svetlozar T. Rachev, Stogan V. Stoyanov, and Frank J. Fabozzi

How to Select Investment Managers and Evaluate Performance

by G. Timothy Haight, Stephen O. Morrell, and Glenn E. Ross

Bayesian Methods in Finance

by Svetlozar T. Rachev, John S. J. Hsu, Biliana S. Bagasheva, and Frank J. Fabozzi

The Handbook of Municipal Bonds

edited by Sylvan G. Feldstein and Frank J. Fabozzi

Subprime Mortgage Credit Derivatives

by Laurie S. Goodman, Shumin Li, Douglas J. Lucas, Thomas A Zimmerman, and Frank J. Fabozzi

Introduction to Securitization

by Frank J. Fabozzi and Vinod Kothari

Structured Products and Related Credit Derivatives

edited by Brian P. Lancaster, Glenn M. Schultz, and Frank J. Fabozzi

Handbook of Finance: Volume I: Financial Markets and Instruments

edited by Frank J. Fabozzi

Handbook of Finance: Volume II: Financial Management and Asset Management

edited by Frank J. Fabozzi

Handbook of Finance: Volume III: Valuation, Financial Modeling, and Quantitative Tools

edited by Frank J. Fabozzi

Finance: Capital Markets, Financial Management, and Investment Management

by Frank J. Fabozzi and Pamela Peterson-Drake

Active Private Equity Real Estate Strategy

edited by David J. Lynn

Foundations and Applications of the Time Value of Money

by Pamela Peterson-Drake and Frank J. Fabozzi

Leveraged Finance: Concepts, Methods, and Trading of High-Yield Bonds, Loans, and Derivatives

by Stephen Antczak, Douglas Lucas, and Frank J. Fabozzi

Modern Financial Systems: Theory and Applications

by Edwin Neave

Institutional Investment Management: Equity and Bond Portfolio Strategies and Applications

by Frank J. Fabozzi

Robust Equity Portfolio Management + Website

Formulations, Implementations, and Properties Using MATLAB

WOO CHANG KIM JANG HO KIM FRANK J. FABOZZI

Copyright © 2016 by Woo Chang Kim, Jang Ho Kim, and Frank J. Fabozzi. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data

Names: Kim, Woo Chang. | Kim, Jang-Ho. | Fabozzi, Frank J.

Title: Robust equity portfolio management + website : formulations, implementations, and properties using MATLAB / Woo Chang Kim, Jang Ho Kim, Frank J. Fabozzi.

Description: Hoboken : Wiley, 2015. | Series: Frank J. Fabozzi series | Includes index.

Identifiers: LCCN 2015030347 | ISBN 9781118797266 (hardback) | ISBN 9781118797303 (epdf) | ISBN 9781118797372 (epub)

Subjects: LCSH: Porffolio management. | Investments--Mathematical models. | Investment analysis--Mathematical models. | BISAC: BUSINESS & ECONOMICS / Investments & Securities.

Classification: LCC HG4529.5 .K556 2015 | DDC 332.60285/53--dc23 LC record available at http://lccn.loc.gov/2015030347

Cover Design: Wiley

Cover Image: © Danil Melekhin/Getty Images, Inc.

WCK

To my daughter, Joohyung

JHK

To my wife, Insun Jung

FJF

To my sister, Lucy

Preface

The mean-variance model for constructing portfolios, introduced by Harry Markowitz, changed how portfolio managers analyze portfolios, especially for managing equity portfolios. The model provides a strong foundation for quantifying the return and risk attributes of a portfolio, as well as mathematically forming optimal portfolios. Following the 1952 publication of Markowitz's mean-variance model, there have been numerous extensions of the original model, particularly starting in the 1990s, that have sought to overcome criticisms of the original model. In this book, we focus on one of these extensions, the construction of robust portfolios for equity portfolio management within the mean-variance framework. We refer to this approach as robust equity portfolio management.

The book will be most helpful for readers who are interested in learning about the quantitative side of equity portfolio management, mainly portfolio optimization and risk analysis. Mean-variance portfolio optimization is covered in detail, leading to an extensive discussion on robust portfolio optimization. Nonetheless, readers without prior knowledge of portfolio management or mathematical modeling should be able to follow the presentation, as basic concepts are covered in each chapter. Furthermore, the main quantitative approaches are presented with MATLAB examples, allowing readers to easily implement portfolio problems in MATLAB or similar modeling software. An online appendix provides the MATLAB codes presented in the chapter boxes (www.wiley.com/go/robustequitypm).

Although this is not the only book on robust portfolio management, it distinguishes itself from other books by focusing solely on quantitative robust equity portfolio management, including step-by-step implementations. Other books, such as Robust Portfolio Optimization and Management by Frank J. Fabozzi, Petter N. Kolm, Dessislava Pachamanova, and Sergio M. Focardi, also introduce robust approaches, but we believe that readers seeking to learn the formulations, implementations, and properties of robust equity portfolios will benefit considerably by studying the chapters in the current book.

Woo Chang KimJang Ho KimFrank J. Fabozzi

Chapter 1Introduction

The foundations of what is popularly referred to as “modern portfolio theory” is attributable to the seminal work of Harry Markowitz, published more than a half a century ago.1 Markowitz provided a framework for the selection of securities for portfolio construction to obtain an optimal portfolio. To do so, Markowitz suggested that for all assets that are candidates for inclusion in a portfolio, one should measure an asset's return by its mean return and risk by an asset's variance of returns. In the selection of assets to include in a portfolio, the Markowitz framework takes into account the co-movement of asset returns by using the covariance between all pairs of assets. The portfolio's expected return and risk as measured by the portfolio variance are then determined by the weights of each asset included in the portfolio. For this reason, the Markowitz framework is commonly referred to as mean-variance portfolio analysis. Markowitz argued that the optimal portfolio should be selected based on the trade-off between a portfolio's return and risk. While these concepts are considered the basis of portfolio construction these days, the development of the mean-variance model shaped how investment managers analyze portfolios and sparked an overwhelming volume of research on the theory of portfolio selection.

Once the fundamentals of modern portfolio theory were established, studies addressing the limitations of mean-variance analysis appeared, seeking to improve the effectiveness of the original model under practical situations. Some research efforts concentrated on reducing the sensitivity of portfolios formed from mean-variance analysis. Portfolio sensitivity means that the resulting portfolio constructed using mean-variance analysis and its performance is heavily dependent on the inputs of the model. Hence, if the estimated input values were even slightly different from their true values, the estimated optimal portfolio will actually be far from the best choice. This is especially a drawback when managing equity portfolios because the equity market is one of the more volatile markets, making it difficult to estimate values such as expected returns.

In equity portfolio management, there has been increased interest in the construction of portfolios that offer the potential for more robust performance even during more volatile equity market periods. One common approach for doing so is to increase the robustness of the input values of mean-variance analysis by adopting estimators that are more robust to outliers. It is also possible to achieve higher robustness by focusing on the outputs of the mean-variance model by performing simulations for collecting many possible portfolios and then finally arriving at one optimal portfolio based on all the possible ones. There are other methods that are based on the equilibrium of the equity market for gaining robustness.2

Although various techniques have been applied to improve the stability of portfolios, one of the approaches that has received much attention is robust portfolio optimization. Robust optimization is a method that incorporates parameter uncertainty by defining a set of possible values, referred to as an uncertainty set. The optimal solution represents the best choice when considering all possibilities from the uncertainty set. Robust optimization was developed for addressing optimization problems where the true values of the model's parameters are not known with certainty, but the bounds are assumed to be known. In 1973, Allen Soyster discussed inexact linear programming; in the 1990s, the initial approach expanded to incorporate a number of ways for defining uncertainty sets and addressing more complex optimization problems. When robust optimization is extended to portfolio selection, the inputs used in mean-variance analysis—the vector of mean returns and the covariance matrix of returns—become the uncertain parameters for finding the optimal portfolio. Since the turn of the century, there have been numerous proposals for formulating robust portfolio optimization problems. Much of the focus has been on mathematical theories behind uncertainty set construction and reformulations resulting in optimization problems that can be solved efficiently; and, as a result, there are many formulations that can be used to build robust equity portfolios.

Even though there has been considerable development on robust portfolio management, most approaches require skills far beyond perfecting mean-variance analysis. For example, it is not an easy task for a portfolio manager without extensive background knowledge in optimization and mathematics to understand robust portfolio optimization formulations. More importantly, being able to interpret robust formulations is only the first step. The second step requires solving the optimization problem to arrive at the optimal decision. Programming expertise, in addition to optimization and mathematics, is necessary in the second step because most robust formulations require complex computations. Thus, while the need and the value of robust portfolio management are apparent, only those with appropriate training will be equipped to explore the advanced methods for improving portfolio robustness.

This book is aimed at providing a step-by-step guide for using robust models for optimal portfolio construction. It is not assumed that the reader has prior knowledge in portfolio management and optimization. In this book, the basics of portfolio theory and optimization, along with programming examples, will allow the reader to gain familiarity with portfolio optimization. Once the fundamentals of portfolio management are outlined, robust approaches for managing portfolios are explained with an emphasis on robust portfolio optimization. Details on robust formulations, implementation of robust portfolio optimization, attributes of robust portfolios, and robust portfolio performance will prepare the reader to utilize robust portfolio optimization for managing portfolios. In this book, we not only review theoretical developments but provide numerous programming examples to demonstrate their use in practice. The programming examples that appear throughout the book illustrate the details of implementing various techniques including methods for constructing robust equity portfolios.

1.1 Overview of the Chapters

The book is divided into three parts. The first part, Chapters 2 through 4, introduces the mean-variance model, discusses its shortcomings, and explains common approaches for increasing the robustness of portfolios. The second part, Chapters 5 and 6, contains an overview of optimization and details the steps involved in formulating a robust portfolio optimization problem. The third part, Chapters 7 through 10, focuses on analyzing robust portfolios constructed from robust portfolio optimization by identifying attributes and summarizing performances.

Chapter 2 begins by describing how portfolio return and risk are measured, which leads to formulating the mean-variance portfolio problem. Mean-variance analysis finds the optimal portfolio from the trade-off between return and risk, and the framework also explains the benefits of diversification. Chapter 3 investigates shortcomings of the mean-variance model, which limit its use as a strategy for managing equity portfolios; improvements can be made with respect to measuring risk, estimating the input variables, and reducing the sensitivity of portfolio weights. In particular, the combination of estimation errors in the input values and high sensitivity of the resulting portfolio is a major issue with the mean-variance model. Therefore, in Chapter 4, practices for reducing the sensitivity of portfolios are demonstrated, including robust statistics, simulation methods, and stochastic programming.

Chapter 5 presents a comprehensive overview of optimization, including definitions of linear programming, quadratic programming, and conic optimization. The chapter also discusses how robust optimization transforms basic optimization problems so as to incorporate parameter uncertainty. The discussion is extended to applying robust optimization to portfolio selection in Chapter 6. While concentrating on the uncertainty caused by estimating expected returns of stocks, two robust formulations are shown with specific instructions provided as to their implementation.

Chapters 7, 8, and 9 analyze portfolio attributes that are revealed when portfolios are formed from robust portfolio optimization. In Chapter 7, we provide empirical evidence that indicates that some uncertainty sets lead to portfolios that favor skewness but penalize kurtosis. The high factor exposure of robust portfolios at the portfolio level is addressed in Chapter 8, and Chapter 9 examines portfolio weights allocated to individual stocks for comparing the composition of robust portfolios with mean-variance portfolios that assume no uncertainty. Chapter 10 illustrates the robustness of robust portfolios by observing their historical performance.

The final chapter, Chapter 11, discusses software packages that can help solve robust portfolio optimization and provides examples for finding robust portfolios.

1.2 Use of MATLAB

Financial modeling often requires computer programs for solving complex computations. The use of powerful computing tools is inevitable in portfolio management because portfolio selection problems are mathematically expressed as optimization problems. Thus, tools that efficiently solve optimization problems give portfolio managers a great advantage; the tools are more valuable for robust portfolio management because approaches such as robust portfolio optimization involve more intense computations.

Therefore, in this book we discuss various aspects of robust portfolio management with examples on how to implement models in MATLAB, which is a programming language and interactive environment primarily for numerical computations.3 MATLAB is widely used in academic studies as well as research in the financial industry, especially for computations that involve matrices such as portfolio optimization. The examples presented use MATLAB mainly because the language provides a straightforward approach for executing portfolio optimization. This high-level language with an extensive list of built-in functions allows beginners to easily perform various computations and visualize their results. Furthermore, the syntax for writing a script or a function is so intuitive that the reader can quickly become familiar with MATLAB even without prior experience. Hence, the MATLAB examples throughout the book will not only supplement understanding the theoretical concepts but will also let the reader apply the examples to construct optimal portfolios that reflect their investment goals.

While MATLAB features an add-on toolbox for financial computations, the examples in this book use built-in functions for solving optimization and not the functions in the financial toolbox that are customized for certain types of financial decision problems. For example, the quadprog function in MATLAB is used for implementing portfolio problems that are formulated as quadratic programming. This gives the reader flexibility since the examples will show how the function parameters can be modified based on different investment assumptions and portfolio constraints. Becoming familiar with the built-in optimization functions is also crucial because robust formulations are not included in the financial toolbox and therefore must be solved with the optimization functions. We also include examples that use CVX, which is a modeling system for convex optimization that runs in the MATLAB environment.4 CVX enhances MATLAB, making it more expressive and powerful for solving optimizations like the mean-variance portfolio problems that are formulated as convex optimization problems. Many examples in this book present MATLAB codes that use the built-in functions of MATLAB as well as CVX in order to demonstrate two approaches for obtaining robust portfolios for a given problem. Since CVX is MATLAB-based, the reader will gain exposure to an additional tool without having to learn a new programming environment.

Notes

1.

 Harry M. Markowitz, “Portfolio Selection,”

Journal of Finance

7, 1 (1952), pp. 77–91.

2.

 An example of improving the robustness of inputs is to use shrinkage estimators, introduced in Philippe Jorion, “Bayes-Stein Estimation for Portfolio Analysis,”

Journal of Financial and Quantitative Analysis

21, 3 (1986), pp. 279–292. Using simulation to gain robustness is illustrated in Richard Michaud and Robert Michaud, “Estimation Error and Portfolio Optimization: A Resampling Solution,”

Journal of Investment Management

6, 1 (2008), pp. 8–28. The Black–Litterman model is an equilibrium-based approach that incorporates an investor's views; it was proposed in Fischer Black and Robert Litterman, “Asset Allocation: Combining Investor Views with Market Equilibrium,”

Goldman, Sachs & Co., Fixed Income Research

(1990). Various robust approaches including the ones mentioned here are detailed in

Chapter 4

.

3.

 MATLAB documentations and a list of functions with examples are available at

http://www.mathworks.com/products/matlab/

4.

 A CVX user's guide and download details can be found at

http://cvxr.com/cvx/

Chapter 2Mean-Variance Portfolio Selection

Before we begin our discussion on robust portfolio management, we briefly review portfolio theory as formulated by Harry Markowitz in 1952. Portfolio theory explains how to construct portfolios based on the correlation of the mean, variance, and covariance of asset returns. The framework is commonly referred to as mean-variance. Despite its appearance more than half a century ago, it is also referred to as modern portfolio theory. The theory has been applied in asset management in two ways: The first is in allocating funds across major asset classes. The second application has been to the selection of securities within an asset class. Throughout this book, we apply mean-variance analysis to the construction of equity portfolios.

Mean-variance analysis not only provides a framework for selecting portfolios, it also explains how portfolio risk is reduced by diversifying a portfolio. Robust portfolio optimization builds on the idea of mean-variance optimization. Thus, the topics introduced in this chapter provide an introduction to the advanced robust methods to be explained in the chapters to follow. Specifically, in this chapter we describe how to: