Frank J. Fabozzi, PhD, CFA, is Professor in the Practice of Finance at Yale University's School of Management and the Editor of the Journal of Portfolio Management.

Petter N. Kolm, PhD, is a graduate student in finance at the Yale School of Management and a financial consultant in New York City. He previously worked at Goldman Sachs asset management where he developed quantitative investment models and strategies.

Dessislava A. Pachamanova, PhD, is an Assistant Professor of Operations Research at?Babson College. Her experience also includes work for Goldman Sachs and WestLB, and teaching management science, probability, statistics, and financial mathematics at MIT and Princeton University.

Sergio M. Focardi is a founding partner of the Paris-based consulting firm, The Intertek Group.

Robust Portfolio Optimization and Management

By Frank J. Fabozzi Petter N. Kolm Dessislava Pachamanova Sergio M. Focardi

John Wiley & Sons

Copyright © 2007 Frank J. Fabozzi
All right reserved.
ISBN: 9780-471-92122-6

Chapter One

Introduction

As the use of quantitative techniques has become more widespread in the financial industry, the issues of how to apply financial models most effectively and how to mitigate model and estimation errors have grown in importance. This book discusses some of the major trends and innovations in the management of financial portfolios today, focusing on state-of-the-art robust methodologies for portfolio risk and return estimation, optimization, trading, and general management.

In this chapter, we give an overview of the main topics in the book. We begin by providing a historical outlook of the adoption of quantitative techniques in the financial industry and the factors that have contributed to its growth. We then discuss the central themes of the book in more detail, and give a description of the structure and content of its remaining chapters.

QUANTITATIVE TECHNIQUES IN THE INVESTMENT MANAGEMENT INDUSTRY

Over the last 20 years there has been a tremendous increase in the use of quantitative techniques in the investment management industry. The first applications were in risk management, with models measuring the risk exposure to different sources of risk. Nowadays, quantitative models are considered to be invaluable in all the major areas of investment management, and the list of applications continues to grow: option pricing models for the valuation of complicated derivatives and structured products, econometric techniques for forecasting market returns, automated execution algorithms for efficient trading and transaction cost management, portfolio optimization for asset allocation and financial planning, and statistical techniques for performance measurement and attribution, to name a few.

Today, quantitative finance has evolved into its own discipline-an example thereof is the many university programs and courses being offered in the area in parallel to the "more traditional" finance and MBA programs. Naturally, many different factors have contributed to the tremendous development of the quantitative areas of finance, and it is impossible to list them all. However, the following influences and contributions are especially noteworthy:

* The development of modern financial economics, and the advances in the mathematical and physical sciences.

* The remarkable expansion in computer technology and the invention of the Internet.

* The maturing and growth of the capital markets.

Below, we highlight a few topics from each one of these areas and discuss their impact upon quantitative finance and investment management in general.

Modern Financial Economics and the Mathematical and Physical Sciences

The concepts of portfolio optimization and diversification have been instrumental in the development and understanding of financial markets and financial decision making. The major breakthrough came in 1952 with the publication of Harry Markowitz's theory of portfolio selection. The theory, popularly referred to as modern portfolio theory, provided an answer to the fundamental question: How should an investor allocate funds among the possible investment choices? Markowitz suggested that investors should consider risk and return together and determine the allocation of funds among investment alternatives on the basis of the trade-off between them. Before Markowitz's seminal article, the finance literature had treated the interplay between risk and return in a casual manner.

The idea that sound financial decision making is a quantitative trade-off between risk and return was revolutionary for two reasons. First, it posited that one could make a quantitative evaluation of risk and return jointly by considering portfolio returns and their comovements. An important principle at work here is that of portfolio diversification. It is based on the idea that a portfolio's riskiness depends on the covariances of its constituents, not only on the average riskiness of its separate holdings. This concept was foreign to classical financial analysis, which revolved around the notion of the value of single investments, that is, the belief that investors should invest in those assets that offer the highest future value given their current price. Second, it formulated the financial decision-making process as an optimization problem. In particular, the so-called mean-variance principle formulated by Markowitz suggests that among the infinite number of portfolios that achieve a particular return objective, the investor should choose the portfolio that has the smallest variance. All other portfolios are "inefficient" because they have a higher variance and, therefore, higher risk.

Building on Markowitz's work, William Sharpe, John Lintner, and Jan Mossin introduced the first asset pricing theory, the capital asset pricing model-CAPM in short-between 1962 and 1964. The CAPM became the foundation and the standard on which risk-adjusted performance of professional portfolio managers is measured.

Modern portfolio theory and diversification provide a theoretical justification for mutual funds and index funds, that have experienced a tremendous growth since the 1980s. A simple classification of fund management is into active and passive management, based upon the efficient market hypotheses introduced by Eugene Fama and Paul Samuelson in 1965. The efficient market hypothesis implies that it is not possible to outperform the market consistently on a risk-adjusted basis after accounting for transaction costs by using available information. In active management, it is assumed that markets are not fully efficient and that a fund manager can outperform a market index by using specific information, knowledge, and experience. Passive management, in contrast, relies on the assumption that financial markets are efficient and that return and risk are fully reflected in asset prices. In this case, an investor should invest in a portfolio that mimics the market. John Bogle used this basic idea when he proposed to the board of directors of the newly formed Vanguard Group to create the first index fund in 1975. The goal was not to outperform the S&P 500 index, but instead to track the index as closely as possible by buying each of the stocks in the S&P 500 in amounts equal to the weights in the index itself.

Despite the great influence and theoretical impact of modern portfolio theory, today-more than 50 years after Markowitz's seminal work-full risk-return optimization at the asset level is primarily done only at the more quantitatively oriented firms. In the investment management business at large, portfolio management is frequently a purely judgmental process based on qualitative, not quantitative, assessments. The availability of quantitative tools is not the issue-today's optimization technology is mature and much more user-friendly than it was at the time Markowitz first proposed the theory of portfolio selection-yet many asset managers avoid using the quantitative portfolio allocation framework altogether.

A major reason for the reluctance of investment managers to apply quantitative risk-return optimization is that they have observed that it may be unreliable in practice. Specifically, risk-return optimization is very sensitive to changes in the inputs (in the case of mean-variance optimization, such inputs include the expected return of each asset and the asset covariances). While it can be difficult to make accurate estimates of these inputs, estimation errors in the forecasts significantly impact the resulting portfolio weights. It is well-known, for instance, that in practical applications equally weighted portfolios often outperform mean-variance portfolios, mean-variance portfolios are not necessarily well-diversified, and mean-variance optimization can produce extreme or non-intuitive weights for some of the assets in the portfolio. Such examples, however, are not necessarily a sign that the theory of risk-return optimization is flawed; rather, that when used in practice, the classical framework has to be modified in order to achieve reliability, stability, and robustness with respect to model and estimation errors.

It goes without saying that advances in the mathematical and physical sciences have had a major impact upon finance. In particular, mathematical areas such as probability theory, statistics, econometrics, operations research, and mathematical analysis have provided the necessary tools and discipline for the development of modern financial economics. Substantial advances in the areas of robust estimation and robust optimization were made during the 1990s, and have proven to be of great importance for the practical applicability and reliability of portfolio management and optimization.

Any statistical estimate is subject to error-estimation error. A robust estimator is a statistical estimation technique that is less sensitive to outliers in the data. For example, in practice, it is undesirable that one or a few extreme returns have a large impact on the estimation of the average return of a stock. Nowadays, Bayesian techniques and robust statistics are commonplace in financial applications. Taking it one step further, practitioners are starting to incorporate the uncertainty introduced by estimation errors directly into the optimization process. This is very different from the classical approach, where one solves the portfolio optimization problem as a problem with deterministic inputs, without taking the estimation errors into account. In particular, the statistical precision of individual estimates is explicitly incorporated in the portfolio allocation process. Providing this benefit is the underlying goal of robust portfolio optimization.

First introduced by El Ghaoui and Lebret and by Ben-Tal and Nemirovski, modern robust optimization techniques allow a portfolio manager to solve the robust version of the portfolio optimization problem in about the same time as needed for the classical portfolio optimization problem. The robust approach explicitly uses the distribution from the estimation process to find a robust portfolio in one single optimization, thereby directly incorporating uncertainty about inputs in the optimization process. As a result, robust portfolios are less sensitive to estimation errors than other portfolios, and often perform better than classical mean-variance portfolios. Moreover, the robust optimization framework offers great flexibility and many new interesting applications. For instance, robust portfolio optimization can exploit the notion of statistically equivalent portfolios. This concept is important in large-scale portfolio management involving many complex constraints such as transaction costs, turnover, or market impact. Specifically, with robust optimization, a manager can find the best portfolio that (1) minimizes trading costs with respect to the current holdings and (2) has an expected portfolio return and variance that are statistically equivalent to those of the classical mean-variance portfolio.

An important area of quantitative finance is that of modeling asset price behavior, and pricing options and other derivatives. This field can be traced back to the early works of Thorvald Thiele in 1880, Louis Bachelier in 1900, and Albert Einstein in 1905, who knew nothing about each other's research and independently developed the mathematics of Brownian motion. Interestingly, while the models by Thiele and Bachelier had little influence for a long time, Einstein's contribution had an immediate impact on the physical sciences. Historically, Bachelier's doctoral thesis is the first published work that uses advanced mathematics in the study of finance. Therefore, he is by many considered to be the pioneer of financial mathematics-the first "quant."

The first listed options began trading in April 1973 on the Chicago Board Options Exchange (CBOE), only one and four months, respectively, before the papers by Black and Scholes and by Merton on option pricing were published. Although often criticized in the general press, and misunderstood by the public at large, options opened the door to a new era in investment and risk management, and influenced the introduction and popularization of a range of other financial products including interest rate swaptions, mortgage-backed securities, callable bonds, structured products, and credit derivatives. New derivative products were made possible as a solid pricing theory was available. Without the models developed by Black, Scholes, and Merton and many others following in their footsteps, it is likely that the rapid expansion of derivative products would never have happened. These modern instruments and the concepts of portfolio theory, CAPM, arbitrage and equilibrium pricing, and market predictability form the foundation not only for modern financial economics but for the general understanding and development of today's financial markets. As Peter Bernstein so adequately puts it in his book Capital Ideas: "Every time an institution uses these instruments, a corporation issues them, or a homeowner takes out a mortgage, they are paying their respects, not just to Black, Scholes, and Merton, but to Bachelier, Samuelson, Fama, Markowitz, Tobin, Treynor, and Sharpe as well."

Computer Technology and the Internet

The appearance of the first personal computers in the late 1970s and early 1980s forever changed the world of computing. It put computational resources within the reach of most people. In a few years every trading desk on Wall Street was equipped with a PC. From that point on, computing costs have declined at the significant pace of about a factor of 2 every year. For example, the cost per gigaflops is about $1 today, to be compared to about $50,000 about 10 years ago. At the same time, computer speed increased in a similar fashion: today's fastest computers are able to perform an amazing 300 trillion calculations per second. This remarkable development of computing technology has allowed finance professionals to deploy more sophisticated algorithms used, for instance, for derivative and asset pricing, market forecasting, portfolio allocation, and computerized execution and trading. With state-of-the-art optimization software, a portfolio manager is able to calculate the optimal allocation for a portfolio of thousands of assets in no more than a few seconds-on the manager's desktop computer!

But computational power alone is not sufficient for financial applications. It is crucial to obtain market data and other financial information efficiently and expediently, often in real time. The Internet and the World Wide Web have proven invaluable for this purpose. The World Wide Web, or simply the "Web," first created by Tim Berners-Lee working at CERN in Geneva, Switzerland around 1990, is an arrangement of inter-linked, hypertext documents available over the Internet. With a simple browser, anybody can view webpages that may contain anything from text and pictures, to other multimedia based information, and jump from page to page by a simple mouse click. Berners-Lee's major contribution was to combine the concept of hypertext with the Internet, born out of the NSFNet developed by the National Science Foundation in the early 1980s. The Web as we know it today allows for expedient exchange of financial information. Many market participants-from individuals to investment houses and hedge funds-use the Internet to follow financial markets as they move tick by tick and to trade many different kinds of assets such as stocks, bonds, futures, and other derivatives simultaneously across the globe. In today's world, gathering, processing, and analyzing the vast amount of information is only possible through the use of computer algorithms and sophisticated quantitative techniques.

(Continues...)

Excerpted from Robust Portfolio Optimization and Managementby Frank J. Fabozzi Petter N. Kolm Dessislava Pachamanova Sergio M. Focardi Copyright © 2007 by Frank J. Fabozzi. Excerpted by permission.
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