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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.

Chapter 2

      Mean-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:

      • Measure return and risk of a portfolio within the mean-variance framework

      • Reduce portfolio risk through diversification

      • Select an optimal portfolio through mean-variance analysis

      • Utilize factor models for estimating stock returns

      • Apply the mean-variance model through an example

      2.1 Return of Portfolios

      In modern portfolio theory, a portfolio that is composed of N assets is expressed as weights that add to one in order to represent the proportion of total investment allocated to each asset,

      where

is the weight allocated to asset

. The rate of return of an asset is the change in the value of the asset in terms of percentage change or proportion of the initial value,

      where

and

are the initial and final values of the asset. For simplicity, rate of return is often referred to as return. From the above definition of an asset's return, the portfolio return can be expressed as

      where asset

has a return of

. In matrix form, portfolio return

is written as

      where

and

are vectors in

.

      Then, the expected return of a portfolio, or the mean of portfolio returns, is

      and the linearity of expected value allows writing the expected return as a weighted average of expectations,

2.1

      In matrices, it is expressed as

      where

is a vector of expected returns of assets,

      The expected returns of assets are typically estimated from historical data. For example, the expected value of the past 10 monthly returns may be used as the expected return for the following month. We include a simple MATLAB demonstration in Box 2.1.

      Box 2.1: Function That Computes Return and Risk of a Portfolio

2.2 Risk of Portfolios

      The risk of a portfolio is measured by the variance of returns. The variance of asset returns measures the variability of possible returns around the expected return and is computed as

      where

is the return for asset

. Higher variability results in higher uncertainty and, thus, is considered to expose an investor to more risk. The standard deviation of asset returns is simply the square root of the variance and basically reflects the same information as the variance:

      The variance of a portfolio is not as straightforward as the expected portfolio return; the variance of portfolio returns is not simply the weighted sum of individual asset variances. Instead, recall the property of the variance,

      where

's are random variables and

is the covariance between

and

. Therefore, for a portfolio with two assets, the portfolio variance is

      and for N assets, it becomes

2.2

      where

when

.

      The extra term with covariance is one of the most important findings of modern portfolio theory, providing a major breakthrough in computing portfolio risk. Covariance measures how much two random variables move together:

      More generally, correlation is quoted to show how closely two assets move up or down at the same time. Correlation is computed by dividing the covariance by the product of the individual standard deviations:

      Correlation is more frequently cited because it takes values between positive one and negative one, where positive one indicates a perfect co-movement in the same direction. Furthermore, since the standard deviation is non-negative, the correlation is negative only when the two random variables have negative covariance.

      In matrix form, portfolio variance is equivalent to

      where

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