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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 is the covariance matrix:

Since when , the diagonal elements of are the variances of each asset. Computing the covariance matrix and portfolio variance from return data is shown in Box 2.1.

We conclude this section by mentioning that there are legitimate criticisms of using variance to represent risk. First, variance counts both upside and downside volatility toward risk, whereas most investors will be pleased with upside deviation. Another shortcoming is that variance alone does not completely measure variability when portfolio return is not symmetric. Nonetheless, as mentioned previously, the use of covariance changes the perception of portfolio risk. An example is its contribution to understanding portfolio diversification.

2.3 Diversification

We often hear the saying, “Don't put all your eggs in one basket.” The same applies when investing in stocks. Even though the concept is intuitively understandable, it was difficult to quantify the benefits until the establishment of modern portfolio theory. Keep in mind how portfolio return and risk are formulated while we consider the following example.

An investor decides to invest in either Twitter, Inc. (TWTR) or Tesla Motors, Inc. (TSLA), or both. The investor believes that the daily returns of the first six months in 2014 are a reasonable estimate for future short-term movement. The stock prices of Twitter and Tesla Motors for this period are shown in Exhibit 2.1. While Tesla Motors' stock has positive expected daily return, both stocks have at least 3 % daily volatility, measured by standard deviation. Let us now look into what happens if the investor holds a portfolio with both stocks. According to the formula for expected return and variance of portfolios given by equations (2.1) and (2.2), respectively, a portfolio that allocates half in Twitter and the other half in Tesla Motors has estimated values as shown in Exhibit 2.2. The 50-50 portfolio has positive expected return and, more importantly, has a standard deviation less than investing only in either one of the two stocks,

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

Diversifying between two stocks indeed reduces portfolio risk.

The reduction in risk is due to the low correlation between the two stocks. In fact, if the correlation between the stock movements of Twitter and Tesla Motors were lower, the investor will be able to enjoy an extra decline in the overall portfolio risk. As presented in Exhibit 2.3, the standard deviation of portfolio returns becomes less than 2 % when the two stocks have negative correlation. In most cases, it is extremely difficult to find stocks with negative correlation. But stocks in different industries or sectors have low correlation. Moreover, dividing the investment among various asset classes, such as fixed income instruments and commodities, is a better approach to expand diversification benefits since they normally reveal less co-movement than the stock market.

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

In Exhibit 2.2, we looked at a simple case of dividing the investment equally between the two stocks. But do all combinations of the two stocks reduce risk? Exhibit 2.4 demonstrates what happens when the portfolio is composed of different proportions between the two stocks. The figure included in Exhibit 2.4 is the mean-standard deviation plane, where each portfolio is located as a point based on its level of expected return and standard deviation. Even with two stocks, the portfolio can have a wide range of return and risk levels as the points on the mean-standard deviation plane form a curve. In general, investors desire higher return and lower risk, so a portfolio on the upper left is preferred. The process of computing and selecting optimal portfolios becomes more complex as the number of candidate stocks increase. Mean-variance analysis presents a framework to help the decision making of investors.

Exhibit 2.4 Portfolio return and risk for various proportions

2.4 Mean-Variance Analysis

Modern portfolio theory is based on some assumptions about the market and its participants. As previously stated, investors seek lower risk and higher return. Investors also make decisions based on the expected return and variance, and all investors have the same information. Finally, the theory assumes that investment decisions are made for a single period. Because investors only analyze the mean and variance of returns, the approach is known as mean-variance optimization.5

The portfolio selection problem is formulated as a minimization problem,

where is the target level of portfolio return. By substituting the formulas from sections 2.1 and 2.2, the portfolio problem is written as

where in the first line is added for calculation convenience, and the last line guarantees full allocation of investment principal. Conventionally, the optimization problem is written in matrix form:

5

The approach was first introduced in Harry M. Markowitz, “Portfolio Selection,” Journal of Finance 7, 1 (1952), pp. 77–91; and also in Harry M. Markowitz, Portfolio Selection: Efficient Diversification of Investments (New Haven, CT: Yale University Press, 1959).