Robust Equity Portfolio Management. Fabozzi Frank J.

, 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:

      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, intr

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