style="font-size:15px;"> 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
and for N assets, it becomes
where
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
Since
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
