Chapter13: Risk, Cost of Capital, and Capital Budgeting
13.4 Beta, Covariance, and Correlation
Now that you know how to calculate beta, we want to give you a deeper understanding of what beta is. Since beta is a statistic, it is worthwhile to compare beta to other statistics. We begin this section by comparing beta to covariance. Next, we compare beta to correlation.
Beta and Covariance
p. 402Consider the following thought experiment. Imagine that, using past data over the last five years and the techniques of the previous section, you estimate beta for each of the 30 securities in the Dow Jones Industrial Index. You then rank these 30 securities from highest to lowest beta. Next, imagine that your friend does the same exercise for covariance. That is, using the same data over the last five years, he estimates the covariance of each of the 30 securities and ranks them from high to low.
How will your ranking on beta and your friend's ranking on covariance be related? You may be surprised to find that the two rankings are identical. Here's why. Consider Formula 13.2, relating beta to covariance, which we reproduce below:
where Cov(R_{i}, R_{M}) is the covariance between the return on asset i and the return on the market portfolio, and Var(R_{M}) is the variance of the return on the market. The formula tells us that we go from covariance to beta by dividing by a constant, the variance of the market. Rankings are always preserved when we divide by a constant. For example, suppose we rank everyone in your finance class by height, as measured in inches. A basketball player might be the tallest at, say, 84 inches. Now we convert everyone's height to feet by dividing by 12. The basketball player would come in at 84/12 = 7 feet and still be tallest. The same principle applies to the above formula for beta. For every stock, the variance of the market is the denominator of the beta calculation. Thus, a stock with a high covariance relative to other stocks must have a high beta relative to other stocks and vice versa. This is an important point because it tells us that beta and covariance, while they are two different statistical terms, measure the same concept.What is that concept? As stated in Chapter 11, beta measures the responsiveness of the return on the security to the return on the market. For example, Figure 13.3 tells us that Microsoft's beta is .86. A 1 percent return on the market would imply an expected return on the security of .86 percent. Because beta is just a transformation of covariance, covariance must measure responsiveness as well.
Which term, beta or covariance, is easier to use? Beta is clearly easier to use because of the above interpretation. Covariance, while it also measures responsiveness, does not lead to the same interpretation. If, for example, the covariance between a security and the market is, say, .0056, we cannot state that the stock is expected to rise .0056 percent for every 1 percent return on the market portfolio. In fact, the covariance number does not lend itself to any easy interpretation. We are better off thinking in terms of beta. For example, when Wall Street firms train new hires in modern portfolio theory, they often teach both beta and covariance but then tell the recruits, “Never use the word, covariance, again. Anything you can say in terms of covariance, you can say more clearly in terms of beta.” While this injunction may be going too far for our tastes, we agree with the sentiment.
Beta and Correlation
Consider the lefthand side of Figure 13.5, which plots the returns on security A for different periods against the returns on the market for the same periods. Now consider the righthand side of this figure, which plots the returns on security B over the same periods against the returns on the market. The characteristic line (or, alternatively, the regression line) is drawn for both securities and, while it may be hard to see from the graph, the two lines are identical. Since beta is the slope of the characteristic line, both stocks must have the same beta.

However, do the two stocks have the same correlation with the return on the market? No, because correlation measures the tightness of fit around the regression line. As can be seen, the points in stock B's graph are closer to the line than are the points in stock A's graph. Thus, the correlation between stock B and the market is higher than the correlation between stock A and the market.
Beta and correlation are two different concepts. Beta measures the responsiveness of a stock to movements in the market portfolio and is represented by the slope of the characteristic line. Correlation measures the tightness of fit around the regression line. Figure 13.5 shows that the two concepts are not the same because, while both securities have the same beta, they have different correlations.
Are the two concepts, beta and correlation, related in any way? As it turns out, yes. A security with a positive beta must have a positive correlation with the market. A security with a negative beta (i.e., a stock with a negatively sloped characteristic line), must have a negative correlation with the market. Otherwise, we should think of beta and correlation as two distinct concepts.
Now, which term, beta or correlation, is more important? There are many important uses for correlation in statistics, as well as in other areas of finance. However, our purpose now is to calculate the cost of capital for capital budgeting projects. As we have seen, beta is an important input for calculation of the cost of capital. By contrast, correlation is not important in this context. Thus, we will not be using the term, correlation, for the rest of the chapter.
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