of each other, their covariance will be zero. Thus, covariance is a statistical measure of the extent to which two variables move together. The formula for covariance is similar to that for variance, except that instead of squaring the deviations of one variable, such as the returns of fund i, the formula cross multiplies the contemporaneous deviations of two different variables, such as the returns of funds i and j:
where Ri is the return of fund i, μi is the expected value or mean of Ri, Rj is the return of fund j, and μj is the expected value or mean of Rj.
The covariance is the expected value of the product of the deviations of the returns of the two funds. Covariance can be estimated from a sample using Equation 4.16:
where Rit is the return of fund i in time t, and
The estimation of the covariance for a sample of returns from a market index fund and a real estate fund is shown in Exhibit 4.2. Column 8 multiplies the fund's deviation from its mean return by the index's deviation from its mean return. Each of the products of the deviations is then summed and divided by n − 1, where n is the number of observations. The result is the estimated covariance between the returns over the sample period, shown near the bottom right-hand corner of Exhibit 4.2.
Exhibit 4.2 Covariance, Correlation, and Beta
Source: Bloomberg.
Because covariance is based on the products of individual deviations and not squared deviations, its value can be positive, negative, or zero. When the return deviations are in the same direction, meaning they have the same sign, the cross product is positive; when the return deviations are in opposite directions, meaning they have different signs, the cross product is negative. When the cross products are summed, the resulting sum generates an indication of the overall tendency of the returns to move either in tandem or in opposition. Note that the table method illustrated in Exhibit 4.2 simply provides a format for solving the formula, which can be easily solved by software. Covariance is used directly in numerous applications, such as in the classic portfolio theory work of Markowitz.
4.3.2 Correlation Coefficient
A statistic related to covariance is the correlation coefficient. The correlation coefficient (also called the Pearson correlation coefficient) measures the degree of association between two variables, but unlike the covariance, the correlation coefficient can be easily interpreted. The correlation coefficient takes the covariance and scales its value to be between +1 and −1 by dividing by the product of the standard deviations of the two variables. A correlation coefficient of −1 indicates that the two assets move in the exact opposite direction and in the same proportion, a result known as perfect linear negative correlation. A correlation coefficient of +1 indicates that the two assets move in the exact same direction and in the same proportion, a result known as perfect linear positive correlation. A correlation coefficient of zero indicates that there is no linear association between the returns of the two assets. Values between the two extremes of −1 and +1 indicate different degrees of association. Equation 4.17 provides the formula for the correlation coefficient based on the covariance and the standard deviations:
where ρij (rho) is the notation for the correlation coefficient between the returns of asset i and asset j; σij is the covariance between the returns of asset i and asset j; and σi and σj are the standard deviations of the returns of assets i and j, respectively.
Thus, ρij, the correlation coefficient, scales covariance, σij, through division by the product of the standard deviations, σi σj. The correlation coefficient can therefore be solved by computing covariance and standard deviation as in Exhibit 4.2 and inserting the values into Equation 4.17. The result is shown in Exhibit 4.2.
4.3.3 The Spearman Rank Correlation Coefficient
The Pearson correlation coefficient is not the only measure of correlation. There are some especially useful measures of correlation in alternative investments that are based on the ranked size of the variables rather than the absolute size of the variables. The returns within a sample for each asset are ranked from highest to lowest. The numerical ranks are then inserted into formulas that generate correlation coefficients that usually range between −1 and +1. The Spearman rank correlation coefficient is a popular example.
The Spearman rank correlation is a correlation designed to adjust for outliers by measuring the relationship between variable ranks rather than variable values. The Spearman rank correlation for returns is computed using the ranks of returns of two assets. For example, consider two assets with returns over a time period of three years, illustrated here:
The first step is to replace the actual returns with the rank of each asset's return. The ranks are computed by first ranking the returns of each asset separately, from highest (rank = 1) to lowest (rank = 3), while keeping the returns arrayed according to their time periods:
This table demonstrates the computation of di, the difference in the two ranks associated with time period i. The Spearman rank correlation, ρs, can be computed using those differences in ranks and the total number of time periods, n:
(4.18)
Using the data from the table, the numerator is 12, the denominator is 3 × 8 = 24, and ρs is 0.5. Rank correlation is sometimes preferred because of the way it handles the effects of outliers (extremely high or low data values). For example, the enormous return of asset 1 in the previous table is an outlier, which will have a disproportionate effect on a correlation statistic. Extremely high or very negative values of one or both of the variables in a particular sample can cause the computed Pearson correlation coefficient to be very near +1 or −1 based, arguably, on the undue influence of the extreme observation on the computation, since deviations are squared as part of the computation. Some alternative investments have returns that are more likely to contain extreme outliers. By using ranks, the effects of outliers are lessened, and in some cases it can be argued that the resulting measure of the correlation using a sample is a better indicator of the true correlation that exists within the population. Note that the Spearman rank correlation coefficient would be the same for any return that would generate the same rankings. Thus, any return in time period 1 for the first asset greater than 0 % would still be ranked 1 and would generate the same ρs.
4.3.4 The Correlation Coefficient and Diversification
The correlation coefficient is often used to demonstrate one of the most fundamental concepts of portfolio theory: the reduction in risk found by combining assets that are not perfectly positively correlated. Exhibit 4.3 illustrates the results of combining varying portions of assets A and B under three correlation conditions: perfect positive correlation, zero correlation, and