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 perfect negative correlation.
Exhibit 4.3 Diversification between Two Assets
The highest possible correlation and least diversification potential is when the assets' correlation coefficient is positive: perfect positive correlation. The straight line to the lower right between points A and B in Exhibit 4.3 plots the possible standard deviations and mean returns achievable by combining asset A and asset B under perfect positive correlation. The line is straight, meaning that the portfolio risk is a weighted average of the individual risks. This illustrates that there are no benefits to diversification when perfectly correlated assets are combined. The idea is that diversification occurs when the risks of unusual returns of assets tend to cancel each other out. This does not happen in the case of perfect positive correlation, because the assets always move in the same direction and by the same proportion.
The greatest risk reduction occurs when the assets' correlation coefficient is −1: perfect negative correlation. The two upper-left line segments connecting points A and B in Exhibit 4.3 plot the possible standard deviations and mean returns that would be achieved by combining asset A and asset B under perfect negative correlation. Notice that the line between A and B moves directly to the vertical axis, the point at which the standard deviation is zero. This illustrates ultimate diversification, in which two assets always move in opposite directions; therefore, combining them into a portfolio results in rapid risk reduction, or even total risk reduction. This zero-risk portfolio illustrates the concept of a perfect two-asset hedge and occurs when the weight of the investment in asset A is equal to the standard deviation of asset B divided by the sums of the standard deviations of A and B.
But the most realistic possibility is represented by the curve in the center of Exhibit 4.3. This is the more common scenario, in which the assets are neither perfectly positively nor perfectly negatively correlated; rather, they have some degree of dependent movement. The key point to this middle line in Exhibit 4.3 is that when imperfectly correlated assets are combined into a portfolio, a portion of the portfolio's risk is diversified away. The risk that can be removed through diversification is called diversifiable, nonsystematic, unique, or idiosyncratic risk.
In alternative investments, the concept of correlation is central to the discussion of portfolio implications. Further, graphs with standard deviation on the horizontal axis and expected return on the vertical axis are used as a primary method of illustrating diversification benefits. Assets that generate diversification benefits are shown to shift the attainable combinations of risk and return toward the benefit of the investor, meaning less risk for the same amount
