Читать книгу Alternative Investments - Hossein Kazemi - Страница 65

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 d_i, 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.