Covariance

The Calder Global Opportunity Fund reports a beta of 1.15 measured against a broad U.S. stock index, together with an R-squared of 42. The fund invests heavily in commodities and foreign currencies. What does the low R-squared tell an analyst about that reported beta?

1. A.The fund captured 42 percent of the index's total return over the measurement period.Wrong. R-squared measures explanatory fit, not capture of return. A capture ratio is a different statistic entirely.
2. B.The fund's returns were 42 percent less volatile than the index's returns.Wrong. Relative volatility is described by beta or by comparing standard deviations, not by R-squared.
3. C.Only about 42 percent of the fund's return variation is explained by that index, so the beta is an unreliable description of this fund's market sensitivity.Correct. Beta is only meaningful when the benchmark actually explains the fund's returns, and R-squared measures that fit.
4. D.The beta remains reliable, because R-squared measures the manager's skill rather than the fit to the benchmark.Wrong. Skill relative to a benchmark is alpha. R-squared measures goodness of fit, and a poor fit undermines the beta.

Why: R-squared measures how much of a fund's return variation is EXPLAINED by movements in the chosen benchmark, on a scale of 0 to 100. Beta is only meaningful to the extent the benchmark actually drives the fund's returns. With an R-squared of 42, less than half of this fund's variation is attributable to the U.S. stock index, so the 1.15 beta is a statistically weak description of its market sensitivity. The practical lesson: always read beta alongside R-squared, and be suspicious of a beta computed against a benchmark the fund does not really track.

Fund Larkspur has a standard deviation of 18% and Fund Meridian has a standard deviation of 12%. The covariance of their returns is 0.0108. The correlation coefficient between the two funds is:

1. A.0.06Incorrect. 0.0108 / 0.18 also divides by only one standard deviation, this time the larger one.
2. B.0.0216Incorrect. 0.0216 is the product of the two standard deviations, the denominator of the calculation, not the answer.
3. C.0.50Correct. 0.18 x 0.12 = 0.0216, and 0.0108 / 0.0216 = 0.50.
4. D.0.90Incorrect. 0.0108 / 0.012 divides by only one standard deviation instead of the product of both.

Why: Covariance tells you the direction in which two assets move together but its scale is meaningless on its own. Dividing by the product of the two standard deviations rescales it into a correlation coefficient that always lies between -1.0 and +1.0. Here 0.18 x 0.12 = 0.0216, and 0.0108 / 0.0216 = 0.50. A correlation of 0.50 means the funds move together only moderately, so combining them still delivers real diversification benefit.