Lets say I have fund A and fund B and both aim to track the S&P500. I want to compare their performance over time and see who did a better job of tracking the Index. Should I compute the standard deviation of the tracking error of both funds compared to their index or execute a linear regression with index returns as the independent variable and fund returns as the dependent variable and look at the R2 of both regressions to see how much the returns of the index can explain the returns of the funds?
Hi: Take the worst case where one fund is 15 bps down ( relative to the index ) every month and another fund flips back and forth between being up 1 basis point to down 1 the next month. Then the second fund will have a standard deviation of tracking error that is much greater than the first fund but you clearly would prefer it anyway. So, I think some MSE type statistic would be more useful. This way you capture the bias also. Inevitably, using some regression approach of fund return on index return should be sort of equivalent to an MSE approach. For example, if the empirical mean of the fund and index were the same, then the R2 of the regression would be equivalent (well, some function of it ) to the standard deviation of the tracking error. But, generally, the empirical means are different which is why empirical MSE is probably a better approach. Others may prefer the regression approach for some other reason so there may not be one answer ?
It seems to me that you you would need to mathematically define the statement: which did a better job of tracking the index, A or B?
@Mark raises the point that a fund that makes a profit is preferred over another that doesn't (perhaps it is consistently generating a little alpha somehow) but that might not be the case if you regard the profitable fund as just a statistical result and it may lose next time. In any case it depends on what and how you want to measure it.
If I was measuring, in isolation, the ability to track the index I would probably take a sample of the absolute values (or MSE like Mark) of the daily return differences between fund and index and perform some statistical test to see if one of those has a higher variance than the other.