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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?

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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 ?

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  • $\begingroup$ This makes quite some sense! I wonder though why some researcher take the standard deviation then. It doesn't make really sense. $\endgroup$ – user9259005 Sep 17 '18 at 10:23
  • $\begingroup$ Hi: I'm glad it may help you. Note though, that as long as the amounts that the two funds are off are generally up and down, then their empirical means may not be that different, in which case, the variance alone can be okay. So, my example was pretty extreme but I still think it's a better way. Keep in mind that nothing is perfect. Whatever you use is still a sample statistic based on some number of observations and it too is a random variable with uncertainty., $\endgroup$ – mark leeds Sep 17 '18 at 16:09
  • $\begingroup$ Would you know another way to measure homogeneity of mutual funds? I want to execute research that compares fee structures of similar mutual funds and the tracking error is the first measure that came to my mind. $\endgroup$ – user9259005 Sep 17 '18 at 16:31
  • $\begingroup$ Like the Attack68 said, I think you need to mathematically define it. Do you care about volatility more than return or vice-versa. If you really want to define it mathematically, it really depends on a risk aversion parameter to trade off risk versus return. Then, once you have that parameter, utility to investor is defined mathematically as $\mu$ - $\lambda \times \sigma^2$. $\endgroup$ – mark leeds Sep 18 '18 at 6:57
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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.

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