Comparing Index Funds with Tracking Error and Mean Squared Error
Summary
The document considers how to compare two funds designed to track the same index. It contrasts the standard deviation of return differences with regression fit statistics such as R-squared, and argues that a tracking metric should reflect the criterion being evaluated. Standard deviation alone can favor a fund whose deviations fluctuate around zero over one that consistently lags, even when the latter may be preferable to an investor.
Mean squared error of fund-minus-index returns includes both average bias and variability, making it a possible measure of total tracking deviation. The discussion notes that regression-based measures can relate to tracking error under particular conditions, such as equal sample means, but need not capture the same thing generally. It also suggests comparing absolute differences or their variance and applying a statistical test. There is no universal winner among metrics: the appropriate choice depends on whether the goal is to measure tracking fidelity, reward, or another fund characteristic.
Key ideas
- Tracking quality needs a clear definition before choosing a comparison metric.
- The standard deviation of return differences measures variability but does not capture persistent tracking bias.
- Mean squared error incorporates both the average deviation and its variability.
- R-squared and tracking-error measures may be related under specific conditions, but they are not generally interchangeable.
- A statistical comparison can help assess whether observed differences in tracking performance are meaningful.
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Full text
# tracking error or R2? # tracking error or R2? 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? ## Answer by mark leeds (score 2, accepted) https://quant.stackexchange.com/a/41761 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 ? ## Answer by Attack68 (score 0) https://quant.stackexchange.com/a/41778 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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