Tracking error measures to what degree the returns of a fund or positions move in synch with or move independently of a benchmark index. More specifically, given the returns to a fund, R, and the returns of an index, RI, we can calculate the following regression equation:
where α and β are constants, and ε is a mean zero error term. The tracking error is then the standard deviation of the error term, ε. The lower the tracking error the more closely the returns of the fund track the index.
The preceding definition is the primary definition for tracking error used by Northstar Risk. Other commonly used definitions of tracking error include the root mean square of R − RI, or the standard deviation of R − RI. The later definition would be equivalent to the definition used by Northstar Risk, when β is equal to 1.
The advantage of the definition based on regression analysis is that it is consistent with how we disaggregate risk and performance into outperformance (alpha), systematic risk (beta), and idiosyncratic risk. While the alternative definitions may work well for long-only funds, the regression approach may be more appropriate for many hedge fund strategies where betas to any benchmark are likely to be significantly less than 1.