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I am here for the first time and read quite a few posts before asking this question. In my class, my finance Professor wrote the formula for Tracking Error $TE$:

$$TE = \sqrt{(1-R^2)} \times \sigma$$

where $\sigma$ stands for the standard deviation.

I don't find this formula in any of the books. Could anyone help me to understand the relationship between $R^2$, $\sigma$ and $TE$?

Thanks!

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The simplest and most common method for finding the Tracking Error of Fund X versus a Benchmark B is to compute the standard deviation of the differences in monthly returns of the Fund and the Benchmark:

$$TE=\text{STDEV}(r_{X,i}-r_{B,i})$$

A slightly more complicated method involves performing a regression of Fund X returns on the benchmark returns. Then we compute $$TE = \sqrt{(1-R_{XB}^2)} \times \sigma_X$$

Where $R^2_{XB}$ is the R-squared of the regression, i.e. the percentage of the variance of the fund returns that is "explained" by benchmark returns (and so $(1-R^2_{XB})$ is the percentage of variance that IS NOT explained), and $\sigma_X$ is the standard deviation of fund returns.

These two definitions are not equivalent. In the first definition "perfect tracking" ($TE=0$) means that the fund returns copy the benchmark returns 1 to 1. The second definition the fund may track a version of the benchmark returns levered up or down (to find the degree of leverage implied, check the Beta of the Regression).

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