Tracking Error: Return Differences Versus Regression Residual Volatility

Summary

The document distinguishes two tracking error calculations. The direct method takes the standard deviation of the periodic return differences between a fund and its benchmark; under this definition, zero tracking error means the fund matches benchmark returns one for one. The post asks how this relates to a formula using the fund’s return volatility and the benchmark regression’s R-squared.

The answer explains that the regression-based expression measures the standard deviation of the portion of fund return variation not explained by the benchmark. It can treat a fund that moves proportionally more or less than the benchmark as well tracked, since regression beta captures the degree of scaling. The two measures are therefore not equivalent: the appropriate choice depends on whether tracking means matching benchmark returns directly or measuring residual variation after a fitted relationship. The document gives definitions, not empirical results, and does not specify choices such as return frequency or annualization.

Key ideas

• Direct tracking error is the standard deviation of fund-minus-benchmark returns.
• A regression-based measure uses fund volatility and the unexplained share of its variance.
• The regression measure can regard leveraged or de-leveraged benchmark exposure as tracking.
• Choose the definition based on whether exact return matching or regression residual risk is intended.

Tags

# What is the formulat to compute Tracking Error?


# What is the formulat to compute Tracking Error?












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!

## Answer by nbbo2 (score 3)

https://quant.stackexchange.com/a/36527

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.