# Tracking error

In finance, tracking error is a measure of how closely a portfolio follows the index to which it is benchmarked. The best measure is the root-mean-square of the difference between the portfolio and index returns.

Many portfolios are managed to a benchmark, typically an index. Some portfolios are expected to replicate, before trading and other costs, the returns of an index exactly (e.g., an index fund), while others are expected to 'actively manage' the portfolio by deviating slightly from the index in order to generate active returns. Tracking error (also called active risk) is a measure of the deviation from the benchmark; the aforementioned index fund would have a tracking error close to zero, while an actively managed portfolio would normally have a higher tracking error. Dividing portfolio active return by portfolio tracking error gives the information ratio, which is a risk adjusted performance measure.

## Definition

If tracking error is measured historically, it is called 'realized' or 'ex post' tracking error. If a model is used to predict tracking error, it is called 'ex ante' tracking error. Ex-post tracking error is more useful for reporting performance, whereas ex-ante tracking error is generally used by portfolio managers to control risk. Various types of ex-ante tracking error models exist, from simple equity models which use beta as a primary determinant to more complicated multi-factor fixed income models. In a factor model of a portfolio, the non-systematic risk (i.e., the standard deviation of the residuals) is called "tracking error" in the investment field. The latter way to compute the tracking error complements the formulas below but results can vary (sometimes by a factor of 2).

### Formulas

The ex-post tracking error formula is the root mean square (RMS) of the active returns,[1] given by:

$TE = \omega =\sqrt{\operatorname{E}[(r_p - r_b)^2]}$

where rp − rb is the active return, i.e., the difference between the portfolio return and the benchmark return.

Nevertheless it is commonly calculated as the standard deviation of the active returns:

$TE = \omega =\sqrt{\operatorname{Var}(r_p - r_b)} = \sqrt{{E}[(r_p-r_b)^2]-({E}[r_p - r_b])^2}$

which in case of large portfolio deviations would lessen TE significantly and mislead its original meaning.

A somewhat improbable but illustrative example that shows the fallibility of the standard deviation formula is a series of equal active returns. (In practice, active returns are unstable.) Such a series has 0 standard deviation, i.e., there is no variability or dispersion in the active returns. However, the portfolio is not truly tracking the benchmark; it is, in fact, increasingly diverging from its benchmark.

Even though the second definition in terms of volatility of active returns is sometimes used as the tracking error in an information ratio, it should actually not be used as, for example, a series of equal negative active returns would yield an infinite information ratio while the deviation of the (under-performing, in this example,) portfolio from the benchmark would be substantial. Early papers[2] labeled the concept discussed in this article 'tracking error volatility' and are a possible source for the second conception of tracking error; the 'volatility' has been excluded in more recent studies.

## Examples

• Index funds are expected to have minimal tracking errors.
• Inverse exchange-traded funds are designed to perform as the inverse of an index or other benchmark, and thus reflect tracking errors relative to short positions in the underlying index or benchmark.

## References

1. ^ Grinold, R. and Kahn, R., "Active Portfolio Management", McGraw-Hill, 1999
2. ^ e.g., Richard Roll, "A Mean/Variance Analysis of Tracking Error," Journal of Portfolio Management, 1992