Tracking error

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In finance, tracking error is a measure of how closely a portfolio follows the index to which it is benchmarked. It measures the standard deviation of the difference between the portfolio and index returns.

Many portfolios are managed to a benchmark, normally an index. Some portfolios are expected to replicate the returns of an index exactly (an index fund), while others are expected to 'actively manage' the portfolio by deviating slightly from the index in order to generate active returns or to lower transaction costs. 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 metric.

If tracking error is measured historically, it is called 'realised' or 'ex post' tracking error. If a model is used to predict tracking error, it is called 'ex ante' tracking error. The former is more useful for reporting performance, whereas ex ante 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.

For active portfolios ex ante tracking error measures are necessarily lower than ex post. This is because future portfolio weights are randomly, but normally distributed around a mean (‘ex post stochastic’). Some studies have suggested a factor of as high as two (see Tracking error : ex ante versus ex post measures, Satchell and Hwang, Journal of Asset Management, Vol 2. No 3, December 2001).

The ex-post Tracking Error formula is the standard deviation of the active returns[1], given by:

T.E. = \sqrt{\frac{\sum_{i=1}^N \left( X_i - \bar{X} \right)^2 }{N-1}}

where Xi is the difference between the portfolio return and the index return for period i, that is, if di is the return for the asset in period i, and bi is the return for the benchmark period in i, then Xi=di-bi. N is the number of observations, and

\bar{X} = \frac{\sum_{i=1}^N X_i}{N}

There is also an incorrect version of the tracking error formula - shown below - that has appeared extensively across internet sources such as [2] and [3]. This erroneous formula frequently appears alongside a claim that it is a standard deviation. Clearly the following formula does not conform to the definition of standard deviation because in general the mean of the difference between portfolio and benchmark return is non-zero:

\text{(INCORRECT) }T.E. = \sqrt{\frac{1}{N-1}\sum_{i=1}^N (d_i- \ b_i)^2 }

Again, the above version of the tracking error formula is incorrect, and is only included here in order to help the reader identify and avoid a common mistake.

[edit] References

  1. ^ Grinold,R. and Kahn, R., "Active Portfolio Management", McGraw-Hill, 1999
  2. ^ Example of Incorrect Definition in: "Tracking Error - A complete definition" (StreetAuthority.com)
  3. ^ Example of Incorrect Definition in: "Tracking Error and the Information Ratio" (jpmorganfunds.com)