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Sample tracking error: deviations of active returns

Sample tracking error measures the dispersion of portfolio-minus-benchmark returns around their own mean.

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Define the paired differences

For each matching period, let aₜ be portfolio return minus benchmark return. CFA Institute calls the standard deviation of active returns active risk or tracking error. The sample formula below estimates that standard deviation from n observed differences.

sa=∑t=1n(at−aˉ)2n−1s_a=\sqrt{\frac{\sum_{t=1}^{n}(a_t-\bar a)^2}{n-1}}

aₜ is an active return, ā is its sample mean, and n is at least two. Returns and the result use the same units.

Use a constant-difference check

Suppose the observed active returns are 1%, 1% and 1%. Their mean is 1%, so every deviation from the mean is zero. Sample tracking error is zero. This does not mean there was no return difference; it means that the observed difference did not vary.

Distinguish dispersion from distance to zero

Squaring the raw differences without subtracting their mean answers a different question. In the constant 1% example, the root mean square of the raw differences is 1%, whereas their standard deviation is zero. State which statistic you are calculating before comparing figures.

State sample and frequency conventions

Using n instead of n − 1 produces a population standard deviation for the finite set rather than this sample estimate. Daily and monthly figures also describe different intervals. A statistic needs its denominator and observation frequency to be interpretable.

Further references