σ2 = 1/(n - 1) Σ(xi - yi)2
Where σ is the tracking error
n is the number of periods over which it is measured
x is the percentage return on the portfolio in period i
y is the percentage return on the benchmark
Some sources claim that the average error should be subtracted from right side of the equations above. That would give us the standard deviation of the tracking error from the tracking error over time. The formula given here appears preferable as it is a measure of deviation from the benchmark itself.
In order to make the tracking error comparable it should be annualised. In order to do the right right of the equation should be multiplied by the number of periods in an year. Equivalents σ multiplied by the root of the number of periods in an year. So if the error is based on monthly returns, it should be multiplied by root 12 to annualise.
Tracking error may be calculated from historical data (as above) or estimated for future returns. The former is called ex-post tracking error, and the latter ex-ante (standard terminology for statistics).
The causes of tracking error
An index tracker has two different causes of tracking error:
- trading and management costs,
- the differences in the composition of the portfolio and the benchmark.
The second of these is a direct result of the need to minimise the first. The simplest way to construct a tracker would be to simply hold every security in an index in proportion to its weighting in the index. The problem with this is that it increases trading costs as it involves holding a large number of securities.
In order to control trading costs, tracker funds hold a selection of securities that is statistically likely to replicate the performance of the index. This requires statistical analysis to construct the portfolio that will most accurately track the index at the lowest cost. This is why tracker funds are run by quants.
A portfolio that is a selected sample of the index will clearly not perform exactly as the index does. It should, however, perform very closely in like with the index.
In addition to the costs of actually trading the portfolio, the fees charged by fund managers also reduce the performance. While these are much lower than the fees charged by active fund managers, they still have a significant effect over time.