Tracking error is used within the finance industry to determine how well one instrument tracks another. Often tracking error is incorrectly defined as the standard deviation of active returns rather, than the square root of the average squared active return. While these terms are similar in many cases, we discuss examples where the definition can significantly differ.

## Defining tracking error

Consider two financial instruments, where one is designed to track the other. The active return is the return difference between the two instruments and the tracking error should quantify the average size of the active returns; that is – how closely they mimic each other. If the tracking error, is zero then you expect them to be a carbon copy of each other.

As Figure 1 shows, tracking error can be modelled by two components – a drift term and a stochastic term. The drift term represents a constant deviation in tracking between two instruments. A real-world example of this might be the management fees charged by an ETF provider. The stochastic term adds noise to the tracking over the short term, but over long periods should average to zero.

## Figure 1: Cumulative active returns and active return distribution by drift term, stochastic term and combined

The tracking error is given by the root mean square of the active returns, and provides an insight into the size of both the drift and stochastic terms [1]. However, it is often mistakenly given as the standard deviation of the active returns, meaning that the drift term being ignored. This is inadequate because an apparently zero tracking error does not guarantee successful tracking.

## Toy example

To illustrate some of the problems with incorrectly defining the tracking error, let’s look at a toy example as in Figure 1. Here, we have three trackers which attempt to track an index. The first tracker has a constant drift term corresponding to -50 basis points annually. The second tracker has a stochastic term, which produces noise in the tracking of about 1% annually, but no long-term drift. And finally, our third tracker has both drift and stochastic terms at -50 basis points and 1% annually.

Table 1 provides the mean and standard deviation of the active returns as well as the tracking error for each of the trackers.

## Table 1: Statistics for three hypothetical index trackers

The danger of incorrectly defining the tracker error is apparent. The standard deviation of the active returns overstates the tracking performance as it only considers the stochastic term and ignores any drift terms. And so, it seems that the first tracker is performing well despite constantly underperforming the index (similarly, we cannot distinguish between the second and third trackers).

The true definition of the tracking error, however, shows that there is a difference between all three trackers. If we increase the size of either the stochastic or drift terms, the tracking error also increases. In order to correctly disentangle the different contributions to the tracking error, it serves to also show the mean and standard deviations of the active returns. All three help an investor understand why a tracker may not be performing well. But any headline measurement in determining how well an instrument is being tracked should use the correct definition of tracking error.

## Conclusion

Defining the tracking error as the standard deviation of the active returns is pervasive throughout the finance industry. This incorrect definition can lead to a false sense of security, as investors expect trackers to perform better than they actually do. The correct definition of the tracking error – the root mean square of the active returns – accounts for both stochastic and drift terms and so is a better measurement to use in judging tracking quality.

### References

[1] The root mean square is calculated by finding the mean of the square of the active returns and taking the square root.