How different are the risk model providers, part three: predicted tracking error

In my third and final post on the question of “How different are APT, Barra, and Northfield?” I’ll shift focus to the change in predicted tracking error over time. It is critical to understand whether the portfolio is becoming more risky or reducing risk to become more like the benchmark (risk direction). So, we should want to understand whether APT, Barra, and Northfield suggest comparable change.

Our framework is mostly similar to the previous blog posts (part one, part two). First, we use the same 300 U.S. equity mutual funds , focusing on the same three U.S. long term equity risk models. Our change over time will consider the six month and twelve month periods ending on 12/31/2008. So, our starting point is 2,700 predicted tracking errors (300 portfolios x 3 risk models x 3 points in time).

Comparing the change across models, we are most interested in both correlation and covariance. The correlation coefficient reveals the strength and direction of a linear relationship between the tracking errors for two models.
We see consistently positive correlations across models, periods, and styles. For the most part, the tracking error changes between the risk models are largely correlated. Clearly, the Small Cap Growth and Small Cap Value correlations are smaller. Also, Model X is less correlated with the other two models.

Hand in hand with correlation, we should consider the covariance. This metric will tell us how likely one tracking error change is to be unexpectedly large when another model’s change is unexpectedly large. Similarly, a negative covariance would tell us that an unexpectedly large change in one tracking error suggests an unexpectedly small change in the other tracking error. Finally, a covariance of zero indicates that an unexpectedly large change in one tracking error leads to no change in expectation of the change in the other tracking error. Covariance is a good complement to correlation because it offers the extra insight of suggesting the magnitude of how two tracking error changes move together.

In our comparison, the key observation is that the covariance of the twelve month change is consistently higher than the six month change. We didn’t observe this type of difference when looking at the correlation. As with correlation, though, the small cap growth and value styles are the most independent.

Overall, coming back to our initial question of “How different are the various risk model providers?” I would say that there are differences in how tracking error has changed between the three models, but there aren’t tremendous differences in the change. The models are least similar in small cap growth and small cap value. Lastly, model X is more different from models Y and Z which is an interesting conclusion considering what we observed when we compared the actual tracking errors in earlier blog posts.

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