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The Role of the Interaction Term in Performance AttributionAs Brian Singer observed, "... the interaction cross-product has economic significance even though it is not typically an explicit or even conscious decision of the fund manager" [Singer 1996, p. 47]. The paper What is this Thing Called Interaction? shows five different examples of attribution analysis done on the same fund. One of the analyses works from the bottom-up. In that analysis, all of the active return is explained by bottom-up stock selection, with no interaction term at all. In the other four analyses, the portfolio is dissected according to a particular sector scheme (for example, by industry, or by size). The attribution is then calculated using the standard Sector Attribution (a.k.a. "Brinson attribution") approach. One important difference to note is that the amount of interaction varies a lot, depending on the sector scheme which is used. In some cases, the interaction terms from the sector-level mostly cancel one another out, while in other cases they don't. Another important difference is that the breakup of sector allocation and stock selection can also be dramatically different, depending on the sector scheme one chooses. For people who don't want to see an interaction term in their attribution reports, one option is to use the bottom-up attribution model (see the Stock Level Attribution page for details). If you are trying to understand a bottom-up investment process, the absence of an interaction term is one attraction of this model. However, the main attraction of this model is that it measures stock selection from the bottom-up, which leads to very different results than the ones obtained from sector attribution. In a bottom-up investment process, sector allocation and interaction are both irrelevant concepts. The stock selection attributes in the bottom-up model are different from the stock selection results in the sector attribution model. This is because bottom-up stock selection is a different concept from stock selection within sub-sectors of the index. The details of all these calculations are available in the spreadsheet interaction.xls. Interaction HappensSometimes, people make comments such as "I don't want to see interaction on the reports", or "I don't like the idea of interaction". Another common assertion is, "Somebody does asset allocation, and somebody else does the stock selection, but nobody creates interaction". Because of sentiments like these, a common solution is to combine interaction with other terms in attribution reports. Most commonly, people combine interaction with the stock selection term. This can be done by calculating the two terms separately and then adding them, or it can be done by calculating stock selection using the portfolio sector weight (rather than the benchmark sector weight). When an attribution report shows no interaction term, this can lead people to the conclusion that the results are free of interaction. However, this is mistaken, as the following example shows (for full details, see the spreadsheet InteractionHappens.xls). The following table shows attribution results for a portfolio containing two sectors. The stock selection attribute has been calculated using the portfolio sector weight (i.e. it includes the interaction term). These results may appear to cleanly divide the effect of asset allocation from the effect of stock selection.
However, it is interesting to see what happens to these results when the asset allocation changes from 40/60 to 35/65. As this is purely a change to the portfolio's asset allocation, one might expect that this would alter the asset allocation attributes, but not the stock selection attributes. After all, if nothing has changed except for the asset allocation decision, surely the only impact should be on the value added attributable to asset allocation, right? Wrong, in fact. The following table shows the results when the asset allocation is changed to 35/65:
As one would expect, the increased asset allocation tilt away from equities (which had the lower benchmark return), toward bonds (with the higher benchmark return), increased substantially the value-added attributable to asset allocation. But, counterintuitively, this attribution report also shows that the equities/bonds tilt decreased the value-added attributable to stock selection. How can this make sense, since the only change was a change to the portfolio's asset allocation? The answer, of course, is that the interaction term has not been eliminated from this calculation — it has only been buried in the stock selection attribute. The outcome is that the stock selection attribute now depends partially on the portfolio's asset allocation positioning, while if the interaction term was reported separately, there would be a clear separation between asset allocation and stock selection. By combining interaction with the stock selection term, we have created a strange outcome: the stock selection term in the attribution depends on what the asset allocators did. Hiding the interaction term inside the stock selection term doesn't make interaction go away: it simply makes the stock selection term a curious blend of "true" stock selection, and other components that are at least partly attributable to asset allocation. This is not a recipe for clarity. Neither is it a recipe for fairness. It is arguably a very opaque and unfair way to deal with attribution information. So, in summary, it's fine for people to present attribution information in ways that are more informative or understandable for their audience. However, everybody should understand that interaction is an intrinsic part of the sector attribution model. If you combine interaction with another term in an attribution report, you are taking away the reader's ability to distinguish between interaction and the other terms. Interaction can be hidden, but it cannot be made to disappear. In short, Interaction Happens. See the spreadsheet InteractionHappens.xls for full details of these examples. ReferencesSinger, Brian "Evaluation of Portfolio Performance: Attribution Analysis" Journal of Performance Measurement, Winter 1996, pp. 45-55.
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