One of the controversial topics in performance attribution has to do with interaction. This effect exists in several models, but we’ll limit our discussion to its presence in the Brinson-Fachler model. Recall that there are three effects in all: allocation, selection, and interaction (the formulas are shown below).
The interaction effect represents the impact from the interacting of the allocation and selection decisions. There have been several good reasons offered why one shouldn’t show interaction; when not showing it we typically change the weight in selection to the portfolio weight, meaning that the selection decision is expanded to include interaction (though this is rarely stated as such).
If we reflect on what the possible results can be with interaction we conclude the following:
- overweighting (positive) times outperformance (positive) = positive result
- overweighting (positive) times underperformance (negative) = negative result
- underweighting (negative) times outperformance (positive) = negative result
- underweighting (negative) times underperformance (negative) = positive result.
One argument for showing interaction that I posit is that to not do so means that the selection effect will be burdened with negative results when they aren’t deserved (e.g., if we have outperformance but underweighting). The response might be “well, in the end it will all work out, because there are times when selection will get a positive interaction effect when it’s undeserved”. We used to almost universally hold this view regarding returns, thinking that the mid-point Dietz method, for example, was perfectly acceptable, because those times when we penalized the manager by assuming a late arriving flow was present for the full period would be counterbalanced at some point in the future by a benefit of an early flow only being counted for half the time. But we’ve wised up and now promote much more accurate methods.
During our recent Trends in Attribution (TIA) conference, one panelist pointed out the “flaw” of underweighting times underperformance: a positive result. What DOES one make of this? I think it’s easy, after some recent reflection: this shows that the allocation decision was a wise one! That is, they underweighted at a time when there was underperformance (would you propose to overweight?).
In an article I wrote on this topic I proposed that if you want to “eliminate” the interaction effect, then create a “black box” to analyze the interaction effect when it shows up, and to allocate in a conscious and methodical way. I still hold to this belief and still hold to the value of the interaction effect, and oppose any arbitrary assignment to selection or allocation. Space doesn’t permit much more at this time, but perhaps I’ll take this up again at a later date.
Spaulding, David. “Should the Interaction Effect be Allocated? A ‘Black Box’ Approach to Interaction.” The Journal of Performance Measurement. Spring 2008