Smoothing and geometric attribution

Carl Bacon, CIPM, when asked to contrast geometric and arithmetic attribution, will no doubt point out the chief advantages geometric offers:

• Proportional: the active return is a ratio, not a difference, as we find with arithmetic.
• Convertible: the active return is independent of the base currency; the geometric active return will be the same whether it's expressed in dollars, euros, pounds, yen, etc.
• Compoundable: the geometric active return multiplies across time; no "smoothing factor" is required, as it is with arithmetic.

Carl is, of course, correct in all of these points; however, he fails to explain that while a smoothing factor isn't needed "across time," as we "link" our single period effects, one is needed "within time." That is, we need a smoothing factor for every single time period.

In his '05 FAJ article* on geometric attribution, Jose Menchero, PhD, CFA discusses the need for a smoothing factor. He points out how one can arrive at a "pure" geometric equivalent of an arithmetic model, but that this will not result in an approach that will fully reconcile the effects to the excess return: a smoothing factor of some sort is necessary.

Jose offers a rather healthy formula to "smooth out" the residual. Contrast this with Carl, who assigns the residual entirely to the selection effect. And so while Jose ensures that the residual is assigned across all effects in some appropriate or proportionate fashion, Carl is content with it residing entirely with selection. It is not my purpose here to justify one over the other: Carl's is clearly much simpler to work with than Jose's, and perhaps the differences are immaterial.

My point is merely to explain, in as clear a fashion as possible, that just like arithmetic, smoothing is needed. It's just that arithmetic attribution has no residual for the single period, but will across periods; while geometric has a residual for single periods, but once they're smoothed out, won't have one across time. It's the attribution equivalent of you can pay me now, or you can pay me later, but surely you will pay (i.e., you must smooth out a residual at some point).

Space does not permit the elaboration necessary to give this topic the attention it needs, so please consider this merely an introduction to what will follow in this month's newsletter.

*Menchero, Jose, 2005, Optimized Geometric Attribution, Financial Analysts Journal 61.