When comparing geometric and arithmetic attribution, the following are often cited as advantages of the former:
- it’s compoundable (meaning, that it links attribution effects over time without creating temporal residuals)
- it’s convertible (that is, when we convert the returns from one currency to another, the excess return will be the same)
- it’s proportionate (meaning, it is sensitive to the differences in return sizes, given greater importance to outperformance for smaller returns than larger).
I am not looking to debate the merits of geometric attribution in this post, as I believe there are very good reasons why arithmetic rules! But the third point speaks to the issue I raised yesterday. This occurred to me recently, and it seemed to make some practical sense, but I thought it appropriate to run some numbers.
Let’s consider the cases show in this table:
To have a threshold of 100 basis points for materiality may seem high, but I know many firms who use it.
As I suggested yesterday, the difference between the original and corrected returns means more when the returns themselves are small than when they are large; but it is impossible to show this when you are constrained by an arithmetic absolute method to test materiality; that is, when you simply take the difference between the original return and the corrected one. In this table we see that in all six cases the threshold of 100 bps is reached: but the difference between 0.25% and -0.75% surely is felt to be more significant than between 27.35% and 26.35 percent.
If we choose to go with an arithmetic relative approach we see how the sense of proportionality appears. The relative difference at the lower end is huge, because the difference between the original return (0.25%) and the corrected (1.25% in one example and -0.75 in the other) is 400 percent. But when we get to higher numbers we see this drop significantly. I am compelled to think that the 400%, though mathematically correct, is a tad hyperbolic, as one would not really think that the difference warrants such a huge score.
I am unaware of anyone who is employing the geometric method, but I think it has merit, and should at least be considered. Because the returns on the left hand side of the table are small, we see the same 1% reported as we do with the arithmetic (though in reality, if it weren’t for rounding they would be a bit less). We see that the other four examples fail to meet this threshold, and I think rightly so.
I have constructed additional examples and will present them, along with additional details on this idea, in this month’s newsletters. In the mean time, feel free to comment if you’d like.
p.s., Regarding the acceptability of a threshold of 100 basis points: I tend to favor lower levels (e.g., 50 bps) but accept this level, especially given that industry colleagues I respect greatly have adopted such a level. In general, I would not find it acceptable for the level to be higher.
p.p.s., My use of the adjective “temporal” in regards to residuals was to distinguish the residuals that can arise from linking arithmetic effects over time, with single period residuals that arise frequently from the use of a holdings-based attribution model.