Existing attribution models have interaction terms that are difficult to understand and inconsistent with symmetry conditions that should ideally exist. The author introduces a new attribution model without interaction terms that meets the symmetry conditions. In addition, he uses a ratio test to illustrate how this new arithmetic model is consistent with a recently developed geometric model under certain conditions.
The author demonstrates two general principles about attribution models: Arithmetic attribution models base securities’ relative returns on weights generated from the initial market values of the securities, whereas geometric attribution models base securities’ relative returns on weights generated from end-of-period market values. In his examination of two popular arithmetic attribution models, he shows that both produce terms for industry selection (or timing) and stock selection with either one or two extra terms, depending on the model. The extra terms are not easily assessed and violate symmetry in two ways: (1) The benchmark has more terms than the portfolio being assessed, and (2) switching the benchmark with the portfolio within the given model does not produce results of the same magnitude but with a different sign.
The new arithmetic attribution model the author suggests is in a canonical form (i.e., only industry selection and stock selection terms exist because interaction terms are eliminated). He also introduces a recently developed existing geometric attribution model (in canonical form) with conditions to make it compatible with the new arithmetic attribution model. Both models meet the symmetry requirements, which existing models do not.
Furthermore, the author devises a ratio test in which both models must have equivalent ratios of industry selection over stock selection for the entire portfolio and within groups within the portfolio. The newer models meet the ratio condition, demonstrating the connection that should exist between arithmetic and geometric models despite their different weighting conditions.
How Is This Research Useful to Practitioners?
Practitioners can benefit from the author’s demonstration of the difference between an arithmetic attribution model and a geometric attribution model. The primary difference is in how security weightings are determined—that is, by either the initial holdings of securities (arithmetic) or the end-of-period holdings of securities (geometric). The two models are connected through the two symmetry conditions and the newly introduced ratio condition.
Furthermore, the author also makes the very logical point that attribution models should not contain cross-product or interaction terms that cannot be easily explained. Practitioners can certainly benefit from such logic and consequently benefit from the arithmetic model that is introduced, which does not have such terms because of its canonical form.
How Did the Author Conduct This Research?
The author demonstrates mathematically how arithmetic and geometric attribution models differ. Using a simple numerical example, he demonstrates that two popular arithmetic attribution models are not symmetrical and have interaction terms that are not easily explained.
He introduces a new arithmetic attribution model that does not have interaction terms because it is in canonical form; he also demonstrates this model mathematically. The new model is symmetrical and connected with a previously developed (but also relatively new) geometric attribution model through a ratio condition. The same numerical example is used to demonstrate the symmetrical properties of the two models and the ratio condition.
I like how the author starts with very basic properties of attribution models and then introduces the new model without interaction terms between the industry selection and stock selection measures. I also like the logical argument regarding what criteria should be used to determine the effectiveness of a given attribution model.