Optimized Geometric Attribution

To address the problem of geometric performance attribution in a recently proposed framework of qualitative characteristics and quantitative properties, the approach reported here is to solve the problem in a two-step fashion. The first step is to define a set of attribution effects in their “pure” geometric form. Because these pure effects will not aggregate in a residual-free manner, however, the second step is to perturb the geometric attribution effects in such a way that they deviate as little as possible, subject to the constraint that there be no residual, from their pure values. The resulting optimized attribution effects represent the most accurate formulation of geometric attribution. Performance attribution analysis is a widely used tool for quantifying the impact of active management decisions on active return. The active return is decomposed into a set of attribution effects that are themselves directly related to active management decisions. The attribution effects, when aggregated, fully account for the active return.Broadly, two formulations of performance attribution—arithmetic and geometric—are in use. The arithmetic approach defines active return as a difference, whereas the geometric approach defines it as a ratio. This article shows that these definitions present unique, yet related, challenges. Arithmetically, defining the attribution effects for a single period is simple but linking them over multiple periods is challenging. Geometrically, the situation is reversed: Linking attribution effects over multiple periods is simple, but defining them for a single period is challenging.A recent article identified a set of qualitative characteristics and quantitative properties for the case of arithmetic attribution. Qualitatively, the attribution methodology should be intuitive, transparent, and robust. Quantitatively, it should be residual free, fully linkable, commutative, and metric preserving. This article applies these characteristics and properties to the case of geometric attribution.I develop a geometric attribution methodology consistent with this framework via a two-step procedure. In the first step, I define the geometric attribution effects in their “pure” metric-preserving form. The attribution effects so constructed exhibit all of the desirable characteristics and properties, except that they do not aggregate in a strictly residual-free fashion. The second step, therefore, is to eliminate the residual. I meet this challenge by applying an optimization procedure that, subject to the constraint that there be no residual, minimizes the perturbation from the pure values. I compare this algorithm with others that have been proposed and argue that the algorithm resulting from my process represents the most accurate formulation of geometric attribution. Several examples illustrate the spurious effects that can arise with alternative approaches to geometric attribution.