Multiperiod Arithmetic Attribution

This article presents a set of qualitative characteristics and quantitative properties for arithmetic multiperiod performance attribution. Such characteristics and properties are essential for ensuring a sound and accurate linking of attribution effects over time. A comparison of various linking algorithms within this framework shows that linking algorithms that are not consistent with this set of quantitative properties can exhibit spurious effects that distort the attribution analysis. Performance attribution is a powerful and widely used analysis for explaining the sources of portfolio return relative to a benchmark. The objective is to quantify the impact of active management decisions and thereby be able to evaluate the effectiveness of an investment strategy. Active returns are decomposed into attribution effects, which are identified with the active management decisions. These attribution effects, when aggregated together, fully account for the active return. For such an analysis to be meaningful, the performance attribution model must reflect the decision-making process. As a consequence, a variety of models have been developed (during roughly the past 20 years) for decomposing active returns. The active returns for fixed-income portfolios, for instance, are commonly decomposed into effects related to yield-curve movements and/or credit spreads. The active return for equity portfolios may be decomposed within the context of a multifactor risk model or according to a sector-based approach that measures the effects of the allocation and selection decisions. These methodologies have also been generalized for the case of multiple currencies with hedging. Although these attribution models decompose active returns in distinct ways, they are similar in that all are based on single-period analyses. Practitioners, however, typically want to explain the sources of active return for an investment period of, say, a quarter or a year. During this longer period of time, portfolios may be rebalanced several times, thereby rendering the single-period assumption invalid. What is needed, therefore, is a means of linking attribution effects across multiple periods. The first articles on the subject of multiperiod attribution appeared only as recently as 1999. Since then, a flurry of papers on the subject have been published. The multiperiod algorithms can be grouped into four classes: (1) linking coefficient approaches, (2) compounded notional portfolio methods, (3) recursive models, and (4) ad hoc smoothing algorithms. But although there is no shortage of methodologies to choose from, what is noticeably lacking is a conceptual framework by which to understand and compare the various approaches. The purpose of this article is to provide such a framework and, in doing so, to provide insight into the underlying structure of multiperiod attribution. At the outset, the article argues that a multiperiod linking algorithm must meet certain qualitative conditions: It should be intuitive, transparent, and robust (meaning that it should yield reasonable results under any set of market conditions). More importantly, the method must satisfy four quantitative properties: The algorithm should be (1) residual free, (2) commutative, (3) metric preserving, and (4) fully linkable. The commutative property implies that the results of the analysis are independent of the ordering of the periods. The metric-preserving property requires that the linking be performed in a manner consistent with the basic measure of relative performance (i.e., arithmetic or geometric) being used. The article describes tests of the arithmetic algorithms with respect to the quantitative properties. Some of the distortions and nonintuitive effects that result when the quantitative properties are violated are documented and discussed. Only one methodology—a linking coefficient approach known as the “optimized linking algorithm”—is consistent with all of the properties.