Rate of Return Dispersion: Equal Weighted versus Asset Weighted
By David Lee, CFA
Richard E. Dahab, CFA
The issue of investment managers’ dispersion of returns has gained importance over the past decade as managers strive to provide a uniform product (or at least the appearance of a uniform product) to their clients. Managers want to report the lowest dispersion number (most consistent performance) they can without distorting the data. Plan sponsors want the most meaningful measure of what they can expect from their managers. Unfortunately, there is not a standard for reporting and different "correct" methodologies give different results. It is our belief that an equal weighted dispersion measure is far more useful than a dollar weighted measure and should be adopted as the standard.
The Association for Investment Management and Research Performance Presentation Standards (AIMR-PPS) gives investment firms guidance in the calculation of investment returns but allows discretion in the presentation of dispersion within their composites. Although there are multiple methods of presenting dispersion, most investment firms use standard deviation as their measure of dispersion. Standard deviation can be presented as equal weighted or asset weighted. From our experience most firms use asset weighted in contradiction to the AIMR-PPS statement that equal weighted is more widely accepted.
While examples can be created that show asset weighting producing a larger dispersion than equal weighting, generally asset weighting produces a lower figure which may explain the propensity for mangers to use this methodology. Furthermore, relating equal weighted dispersion with asset weighted dispersion is impossible. For plan sponsors, this means that when using dispersion as a criterion for manager selection and/or retention, knowing whether or not the composite presents equal weighted or asset weighted dispersion is critical.
The following illustrates the potential for confusion when looking at dispersion. Both an equal weighted dispersion and an asset weighted dispersion are calculated for a hypothetical composite. The composite has 10 portfolios whose total assets are worth $1B. As it is with many composites, for this example, it is assumed that the majority of the composite consists of similar size portfolios with only a handful that are significantly smaller (or in this case larger). Rates of return for the portfolios range from 9% to 11%.
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Asset weighted = 0.50 dispersion Equal weighted = 0.82 dispersion |
The underlying data for the calculation of both dispersions is the same; however, depending on which standard deviation calculation is used, the results are quite different. The formulas for calculating asset weighted and equal weighted dispersion are provided below.†
Asset Weighted Standard Deviation:
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CASSET = asset weighted composite return MVBi = beginning market value of the ith portfolio Ri = unweighted return on the ith portfolio |
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Sc = asset weighted standard deviation WtMEAN (R) = asset weighted mean return Wi = weight of the ith portfolio |
Equal Weighted Standard Deviation:
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MEAN (R) = equal weighted composite return N = number of portfolios |
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Sc = equal weighted standard deviation |
While a difference of .32 between 0.50 and 0.82 appears to be small, it is in fact rather large. Most managers would like the dispersion of their accounts’ performance to be less than 0.20 and would be troubled with a figure of 0.82. Even though composites have a nominal amount of dispersion, reflecting cash flows into and out of composites and imperfect mirroring of each portfolio in a composite, managers eschew large dispersions.
A survey of investment firms affirmed the trend of dispersion discrepancy. For a set of 50 equity composites, each provided by a different investment firm, composites that provided equal weighted dispersion averaged .92 greater than their asset weighted counterparts in a 10-year period, significantly greater than even the example above. In a like group of fixed income composites though, the difference was not meaningful. Fixed income securities are on the whole less volatile than equities and thus composites of fixed income portfolios should produce less dispersion.
Another reason to question the variance between equal weighted and asset weighted standard deviation is the ability to "manage" asset weighted dispersion. In an equal weighted composite, each portfolio has uniform consideration; the inclusion or removal of a portfolio only has a nominal effect on dispersion. Asset weighted dispersion, on the other hand, exhibits great divergence. The selective inclusion or exclusion of a handful of large portfolios can alter asset weighted dispersion significantly. AIMR-PPS only requires that "all actual fee-paying discretionary portfolios must be included in at least one composite;" this allows investment firms the freedom to create as many or as few composites they deem necessary.
With the potential of data skew for asset weighted dispersion and its apparent predominance in investment firm presentations, plan sponsors should take care when using dispersion as a data point in the investment selection process. Although using asset weighted standard deviation is compliant with AIMR-PPS, equal weighted dispersion is more objective and should be adopted as the standard.
† See the second edition of the AIMR Performance Presentation Standards Handbook (Charlottesville, VA: AIMR, 1997).