A Note on Ambiguity in Portfolio Performance Measures

Abstract

IN A RECENT ARTICLE in the Journal of Finance [4], Richard Roll argues that the Security Market Line (SML) criterion gives ambiguous performance signals for portfolio evaluation. This note reports results of an empirical test of robustness of the SML criterion applied. to different indices over two time periods. The evidence suggests the ambiguity issue may be moot. Roll argues that ambiguity results from the inability to decide upon a unique index to be used in the estimation of portfolio betas. When a mean-variance inefficient index is used, portfolios can be ranked on the basis of vertical deviations from an empirically fit SML. However, different inefficient indices will yield different SML's, and thus different rankings. When a mean-variance efficient index is used, all observations will plot on the SML and it will be impossible to assign rankings to the portfolios. As Roll demonstrates, it is possible that rankings can be reversed from one mean-variance inefficient index to another. Mayers and Rice [3] take issue with the findings of Roll. They do not question Roll's proof of the ambiguity of the SML criterion, but instead argue that the SML criterion can be used to identify superior performance. By setting up a model with an informed investor and uninformed investors, they show that the SML criterion will designate inferior portfolio managers correctly, but may incorrectly designate superior portfolio managers. Mayers and Rice claim that their conclusions do not warrant rejection of the SML criterion as a ranking device. As Roll [5] points out in his reply, Mayers and Rice are missing the point because they assume that everyone agrees upon which market index is to be used. An inefficient index is required to obtain rankings of portfolios, but there is no guarantee that the ranking reflects actual preference orderings of investors. Roll goes on to argue that the SML criterion should be abandoned because of the ambiguity issue. As an alternative criterion, he suggests measuring portfolio performance against the efficient frontier in mean-variance space. Since an index does not have to be chosen for this measurement, a degree of ambiguity is removed. Roll suggests that if the SML criterion is to be saved, it needs to be empirically demonstrated that commonly used indices do not rank portfolios very differently. In the section that follows we examine the ambiguity issue empirically. For