Testing The Mean-Variance Efficiency of Well-Diversified Portfolios in Very Large Cross-Sections

We propose a new way of testing the mean-variance efficiency of well-diversified portfolios that exploits the cross-sectional size of typical financial datasets. The methodology consists of a sequence of simple tests, the results of which are aggregated in a statistic. This statistic is shown to be asymptotically standard normally distributed, despite dependence, in cross-section and over time, of the idiosyncratic risk. We investigate theoretically the asymptotic power of our test against the alternative that the well-diversified portfolio is not mean-variance efficient. By construction, our procedure is more powerful than standard tests of mean-variance efficiency that combine the assets in the cross-section into a limited set of (arguably) arbitrary portfolios. Even in cases where the latter has zero power, it can have unit asymptotic power. The incremental power is evidenced in tests of the mean-variance efficiency of the value weighted portfolio of common stock listed on the NYSE and AMEX. Unlike previously thought, however, the selection bias caused by including only continuously traded securities in the test is found to be important. By running the test in a case where it is known to have zero power, we are able to empirically confirm the correctness of the theoretical asymptotic properties of our statistic.