The authors propose a likelihood-ratio test of the hypothesis that the minimum-variance frontier of a set of K assets coincides with the frontier of this set and another set of N assets. They study the relation between this hypothesis, exact arbitrage pricing, and mutual fund separation. The exact distribution of the test statistic is available. The authors test the hypothesis that the frontier spanned by three size-sorted stock portfolios is the same as the frontier spanned by thirty-three size-sorted stock portfolios.

The authors propose a likelihood‐ratio test of the hypothesis that the minimum‐variance frontier of a set of K assets coincides with the frontier of this set and another set of N assets. They study the relation between this hypothesis, exact arbitrage pricing, and mutual fund separation. The exact distribution of the test statistic is available. The authors test the hypothesis that the frontier spanned by three size‐sorted stock portfolios is the same as the frontier spanned by thirty‐three size‐sorted stock portfolios.

A Bayesian approach is used to investigate a sample’s information about a portfolio’s degree of inefficiency. With standard diffuse priors, posterior distributions for measures of portfolio inefficiency can concentrate well away from values consistent with efficiency, even when the portfolio is exactly efficient in the sample. The data indicate that the NYSE–AMEX market portfolio is rather inefficient in the presence of a riskless asset, although this conclusion is justified only after an analysis using informative priors. Including a riskless asset significantly reduces any sample’s ability to produce posterior distributions supporting small degrees of inefficiency.

We characterize the sets of mimicking positions with returns that can serve in place of factors in an exact K ‐factor arbitrage‐pricing relation for a set of N assets. All of the sets are K ‐dimensional nonsingular linear transformations of each other. We interpret three examples of such transformations and discuss empirical considerations. We provide conditions under which the mimicking positions can be expressed as portfolios, and we characterize the relation between mimicking portfolios and the minimum‐variance frontier.

Tests of asset‐pricing models are developed that allow expected risk premiums and market betas to vary over time. These tests exploit the relation between expected excess returns and current market values. Using weekly data for 1963 through 1982 on ten common stock portfolios formed according to equity capitalization, a single‐risk‐premium model is not rejected if the expected premium is time varying and is not constrained to correspond to a market factor. Conditional mean‐variance efficiency of a value‐weighted stock index is rejected, and the rejection is insensitive to how much variability of expected risk premiums is assumed.

We propose a simple approach to account for commonalities in mutual fund strategies that relies solely on information on fund returns and investment objectives. Our approach augments commonly used factor models with an additional benchmark that represents an equal investment in all same-category funds, which we call an active peer benchmark (APB). We find that APBs substantially reduce the average time series correlation of residuals between individual funds within a group when added to a four-factor equity model (or to a seven-factor fixed-income model). Importantly, adding this APB significantly improves the selection of funds with future outperformance.