The question whether a given porfolio is mean-variance efficient is a basic problem of investment analysis. Mean-variance efficiency is also the basis of the Capital Asset Pricing Model. This paper presents the explicit form of the likelihood ratio test of the hypothesis that a given portfolio, or a particular market index, is ex-ante mean-variance efficient in the case where there is no riskless asset. Geometric relations are illustrated to provide intuition about the constrained maximum likelihood estimators and the test statistic, and two simple economic interpretations of the test are given.

Shmuel Kandel, Orthogonal Frontiers and Alternative Mean-Variance Efficiency Tests: Discussion, The Journal of Finance, Vol. 42, No. 3, Papers and Proceedings of the Forty-Fifth Annual Meeting of the American Finance Association, New Orleans, Louisiana, December 28-30, 1986 (Jul., 1987), pp. 620-622

This paper explores geometric relations, in mean‐variance space, among the sample frontier, the maximum likelihood estimator, and two other estimators of the zerobeta return. It is also demonstrated that a partition of the portfolio space is determined by a family of parabolas; the zeros of each parabola are the maximum likelihood estimators associated with all portfolios on the parabola. This observation is the basis for an additional interpretation of the statistic of the Likelihood Ratio Test of portfolio efficiency without a riskless asset.

This paper explores geometric relations, in mean-variance space, among the sample frontier, the maximum likelihood estimator, and two other estimators of the zerobeta return. It is also demonstrated that a partition of the portfolio space is determined by a family of parabolas; the zeros of each parabola are the maximum likelihood estimators associated with all portfolios on the parabola. This observation is the basis for an additional interpretation of the statistic of the Likelihood Ratio Test of portfolio efficiency without a riskless asset.

This paper presents an analysis of the testability of the mean variance efficiency of a market index when the returns on some components of the index itself are not perfectly observable. The results are basically not supportive of the notion that mean variance efficiency is testable on a subset of the assets. Bounding the market share of the missing asset and its expected return is not sufficient to produce a valid test. When the variance of the missing asset is bounded, and the amount of wealth that might be missing is small, it is possible, in principle, to reject correctly the mean variance efficiency of a market index.

A framework is presented for investigating the mean-variance efficiency of an unobservable portfolio based on its correlation with a proxy portfolio. A sensitivity analysis derives the highest correlation between the proxy and a portfolio that reverses the inference of a test of SHarpe-Lintner tangency. For example, the maximum correlation between the value-weighted NYSE-AMEX portfolio and a portfolio inferred tangent ranges from 0.76 to 0.48. We also test whether the correlation between the proxy and the tangent portfolio exceeds a given level. This hypothesis is often rejected for the NYSE-AMEX proxy at a correlation of 0.7.

[We find that conditional means and variances of consumption growth vary through time, and this variation appears to be associated with the business cycle. A pricing model with fluctuating means and variances of consumption growth provides implications about conditional moments of returns for both short and long investment horizons, and these implications are explored empirically. The U-shaped pattern of first-order autocorrelations of returns, as well as business cycle patterns in the price of risk, appears to be consistent with the model, but our exploration suggests that other implications about conditional return moments are at odds with the data.]