The potential performance of an asset set may be obtained by choosing the portfolio proportions to maximize the Sharpe (1966) performance measure. If a portfolio has a Sharpe measure equivalent to the potential performance of the underlying set of assets, then it is efficient. Multivariate statistical procedures for comparing potential performance and testing portfolio efficiency are developed and then evaluated using simulations. Two likelihood ratio statistics are then used to compare stock and bond indices against sets of 20 and 40 portfolios. The procedures are also compared to the Gibbons (1982) methodology for testing financial models.

ABSTRACT A test for the arbitrage pricing theory which employs a multivariate linear regression model is developed. Given a sample of return premiums for a set of N assets which includes a subset of k linearly independent portfolios, the k factor APT hypothesis is accepted if the intercept term is zero in the multivariate regression of the returns on the k portfolios. The test may be carried out simply, by using univariate multiple regression software. The relation of this test to the concept of performance potential and Sharpe's measure of performance is also discussed. If the performance potential of the k portfolios is not significantly less than the performance potential of the complete set of N assets, then the k factor APT hypothesis is accepted.

The purpose of this paper is to provide a link between the various multivariate tests of asset pricing and a performance measure for asset sets. The paper includes a unified summary of various F tests for mean-variance efficiency, intersection, and spanning for sets and subsets of financial assets. Both the risk-free asset and no risk-free asset environments are discussed. These tests are then related to the concept of potential performance for asset sets. The potential performance measure can be viewed as an extension of the Sharpe performance measure for single portfolios. The economic intuition behind the tests is that the multivariate tests of portfolio efficiency, intersection, and spanning are tests of zero potential performance at particular margins between the asset or portfolio subset and the full asset set.

J. D. Jobson, Bob Korkie, Statistical Inference in Two-Parameter Portfolio Theory with Multiple Regression Software, The Journal of Financial and Quantitative Analysis, Vol. 18, No. 2 (Jun., 1983), pp. 189-197

ABSTRACT The marginal performance contribution made by new assets in a portfolio is identified. The maximum change in a portfolio's Sharpe performance from the addition of new assets is a simple function of a generalized Jensen index and the unexplained covariances from a multivariate market model. Deviations from a higher dimension market line may be used to rank the desirability of asset additions to an existing portfolio. Statistical tests for the equality of the performance contributions by new assets is possible.