A Performance Interpretation of Multivariate Tests of Asset Set Intersection, Spanning, and Mean-Variance Efficiency

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.