The Gibbons, Ross, and Shanken Test for Portfolio Efficiency: A Note Based on Its Trigonometric Properties

This study is intended as a note and provides an extension to a much-used and established test for portfolio efficiency, the Gibbons, Ross, and Shanken GRS-Wald test. Tests devised to measure portfolio efficiency are crucial to the theoretical issues related to CAPM (Capital Asset Pricing Model) testing and have applications for the fund manager who seeks to rank portfolio performance. This study looks at the GRS-Wald test for portfolio efficiency and extends it to make it visually more interpretive without any loss of generality in its structure. The geometrically recast statistic draws upon the trigonometric properties of a portfolio in the mean-variance space and a mathematical proof of the equivalence of the two statistics is provided. The GRS-Wald test is a widely used statistic in studies addressing the issue of portfolio efficiency and CAPM deviations. A simulation demonstrates the use of the recast GRS-Wald test in testing for the mean-variance efficiency of a test portfolio. The study also provides a table of the GRS-Wald test, based on a range of mean-variance locations (cosine of portfolio angles) at which the test portfolio and the efficient market portfolio can be placed.