LP tests for MV efficiency

We derive empirical tests for the mean-variance efficiency of a given portfolio. The tests can be computed using straightforward linear programming, and they give substantial flexibility in modeling the investment possibilities. Using this test, we can reject the hypothesis that the S&P 500 index is mean-variance efficient relative to the 25 Fama and French (1993) equity portfolios. Mean-variance analysis (MVA; Tobin (1958), Markovitz (1952, 1959)), is the dominant framework for analyzing investment behavior. It is useful both for positive analysis (where the objective is to analyze the decision rules actually used by decision-makers) as well as in normative analysis (where the objective is to support practical decision making). Applying MVA criteria to empirical data typically requires solving largescale quadratic programming (QP) models. QP problems are solvable in polynomial time using interior-point methods (see e.g. Nesterov and Nemirovskii (1994)). However, for large problem, involving hundreds or thousands of assets and/or investment restrictions, the computational burden can be prohibitive. To reduce the computational burden, various alternatives to MVA use alternative, linear measures of risk. For example, Yamakazi and Konno (1991) present a model based on Mean Absolute Deviation (MAD), and Young (1998) presents a model based on the minimum return (or maximum loss). Both models require straightforward Linear Programming (LP), and hence are computationally more attractive than MVA. This paper extends this literature by presenting LP tests based on the Hanoch and Levy (HL; 1970) definition of mean-variance (MV) efficiency. The standard definition of MV efficiency classifies a portfolio as efficient if and only if no other portfolios exist with a higher mean and a lower variance. This definition is consistent with the expected utility theory (EUT) if the probability distribution of return has a normal shape or, more generally, an elliptical shape (see e.g. Bigelow (1993) for a complete characterization of the necessary and sufficient conditions). By contrast, our tests are derived from an alternative MV efficiency definition by Hanoch and Levy (HL; 1970): a portfolio is efficient if and only if there exists an increasing and concave, quadratic utility function that rationalizes the portfolio. The HL definition is more powerful than the standard definition; the HL efficient set is subset of the standard MV efficient set (see e.g. HL, Levy and Hanoch (1970), and the Corollary to Theorem 2 in Section III). Further, the HL efficient set is a proper subset of the efficient set of secondorder stochastic dominance (SSD; see e.g. Hadar and Russel (1969) and Hanoch and Levy (1969)), which considers the general class of increasing and concave utility functions. HL and Meyer (1979) present necessary and sufficient conditions for testing if a given portfolio is MV efficient in the HL definition. As is true for the standard MV * Post is with the department of Finance of the Erasmus University Rotterdam in the Netherlands. He gratefully acknowledges financial support by Tinbergen Institute and Erasmus Research Institute of Management.