The efficient set mathematics when mean-variance problems are subject to general linear constraints

Abstract

In this paper we develop the efficient set mathematics for the case where mean-variance portfolio problems are subject to general linear constraints. We derive closed-form expressions for the optimal portfolio, the associated multipliers, and the portfolio's expected return and variance in each interval where the active set changes; identify where kinks on the efficient frontier occur; and show that securities plot on a security market hyperplane. The analysis extends our knowledge of portfolio theory, clarifies the zero-beta Capital Asset Pricing Model and the conditions under which securities plot on the Security Market Line (SML), and provides insight into the ambiguities associated with using the SML criterion to measure investment performance.