Analytically Deriving Efficient Surfaces in Portfolio Selection

Abstract

We consider the practical issue - how an investor incorporates multidimensional risks from factor models directly into portfolio selection, and formulate the issue by multiple objective portfolio selection. Then we analytically derive efficient surfaces in multiple objective portfolio selection and demonstrate the properties and financial implications, as an extension of Merton's analytically deriving efficient frontier. Merton's 2-mutual-fund theorem and parabolic structure of efficient frontier are extended into 3-mutual-fund theorem and paraboloidic structure, respectively. The efficient portfolios of traditional portfolio selection are still efficient in the new setting. Furthermore, market portfolio is efficient if all investors hold efficient portfolios. We also show the results hold for general k-objective portfolio selection model. Traditional portfolio selection can be taken as the projection onto mean-variance space from multiple criteria portfolio selection. In the process of projecting an efficient surface, we show the boundary of the projection is exactly Merton's efficient frontier. Interestingly, market portfolio probably loses its efficiency in the projection. The empirical evidence that market indices are deep below efficient frontier can be explained by multiple objective portfolio selection and the projection. As an advantage of multiple objective portfolio selection, the tradeoffs between expected return and multidimensional risks can be clearly and precisely described by the curvature of an efficient surface. A new avenue to incorporate explicitly multidimensional risks into portfolio selection and monitor directly the tradeoffs can be opened. Moreover, this paper can help an investor visualize portfolio selection in 3 dimensional space and even higher dimensional spaces. Meanwhile, we acknowledge that the linear modelling of portfolio liquidity and other risks is overly simplistic, because the risks can be highly nonlinear and time-varying. Our intention is to enrich traditional portfolio selection and delineate efficient surfaces.