The Efficient Frontier for Weakly Correlated Assets

Abstract

For a general Markowitz portfolio selection problem with linear inequality constraints, it is not possible to obtain a closed form solution. The number of parametric intervals and corresponding segments of the efficient frontier is not known a priori. In this paper, we analyze the structure of the efficient frontier under the assumptions of weakly correlated assets and no short sales constraints. By weakly correlated, we mean the off diagonal elements of the covariance matrix are small relative to the diagonal ones. We obtain an explicit approximate solution for the entire efficient frontier. The error in the approximation is the order of the norm squared of the off diagonal part of the covariance matrix. The assumption of weakly correlated assets is restrictive. However, the explicit approximation of the efficient asset holdings in the presence of bound constraints gives insight into the nature of the efficient frontier. We prove that the efficient frontier is traced out in a monotonic fashion whereby assets are reduced to zero and subsequently remain at zero in order of their expected returns and the number of parametric intervals is equal to the number of assets. This generalizes the results of Best and Hlouskova (Math Methods Oper Res 52:195–212, 2000). The derived structure and approximation are illustrated by a 3-asset example.