Numerical Stability of Optimal Mean Variance Portfolios

In this paper we study the numerical stability of the closed form solution of the classical portfolio optimization problem. Such explicit solution relies upon a technical symmetric square matrix $$\mathbf{A}$$ A , whose dimension is 2 independently from the number of assets considered in the portfolio. We observe that the computation of $$\mathbf{A}$$ A involves the problem’s possible sources of instability, which are essentially related to the estimation and the inversion of the covariance matrix and to the relative scaling of the problem constraints equations, namely the budget and the portfolio expected return constraints respectively. We propose a theoretical approach to minimize the condition number of $$\mathbf{A}$$ A by rescaling both its rows and columns with an optimal constant. Using this result, we substitute the original ill-posed optimization problem with an equivalent well-posed formulation of it. Finally, through a simple empirical example, we illustrate the validity of the proposed approach showing considerable improvements in the stability of the closed form solution.