Uncertainty of Efficient Frontier in Portfolio Optimization

Abstract

Portfolio optimization is a large area of investigation both in theoretical and practical setting since the seminal work by Markowitz where a mean-variance model was introduced. From optimization point of view, the problem of optimal portfolio in mean-variance setting can be formulated as convex quadratic optimization under uncertainty. In practice one needs to estimate parameters of the model to find an optimal portfolio. Error in the parameter estimation generates error in the optimal portfolio. It was observed that the out of sample behavior of obtained solution is not in accordance with what is expected. Main reason for this phenomena is related with the estimation of means of stock returns, estimation of covariance matrix being less important. In the present paper we study uncertainty of identification of efficient frontier (Pareto optimal portfolios) in mean-variance model. In order to avoid the estimation of means of returns we use CVaR optimization method by Rockafellar and Uryasev. First we prove, that for a large class of elliptical distributions efficient frontier in mean-variance model is identical to the trajectory of CVaR optimal portfolios with the change of the confidence level. This gives an alternative way to recover efficient frontier in mean-variance model. Next we conduct a series of numerical experiments to test the proposed approach. We show that proposed approach is competitive with existing methods.