Mean-shortfall optimization problem with perturbation methods

Abstract

Many researches have been done on portfolio optimization since Markowitz (1952) published a diversified investment model. Markowitz's mean-variance portfolio optimization problem is established under the assumption that the distribution of returns follows a normal distribution. However, in real life, the distribution of returns does not follow a normal distribution, and variance is not a robust statistic as it is heavily influenced by outliers. To overcome these potential issues, mean-shortfall portfolio model was proposed that utilized downside risk, shortfall, as a risk index. In this paper, we propose a perturbation method that uses the shortfall as a risk index of the portfolio. The proposed portfolio utilizes an adaptive Lasso to obtain a sparse and stable asset selection because it can reduce management and transaction costs. The proposed optimization is easily applicable as it can be computed using an efficient linear programming. In our real data analysis, we show the validity of the proposed perturbation method.