Mean-Extended Gini Portfolios: A 3D Efficient Frontier

Using a numerical optimization technique we construct the mean-extended Gini (MEG) efficient frontier as a workable alternative to the mean-variance efficient frontier. MEG enables the introduction of specific risk aversion into portfolio selection. The resulting portfolios are stochastically dominant for all risk-averse investors. Solving for MEG portfolios allows investors to tailor portfolios for specific risk aversion. The extended Gini is calculated by the covariance of asset returns with a weighing function of the cumulative distribution function (CDF) of these returns. In a sample of asset returns, the CDF is estimated by ranking returns. In this case, analytical optimization techniques using continuous gradient approaches are unavailable, thus the need to develop numerical optimization techniques. In this paper we develop a numerical optimization algorithm that finds the portfolio optimal frontier for arbitrarily large sets of shares. The result is a 3-dimension MEG efficient frontier in the space formed by mean, the extended Gini, and the risk aversion coefficient.