Empirical Mean-Variance-Skewness Efficient Frontiers for Skew-Normal Distributions

For Stein’s Lemma extended to multivariate skew-normal variables (see Adcock, 2007) all rational investors select the optimal portfolio on a single mean-variance-skewness efficient surface. The best (subjective) efficient portfolio for a risk-averse and skewness-prone investor will depend on the expected return and the (subjective) trade-off between variance and skewness. We write the general efficient iso-expected return curves for skew-normal efficient portfolios. Then we prove that: (1) a skewness-neutral investor, will select her best efficient skew-normal portfolio corresponding to that of minimum skewness; (2) vice versa a skewness-prone investor, will chose that giving her best balancing trade-off between variance and skewness. After having checked the goodness-of-fit of skew-normality in asset returns, we empirically plot efficient frontiers for optimal portfolios belonging to the surface mean-variance-skewness, getting a confirmation of our theoretical results.