Third Degree Stochastic Dominance and Mean-Risk Analysis

In their recent article, Ogryczak and Ruszczy\'nski (1999) proved that those portfolios associated with the efficient frontiers generated by mean-lower semi-standard deviation model and mean- (lower semi-)absolute deviation model are efficient in the sense of second degree stochastic dominance. This rather surprising result reveals the importance of lower partial risk models in portfolio analysis. In this paper, we extend the results of Ogryczak and Ruszczy\'nski for second degree stochastic dominance to third degree stochastic dominance. We show that portfolios on a significant portion of the efficient frontier generated by mean-lower semi-skewness model are efficient in the sense of third degree stochastic dominance. Also, we prove that the portfolios generated by mean-variance-skewness model are semi-efficient in the sense of third degree stochastic dominance.