A discussion of stochastic dominance and mean-risk optimal portfolio problems based on mean-variance-mixture models

The classical Markowitz mean-variance model uses variance as a risk measure and calculates frontier portfolios in closed form by using standard optimization techniques. For general mean-risk models such closed form optimal portfolios are difficult to obtain. In this note, assuming returns follow the class of normal mean-variance mixture (NMVM) distributions, we obtain closed form expressions for frontier portfolios under mean-risk criteria when risk is modeled by the general class of law invariant convex risk measures. To achieve this goal, we first present a sufficient condition for the stochastic dominance relation on NMVM models and we apply this result to derive closed form solution for frontier portfolios. Our main result in this paper states that when return vectors follow the class of NMVM distributions the associated mean-risk frontier portfolios can be obtained by optimizing a Markowitz mean-variance model with an appropriately adjusted return vector.