  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, we obtain closed form
expression for frontier portfolios under mean-risk criteria when risk is
modelled by any finite law-invariant convex measures of risk and when return
vectors follow the class of normal mean-variance mixture (NMVM) distributions.
To achieve this goal, we first present necessary as well as sufficient
conditions for stochastic dominance within the class of one dimensional NMVM
models and then we apply them to portfolio optimization problems. 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.
