We introduce the idea of parametric risk measure and discuss properties of robustness of the risk based preferences and of consistency of these with the usual standard deviation based preferences. Furthermore, we analyze robustness and consistency of mean-risk dominance relation and mean-risk efficient set, providing necessary and sufficient conditions for robustness and consistency of the efficient set generated by the parametric risk measure associated to the W-ruin probability. Among other things, this allows an enlightening re-reading, of the ground breaking de Finetti’s paper on variable quota share proportional reinsurance.
