Coherent Risk Measures and Stochastic Dominance
The seminal Markowitz portfolio optimization model uses the variance as the risk measure in the mean-risk analysis. The mean-variance model is, in general, not consistent with stochastic dominance rules. Several other risk measures have been later considered thus creating the entire family of mean-risk (Markowitz type) models. Opposite to the mean-variance approach, for general random variables some consistency with the stochastic dominance rules was shown for the Gini's mean difference, for the mean absolute deviation and for many other LP solvable models as well. In this paper we introduce general conditions for risk measures sufficient to provide the SSD consistency of the corresponding models. Actually, we show that under simple and natural conditions on the risk measures they can be combined with the mean itself into the robust optimization criteria thus generating SSD consistent performances (safety) measures. The analysis is performed for general distributions but we also pay attention to special cases such as discrete or symmetric distributions. Recently, a class of coherent risk measures has been defined by means of several axioms. Again, the coherence has been shown for the MAD model and for some other LP computable measures. We too analyze when our conditions guarantee also the coherence of the corresponding performance functions. (fragment of text)
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