Robust Return Risk Measures

In this paper we provide an axiomatic foundation to Orlicz risk measures in terms of properties of their acceptance sets, by exploiting their natural correspondence with shortfall risk measures, thus paralleling the characterization in Weber (2006). From a financial point of view, Orlicz risk measures assess the stochastic nature of returns, in contrast to the common use of risk measures to assess the stochastic nature of a position's monetary value. The correspondence with shortfall risk measures leads to several robustified versions of Orlicz risk measures and of their optimized translation invariant extensions (Rockafellar and Uryasev, 2000, Goovaerts et al., 2004), arising from an ambiguity averse approach as in Gilboa and Schmeidler (1989), Maccheroni et al. (2006), Chateauneuf and Faro (2010), or from a multiplicity of Young functions.We study the properties of these robust Orlicz risk measures, derive their dual representations, and provide some examples and applications