Generalized Orlicz premia

Abstract

We introduce a generalized class of Orlicz premia based on possibly nonconvex loss functions, extending the classical framework of Haezendonck and Goovaerts (1982). Without the usual convexity requirement on the loss function $\Phi$, the Orlicz framework naturally encompasses quantiles, expectiles and $L^p$-quantiles while preserving the fundamental properties of Orlicz premia. We show that within this framework cash-additivity axiomatizes $L^p$-quantiles, generalizing the classical `collapse-to-the-mean' result for cash-additive convex Orlicz premia into a `collapse-to-$L^p$-quantiles' result, with expectiles as a special case. We focus on two natural classes of nonconvex loss functions: concave-convex Orlicz functions, which mimic the idea of S-shaped value functions in prospect theory, and GA-convex Orlicz functions, which can be described in terms of comparative convexity with respect to a logarithmic reference, and for which the corresponding Orlicz premium is geometrically convex. Finally, we show that a suitable subclass of generalized Orlicz premia coincides with the class of law-invariant, monotone, positive, positively homogeneous, normalized functionals that are weakly lower semicontinuous, continuous from above, and whose level sets are convex with respect to mixtures (the so-called CxLS property).