Preference Robust Distortion Risk Measure and Its Application

Distortion risk measure (DRM) plays a crucial role in risk measuring and managing, especially in insurance pricing. Various DRMs have been introduced but little is discussed about which DRM at hand should be chosen to address a decision maker (DM)'s risk attitude. This paper aims to fill out the gap. Specifically we consider a situation where the true distortion function is unknown either because it is difficult to identify/elicit and/or because the DM's risk attitude is ambiguous. We introduce a preference robust distortion risk measure (PRDRM) which is based on the worst-case distortion function from an ambiguity set of distortion functions to mitigate the impact arising from the ambiguity. The ambiguity set is constructed under well-known general principals such as concavity and inverse S-shapedness of distortion functions (over-weighting on events from impossible to possible or possible to certainty and under-weighting on those from possible to more possible) as well as new user-specific information such as sensitivity to tail losses, confidence intervals to some lotteries, and preferences to certain lotteries over others. To calculate the proposed PRDRM, we use the convex and/or concave envelop of a set of points to characterise the curvature of the distortion function and derive a tractable reformulation of the PRDRM when the underlying random loss is discretely distributed. Moreover, we show that the worst-case distortion function is a non-decreasing piece-wise linear function and can be determined by solving a linear programming problem. Finally, we apply the proposed PRDRM to a risk capital allocation problem and carry out some numerical tests to examine the efficiency of the PRDRM model