Risk sharing with lambda value-at-risk under heterogeneous beliefs

Abstract In this paper, we study the risk sharing problem among multiple agents using lambda value-at-risk as their preference functional, under heterogeneous beliefs, where each agent’s belief is represented by a probability measure. We obtain semi-explicit formulas for the inf-convolution of multiple lambda value-at-risk measures under heterogeneous beliefs and the explicit forms of the corresponding optimal allocations. To show the impact of belief heterogeneity, we consider three cases: homogeneous beliefs, conditional beliefs, and general beliefs with two agents. For those cases, we find more explicit expressions for the inf-convolution, showing the influence of the relation of the beliefs on the inf-convolution. Moreover, we consider in a two-agent setting the inf-convolution of one lambda value-at-risk and a general risk measure, including expected utility, distortion risk measures and lambda value-at-risk as special cases, with differing beliefs. The expression of the inf-convolution and the form of an optimal allocation are obtained. In all above cases, we demonstrate that trivial outcomes arise when both belief inconsistency and risk tolerance are high. Finally, we discuss risk sharing for an alternative definition of lambda value-at-risk.