Lambda Value-at-Risk under ambiguity and risk sharing

Speaker: Alexander Schied, Professor and Munich Re Chair in Stochastic Finance, Dept. of Statistics and Actuarial Science, University of Waterloo

Abstract: We investigate Lambda Value-at-Risk (ΛVaR) under ambiguity, where the ambiguity is represented by a family of probability measures. We establish that for increasing Lambda functions, the robust (i.e., worst-case) ΛVaR under such an ambiguity set is equivalent to ΛVaR computed with respect to a capacity, a novel extension in the literature. This framework unifies and extends both traditional ΛVaR and Choquet quantiles (Value-at-Risk under ambiguity). We analyze the fundamental properties of this extended risk measure and establish a novel equivalent representation for ΛVaR under capacities with monotone Lambda functions in terms of families of downsets. Moreover, explicit formulas are derived for robust ΛVaR when ambiguity sets are characterized by ϕ-divergence and the likelihood ratio constraints, respectively. We further explore the applications in risk sharing among multiple agents. We demonstrate that the family of risk measures induced by families of downsets is closed under inf-convolution. In particular, we prove that the inf-convolution of ΛVaR with capacities and monotone Lambda functions is another ΛVaR under a capacity. The explicit forms of optimal allocations are also derived. Moreover, we obtain more explicit results for risk sharing under ambiguity sets characterized by ϕ-divergence and likelihood ratio constraints.