Lambda Value-at-Risk under ambiguity and risk sharing

TL;DR

This paper extends Lambda Value-at-Risk (ΛVaR) to ambiguous probability measures, unifies it with Choquet quantiles, and explores its properties and applications in risk sharing among multiple agents.

Contribution

It introduces a novel extension of ΛVaR under ambiguity using capacities, providing explicit formulas and analyzing risk sharing and comonotonicity properties.

Findings

Robust ΛVaR under ambiguity equals ΛVaR with capacities for increasing Lambda functions.

The family of risk measures is closed under inf-convolution, enabling risk sharing analysis.

Explicit formulas are derived for ΛVaR under φ-divergence and likelihood ratio constraints.

Abstract

In this paper, we investigate the Lambda Value-at-Risk ($\Lambda$VaR) 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) $\Lambda$VaR under such an ambiguity set is equivalent to $\Lambda$VaR computed with respect to a capacity, a novel extension in the literature. This framework unifies and extends both traditional $\Lambda$VaR 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 $\Lambda$VaR under capacities with monotone Lambda functions in terms of families of downsets. Moreover, explicit formulas are derived for robust $\Lambda$VaR when ambiguity sets are characterized by $\phi$-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 $\Lambda$VaR with capacities and monotone Lambda functions is another$\Lambda$VaR 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 $\phi$-divergence and likelihood ratio constraints. Finally, we explore comonotonic risk-sharing for $\Lambda$VaR under ambiguity.