Risk Sharing with Lambda Value at Risk

In this paper, we study the risk-sharing problem among multiple agents using lambda value at risk ( Λ VaR ) as their preferences via the tool of inf-convolution, where Λ VaR is an extension of value at risk ( VaR ). We obtain explicit formulas of the inf-convolution of multiple Λ VaR with monotone Λ and explicit forms of the corresponding optimal allocations, extending the results of the inf-convolution of VaR . It turns out that the inf-convolution of several Λ VaR is still a Λ VaR under some mild condition. Moreover, we investigate the inf-convolution of one Λ VaR and a general monotone risk measure without cash additivity, including Λ VaR , expected utility, and rank-dependent expected utility as special cases. The expression of the inf-convolution and the explicit forms of the optimal allocation are derived, leading to some partial solution of the risk-sharing problem with multiple Λ VaR for general Λ functions. Finally, we discuss the risk-sharing problem with Λ VaR + , another definition of lambda value at risk. We focus on the inf-convolution of Λ VaR + and a risk measure that is consistent with the second-order stochastic dominance, deriving very different expression of the inf-convolution and the forms of the optimal allocations.