Risk Sharing with Lambda Value at Risk

In this paper, we study the risk-sharing problem among multiple agents using lambda value at risk ($\mathrm{\Lambda }\text{VaR}$) as their preferences via the tool of inf-convolution, where $\mathrm{\Lambda }\text{VaR}$ is an extension of value at risk ($\text{VaR}$). We obtain explicit formulas of the inf-convolution of multiple $\mathrm{\Lambda }\text{VaR}$ with monotone Λ and explicit forms of the corresponding optimal allocations, extending the results of the inf-convolution of $\text{VaR}$. It turns out that the inf-convolution of several $\mathrm{\Lambda }\text{VaR}$ is still a $\mathrm{\Lambda }\text{VaR}$ under some mild condition. Moreover, we investigate the inf-convolution of one $\mathrm{\Lambda }\text{VaR}$ and a general monotone risk measure without cash additivity, including $\mathrm{\Lambda }\text{VaR}$, 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 $\mathrm{\Lambda }\text{VaR}$ for general Λ functions. Finally, we discuss the risk-sharing problem with $\mathrm{\Lambda }{\text{VaR}}^{+}$, another definition of lambda value at risk. We focus on the inf-convolution of $\mathrm{\Lambda }{\text{VaR}}^{+}$ 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.