A new risk measure, the Lambda Value-at-Risk (VaR), was proposed from a theoretical point of view as a generalization of the ordinary VaR in the literature. Motivated by the recent developments in risk sharing problems for the VaR and other risk measures, we study the optimization of risk sharing for the Lambda VaR. Explicit formulas of the inf-convolution and sum-optimal allocations are obtained with respect to the left Lambda VaRs, the right Lambda VaRs, or a mixed collection of the left and right Lambda VaRs. The inf-convolution of Lambda VaRs constrained to comonotonic allocations is investigated. Explicit formula for worst-case Lambda VaRs under model uncertainty induced by likelihood ratios is also given.

This paper introduces a flexible risk decomposition method for life insurance contracts embedding several risk factors. Hedging can be naturally embedded in the framework. Although the method is applied to variable annuities in this work, it is also applicable in general to other insurance or financial contracts. The approach relies on applying an allocation principle to components of a Shapley decomposition of the gain and loss. The implementation of the allocation method requires the use of a stochastic on stochastic algorithm involving nested simulations. Numerical examples studying the relative impact of equity, interest rate and mortality risk for guaranteed minimal maturity benefit (GMMB) policies conclude our analysis.

This paper discusses the accountability gap problem posed by artificial intelligence. After sketching out the accountability gap problem we turn to ancient Roman law and scrutinise how slave-run businesses dealt with the accountability gap through an indirect agency of slaves. Our analysis shows that Roman law developed a heterogeneous framework in which multiple legal remedies coexist to accommodate the various competing interests of owners and contracting third parties. Moreover, Roman law shows that addressing the various emerging interests had been a continuous and gradual process of allocating risks among different stakeholders. The paper concludes that these two findings are key for contemporary discussions on how to regulate artificial intelligence.