Optimal reinsurance under the linear combination of risk measures in the presence of reinsurance loss limit

Optimal reinsurance problems under the risk measures, such as Value-at-Risk (VaR) and Tail-Value-at-Risk (TVaR), have been studied in recent literature. However, losses based on VaR may be underestimated and TVaR allows us to account better for catastrophic losses. In this paper, we propose a new family of flexible risk measures denoted by LVaR, which is a weighted combination of VaR and TVaR. Based on the new risk measures, we deal with the optimal reinsurance problem by minimizing the LVaR of the total risks of an insurer when two types of constraints for reinsurer's risk exposure are considered. The results indicate that the two-layer reinsurance is always an optimal reinsurance policy with both types of constraints. Also, we find that the optimal reinsurance policy depends on the confidence level, the weight coefficient, the safety loading, the tolerance level, as well as the relations between them. Finally, we illustrate the results by numerical examples and compare them with the results in Lu et al.