Optimal Reinsurance Designs Under Several Risk Measures

Nowadays, more and more scholars have focus on "Optimal reinsurance de-signs" which is an important topic in actuarial science. In some classic reinsur-ance models, the optimal ceded loss functions can be characterized accurately. In this thesis, we devote ourself deeply into the conclusions of former scholars, and promote them. Moreover, we propose several optimal reinsurance models which may be of interest in economics.First, for the case of single risk, we study the optimal strategies under the Haezendonck risk measure and the GlueVaR risk measure. The Haezendonck risk measure is generated by the convex function and it has the property of "sub-additivity", meanwhile, the GlueVaR risk measure is a linear combination of VaR and CVaR. In order to minimize the total risk of the insurer under these two risk measures, the optimal ceded loss function should have the form of the surplus reinsurance or the two-layer reinsurance respectively.Second, in the case of multivariate risks, we promote the "minimax model" in Cheung et al. (2014) to make it suitable for the premium principles which satisfy the stop-loss order or have the form of variance related. In insurance practice, the minimax model is able to be understood and applied easily. Furthermore, by the inspiration of multivariate VaR risk measure, we propose a reinsurance model which can minimize the total capital requirements. In this model, not only we find the optimal ceded loss functions for each risk, but also the capital requirements of the insurer can be minimized. All of the ceded loss functions are the layer reinsurance under these two types of models.Third, we study the case of which the insurer cedes the risk to several reinsur-ers. Assume that the insurer and all of the reinsurers apply coherent distortion risk measures. The necessary and sufficient condition for reaching the Pareto optimality is that, every ceded loss function has the form of the multi-layer rein-surance.In conclusion, the research process in this work is from the view of micro to macro, from the reinsurance models for single risk to the overall reinsurance of the insurer and the optimal risk management strategies, until the equilibrium of the global reinsurance market. Consequently, we hope the readers can enjoy the optimal reinsurance design through this work.