A continuous-time theory of reinsurance chains

A chain of reinsurance is a hierarchical system formed by the subsequent interactions among multiple (re)insurance agents, which is quite often encountered in practice. This paper proposes a novel continuous-time framework for studying the optimal reinsurance strategies within a chain of reinsurance. The transactions between reinsurance buyers and sellers are formulated by means of Stackelberg games, in order to reflect the conflicting interests and unequal negotiation powers in the bargaining process. Assuming the variance premium principle and the mean–variance criterion on the surplus processes, we solve the time-consistent optimal reinsurance demands and pricing strategies in explicit forms, which are surprisingly plain. Based on the proposed reinsurance chain models, our in-depth theoretical analysis shows that: (a.) it is optimal to situate more (resp. less) risk averse reinsurers to the latter (resp. former) positions in a chain of reinsurance; (b.) adding new reinsurers will lower the reinsurance prices at all levels in a chain of reinsurance, promoting the existing agents to rationally control their respective risk exposures; and essentially (c.) alleviate the systemic risk in the chain structure.