Mixed Band Control of Mutual Proportional Reinsurance

Abstract

In this paper, we investigate the optimization of mutual proportional reinsurance-a mutual reserve system that is intended for the collective reinsurance needs of homogeneous mutual members, such as P&I Clubs in marine mutual insurance and reserve banks in the US Federal Reserve, where a mutual member is both an insurer and an insured. Compared to general (non-mutual) insurance models, which involve one-sided impulse control (i.e., either downside or upside impulse) of the underlying insurance reserve process that is required to be positive, a mutual insurance differs in allowing two-sided impulse control (i.e., both downside and upside impulse), coupled with the classical proportional control of reinsurance. We prove that a special band-type impulse control   , , , a A B b a with and a A coupled with a proportional reinsurance policy (classical control), is optimal when the objective is to minimize the total maintenance cost. That is, when the reserve position reaches a lower boundary of , the reserve should immediately be raised to level A; when the reserve reaches an upper boundary of b, it should immediately be reduced to a level B. An interesting finding produced by the study reported in this paper is that there exists a situation such that if the upside fixed cost is relatively large in comparison to a finite threshold, then the optimal band control is reduced to a downside only (i.e., dividend payment only) control in the form of   , B b 0, 0; with 0 a A   . In this case, it is optimal for the mutual insurance firm to go bankrupt as soon as its reserve level reaches zero, rather than to jump restart by calling for additional contingent funds. This finding partially explains why many mutual insurance companies, that were once quite popular in the financial markets, are either disappeared or converted to non-mutual ones.

Optimal combined dividend and reinsurance policies under interest rate in Lévy markets

“…It can be shown Liang and Huang, 2011;Liang and Sun, 2011;Schmidli, 2008) that the limiting Lundberg model for large portfolios can be approximated by the diffusion process. Due to the unrealistic nature of the Lundeberg model, several extensions have been proposed and investigated in both the diffusion model setting and the Lundeberg setting, see for example, , Choulli et al (2001), Taksar (1998b, 1998c), Taksar et al (2013), Tsoularis (2014) and Zhang (2012). An excellent survey of recent works on optimal dividend control policies can be found in Asmussen and Taksar (1999), Choulli et al (2001), Hojgaard and Taksar (1998a) and references therein.…”