Research On Reinsurance Contracts With Ambiguous Characteristics

Lately,the rapid development of world economy,the increasing active of financial technology innovation,the continuous emergence of new industries bring about the accumulation of wealth,but also lead to new risks in the making and increasing total risk.Insurance companies meet new opportunities for development,as well as face more complicated external risk factors.In order to spread risk,insurance companies have to sign reinsurance contracts with reinsurance companies.Reinsurance companies need to specify rational reinsurance prices to ensure the feasibility of the contracts.Under the logical framework of given reinsurance prices,changing reinsurance prices as history claims and changing reinsurance prices as demands for reinsurance,this thesis builds rigorous mathematical finance models to study the design p roblems of reinsurance contracts.First,assume that the reinsurance price is given,and we investigate the optimal reinsurance and investment strategies of an insurance company.Investment can make insurance fund preserve and increase value,and reinsuran ce can transfer claims risk.Both of them are indispensable parts of fund management chain of insurance companies.Current literatures deem that combining investment and reinsurance of insurers are necessary.Chapter 2 investigates the insurer's optimal excess loss reinsurance and investment maximizing the expected utility of terminal wealth under a jump claim model and an approximation diffusion claim model,respectively.This study verifies that approximating the jump process with the diffusion process ca uses a large deviation of the reinsurance decisions,for the reinsurance contract with a high relative safety loading of reinsurance.More importantly,the optimal reinsurance and investment strategies are independent of each other,if the claims process is independent of the price process of the risky asset.Therefore,the optimal reinsurance and investment are two independent optimization problems for the in surer.Second,assume that the reinsurance price changes with history claims.The bonusmalus contract is the most common of contract form: a client is offered a reduction in premium in the next phase as a reward,if the money amount of his claims does not surpass some prespecified quantity;otherwise,he is offered an increase in premium in the next phase as a punishment.Chapter 3 of this paper proposes a continuous-time claim model with extrapolative expectation: using the exponential weighting of the already occurred claims and the initial estimated claim expe ctation builds the dynamic updating expectation of the claim model.The extrapolator thinks that if the recent claims increase or decrease,then they will keep this increasing or decreasing trend.This model makes insurance premiums computed by the expected value principle have the characteristics of rewards and punishments,and satisfies the psychological characteristics of behavior individual giving more attentions to recent claims.This study shows that,unlike the standard risk retention without extrapolative bias,the extrapolative risk retention level increases rather than decreasing with respect to decision time.In addition,the extrapolative expectation provides a kind of protection similar to reinsurance for the insurer.When the extrapolative expectation increases,the insurer reduces demand for reinsurance.Third,assume that the reinsurance price changes as the demands and model ambiguity.A reinsurance contract involves the insurer's and the reinsurer's interests.Although the extrapolative expectation introduced in chapter 3 results in changing insurance premiums,it only considers the insurer's profit to some extent,because the price of unit risk does not change as demands.Moreover,the studies of the first two chapters assume that the insurer only faces risk.Risk refers to a situation where the state variable has a known probability distribution.However,ambiguity refers to a situation where the distribution function of the state variable is uncertain.Both of them generate different effects on decision makers' behaviors.Reinsurance commonly underwrites catastrophe risks,which have limited history data.Limited data makes a single probability measure cannot precisely describe the real claim process.Therefore,finding suitable methods to design robust reinsurance contracts,which can effectively protect the insurer and the reinsurer under model uncertain conditions,is critical.Chapters 4,5 and 6 introduce model ambiguity and combine the both sides' profits of the insurer and the reinsurer to study robust proportional and excess loss reinsurance contracts using the principal-agent theory.The principal-agent theory gives the general method to determine the relative safety loading of reinsurance.This thesis adopts the relative safety loading to characterize the reinsurance price.The reinsurer designs robust reinsurance prices to maximize his own utility,subject to the insurer's incentive compatibility constraints.Chapter 4 assumes that claims follow a diffusion process with certain drift,and studies the robust proportional reinsurance contracts,when the insurer and the reinsurer respective face belief distortions and both of them face belief distortions at the same time.Conclusions show that the reinsurer with decision-making power will not passively accept fixed prices and decreasing demands.He dynamically lowers the reinsurance price which induces the insurer's proportional reinsurance demand to be constants.Ambiguity-averse decision makers believe that true claims are greater than the reference values.Thus,the ambiguity-averse reinsurer raises prices,and the ambiguity-averse insurer raises demands.Specifically,when ambiguity aims at a diffusion process,the insurer's and the reinsurer's ambiguity aversions are equivalent to increase their respective effective risk aversions.Moreover,ignoring the effects of model ambiguity on the reinsurance contract will give rise to large utility loss.At last,suppose that insurance claims satisfy a jump process with uncertain jump intensity.We study the robust proportional and excess loss reinsurance contracts,when the reinsurer and the insurer are ambiguity averse,respe ctively.This study finds that,although changes of the insurer's or the reinsurer's risk and ambiguity aversions result in the same trends of the optimal reinsurance contracts,they are no longer equivalent;the prices of the proportional and excess loss reinsurance dynamically decrease,and the demands for reinsurance are not constants but decrease as time;the price of the excess loss reinsurance is higher than that of the proportional reinsurance.When model uncertainty and ambiguity aversion exist,the low risk aversion of the insurer or the high risk aversion of the reinsurer leads the proportio nal reinsurance contract to dominate the excess loss reinsurance contract for the insurer;otherwise,the opposite result holds.