Research On Optimal Reinsurance Problem Of Insurance Companies

Reinsurance and investment are the main ways for the insurers to manage risk.In recent years,optimal reinsurance and investment has became a hot topic in the actuarial field.In reality,insurance companies generally sign a large number of policies with policyholders,which causes the policyholders to transfer the risks to the insurance companies.Thus,insurance companies are often exposed to large risks.Once huge claims arise,insurance companies may be unable to pay the policyholders’ losses,which could lead to ruin.Generally,insurance companies can purchase reinsurance to transfer risks to reinsurance companies.Therefore,it is crucial to allocate losses appropriately between insurance companies and reinsurance companies.Numerous studies have investigated optimal(re)insurance design in both static and dynamic frameworks,which typically assume that the insurer and the reinsurer share the same probabilistic beliefs about the loss distribution.However,this above assumption has long been criticized for its inconsistency with market practice,and empirical evidence shows that the two parties usually have different private information about the underlying loss.In many situations,the insurer and the reinsurer may assign different likelihoods to the realizations of the insurable random loss.Thus,it is natural to take into consideration belief heterogeneity in a decision-making problem.Moreover,most research on investment and/or reinsurance optimization problems assumes that the model is deterministic,that is,the decision maker knows the exact true probability measure for the optimal investment and/or reinsurance problem.However,the controversial aspect is which model is absolutely correct in representing the probability of the real world.In fact,model uncertainty is widespread in financial markets,especially in consumption,asset pricing,and portfolio selection.However,since the intensity of claim arrivals and the distribution of claims cannot be estimated precisely,the insurance market suffers from the same model uncertainty problem.It is very reasonable for decision makers to consider model ambiguity in the insurance market when making decisions.In addition,the optimal reinsurance strategy in most of the literature is unilaterally determined by the insurer.However,in practice,the optimal arrangement is jointly decided by the insurer and the reinsurer when the reinsurance contract is signed.As Borch(1969)indicated: “There are two parties to a reinsurance contract,and an arrangement which is very attractive to one party may be quite unacceptable to the other.” Therefore,it is necessary to consider the interests of both insurers and reinsurers for the choice of reinsurance strategies.How should insurance companies reasonably purchase reinsurance under the above three situations ?To address the above phenomenon,we construct models and consider the reinsurance/reinsurance investment problem under different assumptions.Specifically,the research is carried out from the following four aspects.Firstly,heterogeneous beliefs about the risk process are measured by the MLR orders.In the monotonic likelihood ratio order,we research the reinsurance optimization problem when maximizing the expectation of terminal wealth utility.Secondly,considering the ambiguity aversion of insurers to claim arrival intensity,we adopt a robust approach to study the worst-case optimal reinsurance strategy to minimize the discounted ruin probability.Thirdly,from the perspective of both the insurer and the reinsurer,we study the optimal reinsurance strategy jointly determined by the insurer and the reinsurer to maximize the mean-variance utility under the stackelberg game framework.Fourthly,taking into account the interests of both insurers and reinsurers as well as the larger market share of reinsurance,we study the robust optimal investment reinsurance problem of insurers under the preferred reinsurance levels of the reinsurer.This thesis focuses on the design of optimal reinsurance investment for insurance companies.This thesis consists of 6 chapters.Chapter 1 is the introduction.Chapters 2-5 are the main content of the paper.Chapters 2 and 3 examine the optimal reinsurance strategy from the insurer’s perspective under different model assumptions and objective settings,respectively.Besides considering the interest of insurer,Chapters 4 and 5 take into account the interest of reinsurer,and study the optimal reinsurance strategy from the perspective of the interests of both parties.Chapter 6 is conclusion.The specific structure of the paper is organized as follows.Chapter 1.Introduction.We introduce the research background and significance,innovations,and a literature review related to the reinsurance optimization problem.Chapter 2.Optimal dynamic reinsurance under heterogeneous beliefs and CARA utility.We examines the optimal dynamic reinsurance policy for an insurance company under belief heterogeneity.We assume the reinsurance premium is calculated according to the mean-CVa R principle and impose the incentive compatibility constraint to rule out moral hazard.Under the objective of maximizing the exponential utility function,we obtain the optimal strategies in closed form via a “relaxation and modification” approach.The optimal contracts have more complicated structures than the standard proportional and excess-ofloss reinsurance widely investigated in the literature.In particular,we demonstrate that the insurer may optimally choose to purchase proportional reinsurance in different layers when the reinsurer is more pessimistic about the underlying loss.Our model lends support to the observation that reinsurance contracts in practice often involve proportional reinsurance in multiple layers.We also demonstrate that belief heterogeneity can explain the inverse relationship between the purchase of reinsurance and the size of the insurer’s losses observed in the reinsurance market.Chapter 3.Robust optimal dynamic reinsurance policies under the mean-RVa R premium principle.This work investigates a robust optimal dynamic reinsurance problem for an ambiguity averse insurer(AAI)concerned about potential ambiguity of claim intensity.The study aims to determine a robust optimal reinsurance contract to minimize the discounted ruin probability imposed a penalization owing to model ambiguity,including discounting for the time of ruin.We suppose that the surplus process is modeled by a diffusion model.The AAI purchases reinsurance from the reinsurer to manage risk,subject to the incentive compatibility constraint,to reduce moral hazard.Moreover,the reinsurance premium is calculated based on the mean-RVa R premium principle,which generalizes the expected value premium and the mean-CVa R premium and reflects the different risk preferences of reinsurers.Based on the dynamic programming approach,we obtain the value function and optimal reinsurance policies(the dual excess-of-loss reinsurance).Finally,we show a numerical example to illustrate the effects of premium and ambiguity averse on the discounted ruin probability and the robust optimal contract.Chapter 4.Optimal α-weighted reinsurance strategy under a stochastic Stackelberg game framework.We examines a stochastic Stackelberg differential reinsurance game between an insurer(follower)and a reinsurer(leader),wherein the reinsurance policy is jointly determined by both game players under the mean-variance criterion.To describe the joint decisions of both the insurer and reinsurer,the reinsurance strategy is determined by a weighted average of each player’s risk tolerance,which is called an α-weighted reinsurance strategy.Considering the incentive compatibility for both players,we explicitly develop the optimal time-consistent equilibrium reinsurance strategy,which is a more generalized form than the classical proportional or excess-of-loss reinsurance.Chapter 5.Robust optimal investment-reinsurance strategies with the preferred reinsurance level of reinsurer.We investigates robust equilibrium investment-reinsurance strategy for a mean–variance insurer.With a larger market share,a reinsurer has a greater say in negotiating reinsurance contracts and makes the decision to propose the preferred level of reinsurance and charges extra fees as a penalty for losses that deviate from the preferred level of reinsurance.Once the insurer receives a decision from the reinsurer,the insurer weighs its risk-bearing capacity against the cost of reinsurance in order to find the optimal investmentreinsurance strategy under the mean-variance criterion.The insurer who is ambiguity averse to jump risk and diffusion risk obtains a robust optimal investment-reinsurance strategy by dynamic programming principle.Moreover,reinsurance strategy is no longer stop-loss reinsurance or proportional reinsurance.In particular,the insurer may purchase proportional reinsurance for different ranges of loss and the proportion depends on the extra charge rate,which is more consistent with market practice than the standard excess-of-loss reinsurance and proportional reinsurance.We find the optimal reinsurance policy depends on the degree of ambiguous aversion to jump risk and not on the degree of ambiguous aversion to diffusion risk.Chapter 6.Conclusion.We summarize the main findings of this thesis.The novelties of this thesis are as follows.1.We explore the optimal reinsurance design under belief heterogeneity in the dynamic context.Allowing the insurer and the reinsurer to have heterogeneous beliefs about the risk process,as measured by the MLR orders.Although there is a large literature that explores the optimal reinsurance design under belief heterogeneity in the static context,very few of them investigate this problem under increasing LRs due to the technical difficulty.In this paper,we use the Esscher transform to model the reinsurer’s and the insurer’s heterogeneity beliefs about the underlying loss and derive the optimal solution analytically.2.We allow the insurer to choose reinsurance strategies from a wide class of reinsurance contracts that satisfy the so-called incentive compatibility constraint and includes the proportional reinsurance and excess of loss reinsurance.This constraint can rule out ex moral hazard.In other words,it can avoid insurers from misreporting losses and getting more compensation from reinsurers.Moreover,we adopt the mean-RVa R premium principle that extends the mean-CVa R premium principle proposed by Tan et al.(2020)and satisfies different risk preferences of reinsurers by adjusting the parameters.3.We innovatively adopt a “relaxation and modification” approach to solve the HamiltonJaboian-Bellman(HJB)equation in the presence of belief heterogeneity,mean-CVa R principle,and the incentive compatibility constraint.Specifically,we first relax the incentive compatibility constraint and derive the corresponding “relaxed” solution.We then modify the “relaxed” solution to satisfy the incentive compatibility constraint.This method transforms the original infinite-dimensional optimization problem into a finite-dimensional(three-dimensional)optimization problem.Furthermore,This approach is intuitive and allows for a graphical illustration.4.We propose a new reinsurance strategy called an α-weighted reinsurance strategy.In most of the literature,the reinsurance strategy is controlled by the insurer,and the reinsurance premium is decided by the reinsurer.The safety loading of the reinsurance premium is usually considered as a control variable determined by the reinsurer.However,in practice,the final reinsurance strategy is agreed upon by both parties when the reinsurance contract is signed.Therefore,instead of finding the optimal safety loading,we focus on designing the loss structure of the reinsurance contract.When transferring risks,an insurer cedes part of the losses to a reinsurer according to the insurer’s risk tolerance and insurance demand.However,the reinsurer may not accept losses ceded by the insurer considering its underwriting capacity.To weigh the needs of both parties,we propose a new reinsurance strategy called an α-weighted reinsurance strategy,which considers both the insurer’s insurance demand and the reinsurer’s underwriting capacity.5.Innovatively,we give a scheme for insurers and reinsurers to determine reinsurance contracts.We assume that the reinsurance strategy is determined in the following two steps,taking into account the interests of both parties and the market share of the reinsurer.First,the reinsurer can derive the preferred reinsurance level under mean variance criterion as the benchmark.Second,the insurer,as the other party to the reinsurance contract,cedes the loss to the reinsurer and assign a portion of the premium to the reinsurer.If the ceded loss by the insurer exceeds or falls below the preferred reinsurance level of the reinsurer,the reinsurer will charge additional premiums as a penalty,which depends on the degree of deviation from the reinsurance strategy of both parties.Based on the preferred reinsurance level of the reinsurer,the insurer searches for his own optimal reinsurance strategy with penalty function.