Research On Equilibrium Strategy For Dynamic Mean-Variance Problem In Insurance

Insurers,who operate risk,are special financial institutions.On one hand,they need to purchase reinsurance to spread risk.On the other hand,they want to get more profits by investment.Therefore,the research on the optimal reinsurance and investment problems for insurers is becoming a focus in insurance and actuarial science,with high academic values and broad application prospects.The mean-variance portfolio selection problem proposed by Markowitz in 1952 has become one of the foundations of modern finance theory.The mean-variance portfolio selection problem is to seek a best allocation of wealth among a variety of securities so as to achieve the optimal trade-off between the expected return of the investment and its risk over a fixed time horizon.It is apparent to all that the mean-variance criterion lacks the iterated-expectation property,which results in that continuous time/multi-period mean-variance problems are time-inconsistent in the sense that the Bellman Optimality Principle does not hold and hence the traditional dynamic programming approach cannot be directly applied.The optimal policies to dynamic mean-variance problems considered in most literatures are derived under the implicit assumption that the decision makers pre-commit themselves to follow in the future the policies chosen at the initial time,namely,the decision makers initially choose policies to maximize their objective functions at time 0 and thereafter do not deviate from these policies.Such policies are so called pre-commitment policies,which are time-inconsistent in that they are optimal only when sitting at the initial time.However,time consistency of policies is a basic requirement for rational decision making in many situations.A decision maker sitting at time t would consider that,the optimal policy derived at time t should agree with the optimal policy derived at time t + ?t.Equilibrium strategy is time-consistent.Based on existing literatures,by an extended HJB equation and BSDE methods,this paper carries out the following aspects' research on the equilibrium reinsurance and investment strategy for an insurer in a dynamic mean-variance problem.Since the investment decision for an insurer may involve quite a long period,it is reasonable to take the risk of the interest rate into account.Section 2 considers an equilibrium proportional reinsurance-investment strategy for an insurer under stochastic interest rate model and inflation risk.Besides proportional reinsurance,excess-of-loss reinsurance is another important reinsurance form.Section 3 studies an equilibrium excess-of-loss reinsurance-investment strategy for an insurer under stochastic volatility model.Since in many problems,it is difficult to derive the explicit solutions if reinsurance is assumed to be excess-of-loss,we only consider excess-of-loss reinsurance in Section 3.Moreover,mathematical methods have been applied to quantitative finance domain more and more.Scholars have improved the basic model for insurers,and used complicated technical methods to simulate the psychology and behavior for the insurers in real markets.One issue of these behavior finance is to consider the robustness of the optimal strategies when the insurers lack confidence in the parameters estimation,which guards against the worst-case scenario of the estimation.Therefore,Section 4 and 5 introduce model uncertainty into optimal reinsurance and investment problem.Most of literatures consider the optimal reinsurance and investment problems only from the insurer's point of view,and few of them consider the strategy which is optimal or acceptable for both the insurer and the reinsurer.In practice,if we only consider the insurer,the optimal reinsurance strategy obtained is often unacceptable for the reinsurer,which alarms us that there are two parties in a reinsurance contract,and their interests collide with each other.So Section 6makes a reciprocal reinsurance treaty.Section 7 investigates the optimal investment problem of a special insurance,defined contribution pension plan.Above sections assume a Markovian model,to make the model more general,Section 8 extends the problem to a non-Markovian model.There are mainly two differences between Section 8 and other sections.On one hand,Section 2-7 consider a feedback equilibrium strategy,while Section 8 follows the definition of open-loop equilibrium strategy.On the other hand,Section2-7 adopt an extended HJB equation to derive the equilibrium strategy,while Section 8applies the BSDE method.This paper is devoted to making the model and problem more practical,and trying to give the explicit expressions of the equilibrium strategies.Some numerical examples are presented to illustrate our results.