Research On Optimal Reinsurance And Investment Polices For Insurers Under Complicated Financial Models

The insurer transfers partly it’s claims loss to the second carrier so as to diversification of risk and stability of operation. Meanwhile, investment business play an important role for raising solvency and increasing profits of insurance companies. Therefore, optimal risk management by controls of reinsurance and investment in actuarial science has been a hot topic recently. In past few years, stochastic control theory and techniques have been widely used in applications of investment and reinsurance problems.In this paper, the claims process of an insuer is assumed that follows a compound Poisson process or a Brownian motion with drift. Optimal reinsurance and investment polices under complicated financial models are studied by stochastic process theory and stochastic control method. This paper mainly include the contents below.First, suppose that the insuer faces two lines insurance businesses which have heterogeneous risks. For insurance claims exhibiting large fluctuations, the variance premium principle and proportional reinsurance are more suitable. Whereas, for claims exhibiting central tendency, the expected premium principle and excess of loss reinsurance should work well. Under maximization of adjustment coefficient and utility of terminal wealth respectively respectively, we obtain the optimal reinsurance polices.The results present that the role of adjustment coefficient in expressions of the polices is similar to risk aversion parameter’s.Second, we model the price of risky asset with CEV process which can capture stochastic volatility. The insuer invests it’s surplus in CEV stock, and purchases proportional(or excess-of-loss) reinsurance. Using stochastic theory, we obtain the expressions of optimal polices and value function under maximizing the utility of terminal wealth. The results show that the first fraction of investment policy is analogous to Merton investment policy, and the second term adjusts this investment behavior according to the time, the elasticity and the price level.Third, the price of risky asset is described by exponential mean reversion process which can capture risky asset’s characteristic of mean reversion. The insuer invests it’s surplus in exponential mean reversion financial market, and purchases combining quota-share and excess of loss reinsurance. Under maximizing the utility of terminal wealth, the reinsurance policy is always pure excess of loss, and investment policy is Merton investment policy adding another term which adjusts investment behavior according to the speed of mean reversion, the time and the price.Fourth, we model the price of risky asset with Lévy process which can describe the jumps of the price. Under a generalized mean-variance criterion, which leads to wealth-path dependent optimization, we discuss the optimal investment and reinsurance polices by using stochastic control theory and Lagrange multiplier method. And we get the variance minimizing frontier and efficient frontier in the classical mean-variance model. The results show that investment policy is dependent on the risk premium and wealth level at the present, and reinsurance policy is proportional to intensity of the loss.The contributions and innovations of this paper are below.(1) There are different premium principles and reinsurance polices used for two heterogeneous lines businesses of the insuer. The adjustment coefficient is obtained for diffusion approximation risk model. And the optimal reinsurance polices are found under minimizing the ruin probability and maximizing the utility of terminal wealth respectively. We also have discussed that the roles of the adjustment coefficient and risk aversion.(2) The price of risky asset is modelled with CEV process which can capture stochastic volatility. For a compound Poisson risk model, optimal proportional reinsurance and invetment polices are get in presence of the net profit conditon and variance premium. And for a diffusion approxima- tion risk model, optimal excess of loss reinsurance and invetment polices are obtained in presence of expected value premium. We show that the excess of loss reinsurance is always better than proportional reinsurance.(3) We find the optimal investment and reinsurance policies for the exponential mean reversion model under maximizing the utility of terminal wealth. And we also show that the excess of loss reinsurance is always better than combining quota-share and excess of loss reinsurance.(4) A generalized mean-variance criterion(wealth-path dependent) is introduced for Lévy model which describes jumps of risky asset price. We get the optimal quota-share reinsurance and investment investment. The variance minimizing frontier and efficient frontier are found for the classical mean-variance model.