Research On The Reinsurance And Investment Strategy For An Insurer With Default Risk

With the rapid development of Internet finance, the risk of financial market is becoming more and more complex. Insurance company plays a significant role in the modern financial market, so it is very important to promote the insurance company’s underwriting business and investment business. When faced with fierce competition in the insurance market, on the one hand, insurance company needs to increase the premium income by reducing the premium rate. Furthermore, insurer can improve the ability of risk management by purchasing reinsurance contract, which can transfer the risk suffer from the claim; on the other hand, with the increase of the insurance fund, in response to macroeconomic inflation risk, the insurer should make full use of the idle funds, and invest it in finance market to obtain the investment income, which improve the level of wealth.Nowdays, the reinsurance and investment optimization problem for insurer has become a hot research topic in the actuarial field, which can be solved by apply stochastic control theory. Based on the existing research results, we will put the credit bonds into the insurer’s portfolio, which can satisfy the diversification needs for insurer. We carry out our research on one insuer’s optimal problem first, and then study a game between two insurers. Apply the dynamic programming principle and stochastic control theory; we obtain the optimal reinsurance-investment policies for an insurer under different economic environment. As all above we concerned, we summarize our research results as the following four aspects:Firstly, we consider an optimal reinsurance and investment problem for an insurer with a credit bond, which means that in addition to the claim risks, the insurer also faces the market risk of stock investment and the default risk of credit bond investment. In order to reduce the risk, the insurer can purchase the proportional reinsurance contract from the reinsurer. The goal of the insurer is to maximize the expected CARA utility of the terminal wealth. To solve this optimization problem explicitly, we decompose the original optimization problem into two sub-problems: a pre-default case and a post-default case. We can solve the post-default problem first by using the dynamic principle. Based on the result for the post-default problem, we can solve the pre-default optimization problem similarly. Then we can derive the optimal reinsurance and investment policies in closed forms. Moreover, we compare the pre-default value function with post-default value function, and find that the pre-default value function is larger than the post-default value function, which means it is nesscery to invest into the credit bond for insurer.Secondly, we study the optimal proportional reinsurance and investment problem for an insurer with bounded memory. The insurer can allocate the wealth into three asset: a risk free bond, credit bond and stock asset. Based on the history of the decision-making effect, insurance company’s managers will adjust the best decisions now; the goal of insurer is to pursue the maximum value of the terminal wealth and history of the average wealth. Using stochastic control theory, we obtain the explicit expressions of the optimal reinsurance and investment decisions and the corresponding value function, and find that longer memory or historical experience will lead to more cautious of reinsurance and investment decision for the insurer.Thirdly, with the fierce competition in the insurance industry, we considering the management of insurance company mutual comparisons of psychological behavior, and construct a class of non-zero-sum stochastic differential reinsurance and investment game between two competing insurance companies. Each insurer will purchase the proportional reinsurance contract from a reinsurer, and can invest in a default financial market. The goal of each insurer is not only to maximize their terminal wealth, but also want to maximize the difference between their terminal wealth and rival’s terminal wealth. It also uses the stochastic control theory; we get the explicit expressions of the two insurance companies’ equilibrium proportional reinsurance and investment decisions and corresponding equilibrium value functions. The results show that the competition between the insurance companies will make the insurance companies to imitate each other’s reinsurance and investment decisions, resulting in the herding effect, thereby increasing the risk of the insurance market system.Finally, we construct a class of non-zero-sum stochastic differential reinsurance and investment game between the insurer and the reinsurer with incomplete information. The insurer purchases reinsurance contract from the reinsurer. On the one hand, reinsurance contract make them become a risk sharing and the existence of a stable relations of cooperation, on the other hand, insurer will hide some claim information to the reinsurance company to purchase the “cheap” contract. At the same time, each of them could put their wealth in a default financial market. Their goal is not only to maximize the expected utility of the terminal wealth, but also pursue to maximize the expected utility of terminal wealth of their partner. Applying the stochastic control theory, the equilibrium reinsurance and investment policies for the insurer and reinsurer are derived explicitly. Moreover, we price the reinsurance contract by means of the theory of supply and demand in economics under the asymmetric information. Results show that the more private information the insurer has, the less it will buy the reinsurance contract, which leads to a lower price for the reinsurance contract. Moreover, cooperation relationship will make the insurance and reinsurance companies to cooperate each other’s interests, and take a more conservative reinsurance and investment decision.