关键词: 最优再保险投资, 仿射平方根模型, 损失相依保费原则, 动态规划

Abstract: Insurers are an important part of the financial market. They not only play a special role in providing security for individuals and enterprises, but also improve the stability and liquidity of the whole financial market and promote the normal operation of economic society. Like other financial institutions, the insurer is a for-profit financial institution. On the one hand, in order to avoid the risk caused by excessive or concentrated claims, the insurer can buy reinsurance to divert some risks. On the other hand, the insurer often invests part of their surplus in financial markets in order to increase profits. Therefore, how to choose the optimal reinsurance and investment strategy to diversify risks and increase returns is a practical problem faced by insurers in practice. In recent years, the reinsurance investment strategy of insurers has also become a hot topic in the field of financial mathematics and actuarial research. Therefore, this issue is of great significance in both practical and theoretical research.
In this paper, we study the optimal reinsurance-investment problem for an insurer with the loss-dependent premium principle. Different from existing literature, the premium principle here can be dynamically updated based on past losses and an estimate of future losses, which is an extension of the traditional expected value premium principle. This premium principle is index-weighted and has a memory feature, which includes not only recent losses but also all past losses. We assume that the insurer’s surplus process follows the diffusion approximation of the C-L (Cramér-Lundberg) model. The insurer can purchase proportional reinsurance or acquire new business to hedge risks or increase profits. It is assums that the financial and insurance markets are independent of each other. The financial market is assumed to be composed of a risk-free asset and a risky asset, where the price process of the risky asset is described by an affine square root stochastic model. The affine square root model is a more general stochastic volatility model where the volatility is a random variable. Particularly, when the model parameters of risk assets are taken to some special values, they can be degraded into CEV (Constant Elasticity of Variance) model, Heston’s model and GBM (Geometric Brownian Motion) model.
Based on the goal of maximizing the insurer’s terminal wealth expectation and under the CARA utility function, we derive explicit expressions for optimal reinsurance-investment strategy and the value function by using dynamic programming, stochastic control and other methods. When the model parameters are taken to particular values, we obtain explicit expressions for the optimal investment strategies under the CEV, Heston’s and GBM models. At last, numerical examples are given to analyze the influences of some model parameters on the optimal reinsurance-investment strategies. Through the analysis of the optimal reinsurance strategy, we find that two important parameters in the loss-dependent premium principle: The inferred intensity(β)and the average value of loss weights (s) have a significant impact on the optimal reinsurance strategy, please refer to 3.1 for the detailed analysis. In the analysis of numerical examples of optimal investment strategies, we analyse the impact of the model parameters on the optimal investment strategy under the CEV model as well as the Heston’s model.
The article has some relevance to the issue of optimal insurance investment, but there is some room for further discussion. Further research could be considered in aspects such as model ambiguity and the correlation between insurance and financial markets. Even we can discuss the similar research under the mean-variance criteria.

Key words: optimal reinsurance-investment, affine-form square-root model, dynamic programming, loss-dependent premium principle