Recently, the valuation of variable annuity products has become a hot topic in actuarial science. In this paper, we use the Fourier cosine series expansion (COS) method to value the guaranteed minimum death benefit (GMDB) products. We first express the value of GMDB by the discounted density function approach, then we use the COS method to approximate the valuation Equations. When the distribution of the time-until-death random variable is approximated by a combination of exponential distributions and the price of the fund is modeled by an exponential Lévy process, explicit equations for the cosine coefficients are given. Some numerical experiments are also made to illustrate the efficiency of our method.
In this paper, we assume that the reserve level of an insurance company can only be observed at discrete time points, then a new risk model is proposed by introducing a periodic capital injection strategy and a barrier dividend strategy into the classical risk model. We derive the equations and the boundary conditions satisfied by the Gerber-Shiu function, the expected discounted capital injection function and the expected discounted dividend function by assuming that the observation interval and claim amount are exponentially distributed, respectively. Numerical examples are also given to further analyze the influence of relevant parameters on the actuarial function of the risk model.
The BP neural network model is a hot issue in recent academic research, and it has been successfully applied to many other fields, but few researchers apply the BP neural network model to the field of automobile insurance. The main method that has been used in the prediction of the total claim amount in automobile insurance is the generalized linear model, where the BP neural network model could provide a different approach to estimate the total claim loss. This paper uses a genetic algorithm to optimize the structure of the BP neural network at first, and the calculation speed is significantly improved. At the same time, by considering the overfitting problem, an early stop method is introduced to avoid the overfitting problem. In the model, a three-layer BP neural network model, which includes the input layer, hidden layer, and output layer, is trained. With consideration of various factors, a total claim amount prediction model is established, and the trained BP neural network model is used to predict the total claim amount of automobile insurance based on the data of the training set. The results show that the accuracy of the prediction by using the BP neural network model to both the data of Shandong Province and to the data of six cities is over 95%. Then, the predicted total claim amount is used to calculate premiums for five cities in Shandong Province according to credibility theory. The results show that the average premium of the five cities is slightly higher than the actual claim amount of the city. The combination of BP neural network and credibility theory can perform accurate claim amount estimation and pricing for automobile insurance, which can effectively improve the current situation of the automobile insurance business and promote the development of insurance industry.
Background: How health status affects residents’ purchase of commercial health insurance and whether social insurance will crowd out the demand of commercial health insurance are important issues related to the development of commercial health insurance. Further, this paper also attempts to answer why people with worse health buy less commercial health insurance through mechanism analysis. Methods: We used Logit model to analyze the impact of self-rated health level on the purchase of commercial health insurance and the crowding out effect of participating in social health insurance on the purchase of commercial health insurance. In addition, we also successively introduced the respondents with worse and worse health status for regression analysis and analyzed the reasons why the poor health groups reduce the purchase of commercial medical insurance. Results: When performing regression analysis on whether to purchase commercial health insurance, the regression coefficient of having social health insurance is -0.497 and the regression coefficient of self-rated health level is -0.182, but the health status has no significant impact on whether to have social health insurance. Conclusions: The expansion of social basic health insurance has a restraining effect on the promotion of commercial health insurance, and self-rated health level has a negative impact on the purchase of commercial health insurance. The reason why individuals with poor health buy less commercial health insurance is that commercial insurance companies often refuse to cover individuals in poor health.JEL: I11, G22
In this paper, we model the insurance company’s surplus by a compound Poisson risk model, where the surplus process can only be observed at random observation times. It is assumed that the insurer observes its surplus level periodically to decide on dividend payments and capital injection at the interobservation time having an $\operatorname{Erlang}(n)$ Erlang ( n ) distribution. If the observed surplus level is greater than zero but less than injection line $b_{1} > 0$ b 1 > 0 , the shareholders should immediately inject a certain amount of capital to bring the surplus level back to the injection line $b_{1}$ b 1 . If the observed surplus level is larger than dividend line $b_{2}$ b 2 ($b_{2} > b_{1}$ b 2 > b 1 ), any excess of the surplus over $b_{2}$ b 2 is immediately paid out as dividends to the shareholders of the company. Ruin is declared when the observed surplus level is negative. We derive the explicit expressions of the Gerber–Shiu function, the expected discounted capital injection, and the expected discounted dividend payments. Numerical illustrations are also given to analyze the effect of random observation times on actuarial quantities.
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