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 absolute ruin insurance risk model is modified by including some valuable market economic information factors, such as credit interest, debit interest and dividend payments. Such information is especially important for insurance companies to control risks. We further assume that the insurance company is able to finance and continue to operate when its reserve is negative. We investigate the integro-differential equations for some interest actuarial diagnostics. We also provide numerical examples to explain the effects of relevant parameters on actuarial diagnostics.

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.