Compound binomial risk model in a Markovian environment with capital cost and the calculation algorithm

Highlights：

• MCPRAM-algorithm solves the problem of one-time recursion for many periods, and can quickly monitor the bankruptcy risk for many years. Many other methods discuss the recurrence of a certain period in the future, and use numerical examples for analysis. However, MCPRAM-algorithm needs to recurse multiple periods, and each period recursion needs to use all the data from the first period. It also needs to calculate many ruin probabilities with different initial values at the same time. It is equivalent to the increasing workload of each phase. The previous recursive operation cannot be repeated in each phase, so the data calculated in the preceding operation should be stored. Later, the speed of calling data and operation model will be greatly improved, and the time can be greatly reduced. The data stored in this paper exceeds 500G, so it requires a lot of calculations.

• The MCPRAM-algorithm of the actuarial model requires highly configured computers, multi-core and multi-line equipment and the application of the parallel computing method to reduce the time required by ordinary computers from several months to less than 2 days. In this study, we calculate seven groups of numerical examples, which represent the insurance market and insurance products in different random environments. The results were satisfactory and bankruptcy probability was reduced in about 5%. The parameter data of the example here provided can be used as a reference or action guide for insurance companies.

• If the parameter data changes in the operation of the insurance market, the RMCM model can be calculated in real time. Dynamic monitoring can be achieved by using the efficient MCPRAM-algorithm. It is a better choice to monitor the risk of multiple periods in the future than only monitoring the risk of phase I or phase II. Therefore, the algorithm greatly improves the application value of the actuarial model and is the biggest and most meaningful outcomes of this study.

• Seven groups were used as numerical examples and their details can be seen in Tables 5 and 6. The tables represent the operation status of the insurance market and insurance products in different random environments. This study compares and obtains a better case2 example, so that the bankruptcy probability in the next five years is about 5%, and the worst case is case0 example. See X1... X9 in Table 5 and Y1... Y7 in Table 6. The parameter data of the better example can be used by insurance companies as a reference or action guide for operation management, while the parameter data of the worse example should be used as a reference for pit avoidance.

• This paper establishes a random environment compound binomial risk model, and the compensation times are calculated as 0 or 1. Therefore, it is more in line with accident insurance, natural disaster insurance and other varieties. These varieties have some specific features, e.g., low premium, low compensation probability, large claim, and one-time compensation and so on. However, the insurance varieties (such as automobile insurance) have more compensation times, higher compensation probability, less compensation amount and multiple claims during the insurance period, so it is more suitable to use the compound Poisson risk model in this case.