Liability insurance premium income forecast based on improved GM (1,1) - A case of Shandong Province

Liability insurance premium income is not only an important source of insurance company profits, but also an important aspect used to evaluate the development level of the insurance industry. Therefore, it has important practical significance for accurate prediction. This paper uses improved GM (1,1) to predict the premium income of liability insurance in Shandong Province, and the results show that the fitting effect is very good.


Introduction
Liability insurance is a compensatory insurance, it is a type of insurance based on which the insured may cause losses to the interests of others during the validity period of the insurance. Liability insurance based business can be divided into employers, public, third party, product and professional liability insurance business five categories, it is widely used and can be used for all individuals, families, or various units that may cause personal injury or property damage to others. In 2019, Shandong Province's liability insurance premium income was RMB 4.034 million, which is close to 22.7 times of 2006 premium income and is growing rapidly. In the context of normal rapid economic development and continuous structural optimization, focusing on the development of liability insurance is a necessary choice for the rapid development of my country's insurance industry. Therefore, predicting liability insurance premium income has very important practical significance.
There are many ways to forecast the premium income.  [3]. The results show that the fitting effect of the model is better, but the trend growth and seasonal contrast of the future prediction data are obvious. In response to the above problems, Zhou Hua et al. based on the TEI@I method, constructed a nonlinear integrated forecasting model with correction errors and machine learning capabilities, and selected monthly data of my country's premium income as the research object for forecasting [4]. The results show that this model compared with the SARIMA model and the ordinary linear addition mixed model, its prediction accuracy is higher. Wu Kaibing et al. used the modified Logistic model to predict premium income indirectly, and the results showed that the compound growth rate of premium income was approximately  10.006% [5]. Liu Hongliang separated the four influencing factors of premium income based on the X12 seasonal adjustment method, and then respectively used the ARIMA model to predict, and finally combined them to obtain the final forecast value [6]. He Shujing et al. used the BP neural network to predict the life insurance premium income of my country from 1989 to 2005 [7]. The results show that the model has a higher prediction accuracy than the econometric model, which highlights the strong application value of the model. GM (1,1) proposed by Deng Julong is an important forecasting model, which uses some known and related data generation rules to predict the future development trend of things when there are few data or related information [8]. There have been many successful examples of using GM (1,1) for prediction [9][10][11], but there are also some cases where the deviation of the prediction results is too large. The background value is an important parameter that affects the prediction effect of the model. In order to improve the prediction effect of the model, scholars have conducted indepth research on the construction of the background value from different angles, and have achieved some valuable results [12][13][14][15]. This paper uses the improved GM (1,1) in literature [13] to predict the premium income of liability insurance in Shandong Province.
The data processing in this paper is realized by MATLAB programming.

Model establishment
There are original data columns Among them, (1) ( ) Z k is the background value sequence constructed by (1) ( ) X k , and the coefficients a and b are the development gray number and gray effect.
The solution of formula (1) can be obtained as (1) Treat it as follows: (1) Lagrange interpolation polynomials are constructed by using three points (2) Divide the interval into three equal parts to obtain three subintervals The posterior ratio The accuracy test of GM (1,1) is shown in Table 1.

Liability insurance premium income forecast
From 2010 to 2019, the premium income of liability insurance in Shandong Province is shown in Table  2 and Figure 1. It can be seen that the premium income of liability insurance in Shandong Province is increasing steadily year by year. In 2019, it has reached 4.03456 billion yuan, nearly 8 times the amount in 2010. It shows that people are paying more and more attention to liability insurance.  Table 2 Table 3), and compare the average relative error  , posterior ratio C and small error probability p (see Table 4). It can be seen from table 4 that both the average relative error and the posterior ratio of the improved GM (1,1) are smaller than those of the traditional GM (1,1), and the prediction accuracy and accuracy are greatly improved. Table 3 Table 4 Comparison of prediction effects between traditional GM (1,1) and improved GM (1,1) Traditional GM (1,1) Improved GM (1,1) Average relative error  The premium income forecast of improved GM (1,1) is fitted with the original data as shown in Figure 2, which shows that the forecast effect is very good. According to table 4, the average relative error of the improved GM (1,1) is   Table 5. It can be seen that between 2020 and 2025,