Prediction of Outstanding Claims Liability in Non-Life Insurance: An Application of Adaptive Grey Model

: In order to assess the solvency of non-life insurance companies, the prediction of outstanding claims liability is very important. Prediction of outstanding claims liability is usually done by using a run-off triangle data scheme. However, if data are not available to form the scheme, the prediction of outstanding claims liability cannot be made. Another alternative for predicting of outstanding claims liability is to use time series analysis. This research uses an adaptive grey model that has the advantage of being free of assumptions of data patterns and a minimum amount of data used to predict is small (at least 4 data). To determine the accuracy of the adaptive grey model, we compare the prediction of outstanding claims liability using a grey model classic. Based on the analysis results, the adaptive grey model is better than the classic gray model in predicting outstanding claims liability.


Introduction
In non-life insurance, payment of claims that have occurred and reported to the company sometimes requires quite a long time to settle. This causes companies to have obligations of claims whics are not finalized that must be fulfilled immediately. In insurance, the obligations are often referred to as outstanding claims liability (OCL). OCL prediction is very important to do, because this information is very important for insurance companies in assessing solvency (the ability of an insurance company to fulfill its obligations).
Generally, prediction of OCL is performed using run-off triangle data scheme [1][2][3][4][5]. The run-off triangle scheme is a data scheme that contains an aggregate claim description and is a summary of the individual claim data set [6]. In this scheme, information from the observed claim data is formed into a upper triangular matrix of size N × N with the occurrence period as a column and the development period of payment as a row. However, if information about the observed claim data is not available then a runoff triangle data scheme cannot be formed. Consequently, OCL predictions cannot be made.
Another alternative that can be used to predict OCL is to use time series analysis. Time seri es analysis is an analysis that is used to make predictions by utilizing data collected in a series of particular time periods or intervals. However, the use of classical time series analysis both deterministic and stochastic often has several assumptions that must be fulfilled, such as time series data patterns and minimum amount of data used. For this reason, this paper uses a grey model that does not require the assumption of data patterns and prediction can be made even though the data is very limited (the least 4 data) [7].
The grey model is part of the grey system theory which was introduced by Deng Julong in 1982 [8]. Because of its ease of use, the grey model is widely applied in various fields. For example, in the telecommunications [7], healthcare [9][10][11][12][13][14], electricity demand [15][16][17][18], energy problems [19][20], transportation [21][22], economic fields [23][24], and so on. The basic form of the grey model is GM(1,1). Although GM(1,1) has been widely applied with good prediction accuracy results, this model still has limitations in its use, because it cannot reflect of data trends at different period of time. To solve this problem, the trend and potential tracking method (TPTM) proposed by Li and Yeh (2008) [25] is used in GM(1,1) to construct an adaptive grey model which is AGM (1,1). TPTM is useful for analyzing data behavior, extracting hidden information from the data, and utilizing the trend and potency value to build the AGM(1,1) [25] so that the accuracy of prediction is better than GM(1,1) [26].

Trend and Potency Tracking Methods
TPTM is an analytical method that uses data characteristics to explore possible changes in data behavior at different period of time [26]. TPTM is the key used in AGM(1,1) to improve accuracy of GM(1,1).
The procedure in determining TPTM which is cited from Chang et.al (2013) are as follows [26].
Step 1: Arrange series data = { 1 , 2 , … , } . Let min be the minimum value of and max be the maximum value of .
Step 2: Determine the variations of paired data ( −1 , ) for = 2,3, … , . Next, determine increasing or decreasing potencies according to sequence of the data. If is positive, then the trend of data in the time period increases, and vice versa.
Step 3: Define weights to the latest data. The weight of data in the time period , denoted by , equals = − 1 for = 2,3, … , .
Step 4: Compute = × for = 2,3, … , . The use of is as operator to strengthen the data trend and potency for each time period. If > 0, then there is increasing potency (IP), and if < 0, then there is decreasing potency (DP).
Step 5: Compute the central location (CL) using the equation CL = min + max 2 . CL is used as the main point to expand the data domain range.
Step 6: Calculate the average of increasing potency (AIP) and average of decreasing potential (ADP), then use both to expand the range of data domains. The upper limit of the extended domain range (EDR_UL) is obtained by the equation EDR_UL = max + AIP and the lower limit of the extended domain range (EDR_LL) is obtained by the equation EDR_LL = min + ADP.
Step 7: Form a triangular TP function using the CL, EDR_UL, and EDR_LL. Compute TP value from the corresponding data by using triangle rule. For CL, set the TP value of CL to 1. The following is an example of a triangle for calculating the TP value presented in Figure 1. Adaptive Grey Model AGM(1,1) is a model obtained by integrating TPTM value into the modeling process of GM(1,1). The procedure of AGM(1,1) is described as follows [26].
Step 6: Construct differential equation from (1) and (4), that is, where and are the developing coefficient and the grey input coefficient, respectively. By using the least square method, the estimation of and can be obtained, that is ̂= ( ) −1 .

Model Evaluation
Good prediction accuracy is indicated by the minimal prediction error value. Prediction error can be used for two situation. The first is making a choice between several alternative prediction models, and the other is evaluating the success or failure of the models used. In this paper, we will use mean absolute error (MAE) and mean absolute percentage error (MAPE) as a prediction error measure. The MAE and MAPE are defined as follows [38].

Results and Discussions
The OCL data presented in Table 1 is divided into two parts, namely training data and testing data. Training data is used to form models and testing data is used to evaluate models. Data from 2002 to 2005 is used as training data, and data from 2006 to 2019 is used as testing data.
By using AGM (1,1), the training data from 2002 to 2005 will be used to build the model. The calculation process is shown in the following steps, and the results are summarized in Table 2. Meanwhile, the summary of the OCL prediction results is presented in Table 3  To test the effectiveness of the AGM (1,1) model, we compare the results of prediction value with GM(1,1) model. The prediction results are presented in Figure 2.  Figure 2 shows that the prediction results using AGM(1,1) is closer to the actual data and have the same pattern as the actual data. Meanwhile, the prediction results using GM(1,1) show the opposite. To see a good prediction accuracy, a measure of model evaluation based on MAE and MAPE is calculated. The results of comparing OCL predictions based on MAE and MAPE using AGM(1,1) and GM(1,1) are presented in Table 4. Based on the results shown in Table 4, MAE and MAPE of the AGM(1.1) are lower than MAE and MAPE of the GM(1.1). So, it can be said that the AGM(1,1) gives better results than the GM(1,1) in predicting OCL. Furthermore, the AGM(1,1) can be an alternative model in predicting OCL.