The use of monte carlo method to model the aggregate loss distribution

Abstract


INTRODUCTION
Government attention toward health and quality of the health of its citizens can be seen in the 1945 Constitution article 28 H paragraph (1) which reads "Every person has the right to live in physical and spiritual prosperity, to live and to have a good and healthy life and to have the right to health services".Referring to the law number 24 of 2011, a Social Insurance Administration Organization or BPJS was established as an institution to organize the national social security program.This state program aims to provide certainty of protection and social welfare for all people, namely BPJS Health and BPJS Employment.BPJS Health is a government program on health insurance with the principle of social insurance and the principle of equity.BPJS Employment covers the work accident insurance, old-age insurance, pension insurance, and life insurance.
A general approach to modeling claim data is to separate the claim frequency from the large claims.The claim severity refers to the total number of policyholder claims per time whereas the claim severity is the costs incurred per claim.Many policyholders do not submit claims.Cases like this produce a zero claim number with high probability.Poisson distribution is the archetype of modeling the claim frequency (Antonio et al., 2010) although, in practice, the amount of data observed often displays features such as overdispersion (variance values are greater than expectations) or less spread (variance values are less than expectations) that are distribution.Chi-square test was used to check the goodness of fit-test for the claim frequency distribution and the Anderson -Darling test was applied to the claim severity distribution.
Next, determine the aggregate loss distribution for health insurance companies which is a compound distribution of the claim frequency and the claim severity.The distribution of compounds was determined by the Monte Carlo method.The software used was Mathematica version 12.After obtaining the aggregate loss distribution from the Monte Carlo method, the risk amount was calculated.The measures of risk referred to were the Value at Risk (VaR) and Shortfall Expectations (ES).

Model of Data Distribution of the Claim frequency of BPJS Health in the City of Tangerang in 2017 Data Description
The claim frequency data used in this study was the data on the numbers of inpatient claims of BPJS Health in the City of Tangerang in 2017.There were 1,500.000 claims data activated at the BPJS Health in the City of Tangerang in 2017.The following is the statistical description of the claim frequency data of the BPJS Health in the City of Tangerang in 2017.The positive value shown in the distribution is tilted to the left and has a long tail to the right.This can also mean that many BPJS Health participants did not submit claims or many BPJS participants were not hospitalized.Kurtosis values greater than three indicate that the distribution has a tapered curve.This can be seen in Figure 1 which shows the asymmetrical shape of the histogram indicating that the data are not normally distributed.

Estimator Value of Parameters and Mass Function Opportunities
Based on the data description, the researchers first chose a negative binomial distribution (NB) because the variance value was greater than the mean.The choice of the second distribution was the negative distribution of binomial generalized exponential (NBGE) this was because the data had a probability of no large claims.Furthermore, the estimator value for the parameters was the maximum likelihood estimator (MLE) method, the negative binomial probability distribution (NB), and negative binomial generalized exponential (NBGE) probability function.The parameters can be seen in Table 2.
Furthermore, the researchers modified the distribution class on the negative binomial distribution (NB) and generalized exponential negative binomial distribution (NBGE), namely zero modified negative binomial (ZM-NB), zero one modified negative binomial distribution (Z1M -NB), zero modified negative binomial generalized exponential distribution (ZM-NBGE), zero one modified negative binomial generalized exponential distribution (Z1M -NBGE), and zero one two modified negative binomial generalized exponential distribution (Z12M) -NBGE).

Graphical Model Analysis
The parameters had been estimated with the Probability mass function of each distribution.The researcher wanted to test the claim frequency model graphically.It can be seen in Figure 2.

Mathematical Model Analysis
Zero One Two Modified Negative Binomial Generalized Exponential (Z12M -NBGE) was the most graphically suitable claim distribution model.Furthermore, the distribution was examined mathematically with the chi-square test.The formulated hypotheses were:   : Data on the claim frequency of BPJS Health in the City of Tangerang in 2017 followed a specific distribution.  : Data on the claim frequency of BPJS Health in the City of Tangerang in 2017 did not follow a specific distribution.By using the estimated parameters and the probability mass function formula for each distribution, the results are presented in Table 3.Based on Table 3, it can be seen that the p-value distribution of Zero One Two Modified Negative Binomial Generalized Exponential (Z12M-NBGE) is 0.99, which means the opportunity not to reject the incorrect hypothesis  0 is 0.01. it is certainly not to reject the hypothesis  0 because the probability of error is quite small.Therefore, the hypothesis testing conducted concluded that the distribution that was most suitable for modeling the claim frequency of BPJS Health in the City of Tangerang in 2017 was the Zero One Two Modified Negative Binomial Generalized Exponential (Z12M-NBGE) distribution.

DATA DISTRIBUTION MODEL OF THE CLAIM SEVERITY OF BPJS HEALTH IN THE CITY OF TANGERANG IN 2017 Description of Data
Data on the claim severity used in the study was the claims data of BPJS Health in the City of Tangerang in 2017.They were 173.001 selected data which covered the claim severity of the inpatient coverage.The following is the statistical description of the claim severity data of BPJS Health in the City of Tangerang in 2017.The skewness values indicate that the distribution of skewness is tilted to the left and has a long tail to the right as can be seen in the histogram Figure 3.This can also mean the severity of the claims was mostly at a value less than the average covered.The kurtosis value was greater than three.It indicated that the distribution had a tapered curve.It can be seen in Figure 3 which shows the asymmetrical shape of the histogram so that it indicates that the data was not normally distributed.

Estimator Value of Parameters and Probability Density Functions
The claim severity is modeled using a positive continuous opportunity distribution (Klugman et al. 2012).In this study, the researchers chose the gamma distribution, lognormal distribution, and logistical distribution to estimate the claim severity, distribution model.Furthermore, the estimator values for the parameters were the maximum likelihood estimation method (MLE) and the probability density function of the gamma distribution, lognormal, distribution, and logistical distribution.The data can be seen in Table 5.

Graphical Model Analysis
The researcher tested the model for the claim frequency graphically which can be seen in Figure 4. Figure 4 shows that (b) lognormal distribution was the most appropriate for the plot of the probability density function.This shows that graphically, the lognormal distribution was the most suitable for modeling the distribution of participants' claims of BPJS in the City of Tangerang in 2017.

Mathematical Model Analysis
The lognormal distribution is a distribution model for the claim severity that is most graphically appropriate.Furthermore, the distribution will be examined mathematically using the Anderson Darling test.The formulated hypotheses are:   : Data on the claim severity of BPJS Health in the City of Tangerang in 2017 followed a specific distribution.  : Data on the claim severity of BPJS Health in the City of Tangerang in 2017 did not follow a specific distribution.From the results of the calculations in Table 6, using the Anderson Darling test can be seen that the distribution of lognormal with an AD value of 0.64303 and p-value of 0.60827.The probability of not rejecting the incorrect Ho hypothesis is 0.39173.This gives the decision not to reject the hypothesis  0 because the probability of error is quite small between the gamma distribution and the loglogistic distribution.Therefore, the hypothesis testing conducted concluded that the distribution that was most suitable for modeling the magnitude of the claims of BPJS Health participants in the City of Tangerang in 2017 was the lognormal distribution.

Model Of Aggregate Loss Distribution of Bpjs Health of Tangerang City In 2017
The statistical description of aggregate loss is presented in table 7. The far from zero values can be seen in Figure 5.The distribution has a very long tail to the right.This shows the distribution of aggregate loss mostly at a value of less than Rp508.053.00.The kurtosis values are greater than three, as can be seen in Figure 5, that the data is more pointed to the right.This shows that the distribution of aggregate loss experienced frequent high fluctuations.Based on Table 9, the absolute and relative errors obtained were relatively small.It can be said that the Monte Carlo simulation method estimated the aggregate loss quite well.

Determining the VaR Value and ES Value for the Aggregate Loss Claim Distribution with the Monte Carlo Simulation Method
The final stage was calculating the risk measure namely Value at Risk (VaR) and Shortfall Expectation (ES).The estimation of the amount of risk was done after generating random variables on the distribution of aggregate loss.Furthermore, the estimated results of VaR and ES values on the aggregate loss distribution by the Monte Carlo method are presented in Table 10.It can be seen in Table 10 that the potential claims that can be tolerated at a 95% confidence level are Rp.4,279,005.00.In other words, the amount of reserved funds to cover the claims of one person per year is Rp. 4, 279,005.00.
ES value of IDR 7,353,810.00at a confidence level of 95% shows that the claims that can be tolerated with a 95% confidence level are Rp.7, 353,810.00.The value of ES that is greater than the value of VaR indicates that the maximum amount of reserved funds for the claims of one person per year is Rp. 7, 353,810.00.This explanation applies to other levels of trust.Thus, the Monte Carlo method in its implementation can easily handle many risks with a degree of trust.

CONCLUSIONS
It can be concluded that the data distribution for the claim frequency of BPJS in the city of Tangerang in 2017 is the distribution of Z12M-NBGE with some parameters, namely ̂ = 1.06317,  ̂ = 1.48856 and  ̂ = 12.8549.The data of the claim severity distribution is the lognormal distribution with ̂ parameter equals to 15.11822 and  ̂ 0.58312.The combination of Z12M -NBGE distribution and lognormal distribution with the Monte Carlo method which

Figure 1 .
Figure 1.Histogram of the Claim Frequency of BPJS Health in the City of Tangerang in 2017

Figure 2 .
Figure2shows (b) the distribution of Zero One Two Modified Negative Binomial Generalized Exponential (Z12M-NBGE) that best matched the histogram with the probability mass function plot.This shows that graphically, the distribution of Zero One Two Modified

Figure 3 .
Figure 3. Histogram of the Severity Claim of BPJS Health in the City of Tangerang in 2017

Figure 4 .
Figure 4. Histogram Data on the Claim severity and the Curve of the Probability Density Function for the distribution of (a) Gamma, (b) Lognormal, (c) Logistical

Figure 5 .
Figure 5. Histogram of Aggregate Loss Distribution Data

Table 1 .
Statistical Description of Participants Claims of BPJS Health in the City of Tangerang in 2017

Table 2 .
The Estimated Parameter Values of the Negative Binomial Distribution (NB) and Negative Distribution of Binomial Generalized Exponential (NBGE).

Table 3 .
Expectations of the Claim frequency from the NB Distribution and the Z12M -NBGE Distribution.

Table 4 .
Statistical Description of the Claim Severity of BPJS Health in the City of Tangerang in 2017

Table 5 .
The Estimator Value of Parameters and Probability Density Functions of Gamma Distribution, Lognormal Distribution, and Logistical Distribution.

Table 6 .
Anderson Darling Value and p-value for the Claim severity of BPJS Health in the City of

Table 7 .
The Statistical Description of Aggregate Loss of BPJS Health of the City of Tangerang in 2017

Table 9 .
Expectations, Variances, Absolute Errors, and Relative Errors of the Distribution of Aggregate Loss

Table 10 .
Estimated VaR values and ES values for the Aggregate Loss distribution