Estimation model of life insurance claims risk for cancer patients by using Bayesian method

This paper discussed the estimation model of the risk of life insurance claims for cancer patients using Bayesian method. To estimate the risk of the claim, the insurance participant data is grouped into two: the number of policies issued and the number of claims incurred. Model estimation is done using a Bayesian approach method. Further, the estimator model was used to estimate the risk value of life insurance claims each age group for each sex. The estimation results indicate that a large risk premium for insured males aged less than 30 years is 0.85; for ages 30 to 40 years is 3:58; for ages 41 to 50 years is 1.71; for ages 51 to 60 years is 2.96; and for those aged over 60 years is 7.82. Meanwhile, for insured women aged less than 30 years was 0:56; for ages 30 to 40 years is 3:21; for ages 41 to 50 years is 0.65; for ages 51 to 60 years is 3:12; and for those aged over 60 years is 9.99. This study is useful in determining the risk premium in homogeneous groups based on gender and age.


Introduction
Lately, people with critical illnesses are expanding. One of them cancer. Ironically, health care costs have also increased rapidly over the past few decades. Things need to be aware, unforeseen events can occur in cancer patients, resulting in cancer patients financially are in need of protection. Therefore, the role of insurance companies is very necessary [2], [8].
Critical illness insurance is devoted to protecting customers' patients with critical illnesses, such as cancer, kidney failure, and heart. Critical illness insurance is different with health insurance [9], [3]. Critical illness insurance provides a number of cash when the customer has been diagnosed with a critical illness [2], [5].
In insurance, the insurer (insurance company), provides financial assistance in the form of insurance money that is called a risk premium to the insured (customer). Determining the risk premiums be reckoned with that the company did not experience a loss [10]. With the increasing number of cancer patients, and the cost of cancer treatment is higher, the number of claims from year to year increasing [12]. This is a problem for the insurer in estimating future claims trends to determine the risk premium. Therefore, in this paper do research on the risk of claims, particularly in cancer patients.
(4) By minimizing the quadratic loss function, ߠ value can be expressed as the average (mean) of the posterior distribution as follows: Using the theorem credibility, we can declare the value of Bayesian estimation ߠ in the form: where credibility factor ܼ is as follows: and μ is the average of the prior distribution of beta distributions declared by β α α µ In practice, there are some situations where a prior value is unknown. So, we use the value of noninformative priors. For example, if ߠ is the opportunity of a binomial distribution, and ߠ not have any information about the prior distribution, the distribution of which had been distributed prior to the hose Uniform (0,1) would seem appropriate. In this case, the prior distribution is the beta distribution with parameter α = 1 and β = 1. However, the interval that we take not the interval (0,1), but the interval is more realistic. Set the interval (ߠ , ߠ ௫ ) to get a good estimate. We denote the value of s as the average of the prior distribution of beta which is the centre of this interval [1], [11].
We mark as 0 θ the more distant boundary from the value of 0.5 of the interval (ߠ , ߠ ௫ ). Calculate the error ℎ as follows: (10) and calculate the number of claim q by using formula as follows: where p is number of insured in the insurance company. Then, we estimate the value of the parameter α and β of beta prior distributions as follows:

Individual Risk Model
To determine the risk premium in homogeneous groups according to age and gender, need to be estimated that many insurance policy issued in the following year by extrapolating the trend of the time series using Statistics program Statgraphics Centurion. To select the most appropriate trend function, we use the procedure Comparison of Alternative Models [9], [7]. As for estimating the value of n in the following period (in 2009), we use the procedure Forecast. In this research, there are two models of risk, namely the risk model of collective and individual risk models. In the collective risk model, we let ܺ ଵ, , ܺ ଶ , ܺ ଷ … … . ܺ ே is a random variable that the variable determining the amount of the claim. The total of the amount of the claim, denoted by [4]: While the individual risk model, the total of the amount of the claim can be denoted as ܵ . So we can write as follows: .. (15) where ܻ indicates the number of claims for individual year j, and n indicates the period of observation. However, it is possible some risks will not give rise to a claim. Therefore, the value ܻ , ݆ = 1,2, … , ݊ may be 0. It will be given two assumptions, namely: • The number of claims in the year to j, ܰ is 0 or 1.
• Possible claims in the year to j is ‫ݍ‬ . Based on the above assumptions, ܰ ~‫ܤ‬ ݅(1; ‫ݍ‬ ), thus the distribution of ܻ is compound binomial with individual claims are denoted ܺ . Then we can write it as follows: Where ߤ , and ߪ ଶ is the average and variance of ܺ . Then, the average and variance of ܵ are: In special cases, when is a composite of several distribution and is a random variable. Based on the central limit theorem we can approach ܵ distribution by the normal distribution. Therefore, in this case, we assign the value of the risk premium equal to 95% of the normal distribution with parameters(ߤ = ‫ܵ(ܧ‬ ), ߪ ଶ = ‫ܵ(ܦ‬ )). Large risk premium (RP) can be calculated with the calculation below [6], [4]:

Results and Discussion
The data used is in the form of simulation data in the form of insured claims data that have been diagnosed with a critical illness at an insurance company for ten years, from 2005 to 2014.
• For insured men aged less than 30 years Based on the percentage of the value of Z obtained in Table 1, it can be concluded that the influence of information in previous years to take into account the value of the insured ߠ men • For insured women aged less than 30 years Based on the percentage of the value of Z obtained in Table 4, it can be concluded that the influence of information in previous years to take into account • For insured men aged 40 years to 50 years Based on the percentage of the value of Z obtained in Table 5, it can be concluded that the influence of information in previous years to take into account the value of the insured ߠ men  Based on the percentage of the value of Z obtained in Table 6, it can be concluded that the influence of information in previous years to take into account the value of the insured ߠ on • For insured men aged 50 years to 60 years Based on the percentage of the value of Z obtained in Table 7, it can be concluded that the influence of information in previous years to take into account the value of the insured ߠ men  Based on the percentage of the value of Z obtained in Table 8 • For insured men aged over 60 years Based on the percentage of the value of Z obtained in Table 9, it can be concluded that the influence of information in previous years to take into account the value of the insured ߠ men aged over 60 years • For insured men aged over 60 years Based on the percentage of the value of Z obtained in Table 10, it can be concluded that the influence of information in previous years to take into account the value ߠ insured women over

Conclusion
Bayesian estimation theory provides a great method for estimating the risk premium for the next period of the claim data information in the previous period. In this study, a large risk premium for insured males aged less than 30 years is 0.85, for those aged 30 to 40 years is 3:58, for those aged 40 to 50 years is 1.71, for those aged 50 to 60 years was 2.96, and for age over 60 years is 7.82. Meanwhile, for insured women aged less than 30 years is 0.56, for those aged 30 to 40 years is 3:21, for those aged 40 to 50 years is 0.65, for those aged 50 to 60 years was 3:12, and for those aged over 60 years is 9.99.