Sistem Bonus-Malus dengan Banyak Klaim Berdistribusi Geometrik dan Besar Kerugian Berdistribusi Weibull Terpotong

Abstract

Insurance is one way to denigrate financial loss by channeling the risk of loss of the person to another entity, for example automobile insurance. The risk premium of the automobile insurance is determined by insurance that is known data in the past (experience rating). System experience rating could call as Bonus-Malus system. In the Bonus-Malus system, policyholders who has submitted one or more claims will be subjected to a risk premium increase (malus), whereas for policyholders who did not file a claim would be rewarded with a decrease in risk premium (bonus) in the next premium payment period. Bonus-Malus system is said to be optimal if it is financially balanced for insurance companies and fair for policyholders. That is, the determination of the risk premium is not only based on the claims frequency but also based on the severity. In this study, we used the Bonus-Malus system by Mert and Saykan (2005) and Ni et al. (2014). But, previous research concern with the determination of the risk premium Bonus-Malus system which is applied to all of the severity that guaranteed by the insurance company. In fact, all of the severity proposed by policyholder could not be covered by insurance company. When an insurance company sets a maximum bound of the severity incurred, it is necessary to modify the model of the severity distribution into the truncated severity distribution. The claim frequency is geometric distribution as a combination of Poisson and exponential distribution. While, the severity distribution by Ni et al. (2014) modified to be the severity distribution with maximum bound is truncated Weibull distribution. Truncated Weibull distribution is a combination of truncated exponential and Levy distribution. In this study, we also compared the risk premium between the full severity and the severity with the maximum bound. The result shows that the risk premium equation with maximum bound is           3 2 t 1 1 2 1 2 Premium K K B c N u K n K N u K n  t c B c N u K n                                 , where 𝐾 is the total claim frequency, 𝑢 is the maximum bound that covered by insurance company, 𝑁 is the total severity which is lower than bound 𝑢, 𝑛 is the claim frequency which is lower than bound 𝑢, 𝐵𝐾−1/2 is the Bessel function with index 𝐾 − 1/2, 𝜃 is the parameter of the claim frequency distribution, and 𝑐 is the parameter of the severity distribution. The risk premium equation for the next period shows that the risk premium depends on the claim frequency, time period, and the total severity. The risk premium of Bonus-Malus system with the severity component based on Weibull distribution is equal to the risk premium based on truncated Weibull distribution when the severity component are less than the maximum bound of severity. Whereas, the risk premium based on Weibull distribution are more expensive than that based on truncated Weibull distribution when the severity are greater than the maximum bound of severity. The smaller of the severity, the less of the risk premium to be paid and the greater of the severity, the greater of the risk premium must be paid.