Bonus-Malus System with the Claim Frequency Distribution is Geometric and the Severity Distribution is Truncated Weibull

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1 IOP Conferene Series: Earth and Environmental Siene PAPER OPEN ACCESS Bonus-Malus System with the Claim Frequeny Distribution is Geometri and the Severity Distribution is Trunated Weibull To ite this artile: D N Santi et al 6 IOP Conf. Ser.: Earth Environ. Si. 6 View the artile online for updates and enhanements. This ontent was downloaded from IP address on 8//7 at :5

2 IOP Conf. Series: Earth and Environmental Siene (6) 6 doi:.88/755-5///6 Bonus-Malus System with the Claim Frequeny Distribution is Geometri and the Severity Distribution is Trunated Weibull D N Santi, I G P Purnaba and I W Mangku,, Department of Mathematis, Bogor Agriultural University Jalan Meranti Kampus IPB, Bogor, West Java, Indonesia teresadiiaan599@gmail.om, purnaba@gmail.om wayan.mangku@gmail.om Abstrat. Bonus-Malus system is said to be optimal if it is finanially balaned for insurane ompanies and fair for poliyholders. Previous researh about Bonus-Malus system onern with the determination of the risk premium whih applied to all of the severity that guaranteed by the insurane ompany. In fat, not all of the severity that proposed by poliyholder may be overed by insurane ompany. When the insurane ompany sets a maximum bound of the severity inurred, so it is neessary to modify the model of the severity distribution into the severity bound distribution. In this paper, optimal Bonus-Malus system is ompound of laim frequeny omponent has geometri distribution and severity omponent has trunated Weibull distribution is disussed. The number of laims onsidered to follow a Poisson distribution, and the expeted number λ is exponentially distributed, so the number of laims has a geometri distribution. The severity with a given parameter θ is onsidered to have a trunated exponential distribution is modelled using the Levy distribution, so the severity have a trunated Weibull distribution.. Introdution Bonus-Malus system is a system where in the next payment, poliyholders who submitted one or more laim will be penalized by premium raise (malus), while poliyholders who submitted no laim will be rewarded by premium redution (bonus). Bonus-Malus system is said to be optimal if it is finanially balaned for the insurane ompany (the total amount of bonus is equal to the total amount of malus) and fair to all poliyholders, that is every poliyholder pays premium proportionally with the risk [4]. In [], it is disussed Bonus-Malus system based on the laim frequeny having negative binomial distribution for modelling risk premium. While, [] using laim frequeny following Poisson inverse Gaussian distribution to minimize the risk of the insurer. Furthermore, Bonus-Malus system is not only based on the laim frequeny but also based on the laim severity, were developed in [4] and [5]. This system uses a negative binomial distribution for laim frequeny and Pareto distribution for laim severity. The other researhers, e.g. [] used geometri distribution for laim frequeny and Pareto distribution for laim severity on Bonus-Malus system. In the other hand, [7] is using negative binomial distribution in laim frequeny and Weibull distribution for laim severity. Content from this work may be used under the terms of the Creative Commons Attribution. liene. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal itation and DOI. Published under liene by Ltd

3 IOP Conf. Series: Earth and Environmental Siene (6) 6 doi:.88/755-5///6 Previous researh onern with the determination of the risk premium whih applied to all of the severity that guaranteed by the insurane ompany. In fat, not all of the severity that proposed by poliyholder may be overed by insurane ompany. When the insurane ompany sets a maximum bound of the severity inurred, so it is neessary to modify the model of the severity distribution into the severity bound distribution. Hene, in this paper, it is disussed Bonus-Malus system with the laim frequeny distribution is geometri and the laim severity distribution is trunated Weibull, in this paper also ompared the risk premium between the full severity and the severity with the maximum bound. In Setion, the proess of deriving the risk premium for our models is disussed. The number of laims onsidered to follow a Poisson distribution, and the expeted number λ is exponentially distributed, so the number of laims has a geometri distribution. The severity with a given parameter θ is onsidered to have a trunated exponential distribution is modelled using the Levy distribution, so the severity have a trunated Weibull distribution. The parameters are estimated using the quadrati error loss funtion [6]. The risk premium based on ompound of geometri distribution and trunated Weibull distribution. In Setion, as an appliation, the risk premium is alulated based on the laim frequeny based on geometri distribution and the severity based on Weibull distribution and trunated Weibull distribution. In this setion, we ompared the risk premium between the full severity based on Weibull distribution and the severity with the maximum bound based on trunated Weibull. The onlusion of this paper is presented in Setion 4.. Optimal Bonus-Malus System In this setion we disuss how to format the title, authors and affiliations. Please follow these instrutions as arefully as possible so all artiles within a onferene have the same style to the title page. This paragraph follows a setion title so it should not be indented. The modelling of the laim frequeny is the same as the one in []. The number of laims for given onsidered to follow a Poisson distribution, and the expeted number of laims λ is exponentially distributed with parameter θ, so the number of laims has a geometri distribution with parameter θ(θ + ) -. That is, θ P( k) = P k λ u λ dλ=, =,,,... and < <. θ + θ + θ + Furthermore, by applying the Bayesian approah, the posterior distribution is given by, K K+ λ( θ+ t) λ ( θ + t) e u λ k, k,..., kt =, λ > () ΓK + t k k () where K = i = k irepresents the total of laim frequeny over t period with k i present the number of laims in eah period respetively. Using the quadrati loss funtion, the following hoie for λ t+, the expeted number of laims of a poliyholder with a laim history k, k,, k t, is : K+ λt+ = λu( λ k, k,..., kt) dλ=. () θ+ t Equation () shows that, the risk premium payable at time t + depends on the laim history of the poliyholder (K), time period (t) and the parameter of the exponential distribution (θ). The modelling of the severity in [] is using Weibull distribution. The expeted number of severity for the next period is

4 IOP Conf. Series: Earth and Environmental Siene (6) 6 doi:.88/755-5///6 ˆ θ t+ B M M =, B M where M is the total severity, K is the total of laim, is the parameter of Weibull distribution. (4) So that the risk premium based on ompound refer to () and (4) is given by, B ( M ) K+ M Premium t+ =. θ t B ( M ) (5) + When the insurane ompany sets a maximum bound of the severity inurred, so it is neessary to modify the model of the severity distribution into the severity bound distribution. Now, suppose that the amount of laim x is distributed aording to the trunated exponential distribution with a given parameter θ. The umulative distribution funtion is given by θx e,< x< u F( X = x θ) =. (6), x u The parameter θ is referred as stable Levy distribution. Then the probability density funtion an be desribed as follows: e 4 θ π( Θ= θ) =, θ >. (7) πθ So the unonditional umulative distribution funtion for < x < u is, δ + The next task is solving refer to (8). First assume that, a δ I = e d a δ a a δ δ δ δ a = e dδ + e dδ. δ a δ x F( x) = e dδ, a. π = (8) 4 (9) a dg a = and = + >, then g is a monotonially δ dδ δ inreasing funtion from - to. So that refer to (9) beomes g a I = e + dg π = () δ a + δ Using substitution, let g δ (, ) The form obtained from solving refer to (9) is needed to have a δ a a Letε = anddε = dδ. Sine δ = implies ε = and δ = implies ε =, refer to (9) an be δ δ written as e dδ.

5 IOP Conf. Series: Earth and Environmental Siene (6) 6 doi:.88/755-5///6 a a δ δ δ δ a I = e dδ + e dδ δ a a δ ε δ ε a δ δ δ a = e d + e dε a a δ ε δ ε = e dδ + e dε a δ = e dδ () Beause a δ I = e dδand I = πthen we have desribed as follows δ + δ F( x) = e dδ π = π a a a δ e x dδ a δ δ π e dδ =. Equation (6) an be = e () So we have, x F( x) = e,< x< u (), x u Thus, the umulative distribution funtion above is the trunated Weibull distribution with parameter ( -,.5). If x i, i =,,,K denotes the amount of i-th laim, then the total amount of laimed for a poliyholder over t period will be equal to K xi.. We assumed that there is n laim with < x i < u, and the amount i= of laim for x i u is K n laim. Suppose N is the total of amount laimed for < x i < u that is N n = i i= x. i. Hene, The total amount of K laim beomes x = x + u( K n) = N+ u( n) K n i i= i= the posterior distribution for θ an be obtained by using Bayesian approah as follows + θ( N+ u( n )) 4θ θ e π( θ x, x,, xk) =. + θ( N+ u( n) ) 4θ θ e dθ (4) 4

6 IOP Conf. Series: Earth and Environmental Siene (6) 6 doi:.88/755-5///6 By slightly modifying the variables refer to (4), it an be rewritten as, K ( N u( K n + θ + )) 4θ θ e N+ u( n) π( θ x, x,, xk) =. K N u( K n) θ N u( K n) θ N u( K n) θ N+ u( n) + N+ u( n) e d θ (5) The integral on the denominator an be transformed to a modified Bessel funtion, whose integral representation is given as follow x y + y v Bv x = y e dy, Bv( x) >, x>, v R. (6) Then the posterior distribution refer to (5) an be written below, ( x x x ) K + θ( N+ u( n )) 4θ θ e N+ u n π θ,,, =. K B ( N+ u( n) ) Using the quadrati loss funtion, the optimal hoie for θ t+ for a poliyholder having amount of laim reports x i, i =,,, K over t period is estimated as ˆ θ t+ ( + ( )) ( ) B N u K n N+ u( n) =. B N+ u n The risk premium to be paid at time t + for a poliyholder whose number of laims history is k, k,, k t and whose amount laim history is x, x,, x k an be alulated aording to the net premium as B ( N+ u( n) ) K+ N+ u( n) Premium t+ =. (9) θ + t B ( N+ u( n) ) The risk premium in (9) is ompound of refer to () and (8). It is shows that the risk premium depends on the laim frequeny, time period, and total severity. The alulation of risk premium with Weibull distribution is refer to (5) and the risk premium based on trunated Weibull distribution is show in (9). It is give varying results aording to the proposed huge losses. At the beginning and when the poliyholders have no laim, the risk premium paid are equal. In other words, when the total severity is M N + u (K - n), the risk premium based on Weibull distribution will be equal to the one based on trunated Weibull distribution. Whereas, when the total severity is M > N + u (K n), the risk premium based on the Weibull distribution is more expensive than the one based on the trunated Weibull distribution. Let suppose the ratio of two Bessel funtions is (7) (8) 5

7 IOP Conf. Series: Earth and Environmental Siene (6) 6 doi:.88/755-5///6 v ( ) + ( ) Bv N+ u n QK ( N+ u( n) ) = B N u K n ( + ( )) ( + ( )) B N u K n = B N u K n When K = and B ( x) B ( x) v =, then we have v Beause B ( x) B ( x) B ( x) v+ v v ( ). ( + ( )) ( + ( )) B N u K n Q N+ u n = =. B N u K n v = +, so the reursive funtion Q k is x K = + QK N+ u n Q N+ u n N+ u n K+ ( ) ( ) In (9), when the total severity is N + u (K n) =, it is means that no laim filed (K = ), it shows that the risk premium is undefined. Hene, we redefine the risk premium when K =. Then the risk premium for the following period if no laim filed will be given by, Premium =. t+ N+ u n = () t+ θ This means that when there are no laim reported, the greater of the period time will ause the smaller of the risk premium must be paid.. () () (). Numerial Illustration.. Bonus-Malus System without Trunated Distribution In this set up the number of laims is assumed to follow a geometri distribution with parameter θ (θ + ) -, θ =.5. Total amount of laim is assumed to follow a trunated Weibull distribution with parameter -, =.5. The alulation of the risk premium to be paid by eah poliyholder if no laim filed refer to (), whereas if there is a laim filed refer to (5). Total of laim severity used are 8 and. The risk premium are alulated for t-th period, t =,,,,7 and for the number of laims K =,,,,5. When there is no hange in the number of laims filed, the risk premium gets heaper along with inreasing time. For example, if a poliyholder submits one laim with total loss is 8 in the first period of observation, the amount of risk premiums have to be paid is 75, it an be seen in Table. Furthermore, in the seond period, the poliyholder submits another laim with loss is. It makes the number of laims during the two periods are laims and the total loss is, so the poliyholder have to pay for 765 that an be seen in Table. 6

8 IOP Conf. Series: Earth and Environmental Siene (6) 6 doi:.88/755-5///6 Table. The result of risk premium with total severity is 8 Period Number of laim (K) (t) Table. The result of risk premium with total severity is Period Number of laim (K) (t) Bonus-Malus System with Trunated Distribution In order to see the effet of the trunated distribution on laim severity, the alulation of the risk premium refer to () if no laim filed, whereas if there is a laim filed refer to (9). The maximum bound of premiums that an be overed by insurane ompanies is u =. Claim severity x i is assumed to be equal to the average of the laim size, that is. Table shows that the risk premium pries depend on the laim frequeny variation. For example, in the beginning the poliyholder pays the premium for 49. If in the first period there is no aident, there will be a premium redution to the level 96. But, if a poliyholder have one aident, the poliyholder have to pay 476 if the amount of laim is more than, otherwise have to pay 97. 7

9 IOP Conf. Series: Earth and Environmental Siene (6) 6 doi:.88/755-5///6 Table. The result of risk premium with u = Number of laim (K) Period (t) Number of laim with the severity less than u (n) Period (t) Table. The result of risk premium with u = (ontinue) Number of laim (K) 4 5 Number of laim with the severity less than u (n) CONCLUSION This paper onsider Bonus-Malus system with the laim frequeny distribution is geometri that ompound distribution of Poisson distribution and exponentially distribution and the laim severity distribution is trunated Weibull whih ompound distribution of trunated exponentially distribution and Levy distribution. Based on the results, it an be shown that the risk premium depend on the level of risk. When the total severity are less than the maximum bound of severity or M N + u (K - n), the risk premium based on Weibull distribution will be equal to the one based on trunated Weibull distribution. Where M is the total severity, u is the maximum bound that overed by insurane ompany, N is the total severity whih the severity are lower than bound u, K is the total laim frequeny, and n is the laim frequeny whih the frequeny are lower than bound u. In other hand, when the total of severity are greater than the maximum bound of the severity or M > N + u (K n), the risk premium based on the Weibull distribution is more expensive than the one based on trunated Weibull distribution. 8

10 IOP Conf. Series: Earth and Environmental Siene (6) 6 doi:.88/755-5///6 5. Referenes [] Georges D and Charles V 989 A generalization of automobile insurane rating models: the negative binomial distribution with a regression omponent J. ASTIN Bulletin [] Lu T 99 Using the poisson inverse gaussian in bonus-malus systems J. ASTIN Bulletin [] Mehmet M and Yasemin S 5 On a bonus malus system where the laim frequeny distribution is geometri and the laim severity distribution is pareto J. Haettepe Journal of Mathematis and Statistis [4] Niholas E F and Spyridon D V Design of optimal bonus-malus systems with a frequeny and a severity omponent on an individual basis in automobile insurane J. ASTIN Bulletin -. [5] Rahim M and Hossein H 9 Generalized bonus-malus systems with a frequeny and a severity omponent on an individual basis in automobile insurane J. ASTIN Bulletin [6] Stuart A K, Harry H P and Gordon E W Loss Models (United States of Ameria: John Wiley & Sons). [7] Weihong N, Corina C and Athanasios A P 4 Bonus-malus systems with weibull distributed laim severities J. Annals of Atuarial Siene