PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM

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1 PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics ad Risk Maagemet YSU, Armeia The paper itroduces a alterative approach to the bous-malus system costructio. I the preseted model the premium calculatio is based o the previous premium ad o the claim severity compoet as well. Followig to the cocept of a optimal bous-malus system the ecessary ad sufficiet coditio for the aggregate premiums to be a martigale series have bee foud out. Thus the preseted approach differs totally from the usual bous-malus classes as well as from the systems, where the severity of the claim is omitted. MSC2010: 60G42; 62P05. Keywords: optimal bous-malus, aggregate premium, martigale. Itroductio. Bous-Malus system (BMS) is a premium calculatio system, which pealizes the policyholders resposible for oe or more claims by a premium surcharge (malus) ad rewards the policyholders, who had a claim free year by awardig discout of the premium (bous). The majority of optimal BMS preseted up to ow i the actuarial literature [1] assig to each policyholder a premium based o the umber of his accidets. I this way a policyholder, who had a accidet with a small size of loss is pealized ufairly i the same way with a policyholder, who had a accidet with a big size of loss. From practical poit of view it is well kow several cosiderable disadvatages of existig BMS possess, which are difficult or eve impossible to hadle withi the traditioal theory of experiece ratig [2]. I particular, the existig systems are based o the followig characteristic: the claim amouts are omitted as a posterior tariff criterio. This characteristic leads to the followig disadvatages. i. Regardig a occurred claim the future loss of bous will i may cases exceed the claim amout. ii. I may cases it gives the policyholder a fillig of ufairess, especially whe a policyholder make a small claim ad the other oe a large, they have the same pealty withi the same risk group.

2 16 Proc. of the Yereva State Uiv., Phys. ad Math. Sci., 2015, 1, p iii. Cosequece of (i ad ii) is the well-kow bous huger behavior of policyholders. iv. Bous huger behavior leads to a asymmetric iformatio betwee the policyholders, isurers ad regulators. May authors have focused o the disadvatages metioed above, i particular the problem of bous huger [3, 4], the problem of asymmetric iformatio [5, 6]. The aim of this paper is to itroduce a alterative bous-malus approach, which at least theoretically elimiates the most importat oes of these disadvatages. A Optimal Bous-Malus Premium Costructio Cosideratios. Cocept of a optimal BMS [3] has bee used i this paper. A BMS is called optimal if it is: a) fiacially balaced for the isurer, that is the average total amout of bouses is equal to the average total amout of maluses; b) fair for the policyholder, that is each policyholder pays a premium proportioal to the risk that he imposes to the pool. Optimal BMS ca be divided i two categories: those based oly o a posteriori classificatio criteria ad those based both o a priori ad a posteriori classificatio criteria. The majority of BMS desiged is based o the umber of accidets disregardig their severity. Thus, let us cosider the desig of a optimal BMS based o a claim severity compoet. Notatios ad Defiitios. Based o the probabilistic approach ad followig axioms of probability theory suppose that all the observatios made o a probability space (Ω,F,P), where Ω is the set of elemetary outcomes ω, F is a σ-algebra of subsets of Ω ad P is a give probability measure o F. Time ad dyamics have a sigificat role for the model costructio, so suppose that a sequece of σ-algebra { F is give: } 0 F 0 F 1 F F. So the basic probabilistic model is (Ω,F,(F ) 0,P) filtered probability space. D e f i i t i o 1. Let X 0,X 1,... be a series of radom variables give o (Ω,F,(F ) 0,P). If X is F -measurable for ay 0, the we will say that X = (X,F ) 0 collectio or just X = (X,F ) is a stochastic series. D e f i i t i o 2. If for X = (X,F ) stochastic series X is F 1 -measurable as well, it will be writte as X = (X,F 1 ) assumig F 1 = F 0 ad X will be called predictable series. D e f i i t i o 3. Let X : Ω R, the X = (X,F ) stochastic series will be called martigale, if for ay 0: a) E X < ; b) E (X +1 F ) = X. From properties of coditioal expectatio it is obvious that the secod property of martigale defiitio ca be rephrased by the followig: A X +1 dp = X dp A

3 Gulya A. G. A Alterative Model for Bous-Malus System. 17 for all 0, A F ad specially if A = Ω, the it ca be writte that b*) [7] EX = EX 1 = = EX 1 = EX 0. Aggregate Premium as a Martigale. Let us cosider a portfolio of a isurace product. Suppose that a series of idepedet radom variables Y 1,Y 2,... are yearly aggregate claim losses of that portfolio, give o a (Ω,F,(F ) 0,P) filtered probability space, where F 0 = {/0,Ω} ad F = σ {Y 1,Y 2,...Y }. Suppose that Y 1,Y 2,... radom variables are so that EY < coditio is satisfies for ay Y 0 ad for all 1. Let deote P 0,P 1,... the radom variables, which describe yearly aggregate premium charge for that portfolio, where P 0 =cost is give ad the other members of that series are defied by the followig formulae: P = (1 α )P 1 + β Y, (1) where P is a aggregate premium collected for -th year. Y is a aggregate claim loss for the give portfolio withi ( 1;) time iterval. It is ecessary to ote that Y is idepedet of P 1 for all 1. α = (α,f 1 ) 1 is a predictable series, which will be called a series of bous factors. β = (β,f 1 ) 1 is also a predictable series, which will be called a series of malus factors. L e m m a. The series P = (P,F ) costructed by the formulae (1) is a martigale if ad oly if: α P 1 = β EY (2) P r o o f. Necessity. Let series α ad β be F 1 -measurable ad the series costructed by (1) be a martigale. Let calculate E (P F 1 ) usig F 1 -measurability of α ad β series, idepedece of Y s, properties of coditioal expectatio ad the defiitio of martigale: E (P F 1 ) = (1 α )P 1 + β EY. For the (b) coditio of martigale defiitio it is ecessary that: β = α P 1 EY. It is obvious that this result is equivalet to (2). Sufficiecy. Let α ad β be F 1 -measurable series ad the relatioship (2) holds. Let costruct a series of P accordig to (1). Substitutig (2) i (1) ad makig some rearragemets, we get: It is easy to see that EP = EP 0 <. As Y is idepedet of F 1, the P = P 1 + β (Y EY ). (3)

4 18 Proc. of the Yereva State Uiv., Phys. ad Math. Sci., 2015, 1, p E (P F 1 ) = P 1 + β (E (Y F 1 ) EY ) = 0. Offerig a isurace product, the isurace compay wishes to have a fiacially stable model. For that purpose it states its strategy for that risk portfolio ad defies a premium level. The aggregate premium received for that portfolio must be sufficiet to cover some level of aggregate claim with appropriate probability, which is defied i the compay s strategy. This meas that the compay states some Y c critical value of aggregate claim ad some ε probability ad defies the aggregate premium P, so that it is greater tha Y c critical value with 1 ε probability. It mathematically expresses as: P(P > Y c ) = 1 ε. (4) So Y c is a ε order quatile of radom variable Y. Fidig α ad β preseted i (1) is our mai purpose, but we ca ot do it with the help of Lemma oly, which gives their relatioship (2). Suppose that the distributio fuctio of aggregate claim is give Y F Y (x). Let s fid α ad β with the help of expressios (2), (4) ad fiacially stable ad optimal BMS cocepts. Reformig (4) ad takig ito cosideratio (1) we get: Or usig iverse distributio fuctio: where ( ) Yc (1 α )P F 1 Y =ε. β Y c (1 α )P 1 =F 1 Y β (ε), FY 1 (ε) = if{x R,F Y (x) ε}. Substitutig (2) ad makig some rearragemets, we get: ad β = F 1 Y (ε) EY (5) FY 1 EY. (6) (ε) EY P 1 Some Examples. Example 1. Suppose that the yearly aggregate claims of a isurace compay are distributed expoetially with λ rate: Y Exp(λ). Let s fid α ad β for this case. Usig the characteristics of a expoetial distributio fuctio we get: β = λ (P 1 Y c ) 1 + l(1 ε),

5 Gulya A. G. A Alterative Model for Bous-Malus System. 19 P 1 Y c P 1 (1 + l(1 ε)). Example 2. Suppose that the yearly aggregate claims of a isurace compay have a Pareto distributio with parameters µ ad λ (Y Pareto(µ,λ)). For fidig α ad β we eed the iverse of that distributio fuctio ad the expectatio. So, β = λ(1 ε) 1 µ λ µ µ 1 ). P 1 (λ (µ 1)(1 ε) µ 1 λ µ Coclusio. Formulae (1) preseted i this paper have theoretical ad practical sigificace as well. It ca be used as a model for premium calculatio i bousmalus system which elimiates (i iv) disadvatages of may BMS s. The model costructed accordig to the secod coditio of BMS is optimal. O the other had, the series of bous ad malus factors are calculated accordig to the first coditio of BMS optimality usig the cocept of martigale., Received R E F E R E N C E S 1. Lemaire J., Hogmi Zi. A Comparative Aalysis of 30 Bous-Malus Systems. // ASTIN Bulleti, 1994, v. 24, 2, p Holta J. Bous Made Easy. // ASTIN Bulleti, 1994, v. 24, 1, p Norberg R. Credibility Premium Plas which Make Allowace for Bous Huger. // Scadiavia Actuarial Joural, 1975, 2, p Lemaire J. Automobile Isurace Actuarial Models. Bosto: Kluwer Nijhoff, Doelly C. et al. Asymmetric Iformatio, Self Selectio ad Pricig of Isurace Cotracts: The Simple No-Claims Case. // Jourale of Risk ad Isurace, 2014, v. 81, p Dioe G. et al. A Review of Recet Theoretical ad Empirical Aalyses of Asymmetric Iformatio i Road Safety ad Automobile Isurace. // SSRN, 2012, 7. Shiryaev A.N. Probability 1. M.: MCCME, 2004, p (i Russia).

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