A Bonus-Malus System as a Markov Set-Chain

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1 A Bonus-alus System as a arov Set-Chan Famly and frst name: emec algorzata Organsaton: Warsaw School of Economcs Insttute of Econometrcs Al. epodleglosc Warszawa Poland Telephone number: Fax number: E-mal: Abstract The purpose of ths paper s to analyse a bonus-malus system wthn the framewor of the theory of an ergodc arov set-chan. It s shown that ths type of a arov chan enables to evaluate the system, even n ts steady-state, under the assumpton that ts transton probabltes change n a defnte range. We set up a model that allows for determnng the consequences of changes n the clam frequency of a polcyholder. As a numercal example we examne the bonus-malus system employed by one of the Polsh nsurance company. Keywords: arov set-chan, bonus-malus system, clam frequency

2 . Introducton In the analyss of a bonus-malus system t s commonly assumed that the clam frequency of an ndvdual polcyholder remans unaltered. Ths assumpton ensures constant transton probabltes and enables to model the system wth the use of a homogeneous arov chan (see e.g. emare [995]). However, t s nown that the clam frequency of a drver may fluctuate from tme to tme for varous reasons. Therefore, n ths paper we relax the above mentoned assumpton and apply an ergodc arov set-chan defned by Hartfel [998]. The arov set-chan consttutes the specfc generalsaton of the dea of classcal arov chans. Its fundamental assumpton conssts n allowng for changes of transton probabltes at each step, though these changes are restrcted by some lower and upper bounds. It s assumed that the transton probabltes belong to a gven compact set, usually defned as an nterval, and ther exact values are not nown. Thus, the varablty of the probabltes s possble, whch may broaden the scope of the analyss of phenomena modelled to date n the framewor of the theory of a homogeneous arov chan. Although a arov set-chan can be treated as a nd of a nonhomogeneous arov chan, for the purpose of ts applcaton there s no need to determne rules of transton matrx changes at each step. oreover, t s easy to examne ts long run behavour, whch n case of a nonhomogeneous arov chan s often dffcult and restrcts ts use. Ths paper s organsed as follows. In Secton 2 the model of a bonus-malus system s constructed wthn the framewor of the arov set-chan theory. Secton 3 s devoted to a numercal example, whch shows how the fluctuaton of the clam frequency of a polcyholder may nfluence some asymptotc measures used to evaluate bonus-malus systems. Secton 4 provdes fnal conclusons. Snce the concept of Hartfel [998] s relatvely new and so far has not been extensvely appled, Appendx descrbes brefly the theoretcal bass and propertes of an ergodc arov set-chan. 2. The model of a bonus-malus system As stated n emare [995], a system employed n automoble nsurance s called a bonusmalus system when: 2

3 - all polcyholders of a gven tarff group are dvded nto a fnte number of classes, denoted by C ( =, 2,..., r ), and ther premum depends only on the class they belong to, and - the class of a polcyholder for a gven perod (usually a year) s determned unquely by the class n the precedng perod and the number of clams reported n that perod. C [ ] Such a system s defned by the ntal class, premum scale b,b,...,b, where = b 2 r b s the premum level n class C, as well as transton rules.e. rules governng the transfer of a polcyholder from one class to another when the number of hs or her clams s nown. where The transton rules are represented by means of T = [ () ], t ( ) t = f f T ( ) = T ( ) r r matrces and T ( ) = denotes the transfer of a polcyholder reportng clams from class C nto class C n the next perod. The probablty of movng from C to C for a polcyholder wth clam frequency λ s gven by = ( ) p ( λ ) = p ( λ ) t, where p ( λ ) s the probablty that a drver wth clam frequency λ has clams n one perod. Under the assumpton that the clam frequency of an nsured s statonary n tme, a fnte homogeneous arov chan wth the state space S = {, 2,..., r} and transton matrx [ ] p ( λ ) = = P ( λ ) = p ( λ ) T s a model of a bonus-malus system (see emare [995]). In ths paper we restrct our attenton merely to these bonus-malus systems, whose models are rreducble ergodc fnte and homogeneous arov chans. It s worth mentonng that most exstng bonus-malus systems form such arov chans. The assumpton of the constant clam frequency of a drver, whch s ndspensable for the analyss of a bonus-malus system wthn the framewor of the homogeneous arov chan theory, seems unrealstc. In fact, the clam frequency may change over tme due to nsurance companes actons, changes n the drvng abltes and behavour of a polcyholder as well as external factors such as weather condtons or a state of roads. Irrespectve of the 3

4 reason for these changes, the need for ther evaluaton arses. An ergodc arov set-chan proves to be helpful n ths respect. In settng up the model of a bonus-malus system, beng the arov set-chan, whch enables us to determne the consequences of changes n the clam frequency of a polcyholder, we need the followng assumptons: () the number of clams of a polcyholder charactersed by λ conforms to a Posson dstrbuton; (2) λ and λ such that ( < λ ) < λ < () are the lower and upper bounds on the nterval of the clam frequency varablty. Snce the actual clam frequency hardly ever exceeds the value, condton () s not restrctve n the analyss of exstng bonus-malus systems. However, t s necessary that the followng relatonshp be satsfed ( ) ( ) p(λ ) [ mn{ p(λ ); p(λ )},max{ p(λ ); p(λ )}], (2) ( ) where λ [ λ, λ ] and p (λ ) for =,, 2,... s the probablty from the Posson dstrbuton. It s easy to verfy that p (λ ) and p (λ ) for =, 2,... are respectvely a decreasng and ncreasng functon of λ n the nterval (,), whch ensures that relaton (2) s vald. Under assumptons () (2) we can determne the matrx nterval that comprses all transton matrces of the arov set-chan, the model of a bonus-malus system. ower and upper bounds on that nterval can be expressed as ( ) ( ) = K mn { p,p } T = [ mn{ p,p }], (3) = ( ) ( ) = max { p,p } T = [ max{ p,p }], (4) = where upper ndces () and (2) ndcate that a gven probablty has been calculated for λ and λ 2 respectvely. ote that K and are nonnegatve From relaton (2) and formulas (3) and (4) we obtan ( ) P ( λ ) [ K, for λ [ λ, λ ]. r r matrces such that K. ( Hence, the nterval [ K, contans all, correspondng to each [ ) ( 2 λ λ, λ ) ], transton matrces of rreducble ergodc fnte homogeneous arov chans that are models of the 4

5 same bonus-malus system and dffer merely n the assumed value of the clam frequency of a polcyholder. Theorem et a arov set-chan be a model of a bonus-malus system under assumptons ()-(2). et [ K,, where K and are gven by formulas (3) and (4), be ts transton matrx nterval. Then the arov set-chan s ergodc. Proof: ote that the arrangement of all nonzero elements n each matrx from the nterval [ K, [ K, s dentcal and depends on the transton rules expressed n matrces. The nterval comprses one-step transton matrces of rreducble ergodc fnte and homogeneous T arov chans. Each of these matrces s both rreducble and prmtve and n ther canoncal form the arrangement of zero and postve elements s dentcal. As the same arrangement of zero and postve elements concerns any matrx from the nterval [ K,, ncludng K and, all matrces belongng to ths nterval are rreducble and prmtve. Ths fact s a suffcent condton that the arov set-chan s ergodc (see Defntons A.5-A.6 and Theorem A. n Appendx). The mportant feature of the matrx nterval s ts tghtness. Ths property s partcularly requred whle applyng the H-o method, an algorthm for computng bounds on each step transton probabltes, statonary dstrbuton as well as mean frst passage tmes (see Hartfel [99], [998]). It can be proved that the transton matrx nterval [ K, of the consdered model s tght. To show that let us defne two sets of ndces. Suppose that for, S ( ) ( ) A = { : mn{ p, p } = p }, ( ) B = { : mn{ p, p } = p }. It s easly seen that for the above sets the condtons hold: ( ) A = { : max{ p, p } = p }, ( ) ( ) B = { : max{ p, p } = p }. The sets A and B are dsont. Ther defnton and the fact that K mplcate the followng relatonshps: - f t for S then p s an element of the matrx K and p 2 ) s an element of the A matrx ; t ( t 5

6 - f t for S then p 2 s an element of the matrx K and p ) s an element of the B t matrx. As an mmedate consequence of the above mplcatons we obtan ( 2 p t pt ) for t A and,t S, (5) ( t for t and p t pt B,t S. (6) Furthermore, referrng to the stochastc property of the transton matrces P( λ ) = [ p ( ) ] and P( λ ) = [ p 2 t A + ], we have ( ) ( ) p p = and p p = for,t S. (7) t t B t t A t + t B t Theorem 2 et a arov set-chan be a model of a bonus-malus system under assumptons ()-(2). et [ K,, where K and are gven by formulas (3) and (4), be ts transton matrx nterval. Then the nterval [ K, s tght. Proof: Snce [ K = { P : P [ K, ] for, 2,.., r}, where P, K, are -th rows of, = P, K, respectvely, t suffces to show that for all S -th rows of K and are bounds on tght vector ntervals. Hence, we have to prove that for each vector nterval K = ] and = q ], the followng condtons hold: [ [ [ K, ] + q t and q + t for all. (8) t t p b, where If the probablty taen from the matrx K s a functon of the clam frequency λ then usng relatons (5) and (7) we get ( ) + ( ) ( ) ( ) p b pt pt pb pt pt = for all,b,t S, t A and f t depends on λ t B t A t B then from (6) and (7) we have p b pt pt pb pt pt = for all,b,t S. t A p b t B t A If the probablty taen from the matrx s a functon of the clam frequency λ then relatons (6) and (7) mply that ( ) + ( ) ( ) ( ) p b pt pt pb pt pt = for all,b,t S, t A t B t A t B t B 6

7 and f t depends on λ then relatonshps (5) and (7) lead us to + ( ) p b pt pt pb pt pt = for all,b,t S. t A t B t A Thus, condtons (8) are fulflled, whch means that the vector ntervals bounded by -th rows of K and and, consequently, the matrx nterval [ K, are tght. t B 3. A numercal example As mentoned n the precedng Secton, the arov set-chan theory enables us to examne the consequences of the clam frequency changes wthn a gven nterval. In order to llustrate the applcaton of the model descrbed n Secton 2, we analyse the bonus-malus system currently employed n frst-party coverage nsurance by Powszechny Zalad Ubezpeczen SA (PZU), the Polsh nsurance company. ost calculatons presented n ths Secton were obtaned usng our own programmes devsed n ATAB 6.. The bonus-malus system of PZU conssts of 3 classes. ew polcyholders enter the system n class C 5. The specfc premum levels for each class as well as transton rules are provded n Table. The propertes of the system allow for modellng t as an rreducble ergodc fnte and homogeneous arov chan and therefore - as a arov set-chan. Table. Bonus-malus system of PZU Class number Premum level (n percentage) Class number after or more clams Source: General Condtons for Frst-Party Coverage Insurance of Powszechny Zalad Ubezpeczen SA establshed on 25th Aprl 23 7

8 Snce the average clam frequency n automoble frst-party coverage nsurance n Poland has been close to.5 over the recent years, let us consder a polcyholder wth the clam frequency varyng from. to.2. Havng assumed that the number of clams follows the Posson dstrbuton, by the applcaton of formulas (3) and (4) we get bounds K and on the nterval comprsng all possble transton matrces of the ergodc arov set-chan, the model of the bonus-malus system for the consdered polcyholder. By Theorem 2 the obtaned nterval s tght, whch s essental to apply the H-o method. Usng the method we calculate lower and upper lmt bounds and H, whch are ran one matrces wth the followng rows: l = h = The above vectors are bounds on the nterval of all possble statonary probablty dstrbutons, whose elements can be nterpreted as the probabltes that the polcyholder belongs to a gven class, once full statonarty has been reached. The varablty range of these probabltes for the drver wth the clam frequency n the nterval <.,.2> s dversfed. Generally, n hgher classes (wth lower premums) the steady state probabltes are hgher and more senstve to the clam frequency changes. Thus, the drver has a better chance of beng n a hgh-dscount class, but the probablty of ths event s subect to larger varatons than the probablty of beng n a low-dscount class. The analyss of bounds on mean frst passage tmes may also provde valuable nformaton. In the context of the model of a bonus-malus system each of these tmes, denoted by m, ndcates an average tme needed by the polcyholder from class C to reach C for the frst tme. For our numercal example, the matrces of lower and upper bounds on these tmes l and h are as follows: 8

9 l = h = It can be seen that the varablty range of the mean frst passage tmes s strongly dversfed. It s relatvely narrow for the mean tmes of the frst promoton to hgh-dscount classes as well as of movng between classes:,,,, C. For the rest of tmes the range s wde, ts spread exceeds 5 years and n some cases even 67 years. Such dversty of the varablty ranges of the mean tmes ndcates a dfferent level of ther senstvty to the changes n the clam frequency. The fluctuaton of the clam frequency n the nterval <.,.2> can result n maxmal change of the mean tme equal only to.2 (n the case of transfer from to C ) as well as to over years (n the case of transfer from to C ). C3 C 2 C9 C C C2 3 It s worthwhle payng attenton to the values of the mean frst passage tmes. It should be noted that some transfers of the polcyholder wth the clam frequency n the nterval <.,.2> are practcally mpossble. It s hard to expect that the polcyholder from one class wll move to the other, f the lower bound on the mean tme of such a transfer exceeds 5 years. Such a value s taen by approxmately 4% of the elements of the matrx l. These are manly lower bounds on the mean tmes of downgradng n the class 9

10 herarchy. It means that n the PZU system t s comparatvely dffcult for the polcyholder to reach the class wth hgher premum. On the other hand, n most cases the expected tmes for lowerng a premum are sgnfcantly shorter and hence the transfer to hgher-dscount classes s feasble. For nstance, the mean tme of the frst passage from the ntal class C 5 to the best one C 3 amounts to about years at best and to 8 years at worst, whch s usually shorter than the whole perod of havng and nsurng a car. 4. Conclusons In ths paper we propose the applcaton of an ergodc arov set-chan to the analyss of a bonus-malus system. We relax the assumpton of the constant clam frequency of a polcyholder, whch s necessary whle modellng the system wth the use of a homogeneous arov chan. It s shown that the theory of the presented chan broadens the scope of studes carred out n the framewor of the classcal arov chan theory. It enables to examne the consequences of clam frequency changes wthn a gven nterval. It provdes tools for determnng the varablty range of each step transton probabltes, statonary dstrbuton as well as mean frst passage tmes. Therefore, we can analyse varous characterstcs senstvty and ntensty of ther reacton to the changes n clam frequency. The obtaned nformaton may be crucal to nsurance companes havng nterest not only n system evaluaton but also n predctng changes n ts performance caused by the factor whch they can nfluence to a lmted extent. Appendx The descrpton of a arov set-chan presented n ths Appendx s based on Hartfel s [998] monograph. Defnton A. et be a compact set of r r stochastc matrces. et consder arov chans wth the state space S = {, 2,..., r}, havng all ther transton matrces n. A arov set-chan s the sequence 2 3,,,...,

11 where = { P : P = P P P, where P for all =, 2,..., } for each. The set 2 contans all possble -step transton matrces obtaned provded that transton matrces at frst step belong to the set. Accordng to Defnton A. a arov set-chan may be treated as a nonhomogeneous arov chan havng each transton matrx n step are not determned unquely. Snce the set Therefore, n a partcular case t can be defned as an nterval. Defnton A.2 The matrx nterval s an nterval [ K, = { P : K P },. ote that ts transton matrces at each s complex, t s closed and bounded. where P = [ p ] denotes a r r stochastc matrx and K = ] and = q ] are [ [ nonnegatve r r matrces such that K. As the nterval [ K, can be constructed by rows, t s useful to defne also a vector nterval. Defnton A.3 The vector nterval s an nterval where [, q] { x : x q} x = [ x =, vectors such that q. ] s a r stochastc vector and = ] and q = q ] are nonnegatve r The mportant feature of the descrbed ntervals s ther tghtness. [ [ Defnton A.4 A matrx nterval [ K, s tght f = mn p and q max p P [K, P [ K, = for all and. A vector nterval [, q] s tght f = mn x and q max x x [, q] x [, q] = for all. It s easy to show that for a tght vector nterval the followng condtons hold: + q t and q + t for all. t t If s a tght nterval [ K, then K and are column tght component bounds on. Henceforth we restrct our attenton merely to arov set-chans determned by a matrx nterval [ K,.

12 arov set-chans are classfed nto ergodc, regular and absorbng, analogously to classcal arov chans. Tang nto account the propertes of most bonus-malus systems, n ths paper we focus only on an ergodc arov set-chan. So as to defne ths type of chans, we need frst to ntroduce the term of an ergodc class. The decomposton of the state space S = {, 2,..., r} of a arov set-chan s based on the structure of the upper bound of the matrx nterval [ K,. Through smultaneous permutatons of rows and columns, can be put nto the canoncal form: = n+ s, 22 n+, 2 s 2 O nn n+,n sn n+,n+ s,n+ s,s where n, s a rreducble matrx for all ss, =, 2,..., n and f t > n then for some =, 2,..., t. It s also assumed that all matrces n the nterval [ K, have undergone the same smultaneous row and column permutatons. The defnton of an ergodc class and state follows. Defnton A.5 et consder a arov set-chan determned by a matrx nterval [ K,. et S be the class of states correspondng to and, consequently, to the nterval of submatrces K, ]. Then class S for n and each of ts states are called ergodc f lm [ K, ] exsts and each matrx n the lmt s ran. theorem. t [ Whle determnng f the class s ergodc, t s convenent to refer to the followng Theorem A. Under the assumptons of Defnton A.5, class S for n s ergodc f K s prmtve. ow we are n a poston to present the followng defnton. Defnton A.6 A arov set-chan s ergodc f t has only one class and that class s ergodc. One of the most mportant propertes of an ergodc arov set-chan s ts convergence. 2

13 Theorem A.2 If a arov set-chan wth the compact set of transton matrces s ergodc, then lm =, where s a compact set of ran one matrces. If the set converges to the set and and H are lower and upper bounds on respectvely, then the sequences }, H } are convergent: lm =, { { lm H = H. We call and H lower and upper lmt bounds on. ote that and H are ran one matrces and ther rows consttute bounds on the set of statonary probablty dstrbutons. In order to compute bounds on sets of transton matrces at each step, Hartfel [99, 998] proposed the applcaton of the H-o method. It s an approxmate teratve algorthm that conssts n fndng column tght component bounds on component bounds on on the bass of column tght, where = 2, 3,... Hence, column tght component bounds on produce column tght component bounds on bounds on 3, and so on. 2, whch gve column tght component For an ergodc arov set-chan t s also possble to fnd bounds on mean frst passage tme, defned as m &&& &&& && & ], (A.) = [ P + 2PP np P2 Pn +... where P and & n P & n s the matrx formed from Pn by replacng ts -th row by the row of s. The matrces of lower and upper bounds on mean frst passage tmes are denoted by and h respectvely. As Hartfel and Seneta [994] proved, the sum n (A.) converges and ts lower and upper bound can be obtaned by applyng the algorthm based on the H-o method. l References Hartfel D. J. [99], Component bounds for arov set chan lmtng sets, Journal of Statstcal Computaton and Smulaton, vol. 38, 5 24 Hartfel D. J., Seneta E. [994], On the Theory of arov-set Chans, Advances n Appled Probablty, vol. 26, Hartfel D. J. [998], arov Set Chans, Sprnger Verlag, ew Yor 3

14 emare J. [995], Bonus malus systems n automoble nsurance, Kluwer Academc Publshers, Boston 4

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