DESIGN OF OPTIMAL BONUS-MALUS SYSTEMS WITH A FREQUENCY AND A SEVERITY COMPONENT ON AN INDIVIDUAL BASIS IN AUTOMOBILE INSURANCE ABSTRACT KEYWORDS

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1 DESIGN OF OPTIMAL BONUS-MALUS SYSTEMS WITH A FREQUENCY AND A SEVERITY COMPONENT ON AN INDIVIDUAL BASIS IN AUTOMOBILE INSURANCE BY NICHOLAS E. FRANGOS* AND SPYRIDON D. VRONTOS* ABSTRACT The maory of opmal Bonus-Malus Sysems (BMS) presened up o now n he acuaral leraure assgn o each polcyholder a premum based on he number of hs accdens. In hs way a polcyholder who had an accden wh a small sze of loss s penalzed unfarly n he same way wh a polcyholder who had an accden wh a bg sze of loss. Movaed by hs, we develop n hs paper, he desgn of opmal BMS wh boh a frequency and a severy componen. The opmal BMS desgned are based boh on he number of accdens of each polcyholder and on he sze of loss (severy) for each accden ncurred. Opmaly s obaned by mnmzng he nsurer s rsk. Furhermore we ncorporae n he above desgn of opmal BMS he mporan a pror nformaon we have for each polcyholder. Thus we propose a generalsed BMS ha akes no consderaon smulaneously he ndvdual s characerscs, he number of hs accdens and he exac level of severy for each accden. EYWORDS Opmal BMS, clam frequency, clam severy, quadrac loss funcon, a pror classfcaon crera, a poseror classfcaon crera.. INTRODUCTION BMS penalze he polcyholders responsble for one or more clams by a premum surcharge (malus) and reward he polcyholders who had a clam free year by awardng dscoun of he premum (bonus). In hs way BMS * Deparmen of Sascs, Ahens Unversy of Economcs and Busness, Passon 76, 0434, Ahens, Greece. E-mal for correspondence nef@aueb.gr and svronos@aueb.gr Ths work has been parally suppored by 96SYN 3-9 on Desgn of Opmal Bonus-Malus Sysems n Auomoble Insurance and he General Secreera of Research and Technology of Greece. The auhors would lke o hank he referees for her valuable commens. ASTIN BULLETIN, Vol. 3, No., 200, pp. -22

2 2 NICHOLAS E. FRANGOS AND SPYRIDON D. VRONTOS encourage polcyholders o drve carefully and esmae he unknown rsk of each polcyholder o have an accden. A BMS s called opmal f s:. fnancally balanced for he nsurer, ha s he oal amoun of bonuses s equal o he oal amoun of maluses. 2. Far for he polcyholder, ha s each polcyholder pays a premum proporonal o he rsk ha he mposes o he pool. Opmal BMS can be dvded n wo caegores: hose based only on he a poseror classfcaon crera and hose based boh on he a pror and he a poseror classfcaon crera. As a poseror classfcaon crera are consdered he number of accdens of he polcyholder and he severy of each accden. As a pror classfcaon crera are consdered he varables whose her values are known before he polcyholder sars o drve, such as characerscs of he drver and he auomoble. The maory of BMS desgned s based on he number of accdens dsregardng her severy. Thus frs le us consder he desgn of opmal BMS based only on he a poseror clam frequency componen... BMS based on he a poseror clam frequency componen Lemare (995) developed he desgn of an opmal BMS based on he number of clams of each polcyholder, followng a game-heorec framework nroduced by Bchsel (964) and Bühlmann (964). Each polcyholder has o pay a premum proporonal o hs own unknown clam frequency. The use of he esmae of he clam frequency nsead of he rue unknown clam frequency wll ncur a loss o he nsurer. The opmal esmae of he polcyholder s clam frequency s he one ha mnmzes he loss ncurred. Lemare (995) consdered, among oher BMS, he opmal BMS obaned usng he quadrac error loss funcon, he expeced value premum calculaon prncple and he Negave Bnomal as he clam frequency dsrbuon. Tremblay (992) consdered he desgn of an opmal BMS usng he quadrac error loss funcon, he zero-uly premum calculaon prncple and he Posson- Inverse Gaussan as he clam frequency dsrbuon. Coene and Doray (996) developed a mehod of obanng a fnancally balanced BMS by mnmzng a quadrac funcon of he dfference beween he premum for an opmal BMS wh an nfne number of classes, weghed by he saonary probably of beng n a ceran class and by mposng varous consrans on he sysem. Walhn and Pars (997) obaned an opmal BMS usng as he clam frequency dsrbuon he Hofmann s dsrbuon, whch encompasses he Negave Bnomal and he Posson-Inverse Gaussan, and also usng as a clam frequency dsrbuon a fne Posson mxure. As we see, all he BMS menoned above ake under consderaon only he number of clams of each polcyholder dsregardng her severy..2. BMS based on he a pror and he a poseror clam frequency componen The models menoned above are funcon of me and of pas number of accdens and do no ake no consderaon he characerscs of each ndvdual.

3 DESIGN OF OPTIMAL BONUS-MALUS SYSTEMS 3 In hs way as menoned n Donne and Vanasse (989), he premums do no vary smulaneously wh oher varables ha affec he clam frequency dsrbuon. The mos neresng example s he age varable. Suppose ha age has a negave effec on he expeced number of clams, would mply ha nsurance premums should decrease wh age. Premum ables derved from BMS based only on he a poseror crera, even hough are a funcon of me, do no allow for a varaon of age, even hough age s a sascally sgnfcan varable. Donne and Vanasse (989, 992) presened a BMS ha negraes a pror and a poseror nformaon on an ndvdual bass. Ths BMS s derved as a funcon of he years ha he polcyholder s n he porfolo, of he number of accdens and of he ndvdual characerscs whch are sgnfcan for he number of accdens. Pcech (994) and Sgalo (994) derved a BMS ha ncorporaes he a poseror and he a pror classfcaon crera, wh he engne power as he sngle a pror rang varable. Sgalo developed a recursve procedure o compue he sequence of ncreasng equlbrum premums needed o balance ou premums ncome and expendures compensang for he premum decrease creaed by he BMS ranson rules. Pcech developed a heursc mehod o buld a BMS ha approxmaes he opmal mer-rang sysem. Taylor (997) developed he seng of a Bonus-Malus scale where some rang facors are used o recognze he dfferenaon of underlyng clam frequency by experence, bu only o he exen ha hs dfferenaon s no recognzed whn base premums. Pnque (998) developed he desgn of opmal BMS from dfferen ypes of clams, such as clams a faul and clams no a faul..3. Allowance for he severy n BMS In he models brefly descrbed above he sze of loss ha each accden ncurred s no consdered n he desgn of he BMS. Polcyholders wh he same number of accdens pay he same malus, rrespecvely of he sze of loss of her accdens. In hs sense he BMS desgned n he above way are unfar for he polcyholders who had an accden wh a small sze of loss. Acually as Lemare (995) s ponng ou all BMS n force hroughou he world, wh he excepon of orea, are penalzng he number of accdens whou akng he severy of such clams no accoun. In he BMS enforced n orea he polcyholders who had a bodly nury clam pay hgher maluses, dependng on how severe he accden was, han he polcyholders who had a propery damage clam. The BMS desgned o ake severy no consderaon nclude hose from Pcard (976) and Pnque (997). Pcard generalzed he Negave Bnomal model n order o ake no accoun he subdvson of clams no wo caegores, small and large losses. In order o separae large from small losses, wo opons could be used:. The losses under a lmng amoun are regarded as small and he remander as large. 2. Subdvson of accdens n hose ha caused propery damage and hose ha cause bodly nury, penalzng more severely he polcyholders who had a bodly nury accden. Pnque (997) desgned an opmal BMS whch makes allowance

4 4 NICHOLAS E. FRANGOS AND SPYRIDON D. VRONTOS for he severy of he clams n he followng way: sarng from a rang model based on he analyss of number of clams and of coss of clams, wo heerogeney componens are added. They represen unobserved facors ha are relevan for he explanaon of he severy varables. The coss of clams are supposed o follow gamma or lognormal dsrbuon. The rang facors, as well as he heerogeney componens are ncluded n he scale parameer of he dsrbuon. Consderng ha he heerogeney also follows a gamma or lognnormal dsrbuon, a credbly expresson s obaned whch provdes a predcor for he average cos of clam for he followng perod. Our frs conrbuon n hs paper s he developmen of an opmal BMS ha akes no accoun he number of clams of each polcyholder and he exac sze of loss ha hese clams ncurred. We assumed ha he number of clams s dsrbued accordng he Negave Bnomal dsrbuon and he losses of he clams are dsrbued accordng he Pareo dsrbuon, and we have expanded he frame ha Lemare (995) used o desgn an opmal BMS based on he number of clams. Applyng Bayes heorem we fnd he poseror dsrbuon of he mean clam frequency and he poseror dsrbuon of he mean clam sze gven he nformaon we have abou he clam frequency hsory and he clam sze hsory for each polcyholder for he me perod he s n he porfolo. For more on hs subec we refer o Vronos (998). Our second conrbuon s he developmen of a generalzed BMS ha negraes he a pror and he a poseror nformaon on a ndvdual bass. In hs generalzed BMS he premum wll be a funcon of he years ha he polcyholder s n he porfolo, of hs number of accdens, of he sze of loss ha each of hese accdens ncurred, and of he sgnfcan a pror rang varables for he number of accdens and for he sze of loss ha each of hese clams ncurred. We wll do hs by expandng he frame developed by Donne and Vanasse (989, 992). Pnque (997) s sarng from a rang model and hen he s addng he heerogeney componens. We desgn frs an opmal BMS based only on he a poseror classfcaon crera and hen we generalze n order o ake under consderaon boh he a pror and he a poseror classfcaon crera. 2. DESIGN OF OPTIMAL BMS WITH A FREQUENCY AND A SEVERITY COMPONENT BASED ON THE A POSTERIORI CRITERIA I s assumed ha he number of clams of each polcyholder s ndependen from he severy of each clam n order o deal wh he frequency and he severy componen separaely. 2.. Frequency componen For he frequency componen we wll use he same srucure used by Lemare (995). The porfolo s consdered o be heerogeneous and all polcyholders have consan bu unequal underlyng rsks o have an accden. Consder ha

5 DESIGN OF OPTIMAL BONUS-MALUS SYSTEMS 5 he number of clams k, gven he parameer l, s dsrbued accordng o Posson(l), P ( kl) l -l l k e k! k 0,,2,3,... and l > 0 and l s denong he dfferen underlyng rsk of each polcyholder o have an accden. Le us assume for he srucure funcon ha l ~ gamma(a, ) and l has a probably densy funcon of he form: a- a l exp( - l) u () l, l > 0, a > 0, > 0 G() a wh mean E(l) a/ and varance Var (l) a/ 2. Then can be proved ha he uncondonal dsrbuon of he number of clams k wll be Negave Bnomal (a,), wh probably densy funcon a Pk () k + a -, k a kb k + l b + l mean equal o E(k)a/ and varance equal o Var(k) (a/) ( + /). The varance of he Negave Bnomal exceeds s mean, a desrable propery whch s common for all mxures of Posson dsrbuon and allows us o deal wh daa ha presen overdsperson. Le us denoe as! k he oal number of clams ha a polcyholder had n years, where k s he number of clams ha he polcyholder had n he year,,...,. We apply he Bayes heorem and we oban he poseror srucure funcon of l for a polcyholder or a group of polcyholders wh clam hsory k,...k, denoed as u(l k, k ). I s ha ( + ) l e u( l k,... k ) G ( a + ), + a + a- -( + ) l whch s he probably densy funcon of a gamma(a +, + ). Usng he quadrac error loss funcon he opmal choce of l + for a polcyholder wh clam hsory k,...k wll be he mean of he poseror srucure funcon, ha s l ( k,..., k ) a l a, where l a b l. a + l () From he above s clear ha he occurrence of accdens n years us necessaes an updae of he parameers of gamma, from a and o a + and + respecvely and he gamma s sad o have he mporan propery of he sably of he srucure funcon as he gamma s a conugae famly for he Posson lkelhood.,

6 6 NICHOLAS E. FRANGOS AND SPYRIDON D. VRONTOS 2.2. Severy componen Le us consder now he severy componen. Le x be he sze of he clam of each nsured. We consder as y he mean clam sze for each nsured and we assume ha he condonal dsrbuon of he sze of each clam gven he mean clam sze, xy, for each polcyholder s he one parameer exponenal dsrbuon wh parameer y, and has a probably densy funcon gven by fxy ( ) - x y y $ e for x > 0 and y > 0. The mean of he exponenal s E( x y) y and he varance s var( x y) y 2. The mean clam sze y s no he same for all he polcyholders bu akes dfferen values so s naural our pror belef for y o be expressed n he form of a dsrbuon. Consder ha he pror dsrbuon of he mean clam sze y s Inverse Gamma wh parameers s and m and probably densy funcon, see for example Hogg and lugman (984) gven by gy () - y m $ e m ( ) G() s. m y s $ + The expeced value of he mean clam sze y wll be: Ey (). s m - The uncondonal dsrbuon of he clam sze x wll be equal o: # PX ( x) fx ( y) $ gydy ( ) 0 3 s - s - s$ m $ ( x+ m) whch s he probably densy of he Pareo dsrbuon wh parameers s and m. Thus, one way o generae he Pareo dsrbuon s he followng: f s for he sze of each clam gven he mean clam sze x y ha x y ~ Exponenal(y) and for he mean clam sze y of each polcyholder ha y ~ Inverse Gamma(s,m) hen s for he uncondonal dsrbuon of he clam sze x n he porfolo ha x ~ Pareo(s,m). In hs way, he relavely ame exponenal dsrbuon ges ransformed n he heavy-aled Pareo dsrbuon and nsead of usng he exponenal dsrbuon whch s ofen napproprae for he modellng of clam severy we are usng he Pareo dsrbuon whch s ofen a good canddae for modellng he clam severy. Takng he mean clam sze y dsrbued accordng he Inverse Gamma, we ncorporae n he model he heerogeney ha characerzes he severy of he clams of dfferen polcyholders. We should noe here ha such a generaon of he Pareo dsrbuon does no appear for he frs me n he acuaral leraure. Such a use can be found for example n Herzog (996). To he bes of our knowledge s he frs me s used n he desgn of an opmal BMS.

7 DESIGN OF OPTIMAL BONUS-MALUS SYSTEMS 7 In order o desgn an opmal BMS ha wll ake no accoun he sze of loss of each clam, we have o fnd he poseror dsrbuon of he mean clam sze y gven he nformaon we have abou he clam sze hsory for each polcyholder for he me perod he s n he porfolo. Consder ha he polcyholder s n he porfolo for years and ha he number of clams he had n he year s denoed wh k,by! k s denoed he oal number of clams he has, and by x k s denoed he clam amoun for he k clam. Then he nformaon we have for hs clam sze hsory wll be n he form of a vecor x, x 2,..., x k and he oal clam amoun for he specfc polcyholder over he years ha he s n he porfolo wll be equal o! x k. Applyng Bayes k heorem we fnd he poseror dsrbuon of he mean clam sze y gven he clams sze hsory of he polcyholder x,..., x k and s ha: g( y x,..., ) ( m +! x ) x k $ e k y + s+ ( ) $ G ( + s) ( m +! x ) k k k m +! x k - y whch s he probably densy funcon of he Inverse Gamma(s +, m +! x k ). Ths means ha he occurrence of clams n years wh aggregae k clam amoun equal o! x k us necessaes for he dsrbuon of he mean k clam sze an updae of he parameers of he Inverse Gamma from s and m o s + and m +! x k respecvely and he Inverse Gamma dsrbuon s sad k o have he mporan propery of beng conugae wh he exponenal lkelhood. The mean of he poseror dsrbuon of he mean clam sze wll be: E( x y)! m+ xk k s+ - and he predcve dsrbuon of he sze of he clam of each nsured x wll be also a member of he Pareo famly. k (2) 2.3. Calculaon of he Premum accordng he Ne Premum Prncple As shown, he expeced number of clams l + (k,..., k ) for a polcyholder or a group of polcyholders who n years of observaon have produced clams wh oal clam amoun equal o! x k s gven by () and he expeced clam severy y + (x,..., x ) s gven by (2). k

8 8 NICHOLAS E. FRANGOS AND SPYRIDON D. VRONTOS Thus, he ne premum ha mus be pad from ha specfc group of polcyholders wll be equal o he produc of l + (k,..., k ) and y + (x,..., x ),.e. wll be equal o! m+ xk Premum a k + + $ s + - In order o fnd he premum ha mus be pad we have o know:. he parameers of he Negave Bnomal dsrbuon a and, (see Lemare (995) for he esmaon of he parameers of he Negave Bnomal) 2. he parameers of he Pareo dsrbuon s and m (see Hogg and lugman (984) for he esmaon of he parameers of he Pareo dsrbuon) 3. he number of years ha he polcyholder s under observaon, 4. hs number of clams and 5. hs oal clam amoun! x k. k All of hese can be obaned easly and akng under consderaon ha he negave bnomal s ofen used as a clam frequency dsrbuon and he Pareo as a clam severy dsrbuon hs enlarges he applcably of he model. (3) 2.4. Properes of he Opmal BMS wh a Frequency and a Severy Componen. The sysem s far as each nsured pays a premum proporonal o hs clam frequency and hs clam severy, akng no accoun, hrough he Bayes heorem, all he nformaon avalable for he me ha he s n our porfolo boh for he number of hs clams and he loss ha hese clams ncur. We use he exac loss x k ha s ncurred from each clam n order o have a dfferenaon n he premum for polcyholders wh he same number of clams, no us a scalng wh he average clam severy of he porfolo. 2. The sysem s fnancally balanced. Every sngle year he average of all premums colleced from all polcyholders remans consan and equal o P a s m - In order o prove hs s enough o show, consderng ha he clam frequency and he clam severy are ndependen componens, ha: and ha E E E l k,..., k a L 6L@ 6 E U E E y x,..., x. s m U 6 6 k@@ - A proof of he frs can be found n Lemare (995), and of he second n Vronos (998). (4)

9 DESIGN OF OPTIMAL BONUS-MALUS SYSTEMS 9 3. In he begnnng all he polcyholders are payng he same premum whch s equal o (4). 4. The more accdens caused and he more he sze of loss ha each clam ncurred he hgher he premum. 5. The premum always decreases when no accdens are caused. 6. The drvers who had a clam wh small loss wll have one more reason o repor he clam as hey wll know ha he sze of he clam wll be aken no consderaon and hey wll no have o pay he same premum wh somebody whch had an accden wh a bg loss. In hs way he phenomenon of bonus hunger wll have a decrease and he esmae of he acual clam frequency wll be more accurae. 7. The severy componen s nroduced n he desgn of a BMS whch from a praccal pon of vew s more crucal han he number of clams for he nsurer snce s he componen ha deermnes he expenses of he nsurer from he accdens and hus he premum ha mus be pad. 8. The esmaor of he mean of severy may no be robus and herefore s prone o be affeced by varaon. For praccal use a more robus esmaor could be used. (.e. cung of he daa, M-esmaor). 3. DESIGN OF OPTIMAL BMS WITH A FREQUENCY AND A SEVERITY COMPONENT BASED BOTHONTHEA PRIORI AND THE A POSTERIORI CRITERIA Donne and Vanasse (989, 992) presened a BMS ha negraes rsk classfcaon and experence rang based on he number of clams of each polcyholder. Ths BMS s derved as a funcon of he years ha he polcyholder s n he porfolo, of he number of accdens and of he sgnfcan for he number of accdens ndvdual characerscs. We exend hs model by nroducng he severy componen. We propose a generalzed BMS ha negraes a pror and a poseror nformaon on an ndvdual bass based boh on he frequency and he severy componen. Ths generalzed BMS wll be derved as a funcon of he years ha he polcyholder s n he porfolo, of he number of accdens, of he exac sze of loss ha each one of hese accdens ncurred, and of he sgnfcan ndvdual characerscs for he number of accdens and for he severy of he accdens. Some of he a pror rang varables ha could be used are he age, he sex and he place of resdence of he polcyholder, he age, he ype and he cubc capacy of he car, ec. As already sad one of he reasons for he developmen of a generalzed model whch negraes a pror and a poseror nformaon s ha premums should vary smulaneously wh he varables ha affec he dsrbuon of he number of clams and he sze of loss dsrbuon. The premums of he generalzed BMS wll be derved usng he followng mulplcave arff formula: Premum GBM F * GBM S (5) where GBM F denoes he generalzed BMS obaned when only he frequency componen s used and GBM S denoes he generalzed BMS obaned when only he severy componen s used.

10 0 NICHOLAS E. FRANGOS AND SPYRIDON D. VRONTOS 3.. Frequency Componen The generalzed BMS obaned wh he frequency componen GBM F wll be developed accordng o Donne and Vanasse (989, 992). Consder an ndvdual wh an experence of perods. Assume ha he number of clams of he ndvdual for perod, denoed as, follows he Posson dsrbuon wh parameer l, and are ndependen. The expeced number of clams of he ndvdual for perod s hen denoed by l and consder ha s a funcon of he vecor of h ndvdual s characerscs, denoed as c ( c,,..., ch, ), whch represen dfferen a pror rang varables. Specfcally assume ha l exp( c b ), where b s he vecor of he coeffcens. The non-negavy of l s mpled from he exponenal funcon. The probably specfcaon becomes P` e k - exp ( c b ) k ( exp( c b )). k! In hs model we assume ha he h ndvdual characerscs provde enough nformaon for deermnng he expeced number of clams. The vecor of he parameers b can be obaned by maxmum lkelhood mehods, see Hausmann, Hall and Grlches (984) for an applcaon. However, f one assumes ha he a pror rang varables do no conan all he sgnfcan nformaon for he expeced number of clams hen a random varable e has o be nroduced no he regresson componen. As Goureroux, Monfor and Trognon (984a), (984b) suggesed, we can wre l exp( c b + e ) exp( c b ) u, where u exp( e), yeldng a random l. If we assume ha u follows a gamma dsrbuon wh Eu ( ) and Var( u ) / a, he probably specfcaon becomes R G( k a) exp( c b V ) k R V- ( k+ a) + exp( c b ) P ( k) S W S, k! G( a) a + W S W S a W T X T X whch s a negave bnomal dsrbuon wh parameers a and exp( c b ). I can be shown ha he above parameerzaon does no affec he resuls f here s a consan erm n he regresson. We choose Eu ( ) n order o have E ( e ) 0. Then R V ( ) exp exp( c b ) E ( c b ) and Var ( ) exp ( c b ) S + a W. S W T X The neresng reader can see for more on he Negave Bnomal regresson Lawless (987). The nsurer needs o calculae he bes esmaor of he expeced number of accdens a perod + usng he nformaon from pas experence for he clam frequency over perods and of known ndvdual characerscs

11 over he + perods. Le us denoe hs esmaor as m + (,..., ; c,..., c + ). Usng he Bayes heorem one fnds ha he poseror srucure funcon for a polcyholder wh,..., clam hsory and c,..., c + characerscs s gamma a wh updaed parameers ( a +!, + ). Usng he classcal quadrac exp ( c b ) loss funcon one can fnd ha he opmal esmaor gven he observaon of,..., and C,..., C +, s equal o: + + m (,..., ; c,..., c ) # 0 m (, u ) f `m,..., ; c,..., cdm DESIGN OF OPTIMAL BONUS-MALUS SYSTEMS! R V S a +! W exp S W ( c b ) a exp( c b ), S + W S W T X where! denoes he oal number of clams of polcyholder n perods. When 0, m / exp( C b ) whch mples ha only a pror rang s used n he frs perod. Moreover when he regresson componen s lmed o a consan b 0, one obans he well-known unvarae whou regresson componen model, see Lemare (995), Ferrera (974). Now we wll deal wh he generalzed bonus-malus facor obaned when he severy componen s used. I wll be developed n he followng way Severy Componen Consder an ndvdual wh an experence of perods. Assume ha he number of clams of he ndvdual for perod s denoed as, he oal number of clams of he ndvdual s denoed as and by X k, s denoed he loss ncurred from hs clam k for he perod. Then, he nformaon we have for hs clam sze hsory wll be n he form of a vecor X,, X,2,..., X,, and he oal clam amoun for he specfc polcyholder over he perods ha he s n he porfolo wll be equal o! X k,. We assume ha X k, follows an exponenal dsrbuon wh parameer y. The parameer y denoes he mean or he expeced clam severy of a polcyholder n perod. As we have already sad, all polcyholders do no have he same expeced clam severy, her cos for he nsurer s dfferen and hus s far each polcyholder o pay a premum proporonal o hs mean clam severy. Consder ha he expeced clam severy s a funcon of he vecor of he h ndvdual s characerscs, denoed as d ( d,,..., dh, ), whch represen dfferen a pror rang varables. Specfcally assume ha y exp( d g ), where g s he vecor of he k

12 2 NICHOLAS E. FRANGOS AND SPYRIDON D. VRONTOS coeffcens. The non-negavy of y s mpled from he exponenal funcon. The probably specfcaon becomes P`X x $ e exp( d g ) - x k, exp ( d g ) In hs model we assume ha he h ndvdual characerscs provde enough nformaon for deermnng he expeced clam severy. However f one assumes ha he a pror rang varables do no conan all he sgnfcan nformaon for he expeced clam severy hen a random varable z has o be nroduced no he regresson componen. Thus we can wre y exp( d g + z ) exp( d g ) w, where w exp(z ), yeldng a random y. If we assume ha w follows an nverse gamma(s, s ) dsrbuon wh Ew ( ) and Varw ( ), > s - 2 s 2, hen y follows nverse gamma(,( s s-) exp( c g )) and he probably specfcaon for X k, becomes k, PX ( x) s$ 8( s- ) exp( d g ) B $ ( x+ ( s-) exp( d g )) s. - s - whch s a Pareo dsrbuon wh parameers s and ( s- ) exp( c g ). I can be shown ha he above parameerzaon does no affec he resuls f here s a consan erm n he regresson. We choose E(w ) n order o have E(z ) 0. We also have 2 8( s-) exp( d g ) B E ( X k, ) exp( d g ) and Var( X k, ) s - 2 b s s - l The nsurer needs o calculae he bes esmaor of he expeced clam severy a perod + usng he nformaon from pas experence for he clam severy over perods and of known ndvdual characerscs over he + perods. + Le us denoe hs esmaor as y ( X,,..., X, ; d,..., d + ). Usng he Bayes heorem he poseror dsrbuon of he mean clam severy for a polcyholder wh clam szes X,,..., X, n perods and characerscs d,..., d + s nverse gamma wh he followng updaed parameers:! k, k IG( s +,( s - ) exp( d g ) + X ). Usng he classcal quadrac loss funcon one can fnd ha he opmal esmaor of he mean clam severy for he perod + gven he observaon of

13 X,,..., X, and d,..., d +, s he mean of he poseror nverse gamma and hus s equal o y + +,, ( X,..., X ; d,..., d ) #,, 0 y `X, w f `y X,..., X ; d,..., d dy! ( s- ) exp( d g ) + X k s+ - When 0, whch mples ha only a pror rang s used n he frs perod s y exp( d g ). DESIGN OF OPTIMAL BONUS-MALUS SYSTEMS 3! k, 3.3. Calculaon of he premums of he Generalzed BMS Now we are able o compue he premums of he generalzed opmal BMS based boh on he frequency and he severy componen. As we sad he premums of he generalzed opmal BMS wll be gven from he produc of he generalzed BMS based on he frequency componen and of he generalzed BMS based on he severy componen. Thus wll be! Premum GBM F * GBM S R V S a+! W ( s- )! exp( d g ) +! X S W k exp( c b ) S a+ exp( c b) W s+ - S W T X k,. (6) 3.4. Properes of he Generalzed BMS. I s far snce akes no accoun he number of clams, he sgnfcan a pror rang varables for he number of clams, he clam severy and he sgnfcan a pror rang varables for he clam severy for each polcyholder. 2. I s fnancally balanced for he nsurer. Each year he average premum wll be equal o P exp( c b ) exp( d g ) (7) In order o prove he above equaon and assumng ha clam frequency and he clam severy componen are ndependen s suffcen o show ha E8m + `,...; ; c,..., c + B exp( c + b + )

14 4 NICHOLAS E. FRANGOS AND SPYRIDON D. VRONTOS and ha E8y `,...; ; d,..., d B exp( c g ). 3. All he properes we menoned for generalzed BMS whou he a pror rang varables hold for hs BMS as well. In he begnnng all he polcyholders wh he same characerscs are payng he same premum whch s equal o (7). 4. The more accdens are caused and he more he sze of loss ha each clam ncurred he hgher s he premum. 5. The premum always decreases when no accdens are caused. 6. Ths generalzed BMS could lead o a decrease of he phenomenon of bonus hunger. 7. The severy componen, whch s more crucal han he number of clams for he nsurer, s nroduced n he desgn of he generalzed BMS. 8. The premums vary smulaneously wh he varables ha affec he dsrbuon of he number of clams and he sze of loss dsrbuon Esmaon The premums wll be calculaed accordng (6). We have o know he number of he years ha he polcyholder s n he porfolo, hs oal number of accdens n years and hs aggregae clam amoun n years. For he frequency componen of he generalzed BMS we have o esmae he parameers of he negave bnomal regresson model, ha s he dsperson parameer a and he vecor b. Ths can be done usng he maxmum lkelhood mehod. For more on he negave bnomal regresson he neresed reader can see Lawless (987), Goureroux, Monfor and Trognon (984a) and Goureroux, Monfor and Trognon (984b). For he severy componen of he generalzed BMS we have o esmae s and g. We wll acheve hs usng he quas-lkelhood and accordng o Renshaw (994). Renshaw s usng he generalzed lnear models as a modellng ool for he sudy of he clam process n he presence of covaraes. He s gvng specal aenon o he varey of probably dsrbuons ha are avalable and o he parameer esmaon and model fng echnques ha can be used for he clam frequency and he clam severy process based on he conceps of quas-lkelhood and exended quas-lkelhood. Followng Renshaw (994) consder he followng scheme. The mean clam severy s denoed by y, caegorzed over a se of uns u. The daa ake he form (u, k u, x u ) where x u denoes he clam average n cell u based on n u clams. Independence of n u and x u s assumed. The uns u / (, 2,...) are a crossclassfed grd of cells defned for preseleced levels of approprae covaraes, ofen rang facors. Denong he underlyng expeced clam severy n cell u by m u and assumng he ndependence of ndvdual clam amouns, he cell means are modelled as he responses of a GLM wh E(x u )m u and Var (X u ) fv(m u )/n u. Covaraes defned on {u} ener hrough a lnear predcor, lnked

15 DESIGN OF OPTIMAL BONUS-MALUS SYSTEMS 5 o he mean m u. For hose unfamlar wh he generalzed lnear models we refer o he classcal ex of McCullagh and Nelder (989). In McCullagh and Nelder (983, 989) a re-analyss of he celebraed car nsurance daa of Baxer, Cous and Ross (979), based on ndependen gamma dsrbued clam amouns can be found. Le us focus now on he Pareo dsrbuon wh parameers s and (s ) exp(c g ) and densy k, s - s - PX ( x) s$ 8( s- ) exp`d g B $ `x+ ( s-) exp( d g ) We have ha 2 9( s-) exp( d g ) C E ( X k, ) exp( d g ) and Var( X k, ) s - 2 b, s > 2. s s - l Inroducng he reparameersaon: m exp( d g ) and f s s - 2, a : mappng (,( s s-) exp( c g ))" ( m, f) wh doman R> 2# R> 0and mage se R> 2# R> mples ha we can consruc a GLM based on ndependen Pareo dsrbued clam amouns for whch he mean responses, X u, sasfy mean m u 2 E(X u ), varance funcon V( mu ) mu, scale parameer f > and weghs n u so ha Var( X u) fv( mu)/ nu. Apar from he mld exra consran on he scale parameer, hese deals are dencal o hose of he GLM based on ndependen gamma responses and he wo dfferen modellng assumpons lead o essenally dencal GLMs. They dffer only n he parameer esmaon mehod. In he case of gamma response we use maxmum lkelhood mehod and n he case of Pareo response we use maxmum quas-lkelhood. 4.. Descrpon of he Daa 4. APPLICATION The models dscussed are appled n a daa se ha one Greek nsurance company provded us. The daa se consss of polcyholders. The mean of he clam frequency s and he varance s The a pror rang varables were age and sex of he drver, BM class and he horsepower of he car. The drvers were dvded n hree caegores accordng her age. Those aged beween 28-45, hose beween and hose aged beween 8-27 or hgher han 55. The drvers were also dvded n hree caegores accordng he horsepower of her car. Those who had a car wh a horsepower beween 0-33, beween and beween The drvers were also dvded n hree caegores accordng her BM class. The curren Greek BMS has 6 classes, from 5 o 20. The malus zone ncludes classes from 2 o 20, he bonus zone ncludes classes 5 o 8 and he neural zone ncludes classes from 9 o.

16 6 NICHOLAS E. FRANGOS AND SPYRIDON D. VRONTOS We fed he Negave Bnomal dsrbuon on he number of clams and he Pareo dsrbuon on he clam szes. We wll fnd he premums deermned from he opmal BMS based on he a poseror frequency componen, he premums deermned from he opmal BMS based on he a poseror frequency and severy componen and he premums deermned from he opmal BMS wh a frequency and a severy componen based boh on he a pror and he a poseror crera Opmal BMS based on he a poseror frequency componen We apply he Negave Bnomal dsrbuon. The maxmum lkelhood esmaors of he parameers are â and ˆ We wll fnd frs he opmal BMS based only on he frequency componen followng Lemare (995). The BMS wll be defned from () and s presened n Table. Ths opmal BMS can be consdered generous wh good drvers and src wh bad drvers. For example, he bonuses gven for he frs clam free year are 26% of he basc premum. Drvers who have one accden over he frs year wll have o pay a malus of 298% of he basc premum. TABLE. OPTIMAL BMS BASED ON THE A POSTERIORI FREQUENCY COMPONENT Year Number of clams Opmal BMS based on he a poseror frequency and severy componen Le us see he mplemenaon of an opmal BMS based boh on he frequency and he severy componen. We f he Pareo dsrbuon o he clam szes and we fnd he maxmum lkelhood esmaes of s and m. I s s and m In order o fnd he premum ha mus be pad we have o know he age of he polcy, he number of clams he has done n hese years and he aggregae clam amoun. The seps ha mus be followed n order o fnd hs opmal BMS are:. We fnd he age of he polcy. 2. We fnd he oal number of clams k ha he polcyholder has done n years.

17 DESIGN OF OPTIMAL BONUS-MALUS SYSTEMS 7 3. We fnd he aggregae clam amoun for he polcyholder, x k 4. We compue he premums usng (3). k 5. We go o he able wh he specfc oal clam amoun and we fnd he premum ha corresponds o k clams n years of observaon. The Bonus-Malus Sysem deermned n he above way s presened n he followng ables. Here we wll llusrae only he cases ha he aggregae clam amoun of a polcyholder s equal o drs, and drs. I s obvous ha we can use he above formula wh any value ha he aggregae clam amoun can have. We use hese values of he aggregae clam amoun for brevy. In he followng ables we wll use he acual values, he premums are no dvded wh he premum when 0, as wll be neresng o see he varaon of he premums pad for varous number of clams and clam szes n comparson no wh he premum pad when 0 bu wh he specfc clam szes. Ths s he basc advanage of hs BMS n comparson wh he one ha akes under consderaon only he frequency componen, he dfferenaon accordng he severy of he clam. Of course he percenage change n he premum afer on or more clams could be also neresng. Le s see an example n order o undersand beer how such BMS work. In Table 3 we can see he premums ha mus be pad for varous number of clams when he age of he polcy s up o 7 years. For example a polcyholder wh one accden of clam sze drs n he frs year of observaon wll pay drs (see Table 2). If he second year of observaon he has an accden wh clam sze drs, hen, a surcharge wll be enforced and he wll have o pay drs, whch s he premum for wo accdens of aggregae clam amoun drs n wo years of observaon (see Table 3). If n he hrd year he does no have an accden, he wll have a reducon n he premum because he had a clam free year and he wll pay drs, whch s he premum for wo accdens of aggregae clam amoun drs n hree years of observaon (see Table 3).! TABLE 2. OPTIMAL BMS BASED ON THE A POSTERIORI FREQUENCY AND SEVERITY COMPONENT TOTAL CLAIM SIZE OF Year Number of clams

18 8 NICHOLAS E. FRANGOS AND SPYRIDON D. VRONTOS TABLE 3. OPTIMAL BMS BASED ON THE A POSTERIORI FREQUENCY AND SEVERITY COMPONENT TOTAL CLAIM SIZE OF Year Number of clams I s obvous ha hs opmal BMS allows he dscrmnaon of he premum wh respec o he severy of he clams. Table 4 shows he premums ha mus be pad when he polcyholder s observed for he frs year of hs presence n he porfolo, hs number of accdens range from o 5 and he aggregae clam amoun of hs accdens ranges from o dr. A polcyholder who had one clam wh clam sze wll have o pay a premum of drs, a polcyholder who had one clam wh clam sze wll have o pay a premum of 3395 drs and a polcyholder who had one clam wh clam sze wll have o pay a premum of drs. TABLE 4. COMPARISON OF PREMIUMS FOR VARIOUS NUMBER OF CLAIMS AND CLAIM SIZES IN THE FIRST YEAR OF OBSERVATION. Number of clams Clam Sze For more on such a sysem he neresng reader can see Vronos (998).

19 DESIGN OF OPTIMAL BONUS-MALUS SYSTEMS Generalzed opmal BMS wh a frequency and a severy componen based boh on he a pror and he a poseror classfcaon crera Le us calculae now he premums of he generalzed opmal BMS based boh on he frequency and he severy componen when boh he a pror and he a poseror rang varables are used. As we sad he premums of he generalzed opmal BMS wll be gven from he produc of he generalzed BMS based on he frequency componen, GBM F, and of he generalzed BMS based on he severy componen, GBM S. Implemenng he negave bnomal regresson model we esmae he dsperson parameer a and he vecor b of he sgnfcan a pror rang varables for he number of clams. We found ha many a pror rang varables are sgnfcan for he number of clams. These are he BM class, he age and he sex of he drver and he neracon beween age and sex. In he mulvarae model â s larger, han n he unvarae negave bnomal model where we had â Ths resul ndcaes ha par of he varance s explaned by he a pror rang varables n he mulvarae model. The esmaes of he vecor b can be found n he appendx. The parameers of GBM S, ha s he parameer of he Pareo s, and he vecor parameer g of he sgnfcan for he clam severy a pror rang varables d, are found usng he quas-lkelhood mehod. The sgnfcan a pror characerscs for he clam severy are he age and he sex of he drver, he BM class, he horsepower of he car, he neracon beween age and sex and he neracon beween age and class. The premums are calculaed usng (6). Below we can see he premums for dfferen caegores of polcyholders. Le us examne wo groups of polcyholders whch have he followng common characerscs. They belong n he malus zone, her car s horsepower s beween 67 and 99, and her age s beween 28 and 45. If he polcyholder s a man he wll have o pay he followng premums afer one or more accdens of oal clam amoun n he frs year. TABLE 5. MEN, AGE 28-45, MALUS-ZONE, HORSEPOWER Year Number of clams

20 20 NICHOLAS E. FRANGOS AND SPYRIDON D. VRONTOS If he polcyholder s a woman wh he above characerscs she wll have o pay he followng premums afer one or more accdens of oal clam amoun n he frs year. TABLE 6. WOMEN, AGE 28-45, MALUS-ZONE, HORSEPOWER Year Number of clams We noce ha men have o pay hgher premums han women. We saw an example of premums obaned wh generalzed opmal BMS wh a frequency and a severy componen based boh on he a pror and he a poseror classfcaon crera. Oher combnaons of a pror characerscs could be used and also hgher oal clam amouns. I s neresng o compare hs BMS wh he one obaned when he only he a poseror frequency and severy componen are used. Usng hs BMS we saw from Table 4 ha a polcyholder wh one accden wh clam sze of drs n one year has o pay 3395 drs. Usng he generalzed opmal BMS wh a frequency and a severy componen based boh on he a pror and he a poseror classfcaon crera, a man, age 28-45, who belongs o he malus zone, wh a car wh horsepower beween for one accden of clam sze drs n one year wll has o pay drs, whle a woman, age 28-45, who belongs o he malus zone, wh a car wh horsepower beween for one accden of clam sze drs n one year wll has o pay drs. Ths sysem s more far snce consders all he mporan a pror and a poseror nformaon for each polcyholder boh for he frequency and he severy componen n order o esmae hs rsk o have an accden and hus perms he dfferenaon of he premums for varous number of clams and for varous clam amouns based on he expeced clam frequency and expeced clam severy of each polcyholder as hese are esmaed boh from he a pror and he a poseror classfcaon crera.

21 DESIGN OF OPTIMAL BONUS-MALUS SYSTEMS 2 5. CONCLUSIONS We developed n hs paper he desgn of an opmal BMS based boh on he a poseror frequency and he a poseror severy componen. We dd hs by fng he Negave Bnomal dsrbuon n he clam frequency and he Pareo dsrbuon on he clam severy, exendng he classcal n he BMS leraure model of Lemare (995) whch used he Negave Bnomal dsrbuon. The opmal BMS obaned has all he aracve properes of he opmal BMS desgned by Lemare, furhermore allows he dfferenaon of he premums accordng o he clam severy and n hs way s more far for he polcyholders and s obaned n a very naural conex accordng o our opnon. Moreover, we developed he desgn of a generalzed opmal BMS wh a frequency and a severy componen based boh on he a pror and he a poseror classfcaon crera exendng he model developed by Donne and Vanasse (989, 992) whch was based only on he frequency componen. The BMS obaned has all he aracve properes of he one obaned by Donne and Vanasse (989, 992) and furhermore allows he dfferenaon of he premums ulzng he severy componen n a very naural conex. Ths generalzed BMS akes no consderaon smulaneously he mporan ndvdual s characerscs for he clam frequency, he mporan ndvdual s characerscs for he clam severy, he clam frequency and he clam severy of each accden for each polcyholder. An neresng opc for furher research could be he exenson of he wo above BMS for dfferen clam frequency and clam severy dsrbuons. REFERENCES BAXTER, L.A., COUTTS S.M. and ROSS G.A.F. (979) Applcaons of Lnear Models n Moor Insurance. 2s Inernaonal Congress of Acuares. BICHSEL, F. (964) Erfahrung-Tarfeung n der Moorfahrzeug-hafplchversherung, Melungen der Verengung Schwezerscher Verscherungsmahemaker, BUHLMANN, H. (964) Opmale Pramensufensyseme, Melungen der Verengung Schwezerscher Verscherungsmahemaker COENE, G. and DORAY, L.G. (996) A Fnancally Balanced Bonus-Malus Sysem. Asn Bullen 26, DIONNE, G. and VANASSE, C. (989) A generalzaon of acuaral auomoble nsurance rang models: he negave bnomal dsrbuon wh a regresson componen, Asn Bullen 9, DIONNE, G. and VANASSE, C. (992) Auomoble nsurance raemakng n he presence of asymmercal nformaon, Journal of Appled Economercs 7, FERREIRA, J. (974) The long erm effec of mer-rang plans on ndvdual moorss, Operaons Research, 22, GOURIEROUX C., MONTFORT A. and TROGNON A. (984a) Pseudo maxmum lkelhood mehods: heory, Economerca, 52, GOURIEROUX C., MONTFORT A. and TROGNON A. (984b) Pseudo maxmum lkelhood mehods: applcaon o Posson models, Economerca, 52, HAUSMANN J.A., HALL B.H. and GRILICHES Z. (984) Economerc models for coun daa wh an applcaon o he paens-r&d relaonshp, Economerca, 46,

22 22 NICHOLAS E. FRANGOS AND SPYRIDON D. VRONTOS HERZOG, T. (996) Inroducon o Credbly Theory. Acex Publcaons, Wnsead. HOGG, R.V. and LUGMAN, S.A. (984) Loss Dsrbuons, John Wley & Sons, New York. MCCULLAGH, P. and NELDER, J.A. (983, 989) Generalzed Lnear Models. Chapman & Hall. LAWLESS, J.F. (987) Negave Bnomal Dsrbuon and Mxed Posson Regresson, Canadan Journal of Sascs, 5, 3, LEMAIRE, J. (995) Bonus-Malus Sysems n Auomoble Insurance, luwer Academc Publshers, Massachuses. PICARD, P. (976) Generalsaon de l eude sur la survenance des snsres en assurance auomoble, Bullen Trmesrel de l Insue des Acuares Francas, PICECH, L. (994) The Mer-Rang Facor n a Mulplcang Rae-Makng model. Asn Colloquum, Cannes. PINQUET (997) Allowance for Coss of Clams n Bonus-Malus Sysems, Asn Bullen, 27, PINQUET J. (998) Desgnng Opmal Bonus-Malus Sysems From Dfferen Types of Clams. Asn Bullen, 28, RENSHAW A.E. (994) Modellng The Clams Process n he Presence of Covaraes. Asn Bullen, 24, SIGALOTTI (994) Equlbrum Premums n a Bonus-Malus Sysem, Asn Colloquum, Cannes. TAYLOR G. (997) Seng A Bonus-Malus Scale n he Presence of Oher Rang Facors. Asn Bullen, 27, TREMBLAY, L. (992) Usng he Posson Inverse Gaussan n Bonus-Malus Sysems, Asn Bullen 22, VRONTOS S. (998) Desgn of an Opmal Bonus-Malus Sysem n Auomoble Insurance. Msc. Thess, Deparmen of Sascs, Ahens Unversy of Economcs and Busness, ISBN: WALHIN J.F. and PARIS J. (999) Usng Mxed Posson dsrbuons n connecon wh Bonus- Malus Sysems. Asn Bullen 29, NICHOLAS E. FRANGOS AND SPYRIDON D. VRONTOS Deparmen of Sascs Ahens Unversy of Economcs and Busness Passon 76, 0434, Ahens, Greece E-mal: nef@aueb.gr and svronos@aueb.gr