ABSTRACT KEYWORDS. Bonus-malus systems, frequency component, severity component. 1. INTRODUCTION

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1 EERAIED BU-MAU YTEM ITH A FREQUECY AD A EVERITY CMET A IDIVIDUA BAI I AUTMBIE IURACE* BY RAHIM MAHMUDVAD AD HEI HAAI ABTRACT Frangos and Vronos (2001) proposed an opmal bonus-malus sysems wh a frequency and a severy componen on an ndvdual bass n auomoble nsurance In hs paper, we nroduce a generalzed form of hose obaned prevously EYRD Bonus-malus sysems, frequency componen, severy componen 1 ITRDUCTI Donne and Vanasse (1989, 1992) have presened a bonus-malus sysems (BM) ha negraes rsk classfcaon and experence rang based on he number of clams of each polcyholder Ths BM 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 Frangos and Vronos (2001) exend BM model by nroducng he severy componen They proposed a BM ha negraes a pror and a poseror nformaon on an ndvdual bass based on he boh frequency and he severy componen Ths BM 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 ome 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 * Ths research was n par suppored by a gran (o 8716) from ascal Research and Tranng Cener, Iran Asn Bullen 39(1), do: /AT by Asn Bullen All rghs reserved

2 308 R MAHMUDVAD AD H HAAI ne 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 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 In hs paper, we presen a generalzed form of hose obaned n Frangos and Vronos (2001) 2 EERAIED BU-MAU YTEM Consder an ndvdual wh an experence of perods Assume ha he number of clams of he ndvdual for perod, denoed as, follows he osson 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,1,,c,h), whch represen dfferen a pror rang varables pecfcally assume ha l exp(c b )U, where b s he vecor of he coeffcens If we assume ha he U are ndependen and dencally dsrbued over me and U follows a gamma dsrbuon wh parameers (a, a) and pdf: f U (u ) a a g a u a 1 e au Usng he Bayes heorem one fnds ha he poseror srucure funcon for a polcyholder wh 1,, clam hsory and c 1,,c +1 characerscs s gamma wh updaed parameers J a +! exp`c b 1 a +!,, (1) 1 exp`c b e us consder he suaon n whch he vecor of ndvdual characerscs remans he same from one year o he nex If one assumes ha c 1 c +1 c and b 1 b b, hen (1) s smplfed o J a a +!, +, exp c b 1 ^ (2) h whch concdes wh hose obaned by Frangos and Vronos (2001) Tha s, he obaned resuls here can be consdered as a general form of he resuls obaned prevously, and can be used o calculae bonus-malus sysems wh a

3 frequency and a severy componen on an ndvdual bass n auomoble nsurance, n general form In summary, f he vecor of ndvdual characerscs remans he same from one year o he nex, (1) and (2) concde For example, f c s he sex/ occupaon of he drver, he sex and he mpac of he sex/occupaon on he number of accdens and he severy of each accden does no change over me In hs case he generalzed form (1) s reduced o he smple form (2) herwse, f ndvdual characerscs do vary subsanally from one perod o he nex, he generalzed form (1) should be used For example, some characerscs such as age of he drver, age of he car, and experence of he drver have sgnfcan mpac on he dsrbuon of frequency and severy of accdens and change over me oe ha he updaed gamma parameers n (1) nclude nformaon up o me + 1 a leas for ndvdual characerscs Tha s, he gamma parameers nclude c +1 I should be noed ha we have characersc nformaon for any ndvduals a me + 1 o esmae he parameers The nsurer needs o calculae he bes esmaor of he rue expeced number of accdens a perod + 1 e l +1 ( 1,, ; c 1,,c +1 ) denoe hs esmaor whch s a funcon of pas experence over he perods ( 1,, ) and of known characerscs over he + 1 perods (c 1,,c +1 ) Usng he classcal quadrac loss funcon one can fnd ha: l +1 ( 1,, ; c 1,,c +1 ) J a +! exp9c b C, (3) a +! exp c b ` 1 If one subsues he value of zero for (hen! 0 and! exp(c b )0) 1 1 no (3), he answer would be exp(c 1 b 1 ) whch mples ha only a pror rang s used n he frs perod For severy componen,consder an ndvdual wh an experence of perods Assume ha 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,1,,X 1,, and he oal clam amoun for he specfc polcyholder over he perods ha he s n he porfolo wll be equal o! 1 k 1 EERAIED BU-MAU YTEM 309,k e 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

4 310 R MAHMUDVAD AD H HAAI 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,1,,d,h), whch represen dfferen a pror rang varables pecfcally assume ha y exp(d g ), where g s he vecor of he coeffcens If we assume ha he are ndependen and dencally dsrbued over me and follows a nverse gamma dsrbuon wh parameers (s, s 1) and pdf: f ^w h s s - 1g s 1 w - - sg - s - 1 exp c w m Usng he Bayes heorem he poseror dsrbuon of he mean clam severy for a polcyholder wh clam szes X,1,,X 1, n perods and characerscs d 1,,d +1 s nverse gamma wh he followng updaed parameers: J R V k, k s +, + + s - 1 +! exp d g, ` g 1 exp `d g T X where denoes he oal number of clams of polcyholder n perods mlar o hose dscussed above regardng he frequency componens, f one assumes ha d 1 d +1 d and g 1 g g, hen (4) s smplfed o J s +, s - 1g exp ^d gh +! k,, (5) 1 k 1 (4) whch concdes wh hose obaned by Frangos and Vronos (2001) The nsurer needs o calculae he bes esmaor of he expeced clam severy a perod + 1 usng he nformaon from pas experence for he clam severy over perods and of known ndvdual characerscs over he + 1 perods e us denoe hs esmaor as y +1 Usng he classcal quadrac loss funcon one can fnd ha: y +1 (X 1,1,,X, ; d 1,,d +1 )! k 1 1g! 1 exp `d s - + s + -1 X k, g exp `d g, (6)

5 EERAIED BU-MAU YTEM 311 ow we are able o compue he premums of he generalzed BM based boh on he frequency and he severy componen The premums of he generalzed BM wll be gven from he produc of he generalzed BM based on he frequency componen and of he generalzed BM based on he severy componen Thus wll be +1 y +1 l +1 (7) 3 CMARI e us now consder he dscrepancy beween he obaned premums based on he generalzed form presened here,, and by he smple form of Frangos and Vronos (2001),, by means of an example Here we consder dfferen condons for 10 ndvduals, bu one can consder more condons and also more samples uppose 3,a 1,! 0, s 2,X, k 0 and he hsorcal 1 feaures for 10 polcyholders are hose presened n Table 1 TABE 1 THE BTAIED REMIUM UI AD e c1 b 1 e c2 b 2 e c3 b 3 e c4 b 4 e d1 g 1 e d2 g 2 e d3 g 3 e d4 g The las wo columns of Table 1 represen and, respecvely As appears from Table 1, he premums obaned by and, for each polcyholder, are dfferen for all cases The premums calculaed by smple form are somemes less han he generalzed form, for example cases 1, 4, 7 and 10, and greaer han n oher cases As we menoned above, s calculaed based on a combnaon of he fxed and varable characerscs over me whls s only obaned from he fxed characerscs Therefore, whenever we have a combnaon of he fxed and varable characerscs over me, lke wha are consdered n

6 312 R MAHMUDVAD AD H HAAI Table 1, he resuls are compleely dfferen In hs case, one should use he generalzed form o oban accurae premums herwse, he calculaed premums would be bgger or smaller han he correc value To acqure a beer undersandng of and obaned n Table 1, we examne he resuls presened n he frs row of Table 1 Assume ndvdual characerscs are sex of he drver (), age of he drver (A), age of he car (AC), and average drvng per day (T) and also consder he followng models: Iexp{c b }exp{a a A A + a AC AC + a T T} (8) II exp{d g }exp{q q A A +q AC AC + q T T} (9) where a and q are he parameers of he models I and II, respecvely The value of s se o zero for women and 1 for men oe ha here we use a combnaon of fxed, and T, and varable characerscs, A and AC The frs row n Table 1 was obaned based on he nformaon for a man wh A 43, AC 2, and T 25 a 1 The dealed resuls are presened n Table 2 TABE 2 THE DETAIED REUT F THE FIRT R I TABE 1 characerscs parameers fxed varable I II T A AC a a A a AC a T q q A q AC q T ex, we consder he suaon n whch he premums are greaer/smaller han e us frs assume 0 In hs case we have: b a + b g exp$ c a exp$ c exp$ d k J 1 1 c b a + c b d g (10) Therefore, we ge

7 EERAIED BU-MAU YTEM 313 > 1 [ < 1 f f ) exp$ c b > c b 1 [ ) exp$ d g > d g ) exp$ c b < c b 1 [ ) exp$ d g < d g 1 (11) Recall ha exp{c b }E( ) and exp{d g }E(X,k) Therefore, (11) can be smplfed o > 1 [ < 1 J ) E` > E! 1 f [ J ) E `Xk, > E k, 1 J ) E` < E! 1 f [ J ) E `Xk, < E k, 1 (12) I can be concluded from (12) ha < f severy and frequency componens for he perod + 1 are smaller han he average of perods Ths ndcaon s consderable and drecs o oban opmal BM e us now consder for he general case (! 0): J k, k 1 exp$ c b + d g a a + exp$ c b k s- 1g +! 1 exp$ d g, J J 1 exp c b a exp c b s 1 1! $ +! $ - g d g + X, (13)

8 314 R MAHMUDVAD AD H HAAI where X,! k, mlar o hose obaned above, we have he followng 1 k 1 condons: ) exp$ c b > c b 1 1 > 1 f [ ) exp$ d > $ g [ < g! exp d 1 k, X k 1, )! > exp$ d g exp$ d g ) exp$ c b < c b f [ ) exp$ d g < d g 1 Xk,! X k 1, )! < exp$ d g exp$ d g k 1, k oe ha n (14) s he rao of he observed losses o he expeced exp ' d g 1 losses a perod Thus, f he lef hand sde of condon n (14) s greaer ha he rgh hand sde, hen we conclude ha he drver has been faced wh more losses han he nsurer expeced n perods Thus, hs evdence should be consdered n calculang opmal nsurance for he nex perod Fnally, we should menon ha he above condons can be consdered as suffcen condons However n pracce as was shown n Table 1, we have combnaons of hese condons (14) 4 CCUI e nroduced a generalzed form of opmal BM wh a frequency and a severy componen based boh on he a pror and he a poseror classfcaon

9 EERAIED BU-MAU YTEM 315 crera developed prevously by Donne and Vanasse (1989, 1992) and Frangos and Vronos (2001) There are many sgnfcan ndvdual characerscs on he dsrbuon of frequency and severy of clams These characerscs, and her mpac, change over me The generalzed BM obaned here allows us o use boh fxed and varable ndvdual characerscs The resuls show ha one ges much more accurae premums usng he generalzed form presened here, f a combnaon of he fxed and varable characerscs s consdered ACEDEMET The auhors would lke o hank he edor rof Carns, rof Frangos and rof Vronos for her consrucve commens whch have led o subsanal mprovemens n hs paper REFERECE DIE, and VAAE, C (1989) A generalzaon of acuaral auomoble nsurance rang models: he negave bnomal dsrbuon wh a regresson componen, Asn Bullen 19, DIE, and VAAE, C (1992) Auomoble nsurance raemakng n he presence of asymmercal nformaon, Journal of Appled Economercs 7, FRA, E and VRT, D (2001) Desgn of opmal bonus-malus sysems wh a frequency and a severy componen on an ndvdual bass n auomoble nsurance, Asn Bullen 33, 1-22 RAHIM MAHMUDVAD ascal Research and Tranng Cener (RTC), Tehran, Iran and roup of ascs, ayame oor Unversy of Toyserkan Toyserkan, Iran HEI HAAI ascal Research and Tranng Cener (RTC), Tehran, Iran and Cenre for pmsaon and Is Applcaons, chool of Mahemacs, Cardff Unversy, CF24 4A, U E-Mal: HassanH@cfacuk