Multidimensional credibility model and its application

Finance 170 2014, XVII, 2 DOI: 10.15240/ ul/001/2014-2-013 In oduc ion Ahuge expansion o a compe i i e insu ance ma ke o ms pa o he ansi ion p ocess in he o me Socialis coun ies. New in e na ional ends in secu ing i s inancial s abili y and new ypes o isk demand a high-powe ed de elopmen o ac ua ial science and i s applica ion in insu ance p ac ice. The on-going Eu opean Union Sol ency II p ojec , aimed a imp o ing he quali y o assessmen o he sol ency o an insu e , places emphasis on he modelling o isk and on in e nal models o managing an insu e 's isk. This una oidably leads o a mo e in dep h use o he heo e ical esul s o ac ua ial science and o hei u he heo e ical de elopmen . Imp o ing he quali y o p emium calcula ion me hods is an e ec i e ac o in educing he insu ance echnical isk o an insu e . In doing so i akes in o accoun mo e and mo e a ing ac o s, hanks o which mo e homogeneous a ing a i s eme ge allowing he possibili y o de e mining mo e equi able p emiums. On he o he hand when he insu e de e mines he p emium i has o con end wi h a smalle amoun o in o ma ion conce ning he isk, o which he gi en a i class is in ended. I he insu e has inadequa e pas da a ela ing o he a i class, i can make use o da a om o he isk classes which a e o a gi en ex en simila . The insu ance isk, o which he e is agi en a i class, is called he indi idual isk. Each a i class o ms pa o he whole po olio and hus each indi idual isk is pa o he collec i e, i.e. po olio, isk. On he one hand he insu e in gene al has a i s disposal a ela i ely la ge amoun o s a is ical in o ma ion conce ning he collec i e isk. This in o ma ion is, howe e , mo e o less limi ed o p icing indi idual isks, due o he wide a ie y o a i classes. An insu e usually has li le s a is ical in o ma ion conce ning indi idual isks and he e o e should use bo h sou ces o da a. Tha is he undamen al idea behind c edibili y heo y. Back in 1918 Albe W. Whi ney p oposed ha he c edible p emium Pc ed should be alinea combina ion o he indi idual p emium Pind and he collec i e p emium Pcol Pc ed = z . Pind + (1 – z) . Pcol (1) whe e Pind is he a e age le el o claim pe con ac in he gi en a i class and Pcol is he espec i e mean o he whole insu ance po olio. F om a s a is ical poin o iew he c edible p emium is he weigh ed a i hme ical a e age o he indi idual and collec i e p emiums, whe e zis he ela i e weigh o he indi idual p emium and (1 – z) ha o he collec i e p emium. The weigh zis e e ed o as he c edibili y ac o . I akes alues in he in e al 〈0;1〉and exp esses he le el o con idence in he da a o he gi en insu ance class when de e mining he p emium o ha class. A con idence le el (weigh ) o 0 is assigned o indi idual isk da a which ha e no use o de e mining he p emium and a le el o 1 is gi en o hose indi idual isk da a which comple ely su ice in o de o de e mine he p emium o ha isk class. This means ha , i z= 0, we mus comple ely ely on he da a om compa able isks when de e mining he p emium o he gi en class. I z= 1, hen we can comple ely ely on he da a o he gi en indi idual isk. Bühlmann es ablished he heo e ical ounda ion o mode n c edibili y heo y, p esen ed as dis ibu ion ee c edibili y es ima ion. Bühlmann and S aub (see [3]) gene alized Bühlmann’s classical c edibili y model. Wi hou MULTIDIMENSIONAL CREDIBILITY MODEL AND ITS APPLICATION Vie a Pacáko á, E ik ·ol és, Bohdan Linda EM_02_14_zlom 4.6.2014 8:54 S ánka 170 Finance 1712, XVII, 2014 doub he Bühlmann-S aub model is he mos impo an model in c edibili y heo y. O iginally i was aimed a he de e mina ion o he c edible p emium, bu a e ce ain modi ica ions i can also be used o de e mine a eliable es ima e o claims equency o o he a e age amoun o claim. I s impo ance lies in he possibili y o using i in a wide ange o applica ions in he ac ua ial p ac ice o non-li e insu ance, bu i can also be used in li e insu ance and in einsu ance. Besides ha he Bühlmann-S aub model ep esen s a base o u he mo e speci ic models, such as he hie a chical, mul idimen- sional o eg ession c edibili y models. In his pape we will conside a gene alisa ion o he one-dimensional Bühlmann-S aub c edibili y model o he mul idimensional c edibili y model. 1. Gene alisa ion o he One- Dimensional Bühlmann-S aub Model o he Mul idimensional C edibili y Model The aim o his pa o a icle is o b ie ly desc ibe he ma hema ical and s a is ical basis o one-dimensional Bühlmann-S aub model ( o de ails see [1] o [7]) and poin ou he analogy o mul idimensional Bühlmann-S aub model wi h he o iginal Bühlmann-S aub model. This pa o he a icle also shows he p ac ical use o mul i-dimensional Bühlmann- S aub model in insu ance p ac ice. Le us suppose ha he po olio o con ac s is di ided in o I isk classes, o which we wan o de e mine he ne p emium. Le he i h isk class (i= 1, 2, ..., I) be cha ac e ised by he indi idual isk p o ile ϑ i, which is he ou come o a andom a iable θ i. To calcula e he ne p emium o each indi idual isk class we will make use o he quan i ies: Sij – he agg ega e claim amoun o isk iin yea j(j= 1, 2, ..., n), wij – he numbe o con ac s o isk iin yea j, Sij Xij = –––– – he a e age annual le el o wij claim o isk iin yea j, which a e in gene al known. The one dimensional Bühlmann-S aub c edibili y model s ems om he ollowing assump ions Bühlmann, S aub [3], He zog [6], Pacáko á [8]. BS1: Condi ional on he alue o θ i he Xij o j = 1, 2, ..., na e independen wi h pa ame e s: E[Xij| θ i] = µ ( θ i)(2) σ 2 ( θ i) Va [Xij| θ i] = ––––– (3) wij BS2:The pai s ( θ 1, X1), ( θ 2, X2) ... a e independen and θ 1, θ 2a e independen and iden ically dis ibu ed. I he assump ions BS1aBS2a e me , he c edible es ima e o he ne p emium is (4) whe e he a e age annual le el o claim pe con ac om he i h isk class (5) is he es ima e o he indi idual p emium, µ0is he collec i e p emium and he c edibili y ac o has he o m (6) The es ima ion o he so-called s uc u al pa ame e s µ0, σ2aτ2plays a undamen al ole in c edibili y heo y. De ini ions o he s uc u al pa ame e s and hei in e p e a ion a e as ollows: S uc u al pa ame e In e p e a ion collec i e p emium he a iance o he a e age annual claim amoun s wi hin a g oup he in e -g oup a iance o he a e age annual claim amoun s The c edibili y es ima e (4), in which we eplace he unknown collec i e p emium µ0by he es ima e EM_02_14_zlom 4.6.2014 8:54 S ánka 171 Finance 172 2014, XVII, 2 (7) is he so-called homogeneous c edibili y es ima e wi h he o m (see [4]) (8) The es ima e (7) o he collec i e p emiumis he weigh ed a i hme ical a e age o he indi idual p emiums X — i, whe e he weigh s a e he c edibili y ac o s zi. Assuming ha he e is he same numbe o con ac s in each isk class in each yea , he c edible p emium de ined by o mula (8) should co e he claims ou go o he whole po olio and o he whole pe iod o n p e ious yea s conside ed. This esul is exp essed in [1] by he balance p ope y: (9) Fo some classes o business (liabili y, comme cial i e) a high pe cen age o he o al claim amoun comes om a small numbe o e y la ge claims. In hese cases a ela i ely small numbe o claims, which a e “la ge” compa ed wi h he “s anda d” claim, ha e a la ge weigh when he p emium a e is de e mined. I he e o e makes sense o es ima e he a e age claim amoun sepa a ely o “la ge” and o “s anda d” claims. I in ac we we e o calcula e he c edible p emium using he one-dimensional Bühlmann-S aub model sepa a ely o “la ge” and “s anda d” claims, we would no ake in o accoun any dependency be ween hem. The e o e i is mo e app op ia e o es ima e he ec o o ne p emiums, which in his case is he wo-dimensional ec o The whole ec o o ne p emiums o indi idual isk i can be es ima ed by he mul idimensional c edibili y model, which is a gene alisa ion o he one-dimensional Bühlmann-S aub model. Ano he example o he use o he mul idi- mensional model would be he calcula ion o he c edible p emium o he sepa a e isk classes o a whole po olio and he simila isk classes o o he insu e s. I , he e o e, an insu e disposes o da a, e en om ano he insu e , ei he b oken-down o in o al, i can use he mul idimensional c edibili y model. Thus, hanks o i , we can calcula e he p emium o a gi en isk class using ei he pas da a o ha class o pas da a o he whole o he po olio o e en his o ical da a o he gi en isk class o o he whole po olio o simila isks o ano he insu e . Whe eas in he one-dimensional Bühlmann-S aub c edibili y model we es ima e he ne p emium µ(θi) o he isk i (i= 1, 2, ..., I), in he mul idimensional Bühlmann-S aub c edibili y model we es ima e o isk i he whole ec o o ne p emiums (10) In he one-dimensional Bühlmann-S aub c edibili y model we obse e he quan i ies Xij ( he a e age annual claim amoun o isk iin yea j) wi h weigh s wij ( he numbe o con ac s o isk iin yea j). In he mul idimensional Bühlmann-S aub c edibili y model we obse e he ec o o a e age annual claim amoun s o isk iin yea j (11) and he weigh s a e he ec o o he numbe o con ac s o isk iin yea j (12) The mul idimensional Bühlmann-S aub c edibili y model s ems om he ollowing assump ions ( o de ails see [2]): MBS1: Condi ional on he alue o θ1 he alues Xij o j= 1, 2, ..., na e independen wi h ap-dimensional ec o o mean alues: (13) and wi h a co a iance ma ix o ype p × p . EM_02_14_zlom 4.6.2014 8:54 S ánka 172 Finance 1732, XVII, 2014 (14) MBS2: The pai s , whe e he a e indepen- den and he θ1, θ2, ... a e independen and iden ically dis ibu ed. The co a iance ma ix (14) is diagonal, hus i is assumed ha he componen s o ec o Xij a e independen . In some si ua ions indepen- dence is no assu ed. Fo example, i we wan o use he mul idimensional c edibili y model o es ima e he claim equencies o la ge claims and he claim equencies o s anda d claims, he assump ion o hei independence is no me . The claim equencies o la ge claims and o s anda d claims ela e o he same numbe o con ac s and he e o e he weigh s a e equal. In he case o equali y o he weigh s we can eplace he so- called s anda d assump ion by (see [1]) he so- called al e na i e assump ion: In assump ion MBS1in place o he co a iance ma ix (14) we can conside he co a iance ma ix (15) whe e he weigh s o all he componen s o he ec o Xij a e equal, i.e. The s anda d and al e na i e assump ions do no in ac co e all possible si ua ions, bu hey do include he majo i y o cases which a ise in p ac ice. Subjec o hese condi ions we can gene alise o mulae (4) o (6), ela ing o he one-dimensional c edibili y es ima e, using he mul idimensional c edibili y es ima e de ined la e by o mulae (16) o (21). I assump ions MBS1aMBS2a e me , we can exp ess he c edibili y es ima e o he ne p emium as ollows (16) whe e Biis he indi idual es ima e and he collec i e es ima e o he ec o µ (θi). Ziis he c edibili y ma ix and i can be shown ha , a e gene alising o mula (6) o he c edibili y ac o , he ollowing applies (17) I is clea ha he ec o o a e age annual claim amoun s o he i h isk class ep esen s he indi idual es ima e o he ec o µ (θi) and he e o e (18) he e, o k= 1, 2, ..., p, he k h elemen o he ec o Biis (19) We can he e o e exp ess he ec o Biby he o mula (20) whe e he ma ix o weigh s has he elemen s (21) I s anda d assump ion (14) is me , he ma ix has elemen s (22) EM_02_14_zlom 4.6.2014 8:54 S ánka 173 Finance 174 2014, XVII, 2 I he co a iance ma ix o he ec o mee s he al e na i e assump ion (15), he ollowing applies (23) The mul idimensional c edibili y model has h ee s uc u al pa ame e ec o s , Sand T which a e analogous o he s uc u al pa a- me e s µ 0, σ 2and τ 2o he one-dimensional Bühlmann-S aub c edibili y model. Thei de ini ions a e as ollows: The ma ices o s uc u al pa ame e s is he ec o o collec i e p emiums is he co a iance ma ix o he ec o is he co a iance ma ix o he collec i e p emium ec o We ob ain he homogeneous c edibili y es ima e i in o mula (16) we eplace he unknown ec o µo he collec i e p emiums by i s es ima e (24) Simila o he one-dimensional Bühlmann- S aub c edibili y model, also in he mul idi- mensional model he homogeneous es ima e and he indi idual elemen s o he ec o sa is y he balance p ope y (25) To compa e he accu acy o c edibili y es ima es, collec i e es ima e and indi idual es ima es, he e a e used quad a ic losses. The quad a ic loss o he c edible p emium is zi-mul iple o he quad a ic loss o he indi idual p emium and (1 – zi)-mul iple o he quad a ic loss o he collec i e p emium (see [9]). Since zi and (1 – zi) ake alues om in e al 〈0;1〉, so he quad a ic loss o he c edibili y p emium is less o a he mos equal o quad a ic losses o he collec i e and he indi idual p emium. 2. Es ima ion o he S uc u al Pa ame e s o he One- Dimensional Bühlmann-S aub and he Mul idimensional C edibili y Models In he one-dimensional Bühlmann-S aub c edibili y model i is necessa y o es ima e, no only he collec i e p emium as shown in o mula (8), bu also he s uc u al pa ame e s o a iabili y σ2and τ2. The s uc u al pa ame e σ2is he mean o he wi hin g oup a iances σ2(θi). As he es ima e o he a iance σ2(θi)in he i h isk class we can ake he sample a iance (26) I is possible o es ima e he mean o hese a iances as he a i hme ic mean. As epo ed by Bühlmann and Gisle [1] he op imal weigh s should be p opo ional o . This exp ession depends on he momen s up o he ou h o de . These momen s a e unknown and hei es ima ion equi es u he s uc u al pa ame e s, he e o e such a p ocedu e is no app op ia e. In he special case, whe e Xij a e condi ionally on θino mally dis ibu ed, he abo e men ioned exp ession does no depend on i. The e o e i is op imal o use he equal weigh s o all pa ial a iances. On he one hand, he mos o insu ance da a a e no no mally dis ibu ed, on he o he hand, weigh s, which would be be e , a e no known. Fo his eason, Bühlmann and S aub [3] sugges ed o es ima e he s uc u al pa ame e σ2using he simple a i hme ic mean o he indi idual a iances : (27) The cha ac e is ic is an unbiased and consis en es ima e o he s uc u al pa ame e σ2. A eade can ind p oo s o hese p ope ies in [1] on he page 94. An unbiased and, p o ided he e is no dominan isk class in he po olio, also consis en es ima e o he pa ame e τ2(see e.g. [7]) is gi en by EM_02_14_zlom 4.6.2014 8:54 S ánka 174 Finance 1752, XVII, 2014 (28) whe e (29) (30) And (31) Rema k: I he si ua ion a ises, ha is nega i e, di e ences be ween he isk classes a e no demons able. In his case we se he pa ame e τ2 o 0. The c edibili y ac o s (6) hen equal 1 and he c edibili y p emiums o all he isk classes a e he same and a e equal o he a e age annual claim amoun o he whole po olio (31). So ha he es ima e o he s uc u al pa a- me e τ2can also be used in he si ua ion desc ibed in he ema k, he es ima e o he s uc u al pa ame e τ2is de i ed om he o mula (32) The homogeneous c edibili y es ima e in he mul idimensional c edibili y model al eady has inco po a ed in i an es ima e o he ec o µin acco dance wi h o mula (24). Hence we will concen a e only on es ima ion o he ma ix o he s uc u al pa ame e s Sand T. To es ima e he diagonal elemen s o he ma ices SaTwe can use analogous app oaches o hose in he case o he one- dimensional Bühlmann-S aub model. The diagonal elemen s o he ma ix Sa e es ima ed in a simila way o ha o he pa ame e σ2in he Bühlmann-S aub model, in acco dance wi h o mula (27) hus: (33) The diagonal elemen s o he ma ix T again can be es ima ed simila ly o he s uc u al pa ame e τ2in he Bühlmann-S aub model. Analogously o o mulae (32) and (28) o (31) we hen ge (34) whe e (35) and (36) (37) (38) We will deno e he non-diagonal elemen in he k h ow and he l h column o he ma ix Tby τkl. Fi s we es ima e τkl using he weigh s hus: (39) Simila ly we es ima e τkl using he weigh s : (40) EM_02_14_zlom 4.6.2014 8:54 S ánka 175 Finance 176 2014, XVII, 2 whe e (41) The ma ix Tis he co a iance ma ix, which means ha i s diagonal elemen s a e he a iances τ2 kand i s non-diagonal elemen s a e he co a iances τkl. By aking he a io o he co a iance τkl and he p oduc o he s anda d de ia ions τk. τlwe ge he co ela ion coe icien , which akes alues in he in e al 〈–1; 1〉. On he basis o his i mus be he case ha |τkl|≤τk. τl. I his condi ion is me , he simple a i hme ic mean o he es ima es (39) and (40) is an es ima e o he co a iance τkl. In gene al he ollowing es ima e is p oposed (see [1]) (42) To es ima e he s uc u al pa ame e s o he one-dimensional and also he mul idimensional Bühlmann-S aub c edibili y model we can use s a is ical me hods, o which we can ind p ocedu es in s a is ical p og amming packages. We will concen a e only on he mul idimen- sional c edibili y model. So he k h diagonal elemen (33) o he ma ix Sis he simple a i hme ic mean o he sample a iances (43) whe e (44) in each o he isk g oups, ela ing o he k h elemen o he es ima ed ec o Le us modi y he elemen s o ma ix T. We can ew i e he cha ac e is ics T(k) in he o m (45) whe e is he sample a iance o he a iable B(k) wi h ela i e weigh s : (46) We hen know ha in acco dance wi h o mula (45) we can exp ess he es ima e (35) o he diagonal elemen τ2 ko he ma ix Tby he o mula (47) In he case o he es ima es (39), (40) o he non-diagonal elemen s τkl o he ma ix Ti is again he case ha (48) (49) whe e is he sample co a- iance be ween a iables B(k), B(l) wi h ela i e weigh s : (50) and is he sample co a ian- ce be ween a iables B(k), B(l) wi h ela i e weigh s : (51) Fo k= 1 we ge he sample a iance (46) and he sample co a iance (50) ( o i= 1, 2, ..., p) om he i s ow o he sample co a iance ma ix o a iables B(1), B(2), ..., B(p) making use o he ela i e weigh s . Simila ly o k= 2 . EM_02_14_zlom 4.6.2014 8:54 S ánka 176 Finance 1772, XVII, 2014 we can ead o he sample a iance (46) and sample co a iance (50) ( o l= 1, 2, ..., p) om he second ow o he sample co a iance ma ix o a iables B(1), B(2), ..., B(p) by making use o he ela i e weigh s . Analogously we can ob ain he sample a iance (46) and sample co a iance (50) o k = 3, 4, ..., p. F om hese elemen s we can hen de elop he ma ix (52) whose k h ow is he k h ow o he sample co a iance ma ix o a iables B(1), B(2), ..., B(p) making use o he ela i e weigh s . I we look a o mula (48) and also o mula (49) we see ha he es ima es o he non- diagonal elemen s o he k h ow o he ma ix Ta e a mul iple o he k h ow o he ma ix SB. F om o mula (47) i is ob ious ha he es ima e o he k h diagonal elemen o he ma ix Tis a mul iple o he di e- ence be ween he k h diagonal elemen o he ma ix SBand he k h diagonal elemen o he ma ix . Gi en ha he ma ix is diagonal wi h elemen s we can de ine he ma ix (53) whe e ⊗is he Hadama d ma ix p oduc and ma ix Co ype p × phas elemen s (54) An es ima e o he co a iance ma ix Tis hen he ma ix (55) I any o i s diagonal elemen s a e nega i e, hen in acco dance wi h o mula (34) we eplace hem by he alue 0 and i any o i s non-diagonal elemen s do no mee he condi ion hen in acco dance wi h o mula (42) we eplace hem by he alue sgn 3. Example o Applica ion An unnamed insu ance company di ides i s MTPL po olio in o eigh a i classes. Fo his ype o insu ance he insu e has a ailable no only i s own da a, bu also summa y da a om o he companies. Speci ically o each isk class i has a ailable he ollowing s a is ics: X – i– he a e age claim amoun (in €), pe con ac yea , o a pe iod o nyea s, – he sample s anda d de ia ion o he a e age claim amoun s o he pe iod o n yea s, wi• – he numbe o con ac s o he pe iod o nyea s. These da a a e se ou in Table 1. The ac ua y has he ask o se ing he ne p emium o he nex insu ance yea . Gi en ha one expec s a signi ican dependence be ween he own company da a and ha o he o he insu e s, i is desi able o use he mul idimensional c edibili y model o se he ne p emium o each a i class. In o de o ge he c edibili y es ima e o he ne p emium using he mul idimensional model we need o ca y ou he calcula ions as shown in Table 2. EM_02_14_zlom 4.6.2014 8:54 S ánka 177 Finance 178 2014, XVII, 2 Tab. 1: Own da a and da a om o he insu e s Sou ce: own Tab. 2: Table o calcula ions Sou ce: own calcula ion By aking he simple a i hme ic means o he sample a iances and he simple a i hme ic means o he sample a iances we ob ain es ima es o he diagonal elemen s o he ma ix S An es ima e o he ma ix Sis he e o e he ma ix To a oid he ela i ely leng hy calcula ions o es ima e ma ix Twe will use he adjus men s which we made ea lie in he a icle (exp essions (45) o (52)). To calcula e he elemen s o ma ix (52) we can use se e al o s a is ical so wa e applica ions. To es ima e ma ix Tin his a icle we use co ela ion analysis p ocedu e in SAS En e p ise Guide applica ion. Fi s we es ima e he co a iance ma ix o a iables B(1) aB(2) ( he alues a e shown in Tables 1 and 2), whe eby we make use o he ela i e weigh s . Simila ly we es ima e he co a iance ma ix o a iables B(1) aB(2) making use o he ela i e weigh s . We hus ge he co a iance ma ices shown in Table 3. EM_02_14_zlom 4.6.2014 8:54 S ánka 178