Bayesian Estimations in Insurance Theory and Practice

Transcription

1 Advacs i Mathmatical ad Computatioal Mthods Baysia Estimatios i Isurac Thory ad Practic VIERA PACÁKOVÁ Dpartmt o Mathmatics ad Quatitativ Mthods Uivrsity o Pardubic Studtská 95, 53 0 Pardubic CZECH REPUBLIC virapacakova@upccz Abstract: - This papr xplais th Baysia vrsio o stimatio as a mthod or calculatig crdibility prmium or crdibility umbr o claims or short-trm isurac cotracts usig two igrdits: past data o th risk itsl ad collatral data rom othr sourcs cosidrd to b rlvat Th Poisso/gamma modl to stimat th claim rqucy or portolio o policis ad Normal/ormal modl to stimat th pur prmium ar xplaid ad applid Kyword: - Crdibility prmium, Prior distributio, Postrior distributio, Baysia stimator, Poisso/gamma modl, Normal/ormal modl Itroductio A typical atur o th isurac practic is th d to st prmium at th bgiig o th isurac cotract Numbr o occurrc o claims ad th total claim amouts or isurac compay i th utur ar th radom variabls Thir suicitly, prcis ad rliabl stimat is xtrmly importat to dtrmi th corrct prmium or xt yar i isurac compay Crdibility thory is a tchiqu, or st o tchiqus, or calculatig prmiums or short trm isurac cotracts Th tchiqu calculats a prmium or a risk usig two igrdits: past data rom th risk itsl ad collatral data, i data rom othr sourcs cosidrd to b rlvat Th sstial aturs o a crdibility prmium ar that it is a liar uctio o th past data rom th risk itsl ad that it allows or th prmium to b rgularly updatd as mor data ar collctd i th utur Watrs, 994) A crdibility prmium rprsts a compromis btw th two abov mtiod sourcs o iormatio Th crdibility ormula or stimatio o pur prmium or claim rqucy P c i xt yar is: P = Z P + Z ) whr r c r ) µ P is stimatio basd o ow past data i isurac compay, or risk, ad µ is stimatio basd o collatral data ad Z is a umbr btw zro ad o, kow as th crdibility actor Crdibility actor Z is a masur o how much rliac th compay is prpard to plac o th data rom th policy itsl Crdibility ormula is ot usd i th orm = Z x + Z ) P c ) µ W will prst Baysia approach to crdibility stimatio by two importat modls or isurac practic Th Baysia Irc Th Baysia philosophy 763) ivolvs a compltly dirt approach to statistical irc Suppos x = x, x, x ) is a radom sampl rom a populatio spciid by dsity uctio x / ) ad it is rquird to stimat paramtr Θ Th classical approach to poit stimatio trats paramtrs as somthig ixd but ukow Th sstial dirc i th Baysia approach to irc is that paramtrs ar tratd as radom variabls ad thror thy hav probability distributios Prior iormatio about Θ that w hav bor collctio o ay data is prior distributio Θ ) that is probability dsity uctio or probability mass ISBN:

2 Advacs i Mathmatical ad Computatioal Mthods uctio Th iormatio about Θ providd by th sampl data x = x, x, x ) is cotaid i th = x i / liklihood / ) ) x Bays thorm combis this iormatio with th iormatio cotaid i Θ ) i th orm Θ / ) x ) ) x ) ) x = 3) d that dtrmis th postrior distributio x) So atr collctig appropriat data w dtrmi th postrior distributio that is th basis o all irc cocrig Θ Not that ) x ) ) x = d dos ot ivolv Θ It is just a costat dd to mak it a propr dsity that itgrats to uity A usul way o xprssig th postrior dsity is to us proportioality W ca writ or simply / ) x ) ) x / 4) postrior liklihood prior Th postrior distributio cotais all availabl iormatio about Θ ad thror should b usd or makig dcisios, stimats or ircs Th Baysia approach to stimatio stats that w should always start with a prior distributio or ukow paramtr, prcis or vagu accordig to th iormatio availabl Not that w ar rrrig to a dsity hr implyig that Θ is cotiuous This cocrs most applicatios bcaus v wh X is discrt, as i biomial or Poisso distributios, th paramtrs π or λ will vary i a cotiuous spac 0 ; or 0, + ) rspctivly Thr may b som situatios i which w d o-iormativ prior For xampl i Θ is a biomial distributd ad w hav o prior iormatio about Θ, th uiorm distributio o itrval 0 ; as a prior distributio would sm appropriat W ot hav prior iormatio about paramtrs basd o prvious practic, rspctivly, stimats by xprts Th valus o ths paramtrs rlct th subjctiv opiio o th dcisio makr, so Baysia approach ca b criticizd as subjctiv Θ 3 Th Baysia Estimator I w hav oud postrior distributio o a ukow paramtr Θ, w d to aswr th qustio how do w us th postrior distributio o Θ, giv th sampl data x = x, x, x ), to obtai a stimator o Θ First w must spciy th loss uctio g x) a masur o th loss icurrd wh x), which is g is usd as a stimator o Θ W sk a loss uctio which is zro wh th stimatio is xactly corrct, that is g x) = Θ ad which icrass as g x) gts athr away rom Θ Thr is o vry commoly usd loss uctio, calld quadratic or squard rror loss Th quadratic loss is did by g x) ; ) = [ g x) ] L 5) ad it is rlatd to ma squar rror rom classical statistics W will show that th Baysia stimator that ariss by miimizig th xpctd quadratic loss is th ma o postrior distributio So ad L g x) ; )) = [ g x) ] x) d E E L g x) ; )) g x) = [ g x) ] x) d quatig to zro g x) x) d = x) d Bcaus o x) d =, w gt x) = E x) g 6) W will cosidr two importat xampls o drivatio o th postrior distributio ad th Baysia stimators udr th quadratic loss uctio or crtai stimatio situatios with giv prior distributios, importat or isurac practic 4 Th Poisso/gamma Modl Suppos w hav to stimat th claim rqucy or a risk whos claim umbrs hav a Poisso distributio with paramtr λ W do ot kow th valu o λ but bor havig ay data rom risk itsl availabl, w assum that th prior distributio o λ is a gamma distributio G; β) Th claim rqucy rat or a class o isurac busiss may li aywhr btw 0 ad + A isurr with a larg xpric may quit accuratly stimat th rat ISBN:

3 Advacs i Mathmatical ad Computatioal Mthods Th gamma distributio may b covit or rprstig ucrtaity i a currt stimat o th claim rqucy rat This distributio is ovr th whol positiv rag rom 0 to +, ad th ma /β ca b st qual to th currt bst stimat Ucrtaity is rprstd by variac /β o th gamma distributio G; β) Our objctivs is to stimat th ukow paramtr λ Suppos w hav past obsrvatios x = x, x, x ) Th Baysia stimat o λ, with rspct to a quadratic loss uctio, giv ths data, is λ = B E λ x) 7) that is th ma o th postrior dsity o λ By assumptio th dsity uctio o th prior G; β) distributio is β λ β β λ) = λ = c Γ ) λ λ 8) Th distributio o a umbr o claims is th Poisso with a ixd but ukow paramtr λ, so th liklihood uctio has th orm λ) xi xi λ λ λ i = = = c xi! x / λ 9) By Bays thorm w gt th postrior dsity o λ, giv x = x, x, x ), i th orm λ x 0) λ xi λ β λ β + ) + x / ) λ λ = λ i that is th gamma distributio with w paramtrs = + x i ) β = β + Thus th Baysia stimator o λ usig th quadratic loss is + xi λb = β + which ca b rwritt as + x λb = = x + β + β + + x = β + β β + β ) I w put actor crdibility Z = 3) β + th λ = E λ x = Z x + Z 4) B ) ) µ which is th crdibility ormula or updatig claim rqucy rats It ca b s rom th crdibility actor xprssio, sic is o-gativ ad β is positiv, that Z is i th rag zro to o ad it is icrasig uctio o I o past data rom th risk itsl ar availabl, th = 0 ad Z = 0 too ad th bst stimat o λ is /β, th ma o th prior gamma distributio It ca b s that Z dos ot tak th valu o or ay iit valu o Th valu o Z dpds o th amout o data availabl or th risk, ad th collatral iormatio through β, which rlct th variac /β o th prior distributio 4 Applicatio o Poisso/gamma Modl Crdibility ormula 4) allows asy applicatio o Poisso/gamma modl i isurac practic W try to show it i this applicatio Th aual umbr o claims rsultig rom motor third-party liability isurac i isurac compay i th yars is giv i Tabl, colum lablld as x i I Poisso/gamma modl or claim umbrs w hav assumd our prior kowldg about th ukow paramtr aual claim rt) λ summarizd by gamma distributio G; β) with paramtrs = 8400 ad β = 0,4 Last colum dotd as λ B cotais valus o Bays stimators o aual claim rats x i or ach yar i basd o i-) past obsrvatios by quatio 4) For calculatio o crdibility actors Z i w usd simpl quatio 3) Tabl Procdur to updat Bays stimat o λ Yar i x i Z i λ B , , , , , , Sourc: Ow calculatio 5 Th Normal/ormal Modl Our problm is to stimat th pur prmium, i th xpctd aggrgat claims or a risk So X is a radom variabl dotig total claims rom a risk i a comig yar ad th distributio o X is ormal, dpds o th valu o a ukow paramtr Θ Th coditioal distributio o X/ is ormal ad th ukow paramtr is th ma o this distributio, bcaus o ISBN:

4 Advacs i Mathmatical ad Computatioal Mthods ; ) X ~ N 5) Th prior distributio o is ormal, ~ N µ ; 6) whr µ, ) ar kow Suppos w hav past obsrvatios o X, x = x, x, x ) Our problm is to stimat X ) E ad w us agai th Baysia stimat with rspct to th quadratic loss I was kow, th pur prmium would b E X = 7) ) So th problm o stimatig X ) E is th sam as th problm o stimatig o as a Baysia stimator = B E x) 8) i th postrior ma o giv x W d to kow th orm o th postrior dsity / x uctio ) Suppos w hav data o prvious obsrvatios x = x, x, x ) so w ca xprss th liklihood / x) as / x ) ) xi = x x ) i + i = As w ca s, th liklihood uctio is quadratic i, ad ca b show to b proportioal to a + a + a3 ) Wh igorig trms ot ivolvig, w ca xprss th ormal prior distributio as big proportioal to ) µ ) µ + = π Th postrior dsity / x) proportioal to / x ) ad atr adjustmts by Bays` thorm is x µ + + µ + x + + / ) So th postrior distributio / x) x 9) ~ with paramtrs ~ µ,, i / ) ~ µ ) ~ is a ormal, say ~ µ ~ + ~ x = c 0) W will id th paramtrs ~ µ, ~ by quatig th powr o ad i two dirt xprssio o / x Th w gt ) ~ µ + x µ = ) + ~ = ) + W ca id th Baysia stimatio o pur prmium as th ma o th postrior distributio, i µ + x B = 3) + That ca b rwritt as E x = Z x + Z 4) ) ) µ which is a crdibility stimat o th pur prmium E x with actor crdibility ) Z = = = 5) Applicatio o Normal/ormal Modl Total aggrgat claims i a particular isurac compay ar modlld with a ormal distributio N ; ), whr is ukow ad = Prior iormatio about suggsts that it is distributd by N µ; ) with kow paramtrs µ = ad = Aggrgat claims rom th last sv yars wr ot icorporatd i th prior iormatio ad thy ar i Tabl, colum amd x i Th Bays stimatios o th pur prmiums or ach yar by quatio 4) with crdibility actors calculatd by 5) thr ar i th last colum o Tabl Tabl Bays stimatios o pur prmium i x i Z B , , , , , , , Sourc: Ow calculatio ISBN:

5 Advacs i Mathmatical ad Computatioal Mthods 6 Coclusios Baysia stimatio thory provids mthods or prmatly updatd stimats o th umbr o claims ad o th pur prmium or ach comig yar i isurac compay Baysia approach combi prior iormatio that ar kow bor collctd o ay data ad iormatio providd by th sampl data, which ar umbr o claims or aggrgat claim amouts i prvious yars Th biggst advatag o th Poisso/gamma modl ad Normal/ormal modl or isurac practic is possibility to xprss thm i th orm o crdibility ormulas by 4) or 4) Ths ormulas allow asy applicatio i isurac practic, as s rom th xampls i subsctios 4 ad 5 Howvr, th Baysia approach dos hav a w srious drawbacks ad limitatios This approach ca b criticizd as subjctiv, bcaus w should always start with a prior distributio o stimatd paramtrs Formulas 3) ad 5) ivolv paramtrs, β i th ormr ad, i th lattr, which w hav assumd to b kow Th valus o ths paramtrs rlct th subjctiv opiio o th dcisio makr; thr is o qustio o stimatig ths paramtrs rom data Th problm o stimatio o ukow paramtrs wh som data rom rlatd risks ar availabl solvs th so-calld Empirical Bays Crdibility Thory, which is ot th subjct o this papr Rrcs: [] P J Bolad, Statistical ad Probabilistic Mthods i Actuarial Scic, Lodo: Chapma&Hall/CRC, 007 [] H Bühlma, A Gislr, Cours i Crdibility Thory ad its Applicatios, Brli: Sprigr, 005 [3] R Kaas, M Goovarts, J Dha, M Duit, Modr Actuarial Risk Thory, Bosto: Kluwr Acadmic Publishrs, 00 [4] E Kotlbová, Baysovská štatistická idukcia v koomických aplikáciách Baysia Statistical Irc i Ecoomic Applicatios), Bratislava: Ekoóm, 009 [5] V Pacáková, Th Baysia Irc i Actuarial Scics, Ctral Europa Joural or Opratios Rsarch ad Ecoomics, Volum 5, Numbr 3-4, 997, pp [6] E Šoltés, V Pacáková, T Šoltésová, Vybraé krdibilé rgrsé modly v havarijom poistí Slctd crdibility rgrssio modls i accidt isurac), Ekoomický časopis, Vol 54, No, 006, pp 68-8 [7] Y K Ts, Noli Actuarial Modls, Cambridg: Uivrsity Prss, 009 [8] H R Watrs, A Itroductio to Crdibility Thory, Lodo ad Ediburgh: Istitut o Actuaris ad Faculty o Actuaris, 994 ISBN: