Long-range contagion in automobile insurance data : estimation and implications for experience rating

Abstract

Abstract The purpose of the paper is to use the date of claims in the prediction of risks. Dynamic random,e¤ects models on longitudinal count data are presented, and estimated on the portfolio of a major Spanish insurance company. The estimated autocorrelation coe¢cients of stationary random e¤ects are decreasing. A consequence is that the predictive ability of a claim,decreases with the lag between the date of risk prediction and the date of the claim. Empirical results are presented, which relate to the dynamic of bonus-malus coe¢cients and to the link between the duration of the histories and the information which they provide. Keywords

Full text

Long-rangecontagioninautomobileinsurance
data:estimationand implicationsforexperience
rating¤
JeanPinquet1,MontserratGuillén2,CatalinaBolancé2
2000-43
November,2000
1UniversityParisX
2UniversityofBarcelona
¤Pinquetacknowledges…nancialsupportfromtheFédérationFrançaisedesSociétés
d’Assurance.Guillénand Bolancé thanktheSpanishCICYTgrantSEC99-0693.
1

2

Résumé
L’objectifdu papierestd’utiliserladatedes sinistresdanslaprédiction
desrisques.Desmodèlesàe¤etsaléatoires sontprésentés,etestimés sur
leportefeuilled’unegrande compagnied’assurancesespagnole.Lescoe¢-
cientsd’autocorrélationestiméspourdese¤etsaléatoires stationnaires sont
décroissants surcesdonnées.Une conséquence estquelepouvoirprédictif
d’un sinistredécroîtavec ledécalage entreladatedeprédictionetladate
desurvenance du sinistre.Desrésultatsempiriques sontprésentés,quise
rapportentàladynamiquedescoe¢cientsbonus-malusetàlamesurede
l’informationapportée parleshistoriquesenfonction deleurdurée.
Mots-clés
E¤etsaléatoiresconstantsetdynamiques.Fonction d’autocorrélation
pourdese¤etsaléatoires stationnaires.
Abstract
Thepurposeofthepaperistousethedateofclaimsintheprediction
ofrisks.Dynamicrandome¤ectsmodelsonlongitudinalcountdata are
presented,and estimatedontheportfolio ofamajorSpanishinsurance
company.The estimatedautocorrelationcoe¢cientsofstationaryrandom
e¤ectsaredecreasing.Aconsequence isthat thepredictiveabilityofa
claimdecreaseswiththelag betweenthedateofriskpredictionand the
dateofthe claim.Empiricalresultsarepresented,whichrelatetothe
dynamicofbonus-maluscoe¢cientsand tothelinkbetweentheduration
ofthehistoriesand theinformationwhichtheyprovide.
Keywords
Time-independentand dynamicrandome¤ects.Autocorrelationfunc-
tionforstationaryrandome¤ects.
3

4

1Introduction
Thepurposeofthepaperistousethedateofclaimsintheprediction
ofrisks.Thisissuehasalreadybeenaddressed byactuarial literature.
Solutionsareobtainedfromcredibilitymodelswhichcan beupdated(Ger-
ber,Jones(1975)),and fromcredibilityestimatorswithgeometricweights
(Sundt (1988)).
Theratingmodelspresentedinthispaperareobtainedafterstatistical
inference onlongitudinalcountdata.Letusclarify…rst themotivations
whichled ustoquestiontheassumptionoftime-independence fortheran-
dome¤ects.
Hiddenfeaturesofriskdistributionsvarywithtime,asdoratingfactors.
Therandome¤ectsaddedinanaprioriratingmodelonlongitudinaldata
shouldthen bedynamic.Variationsofratingfactorsbetweentwodates
shouldincreasewiththerelatedlag,and thesameresultisexpectedfor
hiddenfeaturesinriskdistributions.Hence thepredictiveabilityofaclaim
shoulddecreasewiththelag betweenthedateofriskpredictionandthedate
ofthe claim.Besides,economicanalysis suggeststhatoptimal insurance
contractswithmoralhazardshould penalize recentclaimsmorethanolder
ones(Henriet,Rochet (1986)).Astationaryprocess fortherandome¤ects
will relatethepredictivepowerofclaimstotheaforementionedlag.Ifthe
precedingintuitionisveri…ed,the estimatedautocorrelationfunctionofthe
randome¤ects should bedecreasing,apointalreadymentioned bySundt
(1988).
In Section2,wepresentdi¤erentPoissonmodelswithrandome¤ects.
Thevariance ofatime-independentrandome¤ectcan be estimated
fromdisaggregated data orfromnumbersofclaimsand frequencypremiums
whicharesummedacross theperiods.Ifthe estimatedvariance obtained
fromdisaggregated dataisgreaterthanthesecond one,the estimationof
distributionsfordynamicrandome¤ectscan bethoughtof.Thiscondition
isveri…edonourdataset,whichisdrawnfromtheportfolio ofamajor
Spanishinsurance company.
An unconstrainedautocorrelationfunctionforthedynamicrandomef-
fectsisthenestimatedfromaPoissonmodelwithregressioncomponents.
Foreachlag,the corresponding autocorrelationisestimatedfrompaired
o¤productsoflagged number-residualsand frequency-premiums.Inthe
empiricalstudy,wedo…nd adecreasing autocorrelationfunction,witha
decreasingshapewhichis slowerthana geometricone.
5

OptimalBMS designedfromalinearcredibilityapproacharepresented
in Section3fromtherandome¤ectsmodelsdevelopedin Section2.
In Section4,weassess the consequencesofavarying autocorrelation
speci…cationfortherandome¤ectsonthedynamicofbonus-maluscoe¢-
cients.Anoptimalbonus-malus system(later referredto asBMS)designed
fromamodelwith dynamicrandome¤ectsand adecreasing autocorrelation
functionwill behaveinthefollowingway.Forapolicyholderwithafault-
less history,theno-claimdiscountsinduced byaclaimless yeararesmaller
afterafewyearsthanthoseobtainedfromtheusualcredibilitymodel, but
theyaremoreimportantifclaimswerereportedrecently.The explanation
isthesamein bothcases.The credibilitygrantedto a given periodofthe
pastdecreasesrapidlyastimegoesby,duetotheincreaseofriskexposure
butmostlytothediminutionoftheautocorrelationcoe¢cients.
Increasesofpremiumsinduced byclaimsfromthedi¤erentBMSare
notverydi¤erentinthe empiricalstudy.Asaconsequence,the claimless
policyholdersaresubsidized bytheotherpolicyholdersafterafewyearsif
theusualoptimalBMSisapplied,whereastheactualautocorrelationfunc-
tion decreaseswiththelag. Onthewhole,anoptimalBMS derivedfroma
Poissonmodelwithavarying autocorrelationfunctionseemsacceptableto
policyholders, ifthe estimatedcorrelogramisdecreasing.Besides, itwould
entail strongincentivestocarefuldrivingforthedriverswhoreporteda
claimrecently.
Anotherdi¤erence isthatanoptimalBMS designedfromthistypeof
speci…cationreachesitslimitfasterthantheotherones.Numericalre-
sultsonthisissueareofinterestforpublicauthoritiesfacingatrend of
ratingderegulation.IfcompulsoryBMSweredestinedtodisappearinthe
future,manypeoplesuggest thatasummaryofeach policyholder’sclaim
historyshould beavailabletoeverycompetitorofthepolicyholder’sinsur-
ance company.Insurance companiesarenotforced bycompetitiontouse
privateinformationontheirpolicyholdersintheir ratingstructure.For
instance,apolicyholderwithalongclaimless historycould beovercharged
ifhe could notprovehisgood behaviourtothe competitorsofhisinsur-
ance company.Public claimhistories,ascompulsoryBMS,can prevent the
creationofinformationrentsbyinsurance companies(see Kunreutherand
Pauly(1985)forananalysisina generalsetting).In Spainforinstance,
theAssociationofInsurersiscurrentlydesigningasystemforpoolingdata
onall automobileinsurance policies.FromJanuary2002,all insurerswill
haveaccess toinformationonanypolicyholder regardingthevehicle,type
ofcoverageand claimsfortheprevious…veyears, includingtheoccurrence
date,typeofdamageand totalpaidamount.At thesametime, itwill
6

be compulsoryforall insurerstoprovidetheinformationontheirportfo-
lio.As shownin Section4,a…veyearhistoryrevealsmoreinformation
attainablebyexperience thanwhatcould be expectedfromamodelwith
time-independentrandome¤ects.
Finally,theapplicabilityoftheresultsobtainedinthepaperisbrie‡y
discussedin Section5.Apossiblyusefulresultforpractitionersisthe
following.AnoptimalBMSwithconstantrandome¤ectswill probably
overestimatethepredictiveabilityofclaimsifitisestimatedfromshort
historiesand appliedto a longerduration.Thisresultoccursiftheauto-
correlationfunctionoftherandome¤ectsactuallydecreaseswiththelag.
2Moment-basedestimatorsforlongitudinalcount
data
2.1Time-independentrandome¤ects
The…xede¤ectsmodelonlongitudinalcountdataiscomposedofPoisson
distributions,and isdenotedas
Ni;t»P(¸i;tui)i=1;:::;p;t=1;:::;Ti;max
iTi=Tmax;
whereni;tisthenumberofclaimsreportedforthepolicyholderiin period
t.The coe¢cient¸i;tdependsonregressioncomponents(represented bya
line-vectorxi;t)and ontheduration denotedasdi;t.Wewrite
¸i;t=di;texp(xi;ta);a2Rk;
whereaisacolumn-vectorofparametersand wherekisthenumberof
regressioncomponents.
The…xede¤ectuirepresentstheresidualheterogeneityinthenumber
ofclaimsdistribution.Theprecedingdistributionsholdfor real individuals,
and thevariablesNi;taresupposedtobeindependentinthe…xede¤ects
model. Thisistheusualassumptioninactuarialmodels(observedconta-
giononrisk variablesisonlyapparent).Therandome¤ects(Ui)i=1;:::;pare
assumedi.i.d., and thetwo…rstmomentsare
E(Ui)=1;V(Ui)=¾2
U:
Intherandome¤ectsmodel, wehavethen
E(Ni;t)=¸i;t;V(Ni;t)=¸i;t+¸2
i;t¾2
U;E(Ni)=¸i;V(Ni)=¸i+¸2
i¾2
U;
7

withNi=PtNi;tand ¸i=Pt¸i;t.Distributionsinthemodelwith
randome¤ectsaremixturesofPoisson distributions,and theyreferto
genericindividuals,whorepresentclassesofreal individualswiththesame
observable characteristics.
Twoconsistentestimatorsof¾2
Ucan bethoughtof,whichare
c
¾2
U1=
P
i;t
h(ni;t¡b
¸i;t)2¡ni;ti
P
i;tb
¸2
i;t;c
¾2
U2=Pi(ni¡b
¸i)2¡ni
Pib
¸i2;(1)
whereb
¸i;tand b
¸iarethefrequency-premiumscomputedfromlikelihood
maximizationinthePoissonmodelwithout…xedor randome¤ects.The
second estimatorshould bepreferred becauseitsvariance islower.The
intuitionisthat thisestimatorusesmoreinformation. Onourdataset,
thesecond estimatorisinferiortothe…rstone(see Section4).Asigni…-
cantdi¤erence betweentwoconsistentestimatorsmeansthat themodel is
misspeci…ed(see Hausman(1978)foratest).As showninthefollowing
section,theinequality
0<c
¾2
U2<c
¾2
U1
isanecessaryconditionforthe estimationofsomedynamicrandome¤ects
speci…cations.
2.2Poissonmodelswith dynamicrandome¤ects
The…xede¤ectsmodel isdenotedas
Ni;t»P(¸i;tui;t):
Asmentionedintheintroduction,thisformulationisnatural. The…xed
e¤ectsre‡ecthiddenfeaturesinriskdistributionswhichmay varywith
time,asdoratingfactors.Two assumptionsareretainedontherandom
e¤ects,whicharethefollowing.
²Thedistributionofvec
1·t·Ti(Ui;t)dependsonlyonTi.
²Ifthedistributionofvec
1·t·Ti(Ui;t)isthatofvec
1·t·Ti(Ut);thedistribution
ofvec
1·t·Tmax(Ut)is supposedtobestationary.Thisinvariance property
withrespect totimetranslationsimpliesthat thepredictiveabilityof
aneventwill depend onthelag betweenthedateofriskprediction
and thedateofthe event.Wesupposethat thesquaredrandom
e¤ectsareintegrable.
8

Withan unconstrainedautocorrelationfunctionforstationaryrandom
e¤ects,wehave
Cov(Ut;Ut¡h)=½U(h)¾2
U;0·h<t·Tmax;
with¡1·½U(h)·1;½U(0)=1:
Letus specifydistributionswhichmatchthese constraints.Remem-
berthat therandome¤ectsUtarenon-negative,and thatweassume
E(Ut)=18t=1;:::;Tmax.Ifweretainlog-normaldistributionsfor
the(Ut)1·t·Tmax;we canwrite
Ut=exp(Wt)
E[exp(Wt)];Wt»N(0;¾2
W))log(Ut)»N(¡¾2
W
2;¾2
W);
V(Ut)=¾2
U=exp(¾2
W)¡1)¾2
W= log(1+¾2
U):
Supposenowthat(Wt)1·t·Tmaxfollowsamultivariate,stationaryand non
degenerateGaussian distribution.Wehave
Cov(Wt;Wt¡h)=¾2
W½W(h))Cov(Ut;Ut¡h)=¾2
U½U(h)=exp¡¾2
W½W(h)¢¡1
)½W(h)=log(1+¾2
U½U(h))
log(1+¾2
U);½U(h)=(1+¾2
U)½W(h)¡1
¾2
U:(2)
Thematrix³½W(jh0¡hj)´
1·h;h0·Tmaxispositivede…nite,since itisthe
correlationmatrixof(Wt)1·t·Tmax.Forinstance,
1>½W(h)>¡1(h>0))1>½U(h)>exp(¡¾2
W)¡1
exp(¾2
W)¡1=¡exp(¡¾2
W)=¡1
1+¾2
U:
Negativevalueswhichare close enoughto-1cannotbeobtainedascorre-
lationsofthemultiplicativerandome¤ectswithlog-normaldistributions.
Besides,thesetofattainablevaluesdecreaseswiththevariance (see Em-
brechtsetal. (1999)fordevelopmentsonthis subject).
Fromthemomentequationsderivedintherandome¤ectsmodel
E
2
4X
i;t(Ni;t¡¸i;t)txi;t
3
5=0³02Rk´;
9

E
2
4X
i;t(Ni;t¡¸i;t)2¡Ni;t¡¸2
i;t¾2
U
3
5=0;
E
2
4X
i=Ti>h
X
Ti¸t>h(Ni;t¡¸i;t)(Ni;t¡h¡¸i;t¡h)¡¸i;t¸i;t¡h¾2
U½U(h)
3
5=0;
(0<h<Tmax)(3)
weobtainconsistentmoment-basedestimatorsfora;¾2
Uand (½U(h))0<h<Tmax:
The empiricalcounterpartofthemomentequationrelatedtoaleadsto
X
i;t
³ni;t¡b
¸i;t´txi;t=0;b
¸i;t=di;texp(xi;tba):(4)
Hence,themaximumlikelihoodestimatorofainthePoissonmodel
without…xedor randome¤ects(i.e.theaprioriratingmodel)isacon-
sistentestimatorofainthemodelwithrandome¤ects.Fromthesecond
momentequation,aconsistentestimatorof¾2
Uisc
¾2
U1(see (1)).Finally,
the estimatedcorrelogramofUisobtainedfrom
c
¾2
U1b½U(h)=
P
i=Ti>h
P
Ti¸t>h(Ni;t¡b
¸i;t)(Ni;t¡h¡b
¸i;t¡h)
P
i=Ti>h
P
Ti¸t>hb
¸i;tb
¸i;t¡h=Numh
Denh;(5)
for0<h<Tmax.All these estimatorsare consistentand asymptotically
normal. TheyaregiveninZeger(1988),alongwithmodi…edestimatorsfor
theregressionwhich useweightsrelatedto overdispersionand autocorrela-
tioninordertoreduce theasymptoticvariance.
Thesemoment-basedestimatorscan beusedwithlog-normaldistribu-
tionsifand onlyiftheyful…ll thetwofollowingconditions:
a)c
¾2
U1>0;b)³b½W(jh0¡hj)´
1·h;h0·Tmaxpositivede…nite,withb½W(0)=1
and
b½W(h)=log(1+c
¾2
U1b½U(h))
log(1+c
¾2
U1)=log(1+(Numh=Denh))
log(1+c
¾2
U1)
for0<h<Tmax.These conditionsareveri…edonourdata.
We concludethesectionwithanswerstothequestion:Whatcan be
doneifconditionb)isnotveri…ed?Theprecedingspeci…cationmustbe
10

abandoned,atleastwiththelog-normaldistributionsforthe(Ut)1·t·Tmax.
However,restrictedspeci…cationsoftheautocorrelationfunctioncan be
estimated underweakerconditions.
Letusinvestigateforinstance thespeci…cationwithconstantautocor-
relationcoe¢cients(exceptforthetrivialone):½U(h)=½8h¸1;½2
[¡1;1]:AcorrelationmatrixofdimensionTwithconstantcorrelationcoef-
…cientsispositivede…niteifand onlyif1>½>¡1=(T¡1),so only values
of½belongingto[0;1[can beretainedforany valueofT.Ifwesumthe
equations(3)forh=1;::: ;Tmax;weobtain
c
¾2
U1b½=P0<h<TmaxNumh
P0<h<TmaxDenh=
P
i=Ti¸2
P
t6=t0(ni;t¡b
¸i;t)(ni;t0¡b
¸i;t0)
P
i=Ti¸2
P
t6=t0b
¸i;tb
¸i;t0
)b½=P0<h<TmaxDenhb½U(h)
P0<h<TmaxDenh;
withthenotationsofequation(5).Thismodelwithconstantcorrelation
coe¢cientscan be estimatedonthedataif
0·b½<1,0·
P
i=Ti¸2
P
t6=t0(ni;t¡b
¸i;t)(ni;t0¡b
¸i;t0)
P
i=Ti¸2
P
t6=t0b
¸i;tb
¸i;t0<c
¾2
U1)0<c
¾2
U2<c
¾2
U1;
(6)
withthenotationsofequation(1).
Asimplespeci…cation(inaparametricsetting)ofadecreasing auto-
correlationfunctionistheGaussianautoregressiveprocess oforderone(an
AR(1)) forthe(Wt)1·t·Tmax, i.e.
½W(h)=½h,¾2
U½U(h)=(1+¾2
U)½h¡1:(7)
Theparameter½can be estimatedfromonemomentequationobtained by
summingthe equationsrelatedtothelagsin(3).Weobtainthe equation
in½X
0<h<Tmax
Numh¡(((1+c
¾2
U1)½h¡1)Denh)=0:(8)
The equationadmitsauniquesolutionb½,0<b½<1ifthe conditions
obtainedin(6)areful…lled.Indeed,theleftmemberofthepreceding
equationisthenadecreasingfunctionof½;positiveif½=0,negativeif
½=1.
11

2.3 ExtensionofthecorrelogramfromtheYule-Walkerequa-
tions
Theinsurance contractsoftheportfolioinvestigatedinthe empiricalsection
areobservedforamaximumnumberofperiodsequaltoTmax=7.Hence,
sixautocorrelationcoe¢cientscan be estimatedfromthedata.However,
autocorrelationcoe¢cientsforhighervaluesofthelag areofinterest,for
instance ifthelong-termpropertiesofthederivedBMSareinvestigated.
Extensionsofthe correlogramcan beinferredfromtherestrictedspeci…-
cationsinvestigatedin Section2.2,not tomentionthebasicmodelwith
time-independentrandome¤ects.Inthis sectionwepresentanextension
oftheunconstrainedcorrelograminaparametricsetting(withlog-normal
distributionsforthe(Ut)1·t·Tmax),which usesbasicresultsforstationary
timeseries.
LetusconsidertheGaussian process (Wt)1·t·Tmaxde…nedin Section
2.2.Therateofinnovationfora given periodistheproportionofthe
variance whichisnotexplained bythepast.FortheperiodT+1(1·T·
Tmax), itisexpressedas
1¡V[E(WT+1jW1;:::;WT)]
V(WT+1)=1¡t½1;TR¡1
1;T½1;T;(9)
where½1;T=vec1·t·T(½W(t)) and R1;T=(½W(jt¡t0j))1·t;t0·T.The
conditionalexpectationshould bereplaced byana¢neregressionfora
generalstationaryprocess,but therandome¤ectsprocess (Wt)1·t·Tmaxis
assumedtobe centeredand Gaussian.The equation(9)isderivedfrom
thea¢neregressioninL2ofWT+1withrespect to(WT+1¡t)t=1;:::;T.We
obtainforinstance
E(WT+1jW1;:::;WT)=T
X
t=1'tWT+1¡t;'=vec1·t·T('t)=R¡1
1;T½1;T:
(10)
Fromthestationarityoftheprocess, itiseasilyseenthat therateofinnova-
tionisadecreasingfunctionoftheperiodindex. Onourdata, itsestimation
isalmostconstantforthelastperiods,whichmeansthat the…rstvaluesof
therandome¤ectsprovidelittleinformationascomparedtothelastones.
Anaturalwaytoextend therandome¤ectsprocess (Wt)1·t·Tmaxistouse
aconstantrateofinnovationforthefollowingperiods.The corresponding
extensionisanautoregressiveGaussian process oforderTmax¡1, i.e.
E(WT+1jW1;:::;WT)=E(WT+1jWT¡Tmax+2;::: ; WT)8T¸Tmax;
12

whichmeansthat thepastis summarized bythelastTmax¡1valuesofthe
process.Thedynamic equationontherandome¤ectsisthen
WT+1=Tmax¡1
X
t=1'tWT+1¡t+"T+1;(11)
withCov("T+1;WT+1¡t)=08T¸Tmax;8t;1·t·T:
Thesequence ("T+1)T¸TmaxisawhitenoiseGaussian process,witha
variance denotedas¾2
".The constantrateofinnovationisequalto¾2
"=¾2
W.
The extensionofthe correlogramisthen derivedfromtheYule-Walker
equations.Theparameters('h)1·h·Tmax¡1areobtainedfromthe corre-
lationcoe¢cients(½W(h))1·h·Tmax¡1(whichcan be estimatedfromthe
data)byequation(10),withT=Tmax¡1.Letusapplythe covariance
operatorsCov(WT+1¡h;²)T¸h¸Tmaxtobothmembersofequation(11).We
obtainalinear recurrence equationonthe correlationcoe¢cientsofthe
autoregressiveprocess, i.e.
½W(h)=Tmax¡1
X
t=1't½W(h¡t) (8h¸Tmax):(12)
Letus supposethat the characteristicpolynomialoftheshiftoperatorin
thespace ofthesolutionsof(12), i.e.
½Tmax¡1¡Tmax¡1
X
t=1't½Tmax¡1¡t=T1
Y
t=1(½¡½t)
Tmax+T1¡1
2
Y
t=T1+1(½¡½teiµt)(½¡½te¡iµt)
has simplerootsinC.The coe¢cients½tarereal-valued,and thesecond
productexistsonlyifsomerootsdonotbelongtoR.Wethen have
½W(h)=T1
X
t=1at½h
t+
Tmax+T1¡1
2
X
t=T1+1(btcos(hµt)+ctsin(hµt))½h
t(8h¸0):
Thereal-valuedcoe¢cientsat;bt;ctareobtainedfromthe estimations(b½W(h))1;:::;Tmax¡1.
Theautocorrelationcoe¢cientsconvergetowardszeroifthe coe¢cients½t
belongto]¡1;+1[.Thenthe coe¢cientsb½U(h)followfromequation(2).
3Linearcredibilitypredictorsderivedfromthe
precedingmodels
Let(nt)1·t·T·Tmaxbethehistoryofclaimsrecordedonaninsurance con-
tract (wesuppress theindividual indexinordertosimplifythenotations).
13

Alinearcredibilitypredictor(Bühlmann (1967)) forperiodT+1isob-
tainedfromaregression derivedinthemodelwithrandome¤ects.The
predictorisequaltoba+PT
t=1b
btnt;with
(ba;b
b1;::: ;c
bT)=argmin
a;(bt)t=1;:::;T
b
E
2
4ÃUT+1¡a¡T
X
t=1btNt
!23
5;
wherethe expectationisestimatedintherandome¤ectsmodel. Since
E(UT+1)=1;wehaveba+PT
t=1b
btb
E(Nt)=1:Since b
¸t,thefrequency
premiumderivedfromlikelihoodmaximizationintheaprioriratingmodel,
convergestowardsthefrequencyriskE(Nt)computedinthemodelwith
randome¤ects(see equation(4)),wehave
ba+T
X
t=1
b
btnt=1+T
X
t=1
b
bt(nt¡b
¸t);with
(b
bt)t=1;:::;T=argmin
(bt)t=1;:::;T
b
V
2
4ÃUT+1¡T
X
t=1btNt
!23
5=hb
V(N)i¡1d
Cov(N;UT+1):
WewriteN=vec
1·t·T(Nt):Fromthe consistentestimatorsgivenin Section
2.2,the estimatorsoftheindividualmomentsofinterestare
b
V(Nt)=b
¸t+c
¾2
U1b
¸t2;d
Cov(Nt;Nt0)=b
¸tc
¸t0c
¾2
U1b½U(jt¡t0j) (t6=t0);
d
Cov(Nt;UT+1)=b
¸tc
¾2
U1b½U(T+1¡t):
Thebonus-maluscoe¢cientcan bewrittenas
Ã1¡T
X
t=1credt
!+T
X
t=1credtnt
b
¸t;
where³credt=b
btb
¸t´
t=1;:::;Tarethe credibilitycoe¢cients,whicharethe
solutionsofthelinearsystemwitht=1;:::;Tequations
µ1+b
¸tc
¾2
U1¶credt+X
t06=t
c
¸t0c
¾2
U1b½U(jt¡t0j)credt0=b
¸tc
¾2
U1b½U(T+1¡t):
(13)
14

Thislinearcredibilitysystemcan beusedwiththeunconstrainedcorrel-
ogramestimatedin Section2.2,orfromtherestrictedspeci…cationsesti-
matedafterwards.Forexample,fromtheassumptions½U(h)=½8h>0;
b
¸t=b
¸8t=1;:::;T,weobtain
credt=b
¸b½c
¾2
U1
1+b
¸c
¾2
U1(1¡b½)+b
¸b½c
¾2
U1T8t=1;:::;T:(14)
The credibilitycoe¢cientsderivedfromthePoissonmodelwithaconstant
autocorrelationfunctionincreasewithb½,c
¾2
U1and b
¸(all otherthingsbeing
equal).Thetotalcredibilityconvergestowards1ifTgoestoin…nity.
Propertiesofthe credibilitycoe¢cientsderivedfromequation(13)are
notsimpleto obtainina generalsetting.Ifthefrequencypremiumsare
negligiblewithrespect to one,weinferfromthisequation
credt»b
¸tc
¾2
U1b½U(T+1¡t):
Asequence ofcredibilitycoe¢cients should havethesameshapeasthatof
acorrelogramwiththesamelengthand areversedindex.
Totalcredibilitydoesnotconvergeto onewhenTgoestoin…nityif
theautocorrelationfunction decreasesrapidly,and thelimitcan bevery
inferiorto one.LetusconsiderforexampleaGaussianAR(1)process
fortheadditiverandome¤ectsWt= log(Ut):Withthe empiricalresults
obtainedinthenextsection(i.e.c
¾2
U1=1:269;b½W(h)=0:79h) thetotal
credibilityforanaveragerisk(b
¸t=0:09 8t)convergestowards0:214 when
Tconvergestowardsin…nity.Thislimitisobtainedinround …guresafter
twenty years.Itcorrespondstothemaximumbonusappliedtotheapriori
frequencypremiumofapolicyholderwithaclaimless history.
Simpleupdatingformulasdonotseemtobeavailableforthe credibility
coe¢cients.Gerberand Jones(1975)provethatlinearupdatingformu-
lasexistunderconditionswhich di¤erfromthestationarityassumption
retainedinthispaperfortherandome¤ects.
Predictionthroughanexpectedvalueprinciple(Lemaire(1985),Dionne,
Vanasse(1989),Pinquet (1997)) could beobtainedfromamultivariatelog-
normalspeci…cationfortherandome¤ects(Ut)t=1;:::;Tmax.Priorand pos-
teriorexpectationsdonothaveaclosedform,butcan beapproximated by
numerical integrationorbysimulation(see Pinquet,(1997)foranexample
withtworandome¤ects).Wedid notretainthisapproachinthe empirical
study,since wewould needtocomputeintegralsofhigh dimensionwhich
aredi¢cult to approximate.Besides,thereisnostatisticoflowdimension
15

whichsummarizesthehistoryinthe expressionofthebonus-maluscoef-
…cients,suchasthesumofclaimsand the cumulatedfrequencypremium
inthemodelwithconstantrandome¤ects.ThedescriptionoftheBMS
wouldthen bedi¢cultsince the coe¢cientsdonothaveaclosedform.
4Empiricalresults
4.1Thedataset
Theworkingsamplerepresentsten percentoftheportfolio ofamajor
Spanishinsurance company.Weselectedonlypoliciescoveringcarsfor
privateuse.Thedurationofindividualhistoriesrangefromonetoseven
years,hence Tmax=7withthenotationsofthepaper.Policyholderswere
observed between1991 and 1997,and indicatorsofthe calendaryearsare
partoftheregressioncomponentsinorderto allowforatrend inthepast
(see Besson,Partrat (1992)foroptimalBMSwithatrend).As shown
inTable1,which presentstheregressionresults,thefrequencyofclaims
decreasesfrom1991 to 1995 and increasesafterwards.
Inordertohavesimilar ratesofarrivaland attritionintheworking
sampleand intheportfolio,weselectedthepolicyholdersinthefollowing
way.Ten percentofthepolicyholderspresentin1991 wereselectedat
random,and keptintheworkingsampleaslong aspossible.Ten per
centofthenewcomersin1992 wereincludedintheworkingsample,and
so on.Thesize oftheworkingsampleincreasesfrom120,000 in1991 to
200,000 in1997 (inround …gures).Theattritionratevariesbetween8.5%
and 10%.Theworkingsampleisan unbalanced paneldatasetwhichis
composedof269,388 policyholdersand of1,172,701 periods.All theperiod
durationsare equalto oneyear,whichmeansthat the characteristicsofthe
policyholdersareknownonlyateachanniversarydate.Thisinaccuracyin
theobservationoftheregressioncomponentsisofnoconsequence inour
opinion.
4.2Resultsoftheregression
Table1presentstheresultsofaPoissonmodelwhichexplainsthenumber
ofclaimsatfaultbyregressioncomponentswhichareall indicatorsoflevels
ofdi¤erentratingfactors.Theaveragefrequencyofclaimsperyearisequal
to 0.09.
The estimatedexponentialofthe coe¢cients(writteninamultiplicative
way)relatedtothedi¤erentlevelsofeachratingfactorareaveragedto
16

one(columnST.COFF.,forstandardizedcoe¢cient).Two advantagesare
obtained.
²The coe¢cientsdonotdepend onthelevelthatmustbeomitted
intheregressionforeachratingfactorinorderto avoidcolinearity.
Thisisduetothefact that thevectorof frequency-premiumsderived
fromaPoissonmodelwithregressioncomponentsdependsonlyon
thelinearspace spanned bythe covariates.
²These coe¢cientscan be comparedtotherelativefrequencyofeach
level, whichisthefrequencyofclaimsforoneleveldivided bythe
globalfrequency,columnREL.FRE.inTable1.Considerforinstance
the category“30 yearsorless” oftheratingfactor“ageofthepoli-
cyholder”.Therelativefrequencyis1.236,whereasthestandardized
coe¢cientderivedfromthePoissonmodelequals1.058.Fromthe
likelihoodequationsofthePoissonmodel(see (4)),thenumberof
claimsequalsthesumofthefrequencypremiumsforeachlevel. The
ratio 1.236/1.058=1.168 meansthat theyoungpolicyholdershave,
withrespect to other ratingfactors,afrequencyrisklevelwhichis
16.8%higherthantheaverage.
TABLE1
RATINGSCOREFORTHEFREQUENCYOFCLAIMSATFAULT
VARIABLE:YEAR
WEIGHT(%)REL.FRE.ST.COFF.
1991 10.0 1.082 1.071
1992 11.8 1.008 0.983
1993 13.8 0.966 0.939
1994 15.2 0.961 0.938
1995 16.1 0.982 0.978
1996 16.5 1.001 1.023
1997 16.6 1.022 1.072
17

VARIABLE:GENDER
WEIGHT(%)REL.FRE.ST.COFF.
woman17.6 1.061 0.998
man82.4 0.987 1.001
VARIABLE:GEOGRAPHICALAREA
WEIGHT(%)REL.FRE.ST.COFF.
northern provinces18.3 1.162 1.175
intermediateprovinces27.6 0.990 0.953
southern provinces54.1 0.950 0.964
VARIABLE:AGEOFTHEDRIVINGLICENCE
WEIGHT(%)REL.FRE.ST.COFF.
3yearsorless 5.0 1.558 1.385
between4 and 14 years40.0 1.080 1.023
15 yearsormore55.0 0.891 0.948
VARIABLE:SENIORITYOFTHE POLICYHOLDER
WEIGHT(%)REL.FRE.ST.COFF.
2yearsorless 38.0 1.240 1.216
between3 and 5years27.0 0.954 0.952
morethan5years35.0 0.774 0.802
VARIABLE:AGEOFTHE POLICYHOLDER
WEIGHT(%)REL.FRE.ST.COFF.
30 yearsorless 22.5 1.236 1.058
morethan30 years77.5 0.931 0.983
VARIABLE:COVERAGELEVEL
WEIGHT(%)REL.FRE.ST.COFF.
comprehensive,except…re16.2 1.102 1.089
comprehensive33.0 1.032 0.991
third partyliabilityonly50.8 0.946 0.977
VARIABLE:POWEROFTHEVEHICLE
WEIGHT(%)REL.FRE.ST.COFF.
less than55 hp 22 0.910 0.924
55 hp ormore78 1.025 1.021
18

Spainis splitintothree geographicalareasbytheinsurance company.
Areanumber1iscomprisedofthenorthernregions(Galicia,Cantabria,
Asturias,PaisVasco),areanumber3includesthesouthern provincesand
areanumber2referstotheintermediateprovinces.Theinsurance company
thinksthan northern provincesaremorerisky thantheotheronesbecause
itismorerainythere.Forthesamereasons,zonenumber2ismorerisky
thanzonenumber3.Thisisafull Bayesianapproachofinsurance rating,
and thispriorknowledgeonthedatashould berelatedtoafamousresult
ofSpanishclimatology,namelytheAudreyHepburn’stheorem
Therainin Spainstaysmainlyintheplain,
atheoremwhichcaneven besung(see Cukor(1964)foraproof).
4.3 Estimatorsforthecorrelogramoftherandome¤ects
Thetwoconsistentestimatorsquotedin Section2.1forthevariance ofa
time-independentrandome¤ectarerespectively
c
¾2
U1=Pi;t(ni;t¡b
¸i;t)2¡ni;t
Pi;tb
¸2
i;t=118554:78 ¡105655
10167:12 =1:269:
c
¾2
U2=Pi(ni¡b
¸i)2¡ni
Pib
¸i2=144879:33 ¡105655
50359:14 =0:779:(15)
Wehave0<c
¾2
U2<c
¾2
U1;anecessaryconditionforthe estimationofdy-
namicrandome¤ects.
Letusestimate…rstaconstantautocorrelationfunction.Withthe
notationsofSection2.2,wehave
b½c
¾2
U1=PiPt6=t0(ni;t¡b
¸i;t)(ni;t0¡b
¸i;t0)
PiPt6=t0b
¸i;tb
¸i;t0=144879:33 ¡118554:78
50359:14 ¡10167:12 =0:655;
b½=0:655
1:269 =0:516:
19

Letusestimatetheunconstrainedcorrelogramof(Ut)1·t·6,and theAR(1)
speci…cation.Fromthenumericalvalues
h1 2 3 4 5 6
Numh5846:17 3149:86 2015:71 1268:20 618:50 263:82
Denh7293:90 5121:48 3438:37 2291:66 1353:29 597:31 ;
(the coe¢cientsNumhand Denharede…nedin(5)),weobtainthefollow-
ingestimations.
TABLE2
AUTOCORRELATIONCOEFFICIENTS
UNCONSTRAINEDANDLOG-AR(1)RANDOMEFFECTS
h(lag)123456
b½U(h)0:632 0:485 0:462 0:436 0:360 0:348
b½0
U(h)0:718 0:527 0:393 0:297 0:227 0:174
The…rstlineofthetablegivestheunconstrainedautocorrelationcoe¢-
cients.Thelastlineiscomposedofautocorrelationcoe¢cientsderived
fromanAR(1)speci…cationfortheGaussian process (Wt)1·t·Tmax,with
Ut=exp(Wt)=E[exp(Wt)](see Section2.2).Weobtainb½0
W(h)=b½h;with
b½=0:79 fromequation(8),and thenb½0
U(h)from(7).Theunconstrained
correlogramdecreases,butmoreslowlythanthelog¡AR(1)one.
Asaresultofthedecreasingshapeofthe correlogram,thepredictive
abilityofclaimswill decreasewiththeirage. Onereasonisthe exogeneous
interpretationoftheindividualhistoriesusuallyretained bytheactuarial
models(i.e.unobservedvariationsofriskdistributionsbetweentwodates
increasewiththerelatedlag). Otherpossible explanations stemfromen-
dogeneouse¤ects,whichexpress themodi…cationsofriskdistributionsfor
real individualsinduced byclaims. One canthinkofthemodi…cationsof
riskperception behind thewheelafteranaccident.Anegative e¤ectis
expectedonthefrequencyrisk,whichcould decreasewiththeageofthe
claim.Anothernegative endogeneouse¤ectisduetothe…nancial incen-
tivesinduced bytheBMS(see Lemaire,(1977)and Chiappori, Heckman,
and Pinquet,(2000)).Motorinsurance ratingiscompletelyderegulated
in Spain,and alackofpolicyholders’awareness ofthe experience rating
schemesprobablyreducestheincentive e¤ects.
Letusnowcomputethe extensionoftheunconstrainedcorrelogram
fromtheYule-Walkerequations(see Section2.3).Thefollowingtablegives
20

theautocorrelationfunctionoftheGaussian process (Wt)1·t·6;theparam-
etersoftheAR(6)equationand theinnovationfunction.
TABLE3
AUTOCORRELATIONCOEFFICIENTSBETWEENGAUSSIAN ADDITIVERANDOMEFFECTS
COEFFICIENTSOFTHEAR(6)EQUATION
INNOVATIONFUNCTIONFORTHENEXTPERIOD
h(lag)1 2 3 4 5 6
b½W(h)0:719 0:585 0:563 0:538 0:459 0:447
'h0:570 0:003 0:135 0:123 ¡0:085 0:036
Innov(h+1)0:484 0:474 0:453 0:448 0:447 0:443
Fromtheseresults,weobtaintheautocorrelationfunctionwhichextends
the estimatedvalues(b½W(h))h=1;:::;6fromaconstantrateofinnovation,
hence equalto0:443 forourdata(see Section2.3).The characteristic
polynomialrelatedtothelinearequationwhich de…nestheautocorrelation
coe¢cientsisequalto
½6¡6
X
t=1't½6¡t=2
Y
t=1(½¡½t)4
Y
t=3(½¡½teiµt)(½¡½te¡iµt);
with
½1=0:919;½2=¡0:671;½3=0:683;µ3=1:854;½4=0:579;µ4=0:918:
Theunitofmeasurefortheargumentsofthe complexrootsistheradian.
The extendedautocorrelationcoe¢cientsarethen
b½W(h)=2
X
t=1at½h
t+4
X
t=3(btcos(hµt)+ctsin(hµt))½h
t;
with
a1=0:737;a2=0:033;b3=0:114;c3=0:040;b4=0:116;c4=0:039:
(16)
Whenthelag hislarge,wehavethe equivalence b½W(h)»0:737£(0:919)h:
Thisautocorrelationfunction decreasestowards0atanexponentialrate
(itisdivided bytwoeveryeightyears),butslowerthanwhatweobtained
fromtheAR(1)process.
21

4.4 Experienceratingfromthedi¤erentmodels
Inthefollowingtables,credibilitycoe¢cientsforthedi¤erentperiodsare
computedforaninsurance contractwithafrequencypremiumperyear
equaltotheaveragefrequency,whichis0.09.WeusethePoissonmodels
withrandome¤ectspresentedintheprecedingsections.Thenext table
providescredibilitycoe¢cientscomputedfromthree speci…cationsforthe
randome¤ects.Ifrandome¤ectsaretime-independent (typeAinthe
table)orwithaconstantautocorrelationfunction(typeB),the credibil-
itiesarethesameforeach periodif frequencypremiumsdonotvary,as
showninequation(14).TypeCcorrespondstotherandome¤ectsmodel
with unconstrainedcorrelationcoe¢cients.The coe¢cientsare computed
forhistoriesrangingfromonetosix years.Weusedtheusualcredibility
formulafortime-independentrandome¤ects,and thelinearcredibilitysys-
temgivenin Section3forthetwo othermodels.Fortherandome¤ects
modeloftypeA,we estimatedthevariance fromthenumberofclaims
and frequency-premiums summedacross theperiods.Hence,weretained
c
¾2
U2=0:779 insteadofc
¾2
U1=1:269 (see equation(15)).
Rememberthatacredibilitycoe¢cientisabonusifnoclaimisreported.
22

TABLE4
CREDIBILITYCOEFFICIENTSFORAN AVERAGERISK(PERCENTAGE)
BMSOFTYPE A(TIME-INDEPENDENTRANDOMEFFECTS)
ANDOFTYPE B(DYNAMICRANDOMEFFECTSWITHCONSTANTAUTOCORRELATION)
DurationofhistoriesCredibilityperyear(%)Totalcredibility(%)
typeAtypeBtypeAtypeB
1year6.55 5.29 6.55 5.29
2years6.14 5.02 12.29 10.04
3years5.79 4.78 17.37 14.35
4years5.47 4.56 21.89 18.26
5years5.19 4.36 25.95 21.83
6years4.93 4.18 29.60 25.10
BMSOFTYPE C:DYNAMICRANDOMEFFECTS(UNCONSTRAINEDCORRELOGRAM)
DurationofhistoriesCredibilityperyear(%)Totalcredibility(%)
1year6.47 0 0 0 0 0 6.47
2years4.57 6.17 0 0 0 0 10.74
3years4.15 4.32 5.98 0 0 0 14.45
4years3.74 3.94 4.14 5.83 0 0 17.65
5years2.83 3.57 3.82 4.03 5.72 0 19.97
6years2.66 2.68 3.46 3.71 3.94 5.65 22.10
Fora given durationoftheindividualhistory,the credibilitycoe¢cients
ofthelast tabledecreasewiththelag betweentheprediction periodand
the currentperiod,asdotheautocorrelationcoe¢cients.Forexample,the
credibilitygiveninthelast tableforthelastyearofasix yearshistory
outweighsthe credibilityofthetwo…rstyears.
Totalcredibilityislowerinthelast tableforhistoriesof fouryearsand
more.Hence thebonusappliedtoaclaimless policyholderbecomesless
importantafterafewyearsifadecreasing autocorrelationfunctionisused
fortherandome¤ects.
Letuscomparetotalcredibilityforlongdurations.Weneedtoextend
the correlogramforhighervaluesofthelag,whichisobviousfor random
e¤ectsoftypeAand B.Fortherandome¤ectsoftypeC,weretaintwopos-
sible extensions:C(1)setstheautocorrelationcoe¢cientsequaltothelast
23

estimation(i.e.b½U(h)=b½U(6)8h>6),whereasC(2)usestheYule-Walker
equations(see (16)and (2)).Ifthe…rstextensioncertainlyoverestimates
theactualvalues,thesecond oneprobablyunderestimatesthem,since it
assumesaconstantinnovationintheadditiveprocess.Adecreasingweight
isgiventothepastwhenswitchingfromtypeAtotypeC(2).
TABLE5
LONG-TERMBEHAVIOUROFTOTALCREDIBILITY(PERCENTAGE)
DurationofhistoriestypeAtypeBtypeC(1) typeC(2)
10 years41.2 35.8 29.7 27.7
20 years58.4 52.8 43.5 32.6
40 years73.7 69.1 59.4 34.1
Theresultobtainedinthelastcolumnis striking.Afteralmostafull
lifeofdriving,apolicyholderwithaclaimless historyobtainsafrequency-
bonusofonly34 percent,whichisabout…vetimesthebonusafterthe
…rstyear.Notsurprisingly,thelong-termpropertiesoftheBMS depend
stronglyonthe extensionofthe correlogram.
Letusnowperformanimpulse-responseanalysisofthe evolutionofthe
bonus-maluscoe¢cientifone claimisreported duringthe…rstyear,and
noneduringtheyearsthatfollow.We compareBMSoftypeA,Band C
oversix years,foracarwiththeaveragefrequencypremium.Bonus-malus
coe¢cientsaregivenin percentage.
TABLE6
IMPULSE-RESPONSEANALYSISOFBONUS-COEFFICIENTSAFTERONECLAIM
Years123456
modelA166.2 156 147 139 131.7 125.2
modelB153.5 145.8 138.8 132.5 126.7 121.4
modelC165.5 140 131.7 123.8 111.4 107.5
Ascomparedwiththetwo othermodels,anoptimalBMS designedfrom
amodelwith dynamicrandome¤ectsand avarying autocorrelationfunc-
tionwill behaveinthefollowingway.Aclaimreportedentailsa greater
malusthantheoneobtained bytheBMSoftypeB,but thefollowingno-
claimdiscountsaremoreimportant.Thisisduetothefact thatperiods
withoutclaimreportedincreaseriskexposurebutalsotheageofclaims
24

reportedinthepast.ThisBMS providesimportantno-claimsdiscounts
forpolicyholderswhoreportedaclaimrecently,forinstance 15 percent
fromperiod1toperiod2inthelast table.Thisfeature can berelated
tosome clausesfound incompulsoryBMS,which provideimportantdis-
countsforbad driverswithrecentgood behaviour.InFrance forinstance,
adriverwithabonus-maluscoe¢cientgreaterthanthe coe¢cientapplied
tobeginners(less thanthree percentofthedriversare concerned)israted
accordingtothiscoe¢cientaftertwoconsecutive claimless years.Thisfea-
tureofreal-lifeoroptimalBMSentails strongincentivestodrive carefully
forpolicyholderswithabadaccidentrecord.
Letuscomparebrie‡yontheportfoliothethree BMSretainedforthe
precedingtable.Theyare…nanciallybalanced byconstruction,and the
dispersionofthebonus-maluscoe¢cientsisless importantfromthelast
BMSthanfromtheBMSoftypeAand Bafterafewyears,becausethe
totalcredibilityisinferior.Thiscannotbeseenasadrawbackbecausethe
dispersionofthebonus-maluscoe¢cientsisa goodcriterionofe¢ciency
onlyiftheBMSisconsistentwithrespect totime.Nowconsistencyprop-
ertiesdonotmakesenseiftheactualspeci…cationoftherandome¤ectsis
dynamic.
Inthelast table,we comparetherateofriskrevelation bythedi¤erent
BMS.Foranygiven durationofthepolicyholder’shistory,wewant to assess
thegainofe¢ciencyobtainedfromfurtherobservation,and howitdepends
onthe correlogramretainedfortherandome¤ects.Wederivethestandard
deviationofbonus-maluscoe¢cientsfordi¤erentdurationsofthehistories
and forthedi¤erentrandome¤ectsmodels.Anincreaseofthestandard
deviationwiththedurationassessesthegainofe¢ciencyobtainedfrom
supplementaryobservation.
Computationsareperformedinthedi¤erentrandome¤ectsmodelsfor
a genericindividualwithanaveragefrequencyrisk.Theautocorrelation
functionsarethoseusedinTable5.Thevariance ofthebonus-maluscoef-
…cientsarethoseofthelinear regressionwhich de…nesthelinearcredibility
predictor,thatistosay
td
Cov(N;UT+1)hb
V(N)i¡1d
Cov(N;UT+1)
withthenotationsofSection3.Fordi¤erentdurationsofthehistory,we
obtain
25

TABLE7
STANDARD DEVIATIONOFBONUS-MALUSCOEFFICIENTS
duration(years)1 5 10 20 40 5 !40
BMSoftypeA0.226 0.450 0.567 0.674 0.758 +69%
BMSoftypeB0.186 0.378 0.485 0.588 0.673 +78%
BMSoftypeC(1)0.228 0.355 0.407 0.472 0.538 +52%
BMSoftypeC(2)0.228 0.355 0.389 0.398 0.399 +12%
Thelastcolumnexpressesthegainofe¢ciencyobtained byaforty year
record,ascomparedto a …veyearhistory, intermsofstandard deviation.
Thishistoryisavailableto all the competitorsintheFrenchmarketincase
ofswitchingto anotherinsurance company.The…rst three BMSsystems
useautocorrelationfunctionswhichoverestimatetheactualvaluesforlags
greaterthan6.Thegainofe¢ciencyisthenoverestimated bytheseBMS,
and probablyunderestimated bythelastone.Thistypeofresultmaybe
ofinterestfor regulating authorities,asmentionedintheintroduction.
5Concludingremarks
ThemainfeaturesofanoptimalBMS derivedfromstationaryrandom
e¤ectswithadecreasingcorrelogramseemacceptabletopolicyholders.The
no-claimdiscountsareless importantafterafewyearsforclaimless drivers
thanthosederivedfromusualoptimalBMS. Ontheotherhand,theycan
bemuchmoreimportant (ifexpressedintermsofpremiumsupdating)
forpolicyholderswhoreportedclaimsrecently.Suchsystemswouldentail
strongincentivesforthesedriverstodrive carefully.
Ausefulresultfortheapplicationofactuarialmodelsisthefollowing.
Iftheautocorrelation betweenstationaryrandome¤ectsdecreaseswiththe
lag,thevariance ofatime-independentrandome¤ect (estimatedfromag-
gregated numbersand frequency-premiums)will decreasewiththeaverage
durationofthehistoriesusedinthe estimation.Forinstance,thevariance
estimatedfromtheobservationsofthe…rstperiodisequalto1:09,whereas
thevariance obtainedfromthefull historiesisequaltoc
¾2
U2=0:78 (see
Section4.3).Inthisframework,anoptimalBMSestimatedfromshort
historiesand appliedtoalongerdurationwill overestimatethepredictive
abilityofclaims.Thisresultprovidesasupplementaryreasontousethe
wholehistoryofthepolicyholdersininsurance rating.
26

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JEAN PINQUET
U.F.R.deSciencesEconomiques
UniversitédeParisX
200,avenuedelaRépublique
92001 NanterreCedex
France
email: pinquet@u-paris10.fr
MONTSERRATGUILLÉN
Departamentd’Econometria,EstadísticaiEconomiaEspanyola
UniversitatdeBarcelona
Diagonal, 690
08034 Barcelona
Spain
e-mail: guillen@eco.ub.es
CATALINA BOLANCÉ
Departamentd’Econometria,EstadísticaiEconomiaEspanyola
UniversitatdeBarcelona
Diagonal, 690
08034 Barcelona
Spain
e-mail: bolance@eco.ub.es
28