PARAMETER UNCERTAINTY IN THECOLLECTIVE RISK MODEL 参数不确定性 集体风险模型

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1、111PARAMETERUNCERTAINTYINTHECOLLECTIVERISKMODELGLENNMEYERSNATHANIELSCHENKERAbstractThispaperproposesanewversionofthecollectiveriskmodelthatallowsforuncertaintyinselectingtheexpectednumberofclaimsandtheclaimseveritydistribution.Weprovidetwodifferentmethodsofestimatingtheparamet

2、ersofthismodel.Itisdemonstratedbycomputersimulationthatonemustcombinetheexperienceofseveralinsuredsinordertoaccuratelyquantifyparameteruncertainty.Testsonaverylargesampleofindividualinsureddatashowasignificantimprovementintheaccuracyofthecollectiveriskmodelwhenparameteruncerta

3、intyistakenintoaccount.Thetestsdonotshowperfectagreementbetweenthemodelandtheempiricaldata,buttheagreementiscloseenoughtobeusefulinmanyapplications.112PARAMETERUNCERTAINTY1.INTRODUCTIONThispaperdiscussestheroleofcollectiverisktheoryinmakinginsurancepricingdecisions.Collectiver

4、isktheoryprovidesawayofcalculatingtheprobabilitythatalossarisingoutofaninsurancecontractwillexceedagivenamount.Thecalculationisdoneintermsoftheunderlyingclaimseverityandclaimcountdistributions.Relatedtothisistheexcesspurepremium,whichisthecostofinsuringalllossesaboveagivenamou

5、nt.Alsoofinterestistheexcesspurepremiumratio,whichistheexcesspurepremiumdividedbytheexpectedloss.Ofprincipalconcernistherelationshipbetweenthevarianceofthelossratioandthesizeoftheinsured.AcommonassumptionwasstatedbySimon[lo,p.441asfollows:Astherisksizeincreases,weexpectthevari

6、ance(ofthelossratio)toapproachzero.Largeinsuredsaretypicallywrittenonaretrospectiveratingplanoranaggregateexcesscontract.Theeffectofthisassumptionwouldbethatforalllossamountsgreaterthantheexpectedloss,theexcesspurepremiumratiowouldapproachzeroasthesizeoftheinsuredbecomeslarge.

7、Thepracticalunderwriterwouldfeelveryuncomfortablewithanagreementtoprovidecoverageforalllossesabovetheexpectedlossforazeroornominalpremiumforevenaverylargeinsured.Hiscomplaintwouldbethattheexpectedlosscannotbeestimatedwiththenecessaryprecision.Ofinterestisthedistributionofactua

8、llossesaboutanunbiasedestimateoftheexpectedlosses.Estimatesof