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1、2014-CAS8§8LeeDiagrams,LossDistributionsHCM5/13/14,Page419Section8,LeeDiagrams,LossDistributions324325In“TheMathematicsofExcessofLossCoveragesandRetrospectiveRating--AGraphicalApproach,”PCAS1988,Yoong-SinLeeusesgraphstoillustratemathematicalconceptsthatappl
2、ytobothsizeoflossdistributionsandaggregatedistributions.InthissectionarecoveredLeeʼsfirstthreesections,whichcontainlotsofveryusefulideas.MoreofthequestionsonrecentadvancedratemakingexamshavedealtwithLeeʼsfourthsectiononRetrospectiveRating,whichIcoverinasubs
3、equentsection.ThosewhoaregraphicallyorientedwillfindthatLeeDiagramshelpthemtorememberorquicklyderivemanyimportantformulas.Notation:Leeuseswhatisnowoldernotation,whichdiffersfromwhatiscurrentlyusedonExam4:G(x)⇔S(x)=1-F(x)⇔SurvivalFunctionE[g(X;b)]⇔E[X∧b]⇔Lim
4、itedExpectedValueatb.326324“TheMathematicsofExcessofLossCoverageandRetrospectiveRating---AGraphicalApproach”,byY.S.Lee,PCASLXXV,1988.“OtherApplications”onpp.75-76arenotonthesyllabus.Leecites“APracticalGuidetotheSingleParameterParetoDistribution”,byStevenPhi
5、lbrick,PCAS1985.PhilbrickinturnpointsoutthesimilaritytothetreatmentofInsuranceChargesandSavingsin“FundamentalsofIndividualRiskRatingandRelatedTopics”,byRichardSnader.325Sections1-3coverCASLearningObjectiveB1.326AlsousedinthesyllabusreadingsofClark,Miccolis,
6、andMahler.2014-CAS8§8LeeDiagrams,LossDistributionsHCM5/13/14,Page420LossDistributions:Anumberofimportantlossdistributionconceptscanbedisplayedgraphically.ImportantconceptswillbeillustratedusingthisgraphicalapproachofLee.BelowisshownaconventionalgraphofaPare
7、toDistributionwithα=4andθ=2400:Exercise:ForaParetoDistributionwithα=4andθ=2400,whatisF(1000)?⎛θ⎞α[Solution:F(x)=1-⎜⎟.F(1000)=1-(2400/3400)4=0.752.]⎝θ+x⎠Intheconventionalgraph,thex-axiscorrespondtosizeofloss,whilethey-axiscorrespondstoprobability.Thusforexam
8、ple,theabovegraphofaParetoincludesthepoint(1000,0.752). Incontrast,“LeeDiagrams”havethex-axiscorrespondtoprobability,whilethey-axiscorrespondstosizeofloss.2014-CAS8§8LeeDiagrams,LossDistributionsHCM5/1
