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1、2014-CAS8§14Mahler,W.C.ExcessRatiosHCM5/13/14,Page861Section14,Mahler,WorkersCompensationExcessRatios582ThispaperusesideasfromExam4toestimateexcessratios.ExcessRatio:Mahlerusesanoldernotationforthelimitedexpectedvalue,E[X;x]ratherthanE[X∧x].TheExcessRatioisthelossesexcessofthegivenlimitdividedbyth
2、etotallosses.ExcessRatioatx=R(x)≡(lossesexcessofx)/(totallosses)∞∞=E[(X-x)+]/E[X]=(E[X]-E[X∧x])/E[X]=∫S(x)dx/∫S(x)dx.x0Theexcessratioisoneminusthelosseliminationratio:R(x)=1-LER(x)=1-E[X∧x]/E[X]. Exercise:Aninsuredhaslossesofsizes:$300,$600,$1,200,$1,500and$2,800.Whatisthe(empirical)excessratioat
3、$1000?[Solution:R(1000)=(200+500+1800)/6400=39.1%=1-60.9%=1-LER(1000)=1-E[X∧1000]/E[X]=1-780/1280.]ExcessLossFactors:TheestimatedExcessRatioswouldbeusedtodetermineExcessLossFactors(ELFs)tobeusedinRetrospectiveRating.ThepaperbyMahleronlydiscussesthedifficulttaskofestimatingtheExcessRatios.Forexampl
4、e,inMahlerʼsExhibit8,forHazardGroup3,for$1millionAccidentLimit,theestimatedexcessratiois0.0256.Letusassumeforexampleanexpectedlossratioof65%(withoutLAE).Then(0.65)(0.0256)=0.017wouldbetheindicatedELFwithoutriskload.ForaninsuredinHazardGroup3with$2millioninStandardPremium,weexpect(65%)($2million)=$
5、1.3millioninlosses.Oftheselosses,2.56%areexpectedtobeexcessofa$1millionaccidentlimit.Thusweexpect(2.56%)($1.3million)=$33,280inexcesslosses.NotethattheELF(withoutriskload)timesStandardPremiumis:(0.017)($2million)=$34,000,matchingtheexpectedexcesslossessubjecttorounding.582“WorkersCompensationExces
6、sRatios:AnAlternateMethodofEstimation,”byHowardC.Mahler.Sincethisreadingwasaddedtothesyllabusin2008,therearefewpastexamquestions.CASLearningObjectiveB2.2014-CAS8§14Mahler,W.C.ExcessRatiosHCM5/13/14,Page862Insomestates,ariskloadwouldbeadded.Forexample,onemightadd0.005subjecttoamaximumofonehalfofthe
7、otherwiseindicatedELF.583Inthatcase,theindicatedELFwouldbe:0.017+0.005=0.022.ELF=(ExcessRatio)(ExpectedLossRatio)+(possibleriskload).ExcessLossandAllocatedExpenseFactors:584ExcessLossandAllocatedExpenseFactors(EL
