Casualty Actuarial Mathematics2

1Math490CasualtyActuarialMathematics非寿险精算数学Spring2007UniversityofIllinoisatUrbana-Champaign伊立诺伊大学ProfessorRickGorvettSession5:LossReservingII损失准备金January30,2007天津财经大学统计系2004级张琪翻译2Agenda纲要•Reviewofbasiclossdevelopmenttechniqueandessentialmetrics/quantities•损失准备金进展法及其基本规则的回顾•Otherlossreservingmethods其他方法-Expectedlossratio预期赔付率法-Bornhuetter-FergusonB-F法•Diagnostics趋势法•Reservingissues各方法所遇问题3ReviewofLDFTechnique链梯法复习•Generalapproachoflossdevelopmenttechniques–Lossdevelopmenttriangle(cumulativedata)损失流量三角型（累积）–Age-to-agefactors(linkratios)进展因子•Approachestoselectingfactors–e.g选择进展因子的方法–Straightaverage简单平均–Averageoflastthreeyears取近三年的数据进行简单平均–(Middle)3of5years中位数法–Weightedaverage(bydollars)加权平均–To-ultimatelossdevelopmentfactors估计最终进展因子–Foreach(accident)year:(actualloss)×(LDF)=estimatedultimateloss实际损失*进展因子=该年最终损失额的估计值–Reserve=(ultimateloss)–(to-datepaidloss)准备金=最终损失-已付赔款–IBNR(broaddef’n.)=(ult.loss)–(caseincurredloss)–已发生未报案赔款准备金=最终损失-已发生的损失4ReviewofLDFTechnique(cont.)•Possiblebases:–AccidentYear(AY)事故年–PolicyYear(PY)保单年–ReportYear(RY)•Possibletypesofdata:数据的类型–Paidlosses(withorwithout(A)LAE)已付赔款（包含或不包含可摊费用）–Caseincurredlosses(withorwithout(A)LAE)个案已发生的损失–Closedclaimcounts已结案索赔次数–Reported/incurredclaimcounts已报案/已发生的索赔次数–Paidlossseverity已付赔款损失幅度–Incurredlossseverity已发生赔款损失幅度5ReviewofLDFTechnique(cont.)•Datarelationships数据间的关系–Cumulativeincurredloss=cumulativepaidloss+“snapshot”casereserve已发生赔款的累积额=已付赔款的累积额+未决赔款准备金–Cumulativereported/incurredclaimcount=cumulativepaidclaimcount+“snapshot”openclaimcount已报案/已发生索赔次数的累积量=已付索赔次数的累积量+未结案索赔次数–Paidseverity=cumulativepaidloss÷cumulativepaidclaimcount已付赔款的幅度=已付赔款的累积量/已发生索赔次数的累积量–Reported/incurredseverity=cumulativeincurredloss÷cumulativereportedclaimcount已报案/已发生赔款的幅度=已发生损失额的累积量/已报案索赔次数的累积量6MetricsandQuantitiesofInterest准备金公式•Paidloss+casereserve=(case)incurredloss已付赔款+未决赔款准备金=已发生的损失•UltimateAYorPYloss=paidloss+casereserve+IBNR事故年或报单年的最终损失=已付赔款+已报案未决赔款准备金+已发生未报案的未决赔款准备金•IBNR(IncurredButNotReported)loss=ultimateloss–(case)incurredloss已发生未报案的索赔额=最终索赔额-已发生的索赔额•Totallossreserve=ultimateloss–paidloss总准备金=最终索赔额-已付索赔额•Totallossreserve=casereserve+IBNR总准备金=已报案的未决赔款准备金+已发生未报案的未决赔款准备金•Loss=frequency×averageseverity索赔额=索赔频率*索赔幅度7OtherBasicLossReservingTechniques其他损失准备金方法•Expectedlossratio(ELR)method赔付率法–Lossratio=loss÷premium赔付率=赔付额/保费–Ultimateloss=premium×ELR最终损失额=保费*赔付率–ELRperbestestimate(e.g.,originallossprovisioninpremiumperratemakingprocess)•Bornhuetter-FergusonmethodB-F法–A“middle-ground”betweentheLDFandELRmethods它是链梯法与赔付率法的结合8ExpectedLossRatioMethod赔付率法ProjectedPaidProjectedAccidentEarnedUltimateLoss@LossYearPremiumELRLoss12/06Reserve20025000.65325380(55)20039000.655855008520041,2000.6578060018020051,5000.6597550047520061,5000.65975290685====================Total5,6003,6402,2701,3709Bornhuetter-Ferguson(B-F)Method•MergestheLDFandELRtechniques链梯法与赔付率法的规则•Steps(e.g.,forcase-incurredB-F):B-F法的步骤(1)Determinetheto-ultimateLDF确定链梯法中的最终进展因子(2)Determinetheproportionofultimatelossunreportedto-date=1–{1/(1)}已发生未报案的赔款额占最终赔款的比例=1–{1/(1)}(3)Earnedpremium×ELR=(“apriori”)expectedultimateloss已赚保费*赔付率=最终赔款(4)Expectedunreportedloss=(2)×(3)已发生未报案的准备金=(2)×(3)(5)B-Fultimateloss=(4)+(caseincurredlossto-date)B-F法计算出的最终未决赔款准备金=（4）+已报案的未决赔款10Bornhuetter-FergusonMethodProjectedAccidentEarnedUltimateYearPremiumELRLoss20025000.6532520039000.6558520041,2000.6578020051,5000.6597520061,5000.65975==========Total5,6003,64011Bornhuetter-FergusonMethod(cont.)ProportionProportionProjectedAccidentTo-UltofUlt.ofUlt.ProjectedUltimateYearLDFPaidUnpaidReserveLoss20021.0001.0000.000-38020031.0500.9520.0482852820041.2600.7940.20616176120051.8900.5290.47145995920063.7800.2650.7357171,007==========Total1,3653,63512Diagnostics分析•RelationshipofprojectedultimateL/RtoELR推断出的赔付率与真实值间的关系•LDF/payoutpatternscomparedwithindustryorothersources进展因子/索赔方式的比较–“Internalvsexternal”增量与累积•Paid-to-incurred(or–to-reserve)ratios支付率•Closed-to-reportedclaimcountratios结案率•Averagepaidclaim平均索赔额•Reserveaspercentageofultimateloss准备金占最终索赔额的比例13LossReservingIssuess损失准备金所遇问题•Consideringmultipleresults多种方法的总结–LDFvsELRvsB-F–链梯法与赔付率法以及B-F法–Paidvsincurredvsfrequency-severity–已付赔款与已发生索赔额;索赔频率与索赔强度•Pointestimatesvsreserveranges点估计与区间估计Lossreservediscounting准备金的贴现–Regulatoryconsiderations–基本假设–Paymentpatterns赔付方式–Interestrates利率14CASExam6,Fall2005,#1015CASExam6,Fall2005,#10(cont.)2005年秋财险精算协会考试16NextTime•LossReservingIII损失准备金3•Lossreservingprinciples损失准备金的基本原则•CASStatementofLossReservingPrinciples财险精算协会对损失准备金的基本原则的要求•Accountingissues财务问题•BlanchardstudynoteBlanchard笔记共享