商业信贷风险参数计算(1)

CalculatingBaselIIRiskParametersforaPortfolioofRetailLoansDrPeterGl¨oßnerKelloggCollegeUniversityofOxfordAthesissubmittedinpartialfulfilmentoftherequirementsfortheMScinMathematicalFinance12thofApril,20032AbstractUndertheBaselIIregime,bankscanchooseamongdifferentapproachestomeasuretheregulatorycapitaltounderpintheirriskyassets.Fromthepointofviewoftheamountofcapitalrequired,theRetailIRBApproachcanbeveryadvantageous.Tosatisfyitsrequirements,bankshavetoestimatesensiblevaluesfortheriskparametersProbabilityofDefault(PD)andLossGivenDefault(LGD)onthebasisoftheirowndefaultandlossdata.InpartduetothesegmentationrulesparticulartotheRetailIRBApproach,thisisverydifficult,andthesimplecalculationofrelativefrequencieswillnotdoingeneral–thesampledatadonotallowonetomakeasensibledistinctionbetweenthestructureofthedefaultandlossdensitiesandtherandomnessofthesampledata,asweseeinthisthesis;allmethodswederiveforcomputingriskparametersaredevelopedusingrealbankdata.WedescribeamethodtoestimatePDusingtheconstructionofaLorenzcurvebasedonscoringresults.WhileLorenzcurvesusuallyaremeanstocomputeefficiencyratios,weshowhowaLorenzcurvecanserveasavehicletodefinetheborderlinebetweenstructureandrandomness.ValuesforPDcanbeobtainedfromitinadirectway.Whatmakesitspecificallysuitableforthispurposearesomeinvariancyproperties;weshowthisingeneralandbywayofsampledataofarealretailportfolio.Wefurthercomparethismethodtomultivariatemethods,andproposeamulti-componentsystemtobalancethecomplementaryadvantagesanddisadvantagesofbothapproaches.Veryoften,thereisno“LGDratingsystem”analogoustothePDrating,andsowederivevaluesforLGDbyobservingso-calledspecialprovisioncohortsovertime.Makingaspecialprovisionispartofthedefaultdefinition,andbyexponentiallymodellingthetimebehaviourofspecialprovisionvolumesonecanestimatevalidLGDnumbers,asweshowintherelevantChapter.ThelastChaptergoesbeyondBaselII.Weassumethevalidityofthe“LossofMemoryProperty”foratypicalretailportfolio,andshowthatborrowerdefaultunderthisassumptioncanbecomparedtoradioactivedecay.Themathematicalmodellingofdecayingnucleiistransportedtodefaultingborrowers,fromwheresomeexplicitformulaeforUnexpectedLossarederived.Asalltermsoftheseformulaecanbeestimatedfromoursampleportfoliodata,thismodelcanserveasavalidationtoolforaninternalportfoliomodel.ThisthesissprangfromBaselIIprojectworkinamedium-sizedgermanbank.Fourcol-leaguesofmineandmyselfcollaboratedinproblemscloselyrelatedtothesubjectofthisthesis,andsothethesisowesmuchtothemanydiscussionswehad.Itisapleasuretometoexpressmythankstoallofmycolleagues:DrUlrikeVolmar,DrVesselkaIvanova,DrChristianOehler,DrAchimSteinbauer.Freiburg,Germany,12thofApril2003.DrPeterGl¨oßner5ContentsIntroduction11EstimatingProbabilityofDefault61.1TheDefinitionofProbabilityofDefault....................71.2ConstructionoftheLorenzCurve........................111.3PropertiesoftheLorenzCurve.........................181.4ExamplesofInvariancies.............................211.5Implementation..................................271.6FittingtheLorenzCurve............................311.7MeasuringErrors.................................332EstimatingLossGivenDefault352.1DefinitionofLossGivenDefault........................362.2ModellingtheDecayofSpecialProvisionCohorts...............372.3AveragingoverSeveralCohorts.........................412.4ErrorCalculation.................................443TowardsEstimatingUnexpectedLoss473.1TheLossofMemoryPropertyofRatings...................473.2TheLossofMemoryPropertyExploited....................493.3TheRetailPoissonModel............................523.4ApplicationoftheModel............................583.5CalculatingLossDistributions..........................59SummaryandOutlook67Bibliography67iListofFigures1Riskweightsdependentondefaultprobabilityascalibratedforthe“Quan-titativeImpactStudy”QIS3(October2002).Line1:corporates(LGD=50%,maturity7years);line2:retailmortgagelendingbusiness;line3:otherretailbusiness;line4:retail-so-called“qualifiedrevolvingclaims”.....31.1Histogramofallborrowersasafunctionofthescore,measuredinabsolutenumbers.Thegreylineisthe(problematic!)resultofaleastsquaresfitbyalognormaldistribution.............................91.2Histogramofdefaultedborrowersasafunctionofthescore,measuredinabsolutenumbers.Again,thegreylinegivesaleastsquaresfitbyalognormaldistribution.Observethescaleofthey-axisincontrasttoFigure1.1....101.3Histogramofdefaultedborrowersasafunctionoftwoofthecharacteristics,measuredinabsolutenumbers..........................111.4Distributions(smoothedhistograms)ofallborrowers(linea)anddefaultedborrowers(lined)inabsolutenumbers.....................121.5Cumulativedistributionsofallborrowers(linea)anddefaultedborrowers(lined)inabsolutenumbers...........................131.6Normalizedcumulativedistributionsofallborrowers(linea)anddefaultedborrowers(lined).................................141.7LorenzcurveforthenormalizedcumulativedistributionsofFigure1.6....151.8Initialpairofdistributions(1.000borrowersintotaland200defaultedbor-rowers),andthesamepairunderthequadraticre-