FRM极值理论课件

FinancialRiskManagementHaibinXieSchoolofBankingandFinance,UniversityofInternationalBusinessandEconomicsOffice:Boxue708E-mail:Tel:FRM极值理论ExtremeValueTheoryEVTandVaR1BaselRulesforBacktesting2ExtremeValueTheoryandVaRFRM极值理论BaselRulesforBacktestingTheBaselCommitteeputinplaceaframeworkbasedonthedailybacktestingofVaR.Havinguptofourexceptionsisacceptable,whichdefinesagreenzone.Ifthenumberofexceptionsisfiveormore,thebankfallsintoayelloworredzoneandincursaprogressivepenalty,whichisenforcedwithahighercapitalcharge.Roughly,thecapitalchargeisexpressedasamultiplierofthe10-dayVaRatthe99%levelofconfidence.Thenormalmultiplierkis3.Afteranincursionintotheyellowzone,themultiplicativefactor,k,isincreasedfrom3to4,orplusfactordescribedintheTableinthenextslideFRM极值理论TheBaselPenaltyZonesZoneNumberofExceptionsPotentialincreaseinKGreen0to40.00Yellow50.460.570.6580.7590.85Red≥101FRM极值理论Appendix1WhynormalmultiplierK=3ByChebyshevinequality:P(|x-μ|>λσ)≤1/λ2.Supposesymmetricdistribution,wegetP(x-μ<-λσ)≤1/2λ2,whichdeterminestheMaxofVaR,VaRmx=λσ.Lettheconfidencelevelbe0.99,weget1/2λ2=0.01,fromwhich,wegetλ=7.071.SupposetheusualVaRiscalculatedundertheassumptionofnormaldistribution,wegetVaRN=2.326σ.Thus,weneedamultiplierifnormaldistributionisnotsatisfied.Themultiplier,K=λσ/2.36σ=3.03FRM极值理论Appendix2VaRParameters:TomeasuretheVaR,wefirstneedtodefinetwoquantitativeparameters:theconfidencelevelandthehorizonConfidenceLevel:Thehighertheconfidencelevel,thegreatertheVaRmeasure!Itisnotclear,however,atwhatconfidencelevelshouldonestopHorizon:Thelongerthehorizon,thegreatertheVaRmeasure.Itisnotclear,however,atwhathorizonshouldonestop.VaRParameters:SomerulesforconfidencelevelandhorizonselectionThechoiceoftheconfidencelevelandhorizondependontheintendedusefortheriskmeasures.Forbacktestingpurposes,alowconfidencelevelandashorthorizonisnecessary;forcapitaladequacypurposes,ahighconfidencelevelandalonghorizonarerequired.Inpractice,theseconflictingobjectivescanbeaccommodatedbyacomplexrule,asisthecasefortheBaselmarketriskchargeFRM极值理论ExtremeValueTheoryVaRisallaboutthetailbehavioroflossdistribution,A.K.A,weareonlyinterestedinsomeextremevalueofadistribution.D.V.GnedenkoandEVT7Бори́сВлади́мировичГнеде́нко;January1,1912–December27,1995FRM极值理论GeneralizedParetoDistributionThishastwoparametersx(theshapeparameter)andb(thescaleparameter)Bydefinition,weexpectbtobepositive.ThecumulativedistributionisFRM极值理论GeneralizedParetoDistributionWhenunderlingdistributionofvisnormal,wehave.increasesasthetailofvgetsheavierFormostfinancialdata,in[0.1,0.4]Thek-thmomentofunderlingr.v.isfiniteifFRM极值理论MaximumLikelihoodEstimatorTheobservations,xi,aresortedindescendingorder.SupposethattherearenuobservationsgreaterthanuWechoosexandbtomaximizeFRM极值理论MaximumLikelihoodEstimatorConstraintsxandbaresupposedtobepositive,althoughxnotrequiredtobepositivebythedefinitionofGPD.Negativexindicates:LightertailoftheunderlingdistributioncomparedwithnormalInappropriatevalueofuischosenFRM极值理论FromparameterstotailofvBydefinition:ThereforeAgainsemi-parametricFRM极值理论Whypowerlaw?FRM极值理论ExtremeValueTheory——VaR

FRM极值理论ExpectedShortFallFRM极值理论BlockMaximaModelsDistributionofthelargestvariableAsngoestoinfinity,andthesupportofris[-inf,inf]WeneedtoblowupthevariablewithanormalizationThelimitingdistributionisGeneralizedExtremeValueDistributionFRM极值理论BlockMaximaModelsGeneralizedExtremeValueDistributionVaRunderGEVdistributionAnythingwrong?FRM极值理论BlockMaximaModelsisthedistributionofthelargestvariablenotthevariableitself.The(1-q)thquantileofrisequivalentto(1-q)^nthquantileofr(n)ThecorrectVaRis18FRM极值理论BlockMaximaModelsEstimationBydefinitionofF*,weonlyhaveONEobservationtoestimatethreeparametersWay-outApplyGEVdistributiontomaximumreturnswithineachblockMLESelectionofnGEVisalimitproperty,naslargeaspossibleForgivenT,g=T/nwheregistheeffectivenumberofobservationsforparameterestimationBalance19FRM极值理论MultipleperiodVaRUnderEVTthemultipleperiodVaRisnotjustsquarerootoftimehorizon.Whysquarerootoftimehorizon?Underpowerlaw Fellershowsthattailriskisapproximatelyadditive,therefore:Itiseasytoseethat 20FRM极值理论CoherentRiskMeasures1Monotonicity:ifX1<X2,2Translationinvariance:3Homogeneity:4Subadditivity:FRM极值理论ExerciseBasedona90%confidencelevel,howmanyexceptionsinbacktestingaVaRwouldbeexpectedovera250-daytradingyear?a.10b.15c.25d.50FRM极值理论Alarge,internationalbankhasatradingbookwhosesizedependsontheopportunitiesperceivedbyitstraders.Themarketriskmanagerestimatestheone-dayVaR,atthe95%confidencelevel,tobe$50million.Youareaskedtobeevaluatehowgoodajobthemanagerisdoinginestimatingtheone-dayVaR.Whichofthefollowingwouldbethemostconvincingevidencethatthemanagerisdoingapoorjob,assumingthatthelossesareidenticalandindependentlydistributed(i.i.d)?a.Overthepast250days,thereareeightexceptionsb.Overthepast250days,thelargestlossis$500millionc.Overthepast250days,themeanlossis$60milliond.Overthepast250days,thereisnoexceptionFRM极值理论WhichofthefollowingproceduresisessentialinvalidatingtheVaRestimates?a.stress-testingb.scenarioanalysisc.backtestingd.Onceapprovedbyregulators,nofurthervalidationisrequiredFRM极值理论TheMarketRiskAmendmenttotheBaselCapitalAccorddefinestheyellowzoneasthefollowingrangeofexceptionsoutof250observationsa.3to7b.5to9c.6to9d.6to10FRM极值理论Extremevaluetheoryprovidesvaluableinsightaboutthetailsofreturndistributions.WhichofthefollowingstatementsaboutEVTanditsapplicationsisincorrect?a.Thepeaksoverthreshold,whichthendeterminesthenumberofobservedexceedances;thethresholdmustbesufficientlyhightoapplythetheory,butsufficientlylowsothatthenumberofobservedexceedancesisareliableestimate.b.EVThighlightsthatdistributionsjustifiedbycentrallimittheoremcanbeusedforextremevalueestimationc.EVTestimatesaresubjecttoconsiderablemodelrisk,andEVTresultsareofenverysensitivetothepreciseassumptionsmaded.Becauseobserveddatainthetailsofdistributionislimited,EVestimatescanbeverysensitivetosmallsampleeffectsandotherbiasesFRM极值理论Whichofthefollowingstatementsregardingextremevaluetheoryisincorrect?a.IncontrasttoconventionalapproachesforestimatingVaR,EVTconsidersonlythetailbehaviorofthedistributionb.ConversationalapproachesforestimatingVaRthatassumethatthedistributionof