FinancialRiskManagementHaibinXieSchoolofBankingandFinance,UniversityofInternationalBusinessandEconomicsOffice:Boxue708E-mail:Tel:FRM極值理論ExtremeValueTheoryEVTandVaR1BaselRulesforBacktesting2ExtremeValueTheoryandVaRFRM極值理論BaselRulesforBacktestingTheBaselCommitteeputinplaceaframeworkbasedonthedailybacktestingofVaR.Havinguptofourexceptionsisacceptable,whichdefinesagreenzone.Ifthenumberofexceptionsisfiveormore,thebankfallsintoayelloworredzoneandincursaprogressivepenalty,whichisenforcedwithahighercapitalcharge.Roughly,thecapitalchargeisexpressedasamultiplierofthe10-dayVaRatthe99%levelofconfidence.Thenormalmultiplierkis3.Afteranincursionintotheyellowzone,themultiplicativefactor,k,isincreasedfrom3to4,orplusfactordescribedintheTableinthenextslideFRM極值理論TheBaselPenaltyZonesZoneNumberofExceptionsPotentialincreaseinKGreen0to40.00Yellow50.460.570.6580.7590.85Red≥101FRM極值理論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極值理論Appendix2VaRParameters:TomeasuretheVaR,wefirstneedtodefinetwoquantitativeparameters:theconfidencelevelandthehorizonConfidenceLevel:Thehighertheconfidencelevel,thegreatertheVaRmeasure!Itisnotclear,however,atwhatconfidencelevelshouldonestopHorizon:Thelongerthehorizon,thegreatertheVaRmeasure.Itisnotclear,however,atwhathorizonshouldonestop.VaRParameters:SomerulesforconfidencelevelandhorizonselectionThechoiceoftheconfidencelevelandhorizondependontheintendedusefortheriskmeasures.Forbacktestingpurposes,alowconfidencelevelandashorthorizonisnecessary;forcapitaladequacypurposes,ahighconfidencelevelandalonghorizonarerequired.Inpractice,theseconflictingobjectivescanbeaccommodatedbyacomplexrule,asisthecasefortheBaselmarketriskchargeFRM極值理論ExtremeValueTheoryVaRisallaboutthetailbehavioroflossdistribution,A.K.A,weareonlyinterestedinsomeextremevalueofadistribution.D.V.GnedenkoandEVT7Бори?сВлади?мировичГнеде?нко;January1,1912–December27,1995FRM極值理論GeneralizedParetoDistributionThishastwoparametersx(theshapeparameter)andb(thescaleparameter)Bydefinition,weexpectbtobepositive.ThecumulativedistributionisFRM極值理論GeneralizedParetoDistributionWhenunderlingdistributionofvisnormal,wehave.increasesasthetailofvgetsheavierFormostfinancialdata,in[0.1,0.4]Thek-thmomentofunderlingr.v.isfiniteifFRM極值理論MaximumLikelihoodEstimatorTheobservations,xi,aresortedindescendingorder.SupposethattherearenuobservationsgreaterthanuWechoosexandbtomaximizeFRM極值理論MaximumLikelihoodEstimatorConstraintsxandbaresupposedtobepositive,althoughxnotrequiredtobepositivebythedefinitionofGPD.Negativexindicates:LightertailoftheunderlingdistributioncomparedwithnormalInappropriatevalueofuischosenFRM極值理論FromparameterstotailofvBydefinition:ThereforeAgainsemi-parametricFRM極值理論Whypowerlaw?FRM極值理論ExtremeValueTheory——VaR

FRM極值理論ExpectedShortFallFRM極值理論BlockMaximaModelsDistributionofthelargestvariableAsngoestoinfinity,andthesupportofris[-inf,inf]WeneedtoblowupthevariablewithanormalizationThelimitingdistributionisGeneralizedExtremeValueDistributionFRM極值理論BlockMaximaModelsGeneralizedExtremeValueDistributionVaRunderGEVdistributionAnythingwrong?FRM極值理論BlockMaximaModelsisthedistributionofthelargestvariablenotthevariableitself.The(1-q)thquantileofrisequivalentto(1-q)^nthquantileofr(n)ThecorrectVaRis18FRM極值理論BlockMaximaModelsEstimationBydefinitionofF*,weonlyhaveONEobservationtoestimatethreeparametersWay-outApplyGEVdistributiontomaximumreturnswithineachblockMLESelectionofnGEVisalimitproperty,naslargeaspossibleForgivenT,g=T/nwheregistheeffectivenumberofobservationsforparameterestimationBalance19FRM極值理論MultipleperiodVaRUnderEVTthemultipleperiodVaRisnotjustsquarerootoftimehorizon.Whysquarerootoftimehorizon?Underpowerlaw Fellershowsthattailriskisapproximatelyadditive,therefore:Itiseasytoseethat 20FRM極值理論CoherentRiskMeasures1Monotonicity:ifX1<X2,2Translationinvariance:3Homogeneity:4Subadditivity:FRM極值理論ExerciseBasedona90%confidencelevel,howmanyexceptionsinbacktestingaVaRwouldbeexpectedovera250-daytradingyear?a.10b.15c.25d.50FRM極值理論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極值理論WhichofthefollowingproceduresisessentialinvalidatingtheVaRestimates?a.stress-testingb.scenarioanalysisc.backtestingd.Onceapprovedbyregulators,nofurthervalidationisrequiredFRM極值理論TheMarketRiskAmendmenttotheBaselCapitalAccorddefinestheyellowzoneasthefollowingrangeofexceptionsoutof250observationsa.3to7b.5to9c.6to9d.6to10FRM極值理論Extremevaluetheoryprovidesvaluableinsightaboutthetailsofreturndistributions.WhichofthefollowingstatementsaboutEVTanditsapplicationsisincorrect?a.Thepeaksoverthreshold,whichthendeterminesthenumberofobservedexceedances;thethresholdmustbesufficientlyhightoapplythetheory,butsufficientlylowsothatthenumberofobservedexceedancesisareliableestimate.b.EVThighlightsthatdistributionsjustifiedbycentrallimittheoremcanbeusedforextremevalueestimationc.EVTestimatesaresubjecttoconsiderablemodelrisk,andEVTresultsareofenverysensitivetothepreciseassumptionsmaded.Becauseobserveddatainthetailsofdistributionislimited,EVestimatescanbeverysensitivetosmallsampleeffectsandotherbiasesFRM極值理論Whichofthefollowingstatementsregardingextremevaluetheoryisincorrect?a.IncontrasttoconventionalapproachesforestimatingVaR,EVTconsidersonlythetailbehaviorofthedistributionb.ConversationalapproachesforestimatingVaRthatassumethatthedistributionof