VaR-APARCH模型应用于证券投资风险分析

Value-at-RiskMeasuresForStockMarketUsingAPARCH-GEDModelChenxue-hua(InstituteofQuantitativeEconomicsGuangzhouUniversity,Guangzhou,510405)Email:[e-mail],weproposeanAPARCHmodelwiththreedifferentdistributionsassumptiontoestimateconditionalVaR.ThismodelisthencomparedwiththeGARCHmodelwiththecorrespondingthreedistributionsassumption.Usingback-testingofhistoricaldailyreturnseriesweshowthattheAPARCHmodelyieldsstatisticallyvalidVaRmeasuresandgivesbetterone-dayaheadestimatesthantheGARCHmodel.Keywords:ValueatRisk;APARCHModel;GARCHmodel;FattailsVaR-APARCH510405APARCHVaRGARCHAPARCHVaRGARHCAPARCHGARCHF830.59F830.911IntroduceAprimarytoolforfinancialriskassessmentistheValue-at-Risk(VaR)methodologywhereVaRisdevinedasanamountlostonaportfoliowithagivensmallprobabilityoverafixednumberofdays.Thatis,mathematically,VaRatthe)1(100q−%confidencelevelisdefinedasthelowerq100percentileoftheprofit-lossdistribution.LetXbetheprofit-lossofagivenportfolio,for10q,anunconditionalVaRisaquantileofthemarginaldistribution)(XFXofXdenotedby)1(1qFVaRq−=−TheacceptanceandusageofVaRhasbeenspreadingrapidlysinceitsinceptionintheearly1990s.TheVaRissupportedundertheGroupofTenbanks,theGroupofThirty,theBankforInternationalSettlementsandtheEuropeanUnion.ThelimitationsoftheVaRarethatitmayleadtoawidevarietyofresultsunderawidevarietyofassumptionsandmethods;itfocusesonasinglesomewhatarbitrarypoint;itexplicitlydoesnotaddressexposureinextrememarketconditionsanditisastatisticalmeasure,notamanagerialeconomicone.Inourpaper,weestimateVaRbyusingaasymmetricPowerARCH(APARCH)modelandtheresultiscomparedwithGARCHmodel.AtlastweusebacktestingtovalidatetheuseoftheVaRmodels.2VaryingRiskMeasures2.1CharacteristicsofFinancialTimeSeriesSomewell-knowncharacteristicsarecommontomanyfinancialtimeseries.Volatilityclusteringisoftenobserved(i.e.largechangestendtobefollowedbylargechangesandsmallchangestendtobefollowedbysmallchanges;seeMandelbrot,1963,forearlyevidence).Second,financialtimeseriesoftenexhibitleptokurtosis,meaningthatthedistributionoftheirreturnsisfat-tailed(i.e.thekurtosisexceedthekurtosisofastandardGaussiandistribution,seeMandelbrot,1963,orFama,1965).Moreover,theso-called“leverageeffect”,firstnotedinBlack(1976),referstothefactthatchangesinstockpricestendtobenegativelycorrelatedwithchangesinvolatility(i.e.volatilityishigherafternegativeshocksthanafterpositiveshocksofsamemagnitude).2.2APARCHModelSomewell-knowncharacteristicsarecommontomanyfinancialtimeseries.Volatilityclusteringisoftenobserved(i.e.largechangestendtobefollowedbylargechangesandsmallchangestendtobefollowedbysmallchanges;seeMandelbrot,1963,forearlyevidence).And,financialtimeseriesoftenexhibitleptokurtosis,meaningthatthedistributionoftheirreturnsisfat-tailed(i.e.thekurtosisexceedthekurtosisofastandardGaussiandistribution,seeMandelbrot,1963,orFama,1965).Inaseminalpaper,Engle(1982)proposetomodeltime-varyingconditionalvariancewiththeAutoRegressiveConditionalHeteroskedasticity(ARCH)processesthatusepastdisturbancestomodelthevarianceoftheseries.EarlyempiricalevidenceshowthathighARCHorderhastobeselectedinordertocatchthedynamicoftheconditionalvariance.TheGeneralizedARCH(GARCH)modelofBollerslev(1986)isananswertothisissue.ItisbasedonaninfiniteARCHspecificationanditallowstoreducethenumberofestimatedparametersfrom∞toonly2.Bothmodelsallowtotakethetwocharacteristicsintoaccount.LettXbeastrictlystationarytimeseriesrepresentingdailyobservationsofthenegativelogreturnonafinancialassetpriceandassumethatitfollowsaGARCHprocess.Thenthedynamicsofxandtheconditionalvarianceofthemean-adjustedseriesaregivenby),...,2,1(0),,...,2,1(0,0,0,00121202pjqiqpZXjipjjtjqiitittttt=≥=≥≥≥++=+=∑∑=−=−βαασβεαασσµWheretheinnovationstZareastrictwhitenoiseprocesswithzeromean,unitvarianceandmarginaldistributionfunction)(ZFZ,00α.Unfortunately,GARCHmodelsoftendonotfullycapturethefattailpropertyofhighfrequencyfinancialtimesseries.Ding,Granger,andEngle(1993)introduceanAsymmetricPowerARCH(APARCH)modeltoforecastsconditionalvariance.TheAPARCH(p,q)modelcanbeexpressedas:∑∑=−=−−+−+=qijtpjjitiitit110)(δδδσβεγεαασ(3)Where0),,...,1(0,0,00≥=≥≥ijpjαβδαand).,...,1(11qii=−γThismodelisquiteinterestingsinceitcouplestheflexibilityofavaryingexponentwiththeasymmetrycoefficient(totakethe“leverageeffect”intoaccount).2.3ConditionalVaRThenunderdifferentdistributionassumptionwegettheestimatesoftheconditionalmeanandvariancefordayt+1.AndtheconditionalVaRforthe1-steppredictivedistribution,whichwedenoterespectivelybytqVaRissimplifytoqtttqzVaRσµ+=WhereqzistheupperqthquantileofthecorrespondingmarginaldistributionoftZwhichbyassumptiondoesnotdependont.3Back-testing3.1Likelihood-Ratio-TestThemostrequirementofaVaRmodelisthattheproportionoftimesthattheVaRforecastthatisgeneratesisexceeded(thenumofexceptions)shouldonaverageequalthenominalsignificancelevel,ForTobservations,letΤ0bethenumberofexceptions,Τ1bethenumberofnon-exceptions,andqbetheconfidencelevel.Iftherealprobabilityisnotq,Thenwecanwriteitas=T1/T,and1-representtheproportionoffailures.WecannowuseaLikelihood-Ratio-T