Portfolio value-at-risk with heavy-tailed risk fac

PORTFOLIOVALUE-AT-RISKWITHHEAVY-TAILEDRISKFACTORSPaulGlassermanGraduateSchoolofBusiness,ColumbiaUniversityPhilipHeidelbergerIBMResearchDivision,YorktownHeights,NYPerwezShahabuddinIEORDepartment,ColumbiaUniversityThispaperdevelopsefficientmethodsforcomputingportfoliovalue-at-risk(VAR)whentheunderlyingriskfactorshaveaheavy-taileddistribution.Inmodelingheavytails,wefocusonmultivariatetdistributionsandsomeextensionsthereof.WedeveloptwomethodsforVARcalculationthatexploitaquadraticapproximationtotheportfolioloss,suchasthedelta-gammaapproximation.Inthefirstmethod,wederivethecharacteristicfunctionofthequadraticapproximationandthenusenumericaltransforminversiontoapproximatetheportfoliolossdistribution.BecausethequadraticapproximationmaynotalwaysyieldaccurateVARestimates,wealsodevelopalowvarianceMonteCarlomethod.Thismethodusesthequadraticapproximationtoguidetheselectionofaneffectiveimportancesamplingdistributionthatsamplesriskfactorssothatlargelossesoccurmoreoften.Varianceisfurtherreducedbycombiningtheimportancesamplingwithstratifiedsampling.Numericalresultsonavarietyoftestportfoliosindicatethatlargevariancereductionsaretypicallyobtained.BothmethodsdevelopedinthispaperovercomedifficultiesassociatedwithVARcalculationwithheavy-tailedriskfactors.TheMonteCarlomethodalsoextendstotheproblemofestimatingtheconditionalexcess,sometimesknownastheconditionalVAR.KEYWORDS:value-at-risk,delta-gammaapproximation,MonteCarlo,simulation,variancereduction,importancesampling,stratifiedsampling,conditionalexcess,conditionalvalue-at-risk1.INTRODUCTIONAcentralprobleminmarketriskmanagementisestimationoftheprofit-and-lossdistributionofaportfoliooveraspecifiedhorizon.Giventhisdistribution,thecalculationofspecificriskmeasuresisrelativelystraightforward.Value-at-risk(VAR),forexample,isaquantileofthisdistribution.Theexpectedlossandtheexpectedexcesslossbeyondsomethresholdareintegralswithrespecttothisdistribution.Thedifficultyinestimatingthesetypesofriskmeasuresliesprimarilyinestimatingtheprofit-and-lossdistributionitself,especiallythetailofthisdistributionassociatedwithlargelosses.P.GlassermanandP.ShahabuddinwerepartiallysupportedbyNSFNYIAwardDMI9457189andNSFCareerAwardDMI9625297,respectively.ManuscriptreceivedAugust2000;finalrevisionreceivedJanuary2001.AddresscorrespondencetoPaulGlasserman,403UrisHall,ColumbiaUniversity,NewYork,NY10027;[e-mail],Vol.12,No.3(July2002),239–269
2002BlackwellPublishingInc.,350MainSt.,Malden,MA02148,USA,and108CowleyRoad,Oxford,OX41JF,UK.239Allmethodsforestimatingorapproximatingthedistributionofchangesinportfoliovaluerely(atleastimplicitly)ontwotypesofmodelingconsiderations:assumptionsaboutthechangesintheunderlyingriskfactorstowhichaportfolioisexposed,andamechanismfortranslatingthesechangesinriskfactorstochangesinportfoliovalue.Examplesofrelevantriskfactorsareequityprices,interestrates,exchangerates,andcommodityprices.Forportfoliosconsistingofpositionsinequities,currencies,commodities,orgovernmentbonds,mappingchangesintheriskfactorstochangesinportfoliovalueisstraightforward.Butforportfolioscontainingcomplexderivativesecuritiesthismappingreliesonapricingmodel.Thesimplestandperhapsmostwidelyusedapproachtomodelingchangesinportfoliovalueisthevariance-covariancemethodpopularizedbyRiskMetrics(1996).Thisapproachisbasedonassuming(i)thatchangesinriskfactorsareconditionallymultivariatenormaloverahorizonof,say,oneday,twoweeks,oramonth,and(ii)thatportfoliovaluechangeslinearlywithchangesintheriskfactors.(‘‘Conditionally’’heremeansconditionaloninformationavailableatthestartofthehorizon;theuncondi-tionaldistributionneednotbenormal.)Undertheseassumptions,theportfolioprofit-and-lossdistributionisconditionallynormal;itsstandarddeviationcanbecalculatedfromthecovariancematrixoftheunderlyingriskfactorsandthesensitivitiesoftheportfolioinstrumentstotheseriskfactors.Theattractionofthisapproachliesinitssimplicity.Buteachoftheassumptionsonwhichitreliesisopentocriticism,andresearchintheareahastriedtoaddresstheshortcomingsoftheseassumptions.Onelineofworkhasfocusedonrelaxingtheassumptionthatportfoliovaluechangeslinearlywithchangesinriskfactorswhilepreservingcomputationaltractability.Thisincludes,inparticular,the‘‘delta-gamma’’methodsdevelopedinBritten-JonesandSchaefer(1999),DuffieandPan(2001),Rouvinez(1997),andWilson(1999).Thesemethodsrefinetherelationbetweenriskfactorsandportfoliovaluetoincludequadraticaswellaslinearterms.MethodsthatcombineinterpolationapproximationstoportfoliovaluewithMonteCarlosamplingofriskfactorsareconsideredinJamshidianandZhu(1997),Picoult(1999),andShaw(1999).LowvarianceMonteCarlomethodsbasedonexactcalculationofchangesinportfoliovalueareproposedinCardenasetal.(1999),Glasserman,Heidelberger,andShahabuddin(2000),andOwenandTavella(1999).Anotherlineofworkhasfocusedondevelopingmorerealisticmodelsofchangesinriskfactors.Ithaslongbeenobservedthatmarketreturnsexhibitsystematicdeviationfromnormality:acrossvirtuallyallliquidmarkets,empiricalreturnsshowhigherpeaksandheaviertailsthanwouldbepredictedbyanormaldistribution,especial