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Backtesting Value at Risk: A GMM Duration Based Test

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Niveau: Supérieur, Master
Backtesting Value-at-Risk: A GMM Duration-Based Test Bertrand Candelony, Gilbert Colletazz, Christophe Hurlinz, Sessi Tokpaviz z University of Orléans, Laboratoire d?Economie d?Orléans (LEO), France. Corresponding author: y Maastricht University, Department of Economics. The Netherlands June 2008 Abstract This paper 1 proposes a new duration-based backtesting procedure for VaR fore- casts. The GMM test framework proposed by Bontemps (2006) to test for the dis- tributional assumption (i:e: the geometric distribution) is applied to the case of the VaR forecasts validity. Using simple J-statistic based on the moments de?ned by the orthonormal polynomials associated with the geometric distribution, this new approach tackles most of the drawbacks usually associated to duration based back- testing procedures. First, its implementation is extremely easy. Second, it allows for a separate test for unconditional coverage, independence and conditional cov- erage hypothesis (Christo?ersen, 1998). Third, feasibility of the tests is improved. Fourth, Monte-Carlo simulations show that for realistic sample sizes, our GMM test outperforms traditional duration based test. An empirical application for Nasdaq returns con?rms that using GMM test leads to major consequences for the ex-post evaluation of the risk by regulation authorities. Without any doubt, this paper pro- vides a strong support for the empirical application of duration-based tests for VaR forecasts.

  • portfolio returns

  • main var

  • duration based

  • var violations

  • adequate ex-post techniques

  • generated var

  • coverage hypothesis


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BacktestingValue-at-Risk:AGMMDuration-BasedTestBertrandCandelony,GilbertColletazz,ChristopheHurlinz,SessiTokpavizzUniversityofOrléans,Laboratoired’Economied’Orléans(LEO),France.Correspondingauthor:christophe.hurlin@univ-orleans.fryMaastrichtUniversity,DepartmentofEconomics.TheNetherlandsJune2008AbstractThispaper1proposesanewduration-basedbacktestingprocedureforVaRfore-casts.TheGMMtestframeworkproposedbyBontemps(2006)totestforthedis-tributionalassumption(i:e:thegeometricdistribution)isappliedtothecaseoftheVaRforecastsvalidity.UsingsimpleJ-statisticbasedonthemomentsde…nedbytheorthonormalpolynomialsassociatedwiththegeometricdistribution,thisnewapproachtacklesmostofthedrawbacksusuallyassociatedtodurationbasedback-testingprocedures.First,itsimplementationisextremelyeasy.Second,itallowsforaseparatetestforunconditionalcoverage,independenceandconditionalcov-eragehypothesis(Christo¤ersen,1998).Third,feasibilityofthetestsisimproved.Fourth,Monte-Carlosimulationsshowthatforrealisticsamplesizes,ourGMMtestoutperformstraditionaldurationbasedtest.AnempiricalapplicationforNasdaqreturnscon…rmsthatusingGMMtestleadstomajorconsequencesfortheex-postevaluationoftheriskbyregulationauthorities.Withoutanydoubt,thispaperpro-videsastrongsupportfortheempiricalapplicationofduration-basedtestsforVaRforecasts.Keywords:Value-at-Risk,backtesting,GMM,duration–basedtest.J.E.LClassi…cation:C22,C52,G281TheauthorsthankEnriqueSentanaforcommentsonthepaperaswellastheparticipantsofthe"MethodsinInternationalFinanceNetwork"2008congressinBarcelona.ThepaperwaspartiallyperformedduringthevisitofChristopheHurlinatMaastrichtUniversityviathevisitingprofessorshipprogramofMETEOR.Usualdisclaimersapply.
1IntroductionTherecentBaselIIagreementshaveleftthepossibilityfor…nancialinsti-tutionstodevelopandapplytheirowninternalmodelofriskmanagement.TheValue-at-Risk(VaRthereafter),whichmeasuresthequantileofthepro-jecteddistributionofgainsandlossesoveratargethorizon,constitutesthemostpopularmeasureofrisk.Consequently,regulatoryauthoritiesneedtosetupadequateex-posttechniquesvalidatingornottheamountofrisktakenby…nancialinstitutions.ThestandardassessmentmethodofVaRconsistsinbacktestingorrealitycheckprocedures.Asde…nedbyJorion(2007),back-testingisaformalstatisticalframeworkthatconsistsinverifyingifactualtradinglossesareinlinewithprojectedlosses.Thisinvolvesasystemiccom-parisonofthehistoryofmodel-generatedVaRforecastswithactualreturnsandgenerallyreliesontestingoverVaRviolations(alsocalledtheHit).Aviolationissaidtooccurwhenex-postportfolioreturnsarelowerthanVaRforecasts2.Christo¤ersen(1998)arguesthataVaRwithachosencoveragerateof %isvalidassoonasVaRviolationssatisfyboththehypothesisofunconditionalcoverageandindependence.Thehypothesisofunconditionalcoveragemeansthattheexpectedfrequencyofobservedviolationsispreciselyequalto %.Iftheunconditionalprobabilityofviolationissigni…cantlyhigherthan %,itmeansthatVaRmodelunderstatestheportfolio’sactuallevel2OrtheoppositeofVaRifthelatterisde…nedasalossinabsolutevalue.2
ofrisk.Theopposite…ndingoftoofewVaRviolationswouldalternativelysignalanoverlyconservativeVaRmeasure.ThehypothesisofindependencemeansthatifthemodelofVaRcalculationisvalidthenviolationsmustbedistributedindependently.Inotherwords,theremustnothaveanyclusterintheviolationsequence.AsnotedbyCampbell(2007),theunconditionalcoveragepropertyplacesarestrictiononhowoftenVaRviolationsmayoccur,whereastheindependencepropertyrestrictsthewaysinwhichtheseviolationsmayoccur.ButbothassumptionsareessentialtocharacterizeVaRforecastvalidity:onlyhitsequencesthatsatisfyeachoftheseproperties(andhencetheconditionalcoveragehypothesis)canbepresentedasevidenceofausefulVaRmodel.Eveniftheliteratureaboutconditionalcoverageisquiterecent,varioustestsonindependenceandunconditionalcoveragehypotheseshavealreadybeende-veloped(seeCampbell,2007forasurvey).Mostofthemdirectlyexploittheviolationprocess3.Howeveranotherstreamlineoftheliteratureusesthesta-tisticalpropertiesofthedurationbetweentwoconsecutivehits.Thebaselineideaisthatiftheone-periodaheadVaRiscorrectlyspeci…edforacoveragerate ;thenthedurationsbetweentwoconsecutivehitsmusthaveageo-metricdistributionwithasuccessprobabilityequalto %:OnthesegroundsChristo¤ersenandPelletier(2004)proposedatestofindependence.Thegen-3Forinstance,Christo¤ersen’stest(1998)basedonMarkovchain,thehitregressiontestofEngleandManganelli(2004)reliesonalinearauto-regressivemodel,orthetestsofBerkowitzandal.(2005)basedontestsofmartingaledi¤erence.3
eralideaoftheirduration-basedbacktestingtestconsistsinspecifyingadura-tiondistributionthatneststhegeometricdistributionandallowsfordurationdependence,sothattheindependencehypothesiscanbetestedbymeansofsimplelikelihoodratio(LR)tests.AsnotedbyHaas(2007),thisgeneralduration-basedapproachofbacktestingsoundsveryappealing.Itiseasytoapplyandprovidesaclear-cutinterpretationofparameters.Nevertheless,itmustbenotethatonehavetospecifyaparticulardistributionundertheal-ternativehypothesis.Moreover,LRtestturnsouttosu¤erfromtherelativescarcityofviolations:evenwithoneyearofdailyreturns,theassociatedseriesofdurationsislikelytobeshort,inparticularfora1%coveragerate(thevaluerecommendedbysupervisionauthorities).Consequentlyduration-basedback-testingmethodshaverelativelysmallpowerforrealisticsamplesizes(Haas,2007)anditevenoftenhappensthatstandardLRduration-basedstatisticscannotbecomputed4.Forthesereasons,actualduration-basedbacktestingproceduresarenotverypopularamongpractitioners.Howeverweshowinthispaperthatitispossibletosigni…cantlyimprovetheseprocedures.RelyingontheGMMframeworkofBontempsandMeddahi(2005,2006)wede-riveteststatisticssimilartoJ-statisticsbasedonparticularmomentsde…nedbytheorthonormalpolynomialsassociatedwiththegeometricdistribution.Alsoourduration-basedbacktestconsidersdiscretelifetimedistributions:we4TheLRtestrequiresatleastonenon-censoreddurationandanadditionalpos-siblycensoredduration(i.e.twoviolations)tobeimplemented.AsexperiencedbyBerkowitzetal.(2005)withoneyearoftradingdays(T=250)and =0:01thetestcanbecomputedonlyinsixcasesoutoften.4