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ANewStrategyforDynamicallyHedgingMortgage-BackedSecuritiesJacobBoudoukh,MatthewRichardson,RichardStantonandRobertF.WhitelawMay1995AbstractThispaperdevelopsanewstrategyfordynamicallyhedgingmortgage-backedsecurities(MBSs).TheapproachinvolvesestimatingthejointdistributionofreturnsonMBSsandT-notefutures,condi-tionaloncurrenteconomicconditions.Weshowthatourapproachhasasimpleintuitiveinterpretationofformingahedgeratiobydi erentiallyweightingpastpairsofMBSandT-notefuturesreturns.Anout-of-samplehedgingexerciseisperformedfor8%,9%and10%GNMAsoverthe1990-1994periodforweeklyandmonthlyreturnhorizons.Thedynamicapproachisverysuccessfulathedgingoutthein-terestrateriskinherentinalloftheGNMAs.Forexample,inhedgingweeklyreturnson10%GNMAs,ourdynamicmethodreducesthevolatilityoftheGNMAreturnfrom41to24basispoints,whereasastaticmethodmanagesonly29basispointsofresidualvolatility.Moreover,only1basispointofthevolatilityofthedynamicallyhedgedreturncanbeattributedtoriskassociatedwithU.S.Treasuries,whichisincontrastto14basispointsofinterestrateriskinthestaticallyhedgedreturn.SternSchoolofBusiness,NYU;WhartonSchool,UniversityofPennsylvania;HaasSchoolofBusiness,UCBerkeley;andSternSchoolofBusiness,NYU.WewouldliketothanktheQGroupfor nancialsupport.IntroductionInstitutionsholdsigni cantpositionsinmortgage-backedsecurities(MBSs)foravarietyofrea-sons.Whetherthesepositionsreflecttradesonrelativevalueorinvolveinventoryholdingsduetocorebusinesses,hedgingtheinterestrateriskofthesesecuritiesisanimportantconcern.Thisisespeciallytruegiventhewell-documentedcasesofhugemonetarylossesincurredby nancialinstitutionsandinvestmentgroupswithrespecttotheirMBSportfolios.MBSvaluation,andbyextensionhedging,isnotastraightforwardexercise.While xed-rateMBSsissuedbygovernmentagenciesrepresentdefault-freeclaimstotheinterestandprincipaloftheunderlyingmortgages,thetimingofthesecashflowsdependsontheprepaymentbehaviorofthepool.Inparticular,asinterestratesfall,individualshaveanincentivetore nanceexistingmortgagesatthenewlowerrates.Thus, xed-rateMBSinvestorsareimplicitlywritingacalloptiononthecorresponding xed-ratebond.Eventhoughprepaymentscanoccurforreasonsnotassociatedwithinterestratemovements,interestratesarethepredominantfactorinvaluingMBSs.Becauseofthispredominance,U.S.Treasurysecurities,or,morespeci cally,Treasurynote(T-note)futures,areoftenusedtohedgeMBSs.Thereasonsaretwofold:(i)T-notefuturesareveryliquidderivatives,and(ii)thepricesoftheseinstrumentsaredeterminedbytheunderlyingtermstructureofinterestratesandthusrelatedirectlytothevalueofMBSs.TherearetwocommonapproachestohedgingMBSsusingT-notefutures.The rstispurelyempiricalandinvolvestheregressionofpastreturnsonMBSsagainstpastreturnsonT-notefutures.TheresultingrelationcanthenbeusedtohedgetheinterestrateriskofMBSsusingtheriskinT-notes.Theadvantageofthismethodisthatitdoesnotinvolvestrongassumptionsregardingtheunderlyingmodelfortheevolutionofinterestratesorprepayments.Thedisadvantage,however,issevere.Thismethodisstaticinnature.Itdoesnotexplicitlyadjustthehedgeratioforchangesininterestratesandmortgageprepayments.1Thatis,theobservationsusedintheregressionrepresentanaverageoftherelationbetweenMBSsandT-notefuturesonlyoverthesampleperiod,whichmayormaynotrepresentativeofthecurrentperiod.Asanalternative,thesecondapproachismodel-based.Itinvolvesspeci cationoftheinterestrateprocessandaprepaymentmodel.TheseassumptionsthenhelpmapanMBSpricingfunc-tionaltointerestratesandpossiblyotherfactors.2TheapproachrepresentsadynamicmethodfordeterminingcomovementsbetweenMBSpricesandT-notefuturesprices.Conditionaloncurrentvaluesoftherelevanteconomicvariablesandonparticularparametervalues,thesecomovementsarecompletelyspeci ed.Theproblemswiththisapproacharetwofold.First,thereisnoconsensusregardingwhatisareasonablespeci cationofhowthetermstructuremovesthroughtime,andhowthesemovementsrelatetoprepaymentbehavior.Anymodelpriceisgoingtobetieddowncloselytothesepossiblyadhocassumptions.3Second,andmoresubtle,istherecognitionthat1theparametervaluesthemselvesmayoftenbe\chosenorestimatedfromastaticviewpoint.Forexample,empiricalprepaymentmodelsoftenreflectadhocprojectionsofprepaymentratesonsetsofhousingandinterestratefactors.Dotheresultingcoecients,whichrepresentanaverageoftherelationinthepast,havethesamelinktothevariablesdescribingthecurrentperiod?Manyofthewell-documentedMBS-hedging ascoeswouldimplythatthisisnotthecase.Inthispaper,weprovidea rstpassatbridgingthegapbetweentheregressionandthemodel-basedapproaches.Ourgoalistomaintainsomeofthedistribution-freepropertiesofthepurelyempiricalmethod,whilerecognizingtheimportantdynamicpropertiesofMBSsandother xedincomeinstruments.Inparticular,weproposeamethodbasedonestimatingtheconditionalproba-bilitydensityofMBSreturns,T-notefuturesreturns,andrelevantcurrentinformation(suchasthelevel,slopeandcurvatureofthetermstructure,interestratevolatility,andprepaymenthistory).ThismethodallowsustohedgeMBSswithT-notefutures,conditionaloncurrentinformation.Thus,thehedgeratioisderivedinasimilarwaytoonesfromexistingempiricalmethods,buti