Macroeconomic State Variables as Determinants of Asset Price Covariances

BoardofGovernorsoftheFederalReserveSystem InternationalFinanceDiscussionPapers Number553 June 1996 MACROECONOMICSTATEVARIABLESASDETERMINANTSOF ASSETPRICECOVARIANCES JohnAmmer NOTE: InternationalFinanceDiscussionPapersarepreliminarymaterialscirculatedto stimulate discussionandcriticalcomment. Referencesinpublicationsto InternationalFinanceDiscussion Papers(otherthanan acknowledgmentthatthewriterhashadaccesstounpublishedmaterial)should beclearedwiththeauthoror authors.

Thispaperexploresthepossibleadvantagesof introducingobservablestatevariablesintorisk managementmodelsas astrategy formodelingtheevolutionofsecondmoments. A simulation exercisedemonstratesthatifasset returnsdependupona setofunderlyingstatevariablesthatare autoregressivelyconditionallyheteroskedastic(ARCH),thena riskmanagementmodelthatfailsto takeaccountof thisdependencecanbadlymismeasureaportfolio’s “Value-at-Risk”(VaR),evenifthe modelallowsfor conditionalheteroskedasticityin assetreturns. Variablesmeasuringmacroeconomic newsare constructedas theorthogonalizedresidualsfromavector autoregression(VAR). These newsvariablesare foundto havesomeexplanatorypowerforassetreturns. Wealsoestimatea model ofasset returnsinwhichtimevariationinvariance andcovariancesderivesonlyfromconditional heteroskedasticityintheunderlyingmacroeconomicshocks. Althoughthedatagivesomesupportfor severalofthespecificationsthatwetried, neitherthesemodelsnorGARCHmodelsthatusedonly assetreturnsappearto havemuchabilityto forecastthesecondmomentsof returns. Finally,we allowassetreturnvariancesandcovariancesto dependdirectlyonunemploymentrates--proxying for thegeneralstateoftheeconomy-- andfindfairlystrongevidencefor thissortof specificationrelative to a nullhypothesisofhomoskedasticity.

MacroeconomicStateVariablesasDeterminantsof AssetPrice CoVarianCeS JohnAmmerl 1. Introduction In recentyears, thetradingactivitiesofmajorfinancialinstitutions(hereinreferredto collectivelyas “banks”)havegrownrapidly,andtheyhavebecomeincreasinglyconcernedwith managingtheriskassociatedwithexposuretoumnticipated changesinthemarketpricesoftraded assets. Seniorbankmanagersgenerallyprefertohavethisriskquantifiedasthesumof moneythat couldconceivablybelostoversomeintervaloftime, giventheportfoliocurrentlyheld. To address thisgoal,bankshavedevelopedso-called“riskmanagementmodels”. Becausethevolatilityofan assetpriceoftendominatesitsdriftovera relativelyshortperiodoftime, riskmanagementmodelsare typicallyprincipallyfocusedonforecastingsecond(andpossibiyhigher)momentsofassetreturns, ratherthanforecastingrelativemeanreturns.2 A riskmanagementmodelhastwoessential components: apricing functionanda statevariableprobabilitydistribution. Thepricingfunctionisa mappingfromthepricesor returnsoftradedassetsto achosen setofunderlyingstatevariables. (In 1 Theauthorisan EconomistintheDivisionof InternationalFinance,Boardof Governorsofthe Federal Reserve System. This paper was prepared for the joint central bank conference “Risk MeasurementandSystemicRisk’’(Washington,DC, November16-17,1995).Ithank AmnonLevyfor abieresearchassistanceandMicoLoretanforhelpfulcomments. Thispaperrepresentstheviewsofthe authorandshouldnotbe interpretedasreflectingthoseoftheBoard ofGovernorsoftheFederalReserve Systemor othermembersofits staff. Iamresponsible for anyerrors. 2Thefocus onsecondmomentsmakesriskmanagementmodelsusefulformeasuring,andperhaps designinga strategyto reduce(i.e., hedge)risk. AssefaZlocation models, whichareused to devise trading strategiesthat maximize(possiblyrisk-adjusted)returns, have a relativelygreater focus on forecastingmeanreturns.

-2thesimplestcase, thesetof reievantassetreturnsandthesetof statevariablesare identical,but it maybe advantageousto choosea smallerset of statevariables.) Giventhepricingfunctionandthe jointprobabilitydistributionof theunderlyingstatevariablesoverthe relevantintervaloftiie, itis possibleto computethedistributionofpossible outcomesfor anyportfolio. Riskmamgementmodelstypicallyuse asetof financialpricesandindices(thatis smallerthanthesetofassetsthatcouldpotentiallybe includedintheportfolio)as theunderlyingstate variables. Mostoften,historicalsamplecovariancesof assetpricesareusedas estimatesofthelikely fbturepattern ofcomovements. However,thismodeliig strategybegsthequestionof whatunderlying economicforcesare drivingassetreturns. Alternatively,onecouldincludemacroeconomicvariables suchas activitymeasures,goodsprices, policyvariables,or businesscycleindicatorsamongthestate variablesinarisk managementmodel.3 Sucha strategyhas severalpotentialadvantagesinthe implementationofrisk managementmodels. First, byputtingmoreeconomicstructureonthemodel, itmayhelppractitionersgaininsightintowhatsortofdevelopmentscausethecovariancestrqctureof financialpricesto changeovertime,so thattheirestimatesof secondmomentsincorporateall availableinformation. If macroeconomicmeasuresare relevantstatevariables,theymaypermita moreparsimoniousrepresentationofthestatespaceandthusmorepreciseestimatesof the relationshipsbetweenstatevariablesandthepricesof tradedassets. Inaddition,changesinthe conditionalvolatilityofa macroeconomicstatevariablewouldtendto affectthecovarianceofasset pricesinfluencedby it-- thissort ofmodeling strategycouldallowfor sucheffectsexplicitly. Altemativeiy,the secondmomentsofasset retumscouldbepermitted to dependdirectlyonthe ZeveZ 3A numberofpapers haveexploredthesensitivityofpanels ofstockretum datatomacroeconomic riskfactors-- see,forexample, Chen,Roll,andRoss(1986),King,Sentana,andWadhwani(1990),and Ammer(1993). However,thatbranchof the literaturehasbeenmore focusedon testingthearbitrage pricingtheory(APT)and seekingoutnon-zerorisk premiumsin the contextof theAPT, ratherthan measuringassetreturncovariancesasan endinthemselves. Campbelland Ammer(1993),incontrast, decomposethecovariation in U.S. stockandbondreturnsinto “proximateacuses”, buttheirempirical exercisesonlyinvolverelationshipsamongfinancialdata.

-3of amacroeconomicstatevariable. Thispaperinvestigatesseveraloftheabovepossibilities,withthegoalofassessingthe usefulnessofthe inclusionofmacroeconomicstatevariablesinriskmanagementmodels. Thenext sectionofthepaperusesa simulationexerciseto assessthe importanceofmodelingassetreturnsin termsof thefundamentalvariablesdrivingthem. In thefollowingsection,weexaminestatistical relationshipsbetweenvariouspotentialmacroeconomicstatevariablesandmonthlyreturnson fixed-incomesecurities,equities,andforeignexchangeinstrumentsof theUnitedStates,Japan, Germany,andtheUnitedKingdom. Thefourthsectionofthepaperlooksintotheempirical contributionofstate-dependencetotimevariationinassetreturnvariancesandcovariances. Inthe subsequentsection,weestimatemodelsinwhichsecondmomentsof returnsdependdirectlyon unemploymentrates. Thesixthsectiondiscussespossiblerefinementsto themethodsusedandsome ideasfor futureresearch. 2. DoesFactor-DependenceMatter? Supposethata vectorof N assetreturns(Z) isa linearfunctionofa setofK (contemporaneouslyrealized)observableeconomicfactors(W): (1) The(KxN)matrixBcontainsthefactor“loadings”. For thetimebeing,wewillassumethatthe conditionalmeansof (Z)and(W)are constantovertime,hence,withoutlossof generality,wecan proceedas ifthesemeansare zeroandomitintercepttermsto simpli~ equationssuchas (l). Further supposethatthefactors(W)andthereturnresiduals(u)are bothhomoskedastic,sothat

I -4- (2) Vfzrt-l(w,) = %l(wfr) ‘ Q, Vt and V’rt-l(ut) = u, vt (3) If thefactorloadingmatrix(B)isalsoconstantovertime,thentheassetreturns(Z) willalsobe homoskedastic. In particular: (4) Underthesecircumstances,particularlywithoutknowledgeof thetrue valuesofthe factorloadings (B), itis nothelpfulto usedataon (W)to estimatethe(constant)covariancematrixof (Z) -- itwould be moreefficientto simplycomputethesamplemomentsof (Z). However,nowsupposethatthesecondmomentsof the factors(W)vary overtime. For example,theymighteachfollowindependentgeneralizedauto-regressiveconditionally heteroskedastic(GARCH)processes.4 For a GARCH(1,1)process, theconditionalvarianceof a factorwouldevolveas follows: ‘&-l v = Var,-l(w”,) = Ck+ ak vu-l + ~k k~ Because(Z) isa functionof (W)throughequation(l), thesecondmomentsoftheassetreturns(Z) willnowvary overtime. For exampleiftwoassetreturnsbothdependpositivelyon a particular factor,theircorrelationwilltendto rise (orbecomelessnegative)whenthe conditionalvarianceof 4SeeBollerslev(1986)for detailsonthepropertiesof GARCHmodels.

-5thatfactorisrelativelyhigh. Simulatedvaluesof(W)and(Z)werecreatedfor a versionofthemodeldescribedby equations(5), (3), and(1)withthehelp ofa randomnumbergenerator. Themodelhastwo observablefactorsandtwoassetreturns, Theparameters,whichare giveninthecentral columnof panelAof Table I,were chosento producea highdegreeofheteroskedasticity(viathecoefficientsa andd) andtimevariationinthecorrelationofthetwoassetreturns(throughB). Theresiduals(u) are independentandidenticallydistributed,andthe(W)andthe(u)are allmutuallyorthogonaland distributednormal. Thedataweresimulatedfor 2000periods. Estimatedparametersof thetrue modelappearintherightmostcolumnofpanel A, Theywerecomputedviaa numericalmaximum likelihoodprocedureappliedto thefirst 1000observations. Twoalternativemodelsof thesecondmomentsof (Z)wereestimatedoverthesame sample-- eachignoresinformationthatmightbe in(W). Estimatesarein PanelB of thetable. The firstisabivariate GARCH(l,l) specificationin(Z) withaconstant correlationbetweenthetwoasset returns.s Thesecondis aconstant variance-covariancematrix. For eachmodel,estimatedconditionalsecondmomentswerecomputedoverthesecond halfofthesimulatedsample. Thefitof themodelswasassessedbyusing themto computethe value-at-risk(VaR)foreachperiodof foursampleportfolios,andcomparingtheVaRmeasuresbased onestimatedmodelsto thetrue VaR(from thedatageneratingprocess). VaRisdefinedhereasthe fivepercentlefttailof theone-periodreturn. Theresultsarein thefourpanelsofTable2. Notethesignificantdegreeoftiie variationinthetrue VaR. Notsurprisingly,theVaRmeasurebasedon estimatingthetruemodel performssuperbly. ThecorrelationwiththetrueVaRexceeds99 percentinallfourcases. TheVaR ‘Alternatively, we couldhaveestimateda lessparsimoniousbutmore generalbivariateGARCH model,suchasthatof Chan,Karolyi.andStulz(1994).

-6basedontheGARCHmodelinassetreturnsfares fairlywellmuchofthetime,but isoccasionallyoff by amountson theorderof 100percentof thecorrectVaR.b Thehomoskedasticmodelcannot,of course, captureanytimevariationinmoments,and itsVaRtendsto exceedtheaveragetrueVaR somewhat. 3. ToWhatExtentDoesMacroeconomicNewsDriveAssetReturns? A necessarystepin implementingthemodeloftheprevioussectionwithrealdataisto estimatefactorloadings(B). Onemustfirstacquiresomeobservablemacroeconomicfactors(W). Thiswasaccomplishedby estimatinga VAR(6)inmonthlydataintwelvecandidatemacroeconomic variables-- inparticular,CPI inflation,industrialproductiongrowth,andtheend-of-monthovernight callmoneyratefor eachoffourcountries: the UnitedStates,Japan, Germany,andtheUnited Kingdom.’ Innovationsinovernightinterestrateslikelyreflectnewinformationaboutmonetary policy. Assetpricedatawere collectedfor thesamecountries,mostlyfixedincomeinstruments in thefour currencies. Theappendixdescribeshowdataon interestratesandyieldswereconverted intoone-monthholdingperiodreturns. Thereturndatausedare end-of-month,inpercent-per-month units,andmeasuredindollars. Table3 showstheR2for eachreturn regressedon the 12factorvariables. TheseR* bWhentheestimationsamplewasextendedto 10,000periods,theVaRmeasurebasedonthetrue modeltrackedthetrueVaRevenmoreclosely,buttheVaRbasedontheGARCH-in-returnsmodelwas no moreaccuratethanwhenitwascomputedfromestimatesobtainedfromtheshortersampleperiod. 7Theresidualsfromtheseregressionswerethenorthogomlizedinthefollowingorder: U.S. (CPI) inflation,Japanese inflation,German inflation,U.K. inflation,U.S. industrialproduction,Japanese industrialproduction,Germanindustrialproduction,U.K. industrialproduction,theU.S.callmoneyrate (federalfunds),theJapanesecallmoneyrate,theGermancallmoneyrate,andtheU.K. callmoneyrate. The pointof theorthogomdizationisto facilitateinterpretationof estimatedfactorloadings. Notethat thistransformationhasnoeffectonthe R2for eachassetthatisreportedinTable3.

-7are generallyon theorderoftento fifteenpercent,implyingthatmacroeconomicshocksaccountfor a small.butnon-zeroportionof assetreturnvariation. Chen,Roll,andRoss(1986),King,Sentana, andWahdwani(1990),andRodrigues(1995)havedocumentedsubstantiallygreaterexplanatorypower formonthlyreturns. Howeverthesepapersuseabroadersetof “economicvariables”,includingsuch measuresthatare basedon eithermonthlyassetreturnsor verysimilarvariables,suchas aggregate stockreturns,interestratespreads,andchangesincommodityprices. Becausethegoalhere isto introducevariablesthatare notalreadybeingemployedinriskmanagementmodels,wedonot considerassetreturnsandfinancialpricesforthesetof candidatestatevariables. Anexceptionis madefor thecallmoneyrate,becauseitisarguablyan instrumentof monetarypolicyratherthan freelydeterminedby marketforces. (In addition,itsovernightmaturityissubstantiallyshorterthan ourmonthlysamplinginterval,) 4. EstimatingFactorHeteroskedasticityandConditionalMoments Here, weestimatethemodelthatwassimulatedinsection20fthe paper. Recallthatit allowsreturnsto dependunconditionallyheteroskedasticobservablefactors,buthasnoothersource oftime variationin secondmoments,gThefirststepisto estimateequation(5), theGARCH(1,1) modelsoftheobservableunderlyingfactors,usingthesame(ortho-normalized)macroeconomicnews proxiesas intheregressionsdescribedintheprevioussection. Theparameterestimatesappearin Table4. Therightmostcolumnshowstheresultsofa likelihoodratiotestagainstthenullhypothesis ofhomoskedasticity.Notethatonlyinthree cases(theJapaneseinflationshockandtheU.S, and Japanesemonetarypolicyshocks)isthenullrejectedwith95 percentconfidence. 8Becausethe focusof this paper is on secondmoments,we will ignorethe possibilityoftime variationinmeanreturns,throughout,

-8- Next.equation(1) isestimated.9PanelA of Table5 showsresultsfor a system includingreturnson two assets-- theUK20-yearbondanda Japaneseequityindex. Mostof the estimatesare notsignificantlydifferentfromzero PanelBshowsestimatesofthetwo alternative modelsalsodiscussedinsection2: GARCH(l,l) inthereturnsthemselvesandhomoskedasticity. PanelC showstheloglikelihoodfunctionfor thethreeestimatedmodels. Notethatthefactor-based modelhasa substantiallyhigherloglikelihoodthanthe “GARCHin returns”model-- albeit,using27 parametersinsteadof 7-- butneithermodelcausesthenullof homoskedasticityto be rejectedina likelihoodratio(LR)test.’” PanelD comparestheprojectedconditionalmomentsfromtheestimated modelsto thecrossproductsof subsequentlyrealizedreturns. Thecorrelationcoefficientsimplythat neitherestimatedmodelhas anyabilityto forecastchangesinsecondmoments. Apparently,to the extentthereisconditionalheteroskedasticityinthesetwoassetreturns, neitherspecificationhasbeen very successfulincapturingit. Table6 runsthroughtheanalogousempiricalexercisesfor a l-year German governmentbondand a 7-yearGermangovernmentbond,the(dollar)returnson whichare very highlycorrelated. As inTable5, neitherof thetwomodelsoftimevariationinassetreturnmoments appearsto fitthedatawell. Onecannotrejectthenullhypothesisof homoskedasticityagainsteither alternative.and neithermodel(PanelD)exhibitsanyabilityto forecastvariancesandcovariancesof returns. The failureto rejecthomoskedasticityinfavorof thefactor-basedheteroskedastic modelmaybe inpart becausethe factor-basedmodelhastoomanyparameters-- mostofthefactor gNotethatestimating(5)and(1)sequentiallyyieldsthesameresultsasestimatingthetwoequations simultaneously,becauseu and W are orthogonalby construction. It wouldhavebeen betterto have estimatedtheorthogonalizationofW, (5),and(1)allsimultaneously,althoughitwouldhavebeenmuch moredifficultcomputatiomlly. 10Itispossiblethatamoreparsimoniousfactor-basedmodelwouldhaveledto rejectionofthenull hypothesisof homoskedasticity,butwewantedto avoiddatamining. 1

-9loadingsreportedinpanelA oftables5 and6are notsignificantlydifferentfromzero. In table5, onlythefactorsassociatedwithU.S. inflation,U.S. andJapaneseoutputgrowth,andtheJapanesecall moneyrateare significantwith95percentconfidence(fora one-sidedtest)for oneof thetworeturns. Accordingly,table7 reportsresultsfroma moreparsimoniousfactor-basedmodeloftheUK20-year bondandtheJapaneseequityindex,usingonlythesefourfactors.’] As canbe inferredfrompanelB ofthetable,an LRtestnowrejectshomoskedasticityin favorof thefactor-basedmodel. However,as showninpanelC. themodelstilldoesa poorjob ofexplainingtimevariationin secondmoments-itsforecastoftheUKbondreturnvarianceisstillnegativelycorrelatedwithsquaredreturns,andthe otherreportedcorrelationsare stillcloseto zero. Thus,our LRrejectionappearsto derivefromthe estimatedconditionalvarianceinthefactormodelbeingonaveragelowerthantheunconditional variance,ratherthanthetimevariationinthemodel’sconditionalvariancetrackingthetrue conditionalvariance. In otherwords,theintroductionofthemacroeconomicvariablesintothemodel contributesby improvingourestimatesoftheconditionalmeansof thereturns,nottheconditioml secondmoments. Table8 showstheresultsofan analogousmodelingstrategyfor theGermanl-year and 7-yearbonds. Homoskedasticityisrejectedinfavorofa moreparsimoniousfactor-basedmodelthat usesonlythefactorsassociatedwithU.S. inflationandindustrialproduction. Thisfactor-basedmodel producesforecastsofsecondmomentsthatare weaklycorrelatedwiththecross-productsof returns, butnoneofthesecorrelationsare significantlydifferentfromzerowith95percentconfidence.12 11Notethattheothereightmacroeconomicvariablesareinvolvedintheconstructionofthesefactors, boththroughtheVARequationsandthroughtheorthogonalization. 12Givenasamplesizeof 120,acorrelationcoefficientof .15orgreaterwouldbesignificantwith95 percentconfidence.

-10- 5. DirectState-Dependenceof SecondMoments Accordingly.itmaybe worthexploringothertypesof conditionalheteroskedasticity involvingmacroeconomicvariables. Anotherwayto modelstatedependentheteroskedasticity involvescreatinga lowertriangularmatrixL whoseelements(onor belowthediagonal)are linear finctions ofparticularstatevariables:]3 (6) and m L =0 for i<j (iJ-)J AssumingL isfullrank,thematrixLL’ issymmetricandpositivedefinite,andthusadmissibleas the variance-covariancematrixof returns: (8) Var,-l(Zt)= L,L: TheparametersenteringLcan beestimatedby numericalmaximumlikelihoodmethods.t4 One advantageofa specificationof thissortisthatit ismoreflexiblethanspecificationsbasedon 13Thisstrategydifferssomewhatfromtheapproachof McQueenandRoley(1993),whousedthe levelof industrialproductionrelativeto trend to define three discreteeconomicstates. They found statedependence in the sensitivityof U.S. stock returns to news containedin consumerprice index announcements. 14NotethatthematrixL isnotauniquelowertriangulardecompositionofthevariance-covariance matrix. In particular,anyrow ofL couldbemultipliedby -1 and(8)wouldstillhold. (In fact, itcan be shown that this type of transformationyields all possible lower triangulardecompositionsof a symmetricpositive-definitematrix.) However,sincetheparametersofinterestarethevariance-covariance matrixitself.ratherthanL. the indeterminacyof L isnotimportant.

-11univariateGARCHmodelsinthenatureof state-dependenceincovariancethatitpermits. The biggestdrawbackisthenumberofparametersthatmustbe estimated. Fortunately,thisdisadvantage wastemperedintheapplicationswetriedby a well-behavedlikelihoodfimctionthatwaswell-suited for numericalestimationviaquadraticapproximation. Tables9 and 10showestimatesof modelsof thissort, usingunemploymentratesas the statevariables. Thestatevariablesare laggedtwomonths,so thatthedataispublicatthebeginning of theperiodoverwhichreturnsaremeasured.15Theupperpanelof eachtablecontainstheestimated secondmomentsoftheassetreturns,whenthestatevariablesattheirsamplemean. Subsequently,the tablesshowtheincrementaleffect,relativeto thebaselinecaseof PanelA, of raisingoneofthe unemploymentratesby onepercentagepoint. (Becausetherelationbetweenthemomentsandthe statevariablesisquadratic.theeffectof a twopercentagepointincreasegenerallyisnottwiceas much.) In bothcases,thenullhypothesisofhomoskedasticityis rejectedagainstthisalternative. NotethatU.S. bondreturns-- bothshortandlong-- are estimatedtobebothmorevolatileandmore highlycorrelatedwithotherassetreturnswhentheU.S. unemploymentrateishigh. The increasein thevariancesis significantwith95percentconfidence. Thissuggeststhat, whentheeconomyis weak,thereismoreuncertaintyaboutthefuturecourseof realinterestrates, inflation,or both-- bad timesareuncertaintimes. PanelC of Table 10showsa similarbutweaker(andnotstatisticallysignificant)effect onthevarianceoftheU.K. 10-yearbondreturn(butnoton itscorrelationwithotherassetreturns) whenBritishunemploymentishigh. Interestingly,theeffectsofunemploymentinthetwocountries onthevolatilityof thebilateralexchangerate(reflectedinthedollarreturnonthe l-periodsterling Euro-instrument)are estimatedto beofoppositesign,althoughneitherissignificantlydifferentfrom ‘5Likeresultswereobtainedwithaone-monthlagandwithnolag. Thesimilarityintheestimates underthesealternativesisnotsurprising,giventhattheunemploymentratemovesslowlyovertime.

-12zero. Notealsothatnoneof theoff-diagonalelementsinthelower panelare significantlydifferent fromzero (inthestatisticalsense). Ifthetrue parametersare zero. thisimpliesthatan adequate measureoftheeffectof a changeintheunemploymentrate onthecovariancebetweentwoasset returnscanbe computedfromtheeffectson thevariancesofthetwoassetreturns. Anotherwaythatonecancomparetheperformanceof themodelswithdirect state-dependenceinsecondmomentsto theGARCHmodelsisthroughthecorrelationbetweenthe conditionalsecondmomentsthemodelsproduceandtheactualcross-productsof returns. In general, wemeasurecorrelationssomewhatlargerthanthosereportedinTables5, 6, 7, and 8. For the 3-returnand l-statemodelof Table9, all6 correlationsare positive,rangingfrom5 percent(forthe varianceof the l-year bondreturn)to 11percent(forthevarianceof the 10-yearbondreturn). For the6-returnand2-statemodelofTable 10,20 of21 correlationsare positive,ruining ashighas 28 percentfor thecovariancebetweentheU.S. l-year bondand 10-yearbondreturns. 6. ConclusionsandFuturePossibilities Theresultsdescribedintheprevioussectionimplythat, insomecases, modelsthat allowtheconditionalvarianceof onesetof variablesto dependdirectlyon laggedvaluesof a second setof variablescanprovidebetterforecastsof secondmomentsthanGARCHmodels. Thesemodels are non-linearand involvea largenumberofparameters. Thus, futureworkmightexplorealternative estimationmethodsto thestraightforwardnumericalmaximumlikelihoodmethodusedhere. In addition,itwouldbe interestingto explorewhetherour findingthatbondreturnvolatilityis increasing intheunemploymentrate is robustacrossa broadersampleof countries. It alsowouldbe worthwhile to considerwhatsortof economicmodelswouldbeconsistentwiththisstylizedfact. Wefoundlesssupportfor modelsthatinvolvelinearrelationsbetweenmacroeconomic newsvariablesandassetreturns. Althoughthemoreparsimoniousmodelswe triedfit thedata

I -13somewhatbetterthana simplehomoskedasticmodelof returns,wefoundlittleor no forecasting powerfor secondmoments. Theweakresultsdo notnecessarilymeanthatobservablestatevariables are notan importantdeterminantof assetpricecovariances-- itmaybe simplythata VARat a monthlyfrequencyisan inferiorwayto measurenews. Analternativewouldbe to measureshocksin themacrovariablesrelativeto surveyexpectationsoftheirvaluessometimebeforetheannouncement. Thiswouldhavethefurtheradvantageofmakingitfeasibleto applythetechniqueto higherfrequency data.

-14- Appendix: Constructionof MonthlyBondReturnsfromYieldData onboti We assumethatthe reportedannualizedn-monthyieldto maturity(Y~,~appliesto a hypothetical n-monthbondwithmonthlycouponsthatistradingatpar. For a bondwitha facevalueofone, the monthlycouponwouldbe: ) Y1 Z - ~ c= l+———n,t nJ 100 Alsodefinethemonthlydiscountfactorpertainingto thestatedannualizedyieldtomaturity: [1 Y+ &inJ= 1 + — 1;, Becausethebondistradingatpar andit isassumedto haveunitfacevalue,itspriceisone. The presentvalueofthecashflowfromthebond,discountedat itsyield,mustequaltheprice: n 1 = h!fntn+ cnJ ~M k 9 nJ k=] In orderto computethecapitalgaincomponentof theone-monthreturnon thebond?we will approximatethecurrentyieldto maturityofthepreviousmonth’shypotheticaln-monthparbondwith monthlycoupons(nowa bondwitha maturityof n-1months)withthereportedn-monthyield(Y.,t). Thusitspricemaybe writtenas: Thereturnon thebondbetween(t-1)andt takesintoaccountboththecouponpaymentandthechange intheprice. Expressedinpercent,thereturn is: R = 100(CnJ-~+ ‘n-1,1- 1, n,t Notethatifthen-monthyieldisunchanged,the returnwillconsistonlyof itsincome(coupon) component. l

-15- .. . we dls-t bonds Givenitsamuaiizedyieldto maturity,thepriceofan n-monthpurediscountbond(withunitface value)at (t-1)is [) Y P =1+ n,r-1 + n,t-1 100 Assumingthatithasthesameyieldto maturityas then-monthbond,a discountbondmaturinginn-1 monthsattimet hastheprice: [1‘%)‘%%=);[’ P 11-l,t = Thereturnonthebondbetween(t-1)andt reflectsonlythechangein theprice. Expressedin percent,thereturnis: P - Pnt-l R = 100 ‘-*” ‘ n,t P n?-1 Givenannualinterestpaymentsof X, thepriceofa consolofunitfacevalueisan inversefunctionof itsquotedannualyield: P =+ -,t t Thereturn(inpercentunits)frommontht-1tomontht comprisesa pricechangecomponentanda monthlyinterestpayment: Pt - Pt-l)-Yt-l Y - Yt Yt-l Rt = 100 = 100 ‘-1 P 12 Yt + 12 [ t-1 + ( )-

-16- References Amrner.John(1993),“MacroeconomicRiskandAssetPricing: Estimatingthe APTwithObservable Factors,”~e D-n P~ 448. FederalReserveBoard. Bollerslev,Tim(1986), “GeneralizedAuto-RegressiveConditionalHeteroskedasticity”, ~ tric~31:307-327. Campbell,JohnY. andJohn Ammer(1993),“WhatMovestheStockandBondMarkets? A Variance Decompositionfor Long-TermAssetReturns,”-I of FI_ 48:3-37. Chan, K.C., AndrewKarolyi,andReneStulz(1994), “GlobalFinancialMarketsand theRisk Premiumon U.S. Equity,”~ 4074. Chen, Naifu. RichardRoll.andSteveRoss(1986), “EconomicForcesandtheStockMarkets,” . ~ 59:383-403. MervynKing,EnriqueSentana,andSushilWadhwani(1990), “AHeteroskedasticFactorModelof AssetReturnsandRiskPremiawithTime-VaryingVolatility: An Applicationto SixteenWorldStock . . Markets,”~1 _ts GroupDisc-n P~ 80, LondonSchoolof Economics. MeQueen,GrantandVanceRoley(1993), “StockPrices, News, and BusinessConditions”~view of 1Studiw 6:683-707. Rodrigues,Anthony(1995),“WhyDo VolatilitiesSometimesMoveTogether?”paperpreparedfor joint centralbankconference“RiskMeasurementandSystemicRisk,”Washington,DC, November 16-17,1995.

-17- Table1: SimulatedGARCHin ObservableFactorsandModelEstimates A. truemodel: GARCH(1,1)in2 observablefactors I parameter truevalue estimated c, 0.20 0.18 al 0.40 0.36 I d, 0.40 0.47 I C* 0.20 0.27 I I I az I 0.40 0.29 I dz 0.40 I 0.47 a II I I B 1,00 1.05 11 I B 1.00 1.02 1,2 I I B 1.00 1.01 2,1 I B -1.00 -1.02 2,2 I I~ 02”, I 0.25 0.23 I CJ2U2 I 0.25 0.26 I B. alternativemodels: GARCH(1,1)inreturnsandhomoskedasticreturns parameterestimate GARCHinreturnsmodel homoskedasticmodel c, 0.45 2.47 al 0.61 d, 0.21 C2 0.47 2.34 az 0.57 dz 0.22 -0.02 0.01 P1,2 Note: Seetextformodeldefinitions.

-18- Table2: Actual EstimatedPortfolio“Value-at-Risk”forSimulatedData A. portfolioweights= (O100)’ statistic trueVaR I GARCHin factors GARCHinreturns homoskedastic meanVaR 230.1 I 236.1 239.0 251.5 std.dev. of VaR I 63.7 67.8 53.7 0 p withtrueVaR I 100.0% I 99.5% 83.5% o mean error , 7.3 27.3 50.8 maximumerror 52.7 193.2 513.3 B. portfolioweights= (100O)’ statistic true VaR I GARCHin factors GARCHinreturns homoskedastic meanVaR 230.1 237.0 242.7 258.8 std.dev.ofVaR I 63.7 68.5 51.0 0 p withtrueVaR I 100.0% 99.4% 82.1% o mean Ierror I 8.2 30.2 54.8 maximumerror I 53.8 216.2 506.1 c. portfolioweights= (100100)’ I statistic trueVaR I GARCHin factors I GARCHin returns I homoskedastic meanVaR 312.9 319.8 339.3 362.9 std.dev. of VaR 86.0 98.8 66.0 0 p withtrueVaR 100.0% 99.9% 41.5% o mean error I 9.9 60.3 85.4 maximumerror I 80.0 616.8 465.7 .

-19- Table2 (continued) D. portfolioweights= (100-100)’ statistic I trueVaR I GARCHinfectors I GARCHinreturns I hornoskedastic meanVaR I 325.4 I 335.1 I 345.0 I 358.7 std.dev.ofVaR I 130.1 I 134.6 I 67.0 I o I p withtrueVaR I 100.0% I 99.4% 81.9% I o mean IerrorI I I 13.4 I 60.6 I 93.4 maximumerror I I 120.1 I 619.8 I 1149.3 Note: Modelsare asdefinedinthetext. Eachspecificationwasestimatedoverthefirst 1000 observationsofsimulateddataandthenpropagatedoverthesubsequent1000observations. “Valueat risk”isdefinedasthe5 percentlefttailof theone-periodreturn.

-20- Table 3: LinearRegressionsof AssetReturnsonMacroeconomicFactors proportionof varianceexplained,1/85- 12/94 assetreturn R-squared U.S. equityindex Ill 0.05 U.S. 30-yearbond 0.11 U.S. 10-yearbond o.I1 Ill U.S. 7-yearbond 0.12 U.S. 5-yearbond Ill 0.12 I* U.S. 3-yearbond U.S. 2-yearbond U.S. l-year bond Ill 0.09 U.S. 12-monthEurorate Ill 0.13 Ill U.S. 6-monthEurorate 0.08 U.S. 3-monthEurorate 0.02 U.S. l-monthEurorate 0.01 Japanequityindex 0.13 Japan 10-yearbond 0.07 Japan 12-yearcorporate 0.09 H+ Japan 5-yearcorporate Japan 12-monthEurorate Japan 6-monthEurorate Ill 0.09 Japan 3-monthEurorate 0.09 Japan l-monthEurorate 0.10 Note: Thefactorsare theresiduais (orthogonalizedinthe order listed)froma VAR(6)in U.S. inflation,Japaneseinflation,Germaninflation,U.K. inflation,U.S. industrialproduction,Japanese industrialproduction,Germanindustrialproduction,U.K. industrialproduction,the U.S. callmoney rate(federalfunds),theJapanesecallmoneyrate, theGermancallmoneyrate, andthe U.K. call moneyrate. Returnsare end-of-month,inpercent-per-monthunits,andmeasuredindollars.

-21- Table3 (continued) Ill assetreturn R-squared German equityindex 0.07 German 7-yearbond 0.10 German 5-yearbond Ill 0.10 German 3-yearbond 0.10 German 2-yearbond 0.10 German l-year bond o.11 German12-monthEurorate Ill o.11 German 6-monthEurorate 0.11 German 3-monthEurorate Ill 0.10 German l-monthEurorate 0.10 U.K. equityindex Ill 0.09 Ill U.K. 3.5% 0.15 COIISO] U.K. 20-yearbond Ill 0.14 U.K. 10-yearbond 0.13 U.K. 5-yearbond 0.12 U.K. 12-monthEurorate 0.14 U.K. 6-monthEurorate 0.13 U.K. 3-monthEurorate 0.13 U.K. l-monthEurorate 0.13 Note: Thefactorsare theresiduals(orthogonalizedintheorderlisted)froma VAR(6)inU.S. inflation,Japaneseinflation,Germaninflation,U.K. inflation,U.S. industrialproduction,Japanese industrialproduction,Germanindustrialproduction,U.K. industrialproduction,theU.S. callmoney rate(federalfinds), theJapanesecallmoneyrate, theGermancallmoneyrate, andtheU.K. call moneyrate. Returnsare end-of-month,inpercent-per-monthunits,andmeasuredindollars.

-22- Table4: Estimatesof UnivariateGARCHModelsofMacroeconomicFactors c a d LRtest: X2(2) U.S. CPI 0.86 0.25 -0.11 2.68 inflation (0.38) (0.37) (0.02) JapanCPI 0.67 -0.09 0.46 10.94 inflation (0.19) (0.12) (o.19) GermanCPI 0.67 0.24 0.09 0.79 inflation ~ (0.41) (0.40) (0.11) U.K. CPI 1.14 -0.07 -0.07 0.98 inflation (0.93) (0.93) (0.06) U.S. industrial 1.91 -0.79 -0.09 1.10 outputgrowth (0.35) (o.14) (0.09) Japan industrial 1.34 -0.25 -0.10 2.80 outputgrowth (0.46) (0.45) (0.05) Germanindustrial 0.35 0.78 -0.13 5.31 outputgrowth (o.11) (o.12) (0.03) U.K. industrial 1.58 -0.51 -0.06 0.60 outputgrowth (0.52) (0.47) (0.06) U.S. overnight 1.68 -0.82 0.08 7.59 interestrate (0.25) (0012) (0.02) Japanovernight 1.81 -0.98 0.12 10.17 interestrate (0.25) (0.03) (0.04) Germanovernight 0.52 0.56 -0.07 3.93 interestrate (0.44) (0.44) (0.01) U.K. overnight 1.83 -0.90 0.08 1.91 interestrate (0.28) (0.12) (0.07) (standarderrors inparentheses) .

-23- Table5: ThreeModelsof BivariateAssetReturnVariance-Covariance: UK20-yearbondandJapaneseequityindex,1/85- 12/94 A. linearfunctionof 12univariateGARCH(1,1)observablefactom(homoskedasticresidual) coefficient UK20-year Japaneseequity bond index Us. CPI -0.37 -1.47 inflation (0.26) (0.67) JapanCPI -0.23 -0.64 inflation (0.26) (0.67) GermanCPI -0.38 0.41 inflation (0.26) (0.67) U.K. CPI 0.02 0.58 inflation (0.26) (0.67) U.S. industrial -0.09 -1.54 outputgrowth (0.26) (0.67) Japanindustrial -0.50 -0.15 outputgrowth (0.26) (0.67) Germanindustrial 0.11 -0.76 outputgrowth (0.26) (0.67) U.K. industrial -0.14 -0.55 outputgrowth (0.26) (0.67) U.S. overnight 0.26 0.91 interestrate (0.26) (0.67) Japanovernight -0.77 -0.06 interestrate (0.26) (0.67) Germanovernight -0.19 -0.47 interestrate (0.26) (0.67) U.K. overnight 0,14 -0.77 interestrate (0.26) (0.67) residualvariance 7.98 53.13 residualcovariance 4.08 (standarderrors inparentheses)

I 24- Table5 (continued) B. alternatives:bivariateGARCH(1,1)withconstantcorrelationandhomoskedasticmodel 7 parameterestimate GARCH(1,1) in returns homoskedasticmodel c, 1.67 9.32 al 0.69 d, 0.13 c~ 23.49 61.14 a2 0.39 dl 0.23 0.21 0.21 P1,2 C. loglikelihoodfunctions \ model factor-basedheteroskedastic GARCHin returns homoskedastic in Sf -701.2 -714.9 -718.5 D. correlationsof predictedsecondmomentswithactualcross-productsof assetreturns moment factor-basedheteroskedastic GARCHin returns homoskedastic UKbond -0.18 0.05 0 variance equity 0.01 -0.06 0 variance co- I -0.03 0.06 0 variance

-25- Table6: ThreeModelsof BivariateAssetReturnVariance-Covariance: Germanl-yearbondandGerman7-yearbond,1/85- 12/94 A. linearfunctionof 12univariateGARCH(1,1)observablefactors(homoskedasticresidual) coefficient Germanl-year German7-year bond bond Us. CPI -0.47 -0.61 inflation (0.31) (0.34) JapanCPI 0.23 0.20 inflation (0.31) (0.34) GermanCPI -0.26 -0.43 inflation (0.31) (0.34) U.K. CPI -0.05 -0.09 inflation (0.31) (0.34) U.S. industrial -0.82 -0.82 outputgrowth (0.31) (0.34) Japanindustrial 0.13 0.07 outputgrowth (0.31) (0.34) Germanindustrial 0.40 0.36 outputgrowth (0.31) (0.34) U.K. industrial -0.12 -0.19 outputgrowth (0.31) (0.34) U.S. overnight 0.21 0.25 interestrate (0.31) (0,34) Japanovernight 0.31 0.13 interestrate (0.31) (0.34) Germanovernight -0.06 -0.08 interestrate (0.31) (0.34) U.K. overnight -0.22 -0.17 interestrate (0.31) (0.34) residualvariance 11.87 14.26 residualcovariance 12.57 (standarderrors inparentheses)

-26- Table6 (continued) B. alternatives:bivariateGARCH(1,1)withconstantcorrelationandhomoskedasticmodel parameterestimate GARCH(1,1) in returns homoskedasticmodel c, 16.59 13.27 al -0.34 d, 0.10 c; 28.77 15.82 a2 -0.86 d2 0.06 0.97 0.97 %,2 C. loglikelihoodfunctions model factor-basedheteroskedastic GARCHin returns homoskedastic InSf -486.0 -494.9 498.5 D. correlationsof predictedsecondmomentswithactualcross-productsof assetreturns I I i moment factor-basedheteroskedastic GARCHin returns homoskedastic l-year 0.01 0.02 0 variance 7-year 0.05 0.02 0 variance co- 0.07 -0.02 0 variance

-27- Table7: Factor-BasedModelof BivariateAssetReturnVariance-Covariance: UK20-yearbondandJapaneseequityindex,1/85- 12/94 A. linearfunctionof 4 univariateGARCH(1,1)observablefactors(homoskedasticresidual) coefficient UK20-year Japaneseequity bond index Us. CPI -0.37 -1.47 inflation (0.26) (0.69) U.S. industrial -0.09 -1.54 , outputgrowth (0.26) (0.69) Japanindustrial -0,50 -0.15 outputgrowth (0.26) (0.69) Japanovernight -0.77 -0.06 interestrate (0.26) (0.69) residualvariance 8.34 56.57 residualcovariance 4.29 (standarderrors inparentheses) B. loglikelihoodfunctions model factor-basedheteroskedastic GARCHin returns homoskedastic in g -707.5 -714.9 -718.5 C. correlationsof predictedsecondmomentswithactualcross-productsof assetreturns moment factor-basedheteroskedastic GARCHinreturns homoskedastic UKbond -0.21 0.05 0 variance equity 0.08 -0.06 0 variance co- -0.05 0.06 0 variance

-28- Table8: Factor-Based Modelof BivariateAssetReturn Variance-Covariance: German l-yearbondandGerman7-yearbond, 1/85- 12/94 A. linearfunctionof 2 univanateGARCH(1,1)observablefactors(homoskedasticresidual) coefficient German l-year German7-year bond bond Us. CPI -0.47 -0.61 inflation (0.32) (0.35) U.S. industrial -0.82 -0.82 outputgrowth (0.32) (0.35) residualvariance 12.37 14.78 residualcovariance 13.04 (standarderrors inparentheses) B. loglikelihoodfunctions , model factor-basedheteroskedastic GARCHin returns homoskedastid in $f -493.4 -494.9 -498.5 C. correlationsof predictedsecondmomentswithactualcross-productsof assetreturns moment factor-basedheteroskedastic GARCHin returns homoskedastic l-year 0.11 0.02 0 variance 7-year 0.13 0.02 0 variance co- I 0 0.14 -0.02 variance

-29- Table9: ReturnCovarianceMatrixasDirectFunctionof ObservableStateVariable (1/71-12/94) A. secondmomentsof returnswithstatevariable(USunemployment)atsamplemean variancesondiagonal; us 10-year l-year correlations(iowerL equity USbond USbond A) USequity 19.29 (1,61’) I (%I (wI) 10-yearUSbond 0.19 l-year USbond 0.76 I 0.38 I (0.06) I (0,02) (0.03) B. effectonsecondmomentsof USunemploymentbeing1percentagepointhigher(versusA.) variancesondiagonal; us 10-year l-year (lowerL equity USbond USbond COITdatlOIIS A) USequity -1,63 (1.01) 10-yearUSbond I ::;, I & I II C. likelihoodratiotestversusnullhypothesisofhomoskedasticity LR(distributedXZwith 6 degreesof freedomundernull) = 15.0 Note: Variance-covariancematrixisconstructedas LL’whereeachelementofthelowertriangular matrixL isa linearfi.mctionofthestatevariables. Returnsare measuredindollars.

-30- Table10: ReturnCovarianceMatrixas DirectFunctionof ObservableStates,1/80-12/94 A. secondmomentsof returnswithstatevariables(USandUKunemployment)at samplemeans varianceson diagonal; 10-year 10-year l-month us UK l-year COrdMiOIUi (1OWX L A) USbond UKbond UKEuro equity equity USbond 10-yearUSbond 7.25 (1.06) 10-yearUKbond 0.41 7.42 (0.09) (1.03) l-monthUKEuro 0.15 0.16 13.66 (o.12) (o.12) (1.83) USequity 0.34 0.27 -0.03 18.37 (0.11) (o.10) (o.13) (2.96) UKequity 0.26 0.50 0.55 0.54 35.62 (o.11) (0.10) (0.09) (0.08) (5.23) l-year USbond 0.80 0.33 0.15 0.16 0.14 0.38 (0.05) (o.11) (o.15) (o.14) (0.11) (0.05) B. effectonsecondmomentsof USunemploymentbeing1 percentagepointhigher(versusA.) variancesondiagonal; 10-year 10-year l-month us UK l-year correlations(lowerL USbond UKbond UKEuro equity equity USbond A) 10-yearUSbond 1.77 (1.75) I (wI (%I I I 10-yearUKbond I ::::I)i::)I H)I I l-monthUKEuro USequity 0.02 -0.02 0.05 -1.31 (0.10) (0,08) (0.11) (2,09) UKequity 0.13 -0.05 0.10 -0.15 -1.64 (o.11) (o.10) (0.08) (o.10) (5.94) 0.23 (0.09) l

-31- Table10(continued) C. effectonsecondmomentsofUKunemploymentbeing1percentagepointhigher(versusA.) us variancesondiagonal; 10-year 10-year l-month UK l-year correlations(lowerL A) US bond UKbond UKEuro equity equity USbond 10-yearUSbond -0.77 (0.45) 10-yearUKbond -0.01 0.33 (0.05) (0.61) l-monthUKEuro -0.02 -0.02 1.51 (0.06) (0.06) (1.24) USequity 0.01 -0.05 -0.06 0.76 (0.07) (0.06) (0.07) (1.87) UKequity 0.03 -0.02 -0.05 0.01 -0.19 (0.07) (0.06) (0.06) (0.04) (3.16) l-year USbond -0.03 -0.00 -0.05 -0.01 -0.01 -0.12 (0.02) (0.07) (0.07) (0.09) (0.07) (0.03) D. likelihoodratiotestversusnullhypothesisofhomoskedasticity LR(distributed~zwith42degreesof freedomundernuIl)= 131.0 Note: Variance-covariancematrixisconstructedas LL’whereeachelementofthelowertriangular matrixL isa linearfunctionofthestatevariables, Returnsaremeasuredindollars.