欧洲央行-投资者异质性与大规模资产购买（英）

EUROPEANCENTRALBANK

EUROSYSTEM

WorkingPaperSeries

JohannesBreckenfelder,VeronicaDeFalcoInvestorheterogeneityandlarge-scaleassetpurchases

No2938

Disclaimer:ThispapershouldnotbereportedasrepresentingtheviewsoftheEuropeanCentralBank(ECB).TheviewsexpressedarethoseoftheauthorsanddonotnecessarilyreflectthoseoftheECB.

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Abstract

Large-ScaleAssetPurchasescanimpactthepriceofsecuritiesdirectly,whensecuri-tiesaretargetedbythecentralbank,orindirectlythroughportfoliore-balancingofprivateinvestors.Wequantifyboththedirectandtheportfoliore-balancingimpact,emphasizingtheroleofinvestorheterogeneity.Weuseproprietarysecurity-leveldataonassetholdingsofdifferentinvestors.Wemeasurethedirectimpactonsecuritylevel,findingthatitissmallerforsecuritiespredominantlyheldbymoreprice-elasticinvestors,fundsandbanks.Comparingasecurityatthe90thpercentileoftheinvestorelasticitydistributiontoasecurityatthe10thpercentile,thepriceimpactisonlytwo-thirdsaslarge.Toassesstheportfoliore-balancingeffects,weconstructanovelshift-shareinstrumenttomeasureinvestors’quasi-exogenousexposuretocentralbankpurchases,basedoninvestors’holdingsofeligiblesecuritiesbeforetheQEprogramwasannounced.Weshowthatfundsandbanksselleligiblesecuritiestothecentralbankandre-balancetheirportfoliostowardsineligiblesecurities,withinvestorsex-antemoreexposedtocentralbankpurchasesre-balancingmore.Usingdetailedholdingsdataofmutualfunds,weestimatethatforeacheurosoldtothecentralbank,theaveragefundallocates88centstoineligibleassetsand12centstoothereligibleassetsthatthecentralbankdoesnotbuyinthattimeperiod.Thepriceofineligiblesecuritiesheldbymoreexposedfundsincreasescomparedtothoseheldbylessexposedfunds,underscoringtheportfoliore-balancingchannelatwork.

Keywords:financialintermediaries,mutualfunds,centralbank,assetpricing

JELClassification:E52,E58,G11,G12,G23

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Non-technicalsummary

TocombattheGreatRecessionfollowingthe2008financialcrisisandthemorerecentPan-demicRecession,centralbanksaroundtheglobepurchasedunprecedentedquantitiesofsecurities.Largelyasaresultofassetpurchases(alsoknownasQuantitativeEasingorQE),balancesheetsofmajorcentralbanksballoonedinsize,constituting-atthepeak-between40%(USFederalReserve,BankofEngland)and130%oftheircountry’sGDP(BankofJapan).Bypurchasinglargequantitiesofassets,thecentralbanksaimtoaffectassetpricesthroughouttheeconomy.Indeed,accordingtotheso-calledportfoliore-balancingchannel,QEoperateswellbeyondtheeffectonpricesthecentralbankpurchasesdirectly:italsoaffectsotherassetpricesasinvestorssellingassetstothecentralbankre-balanceintoothersecuritiesnoteligibleforcentralbankpurchases.Atthesametime,theeffectivenessofQEasapolicyisthesubjectofintensedisputeamongeconomistsasitisdifficulttoseparatetheeffectsofQEfromothercontemporaneouseventsandpolicymeasures.Theidentifica-tionandquantificationoftheparticularchannelsthroughwhichQEoperatesissubjecttoseveralchallenges,asthedecisionsofinvestorstosellsecuritiestothecentralbankandto

re-balancetheirportfoliosareaffectedbyahostoffactors.

Inthispaper,wesetouttodealwiththesechallengesandtoquantifyboththedirectandindirecteffectsoflarge-scaleassetpurchases.Weuseproprietarysecurity-leveldataonassetholdingsofmajorinvestors(banks,insurances,pensionfunds,mutualfunds)toanalyzehowdifferentinvestorsadjusttheirholdingsofthesamesecurityinthesametimeperiod,inresponsetocentralbankpurchases.WearguethatQEeffectsdependcruciallyonthecompositionofinvestorssellingassetstothecentralbank.Ourdataallowsustomeasurethedirecteffectsonsecuritylevelandweshowthatcentralbankpurchaseshavesmallereffectsonyieldsforsecuritiespredominantlyheldbymoreprice-elasticinvestors-mutualfundsandbanks.Comparingasecurityatthe90thpercentileoftheinvestorelasticitydistributionto

asecurityatthe10thpercentile,thepriceimpactisonlytwo-thirdsaslarge.

Toassesstheportfoliore-balancingeffects,weconstructanovelshift-shareinstrumenttomeasureinvestors’quasi-exogenousexposuretocentralbankpurchases.Theinstrument

isbasedoninvestors’holdingsofsecuritieseligibleforpurchasesbeforetheQEprogram

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wasannounced.Weshowthatfundsandbanks–majorinvestorsinthebondmarkets-selleligiblesecuritiestothecentralbankandre-balancetheirportfoliostowardsineligiblesecurities,withinvestorsex-antemoreexposedtocentralbankpurchasesre-balancingmore.Incontrast,insurancecompaniesandpensionfundsarelesselastic,andrespondlesstocentralbankpurchases,andthereforecontributelesstotheportfoliore-balancingchannel.Usingdetailedholdingsdataofmutualfunds,weestimatethatforeacheurosoldtothecentralbank,theaveragefundallocates88centstoineligibleassetsand12centstoothereligibleassetsthatthecentralbankdoesnotbuyinthattimeperiod.Thepricesofineligiblesecuritiesheldbyfundsmoreexposedtocentralbankpurchasesincreasecomparedtothose

heldbylessexposedfunds,underscoringtheportfoliore-balancingchannelatwork.

WereachtheseconclusionsbyanalyzingpurchasesconductedbytheEuropeanCentralBank(ECB)from2015to2022.Oursampleencompassestwomajorprograms:theAssetPurchaseProgram(APP)andthePandemicEmergencyPurchaseProgram(PEPP),whichcombinedamountedtonearly5trillioneurosofpurchases(60percentofeuroareaGDP)bytheendof2022.Assetstargetedundertheseprogramsincludegovernmentbonds,corporatebonds,asset-backedsecurities,andcoveredbonds.WeuseSecuritiesHoldingsStatistics(SHS),anadministrativedatabaseofportfolioholdingsofprivateinvestorscollectedbytheECB.Portfolioholdingsarereportedatthesecurity(ISIN),quarter,andinvestor-typelevel.Inaddition,wemergeSHSwithconfidentialinformationonECBpurchasesatthesecuritylevel.Tostudyspillovereffects,wealsoemploydetailedmutualfund-levelportfolioholdings

dataprovidedbyLipper,amarketdataprovider.

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1Introduction

TocombattheGreatRecessionfollowingthe2008financialcrisisandthemorerecentPan-demicRecession,centralbanksaroundtheglobepurchasedunprecedentedquantitiesofsecurities.Largelyasaresultofassetpurchases(alsoknownasQuantitativeEasingorQE),balancesheetsofmajorcentralbanksballoonedinsize,constituting-atthepeak-between40%(USFederalReserve,BankofEngland)and130%oftheircountry’sGDP(BankofJapan).Bypurchasinglargequantitiesofassets,thecentralbanksaimtoaffectassetpricesthroughouttheeconomy.Indeed,accordingtotheso-calledportfoliore-balancingchannel,QEoperateswellbeyondtheeffectonpricesthecentralbankpurchasesdirectly:italsoaffectsotherassetpricesasinvestorssellingassetstothecentralbankre-balanceintoothersecuritiesnoteligibleforcentralbankpurchases.Atthesametime,theeffectivenessofQEasapolicyisthesubjectofintensedisputeamongeconomistsasitisdifficulttoseparatetheeffectsofQEfromothercontemporaneouseventsandpolicymeasures.Theidentifica-tionandquantificationoftheparticularchannelsthroughwhichQEoperatesissubjecttoseveralchallenges,asthedecisionsofinvestorstosellsecuritiestothecentralbankandto

re-balancetheirportfoliosareaffectedbyahostoffactors.

Inthispaper,wesetouttodealwiththesechallengesandtoquantifyboththedirectandindirecteffectsoflarge-scaleassetpurchases.Weuseproprietarysecurity-leveldataonassetholdingsofmajorinvestors(banks,insurances,pensionfunds,mutualfunds)toanalyzehowdifferentinvestorsadjusttheirholdingsofthesamesecurityinthesametimeperiod,inresponsetocentralbankpurchases.WearguethatQEeffectsdependcruciallyonthecompositionofinvestorssellingassetstothecentralbank.Ourdataallowsustomeasurethedirecteffectsonsecuritylevelandweshowthatcentralbankpurchaseshavesmallereffectsonyieldsforsecuritiespredominantlyheldbymoreprice-elasticinvestors-mutualfundsandbanks.Comparingasecurityatthe90thpercentileoftheinvestorelasticitydistributionto

asecurityatthe10thpercentile,thepriceimpactisonlytwo-thirdsaslarge.

Toassesstheportfoliore-balancingeffects,weconstructanovelshift-shareinstrumenttomeasureinvestors’quasi-exogenousexposuretocentralbankpurchases.Theinstrument

isbasedoninvestors’holdingsofsecuritieseligibleforpurchasesbeforetheQEprogram

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wasannounced.Weshowthatfundsandbanks–majorinvestorsinthebondmarkets-selleligiblesecuritiestothecentralbankandre-balancetheirportfoliostowardsineligiblesecurities,withinvestorsex-antemoreexposedtocentralbankpurchasesre-balancingmore.Incontrast,insurancecompaniesandpensionfundsarelesselastic,andrespondlesstocentralbankpurchases,andthereforecontributelesstotheportfoliore-balancingchannel.Usingdetailedholdingsdataofmutualfunds,weestimatethatforeacheurosoldtothecentralbank,theaveragefundallocates88centstoineligibleassetsand12centstoothereligibleassetsthatthecentralbankdoesnotbuyinthattimeperiod.

1

Thepricesofineligiblesecuritiesheldbyfundsmoreexposedtocentralbankpurchasesincreasecomparedtothose

heldbylessexposedfunds,underscoringtheportfoliore-balancingchannelatwork.

WereachtheseconclusionsbyanalyzingpurchasesconductedbytheEuropeanCentralBank(ECB)from2015to2022.Oursampleencompassestwomajorprograms:theAssetPurchaseProgram(APP)andthePandemicEmergencyPurchaseProgram(PEPP),whichcombinedamountedtonearly5trillioneurosofpurchases(60percentofeuroareaGDP)byend2022.Assetstargetedundertheseprogramsincludegovernmentbonds,corporatebonds,asset-backedsecurities,andcoveredbonds.WeuseSecuritiesHoldingsStatistics(SHS),anadministrativedatabaseofportfolioholdingsofprivateinvestorscollectedbytheECB.Portfolioholdingsarereportedatthesecurity(ISIN),quarter,andinvestor-typelevel.Wegroupinvestortypesinthefollowingcategories:Noneuroareainvestors,plusfiveeuroareainvestortypes-banks,mutualfunds,insurancecompanies,pensionfunds,and“other”,whichcorrespondstosmallerresidualinvestors(government,households).Inaddition,wemergeSHSwithconfidentialinformationonECBpurchasesatthesecuritylevel.Tostudyspillovereffects,wezoominonetypeofinvestors-mutualfunds-forwhichwecanuse

fund-levelportfolioholdingsdataprovidedbyLipper,amarketdataprovider.

Inthefirstpartofthepaper,weassesstheimportanceofinvestorheterogeneityforthedirecteffectsofcentralbankpurchases.Large-scaleassetpurchasesareisomorphictonegativechangesinsupply,astheyeffectivelyreducethesupplyofbondsavailabletoprivate

investorsinthemarket.Inademand-supplyframework,theeffectofagivenpurchaseonthe

1Toarriveatthissplit,wetakethere-balancingcoefficientsfromourregressionanalysisforbotheligibleassetsnotpurchasedandineligibleassets.Weweightheestimatesbytheportfoliocomposition(numberofsecuritiesheld)oftherespectiveassettypesfortheaveragefund.

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priceofonesecuritydependsontheslopeofthedemandcurveforthatasset.Inasettingwithinvestorswithdifferentprice-elasticity,thisslopecandependonthecompositionofinvestorsholdingthebond.Empirically,toestimatetheslopeofthedemandcurve,weneedaplausiblyexogenoussupplyshock.However,inoursetting,supplychangesarenot

necessarilyrandomlyallocated,aspurchasescouldbecorrelatedwithassetcharacteristics.

Toaddressthischallenge,weisolatearandomcomponentofpurchasesand,withthat,estimatethepriceelasticityofeachtypeofinvestor.OuridentificationstrategycomparescentralgovernmentbondspurchasedbytheECBissuedbythesamecountryandwithsimilarresidualmaturity.Byincludingsecurityfixedeffectsandquarter-country-residualmaturityfixedeffects,weabsorbvariationinECBpurchasesthatcouldbecorrelatedwithsecuritycharacteristics.Evenwhencomparingpurchasedbondstoveryclosesubstitutes,purchasesbytheECBreduceyields.WeusethissupplyshockgeneratedbytheECBpurchasestoestimateinvestor-typelevelpriceelasticity.Foreachinvestortype,weregressthesecurityholdingsonthatsecurity’syield,instrumentingtheyieldwithECBpurchases.Investorsoutsideoftheeuroarea,alongwitheuroareabanksandmutualfunds,exhibitapositiveprice-elasticitystatisticallydifferentfromzero.Incontrast,insurancecompanies

andpensionfundsarerelativelymoreinelastic.

Giventheestimatedinvestorelasticities,weconstructasecurity-levelmeasureofthepriceelasticityoftheinvestorbase.Foreachsecurity,wecomputeaweightedaverageoftheinvestor-typelevelelasticityestimates,wheretheweightsarethesharesoftheamountoutstandingheldbyeachinvestortype.Investorcompositionmightbeendogenoustoassetpurchases.Toaddressthisconcern,weexploitquasi-exogenouscross-sectionalvariationininvestorbasecompositionacrosssecurities,measuringtheshareheldbyeachinvestortypepriortotheannouncementandimplementationofassetpurchases.Theweightedelasticitymeasureisbyconstructionhigherforassetsheldex-antemorebymoreelasticinvestors.Withthismeasure,wecanfinallytesthowinvestorcompositionaffectsthedirectimpactof

centralbankpurchases.

Wefindthattheeffectsofpurchasesonpricesaresmallerforsecuritiesthatarepredom-inantlyheldbymoreelasticinvestors,butalsothattheeffectsofpurchasesarenon-linear:

theyincreaseasthestockofaparticularsecurityheldbythecentralbankincreases.Our

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regressionsofbondyieldsonECBpurchasesinteractedwiththeweightedelasticitymeasureshowthatECBpurchasesreduceyieldsbutlesssoforsecuritieswithhigherweightedelas-ticity.Furthermore,weshowthatthehigherthestockofagivensecuritytheECBholds,thelowertheelasticitiesoftheremaininginvestorsholdingthatsecurityareandthelargerthepriceimpactofanadditionalECBpurchaseis.Theseresultshighlightthatheterogeneouspriceelasticitiesofinvestorsmatterforassetprices.Quantitatively,sortingsecuritiesbythepriceelasticityoftheirinvestorbase,wefindthatthedirecteffectsofcentralbankpurchasesforasecurityatthe90thpercentileoftheinvestorelasticitydistributionareonlytwo-thirds

aslargeasforasecurityatthe10thpercentile.

Inthesecondpartofthepaper,weexaminetheindirecteffectsoflarge-scaleassetpur-chases:howthepricesofineligibleassetsareaffectedbyportfoliore-balancing.Intuitively,asthepriceofeligiblesecuritiesincreases,privateinvestorsmayre-balancetheirportfoliotowardsineligibleassets,generatingspillovereffects.Wefirstfocusonquantityre-balancingattheinvestor-securitylevelandthenshowtheeffectsonpricesofineligiblesecurities.Inthispartoftheanalysis,wefocusonmutualfunds,oneinvestortypeweestimatedtobeelastic.Theadvantageofzoominginmutualfundsishavingfund-levelportfolioholdings

data.

Sincemutualfundscouldendogenouslychoosetoholdaportfoliohighlyexposedtocentralbankpurchases,weconstructashift-shareinstrumentofmutualfunds’exposure.Foreachfund,ineachquarter,weconstructameasureofpredictedexposuretoECBpurchases.ThisvariabledependsonhowmuchtheECBbuysofeachsecurityineachquarterandhowmucheachfundownedofthatsecurityattheendof2014(beforethebeginningofthecentralbankpurchaseprogram).Theexogeneityoftheinstrumentreliesontheex-antesharestobequasi-exogenous.Weusethispredictedexposuretoinstrumentfortheamountofeligiblesecuritiesthefundssell.Between2015and2022,weestimatethattheaveragefundsells0.5%ofitseligibleportfolioholdingsduetocentralbankpurchases.Atthesametime,weestimatesubstantialheterogeneityacrossfunds.Weexploitthecross-sectionalheterogeneityacrossfundsinexposureandportfolioallocationofineligiblesecuritiestoestimatespillover

effects.

Forthesameineligiblesecurityinthesamequarter,fundsexantemoreexposedtolarge-

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scaleassetpurchasesre-balancetheirportfoliosmoretowardssuchsecurity.Weregresstheportfolioweightofeachineligiblesecurityeachfundholdsonthefund’sexposuremeasure,instrumentedbythepredictedexposure.Oneconcernisthatineligiblesecuritiesheldbymore-exposedfundscoulddifferfromineligiblesecuritiesheldbyless-exposedfunds:toaddressthis,weuseawithin-securityestimatorbasedon

KhwajaandMian

(2008),including

securitybytimefixedeffectsinouranalysis.Intuitively,by“holdingthesecurityfixed”ateachpointintime,weensurethatsecurity-time-varyingfactorsdonotdrivere-balancing.Weestimatethatforeacheurosoldtothecentralbank,anaveragefundallocates88centstoineligibleassetsand12centstoothereligibleassetsthatthecentralbankdoesnotbuy

inthattimeperiod.

Next,westudythepriceimpactoftheestimatedquantityre-balancing.Sincewefindthatfundswithhigherexposurere-balancemore,wetestwhetherthepricesofsecuritiesheldbymoreexposedfundsincreasecomparedtosecuritiesheldbylessexposedfunds.Theintuitionbehindthishypothesisisthatfundsaremorelikelytore-balancetowardsexist-ingsecuritiesintheirportfolio,soheterogeneityinpreferencesthatgenerateheterogeneous

portfolioallocationsacrossprivateinvestorsinfluencesthepass-throughofthepolicy.

ToidentifythecausaleffectofECBassetpurchasesonthepriceofineligiblesecurities,weconstructashift-shareinstrumentoftheamountpurchasedofeachineligiblesecuritybyfunds.Wecomputethepredictedchangeinthenominalamountforeachsecuritybasedontheinteractionoftwoterms:theportfolioweightofeachineligiblesecurityforeachfundandthequasi-exogenousexposuresoffundstocentralbankpurchases.Theexogeneityoftheshift-shareinstrument,inthiscase,reliesonthe“shifter”,theexposureoffunds,beingexogenous.Theintuitionbehindthisinstrumentisthatweuseapredictedversionofhowmuchineligiblesecuritieswouldbeimpactedbysecond-roundpurchaseflowsiffundswere

re-balancingaccordingtopre-existingportfolioweights.

Inouranalysis,wealsoincludequarter-by-securitycharacteristicsfixedeffectstoaddressthepotentialsortingbetweenfundsandsecurities.Inotherwords,weseeknottocomparesecuritiesforwhichweexpectpricedynamicstodiffer.Intheassessmentofquantityre-balancing,weuseawithin-securityestimatortoaddressthisissue.Tostudytheeffectson

prices,weidentifyasetofobservablecharacteristicsthatyieldsimilarresultstothewithin-

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securityestimator(

KhwajaandMian

(2008),

Chodorow-Reich

(2014b))andincludethose

ascontrolsintheestimationofeffectsonprices.Giventhisstrategy,weestimatetheeffectonpricesofineligiblesecuritiesofsecond-rounddemandshocksinducedbycentralbank

purchases.

Wefindevidenceofspillovereffectsasthepricesofsecuritiesthatmoreexposedfundsholdincreasecomparedtothoseheldbylessexposedfunds.Quantitatively,thepriceimpactonineligibleassetsissmallerbutonthesameorderofmagnitudeasthepriceimpactoneligibleassets.Thisresultshowsthatcentralbankpurchaseshavethepotentialtoaffectthepricesofsecuritiesthattheydonotdirectlytarget.Thisinsightcanhelpassessthecal-ibrationandtargetingofassetpurchaseprogramsbycentralbanksbasedontheirobjectivefunction.Ourresultsshowthatprivateinvestors’preferencesmediatethepropagationofthepolicy.Sinceourestimationreliesonrevealedpreferencesbymeasuringinvestors’ex-anteportfolioallocation,ourmethodologycanbeusedtopredicttheeffectsofpurchasesacross

ineligibleassetsforagivensetoftargetedassets.

RelatedLiterature.Thispaperrelatestoseveralstrandsofliteratureonunconven-

tionalmonetarypolicy,intermediaryanddemand-systemassetpricing.

First,werelatetotheliteratureonlarge-scaleassetpurchasesintheEuroArea,study-ingannouncementeffectsofpurchases(

Andradeetal.

(2016),

Altavillaetal.

(2015))and

portfoliore-balancing(

Bergantetal.

(2020),

Albertazzietal.

(2021)),andestimatingprice

elasticities(

KoijenandYogo

(2019),

DeSantisandHolm-Hadulla

(2020))

.InthecontextoftheFederalReserveoperations,

D’AmicoandKing

(2013)studyflowandstockeffectsof

purchases,

LuandWu

(2023)studyportfoliore-balancingduetotraditionalinterestrate

policy,and

Acharyaetal.

(2022)studyhowFedpurchasesinducedlong-durationIG-focused

investorstorebalancetheirportfoliostowardshigher-yieldingIGbonds.Wecontributetothisliteraturebyestimatingspillovereffectsonassetsthatarenoteligibleforcentralbankpurchaseswhileemphasizingtheroleofinvestorheterogeneityanddevelopinganoveliden-

tificationmethodology.

Wesharewith

Koijenetal.

(2021)thefocusoninvestorheterogeneity.

Koijenetal.

(2021)studythedirecteffectsofQEandestimatedemandelasticitiesofinvestorsfordif-

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ferentnationalissuersofgovernmentbondsintheeuroarea(e.g.,GermanBunds,Frenchgovernmentbondsetc.),followingtheapproachof

KoijenandYogo

(2019).Theyfindthat

demandelasticitiesareheterogeneousacrossinvestors,withforeigninvestorshavingthemostelasticdemand,andinsurancecompaniesandpensionfundshavingtheleastelasticdemand.Weestimatethedirecteffectsofpurchasesonasecurity-ratherthannational-issuer-levelandweshowthatthedirecteffectsofQEarenon-linear,andbecomelargerasthestockof

agivensecurityheldbythecentralbankincreases.

Second,thispapercontributestotheintermediaryassetpricingliterature(

Greenwood

andVayanos

(2010),

GreenwoodandVayanos

(2014),

VayanosandVila

(2021))andgrowing

workondemandsystemassetpricing(

KoijenandYogo

(2019),

GabaixandKoijen

(2021))

thathashighlightedtheroleofinvestors’preferencesindeterminingequilibriumassetprices

Coppola

(2021)

.Ourcontributiontothisliteratureistoanalyzetheroleofinvestors’

preferencesforthepropagationoflargescaleassetpurchases.

Finally,werelatetoabroaderliteratureinmonetarypolicy(

FriedmanandKuttner

(2010)

EggertssonandWoodford

(2003)

Stein

(2012)studyinglargescaleassetpurchases

asapolicytool(Bernanke

(2020),

Vissing-JorgensenandKrishnamurthy

(2011)

Chodorow-

Reich

(2014a)

Woodford

(2016))andtheirrelationshiptotraditionalinterestratepolicy

(Greenwoodetal.

(2023))

.

Theremainderofthepaperisorganizedasfollows.In

section2

,weprovideastylizedsupplyanddemandmodelforoneassetandKheterogeneousinvestors,whichwesubsequenlyextendtoNassets.In

section3

,wedescribethedataweuseintheempiricalsections.In

section4

,wetesthowinvestors’compositionmattersforthedirecteffectsofpurchases.In

section5

,westudyspillovereffectsdrivenbyportfoliore-balancingofmutualfunds.In

section6

,weanalyzeportfoliore-balancingofheterogenousinvestortypesbeyondmutual

funds.

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2Atheoreticalframework

2.1Astylized1assetmodelwithKheterogeneousinvestors

Inthissection,wepresentasimpleoneassetmodeltothinkabouttheeffectsofpurchasesonpriceswhenanassetisheldbymultipleinvesto