美联储-公司债券市场的相互关联性（英）

FinanceandEconomicsDiscussionSeries

FederalReserveBoard,Washington,D.C.

ISSN1936-2854(Print)

ISSN2767-3898(Online)

InterconnectednessintheCorporateBondMarket

CelsoBrunetti,MatthewCarl,JacobGerszten,ChiaraScotti,ChaeheeShin

2024-066

Pleasecitethispaperas:

Brunetti,Celso,MatthewCarl,JacobGerszten,ChiaraScotti,andChaeheeShin(2024).“InterconnectednessintheCorporateBondMarket,”FinanceandEconomicsDiscus-sionSeries2024-066.Washington:BoardofGovernorsoftheFederalReserveSystem,

/10.17016/FEDS.2024.066

.

NOTE:StafworkingpapersintheFinanceandEconomicsDiscussionSeries(FEDS)arepreliminarymaterialscirculatedtostimulatediscussionandcriticalcomment.TheanalysisandconclusionssetfortharethoseoftheauthorsanddonotindicateconcurrencebyothermembersoftheresearchstafortheBoardofGovernors.ReferencesinpublicationstotheFinanceandEconomicsDiscussionSeries(otherthanacknowledgement)shouldbeclearedwiththeauthor(s)toprotectthetentativecharacterofthesepapers.

InterconnectednessintheCorporateBondMarket

CELSOBRUNETTIMATTHEWCARLJACOBGERSZTENCHIARASCOTTICHAEHEESHIN*

April2024

ABSTRACT

Doesinterconnectednessimprovemarketquality?Yes.

Wedevelopanalternativenetworkstructure,theassetsnetwork:assetsareconnectediftheyareheldbythesameinvestors.Weuseseverallargedatasetstobuildtheassetsnetworkforthecorporatebondmarket.ThroughcarefulidentificationstrategiesbasedontheCOVID-19shockand“fallenangels,”wefindthatinterconnectednessimprovesmarketqualityespeciallyduringstressperiods.Ourfindingscontributetothedebateontheroleofinterconnectednessinfinancialmarketsandshowthathighlyintercon-nectedcorporatebondsallowforrisksharingandrequirealowercompensationforrisk.

Keywords:financialstability,interconnectedness,institutionalinvestors,bigdataJELClassificationCodes:C13,C55,C58,G1

*CelsoBrunettiandChaeheeShinarewiththeFederalReserveBoard.ChiaraScottiisattheBankofItaly.MatthewCarlisaPh.D.studentattheUniversityofWisconsin-Madison.JacobGersztenisattheUniversityofMichiganLawSchool.Theauthorscanbereachedviaemailat

celso.brunetti@,

mcarl@,

gersztenj@,

chiara.scotti@bancaditalia.it,

and

chaehee.shin@

.WethankNathanFoley-FisherandseminarandconferenceparticipantsattheFederalReserveBoard,QatarCenterforGlobalBankingandFinance,King’sCollegeLondon,UniversityofWisconsin,2022IAAEconference,2022EEA-ESEMconference,and2022InternationalRiskManagementConferenceforhelpfulcomments.TheviewsexpressedinthisarticlearethoseoftheauthorsandnotnecessarilyoftheFederalReserveSystem.WethankGeneKangforexcellentresearchassistance.

1

1.Introduction

Thenotionof“interconnectedness”becamepopularwiththeGreatFinancialCrisis(GFC).Linkagesbetweenmarketsandinstitutionsaswellastheramificationsoffinancialdistresstotherealeconomyputinterconnectednessinthelimelight.Infact,interconnectednessisnowpartoftheregulatoryframework

.1

Interconnectednessisasophisticatedconcept:toolittleinterconnectedness(sparsenetwork)mayimpedemarketfunctioning,andtoomuchinterconnectedness(densenetwork)mayexacerbatetheefectsofashock.Thegoalofthispaperistostudythelinkagesbetweeninterconnectednessandmarketquality.

Wechoosethecorporatebondmarketasoursandbox.Thismarkethasgrownsubstan-tiallyinrecentyearsandrepresentsanimportantsourceoffundingforthecorporatesec-tor

.2

Itisdominatedbyinstitutionalinvestors,whichallowsustomaplinkagesamongthelargestmarketplayerssuchasinsurancecompaniesandmutualfunds.Comparedtoequitymarkets,itsliquidityandmarketfunctioninginthecorporatebondmarkethavebeenundermuchscrutiny,leadingtoarapiddevelopmentoftheliterature(see,

Boyarchenkoetal.,

2021;

Dick-NielsenandRossi,

2019;

TrebbiandXiao,

2019

).Finally,thecorporatebondmarketex-periencedsignificantdisruptionsinMarch2020becauseoftheCOVID-19pandemic(see,

HaddadandMuir,

2021

).Hence,studyinghowinterconnectednessrelatestomarketqualityinbothtranquiltimesaswellasintimesofdistressisparticularlyinformative.

Inthispaper,wedevelopanalternativeandcomplementarynetworkstructure—theas-setsnetwork—whichmirrorsthetraditionalnotionofaportfoliosimilaritynetwork.Thisnewnetworkconstructisderivedattheassetlevelandisbasedontheideathatassetsareinterconnectediftheyareheldbythesameinvestors.

Themoretraditionalportfoliosimilaritynetworkcapturesspilloverefectsduetoover-lappingportfolios:twofinancialinstitutionswithsimilarportfoliosarelinkedbecauseashocktoonefinancialinstitutionhasrepercussionsontheotherfinancialinstitutionthrough

1InterconnectednessisoneofthecriteriausedbytheFinancialStabilityBoardtodesignateGlobalSystemi-callyImportantBanks(G-SIBs).IntheU.S.,interconnectednessisalsousedbytheFinancialStabilityOversightCouncil(FSOC)todesignatenonbankSystemicallyImportantFinancialInstitutions(SIFIs).

2Ithasreachedover$15trillionasofQ42023–seeFinancialAccountoftheU.S.

2

theircommonassetholdings(see,

Cacciolietal.,

2015

).Incontrast,ournetworkconstructcaptureslinkagesamongassetsgiventhattheseassetsareheldbyseveralfinancialinsti-tutions.Theemphasisofournetworkisontheassetsasopposedtofinancialinstitutions.Studyingthenetworkofassetsisfundamentallyimportantforseveralreasons.First,itallowsustoinvestigatehowinterconnectednessoffinancialassetsislinkedtoasset-specificcharac-teristicssuchasliquidityandvolatilityand,moregenerally,tomarketquality.Second,thereisagrowingliteratureoninstitutionalassetpricing;ournetworkstructureprovidesanotherlensthroughwhichtostudyhowassetsheldbyseveralinstitutions—ourassetsnetwork—impactthepricingprocess.Thisisparticularlyrelevantinourframeworkwhichanalyzescorporatebondholdingsbylargeinvestors.Third,assetsinterconnectednessprovidesanal-ternativeanduniqueperspectiveonhowfinancialassetsarelinkedincontrasttocorrelationanalysis.

DieboldandYılmaz

(2014)and

Billioetal.

(2012)constructassetsnetworksbased

onthevariance-covariancematrixofreturns.Ournetworkbuildsedgesbasedonwhetherassetsareheldbycommoninvestors,andis,therefore,potentiallymoreaccuratebecauseitdoesnotrequireestimatinganymomentofthereturnsdistribution(see,

Adamicetal.,

2017

).Finally,thetraditionaloverlappingportfolionetworkputsemphasisonfinancialinstitutionsandismoresuitedforanentity-basedsupervisoryapproach,whileourassetsnetworkmayprovideusefulforanactivity-basedapproachforregulation

.3

Wefirstfocusontheinterconnectednessofthecorporatebondmarket,leveragingtherichinformationavailableintheThomsonReuterseMAXXdatabase,whichcontainsdataoncorporatebondholdingsattheinstitutionalinvestor-bond-year-quarterlevel.Webuildanetworkofcorporatebondsandmeasuretheirinterconnectednessusingcosinesimilarity.Asexpected,wefindthatbondsissuedbylargefirmsarepartoftheportfolioofmanyinvestorsandformthecoreofournetworks,whilesmallerbondissuerscomprisetheperiphery—implyingthatonlyafewinvestorsholdthesebonds.WealsomatchtheinterconnectednessmeasuresofcorporatebondswiththeTRACEdatabasethathassecurity-leveldataoncor-poratebondtradingvolume,liquidity,andvolatility.

Thenewinterconnectednessconstructandthecomplexityofourdataallowustousearichpanelregressionanalysistoinvestigatethelinkbetweeninterconnectednessandspread,

3See,

Borioetal.

(2022)

.

3

liquidity,andvolatilityofcorporatebonds.Wefindthatthehighertheinterconnectednessofanasset—meaningthattheassetiscommontomanyinvestors’portfolios—theloweritsspreadandthehigheritsliquidity.Thisresulthighlightsthat,asexpected,corporatebondsthatareheldacrossseveralportfoliosrequirealowercompensationforriskandaremoreliq-uid.Thisrelationis,however,afectedbymarketconditions.Weexploretheheterogeneousefectsofinterconnectednessthroughouttheconditionaldistributionoftheresponsevari-ables(spreads,liquidity,andvolatility),whilecontrollingforbondcharacteristics,throughapaneldataquantileregression.Wefindthattherelationwehavejusthighlightedisstrongerwhenafinancialassetisunderstress,i.e.thespreadandliquidityofanassetareintheuppertailoftheirconditionaldistributions.Altogether,higherinterconnectednessisassociatedwithlowerspreadsandvolatility,andhigherliquidityinnormalmarketconditions(meanefect)andtheseresultsarestrongerwhenmarketsaredistressed(asshownbyquantileregressions)

.4

Whiletheanalysisthusfardocumentslinkagesbetweeninterconnectednessandmarketqualitymeasures,weareinterestedindeterminingcausality.Thatis,weareinterestedinunderstandingwhetherhigherinterconnectednessreducesspreads,increasesliquidity,andtamesvolatility.Thisisafundamentalissue.Ontheonehand,

AllenandGale

(2000)develops

amodelinwhichcompletenetworks(highinterconnectedness)helpmitigatetheefectsofashockthroughrisksharingand,therefore,arebeneficialtofinancialstability.Ontheother,

Acemogluetal.

(2015)showsthatiftheshockistoolarge,highinterconnectedness

propagatestheshockleadingtoamorefragilefinancialsystem.TheCOVID-19outbreakrepresentsalargeexogenousshock.Following

Hassanetal.

(2023),weseparatebondsissued

byfirmsafectedbytheshockfrombondsissuedbyfirmsnotafectedbythepandemic.Wefindthattheefectsoftheshockaremitigatedwhenbondsissuedbyfirmsexposedtothepandemicarehighlyinterconnectedtobondsissuedbyfirmsnotexposedtotheshock—spreaddecreasesandliquidityincreases.Ourresultsindicatethatinterconnectednessenablesrisksharingand,onnet,isbeneficialtothecorporatebondmarket.

4Ourresultsarerobusttodiferentmodelspecificationsandtoseveralcontrolsthatareknowntoafectcorporatebondpricingdynamics,suchasinvestorconcentrationandthenumberofuniqueinvestors.

4

Tocorroboratetheseresultswealsolookat“fallenangels:”bondsdowngradedfrominvestmentgradetohighyield.WeselectbondswithsimilarcharacteristicsandacreditratingofBBB-(thelowestcreditratingintheinvestmentgradecategory).Onlysomeofthesebondsaredowngradedinthenextperiod.Sincethebondsweconsiderinthisexercisehavesimilarcharacteristics,thebifurcationbetweenfallenangelsandnon-fallenangelsisplausiblyexogenouswithintheshorttimewindowweareconsidering—theanalysisonlyconsiderstwoperiods,beforeandafterthedowngrade

.5

Ourresultsshowthataonestandarddeviationincreaseininterconnectednessofafallenangelsubstantiallydecreasesspreadsandincreasesliquidity.

Overall,ourfindingsestablishthathigherlevelsofinterconnectednessarepositivelylinkedtomarketquality.Moreover,thelinkbetweeninterconnectednessandmarketqual-itychangesovertimewhenmarketconditionsalsochange.Importantly,thislinkisstrongerduringperiodsofmarketdistress.Finally,interconnectednessisparticularlyimportantwhenlargenegativeshockshitfinancialmarkets(COVID-19)andwhenmajorcorporateeventsoccur(fallenangels).Inthesecrisissituations,interconnectedness,throughrisksharing,promotesmarketfunctioning.

Ourpapercontributestoseveralstrandsoftheliterature.First,wecontributetothein-terconnectednessliterature.Networksinfinancehavebeenmappedusingthreemaintech-niques:(i)correlationnetworks,inwhichedgesbetweenfinancialinstitutionsarebasedonestimatesofthevariance-covariancematrixofpubliclyavailabledata,suchasassetreturns(see,

Billioetal.,

2012;

DieboldandYılmaz,

2014

);(ii)physicalnetworks,inwhichedgescapturecontractualagreementsamongcounterparties,suchasinterbanktransactions(see,

Brunettietal.,

2019

);and(iii)commonholdingsnetworks,inwhichinvestorsareconnectediftheyholdsimilarportfolios(see,

Cacciolietal.,

2015;

Greenwoodetal.,

2015;

Cetorelli

etal.,

2023

).Inthispaper,weproposeanewapproachofmappingfinancialnetworkswhichmirrorsthenotionofoverlappingportfolios,andwhichwecalltheassetsnetworkorin-vestorsimilaritynetwork.Similartoourapproach,

AntónandPolk

(2014)connectstocks

commonlyheldbymutualfunds.Theirgoalistostudyhowcommonownershipafectsthe

5Känzig

(2021

)proposesanidentificationstrategybasedonpreciselyselectingthetimeframeofspecificevents,whichforusisthedowngrade.

5

cross-sectionalcorrelationintherateofreturns.Ourfocusisinsteadonthenetworkstruc-tureanditsproperties.Weareinterestedinfullyunderstandingtheinterconnectednessofthenewnetworkandhowitevolvesbothovertimeandindiferentmarketconditions.Infact,ourgoalistoprovideanewandalternativemappingforfinancialnetworks.

Second,weconnecttotheemergingliteratureoninstitutionaldemand-basedassetpric-ing.Onestrandofthisliteraturestudiestheroleofintermediariesinassetpricing,suchasin

HaddadandMuir

(2021)and

Heetal.

(2017)

.Anotherstrandoftheliteratureexaminesthe

roleofinstitutionalholdersinassetpricingand,inparticular,thecompositionofinstitutionalinvestorsasapotentialstatevariableinthecorporatebondmarket.Forinstance,

Ben-David

etal.

(2021)showhowtherisingconcentrationofholdingsbyinstitutionalinvestorsafects

stockvolatilityandpriceine般ciency,

LiandYu

(2022)findthatinvestorconcentrationis

relatedtobondliquidity,and

LiandYu

(2021)and

Bretscheretal.

(2022)analyzehowthe

compositionofinstitutionalinvestorsrelatestocorporatebondmarketqualities.

Corelletal.

(2023)alsolookatEuropeancorporatebondstofindhowconvenienceyieldscouldvaryby

diferingdemandsfromvariousinstitutionalinvestors.Overall,thisliteraturetracksbacktothedemand-basedassetpricingapproachof

KoijenandYogo

(2019)

.Wecontributetothisemergingareabyshowingthattheinterconnectednessofanassetplaysanimportantroleincorporatebondmarkets.

Finally,werelatetotherecentfinancialstabilityliteraturethattriestodeterminewhetherhighinterconnectednessisavulnerabilityoravirtueofthefinancialsystem.Con且ictingviewsexistintheliterature,from

AllenandGale

(2000),whofindinterconnectednesstobe

avirtue,tomorerecentempiricalworksfindingevidenceforfinanciallinkagesandoverlap-pingholdingsofassetstobeacontagionorfiresalesmechanism(

Allenetal.,

2012;

Duarte

andEisenbach,

2021;

Falatoetal.,

2021;

Greenwoodetal.,

2015

,amongothers).Somewhereinbetweenthesetwocon且ictingviews,manyrecentworksstudythenon-monotonictradeofbetweencontagionandrisksharing,socialoptimalityofinterconnectedness,andconditionsforwhichonetypeofnetworkisbetterthananother(

Acemogluetal.,

2015;

Cabralesetal.,

2017;

Elliottetal.,

2014,

2021;

Gofman,

2017

,amongothers).Ourresultsprovideevidenceofacausalefect:interconnectednessimprovesmarketquality.

6

Thepaperisorganizedasfollows.Section

2

describesournovelnetworkapproach,illus-tratingthebuildingblocksoftheasset-basednetworkofinvestorsimilarity.Section

3

sum-marizesthewealthofdatathatweuseintheempiricalinvestigation.Section

4

describestheresultingmeasuresthatweuseintheanalysis.Section

5

explainstheregressionframeworkanditsresults,includingthoseforthequantileregressions.Section

6

examinesthecausallinkagesbetweeninterconnectednessandmarketmarketquality.Section

7

concludes.

2.NetworkApproach

Thereareseveralwaystoconstructnetworksinfinance.Thethreemainapproachescanbebrie且ydescribedas:(i)correlationnetworks,whicharebasedonestimatesofthevariance-covariancematrixofpubliclyavailabledatasuchasassetreturns(see,

Billioetal.,

2012;

DieboldandYılmaz,

2014

);

6

(ii)physicalnetworks,whichre且ectcontractualagree-mentsbetweencounterpartiesandcaptureimportantaspectsofrisksuchasconterpartyrisk(see,

Brunettietal.,

2019

);and(iii)overlappingportfoliosnetworks,whichconnectinvestorsthroughtheircommonholdings(see,

Cacciolietal.,

2015;

Greenwoodetal.,

2015

).Inthispaper,weproposeanewapproachofmappingfinancialnetworkswhichparallelsthenotionofoverlappingportfolios,butthatdrawsedgesbetweenassetsratherthaninstitutions.

Thestartingpointisabipartitenetworkwithtwosetsofnodes:financialinstitutionsorinvestors(퐼s)andfinancialassets(퐴s).AsshowninFigure

1a,

ifafinancialinstitutionholdsanassetinitsportfolio,thereisanedgebetweenthatassetandthatfinancialinstitution.Forexample,becauseinvestor퐼1holdsasset퐴1,thereisanedgebetween퐼1and퐴1.Thetraditionalnetworkofoverlappingportfolios,orcommonassetholdings,impliesthatsince퐴1isheldalsoby퐼2and퐼3,allthreeinvestorsareinterconnectedthroughtheircommonholdingsof퐴1.Similarly,because퐴2isheldby퐼2and퐼3,thereisalinkbetweenthesetwoinvestors(see

Baruccaetal.,

2021

).

Wederiveanalternativenovelnetworkstructureattheassetlevel,basedontheideathattwoassets,퐴1and퐴2,areinterconnectediftheyareheldbythesameinvestor.InFigure

1b,

6Arelatedapproachadoptsquantileregressionanalyses,see

Andoetal.

(2021

)and

Härdleetal.

(2016)

.

7

퐴1and퐴3areinterconnectedbecausebothassetsareheldintheportfolioofinvestor퐼3.Similarly,퐴1and퐴2arealsointerconnectedsincetheyareheldbyinvestors퐼2and퐼3.Infact,퐴1and퐴2areinterconnectedtoahigherextentthan퐴1and퐴3becausetheseassetssharetwooverlappinginvestors.

Thisasset-basednetworkallowsustoexamineimportantefectsofinterconnectedness

acrossfinancialassets.InFigure

1b,

interconnectednessbetween퐴1and퐴3capturesand

quantifiesthefollowingmechanism.Supposeashockhits퐴1(e.g.,downgradetojunk)andreducesitsmarketvalue.Thisshockwillthennegativelyimpacttheperformanceoftheportfoliosofallinvestors,퐼1,퐼2,and퐼3sincetheyallhold퐴1.Investorswillbeforcedtore-balancetheirportfoliostoraisemorecapitalorliquidity(e.g.,inthecaseofmutualfunds,tomeetredemptions)andthere-balancingwilltriggerachangeinholdingsofboth퐴2and퐴3becausethere-balancinginvestorsalsohold퐴2(퐼2and퐼3)and퐴3(퐼1).

InFigure

1b,

ourmeasureofinterconnectednessbetween퐴1and퐴2isstrongerthanthatbetween퐴1and퐴3becausetwoinvestors(퐼2and퐼3)holdtheseassetsasopposedtojustoneinvestorfor퐴1and퐴3.Thisnetworkfeatureimpliesthatthesameinitialshockon퐴1(and/or퐼2and/or퐼3)willlikelyspilloverto퐴2toagreaterextentthanitwillto퐴3,sinceboth퐼2and퐼3willre-balancetheirportfoliosasopposedtojustoneinvestor(퐼3)re-balancinginthecaseof퐴1and퐴3.

Noticethatthenotionofoverlappinginvestorsforabondis,however,diferentthanthesheernumberofinvestorsholdingthebond.InFigure

1b,

퐴1isheldbythehighestnumberofinvestors(퐼1,퐼2,and퐼3),followedby퐴2,whichisheldbytwoinvestors(퐼2and퐼3).However,퐴1hasthesamenumberofoverlappinginvestors—anddegreeofinterconnectedness—as퐴2.Thisarisesbecauseoutofthethreeinvestorsholding퐴1,oneinvestor(퐼1)doesnotinvestinanyotherassets,therebyeliminatingitspropensityto“overlap”withotherinvestors.Ingeneral,aswehaveillustratedinthisexample,itispossiblethatassetswithfewerinvestorsaremoreinterconnected(havemoreoverlappinginvestors)thanotherassetswithmorein-vestors.

Inwhatfollows,wedescribeournotionoftheasset-basednetworkinmoredetailandhighlightthenetworkmeasureusedintheanalysis.

8

2.1.NetworkofFinancialAssetsandInstitutions

Westartbydenotingthenetworkoffinancialassetsandfinancialinstitutionsas푄=(퐴,퐼,E),where퐴=퐴1,퐴2,...,퐴푆isthesetofnodescorrespondingtofinancialassets(corpo-ratebondsonly,inourcase),퐼=퐼1,퐼2,...,퐼푁representsthesetoffinancialinstitutions,andEisa푆×푁matrixrepresentingtheamount,퐸푖,푘,heldby퐼푘in퐴푖:

퐼1

퐼2

···

퐼푁

퐴1

퐸11

퐸12

···

퐸1푁

1

푉퐴

(1)

퐴푆

퐸푆1

퐸푆2

···

퐸푆푁

푆

푉퐴

1

푉퐼

2

푉퐼

···

푉퐼

푁

Summingacrosscolumnsgivesthetotalamountofsecurity푖heldbythesystem(in-vestorsinourdata),푉푖퐴,knownasthestrengthofthenetwork:

푉푖퐴=퐸푖,푘,(2)

andsummingacrossrowsproducesthetotalamountinvestedbyinvestor푘inallassets,푉.

Dependingonthescopeoftheanalysis,퐸푖,푘couldbenormalizedbythetotalissuedamountofasset푖outstandingorby푉푖퐴.

O

WedefineasEthecorrespondingadjacencymatrix

9

퐼1

퐼2

···

퐼푁

퐴1

O

퐸11

O

퐸12

···

O

퐸1푁

퐷퐴

1

(3)

퐴푆

퐸푆1

퐸푆2

···

퐸푆푁

퐷퐴

푆

퐷퐼

1

퐷퐼

2

···

퐷퐼

푁

O

wherethegenericelement퐸푖,푘=1if퐸푖,푘>푞andzerootherwise.Theparameter푞denotesathresholdandintraditionalnetworkanalysis푞=0

.7

Similartobefore,thesumacrosscolumnsgivesthetotalnumberoffinancialinstitutions

holdingsecurity푖,퐷,knownasnetworkdegree,

퐷퐸푘,(4)

andthesumacrossrowsgeneratesthetotalnumberofassetsinvestor푘hasinvestedin,퐷.

2.2.Asset-basedNetworkofInvestorSimilarity

Thenetworkwefocusoninthispaperisderivedfromthenetworkoffinancialassetsandfinancialinstitutions푄describedintheprevioussectionandcapturesinterconnectednessamongassetsbasedonwhethertheassetsbelongtothesameportfolios.

Wedefinethenetworkoffinancialassetsas푂퐴=(퐴,P퐴),where퐴={퐴1,퐴2,...,퐴푆}representsthesetofassets,andP퐴isthematrixmeasuringsimilaritiesofassetsintermsofinvestors.Severaldistancemeasuresexisttoquantifysimilarities(see,

Newman,

2010;

Delpinietal.,

2013;

Baruccaetal.,

2021;

Brunettietal.,

2023)

.Inthispaper,weusethenotion

7Giventherichnessofourdata,wecouldalsoadopt푞>0toselectthestrongestlinksamongnodes.

10

ofcosinesimilarity(ordistance)tomeasureinterconnectednessbetweenanypairofassets푖

and푗∈{1,...,푆}:

푁OO

Ⅱ퐸푖ⅡⅡ퐸푗Ⅱ

푃(5)

O

whereⅡ퐸푖Ⅱisthenormofthevectorofinvestorsholdingasset푖(see,Getmanskyetal.,2016;

Baruccaetal.,

2021

)and푃,thecosinesimilarity,capturesthedistancebetweentwonon-

zerovectorsofaninner-productspace

.8

Finally,foreachasset푖,weaggregateitspair-wiseinterconnectednesswithallotherassets푗in푆where푗≠푖and푖,푗∈{1,...,푆}toproduceanasset-levelmeasureofintercon-nectednessinthisnetwork:

Wenormalizeasset-levelinterconnectednessby(푆一1)*푁,where푆isthetotalnumberofassetsand푁isthetotalnumberofinvestors,toaccountforthefactthatthenumberoffinancialassetsandinstitutionschangeovertimeinourdata.

8Therecanbealternativedefinitionsofsimilarity.Oneoptionistousesimplecountsofthenumberof

OO

portfoliostwoassetsarepartofandhenceusethefollowingdefinitionfor푃퐴:푃퐴=퐸(퐸)T.Anotheroptionistocomputethesemeasuresusingtheparamountsheldbyinvestors푘asafractionoftheamountoutstandingofassets푘,therebycapturinganintensivemarginmeasureofinvestorsimilarity.Inthiscase,wedivideeachelement퐸푖,푘from(

3

)by퐼푠푠푢푒푎푚표푢푛푡표푢푡푠푡푎푛푑푖푛푔푖,andusethisnewadjacencymatrixdirectlytocomputesimilaritymeasures푃퐴.Wetestedtheaforementionedtwoalternativemeasuresandfoundthattheresultsweresimilartothoseusingcosinesimilarityontheextensivemarginofinvestors’holdings.Yetanothermeasure

ofsimilaritycanbederivedfromthenotionofEuclideandistance,namely,푃퐸푘一퐸,푘

However,wedidnotusethismeasureinouranalysisduetothesparsityofthenetworkinoursample.

11

2.3.AnExample:HowShocksPropagateThroughanAssetsNetwork

Anexamplemayhelptoexplaintheseconcepts.Considerthenetworkbelowconsistingofonlythreeassetsandthreeinvestors,wheretheentriesintheleftmatrixrepresentthe

dollaramountofeachassetheldbyeachinvestor.Thisnetworkcanberepresentedbythe。

adjacencymatrixE푒푥푎푚푝푙푒ontheright:

퐴3001퐴3001

。

Thetop-leftcellofthematrixE푒푥푎푚푝푙푒isequalto1becauseinvestor퐼1hasasset퐴1inherportfolio,while0in푐푒푙푙(2,1)indicatesthatinvestor퐼1hasnotinvestedinasset퐴2.Usingequation(

5

),wecanthencomputethecosinesimilaritymetricforanytwopairsofassets:

퐴30.580.71-

Accordingly,followingequation(

6

),thevectorofinterconnectednessmeasurescorrespond-

ingtoPis:

Themagnitudesofasset-levelinterconnectednessshownin퐼퐶indicatethat퐴2,

hasthehighestlevelofinterconnectednessinthenetwork,followedby퐴1and퐴3,whichhasthelowestinterconnectedness.Wehighlightthatthattheinterconnectednessmeasurecapturesanon-linearaspectofthenetworkbeyondthesimplenumberoffirmsinvestingin

12

eachasset,i.e.,theassets’degreeinthebipartitegraph.Forexample,although퐴1isheldbyallinvestorsand퐴2isonlyheldbytwoinvestors,퐴2isthemostcentralnodeinthisnetwork.퐴2’scentralitygivesrisetoahigherasset-levelinterconnectednessrelativeto퐴1.

Whichassetexperiencestheinitialshockplaysafundamentalrole