美聯(lián)儲(chǔ)-公司債券市場的相互關(guān)聯(lián)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.Throughcarefulidenti?cationstrategiesbasedontheCOVID-19shockand“fallenangels,”we?ndthatinterconnectednessimprovesmarketqualityespeciallyduringstressperiods.Our?ndingscontributetothedebateontheroleofinterconnectednessin?nancialmarketsandshowthathighlyintercon-nectedcorporatebondsallowforrisksharingandrequirealowercompensationforrisk.

Keywords:?nancialstability,interconnectedness,institutionalinvestors,bigdataJELClassi?cationCodes: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).Linkagesbetweenmarketsandinstitutionsaswellastherami?cationsof?nancialdistresstotherealeconomyputinterconnectednessinthelimelight.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-periencedsigni?cantdisruptionsinMarch2020becauseoftheCOVID-19pandemic(see,

HaddadandMuir,

2021

).Hence,studyinghowinterconnectednessrelatestomarketqualityinbothtranquiltimesaswellasintimesofdistressisparticularlyinformative.

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

Themoretraditionalportfoliosimilaritynetworkcapturesspilloverefectsduetoover-lappingportfolios:two?nancialinstitutionswithsimilarportfoliosarelinkedbecauseashocktoone?nancialinstitutionhasrepercussionsontheother?nancialinstitutionthrough

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

2Ithasreachedover$15trillionasofQ42023–seeFinancialAccountoftheU.S.

2

theircommonassetholdings(see,

Cacciolietal.,

2015

).Incontrast,ournetworkconstructcaptureslinkagesamongassetsgiventhattheseassetsareheldbyseveral?nancialinsti-tutions.Theemphasisofournetworkisontheassetsasopposedto?nancialinstitutions.Studyingthenetworkofassetsisfundamentallyimportantforseveralreasons.First,itallowsustoinvestigatehowinterconnectednessof?nancialassetsislinkedtoasset-speci?ccharac-teristicssuchasliquidityandvolatilityand,moregenerally,tomarketquality.Second,thereisagrowingliteratureoninstitutionalassetpricing;ournetworkstructureprovidesanotherlensthroughwhichtostudyhowassetsheldbyseveralinstitutions—ourassetsnetwork—impactthepricingprocess.Thisisparticularlyrelevantinourframeworkwhichanalyzescorporatebondholdingsbylargeinvestors.Third,assetsinterconnectednessprovidesanal-ternativeanduniqueperspectiveonhow?nancialassetsarelinkedincontrasttocorrelationanalysis.

DieboldandY?lmaz

(2014)and

Billioetal.

(2012)constructassetsnetworksbased

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

Adamicetal.,

2017

).Finally,thetraditionaloverlappingportfolionetworkputsemphasison?nancialinstitutionsandismoresuitedforanentity-basedsupervisoryapproach,whileourassetsnetworkmayprovideusefulforanactivity-basedapproachforregulation

.3

We?rstfocusontheinterconnectednessofthecorporatebondmarket,leveragingtherichinformationavailableintheThomsonReuterseMAXXdatabase,whichcontainsdataoncorporatebondholdingsattheinstitutionalinvestor-bond-year-quarterlevel.Webuildanetworkofcorporatebondsandmeasuretheirinterconnectednessusingcosinesimilarity.Asexpected,we?ndthatbondsissuedbylarge?rmsarepartoftheportfolioofmanyinvestorsandformthecoreofournetworks,whilesmallerbondissuerscomprisetheperiphery—implyingthatonlyafewinvestorsholdthesebonds.WealsomatchtheinterconnectednessmeasuresofcorporatebondswiththeTRACEdatabasethathassecurity-leveldataoncor-poratebondtradingvolume,liquidity,andvolatility.

Thenewinterconnectednessconstructandthecomplexityofourdataallowustousearichpanelregressionanalysistoinvestigatethelinkbetweeninterconnectednessandspread,

3See,

Borioetal.

(2022)

.

3

liquidity,andvolatilityofcorporatebonds.We?ndthatthehighertheinterconnectednessofanasset—meaningthattheassetiscommontomanyinvestors’portfolios—theloweritsspreadandthehigheritsliquidity.Thisresulthighlightsthat,asexpected,corporatebondsthatareheldacrossseveralportfoliosrequirealowercompensationforriskandaremoreliq-uid.Thisrelationis,however,afectedbymarketconditions.Weexploretheheterogeneousefectsofinterconnectednessthroughouttheconditionaldistributionoftheresponsevari-ables(spreads,liquidity,andvolatility),whilecontrollingforbondcharacteristics,throughapaneldataquantileregression.We?ndthattherelationwehavejusthighlightedisstrongerwhena?nancialassetisunderstress,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,arebene?cialto?nancialstability.Ontheother,

Acemogluetal.

(2015)showsthatiftheshockistoolarge,highinterconnectedness

propagatestheshockleadingtoamorefragile?nancialsystem.TheCOVID-19outbreakrepresentsalargeexogenousshock.Following

Hassanetal.

(2023),weseparatebondsissued

by?rmsafectedbytheshockfrombondsissuedby?rmsnotafectedbythepandemic.We?ndthattheefectsoftheshockaremitigatedwhenbondsissuedby?rmsexposedtothepandemicarehighlyinterconnectedtobondsissuedby?rmsnotexposedtotheshock—spreaddecreasesandliquidityincreases.Ourresultsindicatethatinterconnectednessenablesrisksharingand,onnet,isbene?cialtothecorporatebondmarket.

4Ourresultsarerobusttodiferentmodelspeci?cationsandtoseveralcontrolsthatareknowntoafectcorporatebondpricingdynamics,suchasinvestorconcentrationandthenumberofuniqueinvestors.

4

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

.5

Ourresultsshowthataonestandarddeviationincreaseininterconnectednessofafallenangelsubstantiallydecreasesspreadsandincreasesliquidity.

Overall,our?ndingsestablishthathigherlevelsofinterconnectednessarepositivelylinkedtomarketquality.Moreover,thelinkbetweeninterconnectednessandmarketqual-itychangesovertimewhenmarketconditionsalsochange.Importantly,thislinkisstrongerduringperiodsofmarketdistress.Finally,interconnectednessisparticularlyimportantwhenlargenegativeshockshit?nancialmarkets(COVID-19)andwhenmajorcorporateeventsoccur(fallenangels).Inthesecrisissituations,interconnectedness,throughrisksharing,promotesmarketfunctioning.

Ourpapercontributestoseveralstrandsoftheliterature.First,wecontributetothein-terconnectednessliterature.Networksin?nancehavebeenmappedusingthreemaintech-niques:(i)correlationnetworks,inwhichedgesbetween?nancialinstitutionsarebasedonestimatesofthevariance-covariancematrixofpubliclyavailabledata,suchasassetreturns(see,

Billioetal.,

2012;

DieboldandY?lmaz,

2014

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

Brunettietal.,

2019

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

Cacciolietal.,

2015;

Greenwoodetal.,

2015;

Cetorelli

etal.,

2023

).Inthispaper,weproposeanewapproachofmapping?nancialnetworkswhichmirrorsthenotionofoverlappingportfolios,andwhichwecalltheassetsnetworkorin-vestorsimilaritynetwork.Similartoourapproach,

AntónandPolk

(2014)connectstocks

commonlyheldbymutualfunds.Theirgoalistostudyhowcommonownershipafectsthe

5K?nzig

(2021

)proposesanidenti?cationstrategybasedonpreciselyselectingthetimeframeofspeci?cevents,whichforusisthedowngrade.

5

cross-sectionalcorrelationintherateofreturns.Ourfocusisinsteadonthenetworkstruc-tureanditsproperties.Weareinterestedinfullyunderstandingtheinterconnectednessofthenewnetworkandhowitevolvesbothovertimeandindiferentmarketconditions.Infact,ourgoalistoprovideanewandalternativemappingfor?nancialnetworks.

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)?ndthatinvestorconcentrationis

relatedtobondliquidity,and

LiandYu

(2021)and

Bretscheretal.

(2022)analyzehowthe

compositionofinstitutionalinvestorsrelatestocorporatebondmarketqualities.

Corelletal.

(2023)alsolookatEuropeancorporatebondsto?ndhowconvenienceyieldscouldvaryby

diferingdemandsfromvariousinstitutionalinvestors.Overall,thisliteraturetracksbacktothedemand-basedassetpricingapproachof

KoijenandYogo

(2019)

.Wecontributetothisemergingareabyshowingthattheinterconnectednessofanassetplaysanimportantroleincorporatebondmarkets.

Finally,werelatetotherecent?nancialstabilityliteraturethattriestodeterminewhetherhighinterconnectednessisavulnerabilityoravirtueofthe?nancialsystem.Con且ictingviewsexistintheliterature,from

AllenandGale

(2000),who?ndinterconnectednesstobe

avirtue,tomorerecentempiricalworks?ndingevidencefor?nanciallinkagesandoverlap-pingholdingsofassetstobeacontagionor?resalesmechanism(

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

Thereareseveralwaystoconstructnetworksin?nance.Thethreemainapproachescanbebrie且ydescribedas:(i)correlationnetworks,whicharebasedonestimatesofthevariance-covariancematrixofpubliclyavailabledatasuchasassetreturns(see,

Billioetal.,

2012;

DieboldandY?lmaz,

2014

);

6

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

Brunettietal.,

2019

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

Cacciolietal.,

2015;

Greenwoodetal.,

2015

).Inthispaper,weproposeanewapproachofmapping?nancialnetworkswhichparallelsthenotionofoverlappingportfolios,butthatdrawsedgesbetweenassetsratherthaninstitutions.

Thestartingpointisabipartitenetworkwithtwosetsofnodes:?nancialinstitutionsorinvestors(?s)and?nancialassets(?s).AsshowninFigure

1a,

ifa?nancialinstitutionholdsanassetinitsportfolio,thereisanedgebetweenthatassetandthat?nancialinstitution.Forexample,becauseinvestor?1holdsasset?1,thereisanedgebetween?1and?1.Thetraditionalnetworkofoverlappingportfolios,orcommonassetholdings,impliesthatsince?1isheldalsoby?2and?3,allthreeinvestorsareinterconnectedthroughtheircommonholdingsof?1.Similarly,because?2isheldby?2and?3,thereisalinkbetweenthesetwoinvestors(see

Baruccaetal.,

2021

).

Wederiveanalternativenovelnetworkstructureattheassetlevel,basedontheideathattwoassets,?1and?2,areinterconnectediftheyareheldbythesameinvestor.InFigure

1b,

6Arelatedapproachadoptsquantileregressionanalyses,see

Andoetal.

(2021

)and

H?rdleetal.

(2016)

.

7

?1and?3areinterconnectedbecausebothassetsareheldintheportfolioofinvestor?3.Similarly,?1and?2arealsointerconnectedsincetheyareheldbyinvestors?2and?3.Infact,?1and?2areinterconnectedtoahigherextentthan?1and?3becausetheseassetssharetwooverlappinginvestors.

Thisasset-basednetworkallowsustoexamineimportantefectsofinterconnectedness

across?nancialassets.InFigure

1b,

interconnectednessbetween?1and?3capturesand

quanti?esthefollowingmechanism.Supposeashockhits?1(e.g.,downgradetojunk)andreducesitsmarketvalue.Thisshockwillthennegativelyimpacttheperformanceoftheportfoliosofallinvestors,?1,?2,and?3sincetheyallhold?1.Investorswillbeforcedtore-balancetheirportfoliostoraisemorecapitalorliquidity(e.g.,inthecaseofmutualfunds,tomeetredemptions)andthere-balancingwilltriggerachangeinholdingsofboth?2and?3becausethere-balancinginvestorsalsohold?2(?2and?3)and?3(?1).

InFigure

1b,

ourmeasureofinterconnectednessbetween?1and?2isstrongerthanthatbetween?1and?3becausetwoinvestors(?2and?3)holdtheseassetsasopposedtojustoneinvestorfor?1and?3.Thisnetworkfeatureimpliesthatthesameinitialshockon?1(and/or?2and/or?3)willlikelyspilloverto?2toagreaterextentthanitwillto?3,sinceboth?2and?3willre-balancetheirportfoliosasopposedtojustoneinvestor(?3)re-balancinginthecaseof?1and?3.

Noticethatthenotionofoverlappinginvestorsforabondis,however,diferentthanthesheernumberofinvestorsholdingthebond.InFigure

1b,

?1isheldbythehighestnumberofinvestors(?1,?2,and?3),followedby?2,whichisheldbytwoinvestors(?2and?3).However,?1hasthesamenumberofoverlappinginvestors—anddegreeofinterconnectedness—as?2.Thisarisesbecauseoutofthethreeinvestorsholding?1,oneinvestor(?1)doesnotinvestinanyotherassets,therebyeliminatingitspropensityto“overlap”withotherinvestors.Ingeneral,aswehaveillustratedinthisexample,itispossiblethatassetswithfewerinvestorsaremoreinterconnected(havemoreoverlappinginvestors)thanotherassetswithmorein-vestors.

Inwhatfollows,wedescribeournotionoftheasset-basednetworkinmoredetailandhighlightthenetworkmeasureusedintheanalysis.

8

2.1.NetworkofFinancialAssetsandInstitutions

Westartbydenotingthenetworkof?nancialassetsand?nancialinstitutionsas?=(?,?,E),where?=?1,?2,...,??isthesetofnodescorrespondingto?nancialassets(corpo-ratebondsonly,inourcase),?=?1,?2,...,??representsthesetof?nancialinstitutions,andEisa?×?matrixrepresentingtheamount,??,?,heldby??in??:

?1

?2

···

??

?1

?11

?12

···

?1?

1

??

(1)

??

??1

??2

···

???

?

??

1

??

2

??

···

??

?

Summingacrosscolumnsgivesthetotalamountofsecurity?heldbythesystem(in-vestorsinourdata),???,knownasthestrengthofthenetwork:

???=??,?,(2)

andsummingacrossrowsproducesthetotalamountinvestedbyinvestor?inallassets,?.

Dependingonthescopeoftheanalysis,??,?couldbenormalizedbythetotalissuedamountofasset?outstandingorby???.

O

Wede?neasEthecorrespondingadjacencymatrix

9

?1

?2

···

??

?1

O

?11

O

?12

···

O

?1?

??

1

(3)

??

??1

??2

···

???

??

?

??

1

??

2

···

??

?

O

wherethegenericelement??,?=1if??,?>?andzerootherwise.Theparameter?denotesathresholdandintraditionalnetworkanalysis?=0

.7

Similartobefore,thesumacrosscolumnsgivesthetotalnumberof?nancialinstitutions

holdingsecurity?,?,knownasnetworkdegree,

???,(4)

andthesumacrossrowsgeneratesthetotalnumberofassetsinvestor?hasinvestedin,?.

2.2.Asset-basedNetworkofInvestorSimilarity

Thenetworkwefocusoninthispaperisderivedfromthenetworkof?nancialassetsand?nancialinstitutions?describedintheprevioussectionandcapturesinterconnectednessamongassetsbasedonwhethertheassetsbelongtothesameportfolios.

Wede?nethenetworkof?nancialassetsas??=(?,P?),where?={?1,?2,...,??}representsthesetofassets,andP?isthematrixmeasuringsimilaritiesofassetsintermsofinvestors.Severaldistancemeasuresexisttoquantifysimilarities(see,

Newman,

2010;

Delpinietal.,

2013;

Baruccaetal.,

2021;

Brunettietal.,

2023)

.Inthispaper,weusethenotion

7Giventherichnessofourdata,wecouldalsoadopt?>0toselectthestrongestlinksamongnodes.

10

ofcosinesimilarity(ordistance)tomeasureinterconnectednessbetweenanypairofassets?

and?∈{1,...,?}:

?OO

Ⅱ??ⅡⅡ??Ⅱ

?(5)

O

whereⅡ??Ⅱisthenormofthevectorofinvestorsholdingasset?(see,Getmanskyetal.,2016;

Baruccaetal.,

2021

)and?,thecosinesimilarity,capturesthedistancebetweentwonon-

zerovectorsofaninner-productspace

.8

Finally,foreachasset?,weaggregateitspair-wiseinterconnectednesswithallotherassets?in?where?≠?and?,?∈{1,...,?}toproduceanasset-levelmeasureofintercon-nectednessinthisnetwork:

Wenormalizeasset-levelinterconnectednessby(?一1)*?,where?isthetotalnumberofassetsand?isthetotalnumberofinvestors,toaccountforthefactthatthenumberof?nancialassetsandinstitutionschangeovertimeinourdata.

8Therecanbealternativede?nitionsofsimilarity.Oneoptionistousesimplecountsofthenumberof

OO

portfoliostwoassetsarepartofandhenceusethefollowingde?nitionfor??:??=?(?)T.Anotheroptionistocomputethesemeasuresusingtheparamountsheldbyinvestors?asafractionoftheamountoutstandingofassets?,therebycapturinganintensivemarginmeasureofinvestorsimilarity.Inthiscase,wedivideeachelement??,?from(

3

)by???????????????????????,andusethisnewadjacencymatrixdirectlytocomputesimilaritymeasures??.Wetestedtheaforementionedtwoalternativemeasuresandfoundthattheresultsweresimilartothoseusingcosinesimilarityontheextensivemarginofinvestors’holdings.Yetanothermeasure

ofsimilaritycanbederivedfromthenotionofEuclideandistance,namely,???一?,?

However,wedidnotusethismeasureinouranalysisduetothesparsityofthenetworkinoursample.

11

2.3.AnExample:HowShocksPropagateThroughanAssetsNetwork

Anexamplemayhelptoexplaintheseconcepts.Considerthenetworkbelowconsistingofonlythreeassetsandthreeinvestors,wheretheentriesintheleftmatrixrepresentthe

dollaramountofeachassetheldbyeachinvestor.Thisnetworkcanberepresentedbythe。

adjacencymatrixE???????ontheright:

?3001?3001

。

Thetop-leftcellofthematrixE???????isequalto1becauseinvestor?1hasasset?1inherportfolio,while0in????(2,1)indicatesthatinvestor?1hasnotinvestedinasset?2.Usingequation(

5

),wecanthencomputethecosinesimilaritymetricforanytwopairsofassets:

?30.580.71-

Accordingly,followingequation(

6

),thevectorofinterconnectednessmeasurescorrespond-

ingtoPis:

Themagnitudesofasset-levelinterconnectednessshownin??indicatethat?2,

hasthehighestlevelofinterconnectednessinthenetwork,followedby?1and?3,whichhasthelowestinterconnectedness.Wehighlightthatthattheinterconnectednessmeasurecapturesanon-linearaspectofthenetworkbeyondthesimplenumberof?rmsinvestingin

12

eachasset,i.e.,theassets’degreeinthebipartitegraph.Forexample,although?1isheldbyallinvestorsand?2isonlyheldbytwoinvestors,?2isthemostcentralnodeinthisnetwork.?2’scentralitygivesrisetoahigherasset-levelinterconnectednessrelativeto?1.

Whichassetexperiencestheinitialshockplaysafundamentalrole