system? Does the bank risk concentration freeze theinterbank and Finance North American Journal ofEconomics

ContentslistsavailableatScienceDirect

North

American

Journal

of

Economics

and

Finance

Does

the

bank

risk

concentration

freeze

the

interbank

system?

Marcella

Lucchetta

∗

Universita’Ca’FoscaridiVenezia,DipartimentodiEconomia,Italy

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received19May2014

Receivedinrevisedform2April2015

Accepted2April2015

Availableonline17April2015

Keywords:

Interbanksystem

Risksconcentration

a

b

s

t

r

a

c

t

Probably,onetestofthestabilityofthebankingsystemisto evalu-atehowriskyassetsaredistributedacrossbanks’portfoliosandthe implicationsforthecontagionviainterbankrelations.Thispaper explorestheoreticallyabanksectorwithrisksconcentrationand thefunctioningofinterbankmarkets.Itemploysasimplemodel wherebanks areexposedtobothcreditand liquidityriskthat suddenlycorrelateoverthebusinesscycle.Weshowthatrisk con-centrationmakesinterbankmarketbreakdownsmorelikelyand welfaremonotonicallydecreasesinriskconcentration.

©2015ElsevierInc.Allrightsreserved.

1. Introduction

Akeyfeatureofthe2007–2008financialcrisishasbeenthedisruptionandprolonged malfunc-tioningofinterbankmarkets(see,e.g.Acharya&Merrouche,2013;Afonso,Kovner,&Schoar,2011; Ciccarelli,Maddaloni,&Peydró,2013;Heider,Hoerova,&Holthausen,20101_{)sometimesrelatedtothe} interbanknetworkstructure(Georg,20132_{).Thishascomeasasurprisetomostobservers,since} inter-bankmarketshavebeenfunctioningsmoothlyhistorically,eveninthefaceofseverestressepisodes

∗ Tel.:+393356955599.

E-mailaddress:[email protected]

1_{Ciccarellietal.(2013)findthatfinancialintermediariesareextremelyfragileintheEU.TheECBeffectivelypartlysubstituted}

theinterbankmarketand,inturn,inducedasubsequentsofteningoflendingconditions.

2_{Georg(2013)showsthatnetworkswiththecentralbankthatintervene,solvingmarketincompleteness,aremorestable}

thanrandomnetworks.

http://dx.doi.org/10.1016/j.najef.2015.04.002

suchastheLTCMfailure(Furfine,2000)anditattachesimportancetopolicyinterestrateforloan portfolioriskandbankliquidityasrecentfoundinGiulioni(2015).

Theoreticalresearchaddressingwhyinterbankmarketsmaynotfunctionproperlyhasprovided explanationsbasedoninformationalandmarketfrictionssuchas:asymmetricinformationor mar-ketincompleteness(e.g.Acharya&Skeie,2011;Flannery,1996;Freixas&Jorge,2008;Georg,2013; Gale&Yorulmazer,2012;Heideretal.,2010);informationcontagionwherethepoorperformance ofeachbankconveyspotentialbadnewsaboutthecommonfactoraffectingloanreturns(Acharya& Yorulmazer,2008);marketpower(e.g.Acharya,Gromb,&Yorulmazer,2012;Cai&Thakor,2008);and malfunctioningsecondaryassetmarkets(e.g.Diamond&Rajan,2005,2008;Gorton&Huang,2004, 2006).

Since theearly2000s,however, therehasbeenincreasingevidenceof a positiverelationship betweenmeasuresofsystemicriskinmajorbankingsystemsandbankconcentration.3_Thisevidence raisesthequestionofwhetherthecorrelationofriskspossiblyinducedbyhigherbankrisk concen-trationcouldhaveasignificantimpactonthefunctioningofinterbankmarketsandwelfare,andifso, why.Toourknowledge,thisquestionhasnotbeenaddressedintheliterature.Arecentstudydiscuss theinternationalportfoliodiversificationofMiralles-Marcelo,delMarMiralles-Quirós,and Miralles-Quirós(2015),butasidefromtheinterbankmarketandthecorrelationbetweenliquidityriskand creditrisk.

Specifically,inthispaperweexploretheimplicationsofbankingsystemriskconcentrationforthe functioningofinterbankmarketsinamodelwherebanksareexposedtobothcreditandliquidityrisk. Indeed,asshowinCornett,McNutt,Strahan,andTehranian(2011),Brunnermeier(2009),Covitzand Downing(2007)andEricssonandRenault(2006)creditandliquidityriskmaydramaticallycorrelate overthebusinesscycleespeciallyduringfinancialcrisis.Moreover,DaSilvaandDivino(2013)show thatcreditriskispro-cyclicalanddefaultriskdependsonstructuralfeatures,underliningthebanking regulationroleinpresenceofcreditandliquidityshocks.

Thelineofinquiryofthispaperisrelatedto,andbuildson,thecontributionsbyIbragimov,Jaffee, andWalden(2011)andWagner(2010,2011),whoshowthatinefficienciesmayarisefromindividual bankdiversification,whichdoesnotnecessarilyresultinamoreresilientbankingsystem.However, thesepapersdo notconsiderinterbankmarketsand multiplesourcesofrisks,suchascreditand liquidityrisk,aswedo.4

WebuildasimplemodelalongthelinesofDiamondandDybvig(1983)andBhattacharyaandGale (1987)models,wherebanksareexposedtobothcreditandliquidityrisk,andtherearenoinformational ormarketfrictions.Themarketfailureinoureconomyisthatcontractsareincompleteandtherefore, notallriskscanbeinsured.Interbankmarketbreakdownsaredefinedasparameterconfigurations underwhichthereisnointerbankmarketequilibrium,andbanksimplementautarkicallocations.

Weshowthatanincreaseintheconcentrationofrisks,possiblyarisingfromconcentratedmarket structures,makesinterbankmarketsbreakdownsmorelikely.Differingfromthepreviousliterature, ourresultsarenotdrivenbyasymmetricinformation,marketpowerordysfunctionalsecondary mar-kets.Rather,theyareexplainedbycreditandliquidityriskscorrelationassuddenlyhappensoverthe businesscycle.Asanexample,theabilitytodiversifytheserisksmaybepreventedbyriskmanagement diseconomiesassociatedwithlargesizesoffinancialinstitutionsandthewidescopeandcomplexity oftheiractivities.Indeed,systemicallyimportantfinancialinstitutions(SIFIs)areseenasinstitutions

3_{GroupofTen(2001)concludedthat“Evidencesuggeststhat[risk]interdependenciesbetweenlargeandcomplexbanking}

organizationshaveincreasedoverthelastdecadeintheUnitedStatesandJapan,andarebeginningtodosoinEurope.Although

acausallinkhasnotbeenestablished,theseincreasesarepositivelycorrelatedwithmeasuresofconsolidation.”Apositive

relationshipbetweenbankconcentrationandmeasuresofbanksystemicriskisfoundinBoyd,DeNicolò,andLoukoianova

(2009a)andBoyd,DeNicolò,andJalal(2009b).Duringtheperiodsofintenseconsolidationofthelastdecade,DeNicolòand Kwast(2002)foundincreasedriskinterdependenciesamongU.S.largeandcomplexbankingorganizations.Riskprofilesof

largeandcomplexU.S.andEuropeanbankswerealsofoundtohaveincreasedintheU.S.inEurope,andgloballyinDeNicolò,

Hayward,andBhatia(2004a),DeNicolò,Bartholomew,Zaman,andZephirin(2004b),Stiroh(2004),Hartmann,Straetsman, anddeVries(2005),StirohandRumble(2006),andHoustonandStiroh(2006).

4_{ThispapercanbeviewedasalsoindirectlyrelatedtothepapersbyGaietal.(2011)andAnandetal.(2012),astheyanalyze}

numericallyhownetworkconnectionsanddifferentexogenousinterbankstructuresaffectbanks’shorttermfundingbutdo

tobecarefullymonitoredbyregulatorasapotentialsourceofsystemicrisk(seee.g.ECBFinancial StabilityReview,December2009).

InSection2wedescribethemodel.Themodelset-upcapturesinstylizedformtheimportant distinctionbetweendiversificationofabankingfirmanddiversificationofthefinancialsystem empha-sizedanddocumentedempiricallybyDeNicolòandKwast(2002).Bankscanbeperfectlydiversified individually,butaggregateriskinthebankingsystemcanbeeitherperfectlydiversifiedor concen-tratedacrossbanks.Asthesystembecomescomposedoffewerandlargerbanks,eachbankwillbemore diversifiedindividually,butthebankingsystemwillbelessdiversified,sincealargerfractionofbanks isexposedtothesameaggregateshocks.Inthemodel,thedegreeofbankingsystemdiversificationis parameterizedbetweenthetwoextremesofperfectdiversificationandmaximalconcentration.

Section3definestheinterbankequilibrium.Interbankmarketbreakdownsaresimplydefinedas situationsinwhichtheinterbankequilibriumdoesnotexistandisreplacedbytheautarkic equilib-rium,whereeachbankisdisconnectedfromeachother.

InSection4,theexistenceofinterbankequilibriaisestablishedforthetwoextremecasesofa perfectlydiversifiedandamaximallyconcentratedbankingsystem.Then,itisshownthatforalarge setofeconomies,thesizeofthesetofinterbankequilibriaunderadiversifiedbankingsystemisalways strictlylargerthanthatofaconcentratedbankingsystemforanylevelofcreditandliquidityrisk.This resultindicatesthatinthepresenceofaggregaterisk,adiversifiedbankingsystemislikelytobeless pronetointerbankmarketbreakdowns.

Section5definesthewelfarepropertiesofinterbankandautarkicallocations.Whenaninterbank equilibriumexists,itgivesahigherexpectedutilitytodepositorsthantheautarkicequilibrium,but importantly,depositors’welfareismonotonicallydecreasinginthedegreeofriskconcentration.

Section6providesanexampleextendingthemodeltocapturetheShehzadandDeHaan(2013) findings.Theirresultssuggestthatstockpricesoflargebankswereaffectedmoreduringthecrisis thanthoseofsmallbanks.Theyalsofindthatmanagerialefficiency,loanquality,leverage,andthe volumeofoutstandingloansaffectbankstockprices.Thus,inourextension,riskconcentrationand ahighprobabilityofinterbankmarketbreakdownsisduetoriskmanagementdiseconomiesofscale andscopeaffectinglargeandcomplexfinancialfirms.Specifically,whenourmodelismodifiedby introducingariskcontroltechnologywithdecreasingreturnsoveracertaininvestmentthreshold, bankswillchoosealevelofriskconcentrationthatincreasesinsize.Suchlevelishigherthelargeris thecostofriskcontrolarisingfromtheinternalorganizationoflargeandcomplexfirms.Thisresult suggestsimprovementofriskcontroltechnologiesinlargeandcomplexfinancialinstitutionsandin regulatorybodiesmaybepolicyconcernsasimportantasotherpolicesconsideredintheliteratureto minimizetheprobabilityofinterbankmarketbreakdowns.

Section7concludes.ProofsofallpropositionsareintheAppendix.

2. Themodel

Therearethreeperiods,t=0,1,2,andoneriskyassetthatyieldsarandomreturnatdate2perunit investedatdate0.Itcanassumetwovalues,R=Rh_,andR=Rl_{,withtheprobabilitiesspecifiedbelow.}

Ifaportionoftheinvestmentinthisassetisliquidatedatt=1,ityieldsacertainreturnofperunit invested.Thefractionoftheassetthatisliquidatedisdenoted_˛.ItisassumedthatRh_>R>≥1l_,5_so thatifstorageisavailable,thistechnologywillbedominatedinrateofreturninbothdatesandwill beneverused.Thisisnotastrongassumptionsinceinthespecialcase=1,thelongassetmimicsthe storagetechnologyandcanbeliquidatedtosatisfyearlyconsumers,aswedefineshortly,ortolend intheinterbankmarket.

Atdate0consumersareendowedwithoneunitofthedate0consumptiongood,whichisassumed tobeinvestedallinonebank.Consumersareuncertainabouttheirtimepreferences:withprobability theyareearlyconsumers,whowanttoconsumeatdate1only,andwithprobability1−they

5_{Thisimpliesthattheexpectedvalueofthelongassetisgreaterthanone(thereturnofthestoragetechnology).Furthermore,}

arelateconsumers,whowanttoconsumeatdate2only.6_{Weassumethatdepositorshavelimited} participationintheassetsmarkettherebytheyneedbankingintermediation.7_{Theirpreferencesare} representedbyautilityfunctionU(c),twicecontinuouslydifferentiable,increasing,andstrictly con-cave.Thefractionofearlyconsumersisalsorandomandcanassumetwovalues:=h_,and=l_,

with1>_h_>l_>0.

ThebankingsectoriscomposedbyNex-anteidenticalbanksthatinvestconsumer’sendowmentsat date0.Thebankingsectorisperfectlycompetitive,sothatbanks’objectiveistomaximizedepositors’ expectedutility.

Eachbankisexposedtoliquidityandcreditriskshocks.Therealizationofbothshocksisobserved byabankatdate1.Weassumethereisnoaggregateuncertainty,sothatthefractionofbankswhich isexposedtoagivencombinationofcreditandliquidityshocksisdeterministic.However,the prob-abilities(andhencethedistribution)ofthesebanksatdate0dependsonanexogenousparameter ∈(0,1)thatindexesthedegreeofconcentrationofliquidityandcreditriskinthebankingsystem.

Specifically,thefractionofbanksthatexperienceagivenpairofrealizationsofcreditandliquidity risk(R,)isgivenbythefollowingtable,withp∈(0,1)andq∈(0,1),wherepistheprobabilityofa banktohaveahighreturnandalowliquidityshockandqistheprobabilityofabanktohaveahigh returnandahighliquidityshock.Theseprobabilities,pandq,areindependent.

Thus,asofdate1,therearefourtypesofbanks:

Thereafterwerefertoparameterasanindexof“riskconcentration”.

Afractionofbanksp(type1)experiencesalowliquidityshockandahighfinaldatereturnonthe asset;bycontrast,afractionofbanks(1−p)(type4)experiencesthereverse,thatis,ahighliquidity shockandalowfinaldatereturnontheasset.Thus,creditrisksandliquidityrisksareperfectlypositively correlatedfortype1and4banks.

Conversely,afractionofbanks(1_−)q(type2)experiencesahighliquidityshockbutalsoahigh finaldatereturnontheasset,whereasafractionofbanks(1−)(1−q)(type3)alowliquidityshock butalsoalowfinaldatereturnontheasset.Thus,creditriskandliquidityrisksareperfectlynegatively correlatedfortype2and3banks.

If=0,thenthebankingsystemismadeofbanksthatdiversifytheircreditandliquidityrisks.By contrast,if=1,banksconcentratetheircreditandliquidityrisks.Asincreases,thebankingsystem exhibitsahigherconcentrationofrisksforanygivenvaluesof(p,q).

3. Interbankequilibrium

Hereweexaminehowwellaninterbankmarketworksforanylevelofriskconcentration(i.e.any ).

Thereisaninterbankmarketwhereliquiditycanbetradedattheintermediatedate.Theamount offundsthateachbanktradeintheinterbankmarketisdenotedbandthegrossinterbankrateis denotedbyr.

6_{When=1,inthiscaseearlyconsumersreceive1suchthatinexpectationtheincentivecompatibilityconstraintissatisfied:}

i.e.lateconsumersreceiveE[R]>1.

Atdate0,competitivebanksmaximizetheexpectedutilityofdepositors.Theychoosetheamount ofborrowingb(ifpositive,borrowing,ifnegative,lending)andtheamountofassettoliquidate,˛,to solve: Max ˛,b˘1=U(c1)+(1−)U(c2) (1) Subjectto c1=˛+b (2) (1−)c2=R(1−˛)−rb, (3)

whereristhe“interbank”rate,tobedeterminedinthet=1creditequilibrium. Assumeaninterbankequilibriumexists.Substituting(3)in(2)throughb,

c1+(1−)c2

r =˛+ R

r(1−˛). (4)

Thesolution˛*_{willbetheonethatmakesthebank’sbudgetconstraintthelargest,i.e.thatmakes}

therighthandsideof(4)thelargest.Hence,thesolutionisgivenby: ˛∗_{=0 if} R

r > and ˛∗=1 if R

r <. (5)

Observethattheexistenceofastoragetechnologywouldnotaffecttheconditionstoliquidatethe longasset.Indeed,aslongasthelongassetvalueatperiod1islowerthat,itisalwaysoptimalto liquidateit.

Anecessaryconditionfortheexistenceofaninterbankequilibriumisthat r∈



Rl , Rh 



. (6) Thisisbecauseifr≤Rl

,by(4)allbankswillnotliquidatetheinvestmentintheriskytechnology,and

by(2)theywillwishtofinancealldate1consumptionbyborrowing.Thus,therewouldbenolenders, hencenointerbankequilibrium.Likewise,ifr≥Rh

,by(5)allbankswillliquidatetheinvestmentin

theriskytechnology,andby(3)theywillwishtofinancealldate2consumptionbytherepayments onlendingatdate2.Butatdate1therewouldbenoborrowers,hencenointerbankequilibrium.

Using(2),(3)and(5)in(1),bank(R,)solves: Max b ˘1=U



_˛∗_+b 



+(1−)U



R(1−˛∗)−rb 1−



, (1a)

thefirstorderconditionwithrespecttobis U



˛∗+b 



=rU



R(1−˛∗)−rb 1_−



. (7)

Thus,thesolutionofthebankproblemisgivenby(5)and(7).Notethatbanksoptimalchoicesare theliquidationdecision˛*(R),whichdoesnotdependon,andb(R,␮).Theliquidationdecisionin responsetothecreditriskrealizationdoesnotdependontheliquidityshock(in(5)nothingdepends on),buttheborrowingdecisiondependsonbothshocks(by(7)).

Wecharacterizeequilibriaforlogutilitypreferences,i.e.U(c)=ln(c).Thechoiceofthisspecification ismotivatedbysimplicityandbythefactthatforthesepreferences,date1spotmarketallocationsand optimalbankingallocationsgenerallycoincideinliquiditypreferenceframeworkssuchasours(see Allen&Gale,2007).Ourresultsarerobusttomoregeneralconsumerutilityspecifications.Itiseasyto seeinEq.(7)thatitissufficienttoadoptaconsumer’sutilityfunctiontwicecontinuouslydifferentiable, increasing,andstrictlyconcave.Hence,thesepreferencescanbeviewedasausefulbenchmarkto judgedifferencesinequilibriumsandassociatedwelfarepropertiesindiversifiedandconcentrated riskeconomiesindependentlyofefficiencywedgesbetweenmarketandbankingallocations.

Eq.(7)yields:  ˛∗_+b= _R(1r(1_−˛−∗₎)_−rb. (8) Solving(8),wehave b(R,)=1_r(R(1−˛∗_{)−r(1−)˛}∗_{) . (9)} Sincer∈



Rl ,R h 



,by(5),optimalasset’sliquidationis˛*(Rl_)=1and˛*(Rh_)=0.

Insum,thefourbanktypeshavethefollowingborrowing/lendingpositions

b(Rl_,l_)=−(1−l_{) (9a)} b(Rl_,h_)=−(1−h_{) (9b)} b(Rh_,l₎₌1 r



l_Rh



_(9c) b(Rh_,h₎₌1 r



h_Rh



_{, (9d)}

andequilibriumintheinterbankmarketrequires p1_rl_Rh_+(1−)q1

rhRh−(1−)(1−q)(1−l)−(1−p)(1−h)=0. (10) Theaboveequation(10)islinearwithrespecttorandhastheuniquesolution:

r∗₌ Rh



pl_+(1−)qh



[(1−)(1−q)(1−l)+(1−p)(1−h)]. (11)

Eq.(11)saysthattheinterbankequilibriumrateraisesastheliquidityneedsandtheopportunity costofholdingtheasset,Rh_,increase.

4. Comparisonsofequilibria

Inthissectionweidentifyconditionsensuringexistenceofequilibriafortheextremevaluesof, andcomparethesetofparametersforwhichequilibriaexistsforsuchvalues.

Using(11)andr∗_∈



Rl ,R h 



,weobtain 1≥ pl+(1−)qh (1_{−)(1−q)(1−}l)_{+(1−p)(1−}h)≥ Rl Rh. (12)

Weuse(12)toassesstheexistenceofinterbankequilibriaundertwoextremecases,thatofbanking system’sperfectdiversification(_=0),andthatofmaximalriskconcentrationinthebankingsystem (=1).Themainresultissummarizedinthefollowingproposition.

Proposition1. Fortheperfectdiversifiedeconomy(=0)andfortheperfectconcentratedeconomy

(=1),thesetofinterbankequilibriaisnon-emptyforanyparameterconfigurationofcreditandliquidity risk(h_,l_,Rh_,Rl_).

Now,wewishtocomparethesizeofthesetofeconomies,indexedbypandq,forwhichequilibria existfor=0andfor=1.

Weassesstheequilibriadomainsdefiningtwosetsasbelowdetailed.Tothisscope,wemeasure thesesetscomputingthelinearextremes[

v

0,

v

1]forthecase=0and[

v

1,

v

11]for=1.Theseextremes

Usingtherightandthelefthandsideof(12),theequilibriumdomainsfor=0,[

v

0,

v

10],andfor =1,[

v

1,

v

11],arerespectively



v

0= Rl Rh(1−l) h₊ Rl Rh(1−l) ,

v

10= 1−l 1+h_−l

(13) and



v

1= Rl Rh(1−h) l₊ Rl Rh(1−h) ,

v

11= 1−h 1−h_+l

. (14)

Thelargeristheintervalforwhichequilibriaexistsunderdiversificationorconcentration,the largeristhesetofeconomiesthatmaybenefitfromtherisksharingopportunitiesofferedbythe interbankmarket.

Considerthedifferencebetweentheequilibriaintervalwhen=0andwhen=1,definedas:

G≡0−1=(

v

0−

v

10)−(

v

1−

v

11). (15) Computationsgive: 0= h_(1−l₎



₁₋ Rl Rh



(1+h_−l₎



_h₊ Rl Rh(1−l)



, (16) and 1= l_(1−h₎



₁₋ Rl Rh



(1−h_+l)



l₊Rl Rh(1−h)



. (17)

Ofparticularinterestisthecomparisonofthesetofequilibriaforrelativelylargecreditandliquidity shocks.Thiscomparisonismadeclearerbysubstitutingh_=l_,where 1

l ≥≥1,andRl=ˇRhwhere

ˇ≤1.Parameterisameasureoftheliquidityriskandˇisthecreditrisk.Thesmallerisˇ,thelargeris thedifferencebetweenhighandlowreturn.Thelargeris,thelargeristhedifferencebetweenlow andhighliquidityshock.

Thus,(15)canbeexpressedas

G(l_,,ˇ)= l(1−l)

(1+l_−l₎



_l_+ˇ(1−l₎



−

l_(1−l₎

(1−l_+l₎



_l_+ˇ(1−l₎



. (18)

Thefollowingpropositionestablishesarankingofthesizeofequilibriaunderadiversifiedeconomy andarisk-concentratedeconomy.

Proposition2. Thereexistsalsuchthatforl_≥landany(ˇ,)thesetofinterbankequilibrium

underthediversifiedeconomyisalwaysstrictlylargerthanfortheconcentratedeconomy.

Whenliquidityshocksarelarge,thesetofinterbankequilibriaofthediversifiedeconomyislarger thanthecorrespondingsetoftherisk-concentratedeconomyandtheinterbankequilibriumislikely tobreakdownformoremodelparameterranges.Therefore,adiversifiedeconomyoffersabetter insuranceagainsthighliquidityandcreditrisk.Finallynotethatby(18),forˇ=␥=1,whichamounts toabsenceofcreditandliquidityrisk,G(1,1)=0,thenthetwosetsareequivalent.Thismeansthat bankmarketstructureisimportantespeciallywhentheshocksarehigh.

Fig.1showsgraphicallytheresultofProposition2forasetofeconomies.Thesurfacerepresenting thefunctionG(_l_{,,ˇ)isincreasinginliquidityrisk.}

AsFig.1highlights,thesetofequilibriaisincreasinginthedegreeofbanks’riskdiversificationfor aliquidityriskgreaterthat¯l_{.Thisindicatesthataninterbankislesslikelytobreak-downwhenthe}

Fig.1. BehaviorofG(l_,,ˇ).

5. Welfarecomparisons

Wewanttocompareagents’welfareineconomieswithbankingsystemsdifferingaccordingto marketstructureandriskconcentration.Todothat,wecomputedepositors’expectedutility.

Using(2)and(3),theconsumptioninstates1and2is c1= ˛+b

 (19)

and

c2= R(1−˛)−rb

1− . (20)

Substitutingequilibriumvalues˛,bandrin(19)and(20),weobtaintheconsumptionallocation offeredbydifferentbanktypes,Ctype(i)_=(ctype(i)

1 ,c2type(i))fori=1,2,3,4: Ctype(1)₌



(1−)(1−q)(1−l_{)+(1−p)(1−}h₎



_ pl_+(1−)qh ,Rh

(21a) Ctype(2)₌



(1−)(1−q)(1−l_{)+(1−p)(1−}h₎



_ pl_+(1−)qh ,Rh

(21b) Ctype(3)₌

, R h



_pl_+(1−)qh



(1−)(1−q)(1−l)+(1−p)(1−h)

(21c) Ctype(4)=

, R h



_pl_+(1−)qh



(1−)(1−q)(1−l)+(1−p)(1−h)

. (21d)

Theex-antetheexpectedutilityofarepresentativeconsumeristherefore

W≡ 4



i=1 P(type(i))U(c1type(i),c type(i) 2 ) fori=1,2,3,4. (22) Equivalently:

Fig.2.ThebehaviorofWasafunctionofandliquidityrisk. Unfolding(23)weget: W ≡p

l_log



(1_{−)(1−q)(1−}l)_{+(1−p)(1−}h)



_ pl₊(1−)qh

+(1−l_)log(Rh₎

+(1−)q

h_log



(1−)(1−q)(1−l_{)+(1−p)(1−}h₎



_ pl_+(1−)qh

+(1−h_)log(Rh₎

+(1−)(1−q)

l_{log()+(1−}l_)log

Rh



_pl_+(1−)qh



(1−)(1−q)(1−l_{)+(1−p)(1−}h₎

+(1−p)

h_{log()+(1−}h_)log

Rh



_pl_+(1−)qh



(1−)(1−q)(1−l)+(1−p)(1−h)

(24)

Fig.2showsWforgivenreturnparameters.NotethatWisdecreasingbothinandintheliquidity riskparameter.

Fig.2showsthatwelfaredecreaseswhenbankingsystemriskisconcentratedandliquidityriskis high.Thecornerrepresentinghighlevelsofandcanbeviewedasa“crisis”setofriskrealizations withhighwelfarelosses.

Whentheinterbankequilibriumdoesnotexist,theautarkicallocationwillprevail.Depositors’ expectedutilityunderautarkyisgivenbythesolutionofthefollowingproblem:

Max ˛ W A_{=U(c1)+(1−)U(c2) (1b)} Subjectto c1=˛ (2b) (1−)c2=R(1−˛). (3b)

Theoptimalsolutionis˛*=.Therefore,substitutingin(2b)and(3b),theconsumptionallocations foreachbanktypeatanautarkicequilibriumare:

Ctype(1)_=Ctype(2)₌



_,Rh



_(25a)

Fig.3. BehaviorofWandWA_{withdifferentliquidityshocks.}

Correspondingly,theexpectedutilityofanagentattheinitialdateis:

WA≡p[llog()+(1−l)log(Rh)]+(1−)q[llog()+(1−l)log(Rh)]

+(1−)(1−q)[l_{log()+(1−}l_)log(Rl_{)]+(1−p)[}l_{log()+(1−}l_)log(Rl_)]

(26) Fig.3showsW,theexpectedutilitywhentheinterbankequilibriumexists,andWA_,theexpected

utilityunderautarky,asfunctionsof_and_␥fortwosetsofeconomies.Forhighvaluesofrisk con-centrationandliquidityrisktheinterbankequilibriumdoesnotexist,asWA_{>W.Whentheinterbank}

surfaceliesovertheautarkyplane,theinterbankequilibriumexistssinceW>WA_{forallparameter}

values,withWstrictlydecreasinginthedegreeofriskconcentration.

Fig.3illustratesthegeneralfindingsummarizedinthefollowingproposition3.Thefiguredescribes theexistenceoftheinterbankmarketandthewelfarebehaviorfordifferentlevelsofriskconcentration andliquidityshocks.Thecornerwithhighliquidityshocksandhighconcentrationisacrisis.Inthis casetheinterbankmarketwillnotexistsincetheautarkyequilibrium(theplaneWA_)giveshigher

welfarethantheinterbankequilibrium.

Proposition3. (a)Thereexists(,¯ )¯ suchthatWA_{>Wforall(,)>(,¯ ).¯}

(b)WhenW>WA_{theinterbankequilibriumexists,andWisstrictlydecreasinginforany.}

Itisusefultoillustrateexamplesfordifferentsetofeconomies.Fig.4clearlyshowsthatwhen ishighandliquidityrisk,,issufficientlyhigh,theinterbankallocationisdominatedbytheautarky allocation.

Fig.5showsthatwithhighcreditrisk,i.e.lowprobabilityofrealizationofthehighreturn,the interbankmarketismorelikelytoexistthantheautarkyallocation.Thissuggeststhatinterbank marketsinsureagainstcreditriskalso,whileWremainsdecreasingwithrespectto.

Thepictures3,4and5highlightthatwelfarealwaysdecreaseswhentheeconomyexhibitsahigh risksconcentrationandhighliquidityexposure.

6. Anexampleofendogenousdegreeofriskconcentration

Sofarthedegreeofriskconcentrationinthebankingsystem(parameter)hasbeentreatedas exogenous.Wehaveshownthatwelfaredecreasesinthedegreeofmarketconcentration.Forall

Fig.4. WandWA_{withrelativelyhighpandq.}

Fig.5.WandWAwithrelativelylowpandq.

examplesshown,thehighestdepositors’expectedutilityisreachedattheminimumlevelofrisk concentration(=0).

Recallthat_isaparameterthatdeterminestheprobabilitiesofcreditandliquidityshocksasof date0.Therefore,achoiceofcanbeviewedasabankchoiceofcreditandliquidityrisk.Ifachieving perfectdiversificationatasystemleveliscostless,theninaperfectlycompetitivebankingsystem, bankswouldchoosetheminimumlevelofriskconcentrationandtheprobabilityofinterbankmarket breakdownswouldbeminimized.

Inreality,ShehzadandDeHaan(2013)suggestthatstockpricesoflargebankswereaffected moreduringthecrisisthanthoseofsmallbanksandtheyalsofindthatmanagerialefficiency,loan quality,leverage,andthevolumeofoutstandingloansaffectbankstockprices.Literally,achieving diversificationandcontrollingriskiscostly,sinceriskmanagementcanbeviewedasatechnology availabletofirmssimilar,forexample,tothetechnologyunderlyingcreditriskmodels.Asthesize andscopeoftheoperationsofbanksexpands,financialfirms’spanofcontrolovertheirmanyunitscan becomelessefficientincontrollingrisk.Inthislight,alackofsufficientbankingsystemdiversification

andahigherprobabilityofinterbankmarketbreakdownsmaybeinpartduetoriskcontroldiseconomies ofscaleandscope.

Thepotentialforfirstordereffectsofriskmanagementdiseconomiesofscaleandscopeonrisk concentrationinthebankingsystem,andtheirrelationshipwithbanksize,canbeillustratedbythe followingmodificationofourmodel.

Supposeatdate0bankshaveasizeS≥0,andchooseemployingpartofdate0resources. Specif-ically,theyinvestafractionxofdateresourcesinthetechnologyandchooseincurringacostz()S

asafractionofdate0resources,where>0isthescalecostparameter.Theirresourceconstraintat date0istherefore:

x+z()S_{=S (27)}

Assumez()=a0

 +a1,witha0anda1aspositivecoefficients.Thisfunctioncanbeinterpreted

asacostfunctionofariskcontroltechnologythatexhibitsdecreasingreturnstoinvestmentovera certainthreshold.Itsparameterscoulddependonsizeandscopeoffinancialfirmsoperations,aswell asonincentivesarisingfromasymmetricinformation,marketpowerandotherfactorspointedoutin theliterature,whichmayinturnbeaffectedbyfirms’size.

TocomputeW(),wereplace1withS(1−z()S_{)intheconsumptionallocations,}

c1= ˛S(1−z()S _)+b  (28) and c2= R(1−˛)S(1−z()S _)−rb 1− . (29)

Then,optimalborrowing/lendingchoicesintheinterbankmarketare: b(R,)=S(1−z()S)

r (R(1−˛∗)−r(1−)˛∗) , (30)

Withtheliquidationchoices˛unchanged,theequilibriumintheinterbankmarketisthesolution ofthefollowingequation:

pS(1−z()S) r lRh+(1−)q S(1−z()S₎ r hRh−(1−)(1−q)S(1−z()S)(1−l) −(1−p)S(1−z()S_)(1−h_{)=0, (31)} whichyields r∗₌ Rh



pl_+(1−)qh



[(1−)(1−q)(1−l)+(1−p)(1−h)]. (32)

Notethat(32)isequivalentto(11),meaningthattheequilibriuminterbankratedoesnotdepend onthefunctionS(1−z()S_{).Thismeansthatinthemarketforliquidityistheexcessofdemand}

onsupplythatdeterminesinterbankrate.Multiplyingbothdemandandsupplyforthesizefunction wouldnotaffecttheequilibriuminterbankrate.

The consumption allocations for thefour bank typeswhich allowus tocomputedepositors’ expectedutilityare:

Ctype(1)₌



(1−)(1−q)(1−l_{)+(1−p)(1−}h₎



_S(1−z()S₎ pl_+(1−)qh ,RhS(1−z()S)

(33a)

Fig.6. ExpectedutilityWwithcostsofriskcontrol=1andS=4. Ctype(2)₌



(1−)(1−q)(1−l_{)+(1−p)(1−}h₎



_S(1−z()S₎ pl_+(1−)qh ,RhS(1−z()S)

(33b) Ctype(3)₌

S(1−z()S_), Rh



pl_+(1−)qh



_S(1−z()S₎ (1−)(1−q)(1−l)+(1−p)(1−h)

(33c) Ctype(4)₌

S(1−z()S_), Rh



pl_+(1−)qh



_S(1−z()S₎ (1−)(1−q)(1−l_{)+(1−p)(1−}h₎

. (33d)

Fig.6shows(withdifferentangles)anexampleoftheexpectedutilityfunctionWasafunctionof andtheliquidityriskparameter.ItisapparentthatfunctionWisstrictlyconcave,withamaximum forasaninteriorpoint.WeplotthesurfaceforsizeparameterS=4and=1.

Proposition4. Foranyvalueofandforagivenlevelofliquidityrisk,¯ theoptimalbankrisk

concen-trationlevel*isincreasinginS.

Proposition4statesthatthescaleparameteristhedeterminantofthedegreeofdiversification thatabankchooses,andbankdiversificationisinverselyrelatedtoitsassetssize.

Fig.6showstheconcavitypropertyofthewelfarefunctioninthecaseofcostlydiversification.This impliesthechoiceofadifferentfrom0.

Fig.7showstheoptimalfordifferentbanksize.Themaximumoftheexpectedutilityisdifferent accordingtoS.Theconcentrationisgreaterforabanksizelargerthanone.Bycontrast,forasmall banksizetheoptimalriskconcentrationapproacheszero.

Therefore,theoptimallevelofriskconcentrationmightbelargerthantheminimumfeasible.In turn,riskconcentrationislargerforalargerbanksize.Thelevelofa0anda1maydependfundamentally

Fig.7.ExpectedutilityWforS=1andS=4.

financialfirms’size,suchastoo-big-to-failincentives,maycarryhigherriskcontrolcosts,whichin turncouldresultinahigherlevelofriskconcentrationinthebankingsystem.

7. Conclusion

Ourmodelshowsthatriskconcentrationseemsoffirstorderimportanceforthesmoothfunctioning ofinterbankmarkets.Riskconcentrationmayincreaseinbanksizeunderreasonableassumptions abouttheriskmanagementtechnologyavailabletobanks.Differingfrommostliterature,ourresults areobtainedinamodelwithnoasymmetricinformation,nomarketpowerordysfunctionalsecondary markets.

Withregardtopolicy,theliteraturehasprominentlyfocusedontheimportantroleoftheCentral Bankaslenderoflastresortwheninterbankmarketbreakdownsoccur(seee.g.Freixas,Parigi,& Jean-Charles,2000;Freixas,Antoine,&Skeie,2009;Goodhart&Illing,2002;Repullo,2005).Ourmodel suggeststhattheimprovementofriskmanagementtechnologiesinlargeandcomplexfinancial insti-tutions,aswellasinregulatorybodies,maybeasimportanttominimizetheoccurrenceofinterbank

marketbreakdowns.

Acknowledgements

IacknowledgesfinancialsupportundertheprojectSYRTO,fundedbytheEuropeanUnionunder the7thFrameworkProgramme(FP7-SSH/2007-2013–GrantAgreementn◦320270)andtheproject MISURA,fundedbytheItalianMIUR.

The researchleadingtotheseresultshasreceivedfundingfromtheEuropeanUnion,Seventh

FrameworkProgrammeFP7/2007-2013undergrantagreementSYRTO-SSH-2012-320270.

IthankwithoutimplicationstheseminarInstituteandtheDeutscheBundesbankforcomments andsuggestions.

Appendix.

Proposition1. Fortheperfectlydiversifiedeconomy(=0)andfortheperfectlyconcentrated

econ-omy(=1),thesetofinterbankequilibriasisnon-emptyforanyparameterconfigurationofcreditand liquidityrisk(h_,l_,Rh_,Rl_).

Proof. Theleft-handsideinequalityof(12)canbeexpressedas:

(1−)(1+h_−l_{)q≤(1−)(1−}l_)+(1−h_)−p(1−h_+l_{), (A1)}

If=0, then (1+h_−l_)q≤(1−l_{) (A2)}

If=1, then 0≤(1−h_)−p(1−h_+l_{) (A3)}

Theright-handsideinequalityof(12)canbeexpressedas: (1−)(h₊ Rl Rh(1−l))≥ Rl Rh[(1−h)+(1−)(1−l)]−p[(1−h) Rl Rh+l] (A4) If=0, then q



h₊ Rl Rh(1−l)



≥ Rl Rh(1−l) (A5) If=1, then 0≥ Rl Rh(1−h)−p[(1−h) Rl Rh+l] (A6)

Thus,thesetofeconomiesforwhichaninterbankequilibriumexistsisindexedbyq∈[0,1]if=0, andbyp∈[0,1]if=1.

If=0,using(A2)and(A5),weget:

Rl Rh(1−l) h₊ Rl Rh(1−l) ≤q≤ 1−l 1+h_−l. (A7)

Therefore,if=0,thesetofinterbankequilibriaisnon-emptyif 1−l 1+h_−l≥ Rl Rh(1−l) h₊ Rl Rh(1−l) . (A8)

Inequality(A8)impliesthat(1−l_)(h₊Rl

Rh(1−l))≥ R l

Rh(1−l)(1+h−l),whichcanbe

fur-thersimplifiedtoh_≥ Rl

Rhh,whichinturnisalwaysverifiedsince R l

Rh <1.

If=1,using(A3)and(A6)weget: 1−h 1−h_+l≥p≥ Rl Rh(1−h) l₊ Rl Rh(1−h) . (A9)

Inequality(A9)isalwayssatisfied,sinceitreducestol_≥ Rl

Rhl,whichholdsas R l

Rh <1.

Proposition2. Thereexistsalsuchthatforl_≥landany(,ˇ)thesetofinterbankequilibrium

underthediversifiedeconomyisalwaysstrictlylargerthanfortheconcentratedeconomy.

Proof. Differentiating(18)withrespecttol_{,andevaluatingthederivativeat}l_=1:

G_l(0,,ˇ)

(−1)



4(1+ˇ)2

+2_{ˇ(1+5ˇ)−}3_ˇ2_{−(1+7ˇ+8ˇ}2₎



(−2)2(ˇ(−1)−1)2 , (A10)

since(−1)>0and(−2)2_{(ˇ(−1)−1)}2_{>0,(A10)ispositiveif}

4(1+ˇ)2

Observethatforl_{→1,→1,then(A11)reducesto}

3+2ˇ>0. (A12)

Therefore,thereexistvaluesof¯l_{and¯suchthatfor}l_≥l_{andfor≥¯thefunctionG(}l_,,ˇ)

isincreasing.

Proposition3. (c)Thereexists(,¯ )¯ suchthatWA_{>Wforall(,)>(,¯ );¯}

(d)WhenW>WA_{theinterbankequilibriumexists,andW,isstrictlydecreasinginforany.}

Proof. Weprovepart(a)and(b)separately.

(a)Thereisa(,)>(,¯ )¯ suchthattheautarkyallocationdominatestheinterbankequilibrium. Theexpectedutilityunderautarkyfor_=0andfor_=1are

WA0_≡q[l_{log()+(1−}l_)log(Rh_)]+(1−q)[l_{log()+(1−}l_)log(Rl_{)], (A13)}

and

WA1_≡p[l_{log()+(1−}l_)log(Rh_{)]+(1−p)[}l_{log()+(1−}l_)log(Rl_{)]. (A14)}

Hence,wecomparetheexpectedutilitywithinadiversifyeconomybetweeninterbankallocation andautarkyallocation

W0_−WA0_≡q



l_log



(1−q)(1−l_) ql



+(1−l_)log(Rh₎



+(1−q)



l_{log()+(1−}l_)log



Rh_ql (1−q)(1−l)



−



q[l_{log()+(1−}l_)log(Rh_)] +(1−q)[l_{log()+(1−}l_)log(Rl_)]



_⇒q



l



log



(1−q)(1−l_) ql



−log()



+(1−q)



(1−l₎



log



Rh_ql (1−q)(1−l₎



−log(Rl₎



, (A15)

Thedifferenceofexpectedutilityinariskconcentratedeconomybetweentheinterbankallocation andautarkyallocationis

W1_−WA1_≡p



l_log



(1−p)(1−l_) pl



+(1−l_)log(Rh₎



+(1−p)



l_log() +(1−l_)log



Rh_pl (1−p)(1−l)



−



p[l_{log()+(1−}l_)log(Rh_)] +(1−p)[l_{log()+(1−}l_)log(Rl_)]



_⇒p



l



log



(1−p)(1−l_) pl



−log()



+(1−p)



(1−l₎



log



Rh_pl (1−p)(1−l)



−log(Rl₎



. (A16)

Computing(A15)and(A16)for→ 1

l,W0−WA0>0,andW1−WA1<0.Then,weconcludethat

thereexistacouple(,)>(,¯ )¯ suchthattheautarkyallocationgivesahigherexpectedutilitythan theinterbankmarketequilibrium.

(b)Whenwehaveaninterbankequilibrium,theexpectedutilitydecreasesin.Takethederivative of(24)withrespecttoandcomputeitfor→1andfor→ 1

l, W→1



→ 1 l



→−<0. (A17)

Computethederivativealsoforl_→0

W→1



l→0



<0. (A18)

Therefore, as risk concentration increases, the expected utility in the interbank economy is decreasing.Moreover,theexpectedutilityfor=0andfor=1are

W0_≡q



l_log



(1−q)(1−l_) ql



+(1−l_)log(Rh₎



+(1−q)



l_{log()+(1−}l_)log



Rh_ql (1−q)(1−l₎



(A19) and W1_≡p



l_log



(1−p)(1−l_) pl



+(1−l_)log(Rh₎



+(1−p)



l_{log()+(1−}l_)log



Rh_pl (1−p)(1−l)



. (A20) Let→ 1

l,thedifferenceofexpectedutilitiesis

W0_−W1_=q



log



(1−q)(1−l_) q



+(1−q)



l_{log()+(1−}l_)log



Rh_q (1−q)(1−l)



−



p



l_{log( 0)+(1−}l_)log(Rh₎



_{+(1−p)log()}



_{>0, (A21)}

sinceW1



_→ 1 l



<0andW0



_→ 1 l



>0. Letl_→0, W0_−W1_{→0. (A22)}

Weconcludethatforanytheexpectedutilitydecreasesinconcentration,.

Proposition4. Foranyvalueofandforagivenlevelofliquidityrisk,¯ theoptimalbankrisk

concen-trationlevel*isincreasinginS.

Proof. Taketheoptimal*solvingWforagivenlevelofliquidityrisk¯andforp=q=0.5.ForS→0,

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