What is the minimal systemic risk in financial exposure networks?

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Journal of Economic Dynamics & Control

journalhomepage:www.elsevier.com/locate/jedc

What is the minimal systemic risk in financial exposure networks?

Christian Diem

^a,b,e

, Anton Pichler

^b,c,d

, Stefan Thurner

^b,e,f,g,∗

aInstitute for Statistics and Mathematics, WU Vienna University of Economics and Business, Welthandelsplatz 1, A-1020, Austria

bComplexity Science Hub Vienna, Josefstädter Straße 39, A-1080, Austria

cInstitute for New Economic Thinking, University of Oxford, Manor Road, OX1 3UQ, UK

dMathematical Institute, University of Oxford, Woodstock Road, Oxford OX1 3LP, UK

eIIASA, Schlossplatz 1, Laxenburg A-2361, Austria

fSection for Science of Complex Systems, Medical University of Vienna, Spitalgasse 23, A-1090, Austria

gSanta Fe Institute, 1399 Hyde Park Road, Santa Fe, 87501 NM, USA

a r t i c l e i n f o

Article history:

Received 15 May 2019 Revised 10 February 2020 Accepted 8 March 2020 Available online 12 April 2020 Keywords:

Systemic risk-efficiency Interbank market Financial networks Contagion

Network optimization

Mixed-integer linear programming DebtRank

Network topology measures and Systemic risk

a b s t r a c t

Wequantifyhowmuchsystemicriskcanbeeliminatedinfinancialcontractnetworksby rearrangingtheirnetworktopology.Byusingmixedintegerlinearprogramming,financial linkagesareoptimallyorganized,whereastheoveralleconomicconditionsofbanks,such ascapital buffers, totalinterbank assets and liabilities,and averagerisk-weightedexpo- sureremainunchanged.Weapplythenewoptimizationprocedureto10snapshotsofthe Austrianinterbankmarketwhere wefocusonthelargest70 bankscovering 71%ofthe marketvolume.Theoptimizationreducessystemicrisk(measuredinDebtRank)byabout 70%,showingthehugepotentialthatchangingthenetworkstructurehasonthemitigation offinancialcontagion.Existingcapitallevelswouldneedtobescaledupbyafactorof3.3 toobtainsimilar levelsofDebtRank. Thesefindingsunderlinetheimportanceofmacro- prudentialrulesthatfocusonthe structureoffinancialnetworks.Thenewoptimization procedureallowsusto benchmarkactual networksto networks withminimalsystemic risk.We findthat simpletopologicalmeasures,likelinkdensity, degreeassortativity, or clusteringcoefficient,failtoexplainthelargedifferencesinsystemicriskbetweenactual and optimal networks.We findthat ifthe mostsystemicallyrelevant banks aretightly connected,overallsystemicriskishigherthaniftheyareunconnected.

© 2020TheAuthor(s).PublishedbyElsevierB.V.

ThisisanopenaccessarticleundertheCCBY-NC-NDlicense.

(http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

Increasingcapitalrequirementsformarketparticipantsisanobvioussuggestionforimprovingtheresilienceoffinancial systemsand,inparticular, forreducing systemicrisk infinancialmarkets. Examplesofinnovativepolicy proposals,where capitalrequirementsdependonmacroprudentialregulationareContetal.(2010),whoproposecapitalrequirementsinrela- tiontotheContagionIndexvaluesofbanks,Gauthieretal.(2012),whosuggestthatbankcapitalbuffersshouldcorrespond totheircontributionstooverallsystemicrisk, Markose(2012),who proposesacapitalsurchargerelatedtotheeigenvector

∗ Corresponding author at: Section for Science of Complex Systems, Medical University of Vienna, Spitalgasse 23, A-1090, Austria.

E-mail addresses: christian.diem@s.wu.ac.at (C. Diem), anton.pichler@maths.ox.ac.uk (A. Pichler), stefan.thurner@muv.ac.at (S. Thurner).

https://doi.org/10.1016/j.jedc.2020.103900

0165-1889/© 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license.

( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

centralityofbanksinthefinancialnetwork,andAlteretal.(2015),whoshowthatcapitalrequirementsbasedoneigenvec- torcentralitycansaveupto15%oftotalsystemlosses.Moreover,intheclassicalriskmeasureliterature,followingArtzner etal.(1999) andFöllmer andSchied (2002), therisk of an assetismeasured by theamount ofcapital that needs tobe addedtothepositioninordertomakethepositionacceptable totheregulatorortothefirmitself.Thisapproachcanbe extendedtodeterminethecapitalrequirementsforfinancialinstitutionstobringsystemicrisktolevelsthatareacceptable totheregulator;seeforexample,Feinsteinetal.(2017)orBiaginietal.(2018).

Intherecentpast,afterthelast financialcrisis,bankcapitalrequirementshavebeenadjustedupwards.IntheBaselIII AccordtheregulatoryminimumcapitalrequirementsforCommonEquityTier1(CET1)havebeenincreasedfrom2%to4.5%, andTier1Capitalfrom4%to6%(BCBS,2011a).Additionally,acapitalconservationbufferhasbeenintroducedbyincreasing CET1andTier1capitalfurtherto7%and8.5%,respectively.Ontopofthis,nationalauthoritiescansetanadditionalcounter- cyclical buffer in the rangebetween zero and2.5% forphases of excessive credit growth.Global systemicallyimportant institutionshavetomeetadditionalCET1requirementsinarangeof1–2.5%(BCBS,2011c).

Bankcapitallevelshavebeensteadilyincreasingsincetheintroductionofthesenewregulations.Themonitoringreport oftheBaselCommittee(BCBS,2011b)showsthatforasampleof86internationalbankswithTier1Capitallargerthan$3bn, theCET1increasedfrom7.2%to12.7%intheperiodfrom2011to2018(BCBS,2011b,Graph15).ForGermany,Spain,France, andItaly, ECBdatashowsincreasesinTier 1capitalratiosfrom9.2%,8.1%,8.4%and6.9%to16.4%,13.2%,15.3%,and14.4%, respectively, forthe periodfrom 2008to 2017¹. Nonetheless,some indicatorsofsystemic risk suggest that systemic risk levelsarenotdeclining,butarestillsubstantiallyhigherthanbeforethefinancialcrises.AprominentexampleistheSRISK indicatorofBrownleesandEngle(2016),whichshowsthatthesystemicrisklevelinEuropenowistwiceashighasitwas beforethecrises².

However, capital levelsfor absorbing shocksare only one partof thestory inthe context ofsystemic risk. The other essentialcomponentthatdeterminessystemicriskistheexposurenetworkthatisgeneratedbycontractsbetweenfinancial agents.In particular,thesenetworkscapturethe risksofpotential cascadingeventsthat could threatenlarge fractionsof financial marketswithfailure. Thisfactis reflectedina numberofworkssuchasinAllenandGale (2000),Freixasetal.

(2000),EisenbergandNoe(2001),Bossetal.(2004a),Contetal.(2010),GaiandKapadia (2010),Battiston etal.(2012c), Markoseetal.(2012)ThurnerandPoledna(2013)andGlassermanandYoung(2015).

It is therefore natural to ask what contributions to systemic risk originate specifically from networks andhow their topology influences systemicrisk. Indeed,manycontributions to thesystemicrisk literature investigatethe effectofnet- work characteristicson systemicrisk. AllenandGale (2000) compare theeffects ofdifferentnetwork topologies,such as rings, fullyconnectedgraphs, andinterconnectedsubgroups oninterbank market stability.InBoss etal.(2004a)the role ofscale- free networktopologies inthecontext ofsystemic risk andstabilityisdiscussed.In Bossetal.(2004b) thebe- tweennesscentralitymeasureisintroducedasanetwork-basedmeasureforsystemicrisk.Nieretal.(2007)investigatethe effectsofnetworkconnectivityandconcentrationoncontagiousdefaults.GaiandKapadia(2010)employastylizedanalyti- calcontagionmodelandlookatthefractionofdefaultingbanksforgivenaveragedegrees.Puhretal.(2012)employpanel regressionstostudytheeffectsofnetworkmeasureslikeKatzcentralityonthenumberofdefaultingbanks,whichareob- tainedfromasimulation study.Theconceptoftoointerconnectedto fail isalsopartofthisdiscussion andis investigated, forexample,byMarkoseetal.(2012).GlassermanandYoung(2016)dedicateaconsiderablepartoftheir literaturereview tothistopic.Theseandmanymoretheoreticalandempiricalworksindicatethepossibilityofusingnetworksoffinancial connections asaleveragepointforreducingsystemicrisk ina financialsystemasaneffectivealternativetocostlycapital requirementsthathavebeenshowntohavelimitedeffectsonsystemicriskreduction(Polednaetal.,2017).Wealsofindin ourcurrentstudythatthereorganizationoftheinterbanknetworkscanyieldlowerlevelsofDebtRankthanaBaselIII-like equityincrease.Ifsystemicrisk canbeeffectivelyreducedbyalteringtheunderlyingexposurenetworkcharacteristics,this shouldbeprominentlyfactoredintofinancialmarketstabilitypolicies.Itisthereforeessentialtosystematicallyestimatethe fullpotentialfornetwork-basedsystemicriskreduction.

Inthiswork wepropose amethod forquantifying thesystemicrisk reduction potentialin empiricallyobserveddirect exposure networksbyemployingstandard optimizationtechniques.The systemicrisk ofa networkismeasured withthe so-calledDebtRank(Battistonetal., 2012c). The actualoptimizationrelieson anapproximation oftheDebtRank,because theDebtRankiscomputediterativelyandisthushardtouseinoptimizationproblems.Theapproximationisbasedonthe direct impactsof defaultingbankson their neighboringnodesin theexposure network. Weshow how thesystemic risk optimizationcanbesolved asamixedintegerlinearprogram(MILP)by standardreformulationtechniques.The optimiza- tionproblemcanbe solvedbystate-of-the-artoptimizationalgorithmsandcouldthereforealsobe easilyimplementedin practice.Intheempiricalpartofthisstudyweshowtheeffectivenessoftheproposedmethodbyapplyingittoadataset containingtenquarterly observationsofthe Austrian interbankliability networkfrom2006 to2008.Ourfindings forthe 70largestbankssuggestthatthe DebtRankofindividual bankscanbereducedonaverage byafactorof3.5or71%.This meanssizeablereductionsoftheDebtRankforalmostallofthe70banksacrossthetenquarterswithonlyafewexceptions.

Weevaluatetheeffectivenessofouroptimizationbycalculatingthatbankequitywouldneedtobeincreasedbyfactorsof between2.38and4.26toachieve thesamelevelofDebtRankintherespectivequarter.Theaveragescalingfactoris3.32,

1ECB Statistical Data Warehouse: Consolidated banking data set.

2SRISK levels for different regions are provided by https://vlab.stern.nyu.edu/welcome/risk/

thusonaverage232%oftheexistingbankcapitalwouldneedtobeaddedtothebankingsystemtoreducetheDebtRankof theempiricalnetworkstotheleveloftheoptimizednetworks.Forcomparison,theincreaseinTier1capitalfrom4%to8.5%

byBaselIIIcorresponds toascalingfactorof2.125.IncomparisontothisBaselIII scenario,theoptimizationstillreduces DebtRankbyafactorof1.61.

Inpractice,duetothecurrentlackofincentiveschemesforsystemicriskmanagementLeducandThurner(2017),finan- cialnetworksdonotevolve towardsystemicallyoptimalconfigurations,andobviouslythey donot resultinanywayfrom suchoptimizationprocedures.However,ourstudycangive anestimateforthesystemicriskreductionpotentialstemming fromaspecificreorganizationofempiricallyobservednetworks.Thesameoptimizationalgorithmcanbe usedtocompute networkconfigurationsthatyieldamaximumofoverallsystemicrisk.Inthisway,foranyobservedfinancialnetwork,the proposedoptimizationprocedureyieldsa“range” ofnetworkstructures,correspondingtominimalandmaximumDebtRank.

Thisallowsustoidentifynetworkcharacteristicsthataretypicalforlow,medium,andhighDebtRank.

Closelyrelatedstudiesinclude PolednaandThurner(2016) andLeducandThurner (2017),whichinvestigatehow sys- temicriskcanbereducedbychangingtheunderlyingnetworkswhenfinancialagentsareincentivizedtofavortransactions withlow systemicrisk inthenetwork. Theideaofapplyingnetwork optimizationtechniquesthatare commonlyusedin operationsresearchtosystemicriskreductionisrelativelynew.Ithasbeenpioneeredinthespecificcontextofoverlapping portfolioandfiresalesbyPichleretal.(2020)whofindreductionsinsystemicriskofaround50%byrearrangingthenet- workstructureofthe49majoreuropeanbanks’governmentbondportfolios.However,theoptimizationapproachthere– a quadraticallyconstrainedquadraticprogram(QCQP)– issubstantiallydifferentfromtheonepresentedhere.Arecentpaper byKrause etal.(2019)focusesonsmallhomogeneous macroeconomicshocksaffectingthe assetsofallbankssimultane- ouslyandhowtheseshocksareamplifiedinthebankingsystem. Theyshow aMonteCarlo algorithmforfindingminimal andmaximal networkswithrespecttotheamplificationofsuchsmallhomogeneousmacroshocks.Anotherrelatedstudy isAldasoro et al.(2017).The authors employa theoretical modelof theinterbank network where optimizingrisk-averse banksinvestinilliquidassetsandlend toeachother.Intheirmodelthey accountforcontagionoriginatingfromliquidity hoarding,interbankinterlinkages,andfiresales.Theirmodelleadstoaspecificinterbank networkforwhichpropertiesof thenetworktopologyarereported.

Thepaperisorganizedasfollows.Section2presentsourapproachtoquantifyingsystemicrisk.InSection3we derive theoptimization problemforreducing DebtRank.We discussthe data andthe resultsof theapplication to theAustrian interbankmarketindetailinSection4.WeconcludeinSection5.

2. Quantifyingsystemicrisk

Quantificationof systemic risk in financial networksis a non-trivial taskanddepends on specific aspects of interest.

Based on very differentideas, various systemic risk measures have been suggested. Some, such asthose based on net- works,havealready been mentioned above.Other well-known approachesinclude the^{CoVaR, by AdrianandBrunner-} meier(2016),whichmeasuresthetaildependenceofbankassetreturns,thesystemicexpectedshortfall(SES)by Acharya etal.(2017)measuring thetendencyofa bankto beundercapitalizedifthe wholesystemisundercapitalized, theSRISK measureproposedbyBrownleesandEngle(2016),andtheputoptionportfolioapproachbyLehar(2005).Theadvantageof thesemarket-basedmeasuresforsystemicriskmeasurementisthattheydonotrequiredetailed(oftenrestricted)informa- tionfromfinancialnetworksbutestimate systemicrisk fromopenly accessibledata.Thesemodelsare unabletoestimate thecontributionsfromcascadingeffectsthroughfinancialexposurenetworks.Thedifferencebetweenthesetwostrandsof literatureisemphasizedbyBenoitetal.(2017).

Here we choose the network-based measure DebtRank asa wayto quantify systemic risk. The following method for minimizing systemicrisk isthen applicable to all directfinancial exposure networks, whenever DebtRankis used asthe measure for systemic risk. Examples of analyzing systemic risk on networks include interbank networks Battiston et al.

(2012c);ThurnerandPoledna(2013),derivativesandforeignexchangePolednaetal.(2015),andcredit-defaultswapsLeduc etal.(2017).Withoutlossofgeneralityforanykindofdirectexposurenetwork,wedemonstratethemethodforinterbank asset-liabilitynetworks.

WemodeltheinterbankmarketwithNbanksasadirected weightednetworkrepresentedbytheasset-liabilitymatrix, L. The nodes representbanks; linksare the liabilitiesbetween banks. Ifbank j lendsL_ij (monetaryunits) to banki, we representthisasadirectedlinkfromnodeitonodejwithacorrespondingweightofL_ij.L_ijisj’sexposuretowardsi,(i.e., ifidefaultstheamountL_ij isatriskforj).Wedenotethetotalinterbankliabilitiesofbankitoallothersinthenetworkby l_i=N

j=1L_{i j};thesumofallloansfromitootherbanksisa_i=N

j=1L_ji.Theequityofbankiisdenotedbye_i,andthetotal interbankmarketvolumeinthenetworkisL¯=N

i=1l_i=N

i=1a_i.Therelativeeconomic weightofbankiinthenetworkis

v

_i=a_i/L¯.

Inthecaseofthedefaultofi,weassumethatbankj needstowriteoffL_ij ofitsassets.Forsimplicity,weassumezero recovery.Notethatthisassumptionisnot entirelyunrealisticforshorttimescalesandisfrequentlyusedintheliterature.

Ingeneral,asimple wayofassuminga positiverecovery rateistoassume thattheexposure atdefaultissimplyreduced bythisrecoveryrate. Iftherecovery rateisassumedtobe,say, 40%ontheliabilityL_ij,thismeansthat atmost0.6L_ij has tobewrittenoff bybankj,ifbankidefaults(lossgivendefault).Ifweassumeageneralrecoveryofx∈[0,1]thiswould amounttousingtheweightednetwork(¹−x)^{LfortheDebtRank}calculations.DuetothenatureofEq.(1)thisisthesame

asconsideringascaledequityvector(¹/(¹−x))^e,andthiscaseistreatedinAppendixC.Thus,theassumptionofarecovery ratewouldnotaffecttheinterpretationofourresultsqualitatively.

Asabankcannot havenegativeequity,themaximumimpactthat icanhaveonjise_j.Thismotivatesthedefinitionof thedirectimpactmatrix,

W_{i j}=min

Li j

e_j,1

, (1)

whichdenotestheshareofj’sequitylostduetothedefaultofbanki.Asstatedabove,wequantifysystemicrisk byusing DebtRank.DebtRankis arecursive centralitymeasuredesignedspecifically fornetworksofdirectfinancial exposures and quantifiesthe impact ofbank i onthe entirenetwork ifi defaults.Every banki has aDebtRank value, R_i, betweenzero andone;R_i=0meansthatbankihasnoimpactonotherbanks,whereasR_i=1indicates thattheentireinterbankasset- weightedequityofthesystemisatrisk,shouldi default³ Inthatsense,R_iisthefractionoftheaffectedtotalvalueinthe networkthroughi’sdefault.

Definition 1 (DebtRank). DebtRank is definedby an iterativeprocedure that involves two state variables, h ands. h_i(t) measures thelevelofdistressatiterationt; itisthefractionofequity(e_i), lostduetolossesthatoccurredbeforetimet. Consequently,h_i(t) ∈[0,1], whereh_i(^t)=1meansdefault;i.e.100%ofbankisequityislost.Thevariables_j(t) ∈{U,D,I} takesoneofthreestates:undistressed,distressed,andinactive.DefineSasthesetofbanksthataredistressedatinitialtime t=1.Then,variablesareinitializedwithh_i(¹)=

ψ ∀

ⁱ∈Sandh_i(¹)=0

∀

ⁱ∈/S,ands_i(¹)=D

∀

ⁱ∈S,ands_i(¹)=U

∀

ⁱ∈/S.The variable

ψ

^{specifiestheamountofinitialdistresswhere}

ψ

=1meansdefault.Fort≥2firsth_i(t)isupdatedsimultaneously foralli,followedbyanupdateofs_i(t),foralli.Theupdaterecursionisgivenby

h_i

(

^t

)

=min

1,h_i

(

^t−1

)

+

j

W_jih_j

(

^t−1

)

,where j∈

{

^j:sj

(

^t−1

)

=D

}

^{, (2)}

and

si

(

^t

)

=

_{D ifh}

i

(

^t

)

>0;si

(

^t−1

)

=I I ifsi

(

^t−1

)

=D

s_i

(

^t−1

)

^{otherwise. (3)}

TheiterativeprocedureendsafterTstepsatwhichallnodesareeitherundistressedorinactive.TheDebtRankofbankiis definedas

Ri= N

j=1

hj

(

^T

) v

j− N

j=1

hj

(

¹

) v

j=

j=i

hj

(

^T

) v

j. (4)

Thelastequalityholdsbecauseweassumethatonlybankiinitiallydefaults,leadingtoh_i(¹)=h_i(^T)=1.Theinitialdistress issetto

ψ

=1,andtheinitialdefaultsetcontainsonlybanki.Theshareofbankj’sinterbankassets,v_j,canbeinterpreted as therelative economic value ofj in thenetwork. When bank i initially defaults,and asa consequence bank j loses a fractionofitsequity,h_j(T)> 0.Then,thevaluev_jdetermineshowstrongthislossisreflectedintheDebtRankofbankR_i. WedefinethesystemicriskoftheentiremarketasthesumoftheindividualbankDebtRanks,i.e.

R= N

i=1

R_i. (5)

Fora motivationof thisdefinition,seealso PolednaandThurner (2016).Forcomparative purposes,we alsoemploy a variationofthisdefinitionofDebtRankpresentedinBardosciaetal.(2015).We refertothisdefinitionasDebtRank2.For moredetails,seeAppendixG.DebtRank2hasbeensuggestedasamicrofoundationforshockpropagationinnetworksand isdirectlyderived frombankbalancesheetidentities.Bardosciaetal.(2015)acknowledgethat theoriginalDebtRankfor- mulationcanleadtounderestimationsofsystemicriskbecauseshockspropagatethroughanodeonlyforasingletimeand subsequently thenode becomesinactive.If a bankreceives shocks fromdifferent neighborsat sequentialtimes ittrans- mitsonlythefirstshock, asitbecomesinactive afterreceivingthefirstshock.Similarly,whenabankreceivingashockis partofa loopandwill receiveanothershockfromthesame loopata latertime, itwill notforwardthe shocka second time.ThetwoDebtRanksarethesamefortreenetworksandsomeotherspecialstructures.Ingeneral,DebtRankisalower boundto DebtRank2(Bardosciaetal., 2015). However,sinceDebtRank2allowsformultipleshocktransmissionsofanode thisleads(in principle)to an infinite numberofshocks onnetworks that containloops. In practice,the algorithmstops whentheshocksbecomesmallerthanapredefinedvalue

^{.However,intheoriginalDebtRank}formulationBattistonetal.

(2012c) point out that an infinite cyclingof shocks when loopsare present mightnot be desirable.For thisreason and becauseintheliterature theoriginalDebtRankismorewidely used,westick totheoriginal DebtRankforthe restofthe paper.Anotherinteresting generalization ofDebtRank isstudiedby Bardoscia etal.(2016),which relaxestheassumption thatshockspropagatelinearly.

3From the definition of DebtRank, it is obvious that R i= 1 can occur only if the weight vi= 0 . Thus, in most cases R iis strictly smaller than one.

3. Minimizingsystemicriskasanoptimizationproblem

Thissectionproposesan optimizationprocedure⁴ thatrewires agiveninterbanknetwork toobtaina second(optimal) networkthatisclosetotheoptimalDebtRankfortheprevailingeconomicenvironment(i.e.,foragivenlevelofequity,bank lendingandborrowing,andbanks’riskexposuretootherbanks).BecauseofitsrecursivedefinitioninEq.(4),DebtRankis notrepresentableinclosedform.Thismakesitunpracticaltouseasanactualobjectivefunctionofanoptimizationproblem.

EventhoughanoptimizationwithrespecttoDebtRankisofcoursepossibleinprinciple,itwouldbecomputationallycostly oreven infeasible forlarge networks. Instead we propose a practical andeasy- to-implement methodthat is capable of reducingsystemicrisk (DebtRank)substantially inempirical networks.Forthis,weapproximateDebtRank,R,byasumof piecewiselinearconcavefunctionsthatservesasobjectivefunctionintheoptimization.

Definition2(DirectImpact). ThedirectimpactI_iofbankionitsneighbouringbanksisdefinedby Ii=

N

j=1

Wi j

v

j=1 L¯

N

j=1

min

L_{i j} ej,1

aj. (6)

ThesumofalldirectimpactsisI=N

i=1I_i,whichcanbeinterpretedasafirst-orderapproximationoftheDebtRank.

Directimpact, I,is representable inclosedform, Eq.(6),that allowsusto solvetheoptimizationproblemwithMixed IntegerLinearProgramming(MILP)techniques.Theoptimizationrewireslinksinthenetworkinanoptimalwaybutshould leavecertainnetworkcharacteristicsuntouched.Toensureaneconomicallymeaningfulcomparison,wekeeptotalassetsand liabilitiesofbanksunchanged,aswellasthetotalmarketvolume,L¯.Additionally,werequiretheinterbankriskexposureto remainthesameforeverybank.If

κ

jdenotesthecreditriskofbankj,theaveragecreditriskexposureofbankiisgivenby r_i=N

j=1L_ji

κ

j.Weimplementtheserequirementsasconstraintsintheoptimizationanddiscusstheireconomicmeaningin moredetailinSection3.1.Nowtheoptimizationproblemcanbeformulatedas

min

L∈{^M:M∈R^N×N+ ,Mii=0} N

i=1

N

j=1

min

Li j

e_j,1

a_j

subjectto l_i= N

j=1

L_{i j} ,

∀

^{i (7)}

ai= N

j=1

Lji ,

∀

ⁱ

ri= N

j=1

Lji

κ

^{j ,}

∀

^i.

Thevaluesfore, l,a,v,andL¯canbe obtainedfrombankbalancesheets,whereastheinterbanknetwork Lisusually not publiclyavailable.NotethatifthelastconstraintinEq.(7)isomitted,theoptimizationonlyrequiresrowandcolumnsums ofL.Thus, optimalnetworkscanbe obtainedwithoutknowingtheexactempirical network.Theobjectivefunctionisnot linearbutpiecewiselinearandconcavebecauseoftheminimumoperator;thesumofconcavefunctionsisconcave.InEq.

(7)weomitL¯becauseitisjustapositivemultiplicativeconstant.Theresultoftheoptimizationistheoptimalasset-liability matrix,L^∗. Aglobal optimum exists becauseof theconcavity ofthe objectivefunction and dueto thebounded solution space(L_ij∈[0,min(a_i,l_i,a_j,l_j)]

∀

^{ij).However,theoptimumisnot}necessarilyunique.Wefindgloballyoptimalsolutionsby solvinganequivalentMILP,whichisderivedinthefollowing.

Theoptimizationproblemcomprises N²−N freevariables(no self-links),whichturnsevenmoderatelylargeinterbank marketsintolarge-scaleoptimizationproblems.Tosolvethisproblem,welinearizetheobjectivefunctionbyreformulating itasaMILP.Astheminimumfunctionispiecewiselinear,onecanapplystandardtechniquesofmathematicalprogramming torewriteEq.(7)asaMILP.We usetheconceptofspecialordered sets(SOS)and, morespecifically,SOS2constraintsfor thelinearizationoftheobjectivefunction.ThisconceptdatesbacktoBealeandTomlin(1970)andallowsustofindaglobal solution.

We firstprovide some intuitionof the behavior ofthe objective function Eq.(7), andthen explain the reformulation indetail.As alla_j arenon-negative,we canwrite them insidetheminimum function,anda singletermin theobjective functioninEq.(7)reads, min ^a_e^j

jL_{i j},a_j

.Fig.A.6intheappendixshowsitsbehavior.It increasesuntilL_{i j}=e_jbya_j/e_j and remainsconstantafterwards.Below,weshowhowtorelateeachentryL_ij toapairofvariables,(^y2k−1,y_2k)^.y2k−1accounts forthe part ofL_ij, wherethe objective still increases in L_ij; y_2k accountsfor the region, wherethe objective function is constant(w.r.t.L_ij).TheeconomicinterpretationofthetransitionpointatL_{i j}=e_jisthattheliabilityofbanki,withrespect

4We provide the R code for the optimization procedure and an example with simulated data on https://csh.ac.at/vis/code/network _ optimization/

tobankj,isofthesamesizeasbankj’sequity.Inthecaseofi’sdefault, 100%ofj’sequitywouldbedestroyed.However, when L_ij > e_j, y_2k=min(⁰,L_{i j}−e_j) ^{no longer affects the objective function, as more than 100% of j’s equity cannot be} consumed.Notethattheremaininglossofmin(⁰,L_{i j}−e_j)^{isbornebycreditorsofj,whichareoutsidetheinterbanksystem.}

Wenow showmoreformally howtheobjectivefunctioncanbe transformedintoaMILPwiththehelp ofthevariablesy andasetofdummyvariables,

δ

^.

We statedthe optimizationproblem inmatrix terms.In numericaloptimization itis more commonto optimizeover vectors.WethereforerewriteL∈R^N₊^×N intoavectorx∈R^N₊² bystackingthecolumnsofL,

x=vec

(

^L

)

=

(

^L11,...,LN1,L12...,LN2,L1N,...,LNN

)

. (8) NotethatforeaseofnotationandimplementationwekeepthediagonalentriesL_ii,

∀

^{i.Similarly,wedefinevectorsoflength}

N²forrepresentingassets,liabilities,andequities,

¯

e=

(

^e

1,. . .

,e

1 Ntimes

,e2,...,e

N,.

..,eN

Ntimes

)

, (9)

¯

a=

(

^a

1,.

..,a

1 Ntimes

,a2,...,a

N,.

..,aN

Ntimes

)

, (10)

and

l¯=

(

^l,...,l

Ntimes

)

. (11)

NowwecanwritetheobjectivefunctioninEq.(7)as min

x∈R^N+²

N² j=1min

_a¯

j e¯jx_j,a¯_j

. (12)

The elements inx corresponding to the diagonal elementsof L haveto be zero, which can be enforced withadditional constraintsordirectlyintheoptimizationsoftware.Totranslatetheobjectivefunctionintoalinearformcy,everyvariable x_iissplitintotwoparts,y_2i−1andy_2i,withx_i=y_2i−1+y_2i,and

y2i−1 =min

(

^xi,e¯i

)

, (13)

y_2i=min

(

^xi−e¯_i,0

)

. (14)

The first part, y_2i−1, indicates the range of x_i, where an increase by ^xi leads to an increase in the objective function by ^xi(^a¯i/e¯_i)^{. At x}i=e¯_i, the objective function no longer increaseswith x_i. This range of x_i is accounted for by y_2i. To reformulate theobjectivefunction intermsofthenewvariables y,we needto introducea vectorofbinaryvariables

δ

∈

{

⁰,1

}

^{2n2 inthefollowingway}

δ

^j=

1 if y_j>0 0 if yj=0.

With

δ

^{wecanformulatethefollowing}constraintsforthepairs(^y2i−1,y_2i),foralli,

δ

2i−1≥

δ

2i (15)

y2i−1 ≥

δ

^{2ie¯i (16)}

y_2i−1 ≤

δ

2i−1e¯_i (17)

y_2i≤

δ

2imax 0,min a¯_i,l¯_i

−e¯_i

. (18)

Constraints(15)–(18)ensuretheequivalenceofthereformulatedprobleminEq.(20)andtheoriginalprobleminEq.(7).In particular,Eq.(15)enforcesthaty_2icanonlybelargerthanzeroify_2i−1islargerthanzero.Constraints(16)and(17)enforce that y_2i−1 mustbesmallerthane¯_i,andthat ify_2iisbiggerthanzero,y_2i−1 hastobeequal toe¯_i.Finally,Eq.(18)ensures that x_i=y_2i−1+y_2iissmallerthan therespectiverowandcolumnsumofthecorresponding entryinthe liabilitymatrix.

Wefinallydefinethevectorcoflength4N²,whichdeterminestheslopewithwhichtherespectiveentriesinyincrease,

c_j=

⎧ ⎨

⎩

a¯i

¯

ei ifj=2i−1,andi≤N²

0 if j=2i,andi≤N² 0 if2N²<j≤4N².

(19)

Itfollowsthateverysecondentryinc_2iisequaltozero,astheevencomponentsy_2idonotincreasetheobjectivefunction.

Thesecorrespondtothepartofx_i,wheretheobjectivefunctioniscappedtoa¯_i.Theoddparts,c_2i−1,representtheslopes.

Thelast2N²zerosensurethatthebinaryvariables

δ

^{donotaffectthevalueoftheobjectivefunction.Theobjectivefunction}

cannowbewrittenascz,wherez=(^y,

δ

)∈R⁴ⁿ².

Theconstraintsfor

δ

^{andy,Eqs.(15)–(18),arecompactly}reformulatedasA₁z=0,whereA₁∈R^4n2×4N2,and0denotesa zero-vectoroflength4N².Theconstraintsontherowandcolumnsumsoftheliabilitymatrixintheinitialproblem,Eq.(7), canbe writteninstandardmatrixformasA₂z=a,A₃z=l,andA₄z=r.A₂,A₃ andA₄ areN ×4N² dimensionalmatrices consistingofzerosandones⁵.TheexactstructureoftheconstraintmatricesA₁,A₂,andA₃isoutlinedinAppendixA.Finally, theoptimizationproblemofEq.(7)asaMILPreads

min_z∈R4N2 + cz subjecttoA1z≤0

A2z=a A3z=l A4z=r.

(20)

Thismethodisgeneric andisgenerallyapplicableto alldirectfinancialexposurenetworkswheresystemicriskisquanti- fiedbyDebtRank.Ifdifferenttypesoffinancialnetworkare considered,theliabilitymatrixLhastobe replacedwiththe correspondingexposure matrices. Dependingonthe financialnetworktype, various furtherconstraintscan beconsidered toensurethatcertaineconomicpropertiesofindividualbanks(whichdependonthenetworkbutshouldkeptconstant)do indeedremainthesameaftertheoptimization.Wecontinuebydiscussingsuchconstraintsinmoredetail.

3.1. Implementingeconomicconstraints

Asmentioned, theconstraints inEq.(7) not onlyensure that banksretain their sizeafter optimization,butthey also havean importanteconomicinterpretation.Rowandcolumnsumsrepresenttheinterbankliabilitiesandinterbank assets ofeach bank. Keeping these constant impliesthat each bank retainsits amount ofliquidity⁶ from the interbank market afteroptimization.Ifweassumethattheliquidityabankrequiresfromtheinterbankmarketoriginatesfromitsoperational business,itisimportantthatthisactivityisnotdistortedbytheoptimizationprocedure.

Anotherimportanttypeofeconomic constraintisrelatedtoeconomic risk.Indirect exposurenetworks, counter-party creditriskplaysan importantrolewhenlendingdecisionsarebeingmade. Inthecaseofindirectexposurenetworks,such asoverlappingportfoliorisk, risk associatedto thefinancialassetsheld by thefinancialinstitutions play acrucialrole in themaking of investment decisions.The mostimportant typesare credit, market,and interest raterisk. Forthe sake of comparabilityofempiricallyobservedreferencenetworksandoptimizednetworks,itisdesirabletohaveconstraintsinor- dertoensurethat therisksfortheindividual institutionsremaincomparablebeforeandafteroptimization.Forinterbank networksnoneofthelendingbanksshouldendupwithahighercounter-partycreditriskaftertheoptimization.Toachieve this,weintroducedalinearconstrainttoensurethatthecreditriskinallinterbankloanportfoliosisapproximatelymain- tained.Thisconstraintaccountsforindividualeconomicconditionsofbanksthatareaffectedbythenetworkstructure.We aimtomodelthisfeatureby fixingthepredominantcredit-risk-weighted-exposureoftheinterbank loanportfolioofeach bank.

Letthe consideredcreditrisk indicatorofbank ibe

κ

i.ForagivenliabilitymatrixL,therisk weightedinterbankloan exposure of bank j, that is implied by the interbank network L, is then given by r_j=N

i=1L_{i j}

κ

i, or in matrix notation, r=L

κ

^{.TheconstraintintheMILPofEq.(20)isaddedas}

A4z=r. (21)

We explain in Appendix I that the formulation ofconstraint Eq.(21) asequality, andsmaller or equal, yields the same optimalvalueoftheprobleminEq.(20),giventhattherowandcolumnsumconstraintsareinplace.Further,thisconstraint alsokeepsthe earnings fromthe interbank loanportfolio similar beforeandafter optimizationbecause theinterest rate earnedonaninterbankloanshouldstronglyreflectthecreditriskleveloftheborrower.Additionally,theregulatorycapital leviedontheinterbankloanportfolioalsoremainscomparablebecausecapitalrequirementsdependontherisk-weighted assetsoftherespectivebank.Justastherisk-weightedinterbankloanexposureremainsconstantintheoptimization,sothe risk-weightedassetsshouldalsoretaintheirsize.

Forthecaseofoptimizingindirectexposurenetworks,similarriskconstraintscanbeimplemented.Forexample,Pichler etal.(2020)considerMarkowitzmean-varianceconditionsforoptimizingfinancialexposuresemergingfromcommonasset holdingsanddiscussfurtherpossibleconstraints.Other meaningfulconstraintsforfinancialassetnetworksarecreditrisk constraints,suchthattheaveragecreditrisk– of,forexample,abondportfolio– remainscomparable.Tokeeptheinterest

5Note that there is at least one redundant equation in this set of linear constraints, as N column sums and N −1 row sums imply the N th row sum.

6As we deal only with a single liability matrix, L , we implicitly assume in the optimization procedure that all liabilities have the same maturity, which is of course not realistic. If a family of matrices, L 1, . . . , L t, describing the interbank liabilities for various maturities (or maturity buckets) 1 , . . . , tis available, and the optimization procedure is applied to each maturity separately, so that the original maturity structure is unaffected.

rateriskoffixed incomeportfoliossimilaracross theoptimization,anotherlinearconstraintcanaccount forthematurity ordurationoftheassets.Ingeneral,differentfinancialnetworkswillrequiredifferenteconomicconstraints.

4. OptimizationofempiricalAustrianinterbanknetworks

The solution tothe MILPyields a network withminimal direct impacts,I,but not necessarilyone withminimal sys- temicrisk intermsofDebtRank,R.However, ourcomputationsdemonstratethegreateffectivenessofthisapproximation inmassivelyreducingoverallsystemicrisk.

Weapply theoptimizationto adatasetconsistingof10snapshotsofAustrianinterbanknetworksat10quartersfrom 2006to2008.The Austrianinterbanknetworkhasbeenstudiedbeforeby,forexample,Bossetal.(2004a),Elsingeretal.

(2006),Cacciolietal.(2015)⁷.Thesamplecontainsbetween824and846banks.TheAustrianbankingsystemaccommodates manyverysmallcooperativebanks,whichcannotbeconsideredassystemicallyimportant.Weusethe70largestbankswith respecttototalassetsinthecorrespondingquarterfornumericalfeasibility.Theseaccountforabout86%oftotalassets;the 70th largest bankaccountsfor around0.12% oftotal assets.The 70bankswiththe largesttotal assetscover around71%

ofthe interbankmarket. Wechoose thebanks’ totalassetsize astheselection criterion becausetotal assetsshouldbe a morestablequantitythaninterbank marketshares.Wedeal witha fullyanonymizeddataset,whichmakes itimpossible to estimatethe bank’scredit riskindicators,

κ

i.Approaches forestimating

κ

^{are outlinedin AppendixHforcaseswhere}

sufficientdatawouldbeavailable.Fordemonstrationpurposesweapproximate

κ

^{bytheleverageratioofthebanks’}

κ

ⁱ⁼ _{total assets}^{total assetsi}

i−total liabilitiesi. (22)

Weassumethatahigherleverageratioimplieshighercreditrisk.Tosolvetheoptimizationproblemnumerically,weemploy theMILPsolvercplex,availableintheROptimisationInfrastructure(ROI)package(Theußletal.,2019).Theoptimizationcan beperformedonastandardnotebookandtakesfromafewminutestoseveralhours,dependingonthenetworksample.

Tocomparetheeffectivenessofnetworkoptimizationwithactual policieswe alsocompute theDebtRankforincreased valuesofbankequity.Aspointedout intheintroduction,Tier 1capitalrequirementsincreasedunderBaselIIIfrom4%to 8.5%.ECBdatashowsincreasesofTier 1capitalratiosforGermany,Spain,France,andItaly from9.2%,8.1%,8.4%and6.9%

to 16.4%, 13.2%, 15.3%,and 14.4%,respectively, for theperiod from2008 to 2017. The BaselIII increase corresponds to a scalingfactorof2.215,andtheincreasesintheECBdatarangefrom1.63to2.09⁸.Oursamplefrom2006to2008fallsinto thetimeperiodbeforetheBaselIIIregimeandendswhenECBcapitalmeasurementsstart.Therefore,wecanmeaningfully computehowDebtRanklevelswouldhavelookedundertheBaselIIIregime.Inoursamplethemeanequitytototalassets ratio,1/

κ

i,across time andacrossobservations is10.20%; themedian is7.19%. Ourbaseline scenarioforequity increases computestheDebtRankfortheempiricalnetworkswitha hypotheticalBaselIIIequityvector2.125·e.InAppendixCwe show DebtRanklevelsforvarious scalingfactorsup to4,andwe compute the“break-even” capitalincrease atwhichthe samelevelofDebtRankreductionisachievedasintheoptimization.

4.1. Results

ThereductionofsystemicriskobtainedbytheoptimizationprocedureissummarizedinFig.1(a).Thevaluesofthetotal DebtRank,R,afteroptimization(bluetriangles)aresubstantiallylowerthanthecorrespondingempirical(redsquares)ones acrossall quarters.TheaverageDebtRankintheempiricalandoptimizednetworksisaround 12.51and3.54,respectively, meaning that the average totalDebtRank reduction amounts to approximately71%, ora factor ofroughly 3.5. The Basel IIIequityincrease scenario(orange)reducesaverageDebtRanksubstantiallyto5.7, butisnot aseffectiveastheoptimiza- tionprocedure.InAppendixC wefindthat onaveragetheoriginalequitywouldneedtobe increasedby afactorof3.32 to achieve thesamereduction ofDebtRank asthenetwork optimization.Fig.1(b)showstheindividual DebtRanks, R_i, of the 70banksforthe empiricalcase (redsquares) andtheoptimized (blue triangles). The sizeof thesymbolsrepresents thebanks’ interbankliabilities,l_i.Thefigureshowstwofacts. Thefirstisthatintheoptimizednetwork atleastonebank alwaysremains relativelysystemicallyriskywithrespecttothemajorityofbanks,eventhoughtheirDebtRankissubstan- tiallyreduced. Thesecond observation isthat the DebtRankreduction forsmall-andmedium- sized banks, indicated by trianglesize,seemstoworkevenbetterthanforthelargebanks.Fig.D.10showstherelationshipofDebtRankR_iandinter- bankliabilitiesl_iinmoredetail.Intheempiricalnetworkssmallbanksseverely“punchabovetheirweight”,thatis,banks withsmallinterbankliabilitiesfrequentlyhavehighDebtRanks,R_i,and– judging bytheirsize– theirdefaultwouldcause unnecessary systemic events. The optimizationremedies thisproblem andrenders bankswith smallinterbank liabilities systemicallynegligible.

InFig.1(a)itisseenthatfromQ8toQ10theoptimizedDebtRankincreases,whiletheempiricalDebtRankcontinuesits downwardtrend.Tounderstandwhy, inFig.1(c)weshow thetotal interbank marketvolume andthe totalequityinthe systemovertime,relativetothevaluesinQ1.Largerlevelsofequity– allotherthingskeptequal– shouldreduceDebtRank,

7 Caccioli et al. (2015) uses the same data set which originally consisted of 12 quarters. Due to obvious data errors we dropped 2 of the 12 observations.

8We mention here that in the DebtRank framework it does not make a difference if the increase in the capital ratio is realized by increasing equity or shrinking the asset side, since the DebtRank calculation hinges on the matrix W where W i j= min (1 , L i j/e j)and obviously min (1 , 0 . 5 L i j/e j)= min (1 , L i j/ 2 e j).

Fig. 1. (a) Total DebtRank, R , of the empirical Austrian interbank networks across 10 quarters from 2006 to 2008 (squares). For the optimized networks the DebtRank is drastically reduced (triangles). The optimization reduces systemic risk (measured in DebtRank) by a factor of approximately 3.5. (b) Individual DebtRank, R i, of 70 banks for the empirical and optimized networks in the respective quarters. Here symbol sizes are proportional to the banks’ interbank liabilities, l i. We see that typically large banks have high R i, however note that there are many exceptions, with small banks having considerable systemic risk. (c) Total interbank market volume, L ¯, and equity, E ¯= ^N_i=1e i, over the ten quarters. While decreasing in the first eight quarters, the ratio L ¯/ E increases ¯ substantially in Q9 and Q10.

Fig. 2. Systemic risk profile (DebtRank R i) of the 70 banks for the empirical (red) and the minimized (blue) networks in quarter Q1. Banks are rank-ordered with respect to their DebtRank, R i, in the empirical network. It can be clearly seen that systemic risk is drastically reduced for practically all banks, with only one exception. For the 10 most risky banks a reduction of DebtRank, R iby a factor of 2.1 is observed; for higher ranks, the reduction by a factor of 5.1 is even more drastic. Similar results hold for the other quarters. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

andanincreaseinthemarketvolumeshouldincreaseDebtRank.Thus,thesharpincreaseinthemarketvolume,L¯,fromQ8 toQ10couldbetheexplanationfortheobservedincreaseintheoptimizedDebtRank.

Fig.2depictsthesystemicrisk profileforQ1,whereR_iisshownfor70banksoftheempiricalandtheoptimizedcase.

Banksareordered accordingto theirempirical DebtRank;R_i; themostsystemicallyriskyinstitutionisshowntothe very left.Theeffectivenessoftheoptimizationisclearlyseen.DebtRanklevelsaredecreasedsubstantiallyforalmostall70banks, withtwo exceptions,where banks have a slightlyhigher DebtRank after the optimization. Forthe 10 mostrisky banks (shadedgray)DebtRankisreducedbyafactorofaround2.1;forhigherranks,thereductionisevenmorepronounced,and amountstoafactorof5.1.Formostbanks,DebtRankisdecreasedtomarginallevels.Similarobservationsholdtrueforall quarters;insome,aDebtRankreductionisachievedforall70banks.

Fig.3showsthe relationof ourobjectivefunction (direct impacts),I, andDebtRank,R,that serves asour measureof systemicrisk, withwhich we alsojudge theeffectiveness of theoptimization. Onthe network level,the total DebtRank anddirect impacts of theempirical networks are linearlyrelated witha correlation coefficient of

ρ

^emp=0.81, anda p-

Fig. 3. (a) Total DebtRank values, R , versus direct impacts, I , for the 10 quarters of the empirical (squares) and the optimized networks (triangles). The correlation coefficients of R and I for the empirical and optimized cases are ρemp = 0 . 81 , with a p-value of 0.004, and ρopt = 0 . 6 , with a p -value of 0.07, respectively. (b) The same comparison on the individual bank level, R iversus I i, with ρemp= 0 . 90 , and ρopt = 0 . 9 . The associated p -values are smaller than 2.2e −16. The dashed lines are obtained by simple linear regression.

value of pemp=0.004.This confirmsa posteriori that minimizing the directimpacts is indeeda reasonableand effective waytominimizeDebtRank.Intheoptimizednetworksthelinearrelationshipisweaker(

ρ

opt=0.6,andap-valueofpopt= 0.07).ThisindicatesthattheoptimizationachievesastrongerreductionindirectimpactsthaninDebtRank.Fig.3(b)shows thesame situationforthebank’s individuallevels ofDebtRank,R_i,anddirectimpacts,I_i.The linearcorrelationsforboth network typesare higher(

ρ

^emp=

ρ

^opt=0.9) andtheir p-values arebelow 2.2e−16.The respectiveresults forDebtRank2 (Bardosciaetal.(2015)) areshowninAppendix G.Here, theoptimizationachievesan averagereductionof DebtRank2of about15%.

4.2. Hownetworkschangeduringoptimization

Fig.4 (a) showstheoriginal interbank asset-liability network Lbefore theoptimizationfor quarterQ1. Thecaseafter optimizationisseenin(b).Thenodesrepresentbanks;size isthebanks’equity;thecolorsrepresenttheDebtRankvalue (darkredishigh,lighttonesaremedium,darkblueislowR_i).Thereareobviousdifferences.Wenowaskhowthetopology ofinterbank networkschanges duetothe optimizationprocess. Theaverage degreeof theminimizednetwork (fromthe binaryadjacencymatrix) isk¯=3.04 versusthe empiricalnetworkk¯=38.71.The in-andout-degreedistributionsforthe differentnetworktypespooledtogether forall tenquartersareprovided inFig.D.11(a)and(b).Theaveragein-andout- strengthofthenetworksareunchanged,duetotheconstraintsthatkeepa_i(in-strength)andl_i(out-strength)fixed.

Themostprominentobservationisthatnetworksaftertheoptimizationbecomesparser.Fig.4(c)showsthatthemini- mizednetwork(bluetriangles)isextremelysparsewithanaveragelinkdensityofaround4.4%.Everydotrepresentsoneof the10quarters.

The link density of the network (connectancy) is defined as the fraction of links present in the network, d= m/(^N(^N−1)),wheremis the number of presentlinks andN(^N−1)^{is the}number of possible links.In contrast,the empiri- calnetworks(redsquares)exhibitanaveragelinkdensityofapproximately56%.Note,however,thatbyslightlythresholding theempiricalnetworks,linkdensitiesofabout10%areobtained;seeAppendixD.Onecouldbeledtobelievethathighlink densityisrelatedtohighDebtRank.This isnotnecessarily true.Toshow this, wecomputedthe maximumdirectimpact networks(where wemaximizeEq.(20)),whichleadstonetworkswithsubstantiallyhigherDebtRankthantheempirically observedones. Interestingly,thesemaximizednetworks(green diamonds)arealsosparse,withanaveragelink densityof 11%.Themaximizedandthresholdednetworksarevisualized inFig.D.9.Sparsenetworkscanhavelow orhighDebtRank.

InAppendix Fwe furtherinvestigatethisrelationshipby conductingasmall-scalesimulationstudy. Theinitialfindings – whichneedtobecorroboratedinfutureresearch– suggeststhatforsparsenetworksthevarianceinsystemicriskissub- stantiallyhigherthanforoneswithhigherconnectivity.Additionally,therelationshipseemstobe slightlyconcavewhere, onaverage,systemicriskishighestformediumlevelsoflinkdensity.

In Fig. 4 (d) DebtRank R is plotted against the degree assortativity, which is calculated as r=

ρ

(

v

,w) ^where

ρ

^{(., .) is the sample Pearson} correlation coefficient, v, w are m dimensional vectors, and each entry corresponds to a link L_ij. The lth entry corresponding to link, L_ij is

v

l=k^out_i andw_l=kⁱⁿ_j. Compare also Thurner et al. (2018) Eq. (4.10).

Fig. 4. Interbank networks before and after optimization. (a) Empirical asset-liability network, L , as in Q1, in comparison to (b) the optimized network, L ^∗. Node colors of banks represent their DebtRank (large DebtRank is red, small is blue). Node size is proportional to equity, e i. It is obvious that the optimized network is considerably sparser. (c) DebtRank of the empirical, the minimized, and the maximized networks plotted against the networks’ link densities, d . Every symbol represents a quarter. It is clearly seen that sparse networks can have both high and low DebtRank. (d) DebtRank, R , against the degree-weighted assortativity, r . We see a similar level of dis-assortativity in the maximized and minimized networks, while the empirical network is more assortative and the thresholded empirical network is less assortative. (e) DebtRank, R , plotted against the mean local clustering coefficient, ¯c . We see that there is a tendency toward higher local clustering in the empirical network. The minimized and the thresholded networks show similar average clustering. The average clustering in the maximized networks is slightly higher than in the minimized networks. (f) DebtRank, R , plotted against the average weighted nearest neighbor degrees, k ¯^wnn. The smallest values are observed for the minimized network, followed by the maximized networks. The thresholded networks exhibit higher values, the empirical networks substantially higher values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)