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EPJmanuscriptNo. (willbeinsertedbytheeditor) The dynamics of financial stability in complex networks JoaõP.daCruz1,2andPedroG.Lind1,3 1 DepartamentodeF´ısica,FaculdadedeCieˆnciasdaUniversidadedeLisboa,1649-003Lisboa,Portugal 2 CloserConsultoriaLda,AvenidaEngenheiroDuartePacheco,Torre2,14o-C,1070-102Lisboa,Portugal 3 CenterforTheoreticalandComputationalPhysics,UniversityofLisbon,Av.Prof.GamaPinto2,1649-003Lisbon,Portugal 3 Received:date/Revisedversion:date 1 0 2 Abstract. Weaddresstheproblemofbankingsystemresiliencebyapplyingoff-equilibriumstatisticalphysicstoa systemofparticles,representingtheeconomicagents,modelledaccordingtothetheoreticalfoundationofthecurrent n banking regulation, the so called Merton-Vasicek model. Economic agents are attracted to each other to exchange a J ‘economicenergy’,forminganetworkoftrades.Whenthecapitallevelofoneeconomicagentdropsbelowaminimum, theeconomicagentbecomesinsolvent.Theinsolvencyofonesingleeconomicagentaffectstheeconomicenergyofall 3 itsneighbourswhichthusbecomesusceptibletoinsolvency,beingabletotriggerachainofinsolvencies(avalanche). ] Weshowthatthedistributionofavalanchesizesfollowsapower-lawwhoseexponentdependsontheminimumcapital M level. Furthermore, we present evidence that under an increase in the minimum capital level, large crashes will be avoidedonlyifoneassumesthatagentswillacceptadropinbusinesslevels,whilekeepingtheirtradingattitudesand R policies unchanged. The alternative assumption, that agents will tryto restore theirbusiness levels, may lead to the n. unexpectedconsequencethatlargecrisesoccurwithhigherprobability. i f PACS. XX.XX.XX NoPACScodegiven - q [ 1 Introduction 2 v The well-being of humankinddependscrucially on the finan- 7 1 cial stability of the underlying economy. The concept of fi- 7 nancialstability is associated with theset of conditionsunder 0 which the process of financial intermediation (using savings . fromsomeeconomicagentstolendtoothereconomicagents) 3 0 issmooth,therebypromotingtheflowofmoneyfromwhereit 1 isavailabletowhereitisneeded.Thisflowofmoneyismade 1 througheconomicagents,commonlycalled ‘banks’,thatpro- : videtheserviceofintermediationandanupstreamflowofin- v i terest to pay for the savings allocation. Because the flow of X moneythatensuresfinancialstabilityoccursontopofacom- r plexinterconnectedsetofeconomicagents(network),itmust a depend not only in individual features or conditions imposed Fig. 1. Illustration of a bank ‘apparatus’ for money flow. A bank totheeconomicagentsbutalsoontheoverallstructureofthe lendsmoneytodebtorsusingmoneyfromdepositorsandalsoitsown entireeconomicenvironment.Theroleofbankingregulatorsis money, thecapital.Inreturndebtors payinteresttothebank, which toprotecttheflowofmoneythroughthesystembyimplement- keepsaparttoitselfandpaysthedepositorsback. ingrulesthatinsulateitagainstindividualorlocalisedbreaches thathappenwhenabankfailstopaybacktodepositors.How- ever, these rules do not always take into account the impor- the default of a particular bank, each bank holds an amount tanceofthetopologicalstructureofthenetworkfortheglobal ofmoneyas a reserveforpayingbackits depositors.In other financialstability.Inthispaper,wewillpresentquantitativeev- words, a part of the moneyone banksends downstreamis its idencethatneglectingthe topologicalnetworkstructurewhen own money. This share of own money is called ‘capital’ (see implementingfinancialregulationmayhavea strongnegative Fig1).Lookingtoonesinglebank,ifithasalargeamountof impactonfinancialstability. capital,onereasonablyexpectsthatthebankwillalsocovera The event of not paying back the money owed is called proportionally large debtor default, guaranteeing the deposits ‘default’. In order that downstream defaults do not generate madebyits depositors.On the contrary,if thecapitallevel of Sendoffprintrequeststo: thebankissmall,asmalldebtordefaultissufficienttoputthe 2 JoaõP.daCruz,PedroG.Lind:Thedynamicsoffinancialstabilityincomplexnetworks bankwith no conditionsforguaranteeingthe moneyof all its muneratedabovethe moneyfromdepositorswhodonotbear depositors.Loosingsuchconditions,thebankentersasituation these risks. Consequently, more capital means more costs on calledbankruptcyorinsolvency.Usually,bankregulatorsbase theflowofthemoneyand,intheend,moreconstraintstoeco- theirrulesinsucharguments. nomicdevelopment. In1988,agroupofcentralbankgovernorscalledtheBasel Westartinsection2bydescribinganagent-basedmodel[29] CommitteeonBankingRegulationunifiedthecapitallevelrules whichenablesustogeneratethecriticalbehaviourobservedin thatwereappliedineachofthemembercountriesanddefined economic systems. In section 3 we describe the observables aglobalruletoprotectthebankingsystemthatwasbecoming thataccountfortheeconomicpropertiesofthesystem,namely globalatthetime[21].Roughlyspeaking,theserulesimposed the so-called overall product and business level[32]. Further- a minimumcapital levelof 8% withoutany empiricalreason. more, the agent-based model as well as the macroscopic ob- Afewyearslatertheaccordcameundercriticismfrommarket servables,arediscussedforthespecificsituationofanetwork agentswhofeltthatitdidnotdifferentiateenoughbetweenthe ofbanksandtheirdepositsandloans.Oneimportantproperty variousdebtors,i.e.betweentheentitieswhomthebanklends in financial banking systems is introduced, namely the mini- money,andasecondversionoftheaccord[22]wasfinishedin mum capital level, defined here through the basic properties 2004tobecomeeffectivein2008.Inthissecondversion,banks of agents and their connections. In section 4 we focus on the were allowed to use the Merton-Vasicek model[31] based on financial stability of the banking system, showing that rais- theValue-at-Riskparadigm(VaR)[10]toweighttheamountof ing capital levels promotes concentration of economic agents lentmoneyinthecalculationofthenecessarycapitalaccording if the economic production remains constant and it destroys tothemeasuredriskofthedebtor.Thusthe8%percentagewas economicproductionif that concentrationdoes not occur. Fi- nowcalculatedovertheweightedamountandnotovertheto- nally,wepresentspecificsituationswhereeachagentseeksthe talamount.Thisversionoftheaccordbecomeeffectiveatthe stabilityofitseconomicproductionafteraraiseincapitallev- beginningofthe2008financialturmoil;withregulatorsunder els, leading to a state of worse financial stability, i.e. a state severecriticismfromgovernmentsandthemedia,in2010the inwhichlargecrisesaremorelikelytooccur.Insection5we Committeeissuedanewversion[23]tighteningcapitalrules. drawtheconclusions. Atthesametime,sincethebeginningofthe2000stheaca- demic community has been very critical of the capital rules, particularlybecausetheVaRparadigm,onwhichsuchrulesare 2 Minimal model for avalanches of financial based,assumesthatreturnsarenormallydistributedand“does defaults notmeasurethedistributionorextentofriskinthetail,butonly providesanestimateofaparticularpointinthedistribution”[7]. Themodelintroducedinthissectionisbasedonafundamental In fact, there is a huge amount of evidence[2,17,30,3] that feature that human beings have developed in their individual thereturnsofeconomicprocessesarenotnormallydistributed, behaviour,throughnaturalselection,inordertobeabletofight having typically heavy tails. According to the Central Limit environmentalthreatscollectively.Itiscalledspecialisation[16], Theorem[9], if returnsare heavy-tail distributed, then the un- and describes the tendency individuals have, when living in derlyingrandomvariableshaveinfinitevarianceora variance communities,toconcentrateonone,oratmostafew,specific oftheorderofthesystemsize[18].Ineconomicsystems,ran- tasks. Each individual does not need to do everything to sur- dom variables are related to measurements taken from eco- vive,justtoconcentrateonafewtasksthathe/shecandobet- nomic agents. Thus, the infinite variance results from long- terforalltheotherindividuals.Everythingelse heorshewill rangecorrelationsbetweentheeconomicagents.Wewillargue get from other specialised elements of the community. Thus, that this single fact compromisesthe stability of the flow and specialisation leads to optimisation, enabling the entire com- bringsintoquestiontheeffectivenessofcapitallevelrules. munityto accomplish goals otherwise unattainable.However, Physics, and in particular statistical physics, has long in- italsoimpliesthatindividualsnowneedtoexchangewhatthey spiredthe constructionofmodelsforexplainingtheevolution do,so that all have everythingtheyneed forsurvival.The set of economies and societies and for tackling major economic ofalltaskandproductexchangesbetweenindividualsiswhat decisionsindifferentcontexts[29,11].Thestudyofcriticalphe- weusuallycallEconomy.Consequently,whenbuildinganeco- nomenaandmulti-scalesystemsinphysicsledtothedevelop- nomic system, a reasonable approach is to take agents which mentoftoolsthatprovedtobeusefulinnon-physicalcontexts, are impelled to exchange some productthrough a network of particularlyinfinancialsystems.Onereasonforthisisthatfast tradesbetweenpairsofagents.Inthisscope,letusassumethat macroeconomicindicators, such as principalindices in finan- theeconomicenvironmentiscomposedofelementaryparticles cialmarkets,exhibitdynamicalscaling,whichistypicalofcrit- calledagentsandallphenomenaoccurringinitresultfromthe icalphysicalsystems[19]. interactionofthoseparticles.Letusalsoassumethatagentsare In this paperwe will address the problemof the financial attractedtointeract,exchanginganobservedquantitythattakes stabilityusingstatisticalphysicsmodelsthatexplaintheoccur- theformofmoney,labourorothereffectivemeansusedinthe renceoflargecrises,inordertoshowthattheresilienceof the exchange. This type of model where the decision concerning bankingsystem is notnecessarily improvedbyraising capital anexchangeismadebytheexchangingagentsaloneiscalled levels. Our findings have a concrete social importance, since a“free-marketeconomy”. capitalis the mostexpensivemoneya bankcanprovideto its We represent these interactions or trades between agents debtors.Capitalbelongstotheshareholders,whobeartherisk through economic connections, and call the exchanged quan- of the business and keep the job positions. So it must be re- tity ‘economicenergy’.Thoughthe “energy”usedhere is not JoaõP.daCruz,PedroG.Lind:Thedynamicsoffinancialstabilityincomplexnetworks 3 the same as physicalenergy,we will use the term in the eco- k ≫ k andk ≪ k . Furthermore,in thislatter in,j out,i in,j out,i nomical context only. Notice, however, that human labour is limitk ≪k ,thevalueofα couldinprinciplebeany in,j out,i ij assumed to be “energy” delivered by one individual to those finite value largerthan one. However, to guarantee symmetry withwhomhe/sheinteracts,whichrewardtheindividualwith between the situation when agent i profits from agent j, and an energy that he/she accumulates. The balance between the thatwhen agentj profits from agent i, we considerthe range labour(“energy”)producedfortheneighboursandthereward α ∈[0,2],yieldingα =2fork ≪k . ij ij in,j out,i receivedfromthemmaybepositive(agentprofits)ornegative SuchenergytransactionshaveanEconomicsanalogueac- (agentaccumulatesdebt).FordetailsseeRef.[6].Thisanalogy cordingto basic principles[16]:a large(small) k indicates in,j underliesthemodelintroducedinthefollowing,whereweomit alarge(small)supplyforagentj andalarge(small)k in- out,i thequotationmarksandconsiderentitiesmoregeneralthanin- dicatesalarge(small)demandofagenti.Thus,thedifference dividual,which we call agents. Agent-basedmodels of finan- k −k measuresthebalancebetweenthedemandofan out,i in,j cial marketshave beenintensivelystudied, see e.g. Refs. [27, agentiandthesupplyofitsneighbourj.α saturatesforlarge ij 29]andreferencestherein. positiveandnegativedifferencesinordertoguaranteetheprice Economic connections between two agents are in general tobefinite. not symmetric and there is one simple economic reason for Thedefinitionofα inEq.(2)isnotuniquelydetermined ij that: if a connection were completely symmetric there would by these economic requisites. Similar functions such as the be no reason for each of the two agents to establish an ex- arc-tangent or the hyperbolic tangent have been used in this change. In several branches of Economics we have different context[26].Themainfindingsofthismanuscriptarenotsen- examplesoftheseasymmetriceconomicconnectionslikepro- sitivetothechoiceoffunctionaldependencyofα aslongas ij duction/consumption, credit/deposit, a labour relation, repos, itisastep-functionofk −k . in,j out,i swaps,etc.Inthenextsection,wewillfocusonaspecificcon- In the modeldescribed above,we disregardthe economic nection,namelyincredit/depositconnections. details of agents and connections, keeping the model as gen- Since each connection is asymmetric we distinguish the eralas possible. Still, thisgeneralisationis notdifferentin its two agents involved by assigning two different types of eco- essencefromthe oneaccountantsmustuse to providea com- nomic energy. Hence, let us consider two connected agents, monreportforallsortsofbusiness,withthedifferencethatthey i and j, where i delivers to j an amount of energy W and ij usemonetaryunitsandweusedimensionlessenergyunits. receives an amount E 6= W in return. We call these con- ij ij Becauseeachagenttypicallyhasmorethanoneneighbour, nectionstheoutgoingconnectionsofagenti.Theconnections thetotalenergybalanceforagentiisgivenby where agent i receives from j an amount of energy W and ji deliversinreturnE wecallincomingconnections. ji Theenergybalanceforagentiinonesingletradeconnec- U = U = (1−α )+ (α −1) i ij ij ji tionis, froma labourproductionpointofview, Uij = Wij − allneXighbours j∈Xνout,i j∈Xνin,i Eij. Having two differenttypes of energy,we choose Wij as (3) thereferencetowhichtheothertype Eij = αijWij isrelated wherejrunsoverallneighboursofagenti,andνout,iandνin,i throughthecoefficientαij (seeEq.(2)below).Withoutlossof are, respectively the outgoing and incoming vicinities of the generality,weconsiderthatoneconnectioncorrespondsto the agent. deliveryofoneunitofenergy,W =1,yielding: ij This total energy balance U is related to the well-known i financial principle of net present value (NPV)[28]: When an U =1−α . (1) ij ij agentholdsadepositheorshesupposedlypaysforit(bydef- Inordertoimplementthemodel,weneedtodefinetheform inition) and most (but not all) accounting standards [14] as- ofthecoefficientα inEq.(1),whichisaconnectionproperty. sume it as a negative entry on the accounting balance. Here, ij TheamountofenergyE thatoneagentigetsinreturnfora we modeldepositsas a set of incomingconnectionsfromthe ij deliveredamountW canbetakenasapricewhichdependson sameagentinwhichallassociatedcash-flowswerealreadydis- ij therulesofsupplyanddemand.Anagentdeliveringenergyto counted.Inthisway,ifwecouldthinkofabalancesheettotally manyneighbourstendstoimposeahigherpriceonthem.Sim- builtwithNPV’swewouldbenearUi. ilarly, an agent receiving energyfrom many otherneighbours Aswenotedpreviously,economicenergyisrelatedtophys- will induce a reduction in the price imposed by its creditors. icalenergyinthesensethattheagentsmustabsorbfiniteamounts TheseprinciplescanbeincorporatedinasimpleAnsatzas: of physicalenergyfrom the environmentto delivereconomic energy.Consequently,theeconomicenergybalanceU ofagent 2 i αij = 1+e−(kout,i−kin,j) (2) biymausthtrbeeshfionlidtev.aTlhuee,fibneiltoewnewsshoicfhUtihfeoargeeancthiasgneonltoinsgceornatrbolleletod consumeenergyfromitsneighbours,i.e.belowwhichitloses wherek isthenumberofoutgoingconnectionsofagenti out,i all its incoming connections. Furthermore, since this thresh- andk isthenumberofincomingconnectionsofneighbour in,j oldreflectstheincomingconnections,itshoulddependonhow j. For α > 1 the energy provided by agent i to agent j is ij manyincomingconnectionsouragenthas.Withsuchassump- ‘paid’byj abovetheamountofenergyagentidelivers.Thus, tions,weintroducethequantity agentiprofitsfromthisconnectionandgainsacertainamount of energy, U > 0. For α < 1 the opposite occurs. From i ij Eq.(2)oneeasilyseesthatαij isastep-functionwithaverage c ≡ Ui (4) i value one and very small derivatives in the asymptotic limits (α −1) j∈νin,i ji P 4 JoaõP.daCruz,PedroG.Lind:Thedynamicsoffinancialstabilityincomplexnetworks for ascertaining if the agent is below a given threshold c th or not. We call this quantity c the ‘leverage’ of agent i. Un- i like we did previously in Ref. [6], here U is divided by the i total product of the incoming connections solely and not by the ‘turnover’.Thischoiceismade tobe in line with the way banking regulatorsdefine leverage. Still, this alternate defini- tion does not change the critical behaviour observed in our modelandpreviouslyreported[6].Forthecasethatthemean- field approximationα ∼ hαi holds,the leveragec depends ij i exclusively on the network topology, yielding c = kout,i − i kin,i 1[6]. Leveragehas a specific meaningin Economics, which re- latedtothequantityc :itmeasurestheratiobetweenownmoney i andtotalassets[28].Thus,eachagenthasaleveragec which i variesintimeandthereisathresholdc belowwhichtheagent th ‘defaults’ or goes bankrupt, losing its incoming connections with its neighbours. Since the bankrupted agent is connected to other agents, the energy balances must be updated for every affected agent j. Bankruptcy leads to the removal of all Fig. 2. Illustration of a bankruptcy avalanche. Each arrow points to incomingconnectionsofagenti,reducingtheconsumptionof the agent for which it is an incoming connection. Any agent in the the bankrupted agent to a minimum, i.e. keeping one single economicenvironment ispart ofacomplexnetwork (1),andissus- consumptionconnection,k =1.Thissituationimpliesthat in,i ceptibletogobankrupt,whichwilldestroyitsincomingconnections agentianditsneighboursj shouldbeupdatedasfollows: (2). Consequently, new energy balances must be updated forthe af- c →k −1 (5a) fectedneighbours,whoseleveragecangooverthethreshold(3).Since i out,i theseneighboursalsohaveneighboursoftheirown,connectionwill k →k −1 (5b) out,j out,j continuetobedestroyeduntilallagentsagainhavealeverageabove 1 thethreshold. c →c − . (5c) j j k in,j We keep the agent with one consumption connection in the where V is the outgoing vicinity of agent i, with k out,i out,i systemalsotoavoidthedivergenceofciasdefinedinthecon- neighbours.ThequantityUT variesintimeanditsevolutionre- text of financial regulation[21,22,23]. Such a minimum con- flectsthedevelopmentorfailureoftheunderlyingeconomy.In- sduemalipntgiownivthaliunethhaesnneoxtostehcetrioenff.ectontheproblemwe willbe satsea‘rdetoufrUn’Ti,nwaeficnoannsicdiaelrcthoenrteexlatt.ivWeevcaarinataiolsnodmUUTTea,saulrseotkhneoawvn- The bankruptcyof i leads to an update of the energybal- eragebusinesslevelperagent,definedas the movingaverage anceforneighbourj,whichmaythenalsogobankrupt,andso intimeoftheoverallproduct: on,therebytriggeringachainofbankruptcieshenceforthcalled an‘avalanche’.SeeillustrationinFig.2. 1 1 t+TS Theconceptsofleverageandleveragethresholdareusedby Ω = LT Z UT(x)dx (7) R.C.Merton[20]andO.Vasicek[31]intheircreditriskmodels, S t whicharethetheoreticalfoundationfortheBaselAccords[21, whereT isasufficientlylargeperiodfortakingtimeaverages. S 22,23].Namely,Mertonassumedthisthresholdforpricingcor- SimilarquantitiesareusedinEconomicsasindicatorsofindi- porate risky bonds using a limit on debt-equity ratio and Va- vidualaverage standard of living[32]. In the continuumlimit, sicekgeneralisedittoa“debtorwealththreshold”belowwhich the time derivative of the business level Ω gives the overall thedebtorwoulddefaultonaloan. productuniformlydistributedoverallagents. At each time step a new connection is formed, according tothestandardpreferentialattachmentalgorithmofBaraba´si- 3 Macroproperties: overall product and Albert[1]:Foreachconnectioncreatedoneagentisselectedus- business level ingaprobabilityfunctionbasedonitspreviousoutgoingcon- nections,expressedas Let us consider a system of L interconnected agents which k formthe environmentwhereeach agentestablishes its trades. P(i)= out,i (8) L We call henceforththis environmentthe operatingneighbour- l=1kout,l hood. We can measure the total economic energy of the sys- P and one other agent is selected by an analogous probability tembysummingupalloutgoingconnectionstogettheoverall functionbuilt with incoming connections.Such a preferential productU ,namely T attachment scheme is associated with power-law features ob- N servedintheEconomylongago[24,13]andisheremotivated U = (1−α ), (6) byfirstprinciplesineconomicsthatagentsareimpelledtofol- T ij Xi=1j∈XVout,i low:an agenthaving a largenumberof outgoingconnections JoaõP.daCruz,PedroG.Lind:Thedynamicsoffinancialstabilityincomplexnetworks 5 ismorelikelytobeselectedagaintohaveanewoutgoingcon- 6000 nection,andlikewiseforincomingconnections. As connections are being created, a complex network of economicagentsemergesandindividualleverages(seeEq.(4)) 5000 are changinguntileventuallyoneofthe agentsgoesbankrupt (c <c )breakingitsincomingconnectionsandchangingthe i th U leverage of its neighbours, who might also go bankrupt and T 4000 break their incoming connectionsand so on. See Fig. 2. This avalancheaffectsthetotaloverallproduct,Eq.(6),becausethe dissipated energyreleased duringthe avalanche is subtracted. 3000 Reference Thistotaldissipatedenergyisgivenbythetotalnumberofbro- Constant L ken connections, and measures the ‘avalanche size’, denoted belowbys.Sincetheavalanchecaninvolveanarbitrarynum- 2000 22ee++0055 44ee++0055 66ee++0055 8e+05 berofagents,andisboundedonlybythesizeofthesystem,the tt distributionofthereturns dUT willbeheavy-tailed,asexpected UT foraneconomicsystem.SeeFig.5below. Fig. 3. (Colour online) Illustration of the effect of raising the mini- Until now we have been dealing with generic economic mumcapitallevelontheoverallproductUT,atconstantL = 1500. agentsthatmakegenericeconomicconnectionsbetweeneach Raisingcth from −0.71(solidline)to−0.69(dottedline) doesnot other. No particular assumption has been made besides that significantlychangetheoverallproduct. they are attracted to each other to form connections by the mechanism of preferential attachment and that the economic network cannot have infinite energy.From this point onward, we will differentiate some of these agents, labeling them as ‘banks’.Tothisendwefixthenatureoftheirincomingandout- going connections: The incoming connections are called ‘deposits’,theoutgoingconnectionsarecalled‘loans’.Weshould emphasizethatwearenotsinglingoutthiskindofagentfrom the others. Banks are modelled as economic agents like any other.Wehaveonlynameditsincomingandoutgoingconnec- tions,whichwe couldalso doforalltheremainingagents,as consuming/producing,salary/labour,pension/contribution,etc, tomodeleverysinglebusinesswecouldthinkof.Wearechoos- ingthisparticularkindofagentbecausebanksaretheobjectof bankingregulationandtheaimoffinancialstabilitylaws. Thethresholdleveragec foronebankrepresentsits‘min- th imumcapitallevel’.Thecapitalofonebankisreallyanamount ofincomingconnections,whichareequivalenttodeposits,be- Fig.4.(Colouronline)Illustrationoftheeffectofraisingtheminimum causeshareholdersarealso economicagents.Thismeansthat capitalonthebusinesslevelatconstantL = 1500.Raisingcthfrom −0.71(solidline)to−0.69(dottedline)decreasesthebusinesslevel the‘minimumcapitallevel’inthemodelwillbemuchhigher fromΩ =2.88toΩ=2.75. than in real bank markets because we are disregardingshare- holdersand adding the remaining energydeficit to fulfill c . th Therefore, we cannot map directly the levels obtained in the of1.5×106connectionsaregenerated.Wediscardthefirst105 model onto the levels defined in banking regulation. We can, time-stepswhicharetakenastransient. however,uncoverthebehaviourofeconomicagentsinscenar- Figures3and4illustratetheevolutionoftheoverallprod- iosdifficulttoreproducewithoutsuchamodel. uctU andbusinesslevelΩ forasituationinwhichthemin- T imumcapitallevelisraised,whilekeepingthesize oftheop- eratingneighbourhoodconstant. The solid line shows the ini- 4 Raising the minimum capital level tialsituationwithlowerminimumcapitallevelandthedashed linethefinalsituationwithhigherminimumcapitallevel.From In this section we use the model described above in different Fig. 3 we can see that if the size of the operating neighbour- scenarios, i.e. for different sizes of the operating neighbourhoodis keptconstant,the quasi-stationarylevelof the overall hood and differentminimum capital levels. From Eq. (4) one productdoesnotsignificantlychange. sees that the leverage of one agent is always larger than −1. Following this observation we next investigate the evolu- Since we deal with bankruptcy we are interested in negative tion of the returndistributionfor U , consideringan increase T values of c , which reduces the range of leverage values to ofthe minimumcapitallevelat constantsize L ofthe operat- th [-1,0]. Our simulations showed that a representative range of ingneighbourhood.Tothisendwecomputethecumulativesize values for both the threshold and the size L of the operating distributionofavalanches,i.e.thefractionP (s)ofavalanches c neighbourhoodis[−0.72,−0.67]and[500,2000]respectively. ofsizelargerthans.Numerically,thesizesofanavalancheis Foreachpairofvalues(L,c )thesystemevolvesuntilatotal foundbysummingallconnectionsdestroyedduringthatavalanche. th 6 JoaõP.daCruz,PedroG.Lind:Thedynamicsoffinancialstabilityincomplexnetworks Fig.6.(Colouronline)UnlikeinFigs.3and4itispossibletoraisethe minimumcapitallevelc from−0.71(solidline)to−0.69(dashed th line), while keeping the business level constant. In the case plotted, Fig. 5. Avalanche (crises) size distributions for different scenarios Ω∼2.88 of minimum capital level, keeping the operating neighbourhood un- . changed for each agent. The different distributions match at small sizes, in the region where the Central Limit Theorem (CLT) holds, anddeviate fromeach otherforlargercrises(criticalregion). Inthe criticalregiononeobserves(inset)thatincreasingtheminimumcapi- talleveldecreasestheprobabilityforalargeavalanchetooccur,which supportstheintentionsoftheBaselIIIaccords.However,inthissce- nariooneassumesthateachbankwillhaveasimultaneousdecrease oftheirbusinesslevel(seetextandFig.4).Amorenaturalscenario wouldbeonewhereeachbankreactstotheriseintheminimumcap- itallevel insuchawayastokeepitsbusinesslevel constant,which leadstoacompletelydifferentcrisessituation(seeFig.8). ThevalueofP (s)isthenobtainedbyidentifyingtheavalanches c whosesizeisgreaterthans. Figure5showsthecumulativesizedistributionofavalanches for differentminimum capital levels, keeping L = 2000. For small avalanche sizes, the Central Limit Theorem holds[18] Fig. 7. (Colour online) Keeping the business level constant at Ω ∼ andthusallsizedistributionsmatchindependentlyofthemin- 2.88andraisingtheminimumcapitallevelfrom−0.71(solidline)to imum capital level. For large enough avalanches (‘critical re- −0.69 (dashed line) leadstoadecrease of theoperating neighbour- gion’),the size distributions deviate from each other, exhibit- hood,whichisreflectedinaloweroverallproduct. ingapower-lawtailP (s)∼s−mwithanexponentmthatde- c pendsontheminimumcapitallevelc (inset).Asexpected[5], th theexponentfoundfortheavalanchesizedistributiontakesval- wherethesizeoftheoperatingneighbourhoodisadaptedsoas uesintheinterval2<m<7/2. to maintain the business level constant, the distributions plot- AscanbeseenintheinsetofFig.5theexponentincreases tedinFig.5arenolongerobserved.Inparticular,theexponent inabsolutevalueforlargerminimumcapitallevels,indicating m doesnotincrease monotonicallywiththe minimumcapital a smaller probability for large avalanches to occur. However, levelasweshownext. thisscenariooccursonlywhenthesizeoftheoperatingneigh- Figure 8a shows the critical exponent m and the business bourhoodiskeptconstantand,asshowninFig.4,theincrease levelperagentΩasfunctionsoftheminimumcapitallevelcth oftheminimumcapitallevelisalsoaccompaniedbyadecrease andtheoperatingneighbourhoodsizeL.Foreasycomparison, ofthebusinesslevel.Thismeansthateachagenthaslesseco- bothquantitiesarenormalizedintheunitintervalofaccessible nomicenergyor,incurrentlanguage,ispoorer. values. Assumingthatagentsdonotwanttobepoorerdespitereg- The critical exponent shows a tendency to increase with ulatoryconstraints,andthereforetrytokeeptheirbusinesslev- both the minimum capital level and the operating neighbour- elsconstant(Fig.6),anaturalreactionagainstraisingthemini- hood size. The business level, on the other hand, decreases mumcapitallevelistodecreasethenumberofneighbourswith whentheminimumcapitallevelortheneighbourhoodsizein- whomtheagentestablishestradeconnections,i.e.todecrease crease.Consideringareferencestate F0 withcth,0,L0andΩ0 thesizeoftheoperatingneighbourhood(Fig.7).InEconomics thereisoneisolineofconstantminimumcapitallevel,Γ0 ,and cth thisiscalledaconcentrationprocess[8],whichtypicallyoccurs anotherofconstantoperatingneighbourhoodsize,Γ0,crossing L whentheregulationrulesare tightenedup.Insucha scenario at F . Assuming a transition of our system to a larger mini- 0 JoaõP.daCruz,PedroG.Lind:Thedynamicsoffinancialstabilityincomplexnetworks 7 isoline Γf for which the critical exponentis not necessarily cth smallerthanfortheinitialstate. Fromeconomicalandfinancialreasoning,onetypicallyas- sumesthat,independentlyofexternaldirectives,underunfavourable circumstanceseconomicalandfinancialagentstry,atleast, to maintain their business level. This behaviour on the part of agents leads to a situation which contradicts the expectations of the Basel accords and raises the question of whether such regulationwillindeedpreventlargeravalanchesfromoccurring againinthefuture.Toillustratethis,Fig.8bshowsaclose-up ofthem-surfaceplottedinFig.8a. ForthereferencestateF onefindsanexponentm=2.97± 0 0.18.Anincreaseoftheminimumcapitallevelatconstantop- eratingneighbourhoodsize(stateF )yieldsm=3.34±0.09, L whileincreasingtheminimumcapitallevelatconstantbusiness level(state F ), yieldsm = 2.79±0.09,which corresponds Ω to a significantly higherprobabilitythat large avalanches will occur. 5 Discussion and conclusion Insummary,raising the minimumcapitallevelsmay notnec- essarilyimprovebankingsystemresilience.Resiliencemayre- main the same if banks go after the same business levels, as one should expect, according to economic reasoning. Indeed, since business levels are part of the achievement of any economic agent that enters a network of trades, each agent will try,atleast,tomaintainthislevel,independentlyofregulatory constraints. Furthermore,ourfindingscansolvetheapparentcontradic- tionbetweenthecreditriskmodelsthatserveasthetheoretical foundationforbankstabilityaccordsandthedefinitionofcap- ital levels. In fact, bank stability accords impose on banks an adaptedversionofMerton-Vasicekmodel[31]inwhichitisas- Fig.8.(Colouronline)(a)Normalizedcriticalexponentm¯ andbusi- sumedthateachagenthasaleveragethresholdabovewhichit nesslevel Ω asfunctionsof theminimumcapital level c and sys- defaultsoncredit.Theassumptionofthisthresholdcombined th temsizeL.ForaninitialfinancialstateF0,anincreaseofthemini- withafirstprincipleofEconomics–thattheEconomyemerges mumcapital leveltakes thesystemalong oneof theinfinitelymany fromtheexchangesbetweenagents–naturallyleadstoanin- pathsbetweentheinitialandfinalisolinesatconstantminimumcap- terplaybetweenagentsthatcanpropagatetheeffectofonede- ital level, Γ0 and Γf respectively. (b) If such a path follows the faultthroughouttheentireeconomicsystem. isolineatcocntshtantsystcetmh size,ΓL0,thecriticalexponentincreasesand Economicsystemshavelong-rangecorrelationsandheavy- thus the probability for large avalanches decreases. Simultaneously tailed distributions that are not compatible with a linear as- however,itsbusinessleveldecreases(Ωf < Ω0),whichrunsagainst sumptionthatraisingindividualcapitallevelswillleadtostronger thenaturalintentionsoffinancialagents.Onthecontrary, ifthepath stability. Because of the interdependency, this assumption is is along the isoline at constant business level, Γ0, as one naturally Ω probably valid only in two situations: when it is impossible expectsthefinancialagentswoulddo,thecriticalexponent doesnot for an individual to default; and when individuals behave in- changesignificantly,meaningthatlargefinancialcrisesmaystilloccur dependentlyfromeachother(randomtradeconnections).Both withthesameprobabilityasbefore(seetext). situationsdonotoccurinrealeconomicsystems. These findings can inform the recent governmental measures fordealing with the effects of the 2008financial crises. mumcapitallevelatisolineΓf whilekeepingLconstant,i.e. cth Inparticular,governmentshaveshown[23]atendencyforim- alongtheisolineΓL0,onearrivesatanewstateFLwithalarger posingahighercapitalinvestmentfrombanks.Ifthethreshold critical exponent, which means a lower probability for large isincreased,whilethetotalamountoftraderemainsconstant, avalanchestooccur,asexplainedabove.Howeverinsuchasit- there will be fewer trade connections between the banks and uationthenewbusinesslevelΩf islowerthanthepreviousone theirclients,whichleadstosmalleravalanchesintheevolution Ω0. ofthefinancialnetwork.Ontheotherhand,ifthetotalamount On the contrary, if we assume that the transition from F of trade is assumed to grow, following the rise in minimum 0 tothehigherminimumcapitalleveloccursatconstantbusiness capital,theprobabilityofgreateravalancheswillalsoincrease level,i.e.alongtheisolineΓ0,onearrivestoastateF onthe tothelevelwhereitwasbeforeoreventoahigherlevel. Ω Ω 8 JoaõP.daCruz,PedroG.Lind:Thedynamicsoffinancialstabilityincomplexnetworks The scale-free topology of the economic network plays a 6. J.P.daCruzandP.G.Lind.Self-organizedcriticalityinanetwork major role in the determination of the size distribution of the ofeconomicagentswithfiniteconsumption. PhysicaA,inprint avalanches.Atthesametime,thescale-freetopologyemerges 2011. naturally from the rules introduced, which are motivated by 7. J.Dan´ıelsson,P.Embrechts,C.Goodhart,C.Keating,F.Muen- economicreasoning,namelytheprinciplesofdemandandsup- nich, O. Renault, and H. Song Shin. An academic response to ply.Still,onecouldarguethatforbankregulationpurposes,a baselii. (availableatwww.bis.org/bcbs/ca/fmg.pdf),2001. different (imposed) topology for the connections between fi- 8. M.DefondandC.Lennox.Theeffectofsoxonsmallauditorexits and audit quality. InSingapore Management University(SMU) nancial agents would help to prevent large crises. For exam- AccountingSymposium. ple,iftheeconomicnetworkisstructuredasarandomErdo¨s- 9. P. Embrechts, C. Kluppelberg, and T. Mikosch. Modelling Ex- Reńyi network[33], in which every economic agent has the tremeEventsinInsuranceandFinance. Springer-Verlag,1997. sameprobabilityofbeingchosentoformaneconomicconnec- 10. R.FreyandA.McNeil.Varandexpectedshortfallinportfoliosof tion, the system would not have avalanches.In such a model, dependentcreditrisks:Conceptualandpracticalinsights.Journal since connections are equally distributed throughout the sys- ofBankingandFinance,26(7):13171334,2002. tem, all agents would have statistically the same balance. In 11. S.Ghashghaie,W.Breymann,J.Peinke,P.Talkner,andY.Dodge. otherwords,foreachbankruptcytheexpectednumberofchild Turbulent cascades in foreign exchange markets. Nature, bankruptcies in the avalanche would have either zero size or 381(6585):767–770,1996. the size of the system. Thus, with Erdo¨s-Reńyitopology,one 12. A.G.HaldaneandR.M.May. Systemicriskinbankingecosys- expectsstill the dangerof triggeringsuch a large chain of in- tems. Nature,469:351–355,2011. solvenciesabletocollapsetheentiresystem. 13. M.-B. Hu, W.-X. Wang, R. Jiang, Q.-S. Wu, B.-H. Wang, and Directivesmoreorientedtotheconnectiontopologyemerg- Y.-H.Wu. Aunifiedframeworkfortheparetolawandmatthew inginthefinancialnetworkcouldbeagoodalternative.Inter- effectusingscale-freenetworks.Eur.Phys.J.B,53(2):273,2006. estingly,althoughcontroversial,ourclaimspointin thedirec- 14. IFRS Foundation. International financial reporting standards. tionofIMFreportsinNovember2010[25],whereitisargued (Availableathttp://www.ifrs.org/IFRSs/IFRS.htm),2011. that rapid growth in emerging economic periods can be fol- 15. G.Iori,S.Jafarey,andF.Padilla. Systemicriskontheinterbank lowed by financial crises, and also to recent theoretical stud- market. JournalofEconomicBehavior&Organization,61:525– ies onthe risk of interbankmarkets[15,12]. Indeedthe recent 542,2006. 16. R.LipseyandP.Steyner.Economics.Harper&Row,USA,1981. IMFMemorandumonPortugueseeconomicpolicy[4]already 17. B.Mandelbrot andR.Hudson. The(mis)BehaviorofMarkets- includesdirectivesthatrevealIMF’sconcernnotonlywithtun- A FractalView ofRisk,Ruin, andReward. BasicBooks, USA, ingcapitalbuffersandotherlocalpropertiesbutalsowithmon- 2004. itoringthebankingsystemasawhole,andinparticularkeep- 18. R.N.MantegnaandH.E.Stanley. Stochasticprocesswithultra- ing track of the financial situation of the largest banks in the slowconvergence toagaussian:Thetruncatedlvyflight. Phys. network.Webelievethatsuchglobalnetworkingmeasuresare Rev.Let.,73(22):2946,1994. muchmoretrustworthythanlocalones. 19. R.N. Mantegna and H.E.Stanley. Scaling behaviour inthe dy- namicsofaneconomicindex. Nature,376(6535):46,1995. 20. R.C.Merton.Onthepricingofcorporatedebt:Theriskstructure ACKNOWLEDGMENTS ofinterestrates. TheJournalofFinance,29(2):449470,1974. 21. BaselCommitteeonBankingSupervision. Internationalconver- The authors thank M. Haase, F. Raischel, P.I. Teixeira and J. genceofcapitalmeasurementandcapitalstandards. (availableat Jaregoforusefuldiscussionsandforthereadingofthemanus- http://bis.org/publ/bcbs04a.htm),1998. cript. The authors acknowledge financial support from PEst- 22. BaselCommitteeonBankingSupervision. Baselii:International convergence ofcapitalmeasurement andcapitalstandards: are- OE/FIS/UI0618/2011,PGLthanksFundaçaõparaaCieˆnciae visedframework. (availableathttp://bis.org/publ/bcbs107.htm), aTecnologia–Cieˆncia2007forfinancialsupport. 2004. 23. Basel Committee on Banking Supervision. International regulatory framework for banks (basel iii). (available at References http://www.bis.org/bcbs/basel3.htm),2010. 24. V. Pareto. LeCoursd’Economique Politique. Macmillan, Lau- 1. A.-L.Baraba´si andR.Albert. Emergenceofscalinginrandom sanne,Paris,1897. networks. Science,286:509–520,1999. 25. R.Piazza.Growthandcrisis,unavoidableconnection?(Available 2. L.Borland,J.P.Bouchaud,J.F.Muzy,andG.Zumbach. Thedy- atwww.imf.org),2010. namicsoffinancialmarkets-mandelbrot’smultifractalcascades, 26. V.Plerou,P.Gopikrishnan,X.Gabaix,,andH.E.Stanley. Quan- andbeyond. WilmottMagazine,March2005. tifyingstockpriceresponsetodemandfluctuations. Phys.Rev.E, 3. J.P. Bouchaud and M. Potters. Theory of Financial Risk and 66:027104,2002. DerivativePricingFromStatisticalPhysicstoRiskManagement. 27. S.N.D. Queiro´s, E.M.F. Curado, and F.D. Nobre. A multi- CambridgeUniversityPress,2003. interacting-agent model for financial markets. Physica A, 4. EuropeanCommission,IMF,andECB. Portugal:Memorandum 374:715,2007. ofunderstanding(mou)onspecificeconomicpolicyconditional- 28. S.A.Ross,R.W.Westerfield,andJ.F.Jaffe.CorporateFinance, ity,2011. SixthEdition. McGrawHill,2003. 5. J. P. da Cruz and P. G. Lind. Bounding heavy-tailed re- 29. E.Samanidou,E.Zschischang,D.Stauffer,andT.Lux. Agent- turn distributions to measure model risk. (available at basedmodelsoffinancialmarkets. Rep.Prog.Phys,70(3):409, http://arxiv.org/abs/1109.2803),2011. 2007. JoaõP.daCruz,PedroG.Lind:Thedynamicsoffinancialstabilityincomplexnetworks 9 30. D. Sornette. Why Stock Markets Crash? Princeton University Press,2003. 31. O.Vasicek. Probabilityoflossonloanportfolio. KMVCorpora- tion(availableathttp://www.kmv.com),1987. 32. T. Wenzel. Beyond GDP: Measuring the Wealth of Nations. GRIN,2009. 33. P.Erdo¨sandA.Reńyi. OnRandomGraphs. Publ.Math.(Debre- cen)6,290(1959).