Systemic Risk Management in Financial Networks with Credit Default Swaps Matt V. Leduc1, Sebastian Poledna1,2, Stefan Thurner1,2,3 1IIASA, Schlossplatz 1, 2361 Laxenburg, Austria 2Section for Science of Complex Systems, Medical University of Vienna, Spitalgasse 23, 1090 Vienna, Austria 3Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA 6 Abstract. We study insolvency cascades in an interbank system when 1 0 banks are allowed to insure their loans with credit default swaps (CDS) 2 sold by other banks. We show that, by properly shifting financial expo- sures from one institution to another, a CDS market can be designed n to rewire the network of interbank exposures in a way that makes it a J more resilient to insolvency cascades. A regulator can use information about the topology of the interbank network to devise a systemic in- 9 surance surcharge that is added to the CDS spread. CDS contracts are ] thus effectively penalized according to how much they contribute to in- M creasingsystemicrisk.CDScontractsthatdecreasesystemicriskremain untaxed. We simulate this regulated CDS market using an agent-based R model(CRISISmacro-financialmodel)andwedemonstratethatitleads . n to an interbank system that is more resilient to insolvency cascades. i f - q Keywords: SystemicRisk,CreditDefaultSwaps,DebtRank,Agent-BasedMod- [ els, Multiplex Networks, Interbank Systems 1 v 1 Introduction 6 5 1 Financial derivative contracts have been criticized for their role in the 2007- 2 2008 financial crisis. The opacity of such contracts – and the fact that they 0 are often unregulated or traded over-the-counter (OTC) – has drawn criticism, . 1 in particular to credit default swaps (CDSs)1. CDSs were created in the 1990s 0 as a risk management tool by which a loan could be insured against default 6 risk. Their use is however mostly speculative and has drastically increased since 1 : the early 2000s. By 2007, the size of this market (in terms of outstanding CDS v amount)was$62.2trillionand,althoughitdecreasedafterthefinancialcrisisof i X 2007-2008, it remained considerable at about $25.5 trillion in 2012. r Inafinancialsystem,institutionsareinterconnectedthroughacomplexnet- a work of exposures. This complex web of financial exposures creates systemic risk: the insolvency of a particular institution and the resulting default on its 1 See for example: ”ISDA: CDS Marketplace :: Market Statistics”. Isdacdsmarket- place.com. December 31, 2010. Accessed on September 4, 2015. 2 Leduc, Poledna & Thurner loans can precipitate other institutions into insolvency, thereby generating an insolvency cascade. Studying financial systems from a network perspective has thus received a lot of attention in recent years. It is now known that different financial network topologies have different impacts on the probability of sys- temic collapse [9, 12, 4, 3, 18]. In this sense managing systemic risk reduces to thetechnicalproblemofre-shapingthetopologyoffinancialnetworks.Properly used, CDSs have the effect of transferring a financial exposure from one institu- tion to another. They can thus be used to change the topology of the financial network of interbank exposures. To understand whether CDSs can be efficiently used for risk management, it is thus essential to study the impact they have on the topology of financial networks. In this article, we describe how a regulator can use CDSs to restructure the interbank network. Banks are exposed to each other through interbank loans resulting from the conduct of normal banking operations. To guard against the riskofdefaultofacounter-partyonagivenloan,abankcanbuyaCDScontract from another bank. A regulator can then use information about the topology of theinterbanksystemtoimposeasystemicinsurance‘surcharge’thatisaddedto the CDS spread. This surcharge is proportional to the amount of systemic risk created by the contract. This effectively constitutes a mechanism that ’taxes’ CDS contracts according to how they contribute to increasing systemic risk. CDS contracts that decrease systemic risk – by reallocating exposures more efficiently – remain untaxed. This mechanism has the effect of ‘matching’ CDS counter-parties in a way that reduces systemic risk. With an agent-based model (CRISISmacro-financialmodel),wedemonstratehowthismechanismleadstoa self-organized re-structuring of the interbank system that makes it considerably more resilient to insolvency cascades. One of our contributions is to study the financial system as a multi-layer network.Herethedifferentlayersrepresentdifferenttypesofcontractualobliga- tions (loans, derivatives). We show that insolvency cascades can spread through these different types of edges in non-standard ways. This complements recent treatments of systemic risk in multilayered networks (e.g. [6, 5, 16]). Our main contribution is however to propose a mechanism to regulate the CDS market so as to control the formation of the interbank network (modeled as a multi- layer network). The existing literature, on the other hand, is fairly young and hasmainlyfocusedontheintroductionofacentralclearinghouseforderivatives (andtheassociatedeffectsofbilateralnetting)oronsettingcollaterallevels(e.g. [8]). The only other similar mechanisms proposed were in [17, 18, 15]. We also show that an unregulated CDS market, in which banks are allowed to speculate bybuyingCDSonloansthattheydonotown,candrasticallyincreasesystemic risk by creating many contagion channels. Thepaperisorganizedasfollows.InSection2,wepresentamulti-layermodel of the interbank system, which allows us to study how CDSs affect the topology ofexposuresbetweenthebanks.InSection3,weintroduceameasureofsystemic risk that allows us to characterize how a particular CDS contract increases or decreases systemic risk. We then discuss how a regulator can design a systemic Systemic Risk Management in Financial Networks with CDSs 3 surcharge mechanism for CDS contracts, which effectively rewires the interbank system. In Section 4, we simulate this mechanism using an agent-based model (CRISIS macro-financial model) and show that it can considerably improve the resilience of the interbank system to insolvency cascades. 2 A Multi-Layer Network Model for the Interbank System 2.1 Network of Loan Exposures LetusconsiderafinancialsystemcomposedofB banks.Basedontheirliquidity needs, these banks extend loans to each other. Assume that there can be more thanoneloanbetweenbanksiandj.LetL˜ =(cid:80) lk,wherelk >0isthevalue ij k ij ij of the k-th loan extended by j to i. We let lk = −lk so that L˜ > 0 means j ij ji ij has a positive net exposure to i. This quantity then allows us to define the net exposure matrix as L=max(0,L˜). (1) A positive entry L >0 represents the net exposure of j to i while L =0 ij ij means that j is not exposed to i. The net exposure matrix L defines a network of net exposures, as represented in Fig. 1. Fig.1. Network of interbank loan exposures. Here B = 7 and the interbank system is thus composed of 7 banks. Each edge represents the net loan exposure between two banks. Loans carry credit risk: in the event of bank i’s default, bank j may not be able to recover the net amount L that it has lent to bank i. The network ij of loan exposures L creates systemic risk. Indeed, the impact of the default of some borrower i not only affects the lender j itself, but potentially the lenders’ creditors as well as their own creditors and so on. In the example of Fig. 1, the insolvency of bank 4 not only affects bank 5, but may also propagate to banks 4 Leduc, Poledna & Thurner 6 and 7, thus generating an insolvency cascade. Likewise, the lender is not only vulnerable to the defaults of his own borrowers, but also to the defaults of his borrowers’borrowersandsoon.Inaninterbanknetwork,creditriskthusceases to be a local property and becomes systemic. Such insolvency cascades have been extensively studied, e.g. [9, 12, 17, 1, 3, 10, 2]. The size of an insolvency cascade is shown to depend on the topology of the exposures network. Systemic risk must thus be understood as a network property. In order to guard against the risk of default of a borrower, a lending bank can insure its loan using a financial instrument known as a credit default swap (CDS). 2.2 Basics of Credit Default Swaps (CDSs) A credit default swap (CDS) is a financial swap agreement, in which the buyer makes a series of periodic payments (known as ‘spread’ payments) in exchange for a promise that the seller will compensate the buyer in the event of default on some specific reference loan (or bond). If the CDS buyer holds the reference loan, it is thus a form of insurance contract. In the event of default on that reference loan, the buyer of the CDS typically receives an amount equal to the par value of the loan from the CDS seller. This is illustrated in Fig. 2, where a bank i buys CDS protection from bank j on some reference loan lk extended mn to bank m. If, in the CDS contract of Fig. 2, the buyer (bank i) also owns the reference loan lk , then the CDS is used as insurance against the default of m. We will mi refer to such a contract as a ‘covered’ CDS. In reality, the buyer of the CDS need not own the reference loan lk . The latter will then be referred to as a mn ‘naked’ CDS and can allow the parties to speculate on the credit worthiness of the reference entity (bank m). 2.3 Effect of CDSs on the Interbank Network Topology CDS Exposures Let C¯lmkn > 0 denote the promised payment of the CDS ij contract sold by bank j to bank i in the event of default of the reference entity m on the reference loan lk . We can now define C˜m = (cid:80) C¯lmkn as the total mn ij k,n ij promised payments on CDS contracts sold by bank j to bank i in the event of default of m. We let C˜m = −C˜m so that C˜m > 0 means j has a positive net ij ji ij CDSexposuretoi.ThisquantitythenallowsustodefinethenetCDSexposure matrix on the reference entity m: Cm =max(0,C˜m). (2) A positive entry Cm > 0 represents the net CDS exposure of j to i on the ij reference entity m. In the event of m’s default, j will thus have to pay this amount to i. Cm = 0 means that j has no CDS exposure to i on the reference ij entity m. Systemic Risk Management in Financial Networks with CDSs 5 (a) (b) Fig.2.AtypicalCDScontractbetweenbuyeriandsellerjonsomereferenceloanlk . mn (a) Transactions when there is no default of the reference entity m. (b) Transactions in the event of default of the reference entity m. 6 Leduc, Poledna & Thurner A Two-Layer Representation for the Interbank Network Note that the collection of all CDS exposure matrices {Cm} defines a multiplex net- m=1,...,B work, i.e. a network where different types of edges may exist between different banks. Each particular type of edge represents a net CDS exposure on a partic- ular reference entity m. The edge type is thus labeled by m. The interbank system can thus be represented as a two-layer network: The firstlayerrepresentsthenetworkofloanexposuresLbetweentheB banks,while the second layer represents the multiplex network of CDS exposures between the B banks. Such a two-layer representation is similar to that used in [16]. It is represented in Fig. 3. From that figure, we see that a directed edge ij of weight Fig.3. A two-layer network representing the interbank system. Here B = 7 and the interbank system is thus composed of 7 banks. Layer 1 represents net loan exposures betweenthosebanks.Layer2representsnetCDSexposuresonvariousreferenceentities (banks). L onlayer1representsthenetloanexposureofbankj tobanki.Ontheother ij hand, a directed edge ij of weight Cm represents the net amount that bank j ij will have to pay bank i in the event of the default of bank m. Note that the relations between the different layers are highly non-standard. The default of a particular bank will not only affect its creditors, but also the banks that have sold CDS contracts on that reference bank. It may therefore be difficult to measure the impact of the bankruptcy of one bank on the system as it depends on the interaction between the layers. Fortunately, this multi-layer network can be conveniently mapped to a single-layer network, representing the effective net exposures2 of banks to one another. This effective net exposure is 2 Note that we allow here the netting of different asset classes. As a matter of fact there are detailed international agreements on the netting procedure in the case of failureofacounterparty.Typically,fordepositsandloansitiscustomarytocalculate Systemic Risk Management in Financial Networks with CDSs 7 defined, for any banks i and j, as (cid:16) (cid:88) (cid:88) (cid:17) Leff =max 0,L − Ci + Ci . (3) ij ij jz z(cid:48)j z z(cid:48) This mapping of the two layers into a single exposure layer is illustrated in Fig. 4 and Fig. 5. Fig.4.EffectiveExposurewitha‘Covered’CDS:theTopologyoftheLoansExposure Network is ’Rewired’. In this simple example, there are 4 banks (B =4) and only one CDSexposureonreferenceentity2.SincethebuyeroftheCDSC2 (bank1)alsoowns 13 the loan l , the CDS contracts transfers this loan exposure to the seller of the CDS 21 (bank 3). In Fig. 4, we see that ‘covered’ CDS contracts have the effect of rewiring the networkofloanexposuresbytransferringexposuresfromonebanktoanother.In this simple example, there are 4 banks and only one CDS exposure on reference entity (bank) 2. Thus bank 1 has bought a CDS from bank 3 on the reference bank 2. The value of the CDS exposure here is l so that bank 1 has insured 21 its net loan exposure to bank 2 against the possible default of bank 2. This loan exposure l is thus effectively transferred to 3. 21 It is important to note that in Fig. 4, the buyer of the CDS (bank 1) also ownsthereferenceloanl .InFig.5,weshowtheeffectofa‘naked’CDSonthe 21 network of effective exposures. In this case, bank 1 buys a CDS on a loan that it does not own (i.e. l ). The effect is the creation of a new edge in the network 34 of effective exposures. Thus instead of rewiring the network, a naked CDS has the potential to create new contagion channels. the gross exposure instead of the net exposure and derivatives can only be netted by each type of derivative contract. 8 Leduc, Poledna & Thurner Fig.5. Effective Exposure with a ‘Naked’ CDS: an Additional Exposure (Edge) is Created.Inthissimpleexample,thereare4banks(B =4)andonlyoneCDSexposure on reference entity 3. Since the buyer of the CDS C3 (bank 1) does not own the loan 12 l , the CDS contract creates a new exposure: the seller of the CDS (bank 2) is now 34 exposed to the reference entity (bank 3). 3 A Regulated CDS Market 3.1 Effect of CDSs on Systemic Risk As a network property, systemic risk can be quantified by network metrics. One suchmetricisDebtRank[3,18,17],whichisarecursivemethodtodeterminethe systemic relevance of nodes within a financial network. DebtRank has a natural interpretationasthefractionofthetotaleconomicvalueinthefinancialnetwork that is potentially lost as a result of the default (or distress) of a bank (or a set of banks). Considering the network of effective exposures Leff and the capital E =[E ,...,E ] of all banks, the DebtRank R (Leff,E) of bank i can be readily 1 B i computed (see [3] for details). We study an environment in which CDS contracts are guaranteed by some well-capitalized regulating agency. If bank i defaults and cannot honor its obli- gations, then bank j may lose the value L of its net loan exposure to bank ij i. However if some bank z has a CDS exposure Ci to bank j on the reference jz bank i, then it will pay bank j an amount to compensate the loss. If bank z is unable to fully pay this amount because its exceeds its capital, then bank z will go bankrupt. The regulating agency will however step in to guarantee the CDS payment to bank j. The effective net exposure Leff is thus the appropriate quantity tomeasurethe exposureofone institutiontoanother. Theassumption that there is a well-capitalized regulating agency guaranteeing CDS contracts has already been made in the literature. For example, see [8, 6] for papers that study the central clearing of derivative contracts. Systemic Risk Management in Financial Networks with CDSs 9 We can now naturally define the expected systemic loss as B (cid:88) ELsyst(Leff,E)= PdefVR (Leff,E) (4) i i i=1 where Pdef is the exogenous probability of default of bank i and V is the com- i bined economic value of all banks. R (Leff,E) is the DebtRank of bank i, as i introduced earlier. By exogenous probability of default, we mean that Pdef is i the probability that bank i fails due to factors other than the default of other banks to which it is exposed. This allows us to separate exogenous (or non net- work) effects from contagion (or network) effects. To calculate the contribution of a particular CDS contract C¯lmkn to the ex- ij pected systemic loss, we need to adjust layer m by updating the matrix C˜m as follows C˜m(cid:48) =C˜m+C¯lmkn (5) ij ij ij and C˜m(cid:48) =C˜m−C¯lmkn (6) ji ji ij and then recalculating Leff by using equations (2)-(3). We denote the effective netexposurematrixinthepresenceofthatspecificCDScontractbyLeff[+C¯likmj n]. It is now straightforward to compute the marginal effect of that particular CDS contract on the expected systemic loss: ∆[+C¯lijkmn]ELsyst =ELsyst(Leff[+C¯lijkmn],E[+C¯lijkmn])−ELsyst(Leff,E) (7) B =(cid:88)Pidef(cid:16)V[+C¯lijkmn]·Ri(Leff[+C¯lijkmn],E[+C¯lijkmn])−V ·Ri(Leff,E)(cid:17) i=1 where E[+C¯likmj n] is the vector of the banks’ equities in the presence of the CDS contract C¯lmkn. Likewise, V[+C¯likmj n] is the combined economic value of all banks ij in the presence of the CDS contract C¯lmkn. ij Note that ∆[+C¯lijkmn]ELsyst may be either positive or negative. Indeed, a CDS contractmaydecrease systemicriskifitshiftsaloanexposurefromabankwith higher DebtRank to a bank with lower DebtRank. The default of the debtor bank then has a smaller impact on the system. In such a case, ∆[+C¯likmj n]ELsyst will be negative. 3.2 Using CDSs to Rewire the Interbank Network The spread that a buyer i of a CDS contract must pay to a seller j on the referenceloanlk isafunctionofthereferenceentitym’sprobabilityofdefault. mn 10 Leduc, Poledna & Thurner We denote it by s to emphasize the dependence on m. Since a CDS contract m promises a payment equal to the face value of the loan, the total payment will thus be s ·lk . m mn Thecaseforawell-capitalizedregulatingagencytoguaranteethefulfillment ofderivativecontractshasalreadybeenstudiedinthecontextofcentralclearing (e.g. [8], [6]). Here we employ a different approach: a well-capitalized agency regulates the CDS market by incentivizing the CDS contracts that decrease systemic risk and penalizing those that increase it. It can do so by adding a systemic insurance surcharge to the spread that is normally paid on such a contract.Sincearegulatorwithknowledgeofthenetworktopologycancompute themarginaleffectofaCDScontractonsystemicrisk,shecandeviseasurcharge that will be applied to those CDS contracts that increase systemic risk. This systemic surcharge can be defined as τij(lmkn)=ζ·max(cid:104)0,(cid:90) T v(t)·∆[+C¯lijkmn]ELsyst(t) dt(cid:105) (8) 0 where ∆[+C¯lijkmn]ELsyst(t)=(cid:88)pˆi(t)(cid:16)V[+C¯likmj n]·Ri(Leff[+Clijkmn],E[+Clikmj n])−V·Ri(Leff,E)(cid:17) i Here ζ > 0 is some chosen scaling parameter, v(t) is a discount factor and pˆ(t) is the density function for the exogenous default probability. T is the CDS i contract’s maturity. The spread that bank i must pay for a CDS contract sold by bank j on a loan lk now depends not only on the reference entity m, but also on the two mn parties i and j in the CDS contract, i.e. s (lk )=s +τ (lk ). (9) ij mn m ij mn TheeffectofthatsystemicsurchargewillbethatabuyeriofCDSprotection will now choose the seller j with the smallest effective spread s (lk ) and thus ij mn thecontractthatcontributestheleasttoincreasingsystemicrisk.Withoutthat systemic surcharge, the buyer would pay the same spread s to any seller and m thus would be indifferent to which seller it buys from. Indeed, the spread of a CDS depends solely on the reference entity’s probability of default, not on the CDS seller. While the normal spread s is collected by the CDS seller, the m systemicsurchargeτ iscollectedbytheregulatingagencyandplacedinafund ij that will be used to guarantee the fulfillment of CDS contracts if the seller were to become insolvent. 4 Simulation Results and Discussion 4.1 The Agent-Based Model We simulate this regulated CDS market using an agent-based model (CRISIS macro-financial model). This is an economic simulator that combines a well-