How Interbank Lending Amplifies Overlapping Portfolio Contagion: A Case Study of the Austrian Banking Network

Transcription

1 How Interbank Lendng Amplfes Overlappng Portfolo Contagon: A Case Study of the Austran Bankng Network Fabo Caccol J. Doyne Farmer Nck Fot Danel Rockmore SFI WORKING PAPER: SFI Workng Papers contan accounts of scent5c work of the author(s) and do not necessarly represent the vews of the Santa Fe Insttute. We accept papers ntended for publcaton n peer- revewed journals or proceedngs volumes, but not papers that have already appeared n prnt. Except for papers by our external faculty, papers must be based on work done at SFI, nspred by an nvted vst to or collaboraton at SFI, or funded by an SFI grant. NOTICE: Ths workng paper s ncluded by permsson of the contrbutng author(s) as a means to ensure tmely dstrbuton of the scholarly and techncal work on a non- commercal bass. Copyrght and all rghts theren are mantaned by the author(s). It s understood that all persons copyng ths nformaton wll adhere to the terms and constrants nvoked by each author's copyrght. These works may be reposted only wth the explct permsson of the copyrght holder. SANTA FE INSTITUTE

2 How nterbank lendng amplfes overlappng portfolo contagon: A case study of the Austran bankng network Fabo Caccol 1,2, J. Doyne Farmer 1,2, Nck Fot 3, and Danel Rockmore 1,3,4 1 - Santa Fe Insttute, 1399 Hyde Park road, Santa Fe, NM 8751, USA 2 - Insttute for New Economc Thnkng at the Oxford Martn School and Mathematcal Insttute, Oxford Unversty, Eagle House Walton Well Road Oxford OX2 6ED 3 - Department of Computer Scence, Dartmouth College, Hanover, NH Department of Mathematcs, Dartmouth College, Hanover, NH 3755 Abstract In spte of the growng theoretcal lterature on cascades of falures n nterbank lendng networks, emprcal results seem to suggest that networks of drect exposures are not the major channel of fnancal contagon. In ths paper we show that networks of nterbank exposures can however sgnfcantly amplfy contagon due to overlappng portfolos. To llustrate ths pont, we consder the case of the Austran nterbank network and perform stress tests on t accordng to dfferent protocols. We consder n partcular contagon due to () counterparty loss; () roll-over rsk; and () overlappng portfolos. We fnd that the average number of bankruptces caused by counterparty loss and roll-over rsk s farly small f these contagon mechansms are consdered n solaton. Once portfolo overlaps are also accounted for, however, we observe that the network of drect nterbank exposures sgnfcantly contrbutes to systemc rsk. Contents 1 Introducton 2 2 The Data Statstcal propertes of balance sheets Topologcal propertes of the network Stress tests: contagon through drect exposures 1 4 Systemc rsk due to overlappng portfolos 13 5 Concluson 16 1

3 A Appendx: Comparson wth null models: topologcal propertes 17 B Appendx: Comparson wth null model: stress tests 18 1 Introducton The economc and fnancal crses of the early twenty frst century gve strong ndcatons that the hgh level of nterconnectvty characterzng much of the contemporary economc system can amplfy and propagate the stress orgnated n a specfc economc sector or a specfc fnancal nsttuton to other sectors and other nsttutons [5]. Connectons between fnancal nsttutons are of varous knds, rangng from common assets held n balance sheets of dfferent nsttutons to drect lnkages between nsttutons correspondng to specfc transactons. Whle such connectvty can serve as a means of rsk management or ncreased effcency for these nsttutons, t can also provde channels for contagon, thereby creatng potental sources of systemc rsk. It s for ths reason that understandng the nature and structure of connectons between fnancal nsttutons and ther mpact on the system as a whole s of prmary mportance for the assessment of robustness of fnancal systems. In ref. [37], Upper dscusses and categorzes much of the recent work on robustness and stablty n fnancal markets. The basc object of study s a network of banks where edges encode fnancal exposure. Most of these papers (see for nstance [36]) model shocks to the system usng varants of an algorthm developed by Furfne [21]. The three steps of ths teratve algorthm are to 1) create an extncton of a sngle bank, 2) calculate consequent extnctons va exposure to the orgnal bank above an equty threshold, and 3) spread the contagon to other banks that have exposure to the now extnct banks. Thus, Furfne s algorthm also provdes a method for assessng rsk assocated wth counterparty loss. Ths basc threshold extncton model has been modfed n varous ways to nclude safety nets [38], rsk management strateges [18], dfferent llqudty condtons [21], probabltes of contagon (vs. thresholds) [2,31,35], and the ntegraton of the topology of the bankng network [3, 8, 15, 33]. Of partcular relevance s Müller s detaled analyss of the Swss bankng network [31], where n addton to nterbank exposures, the rsk assocated wth the exstence of credt lnes between nsttutons s taken nto account. The man queston we address n ths paper s f networks of drect nterbank exposures play any role as an amplfcaton mechansm for fnancal contagon. Emprcal studes as well as recent theoretcal developments seem to suggest that, n realstc scenaros, networks of drect exposures are not mportant sources of systemc rsk [37, 42]; nstead the man contrbuton comes from common asset holdng [11, 12, 24]. The pont we make n ths paper s that although the network of nterbank lendng may not by tself trgger global cascades of falures n the bankng system, t can n certan regmes amplfy the stress caused by common asset holdngs. Ths ndcates that the nteracton between dfferent 2

4 contagon channels s mportant, creatng rsks that may be much larger than any sngle channel of contagon alone. In ths paper we focus on the Austran nterbank network, for whch we perform stress tests accordng to dfferent protocols. We also characterze the statstcal propertes of the balance sheets n ths system and of the network of mutual exposures. The Austran bankng system has been prevously studed [9, 34]. Ths work dffers from that not only n the tme perod of study (whch allows us to do some comparson of statstcs over the dfferent epochs), but more mportantly, n that our prmary focus s on the analyss of the varous contagon pathways. The natonal nature of the data also places our work n the growng body of lterature and results that are now beng generated for ndvdual natonal bankng systems (e.g., [17, 25, 31]). Collectvely, ths knd of work can ultmately enable a lessons-learned approach to the study of network aspects of the stablty of nterbank systems as well as the role of regulaton n ths settng. Our emprcal nvestgaton draws heavly on a growng corpus of theoretcal and modelng efforts that am to quantfy the relaton between network topology and contagon effects (see for nstance [3, 8, 15, 22, 33]). These knds of network stress tests have ther analogs n other complex systems, most notably, the n slco knockout experments or extncton analyses that have been performed n a varety of network contexts [2] ncludng metabolc networks [27], proten networks [26], food web models of ecosystems (see e.g., Secton 4.6 of [16] and the many references theren as well as [4]) and most recently, even the macroeconomc system that s the World Trade Web (WTW) [19]. Of some relevance s the robust yet fragle (RYF) categorzaton of networks. Ths s shorthand for networks that are reslent (accordng to some basc statstc such as dameter) under the (sutably defned) falure of a random node, but wll experence a rapd degradaton (n terms of the measured statstc) under a judcous targetng of nodes for falure. Ths has been shown to be related to power law degree dstrbutons [2, 6, 14, 41] and/or small world characterstcs [3, 39, 4]. In the case of the WTW, path length-related measures of robustness do not seem to be approprate and the RYF characterzaton s generalzed accordngly [19]. The paper s organzed as follows: In the next secton we ntroduce the data-set consdered here and provde a characterzaton of the statstcal propertes of balance sheets and of the topologcal propertes of the nterbank network. The propertes of the Austran bankng system that we fnd are consstent wth prevous studes concernng other natonal nterbank systems. Ths smlarty suggests that the conclusons of our paper can potentally be extended to nterbank systems other than the specfc one here consdered. In Secton 3 we present results of stress tests where contagon s due to counterparty loss and roll-over rsk. In Secton 4 we consder what happens when overlappng portfolos are also accounted for, and explctly show that the network of drect nterbank exposures can n certan regmes greatly amplfy contagon due to overlappng portfolos. We present our conclusons n Secton 5. 3

5 2 The Data The data studed here 1 contan nformaton on the balance sheets of Austran banks. Data are avalable on a quarterly bass for the years 26, 27 and 28 and consst of the followng: nterbank clams encoded n an exposure matrx L, whose entres L j represent the labltes bank has towards bank j 2 ; total labltes (ncludng non-nterbank labltes) L tot ; total assets (ncludng non-nterbank assets) A tot ; total amount of lqud assets A lq,.e., assets that can be easly lqudated; 3 The number of banks for whch nformaton s provded changes slghtly from quarter to quarter as detaled n the followng table: 1 st quarter 2 nd quarter 3 rd quarter 4 th quarter A smlar data-set for the Austran bankng system concernng the perod has been prevously studed n [9]. Unless otherwse stated, we consder n the followng subsectons results obtaned by aggregatng data from all the quarters at our dsposal. We have performed analyss also for subsets correspondng to sngle quarters and seen that the propertes of the system are stable over tme and consstent wth that observed at the aggregate level. 2.1 Statstcal propertes of balance sheets In Fgure 1 we show complementary cumulatve dstrbutons 4 for the amount of total assets, total labltes and lqud assets held by banks n ther balance sheets. The three quanttes dsplay a smlar pattern characterzed by an ntal regme wth a flat dstrbuton, a typcal regme where the dstrbuton roughly follows a power law dstrbuton, and a subsequent cut-off regme. 1 Our data was made avalable by the Oesterrechsche Natonalbank. We would lke to thank Claus Puhr and Martn Summer for ther help n sharng and processng the data. 2 Reportng data up untl year-end 27 excluded short-term nterbank lendng wth a maturty of less than one month. 3 Data provded by the Oesterrechsche Natonalbank ncludes only hghly lqud assets,.e. cash and reserves held wth the central bank. 4 We defne the complementary cumulatve dstrbuton of a probablty densty functon f(x) the quanty F (y) = R dxf(x),.e., the probablty that a random varable dstrbuted as f(x) s greater than y. y 4

6 1 complementary cumulatve functon complementary cumulatve dstrbuton total assets lqud assets Fgure 1: Statstcs of banks balance sheet. Upper panel: Complementary cumulatve dstrbuton of total assets (blue dashed lne). Data are aggregated from all 12 quarters and are presented on a log-log scale. We observe three regmes: ) banks wth balance sheet sze smaller than 6 M euro correspondng to the flat part of the dstrbuton; ) typcal banks wth a balance sheet sze between approxmately 6 M and 35 B, for whch the dstrbuton can be approxmate by a power law behavor; ) bg banks wth balance sheet sze greater than 35 B, that characterze the cutoff of the dstrbuton. The black sold lne represents a power law of exponent.74 obtaned from a least squares ft of banks n the typcal regme. Bottom panel: Complementary cumulatve dstrbuton of lqud assets. We observe agan three regmes: ) banks wth small amount of lqud assets (less than 1 M) for whch the dstrbuton s flat; ) banks wth an ntermedate amounts of lqud assets (between 1 M and 1 B) for whch the dstrbuton s approxmated by a power law; ) banks wth large amounts of lqud assets (more than 1 B) n the cutoff of the dstrbuton. The black sold lne represents a power law of exponent.67 obtaned from a least squares ft of banks n the typcal regme. As far as total assets and labltes are concerned (top panel of Fgure 1), n the frst regme we have 3253 data-ponts wth total assets smaller than (approxmately) 6 mllon euro, whle n the cutoff regme 49 data-ponts have total assets greater than (approxmately) 35 bllon euro. The remanng 6687 data-ponts consttute the ntermedate typcal regme, that spans an extended range of values (approxmately 3 orders of magntude) and can be descrbed n terms of a power law wth exponent.75. The bottom panel of Fgure 1 shows a smlar structure for the amount of lqud assets held by banks n ther balance sheet. In ths case the ntal regme s made of banks wth less than 5 mllon euro of lqud assets, whle n the cutoff regme banks have more than 1 bllon euro of lqud assets. The ntermedate regme (rangng over three orders of magntude) can agan be characterzed n terms of a power law, wth an exponent.67. Note that banks n the typcal regme for total assets and labltes are not necessarly n the typcal regme for lqud assets (and vce versa). 5

7 leverage total assets Fgure 2: Scatter plot of leverage vs assets. Banks can be roughly clustered nto two dfferent groups: Red squares refer to banks wth leverage smaller than 4.6; the remanng nsttutons are represented by blue dots. Data refer to the frst quarter of 26. An mportant parameter to characterze the nvestment strategy of fnancal nsttutons s the leverage, that s the rato between total assets and equty. Ths measures the level of borrowng that banks use to fnance ther nvestments: λ = A tot A tot L tot. (1) In Fgure 2 we present a scatter plot of total assets vs leverage relatve to the frst quarter of From the plot, two dfferent groups of banks seem to emerge: Regon I: a regon made of 765 banks wth leverage hgher than 4.6 (blue dots); Regon II: a regon at the bottom of the plot made by 71 banks wth leverage smaller than The nave groupng emergng from Fgure 2 appears to be relevant to characterze the pattern emergng n Fgure 3, where we present a scatter plot of total assets vs total labltes. 5 We lmt our study to the frst quarter of 26 for convenence. Analyss of other quarters produces smlar plots. 6 The data sample ncludes a number of specal purpose banks (for nstance penson funds or treasury departments of large corporates) as well as prvate banks. Nether of these engage n tradtonal lendng busness and should therefore have structurally lower leverage. We cannot however clam that these banks correspond to those observed n regon II as we do not have bank dentfers. 6

8 total labltes total assets Fgure 3: Scatter plot of total assets vs labltes n log-log scale. Dfferent symbols and colors correspond to the same groupng of Fgure 2, and also seem to characterze the system. Banks wth leverage hgher than 4.6 seem to share the same lnear relaton between total assets and labltes. Devatons from the lnear pattern are observed for banks wth leverage smaller than 4.6. The black dashed lne represents a lne of slope 1, that n the double logarthmc scale denotes a lnear relaton between total assets and labltes. Data refer to the frst quarter of 26. In Fgure 3 we present a scatter plot of total assets vs total labltes. We see that banks wth leverage hgher than 4.6 (blue dots) are characterzed by a clear lnear relaton between ther total assets and ther total labltes. Banks wth leverage smaller than 4.6 (red squares) represent nstead a devaton from the overall lnear pattern, characterzed by the relaton L tot.91a tot. The overall lnear relaton between total assets and labltes suggests the presence of a typcal value of leverage used by banks λ 11. In summary, we observe the exstence of a typcal leverage targeted by nsttutons wth a qute heterogeneous pattern of balance sheet sze [1]. 2.2 Topologcal propertes of the network We now turn to the characterzaton of the topologcal propertes of the networks of drect exposures between banks. Ths characterzaton s mportant because results of complex network theory show that there s a relaton between topology of networks and propertes of dynamcal processes takng place on them (see e.g. [7]). Showng that the data-set consdered n ths study shares some of the propertes prevously observed for other nterbank networks s then mportant as t suggests the generalty of the contagon dynamcs observed here. 7

9 Gven the exposure matrx L j, we can characterze the topologcal propertes of the network by ntroducng bnary drected and undrected adjacency matrces for the network. Each of these matrces wll use the notaton X. A drected lablty matrx X la can be bult n such a way that Xj la = 1 f bank borrowed money from bank j. Ths s just the exposure matrx thresholded at zero. A complementary asset matrx X asset can be bult by assgnng Xj asset = 1 f bank lent money to bank j. Thus, ( X asset) = X la where the superscrpt prme denotes transpose. Fnally, an undrected matrx X can be bult such that X j = X j = 1 whenever a relaton exsts between banks and j,.e., X j = max {X la j, X asset j }. Prevous emprcal work has shown that nterbank networks are characterzed by heavytaled degree dstrbutons and negatve degree correlatons [9, 13, 25], and theoretcal work has also been carred on to understand ther mpact on contagon dynamcs [1, 23, 28]. We fnd that the same topologcal propertes apply to the data-set consdered here. 1 Complementary Cumulatve Dstrbuton degree (data) n degree (data) out degree (data) degree (random graph) n and out degree (random graph) degree Fgure 4: Degree dstrbuton. Cyan crcle: complementary cumulatve dstrbutons of node degrees, computed from the undrected adjacency matrx X as k = P j Xj. Yellow stars: complementary probablty dstrbutons of node n-degrees, computed from the drected matrx X asset as k n = P j Xasset j. Magenta damonds: complementary cumulatve dstrbutons of node out-degrees, computed from the drected matrx X la as k out = P j Xla j. Black sold lne: complementary cumulatve dstrbuton of node degrees for Erdős-Reny random graphs wth the same average degree. Black dashed lne: complementary cumulatve dstrbuton of node degrees for drected Erdős-Reny random graphs wth the same n/out average degree. Data are aggregated from all 12 matrces. The degree dstrbuton of the real system appears to be heavytaled. In Fgure 4 we show the complementary cumulatve dstrbutons of n-degrees ( j Xasset out-degrees ( j Xla j ) and degrees ( j X j). There s no obvous power law, but n all 8 j ),

10 the three cases the dstrbuton s characterzed by a heavy tal, as can be observed by the comparson wth the correspondng dstrbutons obtaned for Erdős-Reny random graphs 7 wth the same average degree. The presence of heavy tals n the degree dstrbuton s usually assocated wth a hgher overall robustness of the network wth respect to the falure of a random node, but wth a hgher fraglty wth respect to targeted falures of hghly connected nodes [2, 1] The level of correlatons can be measured through the usual degree assortatvty (cf. [32]) that quantfes the extent to whch nodes of a gven degree lnk to one another r = kk l (k + k )/2 2 l (k 2 + k 2 )/2 l (k + k )/2 2. (2) l Here l denotes the average over all lnks and k, k the degrees of two nodes connected through a lnk. We measured the assortatvty for each of the 12 undrected adjacency matrces, obtanng an average assortatvty (averaged over the 12 quarters) of r av.62 ±.3. The negatve value ndcates that n the nterbank network nodes of low degree tend to be connected wth nodes of hgh degree, a structure characterstc of networks that are effectvely of a hub and spoke topology. As for the heavy-taled nature of the degree dstrbuton, the presence of correlatons among degrees affects the probablty of cascades trggered by the falure of a (random) bank [11] wth respect to uncorrelated Erdős-Reny random graphs. Clusterng coeffcents (see [32]) measure the tendency of neghbors of a gven node to be lnked to each other. The propensty of these networks to form trangles (drected or not) s a natural proxy for the fnancal nterdependence of the nsttutons. The local clusterng coeffcent C for a node s defned as C = number of lnks between neghbors of number of possble lnks between neghbors of, (3) and gves a measure of how the neghborhood of s close to be a clque. The average local clusterng s then measured as 8 C = 1 C. N 7 Erdős-Reny random graphs are constructed by fxng a probablty p of connectng any two nodes and performng the ndependent con-tosses edge-by-edge. Note that n ths case for a graph wth n nodes, the expected degree s p(n 1), so gven a graph wth average degree c, the approprate wrng parameter s p = c/(n 1). Smlar networks can be generated for the drected case by assgnng random drectons to lnks. 8 Notce that the average clusterng coeffcent puts more weghts on low connected nodes, and the observed hgh value may be drven by the hub and spoke structure of the network. 9

11 When we also average over the 12 quarters at our dsposal, we fnd that C =.87 ±.2, where the error has been computed as the standard devaton over the dfferent quarters. Ths value can be compared to that of Erdős-Reny random networks of the same average degree, whch have C =.32 ±.2. A comparson wth a null model n whch we randomly rewre lnks whle preservng the n- and out- degrees of each of the nodes (often called the confguraton model ), shows that the degree sequence s enough to reproduce both the negatve correlatons and the hgh local clusterng observed n the data. There are however hgher order topologcal propertes that such null models cannot explan. For nstance, the real networks appear to be characterzed by a hgh number of drected loops when compared wth synthetc networks (see appendx A). 3 Stress tests: contagon through drect exposures In ths secton, we perform dfferent stress tests on the nterbank network to see to what extent falures of sngle banks propagate through the system. Counterparty loss: The frst mechansm of contagon we consder s counterparty loss. Ths s related to the fact that f one nsttuton goes bankrupt ts credtors face a loss and can n turn go bankrupt f ther equty s smaller than the loss. Note n ths respect that nterbank labltes are often bdrectonal, meanng that bank borrows from bank j as well as j borrows from. In the followng we wll consder for smplcty net exposures between banks,.e. we defne a matrx of net exposures L net wth entres L net j = max{, L j L j }, where L j s the amount of money that bank borrowed from j. The mplct assumpton s that f L j > L j and bank defaults on ts oblgatons to j, j wll cover part of the loss by not payng ts debt to. Let us defne for convenence the captal (or equty) of bank as the dfference between ts total assets and labltes: C = A tot L tot, where A tot and represent total assets and labltes of bank. For each bank, we also ntroduce the state varable σ such that σ = 1 f s bankrupt and zero otherwse. The protocol used to probe the stablty of the system wth respect to counterparty loss s the followng: L tot 1. A seed node q s selected and the correspondng bank s shut down. Here σ q = 1, whle σ = for all other banks. 2. Banks revse ther balance sheet. For each bank f the followng condton holds then the bank fals: C < L net j σ j, (4) j where the rght-hand sde represent the loss of bank, and we assume the extreme case where no money s recovered from faled banks.. If fals σ s set to one. 3. If there are new bankruptces return to step 2. Ext otherwse. 1

12 Roll-over rsk: A second contagon mechansm we consder s roll-over rsk. Ths refers to the stuaton n whch a bank that s used to borrowng money n the nterbank market suddenly cannot rase the funds t needs to run ts busness. The recent economc crss was characterzed by the tendency of fnancal nsttutons to hoard lqudty resultng n a correspondng freezng of the nterbank market. Here we perform a stress test to assess the probablty that nsttutons stop lendng n the nterbank market n response to a sngle bank hoardng lqudty 9. In ths case, σ = 1 f bank stops lendng n the nterbank market and zero otherwse The stress test has been performed accordng to the followng protocol: 1. A seed node q s selected and the correspondng nterbank assets are not rolled-over, that s σ q s set to one and σ = for all other banks. 2. For all banks, f the followng condton holds bank stops lendng n the nterbank market and σ s set to one: ( ) L net j σ j > A lq + f A tot A lq. (5) j The left hand sde of the above expresson represents the shortage of fundng that bank s facng due to other banks hoardng lqudty. Here we assume banks are able to lqudate on a short term a fracton g ( g 1) of ther llqud assets. 3. If new banks stopped lendng return to step 2. Ext otherwse. We defne a contagon event as an event where at least one bank goes bankrupt (counterparty rsk) or starts hoardng lqudty (roll-over rsk) as a response to the ntal perturbaton. We performed these tests usng every node of the network as a seed (for a comparson wth smlar results obtaned for the confguraton model, where lnks are randomly swapped, see Appendx B). Results are reported n Fgure 5 (blue dots) for the two observables: ) Contagon probablty, defned as the probablty of observng a contagon event. ) Condtonal extent of contagon, defned as the average fracton of banks affected by the ntal shock f contagon occurs. 9 The assumpton that banks can stop lendng n the nterbank market s a strong one. In general, banks have contractual oblgatons and a bank could not, even f loans are not pad back. Ths s further amplfed n the Austran case due to the nsttutonal tes of the mult-tered Austran bankng sectors (savngs and cooperatve banks, where lqudty s often managed centrally). 11

13 contagon probablty condtonal extent of contagon tme tme Fgure 5: Results of stress tests. Left panel: Contagon probablty due to counterparty loss (blue dots) and roll-over rsk wth f = (red squares) across the dfferent quarters of the perod Rght panel: Average extent of contagon due to counterparty loss (blue dots) and roll-over rsk wth f = (red squares) across the dfferent quarters of the perod Only a small fracton of the system s affected by the ntal shock. In both panels, blue dots refer to contagon due to counterparty loss, whle red squares refer to roll-over rsk wth f = (.e., banks start hoardng as soon as they run out of lqud assets). We see from the plot that, although the probablty of contagon can be a few percent, the condtonal extent of contagon s on average small both n the cases of counterparty loss and roll-over rsk. The robustness of the system does not come as a surprse, as banks have a smple way of reducng the rsk assocated wth drect nterbank exposures. Let us consder, for nstance, the case of counterparty loss. In ths case a prudent bank manager wll make sure that ther bank s not exposed to a sngle other nsttuton by more than the equty of ther own bank (n some countres, lke Germany, ths prudental measure s actually enforced by law). It s trval to see that f most of the banks adopt ths smple measure, for whch no nformaton s requred by banks other than ther own balance sheet, all domno effects are damped at the start. Gven ths consderaton, are networks of drect nterbank exposures mportant at all? Our answer s yes, they can become mportant as an amplfcaton mechansm when other contagon channels are n place. To make ths pont, n the next secton we consder the case of banks wth common asset holdngs, and show that network effects can n certan regmes greatly amplfy the effect of a sudden devaluaton of the common assets. 12

14 4 Systemc rsk due to overlappng portfolos We have consdered so far the network of nterbank clams as a channel of stress propagaton n the system. Now we want to account for a dfferent mechansm of contagon, namely the exstence of overlaps between bank portfolos, as studed n [11, 12, 24]. As a frst step, we consder the effect of banks sharng a common asset n ther balance sheets. For smplcty, we assume that all banks have a fracton c of a common asset n ther balance sheet, so that f A tot s the total assets of bank, ca tot s the amount of the common asset held by bank. Here c s then a measure of the overlap of banks balance sheets. We now ask what happens f the prce of the common asset drops from a reference value 1 to 1 φ. In such a stuaton, banks need to revse ther balance sheet so that A tot A tot (1 c) + A tot c(1 φ) = A tot (1 cφ). (6) Dependng on the level of deprecaton for the asset, some banks can now go bankrupt f ther labltes exceed the new value of ther assets. 1 1 fracton of bankruptces asset deprecaton wthout counterparty loss asset deprecaton wth counterparty loss c Fgure 6: Result of stress tests. Man panel: Fracton of bankruptces n the system due to the deprecaton of the common asset n absence (blue sold lne) or presence (red dashed lne) of contagon va counterparty loss. Inset: rato of the two quanttes plotted n the man panel. The network ntroduces a channel of contagon that can sgnfcantly ncrease the number of falures f cφ <.5. 1 The way overlappng portfolos are mplemented here (through banks nvestng a share of ther total assets n the same common asset) and the effect on banks of the common asset devaluaton s equvalent to the ntroducton of a common harcut on the sze of banks balance sheets, that translates nto a leverage determned harcut on bank captal. We opt however for an nterpretaton n terms of overlappng portfolos because t would apply to more general settngs as those descrbed n [11, 12, 24], where the structure of overlappng portfolos s more granular. 13

15 We show n Fgure 6 the fracton of banks gong underwater as a functon of cφ. From the plot, we estmate a mnmum value cφ.5 resultng n 3% or more of the system gong down 11. Notce that we haven t taken nto account any nteracton among banks, whch are ndependent apart from the presence of correlatons n ther portfolos. We can now ntroduce nteractons nduced by the network structure, and study the effect of such nteractons n combnaton wth the common shock gven by the deprecaton of the common asset. The logc we used s the followng: 1. The common asset s deprecated. 2. Some banks go down because ther labltes exceed the new value of ther assets. 3. These banks cause ther credtors new losses, and contagon propagates due to counterparty loss. Results of smulatons are shown n Fgure 6 (red dashed lne). By accountng for the network structure we ntroduce a further channel of contagon that overall makes thngs worse n terms of extent of contagon. To provde a more quanttatve comparson between the two cases, n the nset of the fgure we plot the rato between the fracton of bankruptces due to the deprecaton of the common asset alone and the fracton of bankruptces when we also account for counterparty loss. In partcular, we observe that the contagon channel provded by the network can sgnfcantly enhance the number of falures f cφ s smaller than.5. Ths amplfcaton mechansm can be understood ntutvely as follows: As we mentoned earler the rsk of default due to counterparty loss by tself can be effectvely controlled by banks f they avod beng drectly exposed to other nsttutons by an amount bgger than ther own equty. Ths prudent measure can become neffectve, however, when the equty of banks s reduced by the devaluaton of a common asset, as n ths case banks may suddenly fnd themselves exposed to other nsttutons by more than ther captal buffer. Therefore, contagon due to counterparty loss sets n f the devaluaton of the common asset s bg enough. Note that, f the common asset s hghly devalued, the network of nterbank exposures does not amplfy contagon smply because most of the banks are already drven out of busness as a result of the ntal shock affectng ther balance sheets. Ths s why the curve plotted n the nset of fgure 6 goes to one for hgh values of cφ. So far we have consdered n ths secton the case of the sudden deprecaton, at tme t =, of an asset common to all banks. Ths deprecaton was not trggered by the dynamcs of the system, but was due to an exogenous shock. We now consder what happens when asset devaluatons are nstead endogenously nduced by banks that lqudate ther portfolos. Let us consder the followng stress protocol: 11 We note n ths respect that even a small value of cφ may quckly become unrealstc. Although a value of 5% s entrely feasble for a bankng system n an adverse scenaro of a mult-year stress test, earnng on the remander of banks portfolos may compensate for part of the loss. 14

16 1. A bank q s ntally selected for bankruptcy. 2. When a bank goes bankrupt, ts nterbank labltes are not repad and ts portfolo of llqud assets s lqudated. 3. The lqudaton process nduces a devaluaton of the common asset proportonal to the relatve sze of bank,.e., A tot / j Atot j All banks suffer a loss that s due to the devaluaton of the common asset. Banks wth drect nterbank exposures to take a further ht due to counterparty loss. 5. If new banks are bankrupt, return to Step 2. Ext otherwse..5 1 contagon probablty wth network wthout network fracton of common asset condtonal extent of contagon fracton of common asset Fgure 7: Results of stress tests. Left panel: Contagon probablty due to lqudaton of the common asset n absence (blue sold lne) or presence (red dashed lne) of contagon due to counterparty loss. Rght panel: Condtonal extent of contagon due to lqudaton of the common asset n absence (blue sold lne) or presence (red dashed lne) of contagon due to counterparty loss. The presence of a network of drect nterbank exposures can substantally amplfy contagon due to lqudaton of common assets. In Fgure 7 we plot the probablty and average extent of contagon measured for the frst quarter of 26 as a functon of c (fracton of total assets nvested by each bank n the common asset). Each panel shows two lnes. Blue sold lnes refer to the case n whch the network of drect nterbank exposures s not accounted for (.e., we gnore losses due to defaulted counterpartes). In contrast, red dashed lnes show what happens when nterbank exposures enter n the pcture. Clearly, the presence of the network ncreases 12 The assumpton that the drop n prce assocated wth a bank lqudatng ts poston on the asset s proportonal to the relatve sze of the bank wth respect to the entre bankng system may be severe as a sgnfcant fracton of the asset may be held outsde the bankng system. 15

17 both the probablty and extent of contagon. The contagon probablty gradually ncreases from about.14 to.49 as c ncreases from to 1. More nterestngly, we observe that the nteracton between contagon due to counterparty loss and contagon due to common asset holdng nduces a sudden jump n the average extent of contagon for c.3. A comment s n order n ths respect. The jump observed n the average extent of contagon s not due to the fact that around c.3 large cascades of bankruptces suddenly become possble because of contagon due to counterparty loss. In fact, large cascades can be observed also f counterparty loss s not accounted for. Wthout counterparty loss no jump s observed because the probablty of observng a large cascade s always small (the probablty of havng more than 1 bankruptces s for nstance less 2%). If counterparty loss s added to the pcture, nstead, the probablty of observng a large cascade becomes sgnfcant around c.3 (the probablty of havng more than 1 bankruptces s for nstance more than 3%). 5 Concluson Ths paper focused on assessng the mportance of networks of drect nterbank exposures for fnancal contagon under dfferent stress scenaros. Recent emprcal and theoretcal work suggests that, n real nterbank networks, drect nterbank exposures by themselves do not contrbute sgnfcantly to fnancal contagon. The pont we address here was that of understandng whether contagon due to drect exposures can nonetheless become relevant n nteracton wth other contagon mechansms, and we fnd that contagon due to counterparty loss can amplfy the stress nduced by the presence of common asset holdngs n bank balance sheets. In ths paper we consdered data concernng the Austran bankng system n the perod In the frst part of the paper we provded a statstcal characterzaton of the system, n partcular for ts balance sheet and network propertes. We found that there s a power law regme n the dstrbuton of banks sze and that the network of nterbank exposures s characterzed by a heavy-taled dstrbuton of node degree and by negatve correlatons between degrees of neghborng nodes. Such propertes have been shown to be relevant n terms of fnancal contagon and have been observed n the past for other data-sets of natonal bankng systems. Ths fact suggests that the results of stress tests presented n ths paper do not apply only to the specfc data here consdered. We performed stress tests of the Austran bankng system accordng to dfferent stress protocols, and usng dfferent contagon mechansms: counterparty loss, roll-over rsk and devaluaton of a common asset. The system s shown to be farly stable f counterparty loss s the only contagon mechansm n place. Roll-over rsk s also shown not to play a major role by tself. The network of drect nterbank exposures becomes an mportant mechansm of stress amplfcaton, however, when t nteracts wth contagon due to overlappng portfolos. If a 16

18 fracton of bank total assets s nvested n a common asset, we showed that contagon due to counterparty loss can greatly amplfy the fracton of bankruptces occurrng n the system after a macroeconomc shock leadng to the devaluaton of the common asset. Moreover, we showed that contagon due to counterparty loss nduces a dscontnuty n the average number of bankruptces when the asset devaluatons due to bankrupted banks lqudatng ther portfolos s also accounted for. A Appendx: Comparson wth null models: topologcal propertes We have mentoned n Secton 2 of the man text that fxng the degree sequence of the nterbank network s enough to explan smple topologcal propertes lke degree correlatons and local clusterng.!" #"!" #" $" $" Fgure 8: These are the two possble 3-clque motfs on 3 nodes n a drected network: On the left s a drected cycle, on the rght, not a cycle and node serves as a source and node k serves as a snk. The connectvty encoded n the local clusterng s nterestng and relevant to the networks, but n addton we should also account for the drectonalty n these trangles. Drectonalty encodes the actual lendng patterns and not just the dependence. A drected cycle j k ndcates a knd of lablty pattern qute dfferent n nature from j k and k (see Fgure 8). In the network lterature such connectvty patterns are often called motfs and are hghly relevant n the understandng of socal and bologcal networks [29]. Consderng all quarters n the data-set, n reference to the motfs shown n Fgure 8, we found an average of drected cycles and cycles of length 3 wth one source and one snk. We report n Fgure 9 a comparson between the number of drected cycles of length 3, 4 and 5 measured n the network correspondng to the frst quarter of 26, and the dstrbuton obtaned for the same quantty for random networks wth same n-degree and out-degree sequences. Gven the relatvely hgh number of short cycles that have been measured n the real network (Fgure 9), t would be nterestng to speculate whether these local patterns serve a purpose.e. specfc to the nature of nterbank lendng. In order to test ths hypothess, smlar analyss should be carred on on dfferent nterbank networks. 17

19 number of 3 loops number of 4 loops x number of 5 loops x 1 6 Fgure 9: Drected loops. Hstograms of number of drected loops of order 3 (upper panel), 4 (mddle panel) and 5 (lower panel) obtaned from 1 3 drected networks wth same n-degree and out-degree dstrbutons of the real network correspondng to the frst quarter of 26. The red dashed vertcal lnes correspond to the values measured for the real network. The densty of these local patterns s much hgher n the real network than n the random ones (the dfference n the three panels s hgher than 8 standard devatons). Ths suggest that these motfs mght serve a purpose specfc to the nature of the nterbank networks. B Appendx: Comparson wth null model: stress tests In ths secton we compare results of stress tests obtaned from real and synthetc networks. We start by lookng at counterparty loss, and we focus on the frst quarter of 26. The null model s the ensemble of networks obtaned by randomly rewrng the lnks of the real network, that s the random ensemble of networks that has the same n-degree and out-degree sequences of the real network. An observaton concernng the rewrng procedure s n order at ths tme. It s now mportant to remember that we are dealng wth a weghted drected network, where each entry of the lablty matrx represents a loan from one nsttuton to another. Durng the rewrng procedure we have to decde what to do wth the weght attached to each lnk that s beng rewred. As far as counterparty loss s consdered, gven that contagon comes from the asset sde of the balance sheet, the weght of each rewred lnk stays wth the bank for whch that specfc lnk represents an asset. Total labltes are then adjusted at the end of the rewrng process n such a way to match the real equty of nsttutons. Results for the frst quarter 13 of 26 are reported n Fgure 1 for three dfferent quanttes: ) contagon probablty fracton of nodes whose ntal falure trggers at least one more falure n the system; 13 Smlar results have been obtaned for the other quarters of 26 as well as 27 and

20 ) condtonal average extent of contagon average number of banks gong down f contagon takes place; ) maxmum extent of contagon maxmum number of bankruptces measured for a network. Frst of all, we observe n the upper panel of Fgure 1 that the networks generated wth the null model we consdered overestmate the contagon probablty measured for the real network. Gven the fact that we defne a contagon event as an event where at least one bank goes bankrupt as a consequence of the ntal perturbaton, and that the exposures assocated wth the lnks are preserved n the rewrng procedure, the dscrepancy between contagon probablty n real and synthetc networks must be explaned n terms of a dfferent arrangement of crtcal lnks, where by crtcal lnk we mean an nterbank exposure that s enough to trgger the falure of one bank should ts counterparty go bankrupt. Indeed, the smaller contagon probablty observed for the real network s due to the fact that the same number of crtcal lnks s spread between fewer nodes than for synthetc networks contagon probablty average extent of contagon maxmum extent of contagon Fgure 1: Stress tests, comparson wth null model. Counterparty loss n the frst quarter of 26: hstograms of contagon probablty (top panel), average extent of contagon (mddle panel) and maxmum extent of contagon (bottom panel) obtaned from 1 3 random networks wth same degree sequence as the real one. The red dashed vertcal lnes are the correspondng quanttes measured for the real network. Synthetc data overestmate the probablty of contagon, whle average and maxmum extent of contagon are correctly predcted. Fgure 1 also shows that synthetc networks largely underestmate average and maxmum extent of contagon. A possble explanaton could be n the dfferent local structure of real and synthetc networks. We now turn to the case of roll-over rsk. Notce that now, gven that contagon propagates through the fundng channel, the weght of each rewred lnk stays wth the 19

21 bank for whch that specfc lnk represents a lablty (.e., a source of fundng). Total assets are then adjusted at the end of the rewrng process n such a way to match the real equty of nsttutons contagon probablty average extent of contagon maxmum extent of contagon Fgure 11: Stress tests, comparson wth null model. Rollover rsk n the second quarter of 26: hstogram of contagon probablty (top panel), average extent of contagon (mddle panel) and maxmum extent of contagon (bottom panel) obtaned from 1 3 random networks wth same degree sequence as the real one. The red dashed vertcal lnes are the correspondng quanttes measured for the real network. Synthetc data overestmate the probablty of contagon, as well as the average extent of contagon. In Fgure 11, we report results obtaned for the second quarter of 26 as an example of the pattern observed for the other perods. In ths case, the null model overestmates the probablty of contagon and the average extent of contagon. As before, vs-a-vs the contagon probablty, the dscrepancy s explaned n terms of a dfferent arrangement of crtcal lnks among nodes. The comparson of real data wth the null model consdered here makes t clear that knowledge of the degree sequence s not enough to correctly estmate probabltes and extent of contagon due to counterparty loss or lqudty hoardng. In partcular, we consdered for both channels of contagon a random ensemble of networks that only preserved the actual network s degree sequence and number of nodes. Such constrants are strong enough to force the network to buld strong negatve degree correlatons and to nduce a hgh average clusterng, but ths s not enough to account for the propertes observed for contagon due to fundng or counterparty loss. At least part of the dscrepances observed between the real system and the null model can been explaned n terms of a dfferent arrangement of crtcal lnks. 2

22 Acknowledgments Ths work was supported by the Natonal Scence Foundaton under grant , by the European Unon Seventh Framework Programme FP7/ under grant agreement CRISIS-ICT and by the Sloan Foundaton. The authors would lke to thank Martn Summer and Claus Puhr for ther help n sharng the data and for useful dscussons. We also warmly thank Stefan Thurner for useful dscussons. D. R. acknowledges the support of the Dartmouth College Neukom Insttute for Computatonal Scence. References [1] T. Adran and H. S. Shn. Lqudty and leverage. Techncal Report 328, Federal Reserve Bank of New York, May 29. [2] R. Albert, H. Jeong, and A.-L. Barabás. Error and attack tolerance of complex networks. Nature, 46: , 2. [3] F. Allen and D. Gale. Fnancal contagon. Journal of Poltcal Economy, 18(1):1 33, 2. [4] S. Allesna and M. Mercedes. Googlng food webs: Can an egenvector measure speces mportance for coextnctons? PLoS Comput Bol, 5(9):e1494, [5] A. Babus and F. Allen. Networks n fnance. In P. Klendorfer and J. Wnd, edtors, Network-based Strateges and Competences, pages Warthon School Publshng, Upper Saddle Rver, New Jersey, 29. [6] A.-L. Barabás and R. Albert. Emergence of scalng n random networks. Scence, 286:59 512, [7] A. Barrat, M. Barthelemy, and A. Vespgnan. Dynamcal processes on complex networks. Cambrdge Unversty Press, 28. [8] S. Battston, D. Gatt, M. Gallegat, B. Greenwald, and J. Stgltz. Lasons dangereuses: Increasng connectvty, rsk sharng, and systemc rsk. NBER Workng Paper 1561, 29. [9] M. Boss, H. Elsnger, M. Summer, and S. Thurner. The network topology of the nterbank market. Quanttatve Fnance, 4:677, 25. [1] F. Caccol, T. A. Catanach, and J. D. Farmer. Heterogenety, correlatons and fnancal contagon. Advances n Complex Systems, 15( supp 2):12558, 212. [11] F. Caccol, M. Shrestha, C. Moore, and J. Doyne Farmer. Stablty analyss of fnancal contagon due to overlappng portfolos. arxv: ,

23 [12] F. Cors, S. Marm, and F. Lllo. When mcro prudence ncreases macro rsk: The destablzng effects of fnancal nnovaton, leverage, and dversfcaton. Avalable at SSRN: [13] H. A. Degryse and G. Nguyen. Interbank exposures: an emprcal examnaton of systemc rsk n the belgan bankng system. Dscusson paper Tlburg Unversty, 24-4, 24. [14] J. C. Doyle, D. L. Alderson, L. L, S. Low, M. Roughan, S. Shalunov, R. Tanaka, and W. Wllnger. The robust yet fragle nature of the nternet. PNAS, 12(41): , 25. [15] M. Drehmann and N. Tarashev. Systemc mportance: some smple ndcators. BIS Quarterly Revew, pages 25 37, 211. [16] J. A. Dunne. The network structure of food webs. In M. Pascual and J. A. Dunne, edtors, Ecologcal Networks: Lnkng Structure to Dynamcs n Food Webs, pages Santa Fe Insttute Studes on the Scences of Complexty Seres: Oxford Unversty Press, New York, 26. [17] E. Bastos e Santos and R. Cont. The Brazlan nterbank network structure and systemc rsk. Workng paper Banco Central do Brazl, 219, 21. [18] H. Elsnger, A. Lehar, and M. Summer. Rsk assessment for bankng systems. Managment Scence, 52: , 26. [19] N. Fot, S. Pauls, and D. N. Rockmore. Stablty of the world trade web over tme an extncton analyss. Journal of Economc Dynamcs and Control, to appear, 212. [2] L. Frsell, M. Holmfeld, O. Larsson, M. Omberg, and M. Persson. State-dependent contagon rsk: usng mcro-data from Swedsh banks. 27. Preprnt. [21] C. H. Furfne. Interbank exposures: quantfyng the rsk of contagon. J. of Money, Credt and Bankng, 35(1): , 23. [22] P. Ga, A. Haldane, and S. Kapada. Complexty, concentraton and contagon. Journal of Monetary Economcs, 58(5):453 47, 211. [23] C. P. Georg. The effect of nterbank network structure on contagon and common shocks. Journal of Bankng & Fnance, 37(7): , 213. [24] X. Huang, I. Vodenska, S. Havln, and H. E. Stanley. Cascadng falures n b-partte graphs: Model for systemc rsk propagaton. Scentfc Reports, 3:1219, 213. [25] C. Iazzetta and M. Manna. The topology of the nterbank market: developments n Italy snce 199. Workng paper Banca d Itala, 711,