An Agent-Based Network Simulation Model for Comprehensive Stress Testing and Understanding Systemic Risk

Transcription

1 College of Wllam and Mary W&M ScholarWorks Undergraduate Honors Theses Theses, Dssertatons, & Master Projects An Agent-Based Network Smulaton Model for Comprehensve Stress Testng and Understandng Systemc Rsk Chrstopher J. Pckett College of Wllam and Mary Follow ths and addtonal works at: Part of the Fnance Commons, and the Fnance and Fnancal Management Commons Recommended Ctaton Pckett, Chrstopher J., "An Agent-Based Network Smulaton Model for Comprehensve Stress Testng and Understandng Systemc Rsk" (2014). Undergraduate Honors Theses. Paper Ths Honors Thess s brought to you for free and open access by the Theses, Dssertatons, & Master Projects at W&M ScholarWorks. It has been accepted for ncluson n Undergraduate Honors Theses by an authorzed admnstrator of W&M ScholarWorks. For more nformaton, please contact scholarworks@wm.edu.

2 An Agent-Based Network Smulaton Model for Comprehensve Stress Testng and Understandng Systemc Rsk A thess submtted n partal fulfllment of the requrement for the degree of Bachelors of Arts n Economcs from The College of Wllam and Mary by Chrstopher Pckett Accepted for (Honors) Lance Kent (Economcs) Alfredo Perera (Economcs) Tll Schreber (Economcs) John Merrck (Busness) Wllamsburg, VA Aprl 24,

3 Abstract Ths paper develops an agent-based network smulaton model that measures systemc rsk n the U.S. bankng system. It s shown that ultmate losses to a bank after an ntal shock to the system s greater than the drect loss they would expect to face. Usng actual balance sheet data and smulatng over randomly-generated nterbank networks, the model captures the feedback effects that arse from a shock to the hghly connected and nterdependent system. In addton to capturng these extra losses that arse from bank nteractons over several perods, ths framework measures the dfferent channels through whch the ntal rsk propagates, amplfes, and transforms. The model s then employed n Monte Carlo smulatons for stress tests, whch are analyzed from both the perspectve of a bank rsk manager and a regulator. It s also mplemented n a Value-at-Rsk framework to demonstrate ts potental to nform exstng VaR models employed by banks. An mportant feature of the bankng system that s often omtted n related models s collateral underlyng the majorty of nterbank transactons. The smulatons reveal that as much as 30% of total losses due to an asset shock are due to strans n the repo market. The regulator stress tests hghlght the extra rsk faced by banks heavly nvolved n the securtes and nterbank markets. A wde varety of dfferent scenaros can be tested n ths framework, and by collectng detaled nformaton throughout the smulaton, the composton of systemc rsk and ts evoluton through the system can be analyzed n dfferent ways. 2

4 1 Introducton A beneft of a well-developed fnancal system s that dfferent rsks faced by ndvduals, corporatons, and governments can be broken down nto peces and allocated effcently across tme and space. Fnancal ntermedares facltate ths process by developng and managng fnancal nstruments and markets that allow ths allocaton to happen. An ndvdual s penson savngs can be nvested n a foregn real estate market as easy as t s to leave t n a savngs account, for example, and the nterconnectedness of the global fnancal system fosters ths lqud allocaton of rsk and captal. However, as fnancal crses have revealed n the past, wth ths connectvty comes a very hgh propensty for these rsks to change and propagate through the system n tmes of crss. Central to these systemc crses are banks, whch both retan many of these rsks and serve as conduts through whch rsk flows. Despte the sophstcaton of ther rsk-management systems, they are often caught off-guard when dsaster strkes. A prmary reason for ths s that ther tools such as Value-at-Rsk (VaR) models and stress tests are based on hstorcal data, whch s of no use when new problems arse, the propensty of whch s ncreasng rapdly wth recent fnancal nnovatons. The relatonshps between nsttutons n the fnancal system are as mportant as the nsttutons themselves, motvatng a growng body of research that models the fnancal system from a networktheoretcal approach. Ths approach s natural when tryng to understand the nature of systemc rsk, as the rsk of contagon depends on the lnkages between nodes n the network. Although much can be gleaned about a system by studyng ts network structure, the fnancal system s also a complex, adaptve system (CAS) as t s composed of many heterogeneous partcpants who behave n dfferent ways n response to ever-changng condtons they face as the system evolves second-by-second. There s a constant feedback process n the system so that even f one could predct how an nsttuton wll react n one stuaton, ther acton wll change the envronment and nfluence the decsons by others, further alterng the state of the system. Modelng phenomena lke ths s the goal of agent-based models (ABMs), whch allow for the emergence of macroscopc 1 system-wde dynamcs from the autonomous actons by the ndvdual agents comprsng the system that would otherwse be ntractable analytcally. Especally n tmes of fnancal turmol t s very dffcult to predct what wll happen next snce the selfsh decsons of the dfferent nsttutons can transform non-lnearly nto system-wde changes, whch trggers further autonomous acton. The model descrbed here ams to synthesze the network and agent-based approaches to study the nature of systemc rsk n the fnancal system. The man contrbutons of ths model are descrbed n detal n secton 1.2, but summarzed below. The model 1 Bookstaber,

5 Includes rsk dynamcs drven by collateral n the nterbank debt markets, a factor that has been largely omtted from exstng research. Measures the contrbuton of dfferent channels that comprse overall systemc rsk. It allows for a summarzed breakdown of the dfferent channels from a Monte Carlo smulaton as well as detaled evoluton of the channel contrbutons through tme n sngle runs of the smulaton. Allows for detaled analyss of these channels from the perspectve of a bank rsk manager as well as regulators, appled n a stress-testng and VaR framework 1.1 Lterature Revew There s an ncreasng number of studes that nvestgate systemc rsk usng network models, a good survey of whch can be found n Upper and Worms (2011). Ths approach has also been used extensvely by central banks (Cont et al., 2012) to assess default and contagon rsk n ther respectve fnancal systems. Some of the earlest work n ths area was by Allen and Gale (2000), who showed the dependence of contagon on the partcular structures of the nterbank network. Smulaton studes usng ths approach have been used to allow for the testng of a wde varety of network structures, assumptons, and counterfactuals, sheddng lght on how certan propertes of the system ncrease the chance of contagon (Elsnger et al, 2006, Upper, 2007). There s not a well-defned consensus regardng the mportance of default contagon the rsk that a default of one bank wll cause a domno effect for systemc rsk as some emprcal studes found that n most cases default contagon s not very mportant (Elsnger et al., 2006, Upper and Worms, 2004), whle others found that t plays a large role (Cont et al., 2012). The majorty of papers n ths area focus on nterbank contagon trggered by a defaulted bank, and make conclusons about the network structure as t relates to contagon. Cont et al. (2012) extend ths methodology to the Brazlan bankng system, whch has rch data on balance sheet compostons and blateral exposures. They study dfferent nsttutonal and network characterstcs that affect the systemc mportance of ndvdual banks, and dfferentate between fundamental defaults and defaults caused by contagon. A related approach of dssectng further the dfferent ways banks can suffer losses s the consderaton of the dfferent ways that contagon can start n a bankng system. Chan-Lau (2010) dentfes four types of shocks to a balance sheet that can trgger spll-over effects n the bankng system: Credt shocks, fundng shocks, rsk transfer shocks, and off-balance sheet exposures. Employng a balance sheet-based network analyss approach makes ths flexblty possble as each shock affects the balance sheet n dfferent ways that can result n a dfferent transmsson of rsk to other banks. 4

6 The model presented n ths paper s a type of balance sheet-based network model and s based on the framework developed n Montagna and Kok (2013), who employ an agent-based approach to study systemc rsk n the EU bankng system. They also use a mult-layered nterbank network that captures a few dfferent ways that banks are nterconnected. The three layers are short-term blateral exposures, long-term blateral exposures, and common exposures to fnancal assets. They show that when actvatng all three exposure networks, the total rsk to the system s greater than the sum of rsks when the sngle layers are actvated ndvdually. They generate random nterbank networks guded by balance sheet and hgh-level data on crosscountry credt exposures. The structure of the network s ther degree of freedom n the smulaton, and am to dentfy relatonshps between contagon and network structure. They also construct an algorthmc measure of the systemc mportance of banks wthn the system. In addton to smulatng a contagon stuaton where a trgger bank defaults, they consder a shock to the bankng system n the form of a large market value declne to a wdely held asset class, whch they show has a greater effect on systemc rsk, captured by the many shared exposures banks have to the shocked securty. The model descrbed here s an extenson of ths framework, where some assumptons are relaxed, behavoral rules altered, and new layers added to the nterbank network. The frst major devaton from Montagne and Kok (2013) s the ncluson of collateral n the nterbank markets. Much of nterbank lendng/borrowng s n the form of repurchase agreements (repo), where the borrower sells securtes to the credtor bank whle agreeng to buy them back at an agreed-upon later tme for a lttle more than they sold them for, whch s the effectve nterest rate on the loan. In other words, t s a loan collateralzed by securtes. The repo market played a very mportant role n the recent fnancal crss and was a prmary source of systemc rsk and the amplfcaton of the mortgage market collapse. There was a marked msmatch between many banks assets and labltes as they reled heavly on the cheap, short term fundng n the repo market to take postons n longer-term, less lqud assets. When uncertanty about the fnancal health of banks and other fnancal nsttutons wth large exposures to the housng market, prmarly n the form of ntrnscally leveraged structured products and dervatves, the repo market froze, reducng access to the form of fundng they all depended on. In addton to ths fundng stran, the myrad varetes of mortgage-backed securtes were also used as collateral for much of ths credt, promptng margn calls that precptated further stran on the system. Includng ths dynamc n the model allows for a dfferent transmsson of losses to a credtor bank n the case that a debtor defaults. Many of the network models descrbed above assume dfferent loss gven default (LGD) levels, whle others, ncludng Montagna and Kok (2013), use Esenberg and Noe s (2001) fcttous default algorthm, whch assumes that upon a default of a 5

7 bank, credtors to that bank are compensated wth an equal share of the lqudaton value of the delnquent banks assets. They show that ther algorthm accomodates nstantaneous contagon snce t ensures that the system settles to the same unque state that t would f the banks took acton smultaneously. In ths model, when a repo counterparty 2 defaults, the credtor sells the collateral assets to make up for ther lost cash. In the short tme-span and crss scenaros that ths model s concerned wth, ths approach reflects observed bank behavor 3, and ams to capture smultaneous behavor algorthmcally. The mportance of the repo market as a source of rsk as well as an amplfcaton channel for contagon s hghlghted n Duarte and Esenbach (2013), Greenwood et al. (2012), and Begalle et al. (2013). Duarte and Esenbach use an emprcal model based on the theoretcal model developed by Greenwood et al. to measure the vulnerablty of the U.S. bankng system to fre-sale spllovers from an asset shock, and categorze losses nto dfferent categores: Intal shock, drect losses, asset sales, prce mpact, and spllover losses. These measurements are lnear and do not allow for the autonomy and multple-tme perod dynamcs. The model presented here ncorporates ths approach of breakng down the systems rsk nto dfferent channels, but s able to do so n an agentbased smulaton framework. Begalle et al. (2013) estmate the rsk of fre sales n the tr-party repo market, whose fndngs nform parameters used n the model descrbed below. 1.2 Contrbuton Montagna and Kok (2013) sgnfcantly contrbuted to the network smulaton approach to studyng systemc rsk n the bankng system, especally by ncorporatng an agent-based approach that s deally suted for ths knd of problem. Usng a framework based ther approach, ths model contrbutes to ths body of research n a few ways. Frst, by relaxng some parameter assumptons used n Montagna and Kok and replacng them wth data-based estmates, we can observe dynamcs that may be closer to those seen n realty. Second, because satsfactory data on long-term bank debt was not avalable, and t has been shown that off-balance sheet over-the-counter dervatves plays a large role n amplfyng systemc rsk, ths model replaces the long-term nterbank exposure network wth an off-balance sheet OTC dervatve exposure network. A major feature of the model not employed n a network framework s the collateral channel that underles much of the nterbank oblgatons n the bankng system. Despte provdng protecton for credtors aganst defaultng counterpartes, these arrangements have been shown to sometmes exacerbate ssues as a source of fre sale rsk. A prmary goal of ths model s to be able to dfferentate between the dfferent channels through 2 In ths case the credtor enters a reverse repurchase agreement 3 See Begalle et al.,

8 whch shocks propagate over the course of the smulaton. The model can make sense of the detaled data collected throughout the smulaton, provdng a rcher story of how dfferent channels contrbute to overall rsk, but also how these channel contrbutons nteract and change over multple perods. A common drawback of agent-based models s that they tend to be black boxes that take nputs and produce results wth very lttle enlghtenng nformaton n between. Ths model ams to open the black box wth careful data collecton, nterpretaton and presentaton, that can provde useful nformaton for banks and regulators who want to better understand the reasons why they are vulnerable to certan shocks and how that vulnerablty arses. Many related models examne systemc rsk from the perspectve of a regulator who s nterested n the vulnerablty of the system to certan shocks and to what degree ndvdual banks and ther nterconnectons contrbute to ths vulnerablty. An mportant feature of ths model s ts applcaton to ndvdual bank-level rsk management. Performng stress tests wth ths model can shed lght on the feedback effects of an ntal shock to the bankng system, allowng them to better antcpate where losses wll come from. In addton to stress tests, ths framework can be used n a VaR analyss settng. An agent-based smulaton approach to VaR models and stress tests decreases ther relance on hstorcal data, and by explctly montorng the propagaton and transformaton of a shock as t moves through the system, can gve a more complete vew of a banks rsk. Even f a stress test accounts for correlatons between securty types n ther portfolos, assessng the expected loss due to a prce shock alone may not tell the whole story. A smple prce drop of an asset class can turn nto fundng problems and spur fre sales n unexpected markets. Lkewse, a counterparty default can have farther-reachng effects than just counterparty credt rsk. In cases lke these, autonomous actons by agents alter the envronment, whch causes further acton, and on aggregate can lead to unexpected consequences to the system. 1.3 Model descrpton and outlne The fnancal system n the model conssts of several randomly generated nterbank networks and ten banks wth heterogeneous balance sheets based on regulatory data from Q At the begnnng of each tral of the smulaton the system s shocked ether by an asset prce declne or a bank default and each bank takes ndependent acton to stay solvent. Ther actons may have drect or ndrect mpacts on others, whch wll necesstate another teraton of ndvdual bank decson-makng n response to the changed envronment. Ths process repeats untl the system reaches an equlbrum where they do not have any more needs to fulfll. The next secton descrbes the model set-up and dynamcs, followed by a descrpton of ts use wthn the Monte Carlo smulaton framework. The data and ts mappng to the model s then dscussed, followed by the results secton. The model s employed to nvestgate the propertes of systemc rsk n 7

9 response to a crss scenaro of large market value declnes of a few asset classes, examned from both the perspectve of a bank rsk manager and a regulator. The model s then ntegrated nto the tradtonal VaR analyss that banks currently use extensvely for rsk-montorng (Bookstaber, 2012). 2 Bankng system set-up 2.1 Balance Sheets and Intal Condtons Bank balance sheets are based on nformaton contaned n Form FR Y-9C, consoldated accordng to the data mappng descrbed n Secton 4. As part of each bank s assets a, they hold cash c, securtes v, short-term nterbank loans/repo d s and other assets oa. Ther labltes comprse deposts p, short-term nterbank borrowng b s, and other labltes ol. Although not explcty accounted for on ther balance sheet, bank also has off-balance sheet over-the-counter (OTC) dervatve contracts wth other banks, denoted by OT C. a = v + c + d s + o a (1) l = p + b s + o l (2) Ther securty holdngs v are composed of sx dfferent types of securtes s k where k {1, 2, 3, 4, 5, 6} and v = 6 k=1 s k pt k, s k, pt k > 0 and pt k s the prce of securty k at tme t. The sx groups are 1) U.S. Treasures (UST) 2) U.S. Government Agency and State securtes (USGA) 3) Mortgage-Backed Securtes (MBS) 4) Asset-Backed Securtes and Structured Products (ABS) 5) Equty securtes 6) Other debt securtes Each securty type has a parameter α k that measures the lqudty of the market for that securty. The lower the value of α k, the more lqud the market s, whch has mportant mplcatons n a fresale scenaro snce t dctates how elastc a securty s prce s to sellng n the market. Each securty also has a harcut h k, whch s how much a borrower must commt above the amount of a collateralzed loan, expressed as a percentage. Based on the FRBNY s tr-party repo 8

10 data, we assume that harcuts for all securtes are fxed at Dependng on the harcut h k of securty k and how much bank has borrowed n the STIB market, a porton of ther holdngs s k wll be pledged to ther credtors as collateral for ther repurchase agreements. In ths case the securtes n custody reman on the bank s balance sheet but they are not avalable for mmedate sale. Thus f a bank wants to lqudate some securtes to stay solvent n the model, they are only allowed to sell securtes not beng used as collateral at tme t. Dfferentatng between these two sets of securtes, we defne where s,a k s k = s,a k s the amount of securty k avalable for sale, and + s,ua k (3) = s k Σ N j=1m sc j (4) s,ua k = p κ (m s j(1 + h κ )) (5) s the unavalable porton. h κ and κ = m c j Smlar to the securtes set up, a porton of bank s cash wll also be ted up as collateral for varous exposures. Frst, when banks face margn calls due to fallng market values of ther securtes held as collateral aganst borrowng, they wll post cash as margn to satsfy ther credtors. Each bank also has off-balance sheet credt exposure to OTC dervatve counterpartes, the majorty of whch s collateralzed by cash (USD and foregn currences). Thus when tryng to meet ther own solvency requrements as well as mmedate demands from credtors durng the stress scenaro, a porton of ther cash holdngs wll be unavalable to spend. So c = c a + cua, where c a s ther dsposable cash and c ua s the nondsposable porton. 2.2 Interbank networks Banks are connected to each other through three dfferent networks n the model, whch are an ntegral part of the system s ntal state and evoluton. Ther exposures and relatonshps wth other banks wll drve much of the behavoral dynamcs of the system followng an ntal shock. Detaled data on these nterbank lnkages n the U.S. does not exst, so these network structures wll be randomly generated and mapped to the model, a detaled dscusson of whch can be found n Appendx A. The frst network s the short-term nterbank lendng (STIB), or repo market, whch s denoted by m s, where m s j 4 Dscussed n Secton 4 represents a collateralzed loan from bank to j. Ths market s a major source 9

11 of fundng for many banks and wll play a large role n the propagaton and transformaton of rsk durng the smulatons. Underlyng each transacton captured n the entres of m s are collateral securtes posted by the borrowng bank. The matrx contanng ths nformaton s denoted by m c, where each entry m c j {1, 2, 3, 4, 5, 6} n the matrx m c of collateral asset types corresponds to the type of securty held aganst a loan from bank to j. Ths network comes nto play when banks assess the avalablty of securtes to lqudate snce a porton of ther holdngs wll be held n custody by repo counterpartes. It wll also be a potental source of fre sales as credtors wll sell collateral held aganst loans to banks that default. Banks are also exposed to each other through the off-balance sheet over-the-counter dervatves market. The network m obs captures counterparty rsk n the large OBS OTC dervatve market, whch, due to the nherent leverage n dervatve transactons and ther exstence off-balance sheet s an mportant channel through whch contagon can spread that s much larger than balance sheet szes n the system would suggest. Each entry m obs j of ths matrx represents the off-balance sheet credt exposure bank has to j. Ths network wll come nto play when bank s assess ther avalable cash to spend as some of t wll be used as collateral for ther OBS OTC dervatve counterpartes, and n the event of a default, ts effect on counterpartes depends on ths exposure matrx. Another mportant network n the fnancal system s the network of common exposures to securtes. As n Montagne and Kok (2013), consder a matrx m v where each entry m v j represents the percentage amount of shared securtes between bank and j s asset holdngs. In other words, t s the degree to whch a shock to the market prce of an asset held by two dfferent banks wll affect the overall system. These exposures are mplct n the model as banks shared exposure to assets s a consequence of ther balance sheet compostons, but ths shows that a connecton between two fnancal nsttutons does not have to be an explct transacton, and s an mportant dmenson to the overall structure and dynamcs of the system. 2.3 Bank behavor and contrants Banks face a number of constrants that dctate how they react to shocks to ther balance sheets and actons by counterpartes. They must keep ther rsk-weghted captal rato (RWCR) above the mnmum requrement θ of 8% 5. Ther RWCR s determned by θ = a l w b d s + Σ6 k=1 w ks k pt k + θo + w obs OT C (6) 5 Basel III, Overvew of Captal Requrements n U.S. Basel III Proposals, Davs Polk & Wardwell, LLP,

12 where w b s the rsk-weght gven to short-term nterbank loans, w k s the rsk-weght of securty k, θ o s the rsk-weght of the remanng assets, and w obs s the rsk-weght of off-balance sheet OTC dervatve contracts. Cash c carres a rsk-weght of zero. 6 They must also mantan a smplfed verson of a mnmum lqudty buffer to reman solvent. We assume that ther only requrement s that they hold a mnmum amount of cash aganst ther potental cash outflows. In ths case we focus on deposts and short-term borrowng, whch are subject to relatvely quck outflows n stress scenaros. Thus they must ensure that c β(p + b s ) (7) where β s the buffer sze. We wll fx β at 5% as n Motagna and Kok (2013), and consstent wth depost runoff and cash outflow assumptons n Basel III 7. It s assumed that banks do not face wthdrawals from ther depost labltes p or ther other labltes o l. On the asset sde, only the other assets oa are assumed to be fxed throughout the smulaton. 3 Model dynamcs Ths secton descrbes the model dynamcs durng a sngle run of the Monte Carlo smulaton, whch evolves over several perods t. Although mportant structural features of the envronment n whch the agent banks nteract n the model such as the nterbank exposure network and collateral matrx are randomly generated, once the exogenous ntal shock s appled, the resultng dynamcs are ultmately determnstc. In an agent-based framework t s dffcult to predct the system outcome n response to a shock, but the constrants and hard-coded behavor of the agents gude the system to a unque equlbrum. Indeed, one of the prmary ams of ths model s openng the black box and creatng a detaled and clear pcture of ts complcated dynamcs. Although each tral lasts several tme-perods, the tme-frame of ths model n the real world would would be several hours to a few days. Ths s because t ams to capture the mmedate actons ndvdual banks take n response to a shock, whch could very well happen smultaneously. To overcome ths computatonally, n each perod a bank takes acton whle assumng that the system remans unchanged from the prevous perod. Once each bank has had ther turn, ther actons are collectvely appled, alterng the system, whch may prompt further acton by banks n the subsequent perod. 6 They could, n prncple, take other actons such as repong out securtes to get fundng to pay back loans that are mmedately due, but n ths framework we are concerned wth rare events where t s lkely that the market s jttery as banks are reluctant to extend new loans to one another. 7 Basel Commttee Revses Basel III Lqudty Coverage Rato, Davs Polk & Wardwell, LLP,

13 The model ams to capture the propagaton, amplfcaton, and transformaton of rsk throughout the fnancal system n a stress scenaro. Due to ts short tme frame and avalable data, t focuses on the more lqud and volatle markets and relatonshps between banks, whch motvates the balance sheet constructon descrbed above, as well as why certan banks wll be more actve and possbly more vulnerable n the stress test smulatons. Banks heavly nvolved n the short-term nterbank lendng (repo) market and securty markets wll be more affected by shocks to market prces or defaults themselves, but wll also be more prone to create rsk for others. Condtonal on parameters and ntal condtons, we have the ntal state of the fnancal system comprsng N banks, or nodes, endowed wth heterogenous balance sheets and nterconnectons. Ther exposures to other banks through the short-term nterbank debt and off-balance sheet OTC dervatves markets wth the correspondng securtes and cash used as collateral are accounted for n ther balance sheets and represented by the matrces m s, m obs and m c, respectvely. A perod s when every bank has a chance to react to system changes n the prevous perod. After all of the banks have had ther turn, the effects of a default (f any) wll be appled and new securty prces wll be calculated. At the begnnng of a sngle nstance/run of the smulaton t = 0, a shock s appled to the system n the form of a sgnfcant negatve market value change to one or more types of securtes held by the banks. Intally, ths wll mmedately depress the value of banks holdngs v of these securtes, and thus ther assets a, whch wll decrease ther RWCR θ. On top of ths, they may have posted the shocked securtes as collateral aganst ther short-term borrowngs b s or dervatve exposures OT C, and may face a margn call from ther credtors. If θ falls below the requred level θ and/or receve margn calls, they wll frst use ther avalable cash c to satsfy mmedate needs. They wll then try to reduce ther short-term nterbank (STIB) loans d s to brng up ther rato and rase cash to bolster ther lqudty buffer, pay off credtors, and post cash margn. Securty prces are determned endogenously n the model, and are a functon of the amount of sellng n the prevous perod. At t = 0, the prce of every securty s k = 1, but n later perods the prce of each securty s recalculated based on sellng actvty drven by banks lqudatng securtes to stay solvent, or by credtor banks sellng collateral assets whch secured ther now-delquent loan to a defaulted bank. The endogenous prce equaton s based on the estmated fre sale elastctes from Begalle et al. (2013). Ther numbers are based on the effect of a 200 bllon dollar lqudaton of an asset class, so the prce equaton scales ther elastcty accordngly. For each securty k, calculate p t k = pt 1 k (1 N =1 sold,t 1 k α k ) (8) 22 where 0 sold k s k s the amount of securty k sold by bank n the prevous perod n ther effort 12

14 to stay solvent, or by credtor banks holdng securty k as collateral aganst a loan to a defaulted bank. a k s securty k s prce senstvty to sellng by market partcpants. These new prces wll drectly affect the value of the collateral securng an nterbank loan, and f the value of the collateral falls below the par value they wll face a margn call. A bank can face a margn call for ther short-term repo borrowng by credtors or by OTC dervatve counterpartes. Bank wll face a margn call from a counterparty j f m s j pt κ(m s j (1 + h κ )) < h κ m s j or f mobs j p t κ(m obs j (1 + h κ)) < h κ m obs j where h κ s the harcut for securty κ, and κ = m c j. In other words, f the value of the collateral, whch starts off greater than the amount of the loan by the harcut amount, drops below the outstandng loan amount, then they wll have to post margn equal to the dfference between the loan and the market value of the collateral. Recall that κ = m c j s the type of securty used as collateral n the repo transacton. Thus the total amount owed by bank due to margn calls s mc = N j=1 max{(m s j + m obs j ) p t κ(m s j(1 + h κ ) + m obs j (1 + h κ )) ; 0} (9) wth k = m c j for each k. The debtor bank wll post cash to fulfll the margn requrement defned above and subsequent downward market prce changes wll be offset by further pledgng of cash to ther credtor. In the event that ther credtor does not rollover a proporton of the repo transacton, the sze of the loan wll decrease accordngly. In ths case, the posted cash margn wll be returned to the borrower n an amount equal to the sze of the loan wthdrawal f the latter amount s less than the total posted cash margn. If the wthdrawn amount s greater than the total posted cash margn, they wll return some securtes. Ths dynamc s mportant snce cash and securtes used as collateral reman on a bank s balance sheet, but cannot be used or sold to fulfll ther varous needs n a stress scenaro snce they are n legal custody of ther credtor. The exact amounts of cash margn and securtes posted and released depends on the sze of loans and types of collateral used n ndvdual transactons between banks. To account for these detals, defne an entry n the STIB/repo matrx at current tme τ as m s,τ j = τ ω j m s j, (10) t=0 where ω t s the amount bank decdes to wthdraw dependng on dfferent oblgatons outlned below. (10) takes nto account any wthdrawals of loans from j to n perods leadng up to τ. The amount of cash c that s beng used for margn calls to varous counterpartes at tme τ s 13

15 c m { τ = max mc t t=0 N j=1 (m s,0 j m s,t j ) ; 0 } Thus the amount of cash that s not avalable for use by bank n perod t s equal to the total amount of cash posted as margn snce perod 0, less the amount returned back to them due to a shrnkage of the loan amount. c ua = N j=1 (11) (m obs j p t κ + h κ m obs j ) + c m (12) In addton to the drect balance sheet losses and new oblgatons due to securty prce changes, a bank can also suffer losses when a bank who they lent to n the short and long-term nterbank market defaults. Ths wll trgger a fresale of the collateral assets, and bank may recoup less cash from sellng the collateral nto a dvng market than the sze of the delnquent loan. Ths dynamc s descrbed n detal below. In an effort to stay solvent n response to ther shocked assets and mmedate oblgatons to credtors, banks n the prevous perod may have wthdrawn short-term loans to rase cash and boost ther RWCR. Thus t s possble that a credtor j n the STIB market wll not roll over a proporton of loans to bank, n whch case bank wll have to pay them back the wthdrawn amount. Ths s yet another oblgaton they must fulfll to stay solvent, and across all s credtors s calculated as b s,t = N j=1 ω t jm s,t 1 j (13) If they stll need to boost ther RWCR, rase ther lqudty buffer, fufll a margn call, or return called-back funds they borrowed n the STIB, then they wll have to take acton. In other words, they wll check f θ < θ, c a < β(d + b s ), bs > 0 or mc > 0. They frst use avalable cash c to pay back b s,t and mc t. If that s not enough, or ca < bs +mc, they wll wthdraw short-term nterbank loans. If θ < θ, they wll frst try to reduce short-term exposure by decreasng ther repo lendng as short-term loans have a hgher rsk-weghtng than cash. The total amount to wthdraw wll depend on how much s needed due to each requrement. Interbank requrement: If b s r b = max{b s c a ; 0} (14) > c a, then c = c ua. If not, then c = c b s. Bank wll use ther avalable cash to pay off short-term debt. If they do not have enough, the dfference between what they owe b s much cash they have c s the amount to wthdraw from the STIB market. and how 14

16 Margn requrement: r m = max{mc c ; 0} (15) Bank wll use ts avalable cash, whch was altered when tryng to meet the nterbank requrement, to cover ther margn requrements. When postng cash margn, each pece wll be pad out to each specfc counterparty j dependng on the sze of the margn call. So each j wll get max{m s j pt k (ms j + h k) h k m s j ; 0} from. Lqudty requrement: r lq = max{β(p + b s ) c ; 0} (16) In other words, f bank has pad off mmedate credtors and has enough lqudty (c r b r m) β(p + b s ), then rlq = 0. For needs drven by captal requrements, bank must determne how much they can possbly restore ther θ to reach θ by decreasng ther short-term lendng d s. The way they do ths n the model s by consderng what would happen to ther θ f they wthdrew from the STIB market n small ncrements untl they reach θ. The resultng loan amount that wll restore ther RWCR s denoted by d s. If they cannot reach θ by wthdrawng all of ther loans d s, they wll have to move on to securtes lqudaton. Captal requrement: r cap = max{(d s max{(r b + r m + r sh + r lq ) c a } d s ; 0}, d s > 0 (17) In other words, they frst use ther avalable cash c a to cover the precedng requrements, and f they use t all up then they wll wthdraw the shortfall from the nterbank market. r cap takes ths nto account because t captures the fact that wthdrawals due to the other requrements could end up restorng θ naturally, whch wll not necesstate further wthdrawals for captal reasons. The amount to wthdraw s thus the total loans d s less the amount used up satsfyng the earler requrements, mnus the deal level d s. If d s 0, then bank cannot restore θ even by wthdrawng all of ther short-term loans. They wll have to contnue to the next stage where they wll lqudate securtes. If, after accountng for the wthdrawals drven by nterbank, margn, and lqudty requrements, ther short term lendng d s s low enough to restore ther requred RWCR to θ, then r cap = 0. If not, they must wthdraw r cap. Thus ther total requrement that wll be fulflled by wthdrawng from the STIB market s 15

17 r = mn{d s ; max{r b + r m + r lq + r cap c a ; 0}} (18) They wll frst use ther avalable cash c a to cover ther requrements, and then resort to decreasng ther STIB lendng. If they use up ther avalable cash and the amount to wthdraw s no greater than ther total amount of short-term lendng, then bank can meet ther mmedate needs by wthdrawng funds from the STIB market. They wll wthdraw evenly across all counterpartes j = 1, 2,...N the fracton ω of the total amount lent m s j for each j where ω = r d s, ω [0, 1] (19) wth d s = ΣN j=1 ms j Ths amount s turned nto cash for bank whle the proportonal amount of collateral securtes become avalable for sellng by bank j. However, f r > b s, then they cannot meet ther mmedate oblgatons and fulfll ther captal and lqudty ratos by solely reducng ther STIB lendng, so they must wthdraw everythng d s = ΣN j=1 ms j. Set ω = 1. Now d s = 0, whch wll have an mpact on both a and m b d s n θ. Lqudatng securtes At ths pont, bank has wthdrawn d s, whch, assumng that none of ts debtors default n the current perod, they wll be able to turn nto cash. The next set of requrements are smlar to the ones outlned above, except that bank now assumes that they have cash to spend equal to ther total short-term loans, and the amounts are computed n terms of quantty of a partcular securty k to be sold at the current market prce whose proceeds wll cover the requrement. The decson of the order n whch the securty types wll be sold depends on how lqud the market s and what requrement they are tryng to fulfll. For example, when tryng to rase cash to pay back credtors and bolster ther lqudty buffer, a bank wll prefer to lqudate the securty k wth the deepest market,.e. wth the lowest α k. However, t may be the case that when tryng to reduce ther exposure to rsky assets, they may choose to lqudate assets wth hgher rsk-weghtngs, whch wll have a greater mpact on ther RWCR per quantty sold. The detals of ths process wll be dscussed below. Another constrant on ther securty lqudaton s how much of each securty k s used as collateral for ther short and long-term borrowngs. Although these securtes wll show up on ther balance sheets, they cannot sell them as long as they stll owe money to ther credtors who hold them n custody. Thus the amount of each securty avalable to lqudate s s k Σ N j=1 msc j where m sc j = p κ(m s j + h κm s j ), h κ and κ = m c j. 16

18 The bank wll determne how much of securty k must be sold to meet the requrements drven by each of the followng oblgatons untl s a k = 0, then move on to the next securty, whch wll be chosen to maxmze the proceeds from the sale. We wll see, however, that when every bank behaves ths way, fresale rsk can ncrease and reduce the effcacy of ths strategy. It s also mportant to note that even f they run out of a securty n perod t, t s possble that some of that securty wll be released by credtors n the next perod who decreased the sze of ther short-term loan (reverse repo) to bank, thus freeng up more of securty k to lqudate. Defne φ as the amount of avalable funds that bank can apply towards fulfllng the below requrements. Ths amount wll equal ther avalable cash c a, how much they wthdrew from the STIB market N, and the amount from these two sources that have been allocated to the j=1 ms,t j prevous oblgatons. φ b = c a + φ m = c a + φ lq = c a + N j=1 N j=1 N j=1 m s,t j m s,t j zb m s,t j zb z m z sh (1) Interbank requrement z b { max{b s = mn (c a + φb ) ; 0} p k } ; s,a k (20) They wll have nterbank needs as long as ther STIB oblgatons are not covered by ther avalable cash and money comng n from others after wthdrawng all of ther STIB lendng n the prevous stage. They assume at ths pont that everythng they wthdraw wll be returned n full. If a debtor defaults n the current perod, t s possble that the realzed amount of cash they wll get back wll be less than the sze of the loan, whch they wll have to make up for. Ths shortfall s captured n z sh below. Subject to the current market prce of securty k, the equaton for z b wll return the amount that must be sold to rase enough cash to meet ther oblgatons towards others. If they cannot meet ths oblgaton by sellng just k, they wll sell everythng they can s,a k 17 and move on to the next

19 securty. In ths case t wll be the securty wth the next most lqud market α k+1. They wll repeat ths process untl nterbank oblgatons z b are satsfed. If they can satsfy z b usng ther cash c and expected nflow Σ N j=1 m j from the STIB market, then they wll not have to sell any securtes. If they must sell securtes, cash wll have been drven to zero and then techncally ncreased by the amount of the proceeds from the lqudaton p t k zb. However, these proceeds cannot be used snce they are owed to credtors. (2) Margn requrement z m { max{mc t = mn φ m ; 0} p µ ; s,a k zb } (21) Ths says that f ther total margn call requrement cannot be covered by ther avalable cash or wthdrawn STIB funds, then they wll sell an amount of securty k equal to that amount. The next requrement captures s oblgatons due to promsng funds to other banks j under the assumpton that ther own nflows would be as expected. If ther realzed nflows that were pledged to counterpartes are less than the amount promsed n the prevous perod, they wll have to make that up n perod t. (3) Shortfall requrement { N shortfall = max ωj t 1 m s,t 1 j + z sh j=1 { shortfall = mn p t k ; s,a k N j=1 (zb } p fs,t κ (m s,t j + h κm s j) ; 0 + z m (22) }, κ = m c j (23) In the prevous perod, bank wthdrew an amount ω t 1 m s,t 1 j and assumed that they would receve t n full when decdng how much of ther securtes holdngs to lqudate. However, f, n that same perod any of ther debtors defaulted, they may receve less than they expected. Snce they held a collateral asset κ aganst loans to the defaulted counterparty, they wll sell all of ther holdngs at the prce p t+1 κ. Because other banks may have lent to the defaulted bank, there wll be a large amount of sellng of securty κ, so t s lkely that p t+1 κ < p t κ. (4) Lqudty requrement z lq { max{β(p + b s = mn ) φlq p k + c ua ; 0} ; s,a k } (zb + z m + z sh ) (24) 18

20 In ths case we assume that even though some of ther cash s posted as collateral for off-balance sheet dervatve postons and as margn for nterbank loans, they can consder ths cash to be part of ther lqudty buffer. The above equaton says that f ther pledged cash c ua are less than the requrement amount β(p +b s φ lq (z b If z lq = s,a k (zb and avalable cash ), then they wll have to lqudate some securtes. + z m), then sell all of t for pt k (s,a µ (z b + z m)) so that ca = ca + pt k (s,a µ s met. + z m )). Then recalculate zlq. Repeat untl ther lqudty requrement z lq (5) Captal rato requrement If bank stll needs to take acton to restore ther RWCR, they can do so by sellng securtes, whch account for a consderable porton of ther rsk-weghted assets. Decreasng that number by convertng those holdngs nto cash, whch holds a rsk-weghtng of zero, wll ncrease ther rato θ. It s possble that n the process of lqudatng securtes to fulfll the precedng oblgatons, the rato was mproved, but f there stll remans a RWCR shortfall, they wll lqudate more securtes. They sell securty k n small ncrements untl θ = θ or ther avalable holdngs of k runs out. They sell each porton s,a k of securty k, effectvely shftng the amount s,a k pt k from ther rsk-weghted assets to cash, so that c = c a + s,a k pt k + cua. The total amount of securty k that s sold n a perod s sold,t k = N =1 (z b (k) + z m (k) + z sh (k) + z lq (k) + z cap (k)) (25) where z cap (k) means that securty k was sold satsfyng that partcular requrement. Default and fresale dynamcs The dynamcs descrbed n ths secton take place after every bank has acted accordng to the foregong process. If bank cannot meet any of the above requrements by wthdrawng from the short-term nterbank market or lqudatng securtes, they default. Throughout the perod each bank assumes that all the other prevously-solvent banks wll reman solvent, but after they have all had ther turn to react to each other s actons from the last perod accordng to the dynamcs descrbed above, they wll have a chance to take acton due to defaults. If bank defaults, each bank j that has short-term nterbank (reverse repurchase agreement) exposure to wll sell all of the collateral securtes m s,t j (1 + h k) and retan the total posted cash margn from to j durng the prevous perods. Ths assumpton s based on fndngs from Begalle et al. (2013), who found that despte the benefts of a partal and orderly lqudaton of collateral assets after a default, regulatons are such that ndvdual banks have strong ncentves to sell everythng, exacerbatng 19

21 the fresale envronment as banks collectvely act n a way that may end up hurtng the market more. To capture ths fresale rsk, each j sells the collateral at a new prce p t+1 k that accounts for the hgh amount of sellng so that ther realzed collateral lqudaton proceeds s p t+1 k (m s,t j (1 + h k)) where p t+1 k s calculated accordng to prce equaton. Bank j s loss λ s j due to ther STIB exposure to a defaulted bank s equal to the dfference between the loan amount and the proceeds from the sale of the collateral at the fresale prce and the retaned cash margn. λ s j = m s j p t+1 k (m s,t j (1 + h k)) + τ mc t j (26) It s also possble that λ s j becomes a slght gan for j f the prce at whch they sell the collateral has not fallen enough to negate the sale of slghtly more collateral (m s,t j + h k) than the sze of the delnquent loan. If a bank j has off-balance sheet OTC dervatve exposure to the defaulted bank, then they retan the cash collateral, whch amounts to 90% of the exposure value. Ther loss gven default (LGD) due to these exposures s therefore 10%. t=0 λ obs j = (0.1)m obs j When a bank defaults t s not just banks that have loans and other asset-sde exposures to that are vulnerable. It s just as lkely that a defaulted bank s a credtor to another j. In that case bank j rsks losng ts collateral securtes or cash held n custody by the defaulted counterparty, whch was a major ssue for many counterpartes to Lehman Brothers n 2008, especally n Europe 8. When a defaulted credtor who holds a borrower j s collateral defaults, those assets are often frozen and rretrevable by the borrower untl lengthy bankruptcy processes are completed. In the tme frame of the stress scenaros faced by the system n ths study, bank j wll consder t a loss f the value of ther collateral s worth more than ther assets whose purchase was funded by the borrowed funds. Ths loss s not drect, but puts stran on ther rsk-weghted captal ratos as the market value of ther assets are smaller whle also havng less avalable securtes to lqudate to restore t. Another rsk that a default poses to the system s through the fre sale channel. When tryng to stay solvent a bank wll contnue to lqudate all ts avalable-for-sale securtes to try to ncrease ther θ. Thus f they end up defaultng, they wll have naturally lqudated all of ther avalable securtes (the rest beng retaned by debtors who held some of s securtes as collateral). Thus n each perod, each bank evaluates losses and oblgatons due to a shock or actvty n the prevous perod. If necessary, they each take acton as f the system s n the same state as t was at the end of the prevous perod. By watng to reflect banks actons durng perod t untl perod t + 1, 8 Begalle et al.,

22 the dynamc of the system as f they were actng smultaneously s preserved. Ths loopng process wll contnue untl the system settles and no more banks have to take acton due to actvty n the prevous perod. 3.1 Channels A prmary goal of ths framework s to provde nsght nto the evoluton of systemc rsk by measurng the contrbuton to total losses of dfferent channels through whch ths rsk propogates. Bank s total loss n each perod s denoted by λ t, whch comprses fve loss components defned below. The ntal shock channel s the drect effect of the prce shock of securty k S on a bank s balance sheet, where S s the set of shocked securtes. Defne the ndcator functon I [x Ω] on a set Ω by 1. Intal shock 1 : x Ω I [x Ω] = 0 : x Ω λ s = (27) 6 (p 0 k s0 k pξ k s0 k )I [k S(0)] (28) k=1 where p ξ k s the shocked securty prce and the set S(0) = {k : k shocked n perod 0} 2. Fre sales λ fs,t = 6 k=1 where p t+1 k > p t k f there was sgnfcant sellng of k n perod t. p t k st k pt+1 k s t k (29) The fre sale channel captures the effect of declnng market prces of securtes that bank holds on ther balance sheet due to sellng n the prevous round. 3. Fundng/lqudty stran λ bs,t = b s,t, (30) recallng that b s s amount that bank owes mmedately due to STIB credtors hoardng lqudty. Banks frst course of acton when tryng to restore ther RWCRs θ s to wthdraw funds from the repo/stib market. If a bank borrowed from, they wll have to come up wth the cash, and ther balance sheet wll shrnk by that amount. Ths s a lkely pont where the rsk n the system changes, snce ths fundng stran may force a bank to lqudate securtes, whch can then cause margn calls and fre sale losses. Cycles lke these contnue untl the system reaches an equlbrum or they all default. 21

23 4. Post-default fre sale λ pdfs,t = N (m s,t j pt+1 κ j=1 (m s,t j (1 + h κ)))i [j D(t)] (31) where κ = m c j s the type of securty used as collateral, and D(t) s the set of banks who defaulted n perod t. Ths channel captures fre sale losses that are specfcally due to wdespread sellng of securtes used as collateral for loans to a defaulted bank. Because these loans are overcollateralzed, however, t s possble that a credtor may proft from ths stuaton f the fre sale loss does not exceed the extra proft from the harcut. 5. OBS OTC dervatve losses λ obs,t = N j=1 (0.1) m b j OT C I [j D(t)] (32) Snce all OBS OTC exposures are collateralzed 90% by cash, the loss due to ths channel wll be 10% of the exposure sze. Thus the total loss for bank n a perod s λ t = λ s + λ fs,t + λ bs,t + λ pdfs,t + λ obs,t (33) and total loss over the course of a tral s T t=1 λt where T s the number of perods t takes for the system to reach equlbrum. To supplement bank rsk-management tools and regulatory stress tests, ths model s concerned wth the feedback effects of an ntal shock, whch are nontrval n the scenaros consdered below. We can separate these effects by subtractng the ntal shock λ s from the total loss T t=1 λt. 3.2 Smulaton dynamcs and data collecton Monte Carlo smulaton s used wth ths model to estmate the effect of a shock on the fnancal system. We have balance sheet data and banks nvolvement n the nterbank markets, but there s not publcally avalable data on the myrad blateral exposures between them. Ths nterbank network s very mportant for how rsk moves throughout the system, so the model smulates over a range of possble network structures. Bank balance sheet compostons and the ntal shock are the same for each tral, whle the base nterbank exposure network and collateral type matrx wll be randomly generated. The 22

24 banks wll behave subject to the framework descrbed above, and the tral wll termnate when the system reaches ts steady state. The varaton between the trals s due to the varyng strengths of nterbank lnkages and the type of securtes used as collateral for the repo (STIB) market. The base nterbank exposure matrx determnes both the proporton of each banks repo lendng and borrowng wth one another, as well as the off-balance sheet over-the-counter dervatve contracts. Contagon due to the ntal shock wll depend on the strength of these nterbank lnkages and the types of securtes used for collateral. A bank who has posted a large amount of ther MBS holdngs as collateral, for example, wll be vulnerable to a shock to MBS prces, as they wll face margn calls, and n the case that they need to lqudate securtes to stay solvent, they wll have less MBS avalable to sell snce some of t wll be held n custody by ther credtor. The model collects detaled nformaton about each bank s balance sheet, constrants, oblgatons to other banks, and ther actons n each perod of a sngle tral. Dependng on the goal of the stress test, the model collects data n dfferent ways. After an entre Monte Carlo smulaton (1000 runs) the model wll summarze the data and produce average losses due to each channel wth confdence bands, as well as the probablty of default under the stress scenaro. To gan more nsght nto the evoluton of the dfferent rsks, the model consoldates data from a sngle tral to demonstrate how the dfferent channels change over tme as banks take acton to tend to ther balance sheets. 4 Data and ts mappng to the model 4.1 Balance Sheet Data Mappng Bank balance sheet data are based on the quarterly regulatory flng, FR Y-9C, by bank holdng companes to the Federal Reserve Board. The FR Y-9C contans consoldated balance sheet nformaton, as well as more detaled nformaton on securty holdngs and off-balance sheet exposures. It also contans nformaton on the banks rsk-weghted assets and regulatory captal. Because an mportant constrant that the agents face n the model that gudes much of ther behavor s the mantenance of ther rsk-weghted captal rato θ, only nsttutons that report ths nformaton n the FR Y-9C are consdered. The top 10 banks by asset sze are used n the model, so the nsttutons that are excluded due to ths are AIG and GE Captal, whch rank 7th and 8th, respectvely. The focus of the model s on the short-term propagaton of rsk due to common exposures to securtes, OTC dervatve contracts, and short-term nterbank collateralzed lendng and borrowng. The balance sheet data n the FR Y-9C s consoldated n order to solate the pertnent tems for the model. The asset-sde of each bank s balance sheet conssts of cash c, short-term loans, or 23

25 Table 1: Balance sheet consoldaton from FR Y-9C Table 2: Securty rsk-weghtngs and fre sale elastctes repurchase agreements d s, securtes s k, and other o a. Total cash and repo numbers are used from the FR Y-9C, whle s k s a consoldaton of ther Tradng Assets and Securtes. The securty types n the FR Y-9C are mapped to sx groups of securtes as n Table 1. The actual balance sheet numbers are scaled down by a factor of E-07, whch was chosen so that Goldman Sachs assets would be $100. GS was chosen because ther assets were roughly the medan sze n the ten bank sample. 4.2 Exogenous Parameters Ths model reles on a number of parameters that reman fxed throughout the smulaton exercses, whch nclude asset rsk-weghtngs, securty market prce elastctes, and harcuts used for repo collateral. The balance sheet data used n the model matches the aggregate numbers from the FR Y-9C, but there s sgnfcant consoldaton of dfferent lne tems nto broader categores. Based on Basel III regulatons 9, each banks assets are weghted accordng to Table See 24

26 Because of the relatvely hgh-level data on rsk-weghted assets n the FR Y-9C, as well as further consoldaton of asset types n the model, the other rsk-weghted assets θ o s chosen so that each bank s RWCR θ matches the actual rato θ act reported n the flng 10. When a bank borrows from the short-term nterbank market, they enter nto repurchase agreements, where they pledge securtes as collateral n exchange for cash. The harcut s the amount of extra securty the borrower wll pledge as protecton to the credtor aganst adverse market movements, expressed as a percentage of the loan amount. The Federal Reserve provdes monthly data on collateral types, volume, and harcuts n the tr-party repo market. Harcuts range from 2% to 8%, but t s assumed that each securty n the model has a harcut of the medan 5% 11. Another mportant set of parameters are the prce elastctes of the types of securtes wth respect to sellng n the market. The degree to whch wdespread sellng of a partcular asset depresses prces drves the fre sale channel of contagon and depends on these parameters. Begalle et. al (2013) study the rsk of fre sales n the tr-party repo market n the U.S., and n examnng the structure, statstcs, and regulatons surroundng the market, they estmate the rsk of fre sales n dfferent asset classes used as collateral. Based on the sze and tradng volume of each market, they use a VaR analyss to arrve at potental shortfalls a bank would face f they tred to lqudate a type of securty used as collateral durng a fre sale stuaton. The elastctes α k are based on the medan percentage shortfalls. 5 Results 5.1 Stress testng Banks hold on ther balance sheets a wde varety of fnancal nstruments whose prces are senstve n dfferent ways to market forces. Understandng as much as possble about ther balance sheet faced wth market uncertanty and carefully gudng ther proft-seekng behavor to reduce the chance of sgnfcant losses s the dffcult goal of rsk management. VaR analyss s wdely used by banks to gve them an dea about the dfferent rsks ther balance sheets face over some perod wth some degree of statstcal confdence. These analyses are often based on hstorcal data, and n normal market condtons perform adequately. In abnormal crcumstances, however, these tools are very poor measures of rsk, whch was made partcularly apparent n the most recent fnancal crss, and even arguably a contrbutng factor. Understandng the rsk of rare events lke these s even more mportant snce that s when the consequences of underestmaton are the most dre. 10 θ o = a l θ act (w b d s + Σ 6 k=1w k s kp t k + w obs OT C ) 11 senstvty analyss to come 25

27 Stress testng s an mportant supplement to tradtonal tools, especally durng perods of relatve quet as the chance of complacency of decson makers s hghest and statstcal models memores of stressed tmes are the fantest. New regulatons and the recent fnancal crss have made stress tests an ntegral part of contnual rsk-montorng by regulators and banks as they are both very nterested n how ther balance sheet wll fare durng tmes of market stress. Stress tests vary wdely n complexty and and how they choose nput rsk scenaros. Under the Dodd-Frank Act, the Federal Reserve requres banks to run very complex stress tests that nvolve dozens of nput varables over long tme perods. However, nstead of a bank focusng on just ther own portfolos and how they expect the value to change due to scenaros, albet very comprehensve ones, the framework outlned above can nform ths approach by consderng the mportant feedback effects durng a stress scenaro. Even f a bank knows how ther portfolo wll change due to some shock, t s just as mportant to have some expectaton of how they wll react and how collectve acton of other players wll subsequently affect ther balance sheet. 5.2 Stress testng wthn the model In addton to provdng nsght nto market feedback effects, the model can attrbute losses to dfferent transmsson mechansms. Understandng the dfferent ways that a seemngly smple shock can eventually affect a balance sheet gves a more holstc pcture of a frms dfferent rsks and gude preventatve acton. For example, due to an ntal downward shock to an asset a bank s holdng, they may take some acton that wll lkely prompt other market partcpants to take acton. The nterconnectedness of the fnancal system all but ensures the propagaton of these shocks throughout the system. The rsk posed to a bank due to an asset shock can manfest nto lqudty hoardng, whch could prompt others to sell securtes to rase lqudty, whch could then return to the trgger bank n the form of further asset prce declnes or margn calls. The stress test below s an example of how a shock to a few securtes can turn nto dfferent types of rsks as t moves around the system, wth the overall damage beng greater than the ntal shock. The model ams to put a number on these knock-on effects and to provde nsght nto the process of rsk propagaton and transformaton. 5.3 Smulaton results - Bank perspectve Large market value declnes n one asset class are often accompaned by market declnes n other, especally related, markets. The stress scenaro consdered s a 30% declne of MBS prces, equtes, and other debt securtes, smulated 1000 tmes. Each tral begns wth the same shock, but the nterbank exposure network and the types of securtes used as collateral for repurchase agreements are randomly generated, drvng the varaton of the outcomes from each mult-perod run. Ths 26

28 Table 3: BoA stress test smulaton results (30% downward shock to MBS, Equtes, and Other Debt) example s from the perspectve of Bank of Amerca (BoA), the second-largest bank by assets n U.S., and the results are summarzed n Table 3. The ntal loss due to the pure market value shock to ther securty holdngs s 5.3%, but ther average total loss s 9.705%. The addtonal 0.66% loss over the pure effect was due to further market value declnes of securtes they held due to fre sales, and 3.7% was due to lqudty/fundng problems. Due to ther short-term credtors not rollng over repo transactons, they were forced, on average, to come up wth cash equal to 3.7% of ther balance sheet. Ths forced reducton of ther labltes reduces the sze of ther balance sheet, but lkely causes BoA to wthdraw funds from the short-term nterbank market and sell securtes themselves, contrbutng to the fre sale asset prces and causng problems for other banks who depend on ther short-term lendng. In ths partcular scenaro, the post-default fre sale and off-balance sheet exposure channels dd not have a very sgnfcant role. The small contrbuton of the OBS OTC exposure s lkely due to the fact that BoA was not heavly exposed to banks that defaulted, and to the fact that those exposures are collateralzed 90% by cash n the model. The post-default fre sale s also small due to the over-collateralzaton of short-term loans, and can even lead to gans as BoA can sell an amount of securtes greater than (harcut h = 0.05) the sze of the loan. In ths scenaro the tmng of these post-default collateral sales are such that the prce of the collateral had not fallen enough to cause shortfalls when the tme came to lqudate. To gve a more detaled pcture of the dynamcs of ths process, we can focus on a sngle tral of the smulaton and watch the dfferent channels nteract and the balance sheet composton 27

29 Fgure 1: Balance sheet composton over tme durng a sngle tral change over each perod. Fgure 1 shows a relatvely hgh-level vew of BoAs balance sheet over tme, plottng total assets, labltes, total securtes holdngs, and cash levels. Fgure 2 plots ther oblgatons and constrants n each perod, ncludng the sze of ther margn calls, how much they must come up wth to pay back short-term credtors (fundng stran), how much cash they have posted as margn, and how much of ther securtes portfolo s beng used as collateral n ther repurchase agreements. Observe n Fgure 1 how ther assets fall due to the ntal shock and subsequent fre sales n perod 1. Ther cash level rses over ths perod as they wthdraw cash from the repo market (do not roll over ther reverse repo transactons) and sell securtes. Fgure 2 gves a more detaled vew of the forces drvng these aggregate changes. The asset shock trggered margn calls, causng more of ther cash to be ted up as margn. At the same tme, other banks wthdrew short-term loans from BoA (Fundng Stran), whch released many of ther securtes from custody makng them avalable for sale. At the same tme, however, ths reduced lqudty lkely caused problems elsewhere as they are forced to convert ther assets nto cash to repay ther mmedate short-term credtors. If they have enough cash on hand that s not ted up as margn, they can use that, otherwse they wll wthdraw from the STIB market themselves or lqudate securtes. Fgure 3 breaks down BoA s losses n each perod by channel. Most of ther loss n the frst perod s due to the ntal shock, but the total loss s a bt hgher do to fre sales by BoA and other banks as they tred to stay solvent. The endogenous prces of each securty n each perod are plotted n Fgure 4, whch mrrors the losses that BoA ncurs n each perod due to market value 28

30 Fgure 2: Constrants and oblgatons over tme durng a sngle tral Fgure 3: Losses, broken down by channels over tme 29

31 Fgure 4: Securty prces n each perod declnes of ther securty holdngs. Also observe that much of ther total loss s drven by other banks wthdrawng short-term lqudty from the market n later perods. Fgure 5 demonstrates how banks make decsons wth respect to one of ther prmary constrants of mantanng a rsk-weghted captal rato θ of at least 8%. θ s plotted over the course of the tral for Bank of Amerca and Goldman Sachs. Dfferences n ther sze, strength of connectons wth other banks, and balance sheet compostons drve the dfferent actons they take to satsfy the constrant. GS holds a large amount of tradng assets on ther balance sheet, ncludng large amounts of the shocked securtes (MBS 5.3% of assets, Equty 10%, Other debt 6.6%, BoA: 11%, 3%, 3.5%). Ths explans the more pronounced effect of the shock on GSs θ ntally, whch causes them to mmedately hoard lqudty, wthdrawng all credt from the nterbank repo market, whch restored ther rato temporarly. Due to the actons taken by other banks n the system, however, ther θ fell back down to 4% at the end of perod 1. They proceeded to sell avalable securtes n perod 2 untl t was restored agan, whch was enough to cushon further feedback losses such as fre sales and short-term loan repayments for the rest of the perods. Bank of Amerca, on the other hand, stayed above the mnmum RWCR requrement θ after the ntal shock, but actons taken by others n the frst perod drove t down below the threshold. They were able to restore t by wthdrawng 45.3% of ther credt from the STIB market (unwnd 30

32 Fgure 5: Indvdual bank behavor to mantan the mnmum RWCR over tme 31

33 Fgure 6: Percentage exposure each bank has to one another (asset and lablty-sde) Fgure 7: BoA s actual ($) exposures (m s ) wth collateral securty type 32

34 reverse repo transactons). Feedback effects brought θ BoA down agan n the next perod, whch they were able to restore wth further wthdrawals. Ths process contnued untl the fourth perod where no acton was necessary subject to ths partcular constrant. It s stll possble that they faced other requrements such as ther buffer sze β(p + d s ), margn calls mc, and fundng strans b s. Fgures 6 and 7 show the structure and collateral composton of the short-term nterbank network for ths partcular tral of the smulaton. Each bar n fgure 6 s the percentage of bank s total short-term loans that s to bank j. Notce that any blateral exposure s capped at 20% of a banks total exposure, as well as the hgh degree of connectvty, but also plenty of asymmetry. Fgure 7 s cross secton of ths nterbank matrx, where the labeled bars are BoAs labltes (repos) and the correspondng collateral securty type. Ths nformaton can be useful for a bank runnng a stress test lke ths, snce t can help explan varaton between the dfferent rsk channels. For example, f much of ther collateral conssted of equtes, then a large market value declne wll ht them harder than f ther posted collateral was more dversfed, and wth knowledge of the rest of the collateral network, they can take preemptve acton to reduce rsk. VaR Analyss The lmtatons of Value-at-Rsk models for rsk management are well-known, a prncple weakness of whch s ther falure to perform n tmes of market stress. Besdes ther relance on hstorcal data, a prmary reason for ths s that they do not take nto account the abnormal stuatons such as lqudty shortages when assessng ther expected losses. Despte ts shortcomngs, VaR analyss remans a ubqutous rsk management tool, and the framework n ths paper can help nform these models. By ntegratng the agent-based network model nto a VaR framework, the model can take nto account the feedback effects from the system. The choce of underlyng dstrbuton drvng the VaR analyss s not the focus of ths paper, but gven a dstrbuton, the model can nclude the knock-on effects that may arse from certan random shocks. In addton to accountng for the feedback effects n the system, the framework descrbed here can allow for a more detaled understandng of the VaR results. On top of the usual expected loss wth a degree of statstcal confdence, the model can decompose the expectaton nto the dfferent rsk channels. The followng example demonstrates how ths model can be ntegrated nto a VaR framework n a very smple case. We assume that equty prce returns follow a normal dstrbuton wth mean 0 and standard devaton σ = The other securtes k n the model are assgned arbtrary correlaton coeffcents ρ k relatve to equty prces, where ρ MBS = 0.2, ρ USGA = 0.2, ρ O.D. = 0.5, ρ ABS = 0.6, and ρ UST = 0.0. The bank used n ths example s Goldman Sachs. The dstrbuton s drawn from 100 tmes, and for each draw, the smulaton over 1000 nterbank 12 Based on S&P 500 prces from July 7, 2008 to July 7,

35 Fgure 8: VaR results for Goldman Sachs Fgure 9: VaR results - probablty of default (MS) 34

36 Table 4: Regulator stress test results for aggregate rsk n entre system networks s run. For each shock the average loss due to each channel and the number of defaults are recorded. The 99 th worst outcome, broken down by channel, s then reported as the potental losses wth 99% confdene. The results are shown n Fgure 6. From ths partcular, smplfed scenaro, the model pcks up feedback effects that account for 46% of the total losses. Agan we see that fundng lqudty s an mportant drver of losses, as well as fre sales. Fgure 7 shows the probablty of default for Morgan Stanley gven the same dstrbuton of equty returns and correlaton coeffcents. Observe that at certan draws the number of defaults ncreases sharply, from 0 to 80, n ths case. 5.4 Smulaton results - Regulator s perspectve Ths framework can also be used by regulators for stress testng, whch offers the advantage of takng nto account the multple channels through whch fnancal nsttutons are connected, and also provdng nsght nto the relatve contrbuton of these dfferent channels as well as ndvdual nsttutons contrbutons to ths rsk factors. A regulator mght be nterested n estmatng how the fnancal system would fare under a severe market downturn lke the one consdered above. Usng the same shock of 30% to MBS, Equtes, and Other Debt prces, we analyze the effect of ths on the system as a whole. Table 4 summarzes the smulaton and Table 5 shows the number of tmes each bank defaulted out of the 1000 trals. Observe that n aggregate t s often the case that the feedback effects from the nteracton of dfferent agents over the course of a tral amplfy the ultmate effect of the ntal shock. In addton to understandng how a shock propagates and amplfes as t spreads throughout the 35

37 Table 5: Regulator smulaton results - number of defaults out of 1000 system, a regulator mght want to know the roles that the market partcpants play n ths transmsson mechansm. Fgure 8 plots each banks relatve contrbuton to the dfferent channels from whch losses manfest as a percentage of total assets n the system. Fgure 9 dsplays contrbutons relatve to each banks own balance sheet. As one would expect, the larger banks n the system contrbute the most to the overall loss to the system n the smulaton, wth JPM, Ct, GS, BoA, and MS sufferng the bulk of the aggregate losses and fundng/lqudty strans. However, controllng for balance sheet sze n the bottom fgure we see that MS and GS have a dsproportonate effect on total losses, exceedng those of JPM, BoA, Ct, and WF. Wells Fargo, n partcular, despte ts sze, fares well n the smulaton. The concentraton of losses on the pure nvestment banks, GS and MS, make sense however, snce the stress scenaros tested and channels examned by the model focus on the lqud, publc fnancal markets and short-term nterbank relatonshps, n whch the nvestment banks partcpate heavly. 6 Concluson As the global fnancal system grows n sze, nnovates, and becomes ncreasngly nterconnected, the need for a more sophstcated understandng of the ever-changng rsks wll become ncreasngly mportant. The model presented here uses an ncomplete sample of the U.S. fnancal system and leaves out some mportant fnancal nsttutons. Lke any theoretcal model, ts applcablty s only as strong as the assumptons and parameters that gude the dynamcs of the system. Combnng the natural platform of networks n fnance wth an agent-based approach has promse, and even the smple results demonstrated here can nform exstng stress testng methodologes, whch, despte 36

38 Fgure 10: Bank contrbutons to rsk channels (% total system assets) Fgure 11: Bank contrbutons to rsk channels (% ndvdual bank) balance sheets 37

39 ther comprehensveness, are also prone to neglect the equally mportant feedback effects durng a stress scenaro. From a bank s perspectve, havng an dea of how they and others wll react n response to a potental shock they may face allows them to prepare for rsks that are not drectly apparent. By trackng these feedback effects through tme a bank can take preventatve acton to reduce ther overall rsk. Smlarly, f regulators have a more granular understandng of rsk propogaton n the fnancal system they can make more nformed and targeted regulatons. The stress tests carred out here showed that the nterbank lendng market s a major source of systemc rsk n the U.S. bankng system, and the collateral underlyng these transactons lnks ths rsk to potental fre sales and margn calls. Observng the dynamcs of a sngle tral of the smulaton revealed that these effects often do not arse untl later perods, after the ntal shock. It was also shown that ths model can be ntegrated nto a VaR analyss framework for rsk management, potentally ameloratng one of VaR s lmtatons of falng to account for feedback effects due to large shocks found n the tals of a chosen dstrbuton. An advantage of ths framework s ts flexblty, snce a wde array of stress test scenaros can be used. Only a few possbltes were dscussed above, but another nterestng scenaro that regulators mght be nterested n s a lqudty crunch scenaro where the nterbank debt market dres up suddenly, whch was shown to be a prmary drver of systemc rsk. Testng the effect of credt rsk transfer products on systemc rsk s also a very pertnent topc for regulators. When ndvdual nsttutons trade these securtes, the rsk n the system can become concentrated, creatng condtons that leave the system vulnerable to crses. In addton to beng easly extentable by usng more nformaton on market structure and bank behavor, a framework lke ths s wellsuted to to be ntegrated nto exstng approaches used by banks and regulators. 38

40 References [1] Allen, Frankln, and Douglas Gale. Fnancal Contagon. Journal of Poltcal Economy (2000): Prnt. [2] Begalle, Bran, Antone Martn, James McAndrews, and Susan McLaughln. The Rsk of Fre Sales n the Tr-Party Repo Market. Federal Reserve Bank of New York Staff Reports (2013). [3] Bookstaber, Rchard. Usng Agent-Based Models for Analyzng Threats to Fnancal Stablty. Offce of Fnancal Research, U.S. Department of the Treasury (2012). [4] Chan-Lau, Jorge A. Balance Sheet Network Analyss of Too-Connected-to-Fal Rsk n Global and Domestc Bankng Systems. IMF Workng Paper (2010). [5] Cont, Rama, Amal Moussa, and Edson Santos. Network Structure and Systemc Rsk n Bankng Systems. Workng Paper (2012). [6] Duarte, Fernando, and Thomas M. Esenbach. Fre-Sale Spllovers and Systemc Rsk. Federal Reserve Bank of New York (2013). [7] Esenberg, Larry, and Thomas H. Noe. Systemc Rsk n Fnancal Systems. Management Scence 47.2 (2001): Prnt. [8] Elsnger, Helmut, Alfred Lehar, and Martn Summer. Rsk Assessment for Bankng Systems. Management Scence 52.9 (2006): Prnt. [9] Montagna, Matta, and Chrstoffer Kok. Mult-layered Interbank Model for Assessng Systemc Rsk. European Central Bank (2013). [10] Ner, Erlend, Jng Yang, Tanju Yorulmazer, and Amadeo Alentorn. Network Models and Fnancal Stablty. Journal of Economc Dynamcs and Control 31.6 (2007): Prnt. [11] Upper, Chrstan. Usng counterfactual smulatons to assess the danger of contagon n nterbank markets, (2007) BIS Workng Papers 234, Bank for Internatonal Settlements. [12] Upper, Chrstan, Worms, Andreas. Smulaton methods to assess the danger of contagon n nterbank markets. European Economc Revew 48.4 (2011):

41 Appendx A Interbank Network Constructon A fnancal system by nature s hghly connected as t fosters lqudty n markets and the creaton and allocaton of dfferent rsks. These connectons, however, are a prmary source of rsk to the system becuase problems for one nsttuton can quckly become problems for others because of ths nterdependence. Despte ts mportance, however, there s no detaled data on these nterbank lnkages n U.S., so ths structure wll be randomly generated n the model. A base nterbank network wll be used to determne the sze of lnkages between counterpartes n the repo market and the off-balance sheet over-the-counter dervatve market (OBS OTC), and s denoted by m b. Ths nterbank network can be represented by the matrx whose entres m b j correspond to an exposure of bank to bank j. In the case of the repo market, m b j would correspond to a collateralzed short-term loan from to j. Smlarly, bank would also be nterested n ther oblgaton towards j, m b j. The value of each entry n the nterbank matrx corresponds to a percentage of s total exposure to j. Ths mposes the mportant constrant that each row and column sum to one. In other words, a bank s proportonal lendng and borrowng to and from all of the other banks cannot exceed 100%. Thus we must have a matrx where N =1 mb j = 1 and N j=1 mb j = 1. Ths s not a trval feature to requre a matrx to have, but a branch of recreatonal mathematcs called magc squares provdes much of the machnery needed to produce matrces conformng to our constrant. A magc square s a square matrx whose rows and columns all add up to the same number whose value depends on the sze of the square and the range of numbers chosen to populate the matrx. To produce an nterbank matrx of proportonal exposures, a random magc square s generated, and each entry s dvded by the row sum to yeld a percentage. Banks cannot lend to themselves, however, so t s mposed on m b that m b = 0 for all. It s assumed that the proportonal exposure that s removed from a row and column by forcng a dagonal entry to be zero represents a bank s exposure to the rest of the market, not explctly treated n the model. Although the ten banks consdered n ths model account for the majorty of actvty n the markets of nterest, ths assumpton acknowedges that the ncluded banks are not entrely dependent on just the other 9 banks. 6.1 Mappng the nterbank network to bank balance sheets The nterbank network s used to construct the short-term nterbank lendng (STIB), or repo market denoted by m s, where m s j represents a collateralzed loan from bank to j. Whereas mb s defned n terms of percentage exposure, m s s defned n terms of actual dollar amounts. Due to the varyng balance sheet szes and compostons of the dfferent banks, t s possble that a large bank 40

42 s dollar exposure to a smaller bank j m s j = mb j ds could exceed even the total amount that the smaller bank has borrowed from the nterbank market. To account for ths, the constructon of m s from m b chooses each m s j so that m s j = d s b s j : d s mb j bs j mb j : d s mb j > bs j mb j (34) In other words, f bank s proposed dollar amount of loans to bank j exceeds the amount that j s supposed to be borrowng from, then the dollar amount s set to the smaller amount, whch n ths case s j s borrowng. The remanng loans d s mb j bs j mb j are allocated to the outsde market. from that are not appled to j A consequence of ths set-up wll be that, as we would expect, actons taken by large banks wll have greater dollar effects on smaller counterpartes despte sharng the same percentage exposure. When m s s constructed accordng to equaton (34), t wll lkely be the case that a small bank such as Captal One s borrowng and lendng wll be fully allocated to a larger bank such as JPM, whereas JPM s proposed percentage lendng to Captal One would be much larger than Captal One s, and be subsequently alloted to the outsde market. When JPM defaults, for example, the percentage exposure of Captal One to JPM wll be much greater than the effect of Captal One s default on JPM. For each run of the smulaton, a dfferent nterbank network s generated. There are three dfferent networks n the bankng system that are derved from the random nterbank matrx. The STIB market matrx m s has a correspondng collateral matrx m c where each m c j s the type of securty used as collateral for the loan m s j. Banks are also connected through off-balance sheet over-the-counter dervatve transactons. The FR Y-9C form reports each bank s net current credt exposure to counterpartes n these postons and the type of collateral they are holdng aganst that exposure. Current credt exposure, or replacement cost, s the market/far value of the dervatve contract f t s postve, and zero otherwse, measurng the cost for bank to replace the poston f ther counterparty were to default. Ths captures counterparty rsk n the large off-balance sheet OTC dervatve market, whch, due to the nherent leverage n dervatve transactons and ther exstence off-balance sheet s an mportant channel through whch contagon can spread that s much larger than balance sheet szes n the system would suggest. The matrx m obs represents ths network where each m obs j s the off-balance sheet credt exposure bank has to j. Snce the magc squares generaton of the random, base nterbank matrx yelded a complete network between the banks n the model and an outsde entty, each m b j s a percentage of bank s total exposure allocated to j. Ths network s statc throughout each run of the smulaton. 41

43 The repo network s derved from ths matrx by dstrbutng each bank s repo lendng and borrowng to other banks accordng to the underlyng m b. Unlke n other nterbank network models, each entry of the matrx m s wll depend on whch bank s usng t, and f they are evaluatng asset-sde exposures or lablty-sde exposures. That s, bank s repo agreement wth j n dollar terms s m s j = mb j ds, where ds s s total lendng n the repo/stib market. Smlarly, f s nterested n ther oblgaton to bank j, they compute m s j = mb j bs. For the off-balance sheet exposure network, bank s nterested n ther current credt exposure to another bank j, whch s smlarly calculated usng the underlyng random nterbank matrx so that m obs j = m b j OT C. 6.2 Generatng the collateral matrx Each entry m c j {1, 2, 3, 4, 5, 6} n the matrx mc of collateral asset types corresponds to the type of securty held aganst a loan from bank to j. Ths matrx s derved probablstcally usng tr-party repo data from the Federal Reserve. The data break down the types of securtes used as collateral n tr-party repo transactons as a percentage of the total. Accordng to the data-mappng descrbed n secton 4, the percentages ψ k from the FRBNY are nterpreted as the probablty that a gven securty type wll be used as collateral for a STIB loan. Thus each entry of the collateral matrx m c j s a random varable wth probablty mass functon f(k) = ψ k (35) where k {1,...6} s the securty type and ψ k s the share of total Tr-Party repo lendng that was collateralzed by securty k n Q Another mportant network n the fnancal system s the network of common exposures to securtes. As n Montagne and Kok (2013), consder a matrx m v where each entry m v j represents the correlaton between bank and j s asset holdngs. In other words, t s the degree to whch a shock to the market prce of an asset held by two dfferent banks wll affect the overall system. These exposures are mplct n the model as banks shared exposure to assets s a consequence of ther balance sheet compostons, ths shows that a connecton between two fnancal nsttutons does not have to be an explct transacton, and s an mportant dmenson to the overall structure and dynamcs of the system. 42

44 Appendx B Interbank matrx generaton algorthm and statstcs The nterbank exposure matrx s generated used technques from whch provdes nformaton on algorthms for creatng magc squares of dfferent szes. The 10x10 matrces used n the model are generated usng four random 5x5 perfect squares and combnng them n a way that preserves the property that all row and columns sums are equal. The numbers n each cell range from 0 to 99 and the magc sum s 495. Thus an entry of m b of 24 corresponds to a 24/495 = 4.8% exposure from to j. There are 275,305,224 dfferent 5x5 magc squares, ncludng rotatons and mrrorngs. The number of possble 10x10 matrces s therefore greater than ths, provdng an ample amount of possble networks used n the Monte Carlo smulaton. The matrces are generated randomly n each tral by producng a random 5x5 matrx, whch nvolves choosng two random rows that form the bases for the eventual magc square. The 5x5 square s then rotated or mrrored randomly. Ths random 5x5 s then turned nto a 10x10 matrx accordng to the technques descrbed n the webste. A fnal random element s ntroduced by randomly rotatng or mrrorng the 10x10 matrx. Each entry of the 10x10 matrx represents a percentage exposure to another bank. The hstogram above s based on 10,000 random 10x10 matrces, categorzed by percentage ranges. There s farly even dstrbuton of exposures, wth some bas towards the upper and lower ranges. The average maxmum exposure of any one back towards another was 18.02% and the average mnmum exposure was 1.78%. 43

45 Appendx C Normalzed Balance Sheets used n the smulatons 44