Confidence-Driven Contagion in Financial Networks (PDF Download ...

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Confidence-Driven Contagion in Financial Networks∗ Modibo K. Camara May 6, 2016

Abstract Two forms of contagion permeate financial markets. The first acts through direct claims on balance sheets, and is well-understood. The second acts through strategic liquidity withdrawal, and is not. This paper proposes a model of interbank lending where loans are represented by edges on a directed acyclic graph. Banks decide whether to roll over or collect outstanding loans. Banks lacking sufficient immediate funding will later default on their interbank obligations. Despite not being supermodular, this game features a unique threshold equilibrium in the presence of higher-order uncertainty. Through Monte-Carlo simulation calibrated to the U.S. federal funds market, we find that confidence-driven contagion causes significant welfare deviations from the optimal coordinating behavior. Network measures common in the empirical literature fail to predict the magnitude of this contagion, and, in its presence, several sensible market interventions actually lead to welfare reductions.



This was prepared for submission as my undergraduate honors thesis at the University of Pennsylvania. I would like to thank my advisors, Rakesh Vohra and Jere Behrman, for their invaluable guidance. In particular, Vohra encouraged me to discipline the paper and introduced me to a sizable swath of the literature on Bayesian and supermodular games. This paper also benefited greatly from discussions with Steven Matthews, Guillermo Ordo˜ nez, and the elder Modibo Camara. The inspiration for this paper is due in part to presentations by Francis Diebold, Laura Liu, and Stephen Morris. Finally, my fellow honors seminar participants endured long presentations and always provided useful feedback. All errors are my own.

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Contents 1 Introduction 1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Preview of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Model of Interbank Lending 2.1 Under Complete Information . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Under Incomplete Information . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Simulation Methodology 3.1 Network Generation and Calibration . . . . . . . . . . . . . . . . . . . . . . 3.2 Balance Sheet Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Simulation Results 4.1 Quantifying Confidence-Driven Contagion 4.2 Identifying Risk Factors . . . . . . . . . . 4.3 Evaluating Pre-Shock Interventions . . . . 4.4 Evaluating Post-Shock Interventions . . .

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5 Conclusion

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6 References

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1

Introduction

Consider an interbank lending market following a shock. Loans are made over-the-counter, linking banks directly and indirectly through shared counterparties. Banks face incentives to withdraw liquidity, both in order to avoid giving loans that will not be repaid and as a precaution against coming illiquidity. That these markets reliably exist (but sometimes fail) attests that market behavior is somehow tied to fundamentals. That these markets fail even in the absence of widespread defaults attests that market confidence, too, plays a role. This paper proposes a model that captures the essence of this situation, and proves that, in the presence of slight idiosyncratic uncertainty, it permits a unique threshold equilibrium. Behavior is strategic, networks and shocks are exogenous, and defaults are endogenous. Its lessons apply more broadly to the study of self reinforcing calamity in financial networks. When it comes to non-strategic contagion (acting through unpaid obligations) Eisenberg and Noe (2001) and many others have presented results that cleanly predict its spread in any arbitrary network. Here, we attempt to do the same for confidence-driven contagion. This is the primary contribution of this paper: a preliminary tool for policymakers to deal with financial networks as they are, not as we would like them to be (as in Allen and Gale, 2000), nor as we expect them to be (as in Erol and Vohra, 2014). We do not stop there, however. With a model of liquidity crises in interbank lending markets, a number of broader questions present themselves. We specify a network generation process and calibrate it to mimic the U.S. federal funds market. Monte Carlo simulations suggest that: • Confidence based contagion causes significant deviations from the optimal behavior. • The market response to a shock is not easily explained by common network measures. • A number of proposed market interventions may actually amplify financial instability. Below, section 1.1 provides a background on financial contagion and the relevant literature. Section 1.2 describes the paper’s methodology and results in greater detail.

1.1

Literature Review

In typical markets for goods, the relationship between two counterparties begins and ends with the initial agreement to transact. By contrast, financial markets are known for their complex, time-staggered transactions. A repurchase agreement involves two transfers of a single asset. An interest rate swap can involve hundreds of transfers that span decades. When an agent makes a purchase in these markets, it not only considers the terms of the 3

contract itself, but also the counterparty with whom the agent will enter into a relationship. At any point in time, we can describe the nature of these relationships by the network of obligations they imply. Recently, such networks have captured the attention of policymakers, particularly in the context of interbank lending markets (see e.g. Bech and Atalay, 2010; Beltran, Bolotnyy, and Klee, 2015; Blasques, Br¨auning, and van Lelyveld, 2015; Brunetti et al., 2015). A primary goal of the financial network literature has been to determine the impact of a potentially localized shock on a system of financial institutions (henceforth, agents). The question is straightforward in contexts where the topology of the obligations network plays an insignificant role; for example, if the set of agents can be partitioned into buyers and sellers of the relevant financial contract. However, more complex patterns of obligation will permit the spread of financial contagion. When an agent’s interaction with one counterparty cannot be disentangled from their interaction with another counterparty, there arises an indirect link between the two counterparties. By virtue of having a mutual trading partner, two agents that did not enter into any financial agreement with one another may nonetheless be affected by each other’s decisions and exposed to each other’s vulnerabilities.1 . Two prominent forms of contagion bear distinction, although in practice they are mutually reinforcing and therefore difficult to distinguish. I refer to the first as balance sheet contagion, thus named because it propagates through direct claims between agents. Simply put, one agents’ failure to meet its obligations will jeopardize the ability of its counterparties to meet theirs. In this manner, a cascade of defaults can ensue. Eisenberg and Noe (2001) provide weak conditions under which this recursive process yields a unique solution, regardless of the order in which debts are written-off. Their fictitious default algorithm simulates succeeding rounds of defaults, in order to determine firms’ balance sheets after all obligations are either paid or defaulted on. A number of papers have extended the Eisenberg-Noe framework to incorporate various market idiosyncrasies, including minimum liquidity requirements (Cifuentes, Ferrucci, and Shin, 2005) and exogenous default costs (Rogers and Veraart, 2013). Others have analyzed its properties, including the sensitivity of the fictitious default algorithm to initial balance sheets (Liu and Staum, 2010), bounds on the magnitude of contagion (Glasserman and Young, 2014), and characteristics of the initial obligation network that tend to facilitate contagion (Elliot, Golub, and Jackson, 2014; Acemoglu, Ozdaglar, and Tahbaz-Salehi, 2015a). The mechanical problem of how balance sheet contagion spreads in a network appears to be well-understood. The other form, confidence-driven contagion, is a special case of a more general phe1

Furthermore, by affecting each agent, it affects the agents’ other counterparties. Indeed, any two agents that are connected by a chain of obligations may be mutually exposed.

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nomenon, where an agent’s optimal decision with respect to a given counterparty depends on the realized decision of the other counterparties. Here, a financial institution withdraws from its typical trading activities because it observes (or expects) a deterioration in the quality of a financial system; of course, this withdrawal further exacerbates (or instigates) the deterioration. For example, a bank facing unexpected deposit withdrawals may become more aggressive in foreclosing on mortgaged properties. The homeowner’s well-being is linked to the choices of the depositors, despite lacking a contract that directly binds them. Typically, confidence-driven contagion is associated with a drop in market liquidity, either through the hoarding of liquid assets or through liquidity-taking behavior such as fire sales. In a seminal work, Allen and Gale (2000) construct a four-region model of interregional risk sharing between banks, where banks interact only with their neighbors on a prescribed network. Contagion results from a choice on the part of banks within a given region to withdraw deposits from other regions. In the case of a complete network, they find that risk sharing is optimal and contagion limited, before continuing to document the trade-offs associated with decreased interconnectivity. Erol and Vohra (2014) follow with a more conceptually abstract model where nodes may undertake joint partnerships in a primary stage, and are presented with the option to strategically default after a random shock is realized. They find that lowering the probability of a negative shock may actually increase the chance of a system-wide failure by encouraging more nodes to connect with one another. Notably, they also find that the networks formed depend significantly on the degree to which edge-level shocks are correlated across the market. A generalized version of their model is used in Erol (2015) to study the effects of bailout policy in financial networks, particularly an associated moral hazard. While both papers offer valuable insights into the structure of financial networks, their analysis of network formation is far more sophisticated than their analysis of behavior after the network is formed. In particular, they assume that behavior corresponds to the optimal iteratively dominant strategy. Additionally, like this paper, Gai, Haldane, and Kapadia (2011) study liquidity hoarding behavior in a fixed network. Yet their procedure of iteratively inducing funding withdrawals appears functionally to Erol and Vohra’s assumption. Real financial markets do not seem to be efficient in this way. Nor do hypothetical financial markets, according to section 4.1. Other papers that model of interbank lending markets generally fail to capture financial contagion. By such markets, we mean those dealing in unsecured, overnight, and over-thecounter loans between banks. These papers typically assume either a continuum of agents (e.g. Freixas and Jorge, 2008; Heider, Hoerova, and Holthausen, 2009; Caballero and Simsek, 2013; Bianchi and Bigio, 2014; Afonso and Lagos, 2012) to avoid explicitly considering the 5

network’s effect on strategic incentives. Others require a hard-coded interpretation of beliefs (Ladley, 2010; Arinaminpathy, Kapadia, and May, 2012). This stands despite a burgeoning literature on games where networks determine which agents interact. That is not entirely surprising: this class of games exhibits a strain of equilibrium multiplicity that traditional refinements are not well-equipped to reconcile. In this paper, however, we propose that one equilibrium refinement may carry some weight. The global games literature2 is predicated on the observation that, in many real life scenarios where players prefer to coordinate on some outcome, noisy information often makes perfect coordination impossible. Instead, we embed the complete information game of interest into a larger space of games, allowing the outcome to vary in imperfectly-observed fundamentals. The compelling result is that as the signal error vanishes, for a large class of games with complementarities, there is in fact a unique iteratively dominant outcome, and hence a unique equilibrium. Even more compelling is that this equilibrium interacts nicely with fundamentals; higher fundamentals always imply higher outcomes. To our knowledge, only one other paper has studied global games on networks: Harrison and Mu˜ noz (2008), which constructs an abstract link formation game.3 In their model, unlike ours, actions are edge-specific and require bilateral agreement. In our model, unlike theirs, the game is generically not supermodular. The follow-up experimental results in Elbittar, Harrison, Mu˜ noz (2008) are promising; the global games approach to network formation outperforms several alternatives found in cooperative game theory. Other experiments (e.g. Heinemann, Nagel and Ockenfels, 2004; Cabrales, Nagel, and Armenter, 2007) have also tended to support the qualitative result that people coordinate on high actions only when presented with sufficiently high fundamentals, even if the particular strategy adopted sometimes deviates from the associated global game prediction. Indeed, this is consistent with actual observations of interbank lending markets under stress. They neither always collapse, nor always succeed, suggesting a high sensitivity to the intensity of the stress. Roukny, Georg, and Battiston (2014) find remarkable stability in the network topology of the German interbank lending market from 2002 to 2012 (although not in the CDS market). Wetherilt, Zimmerman, and Soram¨aki (2010) find that while total activity did not change substantially, the nature of this activity did. During the 2007-08 financial crisis, the number of bilateral relationships decreased and, in particular, shifted away from what earlier constituted the core in a way that the authors interpret as diversification on 2

See Carlsson and van Damme (1993) for the original paper. See Morris and Shin (2006) for a survey of key results and applications. This work has found many applications relevant to financial markets, including currency crises, debt crises, and bank runs. 3 A semantically-proximate but conceptually distinct work is that of Dahleh et al. (2012), which applies a network structure to agents’ information in a global game, rather than to their strategic interaction.

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the part of participants to diminish liquidity risk. Soram¨aki et al. (2007) find that, in the financial turmoil presented in the aftermath of the terrorist attacks on September 11th, 2001, the number of Fedwire transactions shifted dramatically, from an average of 84,786 to 59,640 on that day, as a result of self-reinforcing precautionary liquidity hoarding. Beltran et al. (2015) find a drastic reduction in activity on the federal funds market during the 2007-08 financial crisis. While Ashcraft, McAndrews and Skeie (2009) look less explicitly at the structure of the federal funds network, they proxy a low willingness to lend through excess reserves and high interest rates. Through their model, they interpret this as the result of precautionary liquidity hoarding. Afonso, Kofner, and Schoar (2011) bring reasons to doubt that interpretation, finding that the interbank lending is affected negatively even if it does not freeze entirely, but that lending behavior is better explained by perceptions of counterparty risk than liquidity hoarding. To add more ambiguity, Beltran et al. find that both counterparty risk and liquidity risk are associated with diminished lending. Fortunately, both phenomena are studied in this paper.

1.2

Preview of Results

Section 2 constructs a game-theoretic model of a market’s response to a shock, assuming an initial network of obligations and an immalleable interest rate. Banks are unable to form new connections, and must choose whether to roll-over or withdraw existing loans. Banks face two incentives: (a) they want to lend only to solvent counterparties, requiring coordination with other lenders, and (b) they want to ensure their own solvency by not lending excessively. Defaults can occur due to liquidity shortages, even when all banks are technically solvent. After assuming a vanishing amount of idiosyncratic uncertainty, I draw upon methods from the global games literature to prove that there exists a unique threshold equilibrium. The model underlies several Monte Carlo simulations, described in section 3, where the distribution of markets drawn is calibrated to mimic the Federal Funds market and its network topology (as described by e.g. Bech and Atalay, 2010). Section 4 uses these simulations are used to further study the model. Confidence-driven contagion has a noticeable and statistically significant welfare effect, relative to both the optimal coordinating behavior (assumed in several existing models, as described above) and the maximal coordinating behavior (corresponding to a non-strategic model like Eisenberg and Noe, 2001). Section 4.2 finds that a selection of network measures, largely inspired by the natural sciences, fails to have any predictive power in this model. This stands despite their proliferation in empirical studies of interbank lending markets and even their use in comparative statics for theoretical work. Sections 4.3 and 4.4 evaluate several commonly-discussed counterfactuals for given networks,

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including securitizing loans, breaking up large banks, implementing a limited guarantee fund, and freezing the market. For various reasons, among them moral hazard and the difficulty of coordinating multiple parties, almost all of these interventions are in fact welfare-decreasing.

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2

Model of Interbank Lending

This stylized variant of an unsecured interbank lending market contains a total of n banks, where bank i is endowed with some reserve Bi consisting of cash-on-hand along with its net position in interbank loans. These are interpreted as the bank’s liquid assets, which it may use to pay immediate obligations in order to avoid a default. An adjacency matrix M describes the obligations between banks, which for technical reasons we assume is acyclic. No new relationships may be established; loans may only be rolled over or withdrawn. All banks know the structure of the network, and they know the value of each other’s reserves. Timeline: Conceptually, the model proceeds in three stages. 1. A shock occurs as the existing loans mature. Each bank decides whether to rollover all of its loans (keep Mij = 1, ∀j) or collect all of its loans (Mij = 0, ∀j). 2. Each bank’s short-term reserves are the sum of its reserve Bi with received loans, minus provided loans. Banks with negative reserves are forced into default (Si = 0). 3. The rolled-over loans mature. Banks collect the loaned amount at a fixed interest rate r from a banks j if it has not defaulted (Sj = 1). By now, banks will have been able to liquidate assets in response to the shock; as such, repayment is guaranteed. No amount is received from defaulting counterparties. (This could be interpreted as resulting from a damaging fire sale or a bank run that depletes bank j’s assets.) Banks must still make payments to their defaulting lenders. The precise costs of default are left unspecified, but it is understood that a bank will never intentionally default, and conditional on imminent default, it prefers not to aggravate its situation by rolling over loans. Furthermore, the interest rate r is exogenous, unaffected by market behavior. The restriction that banks either roll over all or none of their loans may seem odd, and indeed, is not ideal. In practice, contracts are bilateral, and may be renewed bilaterally. In terms of the results we provide in this section, extending the model may not be terribly difficult.4 However, the main reason to restrict ourselves to unidimensional actions and unidimensional signals are the computational difficulties associated with identifying fixed points 4 Oury (2005) studies games with multidimensional action spaces, but only finds uniqueness results under the assumption that the game satisfies complementarities in own-actions. That is, higher lending to one counterparty need make higher lending to another counterparty more desirable. Clearly, there would be substitution effects between borrowers that violate this assumption. However, there is evidence in the form of a special case (Fujimoto, 2014) that multidimensional signals (say, on separate interest rates associated with lending to each bank) are able to uniquely select between multidimensional actions that are not selfcomplementary.

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in higher dimensional spaces. The naive extension of this model would prove impractical for even moderate-sized networks. In the crisis scenarios we are most interested in, some bank i will face negative cash reserves in the short term. If a bank j has lent money to i, it will face a choice. If it withdraws the loan, it guarantees itself the principal payment, but cannot invest further in that period. If it rolls over the loan, it trades off interest payments against counterparty risk. This drives one channel of financial contagion. The other common channel, balance sheet contagion, is entirely absent here. It is assumed that banks have their due loans in hand before deciding to roll over; alternatively, one can interpret this as saying that interbank loans have seniority over other obligations, in the event that a bank has negative initial reserves. Likewise, as mentioned earlier, by the time the rolled-over loans are due, we assume banks will have been able to liquidate enough assets to account for the initial, unexpected shock. Accordingly, the fictitious default algorithm of Eisenberg and Noe (2001) is not used. Under complete information, this model will feature a number of plausible equilibria. In the incomplete information version of this game, each agent will receive a private signal of the interest rate, with arbitrarily high accuracy. If it seems bizarre that the interest rate be subject to uncertainty, consider that (a) we need not interpret the interest rate as such a rate, but rather as the benefit of lending over some uncertain outside option, and (b) the results should apply equally well for uncertainty about central bank policies or some other parameter that satisfies certain properties (see footnote 6). With this simple adjustment, we can prove that there exists a unique threshold equilibrium; that is, an equilibrium where banks lend if and only if the interest rate is sufficiently high. This game, however, defies some standard assumptions underlying almost all (if not all) of the global games literature. The uniqueness/existence proof will take some work, as presented in section 2.2.

2.1

Under Complete Information

Consider an n-player normal form game. Each bank i chooses a binary action, ai ∈ {0, 1}, representing the choice to rollover all loans (ai = 1) or collect all loans (ai = 0). Let Bi ∈ R denote bank i’s initial cash reserve. Let M denote an n × n non-negative real-valued matrix, representing the initial obligations network. Specifically, let Mij denote the quantity loaned to bank j by bank i. To avoid trivial cases, assume that all columns of M are nonempty. There are three additional parameters, all of which are common knowledge: • The interest rate r ∈ R+ , received for all loans to non-defaulting counterparties. • The function fi : {0, 1}n → R describes bank i’s payoff when bank i defaults. The following condition ensures that a bank will never intentionally default and that, con10

ditional on defaulting, it prefers to collect outstanding loans. For all a−i ∈ {0, 1}n−1 , Bi > fi (0, a−i ) > fi (1, a−i ) We can interpret fi as representing losses to due fire sales, bankruptcy costs, legal restrictions, and other factors. • The probability of a systemic bailout,  ∈ (0, 1). If such a bailout were realized, all solvent banks with insolvent counterparties would be compensated for their losses. That this parameter exists ensures that loans to any bank become a profitable investment at a sufficiently high interest rate. Generalizing r and  to be bank-specific should not fundamentally affect our results. However it will be important that these two parameters not depend on the in-game actions undertaken by banks.5 While this is not realistic, it is also not unreasonable to consider a world wherein (a) interest rates are contractually fixed and (b) policymakers decide whether or not to intervene before the market has a chance to react. To define the payoff function u : {0, 1}n → Rn , we require one last auxiliary definition. Given an action profile a ∈ {0, 1}n , let the solvency of bank i be  1 B + Pn a M − a Pn M ≥ 0 ij ji i i j=1 j=1 j Si (a) = 0 otherwise

(1)

Then, agent i’s (expected) payoff will be " ui (a) = Si

Bi + (1 − ) ai

n X

#

Mij (rSj − 1) +  ai

j=1

ui (a) = Si

Bi + ai

"

n X

n X

#! Mij (r − 1)

+ (1 − Si )fi

j=1

! Mij (rSj (1 − )) + r − 1

+ (1 − Si )fi

(2)

j=1

An important fact that will complicate our analysis is that this game is not supermodular. Consider the following example. Let banks i and j lend to bank k, where bank k requires loans from both i and j to stay afloat, but is guaranteed to default if it rolls over its own 5

The uniqueness of equilibria in global games depends crucially on the difficulty of coordinating in the absence of common knowledge. When publicly-observable parameters are affected by in-game actions, agents can sometimes use them to infer the private information of their opponents. This issue arises in a variety of models (e.g. Hellwig, Mukherji, and Tsyvinski, 2004; Angeletos, Hellwig, and Pavan, 2006; Angeletos and Werning, 2006). This is an important question that has inspired theoretical exploration of its own; however, exploring its implications here is beyond the scope of this paper.

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loans. Bank i’s best response is certainly increasing in bank j’s action. However, it is also decreasing in bank k’s action. Generally, the game u can be thought of as having two components. The coordinated lending component exhibits the usual complementarities of supermodular games. However, the borrower-lender component is assymetric: the borrower’s best response is increasing in the lender’s action, while the lender’s (preferring prudent to reckless borrowers) is decreasing in the borrower’s action. The saving grace of this model is that the lender’s incentives are closely aligned with the borrower’s. Namely, the borrower will not intentionally default in an equilibrium. We now describe some basic properties of u that will be useful in deriving the more substantial results of section 2.2. Lemma 1 (Partial Dominance Regions) There exists x0 such that r < x0 implies ai = 0 is strictly dominant for all banks i. Likewise, for all banks i where Si (a0 ) = 1, ∀a0 ∈ {0, 1}n , there exists x0 such that r > x0 implies ai = 1 is strictly dominant. Proof. Define x0 := 1 and x0 = 1/. In the former case, r < 1 ensures, for all j ∈ [1, n], rSj (1 − ) + r − 1 ≤ r(1 − ) + r − 1 = r − 1 < 0

(3)

By assumption, at least one Mij will be positive, so that the sum of (3) across all counterparties j will be negative. Setting ai = 0 removes this negative summand but does not decrease Si ; therefore, it is always strictly preferable. In the latter case, rSj (1 − ) + r − 1 ≥ r − 1 > 1 − 1 = 0 For the restricted set of i, by assumption, setting ai = 1 does not affect Si . Following the same logic as above, by enabling a positive summand, aj = 1 is always strictly preferable.  Let Ru , E u denote the set of iteratively dominant action profiles and the set of equilibrium action profiles, respectively, in the game u. If u were supermodular, we could take for granted the result of Milgrom and Roberts (1990), which would provide upper and lower bounds for Ru that are themselves equilibria. This plays a crucial role in Frankel, Morris, and Pauzner’s (2003) uniqueness result for a wide class of global games. Here, we have an analogous result for the upper bound. It remains unclear whether the lower bound holds as well. While this would not materially affect the later analysis, we shall see that a lower bound implies general equilibrium uniqueness, not just uniqueness among threshold strategies.

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Lemma 2 (Maximal Equilibrium) The maximal action profile a ∈ Ru is an equilibrium in u; that is, a ∈ E u . Proof. This proof proceeds in three parts. First, we want to rule out unforced defaults under a. Assume the contrary; that is, ∃i ∈ [1, n] such that ai = 1, Si (0, a−i ) = 1, and u . Si (1, a−i ) = 0. Since Si is non-decreasing in a−i , this implies Si (1, a−i ) = 0 for all a−i ∈ R−i By construction of fi (·), bank i will strictly prefer ai = 0 when default is either guaranteed u or invoked by ai = 1. Accordingly, BRi (a−i ) = 0 for all a−i ∈ R−i . This contradicts our assumption that ai = 1 in an iteratively dominant action profile a. Second, we want to rule out unprofitable lending under a. By the previous part, ai = 1 =⇒ Si (a) = 1 u for all banks i ∈ [1, n]. Since a is iteratively dominant, there must exist some a−i ∈ R−i such that BRi (a−i ) = 1. Compare a−i to a−i . Since a−i ≥ a−i , Si (1, a−i ) ≥ Si (1, a−i ). For any opponents j such that 1 = aj > aj = 0, Sj (a) = 1 guarantees that loans individual counterparties are at least as profitable under a. The remaining opponents’ behaviors are unchanged. These points, along with a quick inspection of u(·), suggest that

BRi (a−i ) = 1 =⇒ BRi (a−i ) = 1 which validates bank i’s behavior, as we intended. Third, having established that ai = 1 =⇒ BRi (a−i ) = 1 it only remains to show that ai = 0 =⇒ BRi (a−i ) = 0 This follows immediately from the maximality of a. Since ai ≤ 0 for all ai ∈ Riu , there exists no action profile under which bank i prefers to lend. Naturally, that includes a. 

2.2

Under Incomplete Information

Consider an n-player normal form game. The parameter r becomes a random variable, uniformly distributed across an interval properly containing [1, 1/]. To each bank i corresponds

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a strategy si : R → {0, 1}, from their private signal xi of r’s realization to an action ai . Its utility is described by vi (xi , s) = E [u(s(x); θ) | xi ] That is, the game v is a modification of the complete information game u where agents have private beliefs about the parameter, facing uncertainty not only on the interest rate r but also on the beliefs of their opponents. Assumption 1 We know the following about the signal xi . 1. Let xi = r + ei , where the random error term ei is independent of r and ej for j 6= i. 2. Let ei have support within (−δ, δ) for some δ > 0. 3. Let ei be uniformly distributed across its support. All results in this section rely on a vanishingly small signal error; that is, we will consider behavior in the limit as δ → 0. Let Rv , E v denote the set of iteratively dominant action profiles and the set of threshold equilibrium strategy profiles, respectively, in the game v. We wish to characterize E v , but cannot apply existing results directly. Nonetheless, we follow the by-now standard approach of Frankel et al. (2003); section 3 of their paper provides a good intuition. The two proofs differ in the following ways. Because Frankel et al. deal with supermodular games, they may take lemmas 3 and 4 for granted as an implication of Milgrom and Roberts (1990). Here, this is less straightforward. As mentioned earlier, we cannot say that best responses are increasing, but will repeatedly abuse the fact that the non-monotonicity arises only when one party is acting against their own interests. Between here and proposition 1, which summarizes the argument of Frankel et al. (2003), we make two additional observations. First, the sort of masochistic behavior we refer to above is ruled out in equilibrium. Second, whether behavior is masochistic or not does not depend on the uncertain parameter.6 Crucially, this provides us with a subset of strategies, T (E v ), that includes all equilibria as well as translations of equilibria across the signal space. Over this subset, we find that the best response function is increasing in strategies. Translation will play an important role in proposition 1. Our results are constrained to a certain class of graphs, as alluded to earlier. Why the following assumption is useful will be made clear in the proof of lemma 3. 6

This would not be the case had we applied private signals to the cash reserve of banks, rather than to the interest rate. Varying the interest rate does not affect whether a bank defaults in this model; it only affects whether investment is profitable.

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Assumption 2 The adjacency matrix M represents a directed acyclic graph. Lemma 3 There exist maximal s ∈ Rv and minimal s ∈ Rv . Both are increasing threshold strategies. Proof. We prove this by induction on the iterated deletion of strictly dominated strategies. More precisely, consider the following inductive hypothesis. Let Rkv denote the set of strategy profiles that survive k rounds of deletion. There exist xi , xi for every bank i with non-trivial preferences such that  1 x > x i si = 0 ow is bank i’s minimal strategy in Rkv and  0 x < x i si = 1 ow is bank i’s maximal strategy in Rkv . We want to show that this holds for the (k + 1)st round of deletion as well. Base Case. Because M is an acyclic graph, we can impose a topological ordering . Let i be the first bank under . Accordingly, i has no incoming loans. If ai = 1 implies Si = 0 in the complete information game u, then ai = 0 is strictly dominant and bank i’s behavior is trivial. Suppose this is not the case. Let x∗ (x∗∗ ) be the minimal (maximal) signal that bank i can receive and still weakly prefer ai = 1 (ai = 0) for some action profile a−i . By lemma 1, x∗ ≥ 1 and x∗∗ ≤ 1/ (having established that bank i cannot default). As such, one of the two variables fails to exist only if bank i’s behavior is trivial; they either always prefer to lend or always prefer to collect. ∗ ∗∗ Consider strategy s∗i (s∗∗ i ) with threshold x (x ). All else equal, the best response is increasing in r, and, therefore, it is increasing in the signal xi of r. Accordingly, 1 ∈ BRiv (a−i ; x∗ ) =⇒ 1 ∈ BRi (a−i ; x) for x > x∗ . (More formally, by a−i in the game v we mean the strategy s−i that always specifies a−i .) Likewise, for the appropriate a−i , 0 ∈ BRiv (a−i ; x∗∗ ) =⇒ 0 ∈ BRi (a−i ; x) for x < x∗∗ . Having established that s−i := a−i induces the appropriate best response behavior in one portion of the interval, and that lemma 1 implies appropriate best response behavior in the other portion, we find that there exist strategies such that s∗i and s∗∗ i are best responses. Therefore, they cannot be strictly dominated. By construction, furthermore, they are minimal and maximal respectively.

15

We extend this result to the other banks by deleting i’s strictly dominated strategies, and moving through agents in the order . Once a given bank j is reached, its predecessors will only have strategies that invoke lending at xj > 1/. (Keep in mind that when the signal xj strictly exceeds some threshold, there exists sufficiently small δ to guarantee that xk exceeds that threshold as well for any bank k.) As such, the result of lemma 1 will apply and a bank j with non-trivial preferences will also find it strictly preferable to lend. We can then establish the existence of a minimal strategy, as above. Meanwhile, the maximal strategy follows from lemma 1 without additional modifications. Inductive case. As above, let x∗ (x∗∗ ) be the minimal (maximal) signal that bank i can receive and still weakly prefer ai = 1 (ai = 0) for some action profile a−i ∈ Rkv that remains undominated by round k. By the inductive hypothesis, x∗ ≥ x and x∗∗ ≤ x. To establish existence of the maximal strategy s∗i with threshold x∗ , we wish to invoke a similar argument as in the base case. There must be some strategy profile s0 ∈ Rkv that supports bank i’s lending at x∗ . We know by construction that bank i will not lend at signals less than x∗ , so we only need to support lending at signals greater than xi . However, since we restrict ourselves to the set Rkv rather than the entire strategy space, we cannot assume that i’s counterparties adopt xi -invariant behavior. Instead, we must support s∗i by using a strategy profile that we know remains in Rkv : one that specifies maximal strategies for each opponent. Take some counterparty j 6= i. If s0j (x∗ ) = 1, recall that bank j follows their maximal strategy profile, which is increasing. Since bank j’s behavior is constant over [x∗ , xi ] and the interest rate is increasing, bank i will prefer to lend to bank j over the specified interval, holding others’ actions fixed. Alternatively, if s0j (x∗ ) = 0, there are two cases to consider. 1. First, i is borrowing from (but not lending to) j. Then for any strategy sj in Rjv , bank i’s incentive to lend is increasing in xi , holding fixed s−ij . 2. Second, i is lending to j. Fixing Sj = 1 and for any strategy sj in Rjv , bank i’s incentive to lend is increasing in xi for a given s−ij . We ask: can Sj = 0 within the relevant interval, [xi , xi ]? Suppose for the sake of contradiction that bank j follows its maximal strategy sj and defaults with positive probability at some signal xi . Notably, the uncertainty around xi ensures that any coordination failure due to the signal error has measure zero, so we can restrict ourselves to actions in the complete information game. A cursory inspection of the utility function uj suggests that bank j will strictly prefer not to lend. Decreasing opponent’s actions will only strengthen this preference. As such, this violates the maximality of sj . 16

Therefore, the threshold strategy based on x∗ will survive the next round iterated deletion. To show the existence of a maximal threshold x∗∗ and minimal strategy s∗∗ , we follow an analogous argument. It is similar enough that we choose to omit it here. Whereas the prime difficulty of establishing a maximal strategy was the possibility for defaults due to over-expenditure, the prime difficulty of establishing a minimal strategy is the possibility of renewed solvency (and hence, greater incentive to lend) after reducing expenditures. We can apply the same argument used in case 2 - that the over-expenditure could not exist to begin with - to bypass this issue. Convergence. Consider e.g. the upper threshold, xk after iteration k. It is non-increasing, by construction. It is also bounded above, by the base case. By the monotone convergence theorem, as k → ∞, this threshold converges to some x∞ . Likewise, the lower threshold will converge to some x∞ . Since xk ≥ xk for all iterations k, x∞ ≥ x∞ holds in the limit. From these limit thresholds, we construct minimal and maximal strategies that survive iterated dominance.  Lemma 4 The maximal iteratively dominant strategy s ∈ Rv is also an equilibrium. Proof. Note that BR(s) ≤ s; otherwise s would not be the maximal iteratively dominant strategy. Consider any bank i with signal xi . As in previous proofs, we can ignore the finite number of cases were x−i coincide with the threshold of some other bank under a given strategy. Under the specified signal error, such cases will arise with probability zero, and therefore not affect payoffs. Accordingly, we can deal with the complete information environment conditional on some interest rate r = xi . Lemma 2 suggests that, in the absence of uncertainty, the maximal iteratively dominant strategy is an equilibrium in the modified game where each bank j’s actions are restricted to be less than or equal to s. (This can be achieved by simply deleting edges on the graph M .) As such, BR(s) ≥ s. Given our first observation in this proof, we conclude that, therefore, s is an equilibrium.  Lemma 5 For any s, s0 ∈ T (E v ), if s ≥ s0 then BR(s) ≥ BR(s0 ). Proof. Let s, s0 be translated threshold equilibria. Suppose s0 > s. At any x that’s not one of (at most n) thresholds, actions s(x) and s0 (x) are adopted with certainty. As such, can consider behavior in the corresponding complete information game. Since s(x) and s0 (x) are translated equilibria, both satisfy the property that no bank is simultaneously defaulting 17

and lending. This property does not depend on the interest rate r, as an inspection of u will reveal. As such, the increase in actions at s0 (x) consists of weakly increased funding to all banks, while preserving their solvency. (Because all banks receive weakly more funding, only strictly more expenditure can impose a default in s that did not exist in s0 . However, we argued that the latter case is impossible in translated equilibrium.) Accordingly, every bank i has weakly stronger incentive to lend in s0 compared to s. The case where x is at some threshold is a zero-probability event given the uncertainty around signals xi . As such, it is payoff-irrelevant. Since the best response is increasing here over the complete information cases, it must be increasing in general.  Proposition 1 The set of threshold equilibria E v is a singleton. Proof. Reinterpret s as the minimal threshold equilibrium, whatever that may be. We want to show that s = s. Define sˆ = min{s ∈ T (s) | s ≥ s}. Define τ ∗ ∈ Rn such that sˆ(x) = s(x + τ ∗ ). Since we established upper and lower bounds for these strategies in the base case 3, such a τ ∗ must exist. Furthermore, note that s ≤ s but sˆ ≥ s. There must exist some signal x∗i for each bank i such that sˆi (x∗i ) = si (x∗i ). Otherwise, we could shift τi∗ further to the right. Consider that: 1. By lemma 4 and the definition of sˆ, {s, sˆ} ⊆ T (E). Therefore, since sˆ ≥ s, lemma 5 implies that BRi (ˆ s(x∗i )) ≥ BRi (s(x∗i )). 2. By construction, s ≥ s. By lemma 4, BRi (s(x∗i )) = s(x∗i ) and BRi (s(x∗i )) = s(x∗i ). Therefore, BR(s(x∗i )) ≥ BR(s(x∗i )). s(x∗i )). 3. By lemma 5 and recalling that τ ∗ > 0, BR(s(x∗i )) ≥ BR(ˆ Transitivity requires that this chain of weak inequalities be a chain of equalities. However, in step 2, that could not be the case if s > s. Therefore, s = s.  At this point, we could demonstrate that the uniqueness property does not depend on the particular distribution in assumption 1. This is entirely irrelevant for our later analysis, however, and a reading of Frankel et al. (2003) would suggest how to go about it if (a) this property were desired to ensure the robustness of the result or (b) a different signal error distribution were used, for some reason. It is also worth noting that while we recreate “limit uniqueness” in this highly asymmetric non-supermodular game, it is extremely unlikely that “belief independence” holds. That is, the equilibrium may vary in the particular distribution of the signal error. We do not view this as a deficiency: it is just one another model parameter that we should attempt to specify in a realistic manner. 18

For games where the set of increasing strategies is closed under best response, Athey (2001) shows that equilibria in a modified game with discrete type spaces will converge to equilibria in the original game with a continuous type space, as the type spaces themselves converge. Strictly speaking, such a result is necessary to justify the method of computing this equilibrium that is adopted in section 3. However, this model fails Athey’s condition, and we have not yet attempted to evaluate that result in this context. Finally, observe that if an analog of lemma 2 for minimal strategies held, then lemma 4 could establish that the minimal iteratively dominant strategy was also an equilibrium. Accordingly, lemma 1 would suggest the existence of a unique equilibrium in general (indeed, a unique iteratively dominant strategy), rather than just a unique threshold equilibrium.

19

3

Simulation Methodology

Having described the model and established the uniqueness of our solution concept, the next step is to study its features and evaluate counterfactuals. This is done in section 4. This presents a problem. The solutions to global games (and games with non-trivial uncertainty in general) tend to be difficult to express analytically, and the game in section 2.2 is no exception. Basic comparative static results exist for supermodular games (e.g. Milgrom and Roberts, 1990) that could conceivably be extended to our case, but they only express direction of change and not magnitude. This is not sufficient for our purposes. More broadly, it is difficult to say much about games on arbitrary networks, since individual networks tend to be too specific to be interesting, well-behaved subspaces of networks are often difficult to delineate, and the full space of possible networks is too diverse for many meaningful results too hold. Some authors limit their analysis to special cases, typically the two extremes of ring and complete networks (e.g. Allen and Gale, 2000; Acemoglu, Ozdaglar, Tahbaz-Salehi, 2015b). In their study of balance sheet contagion, Acemoglu et al. (2015a) employ a clever approach that considers the space of convex combinations of the aforementioned extremal cases. While these exercises offer a valuable starting point, they fall short of capturing the structure and diversity of real financial networks.7 Between the issues addressed above with theoretical approaches and difficulties with studying market collapses empirically (typically, the government will intervene before said collapse), simulation-based studies have proliferated. This will be our approach as well. Upper (2011) provides an excellent review of such work as applied to balance sheet contagion.8 Of the more recent papers, Elliot et al. (2014) is particularly similar to this one, in that it simulates random networks to assess comparative statics in their model. We build an algorithm to solve this model for arbitrarily given networks and rely on a medium-scale simulation to drive our results. In implementing the simulation, we prescribe a network generating process (NGP) in section 3.1 and a balance sheet generating process (BGP) in section 3.2. The quality of the simulation depends on the realism of these processes; while loan-level or balance sheet-level data on banks is generally inaccessible for 7

Amini, Cont, and Minca (2013) take a different approach and prove asymptotic results as n → ∞ for a randomly generated network with n nodes, while preserving the local network structure. This is certainly another viable approach to comparative statics. 8 Interestingly, he argues that most existing simulations face two shortcomings. First, they apply shocks to banks individually, rather than considering systemic shocks. Second, the models lack behavioral foundations. In particular, he contends that explicit modeling of strategic contagion channels would makes these models more applicable towards policy evaluation. We ignore balance sheet contagion, unfortunately, but it is worth noting that both issues he raises are addressed in this paper. The same can be said, of course, for Gai et al. (2011), which also relies on simulations. As mentioned in section 1.1, we find our behavioral foundations to be more convincing.

20

non-policymakers, we do manage to calibrate the NGP to the U.S. federal funds market by using the empirical observations of Bech and Atalay (2010). A set of 75 banks will used to simulate the standard model (sections 4.1 and 4.2). A smaller set of 36 banks will be used to simulate the model after most of the interventions (sections 4.3 and 4.4), to compensate for their longer runtime9 . However, when the NGP is calibrated, only half to a quarter of these banks will actually participate in the market. To compute the model solution, we rely on an analog to best response iteration. That is, for bank i and actions a−i , we find the ex ante utility-maximizing threshold strategy. The space of possible signals (and strategies) is discretized at 0.05 intervals from r = 1 to r = 2. Because the perceived probability of a policymaker intervention is relatively high ( = 0.5), lending is guaranteed to be profitable at r = 2 which permits a fairly restricted signal space. Fixed points of this process are interpreted as fixed points of the true best response function. A maximum of ten iterations are allowed, and a solution is returned once either a fixed point is reached or the last iteration attained. The model possesses several free parameters, which we assign as follows: fi (a) = Bi −

ai 10

r = 1.15 δ = 0.1  = 0.5 ω = 3r where ω is the size of the limited guarantee fund defined in footnote 14. Some of these values (e.g. ) are chosen based on computational considerations. Others (e.g. fi ) are chosen to avoid adding noise to our measurements of welfare. Most often, however, they are chosen to ensure a mix of well- and poorly-functioning markets.  The algorithms underlying the simulation have runtime O max∀i 2ψ(i) where ψ(i) denotes the number of banks whose actions are payoff-relevant to bank i. This gives reasonable performance in sparse networks, as is the case for both real interbank lending markets and the undermentioned NGP. However, we cannot solve true-scale models (with several hundreds of banks) fast enough to permit a sufficiently large number of trials. The problem worsens under two interventions (detailed in section 4). When a limited guarantee is added to the model, ψ(i) must account for all lending banks (whose behavior may affect how much funding remains available for i). Worse, when bank loans are securitized in the spreading intervention, ψ(i) must account for all other banks. 9

21

3.1

Network Generation and Calibration

Network theoretic characterizations of interbank lending markets has been a highly active area of research. Bech and Atalay (2010) find, consistent with Rørdam and Bech’s (2009) study of the Danish interbank lending market, that the federal funds market consists of smaller, peripheral banks providing liquidity to a core of large banks, which then rerout some of that funding to another set of peripheral banks. A core-periphery structure is also observed by Wetherilt et al. (2010), Boss et al. (2006), Iori et al. (2008), Veld and van Lelyveld (2014), and Fernandes and Borges (2013) in the British, Austrian, Italian, Dutch, and Portuguese markets respectively. We will rely on Bech and Atalay (2010) and calibrate the network towards the federal funds market as it existed in 2006. This choice rests on the relative size and importance of the U.S. market, along with the quality of data provided by that paper. While Soram¨aki et al. (2007) also study this market, their analysis is less recent, tied to the first quarter of 2004, and they look at payments across the entire Fedwire Funds Service, whereas Bech and Atalay restrict themselves to federal funds loans. The federal funds market consists mostly of a large connected graph G = (V, E); we ignore the disconnected components since, on average, they comprise only 2% of participating banks. The four components are defined in relation to each other, following the graph theoretic characterization of Dorogovtsev, Mendes, and Samukhin (2001). Soram¨aki et al. (2007) and Roukney et al. (2014) use the same characterization, albeit with different observations. • The giant strongly connected component (GSCC) is the largest strongly connected subset of nodes • The giant in-component (GIN ) consists of nodes u ∈ V such that (u, v) ∈ E for some v ∈ GSCC. • The giant out-component (GOU T ) consists of nodes u ∈ V such that (v, u) ∈ E for some v ∈ GSCC. • The tendrils consist of all remaining nodes, with edges connecting with GIN and GOU T but not GSCC. These four components suggest the use of a stochastic block model to generate our random networks. The stochastic block model is similar to an Erd˝os-R´enyi model, where edges between nodes are formed independently with some fixed probability p. In the stochastic block model, nodes are grouped into categories (i.e. the components described above) and edges are formed independently with probability pab , varying only in the block a of the initial 22

node and the block b of the terminal node. Here, we also require that nodes not have edges to themselves. We describe our stochastic block model by a 4 × 4 matrix P , where each of the entries correspond to the aforementioned probabilities. Pertaining to these blocks, Bech and Atalay (2010) provide useful descriptive statistics that we will make use of in calibrating the NGP. We review them in table 1. Table 1: Descriptive statistics of the U.S. federal funds market.

Number of Banks Proportion of Banks Average Assets Average In-Degree Average Out-Degree Average In-Loan Average Out-Loan

GSCC

GIN

GOUT

Tendrils

57 10% ± 1% $359 billion 19.1 9.3 $310.7 million $552.6 million

303 58% ± 5% $10 billion 0.2 3.8 $810 million $92.4 million

67 17% ± 4% $39.6 billion 5.9 0.2 $110.5 million $26 million

50 14% ± 3% $8.7 billion 2.0 0.7 $82.5 million $35.7 million

Using this, we back out the appropriate parameters from Bech and Atalay (2010). By design, all edges carry weight one (all loans are of a fixed value). As such, average in-degrees and out-degrees will be calibrated without taking into account the variation in loan sizes.10 Specifically, we want to match the following characteristics: AverageOutDegreea =

X

nb Pab

∀b

AverageInDegreea =

X

nb Pba

∀b

where na is the number of banks in component a. These conditions are linearly dependent, so we cannot solve for the values of P directly. Instead, we turn to error-minimization. Let ϕ(P ) denote the average deviation of the P -estimated average node degrees from the empirically-observed degrees in table 1. Under the restriction that the P -estimated degrees be less than or equal to the empirically-observed ones, minimizing the error is equivalent to maximizing the P -estimated degrees. Since these degrees are a linear function of P , our 10

However, the data available in table 1 and the results presented in section 2.2 would certainly be sufficient for a more realistic model that involves variable loan sizes.

23

calibration takes the form of the following linear programming problem. arg min ϕ(P ) subject to . . . ∀P

19.1 ≥ 56P11 + 303P21 + 67P31 + 50P41 0.2 ≥ 57P12 + 302P22 + 67P32 + 50P42 5.9 ≥ 57P13 + 303P23 + 66P33 + 50P43

(In-Degree Constraint)

2.0 ≥ 57P14 + 303P24 + 67P34 + 49P44

9.3 ≥ 56P11 + 303P12 + 67P13 + 50P14 3.8 ≥ 57P21 + 302P22 + 67P23 + 50P24 0.2 ≥ 57P31 + 303P32 + 66P33 + 50P34

(Out-Degree Constraint)

0.7 ≥ 57P41 + 303P42 + 67P43 + 49P44

Pij ≥ 0, ∀i, j

(Well-Definedness)

Pij ≤ 1, ∀i, j We compute the following values for the optimal P :    P =   ∗

 0.0336 0.001 0.104 0  0.056 0 0 0.007   0.004 0 0 0   0 0.002 0 0

This calibration is nearly perfect. Indeed, the only value that is not exactly matched is the average out-degree for banks in GIN ; the P ∗ -estimated value is 3.52 whereas the empirical value is 3.8. Furthermore, the structure of the components is largely preserved. The definitions of components would suggest that • P12 = 0, since if i ∈ GSCC lend to j ∈ GIN then j would be strongly connected to i and therefore j ∈ GSCC. • P31 = 0, since if i ∈ GSCC borrowed from j ∈ GOU T then j would be strongly connected to i and therefore j ∈ GSCC. • P41 = 0, since if i ∈ GSCC borrowed from j ∈ T endrils then j would be connected to 24

GSCC and therefore j ∈ GIN . • P14 = 0, since if i ∈ GSCC lent to j ∈ T endrils then j would be connected to GSCC and therefore j ∈ GOU T . • P23 = 0, since if i ∈ GIN lent to j ∈ GOU T then a path would exist from i to GSCC to j and back to i, implying that i, j ∈ GSCC. ∗ ∗ ∗ ∗ ∗ Indeed, P41 , P14 , P23 = 0 as expected. While P12 P31 are nonzero, they are nonetheless quite small relative to the other probabilities. Finally, there is one conditions on the tendrils: that i ∈ T endrils not lend to j ∈ GIN and borrow from k ∈ GOU T simultaneously. Generically, this cannot be satisfied, because the stochastic block model assumes loans to different counterparties are statistically independent after conditioning on the block. However, since ∗ = 0, i will never borrow from k and the condition is vacuously satisfied. P34 There is another concern. Recall that assumption 2 restricts our uniqueness result to acyclic graphs. The NGP does not preclude cycles. To correct this in a relatively uninvasive way, we find the minimal-value set of loans that, once removed, will clear the network of cycles. This is also known as the minimum feedback arc set problem, and it is NP-complete (roughly speaking, the best-known algorithm features exponential runtime). Fortunately, the properly-calibrated NGP rarely produces cycles on a set of 75 banks; when it does, only two or three banks are involved and the minimum feedback arc set is easy enough to compute. For illustration, a hundred trials of the NGP required an average of 0.14 removed loans per network. Moreover, as figure 1 suggests, directed acyclic graphs exhibit precisely those intricate relationships we have come to expect from our financial system.

25

Figure 1: A sample of initial networks generated for the simulation.11 Finally, Bech and Atalay (2010) observe a number of notable features in the federal funds network. For example, they find that the network is dissortative. As intuition would suggest, banks with a high out-degree (net lenders) are likely to connect with banks that have a high in-degree (net borrowers), and vice-versa. The same applies in the calibrated simulation. They also find that the distributions of in-degrees and out-degrees in the market as a whole are explained much better by negative binomial and power-law distributions, respectively, than the Poisson distribution, which would be asymptotically implied by an Erd˝os-R´enyi process. In our stochastic block model, degrees follow a Poisson binomial distribution; accordingly, the average degrees approach a Poisson distribution under large samples (von Mises, 1921). Here, our NGP fails to coincide with the real federal funds market. 11

Red nodes are net lenders, blue nodes are net borrowers, and black nodes have no net position.

26

3.2

Balance Sheet Generation

We generate the reserve with two random variables, first to generate a reserve under “normal” conditions, and then to apply a negative shock. Let Bi0 denote the pre-shock reserve of bank i, and let Bi1 = Bi0 − Ei denote the post-shock reserve. Lacking empirical guidance, we specify what the generous reader would deem a reasonable distribution. Suppose both Bi0 and Ei are normally distributed conditional on their means, where the mean of Bi is fixed and that of Ei follows a Poisson distribution. Both distributions are independent across banks and identical within each block. As table 1 suggests, the composition of blocks varies in more ways than their position in the federal funds market. This would plausibly reflect itself in different reserve and shock distributions. Recall that the reserve Bit denotes the value of liquid assets of bank i, including both the net position within the interbank lending market as well as any additional reserves. This paper takes the position that banks will work to keep their reserve relatively stable in nonshock conditions, perhaps reflecting regulatory liquidity requirements. We use the following heuristic to prescribe our reserves. Let σi be the standard deviation of Bi0 . 1. The expected reserve E [Bi0 ] = µi should be two standard deviations (2σi ) away from zero. That is, in a well-functioning market where all loans are rolled-over, there is only a 2.5% chance that the bank faces liquidity concerns. 2. Liquidity concerns are not the same as solvency concerns; if said bank has outstanding loans, it can collect them to bolster its financial position. Each bank should expect to have another σi in loans to collect, yielding an (expected) 0.015% chance of default under ideal conditions. 3. Even in the event of a shock, if the market nevertheless presented ideal conditions, then after collecting loans the bank reserve should be two standard deviations away from zero. That is, there is only a 2.5% chance that the bank faces default if the market does not panic. Of course, the actual chance of default may be significantly higher. These two conditions let us specify the initial reserve distribution, namely Bi0 ∼ N (2 · AverageOutDegreeai , AverageOutDegree2ai ) The conditions also allow us to specify shock distribution as Ei ∼ N σi Xai , σi2 Xa2i

27



where X ∼ P oisson(1). To summarize: the shock is normally distributed within each block, but varies in intensity across blocks. Any block may receive a higher or lower intensity shock, and this will vary randomly in the simulation sample. Stronger shocks also exhibit higher volatility in their magnitude. Figure 2 describes the realized distribution of Bi0 and Ei0 , respectively summed and averaged among all banks.

Figure 2: The estimated distribution of pre-shock welfare (left) and the estimated distribution of shock intensity (right).

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4

Simulation Results

The following subsections present the results of the simulations. Section 4.1 compares the outcome under confidence-driven contagion to that under the optimal response. Section 4.2 evaluates several common network measures for their predictive power in this model. Sections 4.3 and 4.4 evaluate various policymaker interventions. While visual aids are provided, most of the analysis comes in the form of linear regressions, either comparing relationships between several variables or establishing that two outcomes are statistically different.

4.1

Quantifying Confidence-Driven Contagion

In the figures below, the blue line follows the maximal solution, where every agent rolls over their loan without regard for counterparty risk or even their own solvency. This corresponds to a complete absence of contagion. The green line follows the optimal rationalizable solution when agents can coordinate perfectly (that is, when δ = 0). This corresponds to the cooperating equilibrium of Erol and Vohra (2014), which is implicitly assumed in several other papers (e.g. Gai et al., 2011). The contagion here is minimal, acting only in response to an existing deterioration of market quality. Finally, the red line follows the unique rationalizable solution to the original model. The contagion here is amplified beyond what is warranted by the shock itself.

Figure 3: The estimated distribution of post-response welfare (left) and the relationship between shock intensity and post-response welfare (right). The differences appear small because the loss associated with a default is far larger than the losses due to missed lending opportunities. In tables 2 and 7, we see that these 29

differences are reliable and statistically significant. Indeed, the market response deviates from the optimal one by nearly the full value of one interbank loan. Table 2: The welfare difference between optimal and market responses.

Dep. Variable: Model: Method: Date: Time: No. Observations: Df Residuals: Df Model:

Change in Post-Response Sum of Utility [Market to Optimal] OLS Least Squares Sun, 01 May 2016 13:10:12 100 99 0 coef

const 0.8835

std err

t

0.093

9.463

Omnibus: 14.789 Prob(Omnibus): 0.001 Skew: 0.954 Kurtosis: 3.544

R-squared:

-0.000

Adj. R-squared: -0.000 F-statistic: -inf Prob (F-statistic): nan Log-Likelihood: -134.52 AIC: 271.0 BIC: 273.7

P>|t| [95.0% Conf. Int.] 0.000

0.698 1.069

Durbin-Watson: Jarque-Bera (JB): Prob(JB): Cond. No.

1.700 16.407 0.000274 1.00

In equilibrium, very few firms will default: indeed, only a subset of those firms that were insolvent to begin with. Firms will not accidentally default, except with a vanishingly small probability, because they understand how fundamentals affect market behavior. What results is a situation where the market collapses, failing to take advantage of profitable opportunities, without necessarily facing widespread defaults. Panic drives a coordination failure that, in the end, does not even warrant the panic. Nonetheless, while confidence-driven contagion has a noticeable welfare impact in this model, it is mild relative to the magnitude of the original shock. This is in agreement with a theme that has emerged in the literature on balance sheet contagion, arguing that although contagion exists, its effects are limited (Upper, 2011; Glasserman and Young, 2014; Fernandes and Borges, 2013).

30

4.2

Identifying Risk Factors

As complex networks have found applications in a wide range of fields, including biology, sociology, and statistical physics, an abundance of measures have developed to characterize these networks. Much of the literature on balance sheet contagion, both empirical and theoretical, has been devoted to understanding how different notions of interconnectedness relate to market outcomes in the presence of contagion. For example, Brunetti et al. (2015) find that interconnectivity in interbank lending markets decreases during a crisis. They write, “physical interbank trading networks serve to identify weakening interconnectedness in the interbank system that may lead to liquidity problems.” Conceptually, interconnectedness has at least two (counteracting) effects on market stability. By increasing connections between nodes, a node experiencing an isolated shock has the opportunity to prematurely collect their outstanding loans, providing liquidity relief that won’t necessarily trigger the default of their borrowing counterparties. On the other hand, deeper or more systemic shocks could trigger a loss of confidence on the part of their lending counterparties, possibly resulting in a liquidity shortage. As such, interconnectedness makes a market more flexible, but the actual flex may be positive or negative. This might provide a functional explanation (in the sociological sense) for the existence of complex interbank market obligations, in that agents acting both as lenders and borrowers increases the flexibility of the market and thereby may alleviate liquidity problems in isolation. This would be consistent with the arguments of Allen and Gale (2000) and Acemoglu et al. (2015a) that an interconnected network will perform better under a weak shock while a sparse network will perform better under a strong shock. In particular, Acemoglu et al. (2015a) refer to this as a phase transition. They also write, rather astutely, that “More broadly, our results highlight the possibility that the same features that make a financial network structure more stable under certain conditions may function as significant sources of systemic risk and instability under other conditions.” Here, we consider four basic measures of a network. First, Bech and Atalay’s (2010) degree of completeness α, defined as Pn Pn α=

Mij0 n(n − 1)

i=1

j=1

where M 0 is the obligation network restricted to banks that participate in the market. This measures the proportion of potential nodes that actually exist. Second, the average clustering coefficient, which measures how closely nodes in the network are connected to one another. Third, the number of separate, disconnected components in the network. Third, the total

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number of loans. Histograms for each measure are presented in figure 4.

Figure 4: Distributions of network measures across the simulated networks. Table 3 presents the results of a reduced-form regression of these measures against the change from post-shock to post-response welfare. This is intended to mimic the experience of an empirical researcher, in the context of this model.

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Table 3: Regressing post-response welfare against network measures.

Dep. Variable: Model: Method: Date: Time: No. Observations: Df Residuals: Df Model:

Post-Response Sum of Utility Minus Post-Shock Sum of Balance OLS Least Squares Sun, 01 May 2016 13:10:13 100 95 4 coef

const 1.0797 Degree of Completeness -16.5947 Avg. Clustering Coefficient -11.3056 Number of Components 0.0077 Number of Loans 0.0089 Omnibus: Prob(Omnibus): Skew: Kurtosis:

5.459 0.065 0.122 2.226

std err

t

2.712 58.806 12.406 0.137 0.030

0.398 -0.282 -0.911 0.056 0.302

R-squared:

0.013

Adj. R-squared: -0.028 F-statistic: 0.3197 Prob (F-statistic): 0.864 Log-Likelihood: -170.27 AIC: 350.5 BIC: 363.6 P>|t| [95.0% Conf. Int.] 0.691 0.778 0.364 0.955 0.763

Durbin-Watson: Jarque-Bera (JB): Prob(JB): Cond. No.

-4.304 -133.340 -35.934 -0.265 -0.050

6.464 100.151 13.323 0.280 0.068

2.002 2.744 0.254 1.81e+04

Collectively, these network measures fail to explain even minor changes in welfare. While no explicit test is presented, variations in shock size and network generation undertaken during the development of this simulation suggest that this is not the result of non-monotonicity in the relationship, but rather a lack of predictive power for confidence-driven contagion. This is not terribly surprising, as individual banks’ incentives do not vary monotonically in the number of their counterparties. Measures without behavioral foundations, as applied by much of the empirical literature on interbank lending markets, may not necessarily be fruitful in predicting strategic behavior. This striking failure also suggests that, when it comes to market panic, the local structure of a network is far more important than its global structure. Policymakers may be better served by adopting a weakest-link type analysis of systemic risk.

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Of course, this evaluation only considers network variation within a fixed NGP. Should we vary the generation process itself, different results may hold. However, most empirical work that does investigate network measures seems to rely on longitudinal rather than crosssectional data. Barring structural change, that would be more comparable to a fixed NGP. And should we vary the parameters or undertake cross-sectional comparisons, it may just as well be that the network measures help us distinguish between the processes that generate the network without necessarily shedding light on their fundamental relationship to contagion.

4.3

Evaluating Pre-Shock Interventions

In the figure below, the red line follows a splitting intervention.12 Here, the very largest banks are split into two. These banks are interpreted as “systemically important”, so that we can consider how breaking them up would affect the efficacy of interbank lending markets. The orange line follows a spreading intervention.13 Here, the loan portfolio of each bank is diversified: every individual loan is transformed into a security that tracks the average repaid value of all outstanding loans. Each bank becomes dependent on a much larger set of counterparties, but is less exposed to any individual counterparty. 12

Banks are ranked by the sum of their in-degree and out-degree. Any banks whose value is more than two standard deviations above the mean are split in half, with their loans split in half as well as their reserves. For illustration, roughly 6% of banks in 75-bank networks are split up in this way. Of course, that would translate to a much larger proportion of banks that actually participate. One unfortunate oversight is that the shock is not applied separately to each of the split banks, although it is not clear how correcting this would affect the results. 13 Instead of bilateral loans, each lending counterparty is connected to each borrowing counterparty in proportion to the borrowing counterparty’s proportion of total borrowing. The total value of the lending counterparty’s claim corresponds to their proportion of total lending. In effect, we securitize interbank loans and distribute those securities. Exposure changes, but under perfect solvency, revenue remains the same.

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Figure 5: The estimated distributions of post-response welfare, following various pre-shock interventions. Reforms that incorporate some component of limiting the size of banks have been prominent in the public discourse since the 2007-08 financial crisis. This can be achieved through a number of measures, both direct and indirect (e.g. taxation or burdensome prudential standards). The Dodd Frank Act of 2010, for example, achieves this by preventing financial institutions from merging or acquiring one another if the resultant firm would be too large. The Riegle-Neal Act of 1994 imposed a similar measure to prevent any particular bank from holding too large a share of nation-wide deposits (FSOC, 2011). A brief discussion is provided in Labonte (2015). We take the simplest case - a direct intervention to break up systemically-important banks - in order to highlight the theoretical impact (through one particular channel) of such policies on market stability. As table 4 shows, the intervention yields no positive benefit. Indeed, it has a very mild but still statistically significant negative effect on welfare. Intuitively, the more banks that are involved in a highly connected market, the more parties need to coordinate in order to avoid a crisis.

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Table 4: The welfare difference between post-splitting and market responses.

Dep. Variable: Model: Method: Date: Time: No. Observations: Df Residuals: Df Model:

Change in Post-Response Sum of Utility [Splitting Intervention] OLS Least Squares Wed, 04 May 2016 21:22:45 100 99 0 coef

const -0.0710

std err

t

0.032

-2.242

Omnibus: 55.680 Prob(Omnibus): 0.000 Skew: -1.560 Kurtosis: 12.427

R-squared:

-0.000

Adj. R-squared: -0.000 F-statistic: -inf Prob (F-statistic): nan Log-Likelihood: -26.412 AIC: 54.82 BIC: 57.43

P>|t| [95.0% Conf. Int.] 0.027

-0.134 -0.008

Durbin-Watson: Jarque-Bera (JB): Prob(JB): Cond. No.

2.181 410.846 6.11e-90 1.00

The spreading intervention can be interpreted as the maximally diversified asset-invariant version of an interbank lending market. The opposite extreme - a market without any diversification - is a market where every organization is in the hands of another; in the absence of cycles, theory would suggest that behavior is efficient (although not necessarily better than before.) On the other hand, by reducing exposure to particular banks and spreading those losses that are inevitable, one might expect a moderated market reaction. In the context of balance sheet contagion, Elliott et al. (2014) also study diversification. They find a non-monotonic relationship with stability: starting at the lower extreme, increasing diversification will first tend to enhance contagion, but after a certain point, more diversification will reduce the risk. Acemoglu et al. (2015a) find a simpler relationship. They interpret complete networks as more diversified than ring networks, and show that convex combinations of the two become more stable as additional weight is placed on the latter. In this paper’s strategic setting, we find the opposite result. Table 5 suggests that the spreading intervention is even more harmful than the splitting intervention, resulting in welfare loss equivalent to roughly one third of an interbank loan.

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Table 5: The welfare difference between post-spreading and market responses.

Dep. Variable: Model: Method: Date: Time: No. Observations: Df Residuals: Df Model:

Change in Post-Response Sum of Utility [Spreading Intervention] OLS Least Squares Wed, 04 May 2016 01:44:14 100 99 0 coef

const -0.3269

std err

t

0.105

-3.101

Omnibus: 0.221 Prob(Omnibus): 0.895 Skew: -0.009 Kurtosis: 2.689

4.4

R-squared:

0.000

Adj. R-squared: 0.000 F-statistic: inf Prob (F-statistic): nan Log-Likelihood: -146.67 AIC: 295.3 BIC: 297.9

P>|t| [95.0% Conf. Int.] 0.003

-0.536 -0.118

Durbin-Watson: Jarque-Bera (JB): Prob(JB): Cond. No.

2.178 0.404 0.817 1.00

Evaluating Post-Shock Interventions

In the figure below, the red line follows an intervention where all repayments are delayed; that is, banks are forced to follow the maximal solution. The orange line follows an intervention where the market is frozen; that is, no loans are rolled-over and the banks are forced to follow the minimal solution. Finally, the yellow line follows an intervention where a central bank provides a limited guarantee of interbank loans.14 The central bank can dis14

Let ω denote the proportion of a defaulted-on loan that the guarantee fund is able to pay. More precisely,     Ω P  ω(a; θ) = min 1, P n n  M θ (1 − S )  a i=1

i

j=1

ij

j

With such a guarantee in place, the utility function becomes ui (a; θ) = ai

n X

Mij (max{θSj , θω} − 1) + αSi

j=1

Note that we have not yet extended the formal analysis of section 2 to this modified utility function.

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perse a limited amount of funds to compensate lenders for loans to defaulting borrowers. If the value of claims exceed the limit, then funds are dispersed proportionally. Implicitly, I assume that all interventions are unanticipated.

Figure 6: The estimated distributions of post-response welfare, following various post-shock interventions. The analog to the limited guarantee is, of course, deposit insurance, and its ability to prevent bank runs was studied famously by Diamond and Dybvig (1983). A more recent paper (Manz, 2009) has a valuable take on these results, which also apply here. The major theoretical case for deposit protection relies on models in the style of Diamond and Dybvig (1983), where deposit insurance enables depositors to reap the full benefits of banking at no cost, by eliminating runs as the only source of failure. However, there is now widespread agreement that most bank failures are driven by weaknesses in economic balance sheets rather than by selffulfilling panics. Thus the adoption of deposit insurance may imply heavy losses and involves a tradeoff: Preventing runs is a welcome result if a bank is solvent, but it is less reasonable if depositors have good reason to run and to enforce closure of an insolvent institution. In a nutshell, deposit protection inhibits both inefficient and efficient bank runs and may encourage banks to engage in imprudent practices. Exploration of this topic within the contagion literature is more limited, precisely because 38

the question is only interesting for its effects on behavior.15 Limited guarantees have received even less attention, even as they appear to be more realistic. In the event of a true systemic banking failure, the guarantee fund would be hard-pressed to cover all loans, and the public loath to finance them. Table 6 presents the results of this intervention. Table 6: The welfare difference between post-guarantee and market responses.

Dep. Variable: Model: Method: Date: Time: No. Observations: Df Residuals: Df Model:

Change in Post-Response Sum of Utility [Guarantee Intervention] OLS Least Squares Sun, 01 May 2016 12:11:29 100 99 0 coef

const -1.4680

std err

t

0.154

-9.547

Omnibus: 21.543 Prob(Omnibus): 0.000 Skew: 0.090 Kurtosis: 1.901

R-squared:

-0.000

Adj. R-squared: -0.000 F-statistic: -inf Prob (F-statistic): nan Log-Likelihood: -184.42 AIC: 370.8 BIC: 373.4

P>|t| [95.0% Conf. Int.] 0.000

-1.773 -1.163

Durbin-Watson: Jarque-Bera (JB): Prob(JB): Cond. No.

2.100 5.171 0.0754 1.00

In this model, a guarantee encourages lending to insolvent counterparties (moral hazard), without necessarily saving them. An average of 1.82 is withdrawn from the fund, suggesting that the welfare benefit due to behavioral change is 0.35. Here, the negative effects of moral hazard (lending to clearly insolvent banks) outweigh the positive effects of avoiding panic. It is worth noting that the simulation clearly exhibits both effects, and at a statistically significant level. 15

Heider et al. (2009) also look at interbank loan guarantees observing that such a policy was put in place by several countries, including by the Banca d’Italia that guarantees timely repayment of loans in the Mercato Interbancario Collateralizzato (a securitized interbank lending market where counterparty risk arose due to uncertain collateral values.). In the context of their model, full guarantees are always more cost-effective than partial guarantees. However, they do not explore whether the guarantee is desirable in itself. Zawadowski (2013) considers the related question of whether it is worthwhile to tax over-the-counter transactions in order to set up a bailout fund.

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Tables 7 and 8 evaluate the two more straightforward interventions. Both simulate a forcibly suspended market, but deal with outstanding transactions in different ways. Table 7: The welfare difference between maximal and market responses.

Dep. Variable: Model: Method: Date: Time: No. Observations: Df Residuals: Df Model:

Change in Post-Response Sum of Utility [Delaying Intervention] OLS Least Squares Sun, 01 May 2016 12:11:29 100 99 0 coef

const -0.6595

std err

t

0.097

-6.802

Omnibus: 2.103 Prob(Omnibus): 0.349 Skew: -0.195 Kurtosis: 2.473

R-squared:

-0.000

Adj. R-squared: -0.000 F-statistic: -inf Prob (F-statistic): nan Log-Likelihood: -138.30 AIC: 278.6 BIC: 281.2

P>|t| [95.0% Conf. Int.] 0.000

-0.852 -0.467

Durbin-Watson: Jarque-Bera (JB): Prob(JB): Cond. No.

1.913 1.788 0.409 1.00

The results indicate that, even in a time of market panic, preventing withdrawals may do more harm than good if it forces banks to maintain investments in clearly insolvent counterparties. Surprisingly, however, preventing renewed loans from being originated has a negligible effect on the market, even if it appears to rid the market of much-needed liquidity. In part, this may reflect excessive liquidity hoarding in the market response, at least for the specified interest rate.

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Table 8: The welfare difference between minimal and market responses.

Dep. Variable: Model: Method: Date: Time: No. Observations: Df Residuals: Df Model:

Change in Post-Response Sum of Utility [Freezing Intervention] OLS Least Squares Sun, 01 May 2016 12:11:29 100 99 0 coef

const 0.0340

std err

t

0.068

0.503

Omnibus: Prob(Omnibus): Skew: Kurtosis:

1.635 0.442 0.290 2.743

R-squared:

Adj. R-squared: -0.000 F-statistic: -inf Prob (F-statistic): nan Log-Likelihood: -102.13 AIC: 206.3 BIC: 208.9

P>|t| [95.0% Conf. Int.] 0.616

-0.100 0.168

Durbin-Watson: Jarque-Bera (JB): Prob(JB): Cond. No.

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-0.000

1.890 1.676 0.432 1.00

5

Conclusion

This paper has presented a model of liquidity crises on arbitrary interbank lending networks, featuring strategic liquidity hoarding and endogenous defaults. It has extended existing results in the global games literature to prove that this non-supermodular game has a unique threshold equilibrium. After specifying a random network, calibrated on the U.S. federal funds market, simulations demonstrate that confidence-driven contagion drives a significant welfare loss relative to the optimal behavior assumed in previous analyses. However, this magnitude of this loss is consistent with previous contentions that contagion’s effects are limited relative to the instigating shock. Network measures inspired by the natural sciences prove useless in predicting the magnitude of confidence-driven contagion, likely due to a lack of behavioral foundations and the relative importance of local network structure in this model, rather than global. Finally, several interventions were assessed for fixed networks, including securitizing loans, breaking up large banks, implementing a limited guarantee fund, and freezing the market. For different reasons, all but one intervention do significantly worse than the free market. Generally, while confidence-driven contagion typically causes inefficiencies, it is nevertheless a product of rational, informed behavior. And while concentrated markets may facilitate the spread of contagion, they also prevent coordination failures. In addressing these questions, it is important to note the critique put forth by Erol and Vohra (2014). They argue that certain obligation networks may be incompatible with a given balance sheet (i.e. shock) distribution. A related critique may have more lasting bite. Under a policy shift or parameter shift, not only would the behavior of banks change conditional on an initial network, but different initial networks may become more or less likely to arise. And, of course, the model presented here is partial. It ignores balance sheet contagion, and restricts itself to one form of interaction between banks. In assessing interventions, it only presents one of many relevant dynamics. Yet, while further work and more robust evaluations may be called for, the dynamic that it does present has needlessly been ignored in much of the existing literature. As our results show, that is not a benign oversight.

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6

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