Contagion in a network of heterogeneous banks

Contagion in a network of heterogeneous banks

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Contagion in a network of heterogeneous banks Ramazan Genc¸ay, Hao Pang, Michael C. Tseng, Yi Xue PII: DOI: Reference:

S0378-4266(19)30298-5 https://doi.org/10.1016/j.jbankfin.2019.105725 JBF 105725

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Journal of Banking and Finance

Received date: Accepted date:

8 August 2018 14 December 2019

Please cite this article as: Ramazan Genc¸ay, Hao Pang, Michael C. Tseng, Yi Xue, Contagion in a network of heterogeneous banks, Journal of Banking and Finance (2019), doi: https://doi.org/10.1016/j.jbankfin.2019.105725

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Contagion in a network of heterogeneous banks

Ramazan Gen¸cay∗, Hao Pang, Michael C. Tseng, and Yi Xue† December 2019

ABSTRACT We consider a financial network where banks are heterogeneous in scale and each bank has only local knowledge regarding the network. Each bank must make counterparty and portfolio decisions while anticipating uncertainty regarding the network structure. Such network uncertainty is an important consideration in banks’ risk management practice, which aims to minimize the effect of exogenous liquidity shocks and hedge against possible fire-sale in asset markets. We show that network uncertainty gives rise to an endogenous core-periphery structure which is optimal in mitigating financial contagion yet concentrates systemic risk at the core of big banks. JEL classification: G01; G22; G28 Keywords: Financial networks; Contagion; Heterogeneous banks; Core-Periphery structure; Network externality. ∗

The late Ramazan Gen¸cay was from Simon Fraser University, Canada. We dedicate the paper to his memory. † Hao Pang is from Duke University, USA. Michael Tseng is from University of Central Florida, USA. Yi Xue is from University of International Business and Economics, People’s Republic of China. Corresponding author: Michael Tseng, E-mail: [email protected], Address: Department of Economics, College of Business, University of Central Florida, P.O. Box 161400, Orlando, Florida 328161400. Michael Tseng gratefully acknowledges the Swiss National Science Foundation (SNSF Starting Grant “Liquidity”) for financial support. Yi Xue gratefully acknowledge financial support from the National Natural Science Foundation of China (NSFC), project No. 71471040 and No. 71971063. We also thank two anonymous referees for helpful suggestions for improving our paper. We are responsible for all remaining errors.

I. Introduction In this paper we study a model of financial crises within a network of financial institutions—“banks” hereafter—which are different in scale and have only imperfect information regarding the network structure.1 The network structure is determined by banks’ optimal counterparty choices. Should a bank become affected by the liquidity shock despite its best efforts in choosing counterparties, it has the ability to liquidate its liquid assets in an asset market. Therefore banks choose counterparties while anticipating a possible portfolio decision involving the trading off between short term liquidity and long term return. When the bank is hit by a large exogenous liquidity shock and become insolvent, the unabsorbed liquidity shock will transmit through endogenously determined financial linkages thus affect other banks’ solvency. In this paper, we are concerned with the characterization of equilibrium network structure and its properties with respect to financial contagion caused by liquidity shock and possible subsequent fire-sale in the asset market. In our model, market outcome is driven by banks’ assessment about their own imperfect knowledge regarding the network structure. In making counterparty choices, a rational bank in our model anticipates that subsequent financial decisions must be made with only local network information available, that is, under network uncertainty. Network uncertainty reflects the fact that, in practice, banks do not have access to the other banks’ balance sheets and thus do not have perfect information regarding the network of interbank cross-holdings. Banks in our model have only local knowledge regarding the financial network once the network is formed—its own balance sheet and the scale of its counterparties.2 We further assume that banks are uncertainty averse. That is, banks in our model make decisions under Knightian uncertainty, where economic agents 1

“Scale” in our model refers to size of asset endowment and amount of cash on hand. See Section II.A. In addition to structural restrictions on the network information available to each bank, lack of global knowledge regarding the network may also be due to unforeseen financial innovations that lead banks to question their view of the network. 2

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are uncertain about the underlying data generating process and makes decisions robust to this uncertainty, using a best-worst case optimality criterion.34 It is not unreasonable to assume that (network) uncertainty aversion is close to the attitude of banks in practice and we show that uncertainty can both aggravate and alleviate different aspects of a financial crisis. In this heterogeneous environment with Knightian uncertainty, we find that the equilibrium network has a core-periphery structure that channels financial contagion along the paths with the most liquidity to absorb a shock. Big banks reside in the core and small banks on the periphery. A shock can therefore transmit only through big banks, who are better able to absorb the shock than small banks, in equilibrium. The endogenously determined network, under imperfect information, therefore achieves the optimal structure in minimizing the domino effect of a liquidity shock. However, this also means that systemic fire-sale risk is also concentrated at the core of big banks. Subsequent to choosing counterparties, banks make liquidity-asset portfolio decisions to hedge against liquidity shock. Network uncertainty may lead to fire-sale which otherwise would not occur in a perfect information environment.5 Recognizing its imperfect knowledge regarding the network, a bank may retrench into liquidity conservation mode. Fire-sale occurs in equilibrium if there is sufficient demand for precautionary liquidity reserve. With respect to contagion and fire-sale, we find that network uncertainty has opposite effects. On the one hand, it drives banks to form a network that minimizes 3

In contrast to risk aversion, when the agent has a fixed prior regarding states of the world, uncertainty aversion means the agent faces a family of priors and is uncertain regarding which prior distribution the states are being drawn from. In our context, a state is specified by a financial network configuration and the location of liquidity shock. As only information available to a bank is information from its own decision making process, multiple data generating processes of the economy are consistent with its local information. 4 Robust decision-making under Knightian uncertainty was first introduced to economics by Hansen and Sargent (2001). It is referred to as “complexity” by Caballero and Krishnamurthy (2008) in the financial networks context. See also, for example, Buraschi et al. (2014) for asset pricing implications of the robust decision making under uncertainty. 5 This is also shown by Brunnermeier and Sannikov (2014), where exogenous uncertainty is amplified in a fire-sale period.

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contagion. On the other hand, it concentrates the risk of fire sale at the core of big banks. Interestingly, network thus formed under imperfect information may partially resolve network uncertainty. This is because equilibrium in our model is a self-fulfilling phenomenon.

As a result, rational banks in our model obtain implied information

regarding the network in equilibrium. This partial resolution of uncertainty may mitigate fire-sale but is not sufficient for banks to provide insurance in the form of, for example, credit derivative contracts. Uncertainty surrounding the financial network is a problem faced by regulators, researchers, as well as the constituent banks of the network. The regulatory difficulty stemming from network uncertainty is exemplified by the episode on the eve of the 2007-2008 Financial Crisis, when regulators with limited information about the degree of Lehman Brothers’ interconnectedness faced the dilemma of letting it fail or save it. For researchers, lack of available network information poses an identification problem. Existing studies have focused on innovation in estimation techniques to overcome the identification problem.6 In this paper, we focus on the network uncertainty faced by the banks themselves and show that it can explain the core-periphery structure, which is a persistent characteristic of the financial network of interbank exposures spanning from the Great Depression to the 2007-2008 Financial Crisis.7 Whereas the empirical literature concerned with estimation makes the core-periphery structure a model assumption, we explain how such a structure may arise endogenously. Due to network uncertainty, “toobig-to-fail” endogenously coincides with “too-connected-to-fail”. We provide a framework that considers network formation and stability jointly, thereby 6

See, for example, Demirer et al. (2018), which employ high dimensional LASSO techniques, Anand et al. (2015), which uses relative entropy techniques, and also the overview in Anand et al. (2018). 7 Calomiris and Mason (1997) and Calomiris and Carlson (2017) document the core-periphery structure of the interbank network dating back to the Great Depression and prior. Similar findings have been obtained for financial networks of the present period. For example, Van Lelyveld (2014) finds a core of highly connected banks intermediating between periphery banks in the Netherlands. Craig and von Peter (2014) provides evidence that the German interbank markets are tiered rather than flat, in the sense that most banks do not lend to each other directly but through money center banks acting as intermediaries.

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tying together two strands of current financial networks literature. In the literature, the stability and contagion implications are typically assessed ex-post after the network has been formed. However, the very same stability implications should also be expected by banks within the network, subject to network uncertainty. We demonstrate that the equilibrium core-periphery structure is “robust-yet-fragile.” That is, even though the systemic consequences of a shock may be catastrophic, the extant network is already optimal with respect to contagion, under network uncertainty.8 This point also goes into our assessment of macro-prudential policies.

Our framework can identify how

macro-prudential policies implemented through other channels interact with the network formation and contagion process.9 The network uncertainty faced by banks in our model has two related components, the uncertainty regarding the counterparties of the counterparty—and counterparties of counterparty of counterparty, etc.—and uncertainty regarding liquidity shock. A natural question is whether the equilibrium core-periphery structure that emerges is due to the counterparty component or liquidity shock component. We show that, regardless of whether the liquidity shock is anticipated, the equilibrium network retains the same characterization. Thus, the counterparty component dominates and, in our setting, the main driver of the results is indeed the banks’ heterogeneity in scale. Heterogeneity in network topology aligns with heterogeneity in scale endogenously. In addition to reflecting bank’s imperfect information in practice, in employing Knightian uncertainty we provide a new framework of financial network formation under asymmetric information. Previous literature has used refinements such as strict Nash equilibrium (e.g.

Bala and Goyal (2000) and Galeotti et al. (2006)) or a stability

criterion (e.g. Jackson and Wolinsky (1996)). By having the banks condition their best responses on only local information according to the Knightian maximin criterion, the 8 9

See Section VI.A for further discussion on the robust-yet-fragile property of a core-periphery network. See Section VII.

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problem of having to “forecast the forecast of others” that arises in a standard rational expectations framework is circumvented. In particular, when considering ex-ante risk in a standard rational expectations framework, solving for equilibrium network is complicated by the fact that risk itself is determined endogenously by the network, which in turn is determined endogenously by the banks’ understanding of risk. In our framework, we obtain a characterization the corresponding equilibrium network which can be analyzed in a tractable way. Our paper relaxes several assumptions made in the literature on financial contagion modeling. Previous papers we are aware of retain all or part of the following features— exogenous or homogeneous network, perfect information, and homogeneous agents. See, for instance, Allen and Gale (2000), Elliott et al. (2014), Acemoglu et al. (2015) and Babus (2016).10 In this paper, we consider endogenous network formation under imperfect information among heterogeneous agents, with the resulting network being incomplete and inhomogeneous. Some recent papers investigate network formation with heterogeneous agents. Chang and Zhang (2018) considers a network of market makers where the agents only choose the counterparties bilaterally therefore incomplete information regarding the entire network plays no role in agents’ decisions. Also, banks differ in need to share risk in their setting whereas we consider inhomogeneity in scale. Farboodi (2014) considers a network formed by financial institutions trying to capture intermediation spreads and thus voluntarily exposing themselves to counterparty risk in a perfect information environment. With the optimal core-periphery structure of the financial network, we find several interesting implications not explored previously by the literature, which emphasizes incentives of link formation with counterparty risk considerations given ex-post, or considers counterparty risk in homogeneous or complete networks. Our model suggests that, in an environment where heterogeneous banks have the objective of minimizing 10

Other related papers include Rochet and Tirole (1996), Upper (2007), Kiyotaki et al. (1997), and Zawadowski et al. (2013). See also the survey Hasman (2013) on factors for determining contagion in network structure.

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loss in a future crisis, the endogenous network structure that emerges is necessarily inhomogeneous and incomplete. Our model predicts that big banks are both too-big-to-fail and too-connected-to-fail in that (only) their failure have systemic consequences for the network. The incompleteness of the network structure in turn gives rise to inhomogeneous network externality. Such inhomogeneous network externality have policy implications in addressing banking crises. The rest of this paper is organized as follows. Section II describes the model environment and characterizes equilibrium network structure. Section IV analyzes the effects of contagion in a equilibrium network. Section V discusses a variation of information structure of our model where the liquidity shock component of network uncertainty is shut down. Section VI discusses extensions of our model and related empirical phenomena. Section VII discusses policy implications of our results. Section VIII concludes. The Appendix contains supplemental discussion on relaxation of our model assumptions, proofs, and an empirical framework in which our model can be taken to data. II. Model A. Basic Environment We consider a four-period—t = −1, 0, 1, 2—economy with single consumption good, which is also the numeraire, and populated by two types of financial institutions— hereafter “small banks” and “big banks”. Small banks and big banks differ in scale in that big banks have larger asset endowment and more cash on hand. Specifically, at the beginning of date −1 banks are endowed with cash and legacy assets. The size of cash and legacy assets endowment differ across big and small banks. Small banks have 1 − ys legacy assets and ys cash, and big banks have x − yb legacy assets and yb cash.11 We assume that ys < 1 and yb < x so each bank has positive legacy asset holding. Big bank are bigger in our model in the sense that they have larger amounts of both cash and 11

The initial total cash of small (resp. big) banks is normalized to 1 (resp. x).

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legacy assets. i.e. yb > ys and x − yb > 1 − ys . Banks of the same scale are identical. At date −1, banks choose their counterparties for interbank claims and thereby form the financial network. In our representation of the interbank claims network as a directed graph, a debtor of a bank is referred to as its forward neighbor. Similarly, a creditor bank is a backward neighbor. Figure 1 shows a circle financial network, which is a special case where each bank has one forward neighbor and one backward neighbor. A circle network has homogeneous topology in that each local neighborhood is identical. In our model, no substantial restriction is placed on network structure and nonhomogeneous structures will arise in equilibrium. Figure 2 illustrates a bank with two forward neighbors and two backward neighbors, with Bi → − Bj denoting that bank Bi has debt claims on bank Bj , i.e. Bj is a forward neighbor of Bi . We assume the amount of debt a bank wants to hold is also exogenously given according to its scale. A small (resp. big) bank has short term debt claims of zs (resp. zb ) dollars on its forward neighbors. Big banks have higher lending capacity than small banks: zb > zs . In the resulting network of interbank claims, bank i in the economy would have total outstanding liability—debt payable to its backward neighbors—denoted by zi0 . This debt must be repaid at date 1. We do not require zi0 to be constant across banks nor do we restrict the number of backward neighbors a bank can have. For example, if bank i has one big bank for backward neighbor, it has zi0 equals to zb , or if it has one big and one small bank for backward neighbor, it has zi0 equals to zb + zs . The cross-debt claims between banks we used in our model are widely found in financial market. Empirically, such financial networks may result from various types of cross exposures in the banking system.12 12 For example, Upper (2007) notes that interbank credits accounted for 29% of total assets of Swiss banks and 25% of total assets of German banks at the end of June 2005.. Another significant source of cross exposures between financial institutions is over-the-counter (OTC) derivative contracts such as interest rate swaps or credit default swaps (see Caballero and Krishnamurthy (2008)). The Bank for International Settlements (BIS) reports that gross credit exposures in the global OTC derivative markets had 2.8 trillion by the end of 2015.

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Bank n − 1

Bank 0

....

Bank 1

Bank 2

Figure 1. A circle financial network. This figure shows a circle financial network with every bank has only one forward neighbor and one backward neighbor.

Fi2

Fi1 Bank i Bi1

Bi2

Figure 2. Forward neighbors and backward neighbors. In this figure, bank i is situated in a network where it has two forward neighbors and two backward neighbors.

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Forward Neighbor Bank

Backward Neighbor Bank

Assets

Liabilities

• short-term debt claim with face value zs • 1 − ys legacy asset (keep or sell) ′ • ys + zs − zs dollars (invest in cash or assets)

• short-term debt claim ′ with face value zs • equity

(a) Balance sheet of a small bank

Forward Neighbor Bank

Backward Neighbor Bank

Assets

Liabilities

• short-term debt claim with face value zb • x − yb legacy asset (keep or sell) ′ • yb + zb − zb dollars (invest in cash or assets)

• short-term debt claim ′ with face value zb • equity

(b) Balance sheet of a big bank

Figure 3. Date 0 balance sheets of big and small banks. This figure shows how the size heterogeneity among banks lead to different date 0 balance sheets consisting of a bank’s cash on hand, legacy assets, and interbank claims and liabilities.

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At date 0, each bank therefore has a balance sheet consisting of cash on hand, legacy assets and interbank claims/liabilities, as depicted in Figure 3. Each bank can choose to hold liquidity by liquidating legacy assets and keeping its date 0 cash or invest in assets which can be transferred to cash at date 2 with rate of return R > 1. Cash can be used to meet date 1 liquidity needs and at date 2 any leftover cash will yield zero rate of return. On the other hand, while asset investment provides higher return, it is completely illiquid at date 1. The asset market is open only at date 0 and banks are not able to sell or buy this asset at date 1.13 The asset is supplied elastically at date 0 at a price of 1, which is also their fundamental value. At date 1, one of the banks experiences a liquidity shock, in the form of liquidity need of amount θ that is senior to interbank claims. At all dates, banks have no information on the realization of location of liquidity shock except that it will occur, or have occurred, to one of the banks at date 1—unless it is the bank itself or its forward neighbor that suffered the shock.14 In addition to the outside asset, legacy asset is also traded in a date 0 spot interbank secondary market at an endogenous price p. The legacy asset has identical liquidityreturn characteristics as the outside asset. While the outside asset is supplied elastically, the supply and demand of legacy asset is determined by banks’ liquidity decisions. The potential buyers of the legacy asset in the secondary market consist (only) of other banks in the network. As banks have access to both the outside market and spot interbank market, the endogenous price p of legacy asset is bounded above by 1, the price of the outside asset. We also assume there are outside parties who have elastic demand for the legacy asset at a constant price pscrap < 1, which gives a lower bound pscrap ≤ p on legacy asset price. When p = 1, legacy assets are traded at their fair value. The endogenous price p reaches the lower bound when there is excess supply of legacy assets, relative to 13

The return-liquidity trade-off is a feature of many financial instruments. Models that incorporate illiquidity premium into asset prices include, for example, Holmstrom and Tirole (1998). 14 We elaborate on the uncertainty faced by banks in the informational structure of our model in Section III.A. An alternative informational setting is considered in Section V.

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the demand of inside buyers. We refer to this scenario as a fire sale of legacy assets. When p lies strictly between 1 and pscrap , the buyers can be viewed as arbitrageurs that exploit asset mis-pricing caused by sellers’ uncertainty regarding liquidity shock. At date 1, a liquidity shock occurs to one of the banks, interpreted as senior outside debt payable immediately. When a shock is so severe that it exceeds the affected bank’s liquidity buffer, the affected bank becomes insolvent. The resulting reduced liquidity available for the backward neighbors may cause insolvency to spread. A domino effect of balance sheet contagion may emerge and passes upstream, i.e. from a bank to its backward neighbors, through the financial network. A bank therefore make its date 0 investment decision conditional on its date 0 information taking into account (to the extent possible) the network structure, the location and severity of the shock, the balance sheet health of its counterparty, that of the counterparties of its counterparty, and so on. Hedging against the liquidity shock shifts banks’ investments decisions at date 0 from assets to cash, which may lead to a fire sale. Date 2 is the consumption period, when returns are realized for solvent banks. We are concerned with characterizations of endogenous network structure at date −1, emergence of fire sale at date 0, and the magnitude of the domino effect date 1 in equilibrium. B. Characteristics of Counterparty Risk The effect of a liquidity shock on each bank depends on its local network topology, which may be nonhomogeneous. The heterogeneity in scale of banks in our model means that the scale of counterparties also plays a role in addition to local network topology. Each bank’s decision at date 0 on whether to build up a liquidity buffer, i.e. to sell or buy legacy assets, is governed by its knowledge of counterparty scales and network topology. More precisely, the following three definitions jointly form sufficient statistics for the effect of a liquidity shock on a bank in the network. Definitions 1 and 2 reflect network topology 11

while Definition 3 corresponds to counterparty scale. The first is distance from original distressed bank. In general, there may exist several different paths a shock can spread from the original bank to another bank. We formally define the distance as the the length of the shortest path: DEFINITION 1: Given a network configuration σ, a liquidity shock of size θ at bank i, d(j; σ, θ, i) is the least number of steps a shock can transmit via a directed path in the network from bank i to bank j, that is, the geodesic path from bank j to bank i through insolvent banks. By definition, d(i; σ, θ, i) ≡ 0. If none of bank j’s forward neighbors is insolvent, d(j; σ, θ, i) = ∞. The number of backward neighbors on the transmission path affects the severity of the domino effect. Backward neighbors act as shock absorbers and more backward neighbors shrinks the affected region. DEFINITION 2: In the event of liquidity shock at bank i and d(j; σ, θ, i) < ∞, N| = {N (1), N (2), ..., N (d(j; σ, θ, i))} denotes the number of backward neighbors of each bank on the geodesic transmission path from i to j. The size of liquidity buffers of banks on the shock transmission path is a factor in determining the length of the transmission path. Larger liquidity buffers absorbs more of the shock as it travels along a possible transmission path. In our environment where scale is the source of heterogeneity, the size of liquidity buffers is determined by the scale of the bank. DEFINITION 3: In the event of liquidity shock at bank i and d(j; σ, θ, i) < ∞, Sj = {S j (1), S j (2), ..., S j (d(j; σ, θ, i))} denotes the size of each bank on the geodesic path from the original distressed bank i to bank j. After the network is formed, banks’ choose optimal date 0 cash-asset portfolio based on their information regarding d, S, N , and the magnitude θ and location ishock of the date 1 liquidity shock. 12

C. Bank’s Investment Problem Under Uncertainty In our model, a bank faces two decisions—a counterparty choice decision at date −1 and an investment decision at date 0. We now define banks’ date 0 informational environment and investment problem.

The model will be solved in Section III via

backward induction. Denote the date 0 choice set of bank j by Aj0 ∈ {S, B}, where, Aj0 = S means selling all of its legacy assets for cash to maintain a completely liquid balance sheet and Aj0 = B means becoming a potential buyer of assets—either legacy or outside assets—at date 0 and retain all its legacy assets.15 At date 1, due to the liquidity shock a bank may be unable to meet its debt obligations in full—possibly despite taking the precautionary action Aj0 = S. A solvent bank j pays its interbank claimants q1j = zj0 and its equity holders q2j ≥ 0. An insolvent bank j pays its interbank claimants q1j < z 0 j and equity holders q2j = 0. A bank in the type of cross exposures network we model typically do not have perfect knowledge regarding network structure. A bank may have very little knowledge regarding parts of the network with high degrees of separation from itself. To reflect this, in our model, a bank only knows the local neighborhood of the network consisting of itself and its own forward and backward neighbors. In other words, banks only possess information on the network that result from its counterparty choices. The formal setting is that of Knightian uncertainty, where a bank may have a number of possible beliefs, or Bayesian priors, regarding the network structure. Let Oj denote the universe of all possible network structures within the economy that is consistent with bank j’s local information. Bank j equips Oj with a family of probability distributions.16 At date 0, banks seek to maximize investment return, i.e. equity value, at date 2, robust to this uncertainty. At date 1, one bank ishock suffers a liquidity shock in the form of an outside debt of 15

We restrict to a binary choice set for simplicity. Each Oj is assumed to be finite but can be arbitrarily large. Similarly, each bank’s family of priors is also finite. 16

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amount θ senior to short term interbank claims. Conditional on ishock and θ, relevant ambiguities regarding the network structure are those defined in Definitions 1, 2, and 3. Using the maximin expected utility representation of Gilboa and Schmeidler (1989) under Knightian uncertainty with multiple priors, bank j’s date 0 problem is:17

max

Aj0 ∈{S,B}

min qj2 (Aj0 , σ; ishock , θ)

(σ)∈Oj

(1)

where qj2 (Aj0 , σ; ishock , θ) is date 2 equity value of bank j given network structure σ, with date 0 choice Aj0 , conditional on ishock and θ. More generally, at date 0 after the network is formed a bank may form certain beliefs regarding the where the shock might occur in the network and its magnitude. Let bank j’s belief be specified by a joint distribution fj (ishock , θ) of ishock and θ. Then each bank j’s date 0 problem under Knightian uncertainty becomes18

max min Ej(i,θ) [qj2 (Aj0 , σ; ishock , θ)]

Aj0 ∈{S,B} σ∈Oj

(2)

where Ej(i,θ) [·] denotes expectation with respect to fj (ishock , θ). At this point we place no restriction on the distribution fj (ishock , θ), allowing for the possibility of heterogeneous beliefs among banks. Banks may believe, for example, magnitude of the shock is correlated with the scale of a bank. In particular, the belief where the marginal distribution of ishock is the maximally uninformative prior— uniform distribution on i—and the marginal distribution of θ is point mass at some θ0 correspond to the case where no information is known regarding ishock but magnitude of shock is known to be θ0 . We do not model what may cause banks to have heterogeneous, or homogeneous beliefs. Our characterization of endogenous network structure is robust to heterogeneous beliefs. 17 Gilboa and Schmeidler (1989) extends von Neumann-Mortgenstern expected utility representation to the multiple Bayesian priors setting. In our context, any preference on lotteries over date 2 equity payments satisfying certain standard axioms admits a maximin representation. 18 Alternatively, one can replace the maximin operator using an equivalent nonlinear “Knightian expectation operator”.

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D. Legacy Asset Market Legacy assets are traded in a date 0 spot market. Given the legacy asset price p, the bank that chooses Ai0 = S sells all of its legacy assets.19 Banks that choose Ai0 = B are potential buyers of legacy assets and expends its cash holding yb (for big banks) or ys (for small banks). If p ∈ (pscrap , 1), buyers spend their cash on legacy assets and the interbank spot market must clear by itself. If p = 1, buyers are indifferent between buying legacy or outside assets and the interbank spot market may clear with additional supply of outside assets. Similarly, if a fire sale occurs and p = pscrap , the interbank spot market may clear with additional demand from outside buyers. The market clearing condition for legacy assets can therefore be written as:

X i

(1 − ys )1{Ai0 =S}

   ≤ 0, if market clearing price p = 1.  + (x − yb )1{Aj =S}  0   j = 0, if market clearing price p ∈ (pscrap , 1). X ys − (z 0 − zs )+  i  1{Ai0 =B}  −   p  ≥ 0, if market clearing price p = pscarp . i 0 X yb − (zj − zb )+ − 1{Aj =B} 0 p j X

(3)

The left hand side of Equation 3 is the excess supply of legacy assets at the date 0 interbank legacy asset market. If excess supply is negative for each p ∈ [pscrap , 1), then legacy assets are traded at maximum value p = 1 and potential buyers buy all legacy assets on the market. If excess supply is zero for some p ∈ (pscrap , 1), then the equilibrium price is p. If excess supply is positive for each p ∈ (pscrap , 1], then the price of legacy assets is the minimum value pscrap for a fire sale. The role of the legacy asset market is to allocate liquidity. 19

In their asset-cash

It makes no material difference to our results if a seller bank just sells enough assets for liquidity or sells them all.

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portfolios, banks take positions on liquidity that are not contingent due to imperfect information. In a fire sale, the mismatch between demand and supply of legacy assets is due to allocation inefficiency caused by information imperfection. When banks can choose portfolios contingent on the liquidity shock, fire sale never arises.20 Aggregate level of liquidity is constant across states and always allocated efficiently to prevent fire sale under public information regarding the shock. Not knowing how liquidity is allocated globally forces uncertainty averse banks into hoarding cash, drives down asset prices, and leads to fire sale.21 E. Equilibrium An equilibrium in our model is therefore a market clearing price p together with each bank’s optimal date −1 and date 0 decisions. Bank j’s action A−1 j at date −1 consists of choosing its forward neighbor and its action A0j at date 0 consists of whether to buy or sell legacy assets. To emphasize that a bank’s actions must be measurable with respect to j its information filtration, we let I−1 and I0j denote bank j’s information sets at date −1 j and date 0 respectively, and Aj−1 (I−1 ) and Aj0 (I0j ) be bank j’s actions. I0j is a refinement j of I−1 , as I0j contains local network information. In subsequent analysis we will specify

whether location of magnitude of shock is contained in I0j . The structure of the economy j is common knowledge among banks and therefore contained in I−1 for each j. An rational

expectations equilibrium in our model is defined as follows: j ), Aj0 (I0j )}j and a DEFINITION 4: An equilibrium is a cross sectional profile {Aj−1 (I−1

legacy asset price p ∈ [pscrap , 1] where j (i) {Aj−1 (I−1 )}j are banks’ optimal choices at date −1 of counterparties that result in

a network of interbank claims. 20

This is shown in Lemma 5 below. In a heterogeneous information setting, asset price may reveal certain information to less informed banks. See Section VII.C. 21

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(ii) {Aj0 (I0j ) ∈ {S, B}}j are banks’ optimal choice at date 0 on whether to buy or sell in the legacy asset market, according to banks beliefs regarding network structure, i.e.

Aj0 (I0j ) ∈ arg max min Ej(i,θ) [qj2 (Aj0 (I0j ), σ; ishock , θ)], σ∈Σj

j where Σj ⊂ Oj (Aj−1 (I−1 )), the set of network structures that is consistent with bank j’s j date 0 network information as a result of its date −1 action Aj−1 (I−1 ).

(iii) {(Aj0 (I0j )j , p} satisfies the market-clearing condition of Equation 3. T (iv) The resulting network σ ∗ is consistent with all bank’s beliefs: σ ∗ ∈ j Σj .

In the Knightian setting, bank j considers Problem 2 and take the support of its prior

j )). to be all network structures consistent with its local knowledge, i.e. Σj = Oj (Aj−1 (I−1

While we take Knightian uncertainty as the appropriate framework in modelling banks’ uncertainty regarding network structure, it turns out that the network structure resulting from banks’ Knightian decisions is not consistent with the Knightian prior. To satisfy rational expectations (Definition 4(iv)), we will shift the supports of bank’s priors slightly in Section III below. More explicitly, one may view equilibrium in our model as an learning process where banks revise their beliefs iteratively to arrive at a fixed point. III. Equilibrium Network Structure The financial network in our model results from banks’ date −1 incentives to choose optimal counterparties. In choosing a counterparty, banks seek to minimize counterparty risk associated with possible liquidity shock while anticipating its date 0 informational environment on local network structure. This section solves the model by backward induction. For simplicity, we assume each bank only has a single forward neighbor.22 ASSUMPTION 1: There are m big banks and n small banks. Each bank can have at most one forward neighbor. 22

This assumption retains the essential heterogeneity feature of our model and will not lead to material difference in our results. Implications of relaxing this assumption is discussed in Section VI.B.

17

A. Date 0 Consider the date 0 investment problem of bank j conditional on local information I0j under Knightian uncertainty. Suppose for the moment that the magnitude of the shock θ lies in bank j’s date 0 information set I0j for all j. At date 2, the equity value of a small bank j is

max{0, 1{Aj =S} · (ls (p) + zs [residual] − zs0 − θ1{j=ishock } ) + 1{Aj =B, zs [residual]=zs } · R 0

0

ys } (4) p

where 1{j=ishock } is the indicator for whether bank j is the one which experiences the shock and therefore requires additional liquidity of θ, zs [residual] is the ex post payment received from its forward neighbor, and

lj (p) = ys + (1 − ys )p

(5)

is the liquidity buffer obtained from choosing Aj0 = S in Equation 2, given price p in legacy asset market.23 In general zs [residual] ≤ zs with strict inequality if the forward neighbor is insolvent. Bank j optimal date 0 action A(I0j )—whether to buy or sell in the legacy asset market—therefore depends on its date 1 liquidity need

0

zj − zs [residual] + θ1{j=ishock } .

(6)

If the liquidity buffer lj (p) obtained at date 0 is greater than the date 1 liquidity need of bank j, then the bank avoids insolvency. Otherwise, it will be insolvent despite taking precautionary action. In considering whether to hoard liquidity or acquire assets, banks must consider characteristics of counterparty risk defined in Section III.A. This in turn requires the characterization of the worst possible network structure, under the maximin 23

Recall that a small bank’s date −1 cash endowment is ys and has 1 − ys of legacy assets.

18

formulation in Problem 2 for a bank in our model:24 LEMMA 1: For Aj0 = S or B, conditional on location ishock of the shock not being itself or j its forward neighbor, the worst network structure σworst ∈ arg min Ej(i,θ) [qj2 (Aj0 , σ; ishock , θ)] σ∈Oj

consistent with bank j’s local information has the following properties: j (i) Shock will happen on its forward neighbor’s forward neighbor, i.e. d(j; σworst , θ, ishock ) =

2. (ii) Both of its forward neighbor and the forward neighbor of its forward neighbor has only one backward neighbor, i.e. Sj = {S j (1), S j (2)} = {1, 1}. (iii) The forward neighbor of its forward neighbor is either a small bank or a big bank j which depends on its own decision on date -1, Aj−1 (I−1 ), i.e. Nj = {N (1), N (2)} = j j {Aj−1 (I−1 ), Aj−1 (I−1 )}. j Lemma 1(i) and (ii) pin down the local network topology of σworst . Lemma 1(iii) shows

that a Knightian bank j’s date 2 payoff depends on the scale of its forward neighbor and rational bank will always believe that his forward neighbor who are facing the same optimizing problem would make the same decision as him. It is clear that the date 2 equity value obtained by the optimal choice of buy or sell is always larger in the case N (1) = N (2) = big, for big forward neighbors can better absorb the shocks, and banks dislike being distressed and forced to liquidate all its legacy assets and forgo potential date 2 return. When the forward neighbor also chooses big bank, the possibility that its forward neighbor be distressed would further decrease. Thus, decision of Nj = {big, big} strictly dominated Nj = {small, small} and becomes the equilibrium network structure. This is also true for cases not covered by Lemma 1. If the shock occurs to the forward neighbor of j j bank j, then σworst has the properties that d(j; σworst , θ, ishock ) = 1, Sj = {S j (1)} = {1}, 24

Lemma 1 also shows that max

min Ej(i,θ) [qj2 (Aj0 , σ; ishock , θ)] =

Aj0 ∈{S,B} σ∈O j (σ)

min

max

σ∈O j (σ) Aj ∈{S,B}

Ej(i,θ) [qj2 (Aj0 , σ; ishock , θ)].

0

Therefore we may consider first the minimax network structure then bank’s buy-sell decision in solving the model.

19

and Nj = {N (1)} = {scale of forward neighbor}. Again having a big bank as forward neighbor always yield equal or better equity value at date 2 ex post. If the shock occurs j to the bank itself, d(j; σworst , θ, ishock ) = 0, Sj and Nj are irrelevant, and the bank is

indifferent between having a big or small forward neighbor. The argument applies to big banks as well as small banks. Integrating over the location i and magnitude θ of the shock as in Problem 2, we obtain: LEMMA 2: For any bank, having a big forward neighbor will always yield equal or better equity value than having a small forward neighbor both ex ante at date 0 according to the maximin Knightian criterion, and ex post at date 2. In other words, choosing a big forward neighbor weakly dominates choosing a small forward neighbor. This describes the network structures that arise under Knightian uncertainty: LEMMA 3: When banks makes decisions under Knightian uncertainty, all banks will choose a big bank as its forward neighbor. In particular, small banks do not have backward neighbors and each small bank has a big bank as forward neighbor. The network structure described by Lemma 3 does not lie in the intersection of Knightian priors of Lemma 1. In particular, the forward neighbor of a forward neighbor is never a small bank. This violates the rationality requirement, Definition 4(iv). If we revise the banks’ priors to be consistent with the network structure of by Lemma 3, choosing a big bank as forward neighbor remains a preferred action for all banks. This leads to a characterization of some equilibrium network structures that arises from bank’s optimal date −1 decision—only big banks can be forward banks of another bank.25 25

Theorem 1 is an existence result. We are mainly interested in meaningful economic implications of the core-periphery network structure covered by Theorem 1 and do not discuss the issue of uniqueness.

20

Bs.

... Bb.

...

...

...

Bs. Bb.

Bb. ...

Bs. ...

... Bb. ... Bs.

Figure 4. Equilibrium network structure. This figure depicts the equilibrium network structure characterized in Theorem 1. Small banks (Bs ) do not have backward neighbors and each small bank has a big bank Bb as forward neighbor. Big banks lend to each other and form in circles or circles with tails.

THEOREM 1: There exist equilibria, as defined by Definition 4, where all banks choose a big bank as its forward neighbor. In particular, small banks do not have backward neighbors and each small bank has a big bank as forward neighbor. Each bank therefore prefers big banks as forward neighbors over small banks and is indifferent among big banks as forward neighbors. The endogenous network structure has certain uniqueness property: COROLLARY 1: In an equilibrium network that falls under Theorem 1, the following hold: (i) Big banks lend to each other and any part of the network that is a cycle must consist of big banks only (see Figure 5). 21

Bs.

...

Bb. ...

...

...

Bs. Bb.

Bb. ...

Bs. ...

...

Bb. ...

...

Bb. Bb.

Bs.

...

(a) Example 1

Bs1

Bs2 Bb1

Bs8

Bs3 Bb4

Bb2

Bs7

Bs4 Bb3 Bs6

Bs5 (b) Example 2

Figure 5. Equilibrium network structure. This figure shows two particular instances of the equilibrium network structure that may emerge. Example 1, big banks form a circle with a tail. In Example 2, big banks form a circle in the core of the network and small banks attach themselves to the core.

22

(ii) A disconnected component of an equilibrium network structure consists of a “core” of big banks and “periphery” of small banks. The periphery must be a tree.26 Along any directed path in the network, the banks on the path are situated monotonically according to scale, from small banks to big banks. Figure 5 gives two examples of equilibrium network configurations. Next we analyze this endogenous resiliency of the equilibrium network with respect to financial contagion. IV. Domino Effect and Fire Sale We now analyze date 1 domino effect given the equilibrium network structure described by Theorem 1 and fire sale conditions in the date 0 legacy asset market in equilibrium. Without loss generality, we assume the network has a single component, as the domino effect cannot spread across disconnected components. To simplify notation, we assume also the core of the network is a circle of big banks. Figure 5(b) depicts one such network.27 En route to full Knightian uncertainty equilibrium characterization, it is useful to consider first best results when the global network structure is public knowledge. ASSUMPTION 2: If a bank has more than one backward neighbor, in the case of insolvency its available liquidity will be evenly distributed among its backward neighbors. Therefore interbank claims have the same seniority. This restriction also has no material impact on our main results, although in practice different creditor may enjoy different levels of seniority. The following lemma is a useful benchmark illustrating how contagion may transmit in an equilibrium network under public knowledge regarding the network structure and the location and magnitude of the shock. 26

A tree is a directed graph that contains no cycles. All the results in this section can be readily extended a general equilibrium network by a componentwise analysis and, within each component, making minor modifications to account for big banks attached to the circle. 27

23

LEMMA 4 (First Best Domino Effect with Given p): Suppose the location ishock and magnitude θ of the shock and network structure is public knowledge and that the network structure satisfies the properties described in Theorem 1 with big banks located only at a core circle. Furthermore, assume that each big bank has 1 big backward neighbor and

m n

small neighbors. At the given the price p of legacy assets, the size D(p) of domino effect can be characterized as follows: i If the shock occurs at a small bank, the resulting domino effect is D(p, θ) = 1. ii If the shock occurs at a big bank, the resulting domino effect is given by

D(p, θ) = arg max { L

L X m θ (1 + )l ≤ b n lb (p) − l=0

c m z n s

}.

(7)

Lemma 4(i) holds because, in the equilibrium network structure, small banks have only forward neighbors and no backward neighbors and therefore cannot spread the domino effect further. Lemma 4(ii) follows from the fact that, in equilibrium, a shock contagion spreads through a path of big banks. When there are no small banks, i.e. m = 0 and the θ c. With network is a circle of big banks, then Lemma 4(ii) specializes to D(p, θ) = b lb (p)

perfect information, domino effect decays linearly. This rate is a upper bound of that of the full Knightian uncertainty case, when banks hoard liquidity when they would not under perfect knowledge. Next we endogenize legacy asset price p and consider fire sale under public knowledge regarding network structure and the shock. We make an assumption that precludes the possibility of a catastrophic shock wiping out the entire excess liquidity of the network. This emphasizes the role of network uncertainty in precipitating fire sale. ASSUMPTION 3: 0

0

θ ≤ m(yb + zs − zs ) + n(ys + zb − zb ). Also, given a shock θ, and D(p, θ) + 1 stressed banks, the contagion mechanism implies that 24

D(p)+1

θ + E(p) =

X

lik (p)

ik =1,Bik stressed

|

{z

,

}

liquidity buffer of stressed banks

where E(p) ≥ 0 is the liquidity of stressed banks in excess of the shock. We are going to restrict to legacy asset price p to be bounded.28 ASSUMPTION 4: θ≤T for some upper bound T . Assumption 4 would imply that θ + E(p) is bounded as well over legacy asset price p. This has the interpretation that the magnitude of the shock and legacy asset price cannot be simultaneously high.29 LEMMA 5 (First Best Fire Sale): Suppose the location and magnitude θ of the shock and network structure is public knowledge, then the unique equilibrium price is p = 1. In particular, there is no fire sale. Lemma 5 in fact is independent of network configurations. As long as the shock is not too large and shock and asset prices jointly satisfy a reasonable constraint, under public knowledge counterparty risk is not a concern and the network adjusts its liquidity demand and supply to prevent a fire sale. In contrast, under uncertainty, fire sale emerges when it would not otherwise. In the general setting, a priori there are three dimensions of uncertainty— geodesic distance toward the original distressed bank (Definition 1), the number of backward neighbors of banks on the geodesic path (Definition 2), and scales of banks on the geodesic path (Definition 3)—faced by each bank at date 0. In the equilibrium network structure, 28

Accordingly, banks’ beliefs regarding θ are assumed to have support restricted to [0, T ]. For example, the magnitude of the shock and asset price tend to be negatively correlated. When the economy is under stress, asset price is unlikely to be high. 29

25

however, the dimension of uncertainty is endogenously reduced to two. A rational bank therefore not only knows that its forward neighbor is a big bank but that the forward neighbor of its forward neighbor is a big bank, and so forth. The equilibrium network structure therefore provides implied information about the scales of banks on the geodesic path. To have a concrete discussion on fire sale in equilibrium, we restrict the date 0 beliefs of banks regarding (ishock , θ) to the following types: ASSUMPTION 5: 1. The marginal distribution of magnitude of shock is a point mass concentrated at θ ≥ maxj (zj − zj0 ) for all banks. 2. One of the following holds: (a) The marginal distribution of ishock is the maximally uninformative uniform prior restricted to big banks. (b) The marginal distribution of ishock is the maximally uninformative uniform prior. THEOREM 2 (Fire Sale in Equilibrium): We have the following conditions for fire sale in equilibrium: i Fire sale does not occur if n(x − yb ) ≤ mys and shock will occur at a big bank (Assumption 5(2.a)). ii Fire sale occurs if n(x − yb )(pscrap ) > mys and shock will occur at a big bank (Assumption 5(2.a)) or banks have no information regarding ishock (Assumption 5(2.b)). As in the first best case of Lemma 5, small banks have no backward neighbors in equilibrium.

Under Assumption 5(2.a), all small bank choose to buy legacy assets.

Whether fire sale occurs depends on whether big banks’ flood the legacy asset market and overtake small banks’ demand. This is a possibility because uncertainty forces big banks into taking precautionary actions they otherwise would not under perfect information. If there is sufficient demand—in particular, enough small banks in the network—there will 26

never be a fire-sale condition regardless of the magnitude of the liquidity shock. If there is sufficient number of uncertain big banks, a fire sale may emerge when θ is large enough. In summary, the equilibrium financial network structure is configured so that small banks are relegated to the periphery and the domino effect is contained in the core of big banks. The system therefore achieves endogenously the optimal structure—up to connections between the core of big banks— that minimizes domino effect. While uncertainty is not completely resolved and fire sale may still occur, in equilibrium partial information regarding the network is revealed to the rational bank. V. Anticipated Shock Thus far both components of network uncertainty—first regarding the counterparties (and counterparties of counterparties, etc) and second regarding the liquidity shock—are both present in our model. A natural question is which effect underpins the core-periphery network structure and characterization of fire sale conditions in the subsequent asset market. For example, in the scenario of Theorem 2(ii) where n(x − yb )(pscrap ) > mys and shock will occur at a big bank, fire sale will take place even when the subnetwork consisting of only big banks is a complete network—the optimal configuration against a shock. This suggests that it is uncertainty regarding the counterparty that drive our results, rather than uncertainty regarding the shock. For a precise identification, we now eliminate uncertainty regarding the shock by assuming the shocks are anticipated and show that our results persist. Specifically, instead of facing uncertainty regarding (ishock , θ) at date 0, suppose now that each bank’s information set I0j at date 0 contains (ishock , θ). That is, at date 0 banks learn that bank ishock will suffer a liquidity shock of magnitude θ. This reduces the uncertainty faced by banks to only that originating from counterparty choice—the three characteristics of counterparty risk defined in Section III.A. If j 6= ishock , bank j’s date 0 buy-sell decision is now determined by it maximin date 1 payment received from its forward neighbor zj [maximin]: 27

   buy legacy assets if zj0 − zj [maximin] ≥ 0   sell legacy assets

if

zj0

.

− zj [maximin] < 0

In turn, zj [maximin] depends on bank j’s maximin view of the its local network structure. Lemma 1 remains true in this informational setting. LEMMA 6: Suppose each bank’s information set I0j at date 0 contains (ishock , θ). For bank j, conditional on location ishock of the shock not being itself or its forward neighbor, the worst network structure consistent with bank j’s local information has the following properties: j (i) Shock will happen to its forward neighbor’s forward neighbor, i.e. d(j; σworst , θ, ishock ) =

2. (ii) Both of its forward neighbor and the forward neighbor of its forward neighbor has only one backward neighbor, i.e. Sj = {S j (1), S j (2)} = {1, 1}. (iii)The forward neighbor of its forward neighbor is either a small bank or a big j bank which depends on its date -1 decision Aj−1 (I−1 ), i.e.

Nj = {N (1), N (2)} =

j j {Aj−1 (I−1 ), Aj−1 (I−1 )}. j However, we do not impose any change to date -1 information set I−1 . Therefore, backward

inducting to date −1 where counterparty choices are made, we see that having a big bank as forward neighbor weakly dominates that from having a small bank as forward neighbor, because a big bank’s liquidity buffer lb (p) = yb + (x − yb )p is larger than that of a small bank ls (p) = ys + (1 − ys )p. Therefore the core-periphery structure persists without uncertainty regarding liquidity shock. THEOREM 3: Suppose each bank’s information set I0j at date 0 contains (ishock , θ). In any network that emerges in equilibrium, all banks will choose a big bank as its forward neighbor. In particular, small banks do not have backward neighbors and each small bank has a big bank as forward neighbor. 28

Therefore the characterization of the equilibrium network structure does not depend on whether banks anticipate shock location and magnitude at date 0. However, this informational setting removes partially banks’ uncertainty regarding ishock —the bank now knows whether itself or its forward neighbor has suffered the shock. This leads to a refinement of the characterization of fire sale occurrence given by Theorem 2. THEOREM 4 (Fire Sale With Anticipated Shock): Suppose each bank’s information set I0j at date 0 contains (ishock , θ). The following characterizations for fire sale hold in equilibrium: i (No fire sale equilibrium) Fire sale does not occur under one of the two following conditions: (a) n(x − yb ) ≤ mys and will occur at a big bank. (b) n(x − yb ) + (1 − ys ) ≤ (m − 1)ys and shock will occur at a small bank. ii If n(x − yb ) > mys , there are following three possibilities: (a) (Unique fair price equilibrium) If θ ≤ lb (pscrap ) + lb (pscrap ), and mys + nyb > 2x then there is a unique equilibrium with price p = 1. (b) (Unique fire sale equilibrium) If θ > lb (1) + lb (1), or if θ ≤ lb (pscrap ) + lb (pscrap ) but mys + nyb ≤ 2x, there is a unique equilibrium with price p = pscrap . There is an unique equilibrium price of p = pscrap and the aggregate amount of new asset purchases equals to 0. (c) (Multiple equilibria) If lb (1) + lb (1) ≥ θ > lb (pscrap ) + lb (pscrap ) and mys + nyb > 2x, then there is a fair price equilibrium and a fire sale equilibrium. When the liquidity component of network uncertainty is shut down and only the counterparty component remains, the self-fulfilling property of equilibria already manifest in the network formation process can now also spill into the asset market, as shown by Theorem 4(ii.c). 29

VI. Discussion A. Robust-yet-fragile Property In making optimal counterparty decisions, banks in our model form a financial network that attenuates systemic fire sale risk, in addition to being optimal with respect to contagion. A similar derivation as that of Section III would show that, compared to a network constructed by a social planner with perfect information, the equilibrium network in our model has lower threshold of asset value below which a fire-sale occurs. This is because implied network information obtained by rational banks is not part of the information disseminated by the social planner. On the other hand, as Theorems 2 and 4 show, systemic fire-sale risk cannot be completely eliminated. In addition to the self-fulfilling property of financial crisis, network uncertainty gives rise to the following stronger property with respect to contagion: the characterization of core-periphery equilibrium network structure remains true even if banks internalize disutility from contagion suffered by other banks, as well as its counterparties.

To

illustrate this, let the magnitude θ of liquidity shock be fixed and consider a game between an additional malicious player im and the banks. The malicious player im ’s objective is to maximize the domino effect D(p, θ)— the number of bankrupt banks—by choosing the location ishock of the shock.30 Modify the bank’s utility from that of Equation 2 to

max minj Ej(i,θ) [qj2 (Aj0 , σ; ishock , θ) − D(p, θ)].

Aj0 ∈{S,B}

σ∈O

The new game is therefore zero-sum in domino effect between the malicious player im and an individual bank. A priori, bank optimization problem in this game is different from the original game. In the original game, banks are merely required to form correct conjectures about other bank’s actions. In this game, because banks are now concerned 30

The informational environment faced by banks in this discussion is therefore between those of Section III and Section Section V.

30

with minimizing contagion throughout the network, banks must be not only rational about other banks’ actions while facing its own uncertainty but also rational beliefs regarding other banks’ beliefs. However, in the original game, because the worst case scenario for bank −i depends on bank i’s actions, banks in the original game already view themselves as being immersed in a zero-sum game. Therefore the equilibrium outcome in the new game remains the same as the original game. There is no “price of anarchy” with respect to contagion in our model. In an environment where all banks face uncertainty, banks implicitly “forecast the forecast of others.” That robustness with respect to contagion is coupled with fragility with respect to systemic fire-sale risk is a consequence of scale heterogeneity among banks. By leveraging their capital to enhance liquidity, big banks also become systemically more risky. From the perspective of the banks’ as well as the regulator, our model suggests that the ideal degree of scale heterogeneity in the financial system involves a trade-off between liquidity risk and systemic risk.31 B. Endogenous Network Incompleteness Under a homogeneous in scale environment (e.g. Allen and Gale (2000)), the network structure that is optimal in containing contagion is homogeneous in network topology—the complete network where each bank diversifies its holding among its forward neighbors. Our model suggests that, in an environment where heterogeneous banks face network uncertainty, the endogenous network structure that emerges is necessarily inhomogeneous and, in particular, incomplete. The scale heterogeneity of banks therefore manifests in the equilibrium core-periphery structure. Empirically, it is reasonable that a bank may prefer to have exposure to those larger than itself when considering counterparty risk. The endogenous network structure constrains the transmission path of a shock along big banks (all such paths lead to the core) who are most able to absorb the shock. This 31 The policy implication of the trade-off between liquidity risk and systemic risk is further discussed in Section VII.

31

minimizes the domino effect caused by a shock. C. Inhomogeneous Network Externality The incompleteness of the network structure in turn gives rise to inhomogeneous network externality. Under the original informational structure of the model or that in Section V, consider a setting where banks have the option to contribute to a bailout fund after choosing counterparties and before the asset market opens. If each bank has θ }, each bank contributes 0 and the same two possible choices of contribution, {0, n+m

characterization Theorem 4 holds in equilibrium. In particular, in the cases covered by Theorem 4(ii.b), where the shock is sufficiently large (θ > lb (1) + lb (1)) that a fire sale emerges under uncertainty. When the number of banks—in particular, the number m of small banks—are large, the contribution of

θ n+m

by any single bank does not change the

domino effect. Under the original informational structure of Section V, if banks know that it is a small bank they have an additional reason to not contribute—they know for certain that there will be no contagion in an equilibrium network. The resulting equilibrium is therefore Pareto inefficient. A social planner might institute a bailout policy of mandatory contribution of

θ n+m

by each bank. This results in a Pareto improvement where all banks choose to be buyers of asset. This would seem to suggest that banks exert a negative externality by not contributing. This externality is driven by network uncertainty and orthogonal to any effect through the asset price channel. This was also shown in the exogenous homogeneous network structure setting of Caballero and Simsek (2013). However, an equal contribution

θ n+m

puts all banks on equal footing in sharing fire sale risk and does

not reflect heterogeneity of banks or network topology. While big and small aligns with core and periphery in equilibrium in the heterogeneous setting, two layers of network externalities—from periphery to the core and from big banks to small banks—exist that act in opposite directions. 32

Due to inhomogeneity in network topology Under setting of Section V, consider a scenario where the banks know at date 0 that a big bank bishock will undergo a liquidity shock of θ.32 Now as above let the banks have choices of contributing 0 or

θ n+m

to a bailout fund and that, in addition, the aggregate

amount of banks’ contribution would be observable to all banks. We assume equal contribution to control for the scale heterogeneity of banks. Suppose lb (1) + lb (1) < θ. Then by Theorem 4, a fire sale emerges without a bailout fund. For the backward neighbors {bik , k = 1, · · · , r} of bishock , under maximin they perceive to become insolvent at date 1 regardless of contribution—the maximin network configuration for each backward neighbor is that it is the only backward neighbor of bishock . Therefore a backward neighbor is indifferent between contributing 0 or a small amount

θ . n+m

In the absence of local

/ {ishock , i1 , i2 , · · · , ir } information possessed by {bik , k = 1, · · · , r}, other banks bj , j ∈ choose optimally to contribute 0 for the same reason as above. If {bik , k = 1, · · · , r} chooses to contribute

θ n+m

rather than 0, the aggregate contribution observable to banks

θ . This reveals to the other banks that bishock has r backward neighbors. If r is is r n+m

sufficiently large so that (r + 1)lb (pscrap ) > θ, the other banks understand that contagion will be localized to bishock ’s immediate backward neighbors, and in turn this may prevent fire sale.33 By contributing, the immediate backward neighbors exert positive externality on the market by disseminating their local information. This positive externality arises due to heterogeneity in network topology and does not exist in, for example, a circle network, which is homogeneous. Due to inhomogeneity in scale In the endogenous network structure, for a bailout policy to be successful, only big banks need to be bailed out in the event of a shock, because contagion cannot spread 32

As pointed out above, if bishock is a small bank, there is no financial contagion or fire sale. Recall that small banks would not have backward neighbor, thus we only consider the worst case that bishock has only one big backward neighbor which can spread the shock. 33

33

from a small bank.34 The endogenous network structure that minimizes contagion also concentrates systemic risk at the core of big banks. In this sense, only big banks exert network externality on other banks. A fire sale may be caused by just a few panicked big banks, in a large network of banks, flooding the asset market. To be more explicit, since only big banks can spread contagion, consider the setting where only big banks the option of contributing θ n

>

θ , n+m

θ n

to the bailout fund, and max{2lb (1), mys } < θ < nyb . Since

cost of contributing is now higher for big banks. On the other hand, a big bank

has maximin equity payoff    R(yb − θ ) + R(x − yb ) if contribute n

2 qbig bank =   0

.

if not contribute

Therefore a big bank will contribute and thereby exerting positive externality on small banks. The same discussion extends to Theorem 2(ii) under the original setting where shocks are unanticipated at date 0. However, under the original setting, a misalignment of beliefs may neutralize the positive externality. If a big bank believes a liquidity shock is unlikely to occur on another big bank or itself, it does not contribute to the bailout fund, while small banks may have insufficient funds to effect a bailout fund by themselves. Erroneous beliefs or overly optimistic priors may result from failure to exercise due diligence in screening assets, such as what occurred prior to the 2008 subprime mortgage crisis. The realization that systemic risk is due to large financial institutions can be seen by governmental intervention on behalf of large financial institutions and subsequent regulations such as Dodd-Franks.35 34

One may consider loss of one small bank in a large network negligible. For example, AIG—a Global Systemically Important Insurer—was bailed out by the U.S. federal government for $180 billion. 35

34

D. Interbank CDS Market Under the setting of Section V, suppose bishock is a big bank and as the asset market opens banks also have the option of writing and purchasing fully collateralized CDS contract. In a equilibrium network, no bank is the backward neighbor of a small bank and therefore only big banks are reference entities of CDS’s. Suppose for each big bank there correspond a CDS having it as the reference entity and notional value of zb0 (face value of interbank liability of a big bank).36 Let pCDS be the endogenously determined unit price. If θ > 2lb (1), big bank would only choose to be buyers of collateralized CDS, as posting the required collateral reduced liquidity.37 A small bank’s problem in optimally choosing amount of CDS to sell , if any, is then

c,

max ys + c1{sell} · (pCDS − sell, not sell

Dmaximin (p) 0 zb ) + R1{not sell} n

s.t. c(zb0 − pCDS ) ≤ ys . (8)

where c is the amount of CDS sold if a decision is made to sell and Dmaximin (p) maxmin domino effect at legacy asset price p. Therefore, in assessing the probability of default

Dmaximin (p) n

of the reference entity, small banks now not only have to make a

maximin decision with uncertainty regarding its local neighborhood but the entire network structure. The constraint c(zb0 − pCDS ) ≤ ys is a small bank’s budget constraint. For each unit of CDS sold, a collateral of (zb0 − pCDS ) is required of the seller. In equilibrium, small banks are indifferent between writing a of CDS and buying asset. This would be the case 36

For simplicity, we assume face value of interbank liability is uniform across big banks in this part of discussion. 37 In the setting of an exogenous and homogeneous circle network configuration (e.g. Caballero and Simsek (2013)), this would hold for all banks. No bank would choose to sell a CDS contract when given the option because putting up the collateral required would reduced the liquidity buffer. In our heterogeneous environment this applies to only the big banks. Also, in Caballero and Simsek (2013), CDS’s are offered by an ad hoc outside agent who does not have knowledge of the network structure. In our setting, that the CDS market remains interbank. The endogenous supply of CDS contracts is therefore completely determined by the network structure. However, network uncertainty gives rise to negative externality of potential CDS sellers that may still render the CDS market nonviable.

35

when

pCDS > and

ys (pCDS zb0 −pCDS

−

Dmaximin (p) 0 zb ) n

Dmaximin (p) 0 zb n

(9)

= R. For a small bank that only knows its forward

neighbor is a big bank and is considering writing a CDS contract, the maximin worst network structure is that its forward neighbor is part of a core circle of big banks. So Dmaximin (p) = d lθp ne.38 With CDS price satisfying Equation 9, each small bank that writes a CDS chooses c =

ys . zb0 −pCDS

The number m# of small banks that chooses to sell

CDS is determined by the market clearing condition for CDS contracts

m#

zb0

lb (p) ys =n . − pCDS pCDS

θ Together with Dmaximin (p) = d lb (p) ne, this gives the necessary condition

n nlb (p) − θ 1 ys < lb (p), n m# θ

(10)

which must hold for the CDS market to clear. The necessary condition that nlb (p) − θ > 0 means the total liquidity of big banks must be able to absorb the shock for small banks to be willing to insure big banks. If θ is too large, the CDS market would fail. Moreover, as θ increases, Equation 10 is violated, and causing small banks to refuse to supply CDS, short of θ reaching maximum aggregate liquidity level nlb (p). Thus, in conjunction with, for example, Theorem 4(ii.b), an interbank CDS market need not prevent fire sale. While big banks exert positive network externality in the form of a bailout fund, there is an additional layer of negative network externality from small banks. Small banks refuse to provide insurance because they must adopt a maximin view of the global network 38 To be more precise, the maximin worst network structure for a small bank in in assessing the default θ risk is the network structure in the hypothesis of Lemma 4. Replacing d lb (p) ne by the domino effect specified in Lemma 4 yields a more cumbersome expression with the same economic implications.

36

structure—not just for their local neighborhood. In practice, CDS’s are traded by both small and large institutions, with large institutions (e.g. sovereign governments and large banks) as reference entities of most liquid CDS’s.39 Our discussion suggests that, during a financial crisis, smaller financial institutions may shun the CDS market. This may describe, for example, the withdrawal of liquidity that occurred in the market for sovereign CDS’s at the beginning of the European sovereign debt crisis. VII. Policy Implications By considering network formation and stability jointly, our framework can identify how macro-prudential policies implemented through other channels interact with the network formation and contagion processes. The policy recommendations one can extract from our analysis support some measures already undertaken by regulators. In equilibrium, those banks whose default would induce the greatest loss for the network are precisely those on the contagion path—and also precisely those best able to absorb a liquidity shock. These are the big banks. Therefore larger banks should, as reflected in current practice, undergo more regulatory scrutiny.40 It is also a corollary of our analysis that big banks are most exposed to other big banks. Mitigating this concentration risk is the aim of “large exposure regulation” introduced by the Basel Committee on Banking Supervision, which sets limits on the maximum exposure a bank can have to a single counterparty.41 In particular, the maximum exposure limit is more stringent in case of exposure of a global systemically important banks (G-SIB) 39

For example, according to the Depository Trust & Clearing Corporation, during the three years ending in June 2013, there were CDS’s traded on only 13 reference entities among U.S. banking firms. 40 Banks which are designated global systemically important banks (G-SIB’s) by the Financial Stability Board and the Basel Committee on Banking Supervision are required to hold additional capital. G-SIB’s are typically among world’s biggest banks. Currently there are 29 G-SIB’s in designated tiers. For lowest to highest, banks in each tier are required to hold 1%, 1.5%, 2%, 2.5%, and 3.5% additional capital. 41 See BCBS executive summary at https://www.bis.org/fsi/fsisummaries/largeexpos.htm.

37

to another G-SIB.42 Extra regulatory scrutiny due to a bank’s larger size and resulting systemic importance may have the effect of dis-incentivizing banks from becoming too large. A bank that does not have to hold additional capital have more cash to return to its shareholders.43 This regulatory disincentive, however, may be overcome by other market and behavioral forces, such as uncertainty aversion. Indeed, in the United States, the five largest banks now control 47% of banking assets, compared with 44% in 2007.44 The concentration of systemic risk at a few big banks is therefore likely to remain a a feature of the banking system in the foreseeable future. The discussion in Section VI.D of the CDS market also suggests that the liquidity and efficiency of the CDS market may attenuate, or exacerbate in case of their absence, financial contagion. Illiquidity of the CDS market is a symptom of banks’ uncertainty.45 On the other hand, an efficient CDS market reduces uncertainty and lowers the cost for counterparties hedging exposures to a bank. The Dodd-Frank Act of 2009 introduced regulations for the CDS market which phased out the riskiest swaps and required the setting up of a clearinghouse. In addition to central clearing, which reduces counterparty risk, a more central and standardized CDS market may reduce adverse selection and improve efficiency. Our analysis also leads to discussions of potential regulatory or supervisory measures not undertaken by regulators thus far. Consider the hypothetical interbank bailout fund used to demonstrate network externalities in Section VI.C. A potential regulatory measure 42

A large exposure is defined as the sum of all exposures of a bank to a single counterparty that are equal to or above 10% of its Tier 1 capital. The limit is set at 25% of Tier 1 capital. However, in the case of exposure of a G-SIB to another G-SIB, a more stringent limit of 15% of Tier 1 capital applies. 43 For example, Royal Bank of Scotland, which briefly became the largest bank in the world before the financial crisis, with a balance sheet the same size as the German economy, has shrunk significantly and, in its own words, become a “much simpler, safer UK-focused bank.” 44 This is documented in the Federal Reserve Economic Data at https://fred.stlouisfed.org/ series/DDOI06USA156NWDB. 45 For example, according to some observers, CDS’s were at points also symptomatic of the euro zone crisis. The CDS spread increased when investors grew more uncertain. The European Union argued that the effect this had on sentiment would drag down the bonds too. As a response, a permanent ban on naked shorting of CDS’s on sovereign debt has been imposed across the European Union since December 1, 2011.

38

would be to realize the network externalities by instituting such a fund. Such a fund could alleviate the moral hazard problem posed by possible public bailout of too-big-tofail banks.46 A mechanism similar in spirit of an interbank bailout fund currently exists for derivatives clearing houses, which have default funds consisting of contributions from all clearing members to share extreme losses that cannot be covered by margin calls. The setting of our model can be extended to one where the banks anticipate the interbank bailout fund, in addition to their local network information, while making counterparty choices. This would result in a interbank network that minimizes their ex-ante expected contributions to the equilibrium bailout plan, as well as maximizing equity return. One would expect the core-periphery property to be robust with respect to this extension. In our setting where banks face network uncertainty, the financial network of crossexposures emerge when uncertainty averse banks decide the benefits of systematic liquidity risk sharing outweigh systemic fire sale risk.47

From this broader perspective, our

discussion is not restricted to curbing contagion risk. Liquidity in the banking system has direct impact on the real economy. In practice, the policymaker faces a similar tradeoff in prudential regulation. It is clear that systemic risk is monotonically decreasing in liquidity requirement. On the other hand, tighter liquidity requirement constricts banks’ intermediation activity.

Given that in our setting banks already makes the

optimal decision with respect to liquidity holding, it is interesting to consider whether incorporating a regulator would have any material impact on our results. For simplicity, consider a social planner, or regulator, who seeks to minimize the ex ante domino effect in our setting. It can be shown that the social planner’s choice in fact coincides with 46

To credibly eliminate the possibility of a public bailout, it is also necessary to have failure resolution procedures for large banks with complex economic, legal, and regulatory relationships. This is a practical challenge outside the scope of this paper. 47 Section A of the Appendix provides further analysis on incentive for network formation within our setting.

39

the equilibrium network structure characterized by Theorem 1.48 Suppose the structure is exogenously chosen by the social planner and banks still has no information regarding the network structure.49 One can show that the legacy asset price threshold for fire sale is higher compared to the de-centralized setting of Theorem 4.50 This is because a bank’s uncertainty and the number of its decision variables are complementary. When banks no longer have to make counterparty choices, they are more willing to hold on to their assets. Therefore, in addition to providing general support for the Basel III approach based on loss absorbency metrics, our setting illustrates that the regulator still has impact on reducing fire sale risk even under ideal liquidity conditions VIII. Conclusion The core-periphery structure has been a consistent feature of network of interbank cross-exposures since the advent of modern banking. In this paper, we explain how such network structure may arise endogenously and analyze its consequences for financial contagion. We showed that the resulting network is robust-yet-fragile. When banks confront risk (counterparty risk) ex-ante while facing uncertainty (network uncertainty) originating from the counterparties network itself, their individual optimal decisions result in a network that is also socially optimal with respect to financial contagion. However, banks’ preference for liquidity in choosing counterparties also concentrates systemic risk at the core of big banks. Big banks are too-big-to-fail because they are too connected—in equilibrium, only their failure has systemic consequences for the network. 48

While it is natural to conjecture that a complete network would be best able to absorb the shock and therefore the social planner’s optimal choice of network configuration, this is not the case. For example, Acemoglu et al. (2015) shows complete networks are least robust with respect to large liquidity shocks. In our setting, the heterogeneity of banks in scale makes the core-periphery structure the social planner’s optimal choice. 49 One may question the reasonableness of such an assumption. The social planner may choose to share information with banks or banks maybe be able to calculate the structure chosen by a social planner. The possible scenarios in the legacy asset market then reduce to those under first best, in which case, domino effect of a social planner’s best structure is still at least as severe as described in this discussion. 50 The precise result is given in Theorem A-2 in Section G of the Appendix.

40

The endogenous structure under imperfect information provides implied information to rational banks, thereby partially resolving network uncertainty faced by banks. However, financial crisis occurs despite partial resolution of uncertainty, which does not extend to the interbank credit derivative market. Thus the concentration of systemic risk is not mitigated by sharing of credit risk. Multiple layers of network externality exist in our heterogeneous setting, with implications regarding macro-prudential regulations. While recent regulatory attention has been focused on capital buffer requirements of large institutions, our model suggests that network externality be taken into account by policy makers and that the ideal network involves a core-periphery trade-off between liquidity risk and systemic risk. Our model admits a number of possible extensions. For example, one may model the banks’ balance sheet with higher granularity. In practice a bank is exposed to many asset classes, and the correlation between asset classes have contagion implications. While one expects the core-periphery structure to be robust, such an extension can shed light on the relationship between asset correlation and contagion with endogenous network formation. Along a different dimension, one can include the regulator as an additional player in the game, which allows for micro-prudential considerations such as regulatory arbitrage. One can also incorporate shadow banks and consider the interaction between the regulator, banks, and non-bank financial institutions and its implications for systemic risk. Our model can also be tested empirically in a suitable framework.51 We leave these considerations for future research.

51

As a bridge to future research, we provide an appropriate empirical framework for our model in Section H of the Appendix.

41

References Acemoglu, D., Ozdaglar, A., and Tahbaz-Salehi, A. (2015). Systemic risk and stability in financial networks. American Economic Review, 105(2), 564–608. Allen, F. and Gale, D. (2000). Financial contagion. Journal of Political Economy, 108(1), 1–33. Anand, K., Craig, B., and Von Peter, G. (2015). Filling in the blanks: Network structure and interbank contagion. Quantitative Finance, 15(4), 625–636. ´ Friedrich, S., Garratt, R., Halaj, G., Fique, J., Anand, K., van Lelyveld, I., Banai, A., Hansen, I., Jaramillo, S. M., Lee, H., et al. (2018). The missing links: A global study on uncovering financial network structures from partial data. Journal of Financial Stability, 35, 107–119. Babus, A. (2016). The formation of financial networks. RAND Journal of Economics, 47(2), 239–272. Bala, V. and Goyal, S. (2000).

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Caballero, R. J. and Simsek, A. (2013). Fire sales in a model of complexity. Journal of Finance, 68(6), 2549–2587. Calomiris, C. W. and Carlson, M. (2017). Interbank networks in the national banking era: their purpose and their role in the panic of 1893. Journal of Financial Economics, 125(3), 434–453. Calomiris, C. W. and Mason, J. R. (1997). Contagion and bank failures during the great depression: The june 1932 chicago banking panic. American Economic Review, 87(5), 863. Chang, B. and Zhang, S. (2018). Endogenous market making and network formation. Available at SSRN 2600242. Craig, B. and von Peter, G. (2014). Interbank tiering and money center banks. Journal of Financial Intermediation, 23(3), 322 – 347. Demirer, M., Diebold, F. X., Liu, L., and Yilmaz, K. (2018). Estimating global bank network connectedness. Journal of Applied Econometrics, 33(1), 1–15. Elliott, M., Golub, B., and Jackson, M. O. (2014). Financial networks and contagion. American Economic Review, 104(10), 3115–3153. Farboodi, M. (2014).

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Hansen, L. P. and Sargent, T. J. (2001). Robust control and model uncertainty. American Economic Review, 91(2), 60–66. Hasman, A. (2013). A critical review of contagion risk in banking. Journal of Economic Surveys, 27(5), 978–995. Holmstrom, B. R. and Tirole, J. (1998). Private and public supply of liquidity. Journal of Political Economy, 106(1), 1–40. Jackson, M. O. and Wolinsky, A. (1996). A strategic model of social and economic networks. Journal of Economic Theory, 71(1), 44–74. Kiyotaki, N., Moore, J., et al. (1997). Credit chains. Journal of Political Economy, 105(21), 211–248. Nevo, A. (2001).

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Appendix The Appendix provides a number of supplements to the main text. Sections A, B and C discuss relaxation of our model assumptions. Sections D, E, and F provide proofs of results in the main text. Section G provides precise formulation on the impact of social planner (regulator) on fire sale risk, as discussed in Section VII. Section H suggests a suitable empirical framework in which our model can be taken to data.

A. Incentive for Network Formation Because we do not model explicitly bank’s choice between autarky, i.e. holding no interbank debt, and being part of a financial network, the amount of debt a bank wants to hold is exogenously given. Therefore our model describes the characteristics of contagion and fire sale conditional on that banks prefer being part of a cross holding network over autarky. The incentive to form financial links is well studied.52 We take these incentives as given and interpret the banks’ choices in our model as the optimal counterparty among those that can fulfill the original aims, such as hedging, that necessitated the search for counterparty. If a bank is hedging the risk of certain position, it can first select counterparties willing to take on the risk and then choose the optimal counterparty as in our model. Alternatively, one may have a model where banks choose to hold interbank debt for risk sharing reasons, as in Allen and Gale (2000). For example, suppose the n big banks in our model believe each big bank has equal probability of experiencing date 1 liquidity shock of magnitude θ in excess of liquidity buffer lb . A risk sharing arrangement in this case would be a complete network of big banks where each big bank i holds

θ−lb n−1

amount of debt of other big bank −i. In our model, one may then take zb =

θ−lb n−1

to be

the amount of interbank debt a big bank wants to hold. In Allen and Gale (2000), contagion is generated by a surprise state that is assigned zero probability when network formation takes place. In particular, banks do not consider counterparty risk and contagion implications are analyzed by exogenously varying risk sharing network configurations with respect to the surprise state. In contrast, in our environment a financial network would emerge endogenously when parameter values are such that liquidity risk sharing benefits outweigh counterparty risk. The liquidity buffer lb = lb (p) in our model (see Equation 4 and Equation 5) depends on legacy asset price p in the secondary market. When multiple equilibria arise as in Theorem 4(ii.c) and fire sale may or may not occur, the probability of counterparty default need not be zero for a bank in the network. In this line of discussion, systemic fire sale risk emerges as a consequence of systematic liquidity risk sharing in the presence of network uncertainty. 52

See, for example, relevant papers mentioned in the Introduction.

45

B. Number of Forward Neighbors The only substantial restriction we have made on network structure is that each bank can only have one forward neighbor. The restriction of only one forward neighbor only serves to simplify the exposition of our model and relaxing this assumption leads to no material change to our results. For example, we may allow banks to have any number of forward neighbors, with the maximum number of forward neighbors bounded by some upper bound M . In practice, there may be a de facto uniform bound on the number of forward neighbors because banks choose optimal diversification in holding debt claims of other banks against various market frictions. Then, for example, conditional on θ, a bank i’s investment problem to optimally distribute debt holding capacity zs or zb among forward neighbors can be rewritten as: min

Fi1 ,Fi2 ,...,FiM −1 ,z1 ,z2 ,...,zM −1

Pm+n−1 j=1

EFi1 [z1 , j, p, θ, σ] + ... +

Pm+n−1 j=1

EFiM [θ − z1 − z2 − ... − zM −1 , j, p, θ, σ]

(A-1)

m+n−1

where • Fiq denotes the qth forward bank of bank i, • EFiq (zq , j, p, θ, σ) is the expected required liquidity buffer for bank i due to its q-th forward neighbor Fiq if it has holds zq of Fiq ’s debt, conditional on the shock occurs on bank j with legacy asset price p and network structure σ. In this more general setting, the characterization of the optimal network structure is the same as the optimal network structure given in Theorem 1. All banks, small or large, will hold debts claims of M big banks. Each small (resp. big) bank has interbank asset of

zs M

(resp.

zb M ).

The main feature of

an equilibrium network structure—small banks will not have backward neighbors and big banks will hold each other debt claims—remains unchanged. This means that Corollary 1, Lemma 5, and fire sale conditions of Theorem 2 also hold. The domino effect of a liquidity shock terminates once it reaches a small bank, although detailed calculations of magnitude of the domino effect may differ.

C. Heterogeneous Information Our results can be extended to cover scenarios where banks have heterogeneous information regarding the shock. In practice, heterogeneity in information possessed by banks may arise because, for example, big banks have better access to information. With heterogeneous information, the equilibrium legacy asset price plays the additional role of aggregating information. The uninformed banks correct their

46

beliefs upon observing the legacy asset price. For example, consider Theorem 2(ii), which gives fire sale conditions in the case of unanticipated shock. Suppose instead of all banks knowing that a shock will occur at a big bank, only big banks know a shock will occur at a big bank and small banks have no information regarding the shock.53 Once the small banks observe the legacy asset price, they deduce that big banks have flooded the market with legacy assets and rationally infer that a big bank will suffer a shock. Small banks then update their beliefs accordingly. Similarly, in the environment with anticipated shock considered in Section V, we may restrict the ability to anticipate the shock to a subset of banks, e.g. the big banks. The information asymmetry between informed and uninformed banks is more severe in this setting. Informed banks know the magnitude θ and location ishock of the shock for certain while the uninformed banks have no information. The interbank market again plays the role of information aggregator. When the uninformed banks observe assets being traded below their fundamental value, they again infer the information possessed by informed banks. In Theorem 4(ii.b), if small banks have no information regarding the shock (with belief given by Assumption 5(2.b)), the interbank market conveys to them that a big bank will suffer a shock, although the small banks would be unable to distinguish whether all big banks are panicking about a potential large shock (θ > lb (1) + lb (1)) or a few big banks are hoarding liquidity for a localized shock (θ ≤ lb (pscrap ) + lb (pscrap ) but mys + nyb ≤ 2x) in our setting.

D. Proof of Lemma 4 (i) In an equilibrium network structure characterized by Theorem 1, a small bank has no backward neighbor and therefore, in the case that it suffers a liquidity shock, cannot transmit the shock. The resulting domino effect is D(p) = 1. (ii) We argue by induction on the distance (d(j; σ, θ, i)) to the original distressed big bank i. We all also make use of the fact that, in the equilibrium configuration σ, shock transmission path consists of only big banks. The liquidity need of a big bank i, situated in an equilibrium network structure, after suffering a shock can be computed as follows. On the asset side, it will receive full repayment zb [residual] = zb from its forward neighbor. Its total liability is zb + m n zs +θ (interbank liability and senior outside debt). Bank i will choose the precautionary action, Ai 0 = S, and sell all of its legacy assets. If zb +

m n zs

+ θ − zb ≤ lb (p),54

bank i remains solvent and the domino effect is D(p) = 1, which is consistent with Equation (8). If 53

In other words, small banks have the maximally uninformative uniform prior of Assumption 5(2.b) and big banks have the uniform prior restricted to big banks specified by Assumption 5(2.a). 54 Recall that lb (p) denotes the liquidity available to a big bank after selling all of its legacy assets.

47

m n zs

+ θ > lb (p), the original distressed bank is insolvent, and makes total payment of zb + lb (p) − θ <

zb +

m n zs .

We have

D(p) ≥

+

1 |{z}

original distressed bank

+

1 |{z}

big backward neighbor

m n |{z}

.

small backward neighbors

According to Assumption 2, which stipulates equal seniority for interbank claims, the shortfall is equally distributed among its backward neighbors. Each backward neighbor therefore receives m n

The

m n zs +θ−lb (p) 1+ m n

less.

small backward neighbors cannot increase D(p) further banks since they do not have backward

neighbors. The big backward bank j of the original distressed bank (with d(j; σ, θ, i) = 1) receives zb −

m n zs +θ−lb (p) 1+ m n

< zb . Then the liquidity need of this bank would be

(zb + Let θ1 =

m n zs +θ−lb (p) 1+ m n

m zs ) − (zb + n

m n zs

+ θ − lb (p) . 1+ m n

, then the same calculation can be carried out with θ1 in place of θ for the big

bank k with d(k; σ, θ, j) = 1. By induction, if contagion spreads to bank k with d(k; σ, θ, j) = l, the distress/shortfall suffered by bank k is θ−Q −Q t −Q t

−Q t ..l times..

where Q = lb (p) −

m n zs ,

t=1+

m n.

(A-2)

We therefore have

D(p) = (arg max{ k

θ−Q −Q t −Q t

−Q m ≤ Q}) · (1 + ) + 1. ..k times.. n t

Solving the continued fraction expression in Equation A-2 proves the lemma.

E. Proof of Lemma 5 The total excess cash in the interbank network is

T =

m(yb − zb0 − zb ) {z } |

+

total excess cash of big banks

n(ys − zs0 − zs ) | {z }

.

total excess cash of small banks

For a given legacy asset price p, let C(p) and C 0 (p) denote total cash endowment of the D(p) + 1 banks who are distressed and the n + m − (D(p) + 1) banks who are unaffected by the shock respectively. We have

48

C 0 (p) + C(p) = T. Also, since a stressed bank sells all of its legacy assets,

D(p)+1

D(p)+1

X

C(p) =

ik =1,Bik stressed

|

−(

lik (p) {z

}

X

ik =1

p(1 − ys ) · 1Bik small bank + p(x − yb ) · 1Bik big bank ).

the liquidity buffer of stressed banks

Therefore the excess demand for legacy asset is

D(p)+1

D(p)+1

C 0 (p) − (

X

ik =1

p(1 − ys ) · 1Bik small bank + p(x − yb ) · 1Bik big bank ) = T −

X

lik (p).

ik =1,Bik stressed

We need to show the excess demand is nonnegative. Now by the definition of the domino effect, D(p)+1

θ≤ Put θ + E(p) =

PD(p)+1 i=1

X i=1

lik (p).

ik =1,Bik stressed

li (p) as in Assumption 4. Then

D(p)+1

C 0 (p) − (

X

D(p)+1

(1 − yb )(if Bi = s) +

X i=1

(x − ys )(if Bi = b)) = (T − θ) − E(p) ≥ 0

by Assumption 4. This proves the lemma.

F. Proof of Theorem 4 (i.a) We first note that, in the equilibrium network structure, small banks have no backward neighbors and therefore always choose to be a buyer of assets if not subjected directly to a shock. (In contrast, a big bank may choose to sell legacy assets and acquire liquidity buffer to meet its obligations even without being subjected directly to a shock.) Therefore when a shock occurs at a big bank, all small banks choose Aj0 = B. The demand for legacy asset is at least mys (recall the maximum legacy asset price is normalized to 1). There is the largest possible supply, n(x − yb ), of legacy assets in the scenario when all big banks choose Aj0 = S to sell their legacy assets. According to market clearing condition in Equation 3, when

49

aggregate supply ≤ n(x − yb ) ≤ mys ≤ aggregate demand fire sale does not occur in this system. (i.b) There would be at least (m − 1)ys demand of assets, with largest possible supply being n(x − yb ) + (1 − ys ). When n(x − yb ) + (1 − ys ) ≤ (m − 1)ys , there is no fire sale. (ii) According to part(i), a fire sale is possible only when n(x − yb ) > mys , or when mys − 1 < n(x − yb ) ≤ mys and the shock occurs on a small bank. Part(i) shows how rational small banks react to a shock. We now consider how big banks react after a shock and the resulting equilibrium in the secondary asset market. Besides the bank originally distressed by the shock and its direct neighbors, all other big banks would behave in the same way, to sell or to buy. There are then three possibilities, which we consider separately. The first possibility is when a shock θ so small that, even at the lowest possible price of legacy assets pscrap , the amount of assets selling on the secondary market is lower than the demand of assets. Formally, when θ < 2lb (p), a big bank who is not the original distressed bank and not a direct neighbor of the original distressed bank, would choose to buy legacy assets. This is because the worst case under Knightian uncertainty for it is when the shock happens just on its forward neighbor’s forward neighbor and both these two banks have only one backward neighbor (thus no small banks as backward neighbor). The small shock will be absorbed by those two big banks that make up its possible transmission path. If a bank is the original distressed bank, it would choose to sell legacy assets according to our assumption. If a bank is the direct backward neighbor of a distressed bank, it would choose to sell only when θ > lb (pscrap ). Thus, the supply side in secondary market should be less than 2(x − yb ) and the demand side would be larger than mys + (n − 2)yb . Thus, with mys + nyb > 2x, in this case, the price of legacy assets will be p = 1 according to Equation (3) and a domino effect size of 1. The second possibility is a shock so large that even at the highest price of legacy assets, p = 1, the demand of legacy assets is less than the supply of legacy assets. Formally, when θ > 2lb (1), all big banks (including the original distressed bank, the direct forward neighbor of the original distressed bank and other banks) would choose to sell legacy assets. The resulting supply would be larger than n(x − yb ) and demand less than mys . Thus, with the condition n(x − yb ) > mys and mys − 1 < n(x − yb ) ≤ mys when

50

shock happens on a small bank, the equilibrium price is p = pscrap and a fire sale emerges. The third possibility is the intermediate case where the liquidity shock, θ, that can be covered by available liquidity of the two big banks on the transmission path at maximum legacy asset price p = 1, but not at minimum price p = pscrap . In this scenario, there are multiple equilibria. If legacy assets trade at p = 1, the available liquidity lb (1) is sufficiently large that no fire sale emerges, similar to the second possibility above. If, instead, the legacy assets price is at the fire sale level, p = pscrap , then the available liquidity, lb (pscrap ) is sufficiently small to lead a fire sale, in equilibrium.

G. Social Planner’s Network As part of our discussion on policy implications in Section VII, it is stated that, in the presence of a social planner (or regulator), the legacy asset price threshold for fire sale is higher compared to the de-centralized setting of Theorem 4. Here we provide a precise statement of this result. The same notation as in Section IV will be used. The social planner seeks to minimize the ex ante domino effect. His problem is therefore: min E(i,θ) [D(p)]

σ∈O

(A-3)

The social planner’s choice of network structure in fact coincides with the equilibrium network: THEOREM A-1: A social planner, subject to the restriction that the network must be connected, will choose a structure characterized by Theorem 1. ASSUMPTION A-1: When the structure is exogenously given by a social planner, banks has no information regarding the network structure. Under Assumption A-1, banks can form (possibly multiple) beliefs on the network structure under Knightian uncertainty, as in the de-centralized setting. The implications on fire sale in the legacy asset market is characterized below. THEOREM A-2 (General Equilibrium Under Social Planner): i Suppose n(x − yb ) ≤ mys —in particular, if all the big banks choose to Ai0 = S, there will not be a fire sale. Then no fire sale results under optimal network structure. ii If n(x − yb ) ≥ mys ,

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(ii.1) (Unique fair price equilibrium) If θ ≤ lb (pscrap ) + ls (pscrap ), then there is a unique equilibrium with price p = 1 where fire sale does not occur, with domino effect of D(p) = 1. (ii.2) (Unique fire sale equilibrium) If θ > lb (1)+ls (1), there is a unique equilibrium price of p = pscrap , where fire sale occurs, and the aggregate amount of new asset purchases equals 0. (ii.3) (Multiple equilibria) If lb (1) + ls (1) ≥ θ > lb (pscrap ) + ls (pscrap ), then there is a fair price equilibrium and a fire sale equilibrium. When comparing with the general equilibrium under endogenous network structure stated in Theorem 4, the legacy asset price threshold for fire sale is higher.

H. Empirical Framework In this section, we provide an empirical framework that could be used to test and verify the main predictions of our model. General Model Counterparty decision problem could be viewed as a demand problem in the interbank borrowing market, and the observed counterparty decisions is the equilibrium outcome of the interbank demand system. Such a setting lends itself naturally to a discrete choice model, where a bank obtains different levels of utility from choosing different counterparties.55 Specifically, let Uijt be the indirect utility bank i would derive from lending to bank j in period t. We consider the model

Uijt = Xjt βi + αi pjt + γi Ljt + ζi Cjt × Ljt + ξjt + εijt ,

(A-4)

where • Xjt is bank j’s characteristics observable from data, • pjt is the price (interest rate) of the loan that bank j accepts in period t, • Ljt is the liquidity measure of bank j in period t, • Cjt is complexity measure of bank j in period t, where larger value means higher network uncertainty complexity (from the perspective of bank i), • ξjt is bank j’s unobservable characteristics, 55

See Berry et al. (1995) and Nevo (2001) for an overview of discrete choice models.

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• εijt is the error term. The bank i specific coefficients (αi , βi , γi ) are be estimated and further modeled as     αi  α      βi   β        =   + ΠDi + Σvi , vi ∼ N (0, I)  γ  γ   i       ζ ζi

(A-5)

where Di (resp. vi )and are observed (resp. unobserved) characteristics of bank i. The error term vi is assumed to have standard multivariate normal distribution. Bank i will choose bank j as counterparty if

Uijt ≥ Uikt , ∀k. This model captures bank heterogeneity and various factors which influence the lending decision of a bank, both observed and unobserved. ξjt captures all those unobserved or hard to measure characteristics of bank j—e.g. the ideal loan amount bank j would like to borrow from the interbank market. εijt may capture effects such as the relationship between bank i and bank j. Di are those observed characteristics that affect bank i’s valuation of different counterparties, such as its own liquidity. vi are those unobserved characteristics that affect bank i’s valuation of different counterparties. Empirical Hypothesis The main empirical hypothesis are the following: Hypothesis 1: γi is positive and statistically significant, for all i. Hypothesis 2: ζi is positive and statistically significant, for all i. Hypothesis 1 corresponds to our model prediction that banks prefer counterparties who possess more liquidity. The interpretation of γi is how much bank i would like to yield to avoid one unit contagion risk, or how much interest rate bank i would like to give up in exchange to one unit of higher liquidity of bank j. Hypothesis 2 corresponds to our model prediction that banks places higher value on liquidity of counterparty when face higher network uncertainty. The coefficient ζi of interaction term Cjt ×Ljt reflects how bank i’s value of bank j’s liquidity change when facing uncertainty regarding bank j’s network.

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Data Requirement The data required to consistently estimate the model consists of loan-level data and balance-sheet data. The loan level data should contain the following variables: loan date, borrower and lender, loan amount and loan price (interest rate). The balance sheet data should contains the following variables: cash and cash equivalent, receivables, deposit, payable, market value, identity of counterparty. Liquidity Measure Ljt Using appropriate data, a suitable liquidity measure Ljt can be constructed as follows: Ljt = Ctj + Rtj − Dtj − Ptj + LiAsjt , where subscript t denotes when the loan is borrowed, C is cash and cash equivalent, R is receivables, D is deposit, and P is payable from bank j’s balance sheet. LiAs denotes “liquidable assets” which is defined as Other Liquidable Assets × Market Price − Liquidation Cost. Consistent with our uncertainty setup in the model, the market price of those liquidable assets should be the expected market prices when shock hits, although book value may be a reasonable proxy. Complexity Measure Cjt As a proxy measure for the uncertainty faced by bank i when considering the financial network originating from a potential counterparty j, we suggest using the number of interbank counterparties of bank j on the balance sheet in period t. Endogeneity of Ljt A potential problem for consistent estimation is that ξjt can be correlated with Ljt , which leads to bias in γˆi . For example, a bank’s liquidity could be correlated with the amount it would like to borrow. A less liquid bank could be undesirable as counterparty because it cannot borrow enough, in addition to being less liquid. To solve this endogeneity problem and isolate the partial effect of liquidity, one needs to consider instruments for the liquidity measure Ljt . A candidate instrument is the average liquidity of other banks in the market. This may be exogenous but not have sufficient variation. More deliberately, one can use the average liquidity of other banks which are not closely linked to bank j.

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Counterfactual Price pjt Unlike others settings where discrete choice models are applied to estimate demand systems, interbank loans are bilateral contracts negotiated between banks and we can only observe the equilibrium price (interest rate) for a loan, not the counterfactual price had a bank i borrowed from a different counterparty k. One potential solution is to use the average of the prices bank k loans to the other banks as a counterfactual price. Or, if more data is available, one can use the average of the prices bank k loans to other banks which have similar characteristics as bank i, the borrower we are looking at. Estimation Assuming availability of data, standard estimation techniques can be adapted to our model and test the specified hypothesis. Estimation of the model can Then, we can resort to the standard solution used in the literature to estimate the model and conduct the hypothesis test.56

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See, for example, Nevo (2001).

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