The formation of financial networks

• Published date: 01 May 2016
• Author: Ana Babus
• DOI: http://doi.org/10.1111/1756-2171.12126
• Date: 01 May 2016

RAND Journal of Economics

Vol.47, No. 2, Summer 2016

pp. 239–272

The formation of financial networks

Ana Babus∗

Modern banking systems are highly interconnected. Despite various benefits, linkages between

banks carry the risk of contagion. In this article, I investigate whether banks can commit ex ante

to mutually insure each other, when there is contagionrisk in the financial system. I model banks’

decisions to share this risk through bilateral agreements. A financial network that allows losses

to be shared among various counterparties arises endogenously. I characterize the probability

of systemic risk, defined as the event that contagion occurs conditional on one bank failing, in

equilibrium interbank networks. I show that there exist equilibria in which contagion does not

occur.

1. Introduction

The recent turmoils in financial markets have revealed, once again, the intertwined nature of

financial systems. In a modern financial world, banks and other institutions are linked in a variety

of ways. These connections often involve trade-offs. For instance, though banks can solve their

liquidity imbalances by borrowing and lending on the interbank market, they expose themselves,

at the same time, to contagion risk. How do banks weigh these trade-offs, and what are the

externalities of their decisions on the financial system as a whole?

In this article, I explore whether banks can commit ex ante to mutually insure each other,

when the failure of an institution introduces the risk of contagion in the financial system. As a form

of insurance, banks can hold mutual claims on one another. These mutual claims are, essentially,

bilateral agreements that allow losses to be shared among all counterparties of a failed bank. The

more bilateral agreements a bank has, the smaller the loss that each of her counterparties incurs.

Contagion does not take place provided each bank has sufficiently many bilateral agreements. I

model banks’ decisions to share the risk of contagion bilaterally as a network formation game.

Various equilibria arise. In most equilibria, the financial system is resilient to the demise of

some banks, but not the others. Equilibria in which there is no contagion can be supported as

well. Moreover, I show that the welfare in equilibrium interbank networks is decreasing in the

probability that contagion occurs. However, more bilateral agreements between banks do not

necessarily improve welfare beyond the point when there is no contagion.

∗Federal Reserve Bank of Chicago; anababus@gmail.com.

I am grateful to Franklin Allen, Douglas Gale, two anonymous referees, and the Editor for very useful comments and

guidance. The views in this articles are solely those of the author and need not represent the views of the FederalReserve

Bank of Chicago or the Federal Reserve System.

C2016, The RAND Corporation. 239

240 / THE RAND JOURNAL OF ECONOMICS

To study these issues, I build on the framework proposed by Allen and Gale (2000). In

particular, I consider a three-period model, where the banking system consists of two identically

sized regions. Banks raise deposits from consumers who are uncertain about their liquidity

preferences, as in Diamond and Dybvig (1983). Each region is subject to liquidity shocks driven

by consumers’ liquidity needs. Liquidity shocks are negativelycor related across the tworegions.

In addition, there is a small probability that one of the banks, chosen at random, is affected by an

early-withdrawal shock and liquidated prematurely.

Banks can perfectly insure against liquidity shocks by exchanging interbank deposits with

banks in the other region. However, the connections created by swapping deposits expose the

system to contagion when the early-withdrawalshock realizes. The loss that a bank induces when

she is affected by the early-withdrawal shock is shared across her counterparties. This implies

that as banks exchange more deposits, the loss on every deposit is smaller. The model predicts a

connectivity threshold abovewhich contagion does not occur. Reaching this connectivitythreshold

may require that banks swap deposits with other banks in the same region. I distinguish between

aliquidity network, that smoothes out liquidity shocks in the banking system, and a solvency

network, that provides insurance against contagion risk.

The distinction between liquidity links and solvency links is useful to study the incentives

that banks have to insure against contagion. In particular, I study whether banks choose to form

solvency links with other banks in the same region. When a bank has at least as many links

as the connectivity threshold requires, then no contagion takes place if she is affected by the

early-withdrawal shock. However, her counterparties still incur a loss on their deposits. When

deciding to form solvency links, banks are willing to incur a small loss on their deposits, if they

can avoid default. However, they are better off if contagion is averted without incurring any cost.

This implies that banks have the incentive to free-ride on others’ links. Because of this, many

network structures can be stable in which full financial stability is not necessarily achieved.

In the main specification of the model, I show that at least half of the banks have swapped

deposits with sufficiently manyother banks in the same region. In other words, a systemic event in

which all banks default if one bank is subject to an early-withdrawal shock occurs in at most half

of the cases. Even then, I find that insuring against regional fluctuations in the fraction of early

consumers through a liquidity network is optimal as long as the probability of the early-withdrawal

shock is sufficiently small.

The equilibria in which all banks have sufficientlymany links and no contagion occurs have

the highest associated welfare. This is intuitive, as banks’ assets are inefficiently liquidated if

a systemic event occurs. However, in interbank networks in which contagion does not occur,

welfare is not necessarily increasing with the number of links that banks have. This is because

when each bank has more links, there are also more banks that incur losses on their deposits.

However, because banks have already sufficiently many links, there is no benefit to offset this

implicit cost.

Thus, increasing the connectivity of the interbank network is beneficial up to the point when

there is no contagion in the financial system. The idea that an interconnected banking system

may be optimal is supported by various other studies. Leitner (2005) discusses how the threat of

contagion may be part of an optimal network design. His model predicts that it is optimal for

some agents to bail out other agents, to prevent the collapse of the whole network. This form

of insurance can also emerge endogenously, and I show that it is an equilibrium in a network

formation game. Linkages between banks can also be efficient in the model of Kahn and Santos

(2008) if there is sufficient liquidity in the financial system. Recently, Acemoglu, Ozdaglar, and

Tahbaz-Salehi (2015) find that the types of financial networks that are most prone to contagious

failures depend on the number of adverse shocks that affect the financial system.

The rationale for why a bank is willing to form solvency links with other banks in the

same region and incur a loss on her deposits is that an early-withdrawal shock to any bank can

have system-wide externalities. In particular, all banks default when a bank that has insufficient

links is affected by an early-withdrawal shock. Banks are willing to pay a premium (i.e., incur a

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BABUS / 241

loss on their deposits) to avoid defaulting by contagion. Thus, a solvency interbank network can

be interpreted as an alternative to formal insurance markets. Moreover, the network formation

approach provides insights about the circumstances in which banks are willing to purchase

protection, as well as about the premia theyare willing to pay. From this perspective,the findings in

this article complement solutions relying on formal insurance arrangements previously proposed

in the literature. For instance, Zawadowski (2013) shows, in the context of over-the-counter

(OTC) traded contracts, that competitive insurance markets fail as banks find the premia for

insuring against counterparty default too expensive. This is because banks do not internalize

that the default of another bank, which is not an immediate neighbor, can nevertheless affect

them in subsequent default waves. Insurance is unattractive in Kiyotaki and Moore (1997) as

well, because of limited enforcement of contracts. Similarly, it has been shown that other formal

arrangements, such as clearinghouses, may increase systemic risk either because they reduce

netting efficiency (as in Duffie and Zhu, 2011) or they reduce dealers’ incentivesto monitor each

other (as in Pirrong, 2009).

Starting with Allen and Gale (2000) there has been a growing interest in how different

network structures respond to the breakdown of a single bank to identify which ones are more

fragile. See, for instance, the theoretical investigation of Freixas, Parigi, and Rochet (2000)

or Castiglionesi and Navarro (2011), and the experimental study of Corbae and Duffy (2008).

In parallel, the empirical literature has looked for evidence of contagious failures of financial

institutions resulting from mutual claims they have on one another and has showns uch interbank

loans are unlikely to lead to sizable contagion in developed markets (Furfine, 2003; Upper and

Worms,2004). Other ar ticles are concerned with whether interbank marketsanticipate contagion.

For instance, in Dasgupta (2004), contagion arises as an equilibrium outcome conditional on the

arrival of negative interim information, which leads to coordination problems among depositors

and widespread runs, whereas Caballero and Simsek (2013) provide a model of market freezes

when the complexity of a financial network increases the uncertainty about the health of trading

counterparties and of their partners. More recently, Alvarez and Barlevy (2014) study mandatory

disclosure of losses at financial institutions which are exposed to contagion via a network of

interbank loans.

This article is organized as follows. Section 2 introduces the model in its generality.Section

3 describes when contagion can occur and the payoffs that banks receive. Section 4 provides the

equilibrium analysis. In Section 5, I present an extension of the model to include small linking

costs and discuss welfare implications. Section 6 concludes.

2. The model

Consumers and liquidity preferences. The economy is divided into 2nsectors, each

populated by a continuum of consumers. Consumers’ preferences are described by a log-utility

function. There are three time periods t=0,1,2. Each agent is endowed with one unit of

consumption good at date t=0. Agents are uncertain about their liquidity preferences: they can

be early consumers, who value consumption only at date 1, or they can be late consumers, who

value consumption only at date 2.

The probability that an agent is an early consumer is q. I assume that the law of large

numbers holds in the continuum, which implies that, on average, the fraction of agents that value

consumption at date 1 is q. However, each sector experiences fluctuations of early withdrawals.

With probability 1/2, in each sector there is either a high proportion, pH, or a low proportion,

pL, of early consumers, so that q=pH+pL

2. In particular, the economy consists of two regions,

A={1,2,...,n}and B={n+1,n+2,...,2n}, such that fluctuations in the fraction of early

consumers are perfectly correlated within each region and negatively correlated across regions.

That is, when sectors in region Areceive a high fraction, sectors in region Breceive a low fraction,

and the other way around.

Aggregate early-withdrawal shocks can affectthe economy with a small, but positive, prob-

ability ϕ. In this case, the average fraction of early consumers is higher than q. For tractability, I

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