Systemic Risk, Interbank Relations and Liquidity
Provision by the Central Bank*
Xavier Freixas
Universitat Pompeu Fabra

Bruno Parigi
University of Padua

Jean-Charles Rochet
Université des Sciences Sociales de Toulouse

Abstract
The possibility of a systemic crisis affecting the major financial markets has
raised regulatory concern all over the world. The contagion from one institution
to another stems mainly from the existence of a network of financial contracts
generated from three sources: the payment systems, the interbank market and themarket for OTC derivatives.
In a model a la Diamond and Dybvig, we introduce liquidity endogenously by
assuming that agents have to make payments in a location different from where
they have their deposits. This provides the need for a payment system or aninterbank market.
We show that, under normal conditions the payment system achieves its taskof reducing the opportunity costs of holding liquid assets. However, we alsoestablish that the payment system is prone to experience gridlocks, which weview as pure liquidity crisis. In this case the Central Bank’s role is simply to act
as a coordinating device.
If returns are uncertain (so that solvency problems can arise) but publicly
observable (so that these solvency problems can be separated from liquidity
problems) lending from the Central Bank becomes crucial for the stability of theentire banking system. Our model allows us to characterize the too-big-to-fail(TBTF) approach often followed by Central Banks in dealing with financial*
Very preliminary and incomplete draft. Comments welcome. Financial support from the
Fondation Banque de France and the Bank of England is gratefully acknowledged. We would like tothank Elu von Thadden, Warren Weber and the audiences at the Universities of Venice, Bologna,
Padua and the Bank of Japan Conference on Systemic Risk for useful comments. The usual
disclaimer applies.

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Xavier Freixas, Bruno Parigi, and Jean-Charles Rochet

distress of money center banks, i.e. banks occupying key positions in the payment
system.

1

Introduction

The possibility of a systemic crisis affecting the major financial markets has
raised regulatory concern all over the world. Whatever the origin of a financial
crisis, it is the responsibility of the regulatory body to provide adequate fire walls
for the crisis not to propagate to other institutions through the credit channel. The
contagion from one institution to another stems mainly from the existence of a
network of financial contracts generated from three sources: the payment systems,
the interbank market and the market for OTC derivatives. Since these contracts
are essential to the financial intermediaries’ function of providing liquidity andrisk sharing to their clients, the regulation of financial markets has to set limits tosystemic risk and patterns for central bank intervention when confronted with asystemic shock. In recent years, the 1987 stock market crash, the Saving andLoans crisis, the Mexican, Asian and Russian crises and the crisis of the hedge
fund Long Term Capital Management have all shown the importance of theintervention of the central banks and of the international financial institutions inorder to reduce the extent, contagion possibilities and effects of the crisis.1
But the intervention of the lender of last resort (LOLR) is costly, both ex ante,
because of the wrong incentives it provides and, ex post, because of the cost oftransferring resources to the institutions in financial distress. For this reason, there
is today a clear consensus on the need to restrict the LOLR intervention to events
that endanger the stability of the financial system.
In contrast with the importance of these issues the theory has not succeeded
yet in providing a convenient framework to analyze systemic risk so as to derive
how the interbank markets and the payment system should be structured and what
the LOLR role should be. This lack of theoretical set up has been an additional
motivation to examine systemic risk in a model with an endogenous demand for
liquidity channeled through the payment system. We analyze payment networks,
focusing on the possible liquidity shortages and the coordinating role of theCentral Bank in avoiding and solving them. More specifically, the main issue wewant to address here concerns the effects of systemic risk and contagion through1
A well known episode of near gridlock and of the coordinating role of the Central Bank isrepresented by the series of events the day after the stock crash of 1987. Brimmer (1989 pp.14-15)
writes that “On the morning of October 20, 1987, when stock and commodity markets opened,
dozens of brokerage firms and their banks had extended credit on behalf of customers to meetmargin calls, and they had not received balancing payments through the clearing and settlement
systems. [...] As margin calls mounted, money center banks (especially those in New York, Chicago,and San Francisco) were faced with greatly increased demand for loans by securities firms. With aneye on their capital ratios and given their diminished taste for risk, a number of these banks became
increasingly reluctant to lend, even to clearly creditworthy individual investors and brokeragefirms.[...] To forestall a freeze in the clearing and settlement systems, Federal Reserve officials(particularly those from the Board and the Federal Reserve Bank of New York) urged key money
center banks to maintain and to expand loans to their creditworthy brokerage firm customers.”

Systemic risk, interbank relations and liquidity provision by the central bank183

the payment system in an environment where both liquidity shocks and solvency
shocks affect the banks’ performance.
Systemic risk has been analyzed in the literature from two different perspectives: one has focused on interbank markets and payment systems, and the otheron the LOLR role. There are by now some contributions to the economics of theinterbank market and payment systems,2 which have shown, in particular, howimportant contagion risk could be (Humphrey 1986, Rochet and Tirole 1996) andwhat are the trade-offs between the different payment systems (Freixas and Parigi1998). The literature on systemic risk has provided an interesting discussion of
the LOLR intervention. First, there is a clear consensus on one point: from ahistorical perspective the creation of central banks has helped to avoid bankpanics (Miron 1986, Eichengreen and Portes 1987, Bordo 1990). However, therole of the LOLR in a developed financial system has been and still is the objectof an intense debate. On the one hand, it is argued that, if the problem is a liquidity shortage, the LOLR functions should be restricted to open market operations(Goodfriend and King 1988, Schwartz 1995). On the other hand, the opposite
view holds that the LOLR should lend to solvent institutions as part of its responsibility to maintain the financial stability, and this is an additional instrument ofcrisis resolution. But even within this view, some would argue that the LOLR
should only intervene to avoid a systemic crisis while others would advocate a far
more active role in preventing any crisis contagion.
The function of the LOLR has not always been exercised by the Central
Banks. In the U.S. before the creation of the Federal Reserve System the Commercial Bank Clearinghouses (CBC) had a key role in providing liquidity duringbank panics. According to Gorton (1985), among the most significant actions of
the CBC during a bank panic was the issuance of loan certificates. Loan certificates were liabilities of the clearinghouse that member banks could use in theclearing process and circulate as currency. These loan certificates, issued up to a
fraction of the market value of the assets of the member bank seeking them, werein effect fiat money of the clearinghouse. In a banking panic the CBC ceased tobehave as an authority regulating competitors in market-like setting, and instead
effectively combined the member banks into a single organization, with the group
accepting corporate liability for the debts of each individual member. Manyobservers (White 1983, Gorton 1985, Timberlake 1978, 1993) pointed out that theFederal Reserve System was a development of the existing clearinghouse associations. An early example of a self-regulating bank clearing system was the
Suffolk System that operated in New England between the 1820s and the 1850s(Calomiris and Kahn 1996). Peer monitoring and peer regulation among participants in the system were important elements in the success of the Suffolk System.
Calomiris and Kahn (1996) report that while saving on transaction costs in checkclearing was the main early motivation of this and other clearinghouses, coordi2
For a survey of the main theoretical issues see Berger, Hancock and Marquardt (1996) and foran analysis of the main institutional aspects see Summers (1994).

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Xavier Freixas, Bruno Parigi, and Jean-Charles Rochet

nation of their actions and provision of mutual insurance during financial panics
later became important as well.
In our paper we introduce liquidity endogenously by assuming that agents
have to make payments in a location different from where they have their deposits.
This provides the need for a payment system or an interbank market.3 In this way
we extend the payment system model of Freixas and Parigi (1998) to more than
two banks, to different specifications of travel patterns and consumers’ preferences and to different commitment possibilities in interbank agreements. The
focus of the two papers is different. Freixas and Parigi consider the trade-off
between gross and net payment systems. In the current paper we concentrateinstead on Central Bank policy issues. This paper is also related to two articles byFreeman (1996a,b). In these articles, demand for liquidity is driven by the mismatch between supply and demand of goods by spatially separated agents thatwant to consume the good of the other location, at different times. If agents’travel patterns are not perfectly synchronized, a centrally accessible institution(e.g. a clearing house) may arise to provide means of payments in order to clear
the debt issued by the agents to back their demand. In our paper instead liquidity
demand arises from the strategies of agents with respect to the coordination of
their actions.
We show that, under normal conditions the payment system achieves its taskof reducing the opportunity costs of holding liquid assets. However, we alsoestablish that the payment system is prone to experience gridlocks, which weview as pure liquidity crises. A gridlock occurs, for example, in an interbank
netting scheme, when the depositors in one bank withdraw all their deposits forfear that the other banks will not be able to honor their netting obligations if theirdepositors have withdrawn all their wealth. Thus, even in the absence of solvency
problems, gridlocks can arise from a coordination failure: depending on thepayment networks that link banks, it might be optimal for all depositors to withdraw, if a sufficiently large fraction of deposits in other banks is withdrawn. In
this case the Central Bank’s role is simply to act as a coordinating device. If
returns are uncertain (so that solvency problems can arise) but publicly observable (so that these solvency problems can be separated from liquidity problems)
lending from the Central Bank may be necessary to avoid systemic risk. In its
Lender of Last Resort capacity the Central Bank can borrow from banks withexcess resources and lend to banks facing liquidity shortages.
When depositors have asymmetric payment needs across space, the role of thelocations where many depositors want to access their wealth (money center
locations) becomes crucial for the stability of the entire banking system. Ourmodel allows us to characterize the too-big-to-fail (TBTF) approach often followed by Central Banks in dealing with financial distress of money center banks,i.e. banks occupying key positions in the payment system.
3
Payment needs arising from agents’ spatial separation with limited commitment and default
possibilities were first analyzed in Townsend (1987).

Systemic risk, interbank relations and liquidity provision by the central bank185

It is worth pointing out that, although the interpretation of the model that weoffer is in terms of payment systems, our model is general enough to encompass
other forms of arrangements between financial intermediaries that fund long term
investments with short term liabilities.
This paper is organized as follows. In section 2 we set up the basic model ofpayment systems. In section 3 we characterize the equilibria when the returns of theinvestment are certain and we describe the intervention of the Central Bank. In
Section 4 we allow for stochastic returns of the investment and we allow for different interbank commitments to credit lines. Section 5 offers some concluding remarks.

2

The model

2.1

Basic set up

We consider an economy with 1 good and N locations with a perfectly competitive bank (that can be interpreted as a mutual bank) in each location. There is acontinuum of consumers of equal mass (normalized to one) in each location.
There are three dates: t = 0, 1, 2. The good can be either stored from one period
to the next or invested. Each consumer is endowed with one unit of the good at
t = 0. Consumers cannot invest directly but can deposit their endowment in thebank in their location which stores it or invests it for future consumption. The
storage technology yields the riskless interest rate which we normalize at 0. The
investment technology yields a gross return R at t = 2, for each unit invested at
t = 0, with R > 1, if not liquidated at t = 1. If a fraction (1 – x) of investment isliquidated at t = 1 the gross return at t = 1 is (1 – x) a, with 0 < a <1, and xR at
t = 2. The bank offers depositors a contract that allows them to choose whether towithdraw at t = 1 or at t = 2. To finance withdrawals at t = 1 the bank liquidates
part of its investment.
Consumers receive utility from consumption at t = 2 only. To introduce
endogenous payment needs across space we assume that the consumers must
consume at t = 2 in other locations in various proportions to be specified later.
The above information is common knowledge. Our model is in the spirit of
Diamond and Dybvig’s (1993) (hereafter D-D). In D-D risk-averse consumers are
subject to a preference shock as to when they need to consume. The risk-neutralbank provides insurance by allowing them to withdraw at t = 1 but exposes itself
to the risk of bank runs since it funds an illiquid investment with demand deposits.
Our model corresponds to a simplified version of D-D where all and only thepatient consumers must consume in the other location(s) and the proportion of
impatient consumers is zero in the limit. This allows us to concentrate on theissue of payments across locations without analyzing intertemporal insurance.
Our focus is on the coordination of the consumers of the various locations, not on
the coordination of the consumers in the same location.
Since we focus on payment systems the good should be interpreted as cash

186

Xavier Freixas, Bruno Parigi, and Jean-Charles Rochet(i.e. Central Bank money). Cash is a liability of the Central Bank that can be
moved at no cost, only by the Central Bank.4
The sequence of events takes place within a 24-hour period. We interpret t = 0
as the beginning of the day, t = 1 as intraday, t = 2 as overnight and R as (oneplus) the interbank (overnight) rate. One can interpret the liquidation cost (1 – a)as the cost of (fire) selling monetary instruments in an illiquid intraday market.5

2.2

General formulation of consumption across space

Travel patterns, that is, which depositor travels to which location, are exogenously determined by nature at t = 1 and can be interpreted as corresponding toconsumption needs arising from other aspects of agents’ economic activities. For
each depositor initially at location i, nature determines in which location j(i ¹ j)
he or she will consume at t = 2. To consume at t = 2 at location j the depositors
at location i can carry the good costlessly by themselves from location i to location j. The implicit cost of transferring the good across space is the foregone
investment return. This motivates the introduction of credit lines between banks
to minimize the amount of good not invested. The credit line granted by bank j tobank i gives the depositor of bank i going to bank j the right to have their deposits
transferred to location j and obtain their consumption at t = 2 as a share of theassets at bank j at date t = 2.
A way to visualize the credit line granted by bank j to bank i is to think that
consumers located at i arrive at location j at t = 2 with a check written on bank iand credited in an account at bank j. Bank i, in turn, gives credit lines to one ormore banks as specified below.6 At t = 2 the banks compensate their claims andtransfer the corresponding amount of the good across space. The technology totransfer the good at t = 2 is available for trades between banks only.
An additional assumption is needed to make explicit the assets and the liabilities resulting from interbank relations. We choose the simplest sharing rule,
namely:Assumption 1. All liabilities have the same priorities at t = 2.
This rule defines how to divide a bank’s assets at t = 2 among the claim
holders. It implies that credit lines are honored in proportion of the amount of theassets of the bank at date t = 2. In particular if Di is the ex post return on a deposit
Total Assets of Bank i. As we will see, more complex priority rules
in bank i, then Di = Total
Liabilities of Bank i
could be more efficient in the resolution of liquidity crises. However, we assume
that they are not feasible in our context: this is a reduced form assumption aiming
at capturing the limitations on the information that is instantaneously available inpayment systems.
4
Notice, though, that this implies that currency crises, which are often associated with systemic
risk, are left out of our analysis. This is so because “cash” is then limited by the level of reserves of
the Central Bank.
5
Since banks specialize in lending to information-sensitive customers, a can also be interpreted
as the cost of selling loans in the presence of an Akerlof’s lemons problem.
6
For a similar characterization of trade credit chains see Kiyotaki and Moore (1997).

Systemic risk, interbank relations and liquidity provision by the central bank187

Let pij be the measure of depositors travelling from location i to j. By definition, we have Si¹j pij = 1. Yet, for the sake of simplicity, unlike otherwise specified,we will impose the additional restrictions Sj¹i pij = 1. In this way we discard the
supply and demand imbalances at a specific location as the cause of a disruption in
the payment system or in the interbank market.
Because of the complexity of the transfers involved in the matrix (pij)i,j, wewill illustrate our findings in two symmetric extreme cases.
· In the first one pij = 1 if j = i + 1 and pij = 0 otherwise.7 To visualize, one
may think that the consumers are located around a circle as in Salop’s(1979) model. Consumers from i arrive at location i + 1, the clockwise adjacent location, where they must consume at t = 2. The payment structure implied by this travel pattern generates what we will define as credit chain interbank funding, when the bank at location i + 1 allows the incoming depositors from location i to withdraw at time t = 2.
1

· In the second travel pattern, pij = N – 1 , with i ¹ j. Each two banks swap N 1– 1
1of their consumers so that at time t = 2 at location j there are N – 1 consumers from each of the other N – 1 locations. We will refer to this perfectly
symmetric case as the diversified lending case.
With credit chain interbank funding, credit flows in the direction opposite totravel. With diversified lending every bank gives credit lines to all other N – 1
banks. In terms of alternative payment mechanisms, the interbank credit described above can be interpreted as a compensation scheme (net system) or a
RTGS with multilateral credit lines.

3

Certain investment returns

We first analyze the equilibria of the game when investment returns are certain, sothat the only issue is coordination among agents. We begin by introducing theplayers of the game, namely the N banks and their depositors. At t = 0 banks
decide whether to extend each other credit lines. In the absence of credit lines, noinvestment is possible, hence consumption at location j at t = 2 equals 1: this is
the allocation in what we call the autarkic situation. On the other hand, in thegeneral case with credit lines, the value of t = 2 consumption at location j isdetermined by a non-cooperative game played by the banks’ depositors. At t = 1
each depositor located at i and travelling to j simultaneously and without coordination determines the fraction xij of illiquid assets L that he wants to maintaininvested. For simplicity we take L = 1, i.e. the bank invests all its deposits at t = 0.
Accordingly, the percentage of investment remaining at the location of destination
j will be given by
7

With the convention N + 1 = 1.

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Xavier Freixas, Bruno Parigi, and Jean-Charles Rochet

X j º å p jk x jk
k¹ j

Because of Assumption 1, each consumer (i, j) now shares the good obtained
from liquidation at bank i (i.e. (1 – xij)a) plus a proportion xij of the t = 2 assets ofbank j. Our objective will be to find the equilibrium solutions of the depositors’
game, that is the matrix of equilibrium strategies x* º (xij)i,j. The expected utilityof the depositors of bank i (before they know their final destination) is thus:
U i ( x) = å p ij D j
j ¹i

(1)

To determine Di, consider the t = 2 balance sheet equation for bank i,öæ
X i Ri + å p ji x ji D j = ç å p ji x ji + X i ÷ Di
÷
ç j
j
ø
è

(2)

where the LHS (RHS) represents the assets (liabilities) of bank i: XiRi is the returnon its investment in place, SjpjixjiDj are the credits due from other banks,(Sjpjixji)Di are its deposits, XiDi is the debt with other banks. By convention, Di = 0
when x i j = 0 for all j. The ra tional beha vio r of each depositor (i, j) is
xij = 1 Û Dj > a. Iti only depends on j. Therefore, to simplify the notation denote
xij = xj and bi = xi x X . Equation (2) becomes
+

i

æ
ö
Di - b i ç å p ji D j ÷ = (1 - b i ) Ri
ç j
÷
è
ø

(3)

or in matrix form (I – BP') D = (I – B)R where P' is the transpose of P = (pij)i,j, B
is a diagonal matrix with coefficients b1, …, bN, D = (D1, …, DN)' and R = (R1, .., RN)'.

3.1

Characterization of equilibria

For a given strategy vector (xij)i,j one can compute the assets in place at bank i (Xi)and the return on a deposit at bank i (Di) . Therefore the optimal strategy, x*ij (i.e.
its best response to Xj) is:ì1 if D j > a
xij* = íî1 if D j < aAs it is intuitive, the best response x*ij depends on the amount of liquidation
experienced by bank j in equilibrium. Disregarding the mixed strategy equilibria
where depositors are indifferent between withdrawing their deposits and transferring them to the recipient banks, we obtain the following result.
Proposition 1 The inefficient situation where for all i,j x*ij = 0 is always anSystemic risk, interbank relations and liquidity provision by the central bank189

equilibrium (Speculative Gridlock Equilibrium). If for all i, Ri > a then theefficient allocation where for all i,j x*ij = 1 is also an equilibrium (Credit Line
Equilibrium).
Several comments are in order. In the Credit Line Equilibrium there is noliquidation while in the Speculative Gridlock Equilibrium all the long run investment is liquidated. Since liquidation is costly, the Credit Line Equilibrium dominates the Speculative Gridlock Equilibrium as well as any other equilibrium
where some banks are liquidated. The Speculative Gridlock Equilibrium arises asa result of a coordination failure like in D-D. If banks rationally anticipated aSpeculative Gridlock Equilibrium, they would not invest in the long run technology but store all the funds available at time t = 0 (autarky).
In the Credit Line Equilibrium with diversified lending, bank i extends credit
lines to all the other banks and receives credit lines from them. In equilibrium thedebt arising from bank i’s depositors at t = 2 using bank i’s credit lines with theother banks, is repaid at t = 2 by bank i serving the depositors from the otherbanks. It is precisely because the behavior of one bank’s depositors is affected by
the expectation of what the depositors at the locations of destination will do, that
this equilibrium is vulnerable to a coordination failure. If the depositors in a
sufficiently large number of banks believe that they will be denied consumption at
the location of destination, it is optimal for them to liquidate their investment,
which makes it optimal for the depositors in all other banks to do the same. The
Speculative Gridlock Equilibrium is related to the notion of Domino Effect that
can arise in payment systems as a result of the settlement failure of some participant. The Speculative Gridlock Equilibrium arises here when investment return iscertain. Notice, however, that in the case of pure liquidity shocks banks do notplay any strategic role: the only agents playing strategically are depositors.
The Credit Line Equilibrium dominates autarky which in turn dominates theSpeculative Gridlock Equilibrium.8 Hence there is a trade-off between a risky
payments system based on interbank credit and a safe payment system which
foregoes investment opportunities.9, 10
Both the Gridlock and the Credit Line Equilibria involve the use of credit
lines. In both equilibria banks extend and honor credit lines up to the amount of
their t = 2 resources. In the Speculative Gridlock Equilibrium it is not the banks
that do not honor the credit lines, rather the depositors that, by forcing the liquidation of the long term investment, reduce the amount of resources available at
t = 2.
There is a clear parallel between these two equilibria in our economy with N
locations and the equilibria in a one-location D-D model. These results are also
8

When a = 1 the last two are equivalent. The cost of the Gridlock Equilibrium is proportional to

1-a.
9

For an analysis of this trade off in a similar set up see Freixas and Parigi (1998).
Autarky is equivalent to a payment system with fully collateralized credit lines like TARGET,(Trans-European Automated Real-Time Gross Settlement Express Transfer), the payment systemdesigned to handle transactions in the Euro area.
10

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Xavier Freixas, Bruno Parigi, and Jean-Charles Rochet

related to the papers by Bhattacharya and Gale (1987) and Bhattacharya andFulghieri (1994) that consider N-location D-D economies without geographic
risks.
The Credit Line Equilibrium can be implemented in several ways: through aCompensation System where credits are netted, by a RTGS (Real Time Gross
Settlement) system with multilateral or bilateral credit lines, through lending by
the Central Bank and through Deposit Insurance.
In this basic version of the model, in the event of a gridlock every bank issolvent although illiquid. Thus no difficulty in distinguishing between insolvent
and illiquid banks arises for the Central Bank.11 The Central Bank has a simple
coordinating role as a LOLR in guaranteeing private-sector credit lines or inproviding fiat money, both backed by the authority of the Treasury to tax thereturn on the investment. The Central Bank can borrow from one bank and lend toanother.
By guaranteeing the value of deposits at the arrival locations, Deposit Insurance eliminates any incentive for the depositors to protect themselves by liquidating the investment, thus making it optimal for banks to extend credit.
Like Deposit Insurance which is never used in equilibrium in the D-D model,
the coordination role of the Central Bank costs no resources (excluding moral
hazard issues), since in equilibrium it will not be necessary for the Central Bank
to intervene.12

3.2

Too-big-to-fail policy

Regulators have often adopted a too-big-to-fail approach (TBTF) in dealing withfinancially distressed money center banks and large financial institutions.13 One of
the reasons is the fear of the repercussions that the failure of a money center bankmight have on the corresponding banks that channel payments through it. Our
general formulation of the payment needs where the flow of depositors going to
the various locations is asymmetric offers a simple way to model this case and to11

For an analysis of this issue see the companion paper by Freixas, Parigi, and Rochet (1998).
The Federal Reserve’s role in facilitating the private-sector rescue of the hedge fund Long
Term Capital Management (LTCM) in 1998 offers an example of the coordinating role of theCentral Bank. The Federal Reserve Bank of New York organized rescue loans by private institutions
to LTCM for fear that a default of the fund on the $80 plus that it had borrowed from some key
international banks and securities firms could jeopardize the stability of the entire financial system(Wall Street Journal 1998). Greenspan (1998) argues that in the refinancing of LTCM “no Federal
Reserve funds were put at risk, no promises were made by the Federal Reserve, and no individual
firms were pressured to participate. Officials of the Federal Reserve Bank of New York facilitateddiscussions in which the private parties arrived at an agreement that both served their mutual selfinterest and avoided possible serious market dislocations. Financial market participants were alreadyunsettled by recent global events. Had the failure of LTCM triggered the seizing up of markets,
substantial damage could have been inflicted on many market participants, including some notdirectly involved with the firm, and could have potentially impaired the economies of many nations,
including our own.”
13
See for example the intervention of the monetary authorities in the Continental Illinois
debacle in 1984 and, to some extent, in arranging the private-sector rescue of Long Term CapitalManagement.
12Systemic risk, interbank relations and liquidity provision by the central bank191

capture some of the features of the TBTF policy. We interpret the TBTF policy as
designed to rescue banks which occupy key positions in the payment system,
rather than simply banks with large size.
Suppose there are three locations (N = 3). Locations 2 and 3 are peripheral
locations and location 1 is a money center location. All the consumers of locations 2
and 3 must consume at location 1, and one half of the consumers of location 1
1
consume at location 2 and the other half at location 3. That is p12 = p13 = 2 andx +x
p21 = p31 = 1, p23 = p32 = 0. This implies that X1 = 2 , and X2 =X3 = x1. Thebalance sheet equations become
2

(

)

3

(

)

1 2
1
ù
é
x + x3 R1 + (D2 + D3 ) x1 = ê x1 + x 2 + x3 ú D1
2
2
û
ë

(4)

[ ]
x R + D x = [x + x ] D

x1R2 + D1x 2 = x 2 + x1 D2
1

3

3

3

1

1

3

Proposition 2 (i) If bank 1 is liquidated, it is optimal to liquidate banks 2 and3 (Too-big-too-fail). (ii) If bank 2 is liquidated, it is not optimal to liquidatebanks 1 and 3.
To prove (i) notice if bank 1 is liquidated, the only way for depositors at
locations 2 and 3 to consume at location 1 is to liquidate their investments. To
prove (ii) notice that when bank 2 is liquidated it is optimal for the depositorsfrom bank 1 going to location 2 to liquidate (x2 = 0, D2 = a). If x3 = x1 =1 theequations (4) become
13
R1 + D3 = D1
2
2
R3 + D1 = 2D3
1

1

from which D1 = 2 (R1 + R3) ; D3 = 4 (R1 + 3R3) . If R1 = R3 = R then D1 = D3 > a.
If Central Bank intervention is costly, either in terms of losing some reputation for tight monetary policy or because taxation entails distortions, the Central
Bank may want to minimize its intervention. The previous example allows us toestablish that the minimum intervention of the Central Bank to avoid a gridlock is
to guarantee the credit lines of bank 2 only.
To summarize the results with certain returns, efficient allocation implies 1)
that the payment system is noncollateralized or is backed by noncollateralizedloans, 2) that the Central Bank intervenes to make noncollateralized loans to
individual banks, and 3) that the Central Bank may minimize its intervention bylending to money center banks only. We now turn to the case of stochastic returns
and determine whether these results carry through.

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Stochastic investment returns

We now assume that the return Ri on the investment at location i can be either Rwith probability p, or 0 with probability 1– p, and with 0 < a < 1 < R. Returns are
i.i.d. across locations. Depositors at bank i observe a signal on the return at location i. This signal could be more or less informative. Stochastic returns introduce
the possibility that a bank is insolvent and that runs are due to fundamental reasons so that liquidation is efficient.14 Obviously, if bank managers have the rightincentives, they would act in the interest of the depositors and would proceed tothe efficient closure of their own bank if they have obtained a bad signal. We willassume, instead, that bank managers prefer the banks to remain in business andthat it is not possible to introduce incentive schemes that could change this. This
limited monitoring assumption is consistent with the observation that forbearance
is a common practice in the banking industry.
If returns are common knowledge, the efficient allocation of resources requires that insolvent banks be liquidated:
ì 0 if Ri = 0.
Xi = í
î 1 if Ri = R(5)Whether this efficient closure rule can be implemented as a Nash Equilibriumof the non-cooperative game, will depend, as we will now see, on two features ofthe payment system. On the one hand it depends on the presence of interbank
commitments to honor credit lines, and, on the other hand, it depends on the
structure of interbank funding.
Unlike the case of deterministic returns, now banks’ ex post returns might
differ. It is thus important to investigate the implication of alternative interbank
commitments. Banks’ possibility not to honor credit lines introduces a new element as also banks may behave strategically. We consider two possibilities. Either
the banks extend credit lines without any commitment, and are able to deny credit
when they observe a bad signal, or the banks have given irrevocable credit linesthat bind them to their counterparts. Let us consider the commitment case first.

4.1

Irrevocable credit lines

A feature of many settlement schemes, typically when transactions are netted, isthat of being fixed price. Basically, banks receiving orders to pay do not have thetime necessary to continuously assess the solvency of the sending banks andconsequently, cannot adjust their credit rates or credit limits. This may be justified
by the cost for the participants of a settlement scheme that involves a large volume of transactions to monitor each other continuously and adjust intraday interest
14
For an analysis of the distinction between fundamental and speculative bank runs see Jacklin
and Bhattacharya (1988). For a model of fundamental runs and Central Bank intervention see Allenand Gale (1998).

Systemic risk, interbank relations and liquidity provision by the central bank193

rates or credit limits to changing conditions.15 One way to model this feature is to
assume irrevocable credit lines among banks.
A first implication of irrevocable credit lines is that there is no mechanismthat disciplines low return banks. A depositor from a low return bank headed for ahigh return destination will obtain the high return there. On the other hand, nodepositor will demand to transfer his or her account to a low return location.
Hence, we obtain that the depositors from low return locations will transfer theirdeposits to the high return locations, while the investment may have been partially liquidated to serve the depositors headed for a low return location. Symmetrically, no depositor transfers his or her deposit to a low return location and there
will be an excess supply of goods at this location. The outcome will be a liquidity
crisis at the high return location and an insolvency crisis at the low return location.

4.2

Non binding credit agreements

Absent commitment to interbank lending, the equilibrium configuration thatarises when returns are stochastic depends on the architecture of the financial
system, namely the matrix (pij)i,j. To simplify we will illustrate this point in theCredit Chain and in the Diversified Lending case, when one bank is insolvent (i.e.
R1 = 0, R2 = ... = RN = R) and N = 3. With Credit Chain the system is vulnerable tocontagion: the liquidation of one bank triggers the liquidation of all the others.
However, the insolvency of one bank (R1 = 0) need not trigger the liquidation of
all the others. The reason is that after liquidation the bank is no longer in businessat t = 2 while an insolvent bank has assets at t = 2 arising from credits from otherbanks. This is established in the following proposition.
Proposition 3 With Credit Chain, at an equilibrium, if x1 = 0 then x2 = 0(contagion). As a result only two equilibria are possible: x = (1, 1, 1) i.e.
forbearance, and x = (0, 0, 0), i.e. gridlock. x = (1, 1, 1) is an equilibrium if
3
a < 7 R.
Recall that by definition Credit Chain p 12 = p 23 = p 31 = 1. To show that3
x = (1, 1, 1) is an equilibrium when a < 7 R we solve the balance sheet equations()

xi -1Ri + xi Di -1 = x i -1 + xi Di(6)

and we obtain1[4 R1 + 2 R2 + R3 ]7
1
D2 = [R1 + 4 R2 + 2 R3 ]7
1
D3 = [2 R1 + R2 + 4 R3 ]
7
D1 =

15
The literature on payment systems has typically neglected the problems related to monitoringbanks’ conditions during settlement intervals. One exception is Rochet and Tirole (1996).

194

Xavier Freixas, Bruno Parigi, and Jean-Charles Rochet

3

6

When R1 = 0, R2 = R3 = R it follows that D1 = 7 R, D2 = D3 = 7 R.
Notice that neither equilibria (forbearance and gridlock) are efficient: theefficient allocation is x = (0, 1, 1) which cannot be supported as an equilibrium
because of contagion. When x = (1, 1, 1) is not an equilibrium, it can be restored if) bailing out
the Central Bank injects an amount of liquidity l C1 = Max(0, 7a–3R4bank 1.
While in the Credit Chain case when an insolvent bank is liquidated it is
optimal for the depositors of all other banks to liquidate as well, this is not necessarily so under Diversified Lending.
Proposition 4 With Diversified Lending Bank 1 can be liquidated without
contagion as soon as R > 2aa.
Recall that by definition of Diversified Lending, when N = 3 it follows thatpij = 1/2, i ¹ j. By liquidating bank 1, depositors from banks 2 and 3 travelling tolocation 1, must liquidate half of their investment to consume at location 1. They
will liquidate only half of their investment if the return on the remaining part issufficiently high, R > 2a.
Like in the Credit Chain case, with Diversified Lending we can characterize
the conditions for x = (1, 1, 1) to be an equilibrium. Solving the balance sheet
equations

(

)

(

)

1
1
1 i +1éù
x + x i -1 Ri + (Di+1 + Di -1 ) x i = ê x i + x i -1 + x i +1 ú Di
2
2
2
ë
û
when x=(1,1,1) and R1=0, R2=R3=R we obtain1[6 R1 + 2 R2 + 2 R3 ]10
1
D2 = [2 R1 + 6 R2 + 2 R3 ]101
D3 = [2 R1 + 2 R2 + 6 R3 ]
10

D1 =

When R1 = 0, R2 = R3 = R, the necessary and sufficient condition for x = (1, 1, 1)
to be an equilibrium is that 2R > 5a. When x = (1, 1, 1) is not an equilibrium the cost). Notice thatof Central Bank intervention to restore it is l D1 = Max(0, 5a–2R
3
7a–3R< 5a–2R
Û a < R which is always satisfied. This leads us to the following
43
result.
When R1 = 0, R2 = R3 = R, the cost of Central Bank intervention to restore theallocation x = (1, 1, 1) is larger under Diversified Lending than under Credit Chain.
However, Diversified Lending is less vulnerable to contagion than Credit Chain:
namely depending on parameter values x = (0, 1, 1) is an equilibrium under Diversi-

Systemic risk, interbank relations and liquidity provision by the central bank

195

fied Lending but for no parameters value it is an equilibrium under Credit Chain.
The gridlock that might arise when returns are stochastic is of a fundamental
nature and not of a speculative one, like when returns are certain.
Furthermore unlike the case with certain returns, where the simple guarantee
of private sector credit lines by the Central Bank is sufficient to avoid a Speculative gridlock, here the Central Bank must intervene at t = 1 to avoid a Fundamental gridlock. The Central Bank intervenes by injecting liquidity in the insolvent bank thus eliminating the incentive to liquidate.

4.3

Imperfect information on banks’ returns

In general, the environment the Central Bank will be confronted with will lie
somewhere between the perfect information case and absence of information case.
This means that the agents will receive a noisy signal on each bank. The Central
Bank will have to act knowing that with some probability it will be lending to(guaranteeing the credit lines of) insolvent institutions and with some probability
it will be denying credit to solvent institutions. Depositors will run on all thebanks which have generated a bad signal.
The consequences are completely different depending on the structure of theinterbank market. In the credit chain funding case, the Central Bank has toprovide credit to all the institutions that need it, and therefore has a high probability of ending up financing insolvent banks. On the other hand, in the diversified lending case, the Central Bank will intervene only with a very small probability. Ex ante, therefore, the Central Bank intervention is much more expensivein the credit chain case, so that in the end a fully collateralized payment system
may be preferred.

4.4

Payments among different countries

Systemic risk is often related to the spreading of financial crisis from one country
to another. Our basic model can be extended to consider various countries insteadof locations within the same country. When depositors belong to different countries, travel patterns that generate a consumption need in another location have thenatural interpretation of demand of goods of other countries, i.e. export demand.
Goods of the other country can be purchased through liquidity (like in autarky inthe basic model) or through a credit line system whereby the imports of a country
are financed by its exports.
The results extend to the model with different countries but the role of themonetary authority is somewhat different now. While the lending ability of thedomestic monetary authority was backed by its taxation power, the lending abilityof an international financial organization is ultimately backed by its capital.
Hence the resources at its disposal are limited and in case of aggregate uncertainty its ability to guarantee banks’ credit lines is limited.16
16

See the role of the I.M.F. in the Asian and Russian crises.

196

5

Xavier Freixas, Bruno Parigi, and Jean-Charles Rochet

To conclude: A synthetic view

Up to now we have examined different environments and determined what shouldbe the Central Bank’s behavior in order to restore the efficient allocation. Thisview is obviously biased because it abstracts from the costs of systematic LOLR
intervention, in particular as they affect individual banks’ behaviors (moral hazard) and also disregards the costs of transferring resources to the private sector aswell as the administrative costs involved.
We now turn to consider the effect of introducing some costs of intervention
and analyze the consequences on the optimal behavior of the Lender of Last
Resort and on the choice of the interbank market structure (unsecured versus fully
collateralized).
To begin with, our analysis has highlighted the role of· connectivity· quality of information
so that these will be the different environments we have to consider in order to geta more clear perspective of the different aspects at stake.
The introduction of a cost of the LOLR intervention implies that for highprobabilities of default the optimal structure of the interbank market is clearly thatof a fully collateralized market which precludes the Lender of Last Resort fromintervening. Yet, in the more general case, the answer will be more complex andhas to depend, as we have mentioned, on both the degree of connectivity and thequality of information.
When information is poor and the degree of connectivity is low, as in thecredit chain equilibrium, the LOLR will intervene often and it will not be accurate,committing type-one and type-two errors. Even in the diversified lending case,given the existence of costs for the LOLR intervention, a fully collateralized
market could be preferred. Still when information improves, the balance changes
in favour of unsecured markets and an active (more accurate) intervention of the
LOLR. Whatever the quality of the information, the LOLR will have to intervene
more actively in the case of low connectivity (credit chain case). Thus, for someintermediate level of costs and quality of information, the fully collateralized
market will be preferred for a low degree of connectivity (credit chain funding)while the unsecured lending will be preferred for a high degree of connectivity(diversified lending).

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