Illiquidity in the Interbank Payment System FollowingWide-Scale Disruptions

Morten L. BechBank for International Settlements∗†

Rodney J. GarrattUniversity of California, Santa Barbara

July 11, 2012

Abstract

We show how the interbank payment system can become illiquid following wide-scale disruptions. Two forces are at play in such disruptions—operational problemsand changes in participants’ behavior. If the disruption is large enough, hits a keygeographic area, or hits a “too-big-to-fail” participant, then the smooth processing ofpayments can break down, and central bank intervention might be required to reestab-lish the socially efficient equilibrium. The paper provides a theoretical framework toanalyze the effects of events such as the September 11 attack. In addition, the modelcan be reinterpreted to analyze shocks to fundamentals which affect the parametersof the intraday liquidity management game. We demonstrate this by showing howprocessing behavior changed in response to heightened credit risk at the time of theLehman Brothers failure.

JEL classification: C72; E58Keywords: Payments, Stag hunt, Network Topology, Moral Suasion, Too Big to

Fail

∗The paper was written while Morten Bech was working at the Federal Reserve Bank of New York. Wethank Wilko Bolt, Paula Hernandez, Thomas Troger, Alan Sheppard, Rafael Repullo, Ned Prescott, twoanonymous referees and participants at the New Directions for Understanding Systemic Risk conferencecosponsored by the National Academy of Sciences and the Federal Reserve Bank of New York, the FourthJoint Central Bank Conference on Risk Measurement and Systemic Risk hosted by the European CentralBank, the Modern Financial Institutions, Financial Markets and Systemic Risk conference at the FederalReserve Bank of Atlanta as well as seminar participants at New York University, Concordia University, theBank of England, De Nederlandsche Bank, and Danmarks Nationalbank for helpful comments. The viewsexpressed in this paper are those of the authors and do not necessarily reflect those of the Federal ReserveBank of New York, the Federal Reserve System or the Bank for International Settlements.†Corresponding author. Centralbahnplatz 2, CH-4002 Basel, Switzerland, (+41) 61 280 8923,

morten.bech@bis.org

1

1 Introduction

The events of September 11, 2001 and to a lesser extent the North American blackout ofAugust 14, 2003 highlighted the fact that parts of the financial system are vulnerable towide-scale disruptions. Moreover, the events underscored the fact that the financial systemconsists of a complex network of interrelated markets, infrastructures, and participants, andthe inability of any of these entities to operate normally can have wide-ranging effects evenbeyond its immediate counter-parties. Not surprisingly, the financial industry and regulatorsare devoting considerable resources to business continuity measures and planning in order tostrengthen the resiliency of the U.S. financial system. The primary objective is to minimizethe immediate systemic effects on the financial system of large scale shocks.

At the apex of the U.S. financial system are a number of critical financial markets thatprovide the means for both domestic and international financial institutions to allocate cap-ital and manage their exposures to liquidity, market, credit and other types of risks. Thesemarkets include federal funds, foreign exchange, commercial paper, government and agencysecurities, corporate debt, equity securities and derivatives. Critical to the smooth function-ing of these markets are a set of wholesale payments systems and financial infrastructuresthat facilitate clearing and settlement.1 Operational difficulties involving these entities ortheir participants can create difficulties for other systems, infrastructures and participants.Such spillovers might cause liquidity shortages or credit problems and hence potentiallyimpair the functioning and stability of the entire financial system.

Wide-scale disruptions may not only present operational challenges for participants of thecore payment and securities settlement system, but may also induce participants to changebehavior in terms of how they conduct their business. The actions of participants have boththe potential to mitigate, but also to augment the adverse effects.2 Hence, understandinghow participants react when faced with operational adversity by their counter-parties iscrucial for both operators and regulators in designing counter measures, devising policy, andproviding emergency assistance if called for.

Despite their immense importance for the financial system, very little research is availableon financial infrastructures and how they and their stakeholders respond to disruptions.Basically, two strands of literature exist. One strand investigates, by way of simulations,the potential contagion effects of a participant failing to meet obligations when due (see e.g.Humphrey (1986), Angelini (1996) and Devriese and Mitchell (2006)). The other strandconsists of event studies of actual disruptions and their effects to the clearing, settlementand payment system. Bernanke (1990) looks at clearing and settlement during the stockmarket crash of 1987 whereas Fleming and Garbade (2002), McAndrews and Potter (2002)and Lacker (2004) all look at the impacts of the attacks of September 11, 2001.

1Examples include the Depository Trust & Clearing Cooperation (DTCC), the Clearing House Inter-bankPayment System (CHIPS), Continuous Linked Settlement (CLS), and the Federal Reserve’s Federal Fundsand Securities Services.

2For a description of the extraordinary cooperative efforts undertaken by participants to overcome theproblems for the payment and securities settlement system caused by the events of September 11, 2001 seethe 2001 annual report of the Federal Reserve Bank of New York.

2

The backbone of the payment and securities settlement system in the U.S. is the Fed-eral Reserve’s Fedwire R© Funds Service (Fedwire). Fedwire is a Real Time Gross Settlement(RTGS) system in which payments are settled individually and with instant finality in realtime. Over 9,500 participants use Fedwire to send or receive time critical and/or large valuepayments on behalf of corporate and individual clients, settle positions with other finan-cial institutions stemming from other payment systems, clearing arrangements or securitiessettlement, submit federal tax payments and buy and sell federal funds. The average dailynumber of payments in the first half of 2011 was 493,000 and the average value transferredwas around $2.7 trillion per day.3 Fedwire continued to operate during the events of Septem-ber 11, 2001 and August 14, 2003, but in both cases the Federal Reserve had to interveneby extending the operating hours and by providing emergency liquidity.

On September 11, 2001, the massive damage to property and communications systemsin lower Manhattan made it more difficult, and in some cases impossible, for many banksto execute payments to one another. The failure of some banks to make payments alsodisrupted the payments coordination by which banks use incoming payments to fund theirown transfers to other banks. Once a number of banks began to be short of incomingpayments, some became more reluctant to send out payments themselves. In effect, bankswere collectively growing short of liquidity. The Federal Reserve recognized this trend towardilliquidity and provided liquidity through the discount window and open market operationsin unprecedented amounts in the following week. Federal Reserve opening account balancespeaked at more than $120 billion compared to approximately $15 billion prior. Moreover, theFederal Reserve waived the overdraft fees it normally charges. On September 14, daylightoverdrafts peaked at $150 billion, more than 60 percent higher than usual. (see Ferguson,2003). Following the blackout in the Midwest and Northeast on August 14, 2003 commercialbanks borrowed $785 million from the discount window compared to the normal range of$100 - $200 million.

Several papers (e.g. McAndrews and Rajan, 2000, Bech and Garratt, 2003, and McAn-drews and Potter, 2002) advocate interpreting the payment decisions of banks participatingin Fedwire as a game. McAndrews and Rajan argue that the timing of payments resemblesa coordination game. This idea emerges in the formal analysis of the intraday liquiditymanagement game of Bech and Garratt where it is shown that the game played by banks isa type of coordination game known as the stag hunt game. In this game there are two Nashequilibria, one which involves early settlement of payments and the other late settlement.Evidence that banks select between these equilibria is found by McAndrews and Potter whodocument the breakdown and reemergence of coordinated payments following the events ofSeptember 11, 2001.

In what follows we shed light on why coordination on early settlement occurs in normaltimes and how operational difficulties for participants in Fedwire are likely to effect equilib-rium selection. These are the central issues in study of systemic risk in payments systems,which is a major concern of central banks around the world.4 Our argument involves con-

3Source: http://www.frbservices.org/operations/fedwire/fedwire funds services statistics.html4Hendricks, Kambhu and Mosser (2007, p. 83) state: “...the key characteristic of systemic risk is the

movement from one stable (positive) equilibrium to another stable (negative) equilibrium for the economy

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structing a potential function for the game played by banks. The potential function is usedto summarize welfare changes that result from movements by individual banks. Changesby individual banks that improve individual welfare necessarily increase potential. Hence,we look to adjustment processes that converge to potential maximizing strategy profiles asindicators of how the system will operate. We are able to characterize circumstances underwhich the system will converge to an early (versus late) payment equilibrium. Moreover, wedescribe the policy options of the central bank and discuss when moral suasion is likely tobe effective. At first, we assume that all banks are identical. Subsequently, we introduceheterogeneity across banks and are able to add new insight into the question, when a bankis too big to fail. In addition, we investigate how different degrees of interconnectednessbetween banks, i.e., network topologies affect resiliency vis-a-vis wide-scale disruptions.

Our analysis of payment system dynamics also applies to shocks to fundamentals whichaffect the parameters of the intraday liquidity management game. Bech (2008) argues thatheightened credit risk raises the expect cost of early processing and lowers the expected costof delay. We examine how payment processing behavior responded to heightened credit riskthat arose during the time of the Lehman Brothers failure.

2 The n-player intraday liquidity management game

Envision an economy with n identical banks using a RTGS system operated by the centralbank to settle interbank claims. Banks seek to minimize their settlement costs. The businessday consists of two periods: morning and afternoon. Banks start the business day with azero balance on their settlement accounts at the central bank. At dawn each bank receivesn−1 requests from customers to pay a customer of each of the other n−1 banks $1 as soon aspossible. Banks either process all requests in the morning period or postpone them all untilthe afternoon period.5 In order to process a payment request a bank must have sufficientfunds in their account at the central bank. Banks without sufficient funds can borrow fundswithin the day from the central bank for a fee. The fee is F > 0 per dollar if the settlementaccount is overdrawn either at noon or at the end of the day.6 The fee can be thought of asan insurance premium reflecting the credit risk assumed by the Federal Reserve.

Banks can avoid fees by delaying payments. However, delaying entails both private andsocial costs. First, delay of settlement might displease customers and counterparties as they

and financial system. According to this view, research on systemic risk should focus on the potential causesand propagation mechanisms for the phase transition to a new but much less desirable equilibrium as well asthe reinforcing feedbacks that tend to keep the economy and financial system trapped in that equilibrium.”

5Of course, in reality, banks receive thousands of separate payment requests each day and are free todecide on the processing time for each of them. However, banks do not, as a rule, manage paymentsindividually. Rather, banks manage their intraday liquidity position during the day by – if necessary –throttling submissions subject to the constraint that all payments have to be made by the end of the day.For simplicity, we break the day into two periods. Delay in the morning in our model is a proxy for slowingdown the processing rate for non-urgent payments. We do not allow delay in the afternoon period, as thisis the period in which all remaining payments must be processed.

6See Coleman (2002) for a discussion of the evolution of the Federal Reserve’s intraday credit policies.

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are left with greater uncertainty or even explicit costs. They might demand compensation ortake their business elsewhere in the future. Second, delaying payments creates an exposureto operational incidents that can prevent the settlement of payments before the close ofbusiness. Third, the process of delaying might it self be costly as additional resources haveto be devoted to managing intraday positions. Fourth, as argued by Kahn et al. (2003) delayincreases the length of time participants may be faced with credit risk exposures vis-a-visone another. We assume that the cost of delaying faced by banks is D > 0 per dollar. Hence,banks have to trade-off the potential cost of mobilizing liquidity against the cost of delay.

Formally, the player set is N = {1, 2, . . . n}. The set of strategies for each bank i isSi = {m, a} where m denotes “all requests processed in the morning” and a denotes “allrequests postponed until the afternoon.”7 Let N−i = N\{i}. The settlement cost of bank idepends on the strategies played by all banks and is given by c(si, s−i) where si ∈ Si is thestrategy played by bank i and where s−i ∈ S−i = ×j∈N−i

Sj is the n−1 dimensional profile ofopponents strategies. The price that banks charge for processing is assumed to be fixed andfor simplicity it is set to zero. The payoff function for bank i is thus equal to the negativeof the settlement cost function, that is

πi(si, s−i) = −c(si, s−i) (1)

The settlement cost a bank incurs by playing morning is the overdraft at noon times theoverdraft fee per dollar. The overdraft at noon is given by the number of other banks playingafternoon, i.e., the number of banks from which the bank does not receive an offsettingpayment (dollar) in the morning. On the other hand, if a bank plays afternoon then thesettlement cost is the value of delayed payments times the delay cost per dollar.

The proposed game satisfies an aggregation property (see Dubey et al. (1980) and Cor-chon (1994)), in that it suffices to keep track of the number of opponents playing the morn-ing strategy.8 Each payoff function πi : ×j∈NSj → R can be replaced with a function π :Si × Z+ → R defined as follows:

π(si, |s−i|m) =

{−(n− 1− |s−i|m) if si = m−(n− 1)D if si = a,

(2)

where |s−i|m is the number of banks playing the morning strategy in the strategy profile s−i.We concentrate on pure strategy equilibria in our analysis. The equilibria of the game aregiven by the following

Proposition 1 If intraday liquidity is relatively cheap, i.e., F < D, then the unique pure-

7The assumption that players choose a single strategy to play against all of the other players in thenetwork is also used in the literature on social coordination games, of which our game is, for some parameterspecifications, a special case; see Goyal and Vega-Redondo (2005) and Goyal (2009).

8Galbiati and Soromaki (2008) find the aggregation property in another ‘liquidity game’ in paymentsystems.

5

strategy Nash equilibrium is (m, . . . ,m). If intraday liquidity is relatively expensive, i.e.,

F ≥ D, then (m, . . . ,m) and (a, . . . , a) are the only pure-strategy Nash equilibria.

Proof. From Eq. (2) we have πi(m,n−1)−πi(a, n−1) = (n−1)D > 0. Hence, (m, . . . ,m)is a Nash equilibrium regardless of the relative magnitudes of F and D. Suppose F ≥ D.Then, we have π(a, 0) = −(n−1)D ≥ π(m, 0) = −(n−1)F . Hence, (a, . . . , a) is a Nash equi-librium if F ≥ D. We now argue that there are no other Nash equilibria for any positive val-ues of F and D. Suppose 0 < |s|m < n then at least one bank i ∈ N plays the strategy m andat least one bank j ∈ N plays the strategy a. For this to occur in a Nash equilibrium it has tobe true that π(m,

∣∣s−{i,j}∣∣m) ≥ π(a,∣∣s−{i,j}∣∣m) and π(a,

∣∣s−{i,j}∣∣m + 1) ≥ π(m,∣∣s−{i,j}∣∣m + 1).

From the bottom part of Eq. (2) we have π(a,∣∣s−{i,j}∣∣m) = π(a,

∣∣s−{i,j}∣∣m + 1). This links

the preceding inequalities into π(m,∣∣s−{i,j}∣∣m) ≥ π(a,

∣∣s−{i,j}∣∣m) = π(a,∣∣s−{i,j}∣∣m + 1) ≥

π(m,∣∣s−{i,j}∣∣m + 1), which contradicts the top part of Eq. (2) (as

∣∣s−{i,j}∣∣m + 1 ≤ n).

Whenever F ≥ D the morning equilibrium (m, . . . ,m) is the efficient equilibrium interms of minimizing both individual and aggregate settlement costs. In the analysis thatfollows we assume that banks initially coordinate on the efficient, morning equilibrium andwe examine the impact of a disruption that forces some banks to switch to the afternoonstrategy. We are interested in the strategic adjustments made by banks in response to thedisruption. In particular, we are interested in how the disruption influences equilibrium. Werestrict ourselves to equilibria in pure strategies.

3 Wide-scale disruptions

The Interagency Paper on Sound Practices to Strengthen the Resilience of the U.S. FinancialSystem issued by the Board of Governors of the Federal Reserve System, the Securities andExchange Commission and the Office of the Comptroller of the Currency defines a wide-scale disruption as an event that causes a severe disruption or destruction of transportation,telecommunication, power, or other critical infrastructure components across a metropolitanor other geographical area and the adjacent communities that are economically integratedwith it; or that results in wide-scale evacuation or inaccessibility of the population withinnormal commuting range of the disruption’s origin.

In the context of our model, we take a wide-scale disruption to mean an event thatprevents a subset of banks from making payments as normal. Specifically, some banksare temporarily forced to play to the afternoon strategy, which takes the system out ofequilibrium. The size of the disruption can be measured by the share of banks that aredisrupted. After the disruption we assume that the disrupted banks again become fullyoperational and that they are, like the non-disrupted banks, free to choose either the morningor afternoon strategy.

The question we ask is: Will the disruption trigger a move to the inefficient afternoonequilibrium or back to the efficient, morning equilibrium? The answer, of course, depends

6

on the size of the disruption, its duration, and what the incentives are for banks to adjusttheir processing decisions following the disruption. In a game with many players this couldbe very difficult to assess. However, here it turns out that the game played by banks isa potential game a la Monderer and Shapley (1996), and as such, the dynamics of thesystem are conveniently and transparently understood by examining the potential function.A potential function is a mapping from strategy profiles (in this case vectors of processingdecisions for the banks) to the reals that summarize all the information needed to computethe Nash equilibria of a game. The Nash equilibria we characterize in Proposition 1 are local(or global) maximizers of the potential function.9

3.1 Potential function

We now turn to a formal characterization of the potential function we utilize to analyze then-player, deterministic intraday liquidity management game.

Definition 1 (Potential Function) A function P satisfying for every si, s′i ∈ Si, s−i ∈

S−i and i ∈ N ,

πi(si, s−i)− πi(s′i, s−i) = P (si, s−i)− P (s′i, s−i)

is a potential function for the n-player, deterministic intraday liquidity management game.

The function P , so defined, is unique up to an additive constant. Existence of a potentialfunction for the game with payoffs specified in Eq. (2) follows from Corollary 2.9 in Mondererand Shapley (1996).10 Moreover, since the game is a special case of a congestion game definedby Rosenthal (1973), its functional form is given by

P (s) =∑

f∈∪ni=1si

|s|f∑`=1

ωf (`), (3)

where ωm(`) = −(n− `)F and ωa(`) = −(n− 1)D; see Eq. (3.2) in Monderer and Shapley(1996). Because the aggregation property applies, we can summarize P with a new functionP : Z+ → R, defined in the following way. For any x ∈ {0, 1, ..., n}, P (x) is equal to thevalue of the function P (·) evaluated at any strategy profile in the set {s ∈ S : |s|m = x}.

9At a local maximizer no unilateral deviation increases the potential of the game. At a global maximizerthere is no alternative strategy profile that has higher potential.

10Following their notation, let A = (m,m, s−{i,j}), B = (a,m, s−{i,j}), C = (a, a, s−{i,j}), and D =(m, a, s−{i,j}). Then the required condition is satisfied since πi(A) = πj(A), πi(C) = πj(C), πi(D) = πj(B),and πj(D) = πi(B).

7

Applying this notation in (3) yields

P (x) = −x∑`=1

(n− `)F −n−x∑`=1

(n− 1)D

= −(nx− 1

2x(x+ 1))F − (n− x)(n− 1)D (4)

where the second equality uses the finite-series expression for a triangular number,∑x

`=1 x =12x(x + 1). P is a second order polynomial in x. The global and local maximizers of P are

important for the analysis that follows and are characterized in the following

Proposition 2 If F < D, then P attains a global maximum at x = n and there are no

other local maximizers. If D ≤ F < 2D, then P attains a local [global] maximum at x = 0

[x = n]. If F = 2D, then P attains a global maximum at both x = 0 and x = n. If F > 2D,

then P attains a local [global] maximum at x = n [x = 0].

4 Adjustment following disruptions

Monderer and Shapley (1996) specify a simple adjustment process that converges to a Nashequilibrium of a potential game in a finite number of steps. In their process it is assumed that,whenever the strategy profile is not a Nash equilibrium, one player deviates to a strategy thatmakes him better off. Unilateral deviations that increase the payoff of the deviator raise thevalue of the potential while unilateral deviations that increase the payoff of the deviator lowerit. Hence, once a Nash equilibrium is reached (there are no more self-improving, unilateraldeviations) the process terminates and the potential function will be at a maximum in thesense that its value cannot be increased by varying any single player’s strategy. Endpointsof the simple adjustment process are local maximizers of the potential function, i.e., Nashequilibria.

We consider a generalization of the simple adjustment process that allows for simultane-ous deviations. Our simultaneous adjustment process (SAP) is defined as follows: Each stepin the adjustment process occurs during one day, which we refer to as a period. Assume thedisruption occurs in period 0 and lasts until period τ . At period 0 the player set is parti-tioned into two groups. Let those who are disrupted and hence forced to play the afternoonstrategy be denoted by Nd and let those who are not disrupted, and hence free to choosetheir processing strategy, be denoted by Nnd. The adjustment process starts in period 1. Atthis time the initial strategy profile is given by s0 = (s0

Nnd , s0Nd) where s0

Nnd = {m, ...,m} has∣∣Nnd∣∣ components and s0

Nd = {a, ..., a} has∣∣Nd

∣∣ components.The period t ≥ 1 adjustment involves two steps.

Step 1. Nature randomly selects a set Dt of eligible deviators. If the disruption is not over

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(i.e., if t < τ), the set of eligible deviators is selected according to the probability functionfnd, which is defined on the elements of 2N

nd, the power set of Nnd. If the disruption is

over, the set of eligible deviators is selected according to the probability function f , whichis defined on the elements of 2N , the power set of N.

Step 2. Banks in the set Dt choose a strategy sti to solve

maxsti∈{m,a}

ui(sti, s

t−1−i ).

Some comments are in order. First, notice that the set of eligible deviators in a perioddepends on whether the disruption is over or not. Moreover, the random process used toselect deviators allows a bank that was already selected as the eligible deviator in periodt′ to be selected again in period t′′ > t′. Thus, the process can switch directions. Banksmake adjustments that are optimal based on the previous period’s strategy profile. Thiscaptures the notion that at any point of time during or after a disruption some (arbitrarilydetermined) set of banks will act, more or less simultaneously, without knowing who else ismaking adjustments to their processing strategy or what adjustments they are making. Theimplicit assumption is that banks behave myopically. That is, they do not anticipate futureadjustments by other banks and they are not allowed to coordinate their adjustments. Thisassumption seems justified in the U.S. system with over 9,000 participants.

We now use the information revealed by the potential function to explain the learningdynamics under the SAP. In the case where F > D, we will use the notion of potentialminimizers. Let µ denote the integer value that minimizes P over the set of integers from 0to n.11 The SAP has the following properties, which are described in

Proposition 3 (a) Suppose F < D. The SAP leads to the equilibrium (m, . . . ,m) from any

starting point. (b) Suppose F = D. The SAP leads to the equilibrium (m, . . . ,m) from any

starting point except (a, . . . , a). At (a, . . . , a), all players are indifferent between playing a or

making a unilateral deviation to m. (c) Suppose F > D. The SAP leads to the equilibrium

(m, . . . ,m) from starting points on the right of µ and to the equilibrium (a, . . . , a) from

starting points to the left of µ.

Proof. (a) If F < D then, by Eq. (4), P (x)− P (x−1) = D(n−1)−F (n−x) > 0 for all1 ≤ x ≤ n. Hence all unilateral deviations from a to m are profitable, but deviations fromm to a are not. Because sooner or later each and every bank becomes an eligible deviator,the adjustment process converges to the morning equilibrium.

11Because P is strictly convex (over continuous x) the minimum defined over discrete points may occur ateither one or two points. However, instances where the minimum occurs at two points are of measure zeroin the parameter space and hence are ignored.

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-16,000

-14,000

-12,000

-10,000

-8,000

-6,000

-4,000

-2,000

0

0 20 40 60 80 100

P(x)

Number of Banks Playing Morning, x

F < D F = D D < F< 2D F = 2D F > 2D

Figure 1: Potential function for different values of F .

(b) If F = D then, by Eq. (4), P (x)− P (x− 1) = (x− 1)F > 0 for all 2 < x ≤ n. Theproof is the same as part (a). If x = 1 then P (x)− P (x− 1) = 0. No one can deviate untilthe disruption is over, and it is not profitable for anyone to deviate once it does end.

(c) If F > D then, by Eq. (4), P (x) − P (x − 1) = D(n − 1) − F (n − x) > 0 for alln ≤ x < µ. Hence all unilateral deviations from a to m are profitable, but deviations fromm to a are not. Likewise, if F > D, then P (x)− P (x− 1) = D(n− 1)−F (n− x) is strictlyless than 0 for all µ > x ≥ 0. Hence, all unilateral deviations from m to a are profitable, butdeviations from a to m are not. By the same reasoning as part (a), the process converges tothe morning equilibrium from the initial state s0 = (s0

Nnd , s0Nd) if

∣∣Nnd∣∣ ∈ {µ + 1, ...n} and

it converges to the afternoon equilibrium if∣∣Nnd

∣∣ ≤ µ− 1.

The different cases outlined in Proposition 1 are illustrated in Figure 1 (the curves fromtop to bottom correspond to labels from left to right). The figure displays the function Pfor n = 100 and different values of D and F. The case of F < D is illustrated by the highestcurve. As indicated by property (a) of Proposition 1 all deviations to m are profitable aftera disruption of any size in this case. The implication is that the system would convergeto the (efficient) morning equilibrium regardless of the size of the shock. In other words,the system is resilient to a disruption of any size in terms of maintaining coordination onearly processing. The same is true for the case where F = D, save the situation where thedisruption hits all n banks, cf. property (b) of Proposition 1.

Three instances of the case F > D (cf. property (c) of Proposition 1) are illustrated bythe bottom three curves in Figure 1. In each instance disruptions that are sufficiently large

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move the system to a point to the left of the minimum of the potential, causing the banks toconverge to the (inefficient) afternoon equilibrium. The figure illustrates an important factwhich we will address more formally in the next section: The higher the cost of liquidity fromthe central bank, F , is relative to the cost of delay, D, the lower is the number of banks thatminimizes the potential, and thus the larger is the “basin of attraction” for the afternoonequilibrium. For example, if F = 2D then the disruption needs to hit half or more of thebanks in order for the system to converge to the inefficient equilibrium. However, if F > 2Dthen a disruption that hits less than half of the banks will cause the system to converge tothe inefficient equilibrium.

The different cases of potential functions shown in Figure 1 make it easy to understandthe Federal Reserve’s response to the terrorist attacks on September 11, 2001 (See Lacker,2004, for a detailed overview). Shortly after the attacks, the Federal Reserve issued severalstatements to the effect that the Fedwire was open and operating and that the discountwindow was available to meet liquidity needs. Moreover, the Federal Reserve began to makeliquidity widely available and for the period from September 11 through September 21, itwaived the daylight overdraft fees for all account holders and eliminated the penalty normallycharged in excess of the effective Federal Funds rate on overnight overdrafts. Both policyresponses are consistent with this model. The statements can been seen as an attemptto encourage the banks to keep coordinating on the efficient equilibrium. Moreover, theinfusion of liquidity and elimination of fees served to lower the cost of liquidity F to virtuallyzero so that deviations to m became profitable. Once the system returned to the morningequilibrium (or had made sufficient progress) the original F was be restored.

5 Banking structure and network topology

So far, we have considered an economy with a homogenous banking structure consisting ofn identical banks. Obviously, this is - in many ways - an over-simplification. For example,in Fedwire the top 100 participants account for 95% of the value and 60% of the volume.Moreover, it is often argued that the financial sector’s resiliency to different types of shocksdepends critically on the interconnectedness of the interbank market cf. Allen and Gale(2000). In the following, we provide an extension of our model that allows us to investigatehow both heterogeneity across size and different degrees of completeness of interbank rela-tionships impact the resiliency of the interbank payment systems to (wide-scale) disruptions.

5.1 Mergers and payment topology

Envision a new economy with one large bank and a number of smaller identical banks.Assume that the large bank is created by merging k > 1 out of the n identical banksin our previous economy.12 For notational convenience, assume that banks 1 through k

12It has been suggested to us by practitioners that some banks have close relationships and do not behavestrategically with respect to one another. An alternative interpretation is that k is the size of the coordinated

11

< n are the ones that are merged. Denote the merged bank by k. The new player set isN = {k, k + 1, . . . n}. A key distinction is that transfers between customers of the mergedbank are handled internally within that bank. Consequently, following a merger, the totalvolume of payments transferred via the RTGS system decreases. The larger the merger thelarger the decrease. Specifically, the total volume (and value) of payments transferred fallsto n(n− 1)− k(k − 1), compared to n(n− 1) before the merger.

In addition, we assume that a merger can lead to a shift in the payment flow. Forexample, the merger might lead to an increase in the volume of transactions involving themerged bank. This would occur, for example, if there are economies of scale or scope inthe processing of payments. We do not formally model payment flows as an endogenousvariable. Rather, we consider different exogenous payment flow effects by including theparameter α ≥ 0, which represents the fraction of each dollar that small banks continue totransact with each other following a merger. The remaining 1−α of each dollar small bankstransacted with each other before the merger now goes to the merged bank. For example,if the merger is size k, each small bank sends α dollars to each of the other small banks andsends k + (1 − α)(n − k − 1) dollars to the merged bank. We keep the total payment flowbalanced, and hence the large bank sends the same amount to each of the small banks.

If α = 1, then there is no shift in payment volume between the large and the smallerbanks, but the analysis is still effected by the merger. Values α < 1 correspond to thescenario discussed above where the small banks are losing market share vis-a-vis the mergedbank. If α > 1, then the small banks are gaining market share vis-a-vis the merged bank.In order to keep payments from the small banks to the merged bank positive we require

α ≤ n− 1

n− k − 1= α. (5)

By varying α, we create different degrees of interconnectedness or network topologies. If0 < α < α, we have a complete graph, where all banks exchange payments with each other.In the two special cases where α = 0 or α = α, we get topologies where not all banks areexchanging payments with each other. If α = 0, then we have a star graph where the smallbanks do not interact with each other, but instead, only interact with the “money center”bank. If α = α, then we have a disconnected graph where the small banks are interactingwith each other, but not with the merged bank, which only process internal transactions.Examples of the payment flows with 0 ≤ α ≤ α in an economy with five banks before andafter a merger of size k = 2 are shown in Figure 2.

sector.

12

1

5 2

34

45

31,2

0 < α < 1

45

31,2

α = 2

45

31,2

α = 0

45

31,2

1 < α < 2

45

31,2

α = 1

Merger with no shift in payment flow

Mergers with shift in payment flow

1

5 2

34

1

5 2

34

45

31,2

0 < α < 1

45

31,2

α = 2

45

31,2

α = 0

45

31,2

1 < α < 2

45

31,2

α = 1

45

31,2

α = 1

Merger with no shift in payment flow

Mergers with shift in payment flow

Figure 2: Payment flows before and after merger (n = 5, k = 2 and α = 2)

special cases where α = 0 or α = α, we get topologies where not all banks are exchanging payments

with each other. If α = 0, then we have a star graph where the small banks do not interact with

each other, but instead, only interact with the “money center” bank. If α = α, then we have a

disconnected graph where the small banks are interacting with each other, but not with the merged

bank, which only process internal transactions. Examples of the payment flows with 0 < α < α in

an economy with five banks before and after a merger of size k = 2 are shown in Fig. 2.

5.2 Potential function for the heterogenous economy

The payoff function of the merged bank k is given by

πk(m, s−k) = −(k + (1− α)(n− k − 1))(n− k − |s−k|m)F (7)

πk(a, s−k) = −(k + (1− α)(n− k − 1))(n− k)D (8)

15

Figure 2: Payment flows before and after merger (n = 5, k = 2, α = 2).

5.2 Potential function for the heterogenous economy

The payoff function of the merged bank k is given by

πk(sk, s−k) =

{−(k + (1− α)(n− k − 1))(n− k − |s−k|m)F if sk = m−(k + (1− α)(n− k − 1))(n− k)D if sk = a

(6)

The payoff function of the i = k + 1, ..., n small banks is given by

π(si, sk, s−{i,k}) =

{−(n− 1− (k + (1− α)(n− k − 1))1sk=m − α

∣∣s−{i,k}∣∣m)F if si = m−(n− 1)D if si = a

(7)where 1sk=m is an indicator function taking value one when sk = m and zero otherwise.

As before, the payoff to a bank, that plays afternoon, is independent of the strategiesplayed by the opponents and the game is an aggregation game. For the small banks itsuffices to keep track of the bank’s own strategy, the strategy of the large bank and thenumber of other small banks playing the morning strategy. Hence, the payoff function π canbe replaced with the function π(·) : Si×Sk×Z+ → R that is specified as π(si, sk,

∣∣s−{i,k}∣∣m) ≡π(si, sk, s−{i,k}). For the merged bank, it suffices to keep track of it’s own strategy and thenumber of small banks playing the morning strategy. Thus, we define πk(·) : Sk × Z+ → R

13

as πk(sk, |s−k|m) ≡ πk(sk, s−k). For each strategy choice of the big bank, the value of thepotential function depends only on the total number of small banks playing the morningstrategy. Hence, we can proceed using a reduced form of the potential function for thegame, P (·) : Sk × Z+ → R, defined to satisfy

P (sk, |(sk+1, . . . , si, . . . , sn)|m)− P (sk, |(sk+1, . . . , s′i, . . . , sn)|m)

= π(si, sk, |s−{i,k}|m)− π(s′i, sk, |s−{i,k}|m) (8)

andP (sk, |s−k|m)− P (s′k, |s−k|m) = πk(sk, |s−k|m)− πk(s′k, |s−k|m) (9)

for all i ∈ {k + 1, . . . n} and si, sk ∈ {m, a}.Once again we can use Eq. 3.2 in Monderer and Shapley (1996) to construct the potential.

For notational convenience we now let x denote the number of small banks playing themorning strategy, so x = |s−k|m. The function is constructed in two parts, corresponding tothe strategy choice of the merged bank. First, suppose sk = m. Then applying Eq. (3), andusing Eq. (7), we have

P (m,x) = −x∑`=1

(n− k − `)αF −n−k−x∑`=1

(n− 1)D

= −((n− k)x− 1

2x(x+ 1))αF − (n− k − x)(n− 1)D, (10)

for x = 1, . . . , n−k. Note that this is the same formula as Eq. (4) with the number of banksreduced to n − k and the weight on payments to small banks changed to α. Second, from(6) and (9), we have

P (a, x) = P (m,x) + (k + (1− α)(n− k − 1))((n− k − x)F − (n− k)D), (11)

for x = 1, . . . , n− k. It is easily verified that Eq. (11) meets the requirement of Eq. (8) forsk = a. That completes the construction.

The function P is a second order polynomial in x for sk ∈ {m, a}, and hence each partof the function has a unique minimum point when evaluated over a continuous domain.As before, the local or global maximizers of the potential function correspond to the Nashequilibria of the game and we summarize them in the following proposition.

Proposition 4 If F < D, then P attains a global maximum at (m,n − k) and there are

no other local maximizers. If D ≤ F < 2D, then P attains a local [global] maximum at

(a, 0) [(m,n− k)]. If F = 2D, then P attains a global maximum at (m,n− k) and (a, 0). If

F > 2D, then P attains a local [global] maximum at (m,n− k) [(a, 0)].

14

If liquidity is relatively cheap (F < D) then our game has a unique equilibrium andcoordination is not needed to sustain the efficient equilibrium. Moreover, the adjustmentprocess always leads to the efficient equilibrium. Hence, early processing of payments willnot be affected by any wide-scale disruption. In contrast, if liquidity is relatively expensive(F ≥ D) then our game has two equilibria, one where all banks play the morning strategyand one where they play the afternoon strategy. We shall focus on the latter case below.

5.3 Convergence following disruptions

Once more, we assume that the system initially is in the efficient morning equilibrium, butthat it is hit by a disruption which forces a set of banks - that may or may not includethe merged bank - to play the afternoon strategy. The adjustment process following adisruption is governed by the function P (sk, x), sk ∈ {m, a} specified in Eqs. (10) and (11).If P (sk, x − 1) − P (sk, x) > 0 for sk ∈ {m, a} then deviations by small banks from themorning to the afternoon strategy increase the potential and are profitable for the smallbanks. Conversely, if P (sk, x− 1)− P (sk, x) < 0 then deviations from the morning strategydecrease the potential and are not profitable for the small banks. For the merged bankthe interesting quantity is the difference between the two potentials, i.e., P (m,x)− P (a, x),which gives the change in potential associated with its deviation. If this change in potentialis positive then it is profitable for a merged bank playing the afternoon strategy to switch tothe morning strategy and vice versa if it is negative. Consequently, the intersection point ofP (m,x) and P (a, x), defined in terms of continuous x, is important in terms of the dynamicsof the adjustment process. We denote this intersection by xc.

The following lemma summarizes key aspects of the function P for each strategy of themerged bank, assuming continuous x.

Lemma 1 P (a, 0) > P (m, 0). Moreover, for F ≥ D, there exists a unique point of intersec-

tion xc of P (m,x) and P (a, x) in the interval [0, n− k], which does not depend on α.

Proof. Solving P (m,x) = P (a, x) one obtains the single solution xc = (n−k)(F−D)F

, whichfalls in [0, n− k] whenever F ≥ D.

Let µm denote the integer value that minimizes P (m,x) over the set of integers from 0 ton−k.13 Potential functions are shown for two qualitatively distinct cases allowed by Lemma1 in figures 3 and 4.

5.3.1 Disruptions to small banks

In this case we are looking at a movement to the left (from the far right) along the curveslabeled P (m,x) in figures 3 and 4. We distinguish three qualitatively distinct cases: self-

13We assume µm is unique. See footnote 11.

15

n-k 0

Self-

perpetuating

Self-

reversing

x

),(^

xmP

),(^

xaP

xc

m

Figure 3: Potential function in a case where µm > xc.

n-k 0

Self-

perpetuating

Indeter-

minate Self-

reversing

x

),(^

xmP

),(^

xaP

xc

m

Figure 4: Potential function in a case where µm < xc.

16

reversing, self-perpetuating and indeterminate. Self-reversing refers to the fact that the sys-tem will converge back to the efficient, morning equilibrium on its own following a disruption.There are always small enough disruptions that are self-reversing. Self-perpetuating meansthe system will converge to the inefficient, afternoon equilibrium in the absence of centralbank intervention. This case exists if and only if F > α

αD, which implies P (m, 0) > P (m, 1).

The scalar αα

is decreasing in α and increasing in k. Hence, if funds are concentrated in themerged bank (i.e., α is low) or the merged sector (k) is large the self-perpetuating disruptionsare only a problem in “high” fee regimes. In the indeterminate case the dynamic adjustmentprocess will converge to one of the two equilibria, but neither is reached with probability1. The existence of indeterminate dynamics depends on the relative positions of µm and xc.The indeterminate case does not exist (except at the single point x = µm) if µm > xc (seefigures 3 and 4).

Self-reversing Any disruption to small banks that keeps the system at a point to theright of the intersection of P (m,x) and P (a, x) where P (m,x − 1) − P (m,x) < 0 (i.e., thecurve labelled P (m,x) is upward sloping) will be self-reversing (see figures 3 and 4). Duringthe disruption none of the unaffected banks will switch to the afternoon strategy, nor willthe merged bank since P (a, x) − P (m,x) < 0 for x > xc. Once the disruption ends thedisrupted banks will switch back to the morning strategy as soon as they are selected todeviate by the adjustment process. This is true because P (m,x− 1)− P (m,x) < 0 impliesP (m,x + 1) − P (m,x) > 0. In these cases the length of the disruption does not affect theoutcome of the adjustment process.

Since self-reversing disruptions do not require intervention by the Federal Reserve torestore efficient processing it is interesting to ask how the cost parameters and the concen-tration of payments in the system affect the range of self-reversing disruptions. We have thefollowing result.

Proposition 5 Suppose D, n, and k are fixed. The largest size of a disruption to the small

banks that is self-reversing decreases or remains unchanged as F or α increases.

Proof. Since µm minimizes P (m,x) over the set of integers from 0 to n − k, min{x ∈{1, ..., n− k}|P (m,x− 1)− P (m,x) < 0} = µm + 1.

Case 1 : xc < µm (see Figure 3). The largest size of a disruption to the small banks thatis necessarily self-reversing is given by dSR = n−k−µm−1. By (10), P (m,x−1)−P (m,x) =α(n− k− x)F − (n− 1)D, hence increases in α or F cause µm to increase or stay the same.Hence, dSR decreases or stays the same as α or F increase.

Case 2: xc > µm (see Figure 4). The largest size of a disruption to the small banks thatis necessarily self-reversing is given by dSR = int(n−k−xc), where int(r) denotes the integerpart or floor of the real number r. By lemma 1, xc does not depend on α. From the proof oflemma 1, xc = (n−k)(F−D)

F, and hence ∂xc

∂F= (n−k)D

F 2 > 0. Hence, increases in α or F cause xc

to increase or stay the same. Hence, dSR decreases or stays the same as α or F increase.

17

The above proposition says that banking systems where more funds are concentrated inthe merged banks (i.e., with smaller α) are more resilient to shocks in the sense that largershocks are self-reversing. This is intuitive because, the coordination problem that leads toinefficient outcomes is reduced when more transfers involve the merged bank. In the extremecase where α = 0, for instance, all transfers are bilateral exchanges between each of the smallbanks and the merged bank. There are no issues of coordination between each pair of smallbanks.

Self-perpetuating Any disruption to small banks that puts the system at a point whereP (m,x − 1) − P (m,x) > 0 will be self-perpetuating (see figures 3 and 4). Our next resultexplains how different concentrations of funds within the banking system affects the minimumsize of disruption that is self-perpetuating.

Proposition 6 Choose F, D, n, k and α with F > ααD. Then, the smallest sized disrup-

tion to the small banks that is self-perpetuating decreases or remains unchanged as α or F

increases.

Proof. Since µm minimizes P (m,x) over the set of integers from 0 to n − k, max{x ∈{1, ..., n − k} : P (m,x − 1) − P (m,x) > 0} = µm. The condition F > α

αD implies that

P (m, 0) − P (m, 1) > 0 and hence it ensures that µm ≥ 1, which is necessary and sufficientfor the self-perpetuating case to exist.

There are two cases to consider. In the first case, µm > xc ≥ 0 (See Figure 3). Then, thesmallest sized disruption to the small banks that is self-perpetuating is dSP = n−k−µm+1.This is true because for integer points x satisfying xc < x < µm−1, P (m,x−1)−P (m,x) > 0,P (m,x + 1) − P (m,x) < 0 and P (m,x) − P (a, x) > 0. Moreover, for points x satisfyingx < xc, P (a, x − 1) − P (a, x) > 0 and P (a, x) − P (m,x) > 0. Hence, all selected deviatorsswitch to the afternoon strategy and the adjustment process converges toward the afternoonequilibrium while the disruption is on and does not reverse direction once the disruptionends. In contrast, disruptions that correspond to points to the right of µm are self-reversingfor the reasons given above. Since, by (10), P (m,x−1)−P (m,x) = α(n−k−x)F−(n−1)D,increases in α or F cause µm to increase or stay the same. Hence, dSP decreases or stays thesame as α or F increase.

In the second case, µm < xc (See Figure 4). Disruptions that affect n−k−µm+1 or moresmall banks are self-perpetuating since P (m,x−1)−P (m,x) > 0, P (m,x+1)−P (m,x) < 0and P (a, x)− P (m,x) > 0 for x ≤ µm− 1. Disruptions that affect fewer than n− k−µm + 1small banks are not (necessarily) self-perpetuating since P (m,x + 1) − P (m,x) > 0 forx ≥ µm and hence disrupted small banks that are selected by the adjustment process mayswitch back to the morning strategy one the disruption is over (provided the large bank hasnot switched to the afternoon strategy). Once again, the result holds because increases in αor F cause µm to increase or stay the same.

18

Indeterminate In cases where xc > µm, any disruption to small banks that places thesystem to the right of µm and to the left of xc (see Figure 4) will have indeterminate dynamics.This is because, for x such that µm < x < xc we have that P (m,x− 1)− P (m,x) < 0 andP (m,x)− P (a, x) < 0. The merged bank has an incentive to switch to a, and the outcomeof the adjustment process depends on whether the disruption ends and enough small banksdeviate back to the morning strategy before the merged bank deviates to the afternoonstrategy (in which case the system converges to the afternoon equilibrium). The likelihoodthat the process will converge to the inefficient equilibrium increases if the disruption islonger.

Proposition 7 Suppose F, D, n, k and α are such that µm < xc. The probability that the

system converges to the inefficient, afternoon equilibrium following a disruption to the small

banks of size n− k −∣∣Nnd

∣∣ , with xc <∣∣Nnd

∣∣ < µm increases as τ increases.

Proof. Let fnd(k) denote the probability the adjustment process selects the merged bankk to deviate from the set Nnd in any round of the adjustment process before the disruptionis over. The probability the merged bank will still be playing the morning strategy after τperiods is

(1− fnd(k))τ .

After period τ the previously disrupted small banks will begin to deviate back to the morningstrategy. The probability that xc −

∣∣Nnd∣∣ small banks in Nd are selected to deviate by the

process before bank k is selected depends on the probability function f but does not dependon the duration τ. Hence, the probability that the process will converge to the afternoonequilibrium is increasing in τ.

5.3.2 Disruptions to the merged bank

A disruption to the merged banks knocks the system from the end (far-right) point of thecurve labeled P (m,x) down to the right end point of the curve labeled P (a, x) in figures 3and 4. The response of the small banks depends on the sign of P (a, n− k)− P (a, n− k− 1).A positive sign means the small banks will continue to play the morning strategy and thesystem will return to the efficient equilibrium once the disruption to the merged bank isover. A negative sign means the small banks will adjust to the afternoon strategy. Wherethe system ends up in the latter case will depend on the length of the disruption to themerged bank and how fast the small banks adjust to the afternoon strategy.

Quick calculations show that the former case occurs if and only if F < αα−αD. The scalar

αα−αD is increasing in α and decreasing in k. Hence, if funds are concentrated in the mergedbank (i.e., α is low) or the merged sector (k) is large disruptions to the merged bank willonly be self-reversing in “low” fee regimes.

We have the following, immediate results

Proposition 8 Disruptions to the merged bank are self-reversing if and only if F < αα−αD.

19

Proof. From Eq. (11), P (a, n−k)−P (a, n−k−1) = D(n−1)−F (n−1−α(n−k−1)) > 0if F < α

α−αD. Hence no small banks will switch to the afternoon strategy. Since P (m,n−k) >

P (a, n − k), once the disruption is over the merged bank will switch back to the morningstrategy.

Corollary 1 If a disruption to the merged bank is self-reversing at 0 < α ≤ α, then it is

also self-reversing at any α′ ∈ (0, α).

In the latter case we can again relate the probability of converging to the inefficientequilibrium to the duration of the disruption.

Proposition 9 Suppose F > αα−αD. The probability that the system converges to the ineffi-

cient equilibrium following a disruption to the merged bank increases as τ increases.

Proof. From Eq. (11), P (a, x)− P (a, x− 1) < 0 for all x ≤ n− k if F > αα−αD. Hence,

along P (a, x) deviations from m to a are profitable from any starting point. Whether or notthe afternoon equilibrium is reached in this setting depends on how many small banks deviateto the afternoon strategy before the disruption to the merged bank ends. The system willconverge to the inefficient equilibrium if the number of banks that deviate to the afternoonstrategy before the disruption ends is greater than n − k − xc = (n − k)D

F. Since all banks

playing m that are selected by the SAP in each period deviate to a (and banks playing ado not deviate back to m) the probability that (n − k)D

Fsmall banks deviate from a to m

in τ rounds is increasing in τ .14 Once more than (n − k)DF

banks have deviated to a andthe disruption ends the merged bank will, when selected, not deviate back to m. Hencethe situation is the same as the homogeneous case covered by Proposition 3 (c) where thenumber of banks is n− k and the starting point is to the left of µ. Hence, the SAP leads tothe inefficient afternoon equilibrium.

The system may converge to the inefficient equilibrium even if the disruption ends withxτ > xc, where xτ = |sτ |m is the number of banks playing the morning strategy whenthe disruption ends. Then convergence depends on whether the merged bank is selected todeviate before the number of small banks that deviate to a under the SAP reaches (n−k)D

F.

However, here we can apply the fact that xt < xt′

for t < t′. That is, the longer thedisruption occurs, the more banks there will be that are playing a, and hence the fewerthe number of additional deviations that are required for x to fall below xc (at which pointthe system converges to the inefficient equilibrium for sure by the argument given in thepreceding paragraph). Hence, the probability that enough remaining banks are selected bythe distribution function f to switch from m to a before k is selected to switch from a to mincreases as τ increases.

14In this proof we do not compute probabilities explicitly as this is not necessary for the stated result.However, explicit computations are possible using the distribution function fnd defined on the power set

2Nnd

.

20

5.4 Role for moral suasion

The appropriate policy response of the central bank is different across the many cases andthe challenge for the central bank is to quickly identify the nature of the disruption. Ceterisparibus, a central bank prefers not to change its (intraday) liquidity regime because doing sohas implications for the credit risk it assumes and its monetary policy stance.15 In the self-reversing cases, the central bank obviously need take no action. In the self-perpetuating casesthe central bank can drive the cost of liquidity towards zero in order the change the natureof the game or it can attempt to impact behavior through moral suasion. Moral suasioncan work in this instance because the central bank facilitates a correlated equilibrium (asdefined by Aumann (1974, 1987)), in which the correlating device recommends the morningprofile (m, ...,m) with probability one. It is in the best interests of any given bank to followthis recommendation given the belief that each of the other banks will follow. In systems aslarge as the US it is not clear whether the central bank can affect the equilibrium outcomethrough moral suasion alone.

In the indeterminate case discussed above the outcome of the adjustment process wasshown to depend crucially on the reaction (or lack thereof) of the large bank. Here there is amore hopeful role for the central bank to perform moral suasion. The inefficient equilibriumcan be avoided if the central bank is able to convince the large bank to be patient and notswitch to the afternoon strategy. Patience would be rewarded because after the disruptionends the system would converge back to the morning equilibrium.

In cases with F > αα−αD where the disruption affects the large bank, the ability of

the large bank to recover quickly is instrumental in terms of achieving coordination on theefficient equilibrium. A situation where a large bank is forced to delay may progressivelydeteriorate as small banks switch to later processing. Eventually even a recovered large bankmay not want to process payments early. Ex ante, stakeholders in the interbank paymentsystem may push for stronger resiliency requirements for larger banks.16 The pivotal role ofthe merged bank, in these instances, suggests that a bank might be considered “too big tofail” in part due to its role in terms of maintaining coordination on smooth processing ofpayments.17

15“On Monday morning September 17, the Federal Open Market Committee (FOMC) met by conferencecall at 7:30 a.m. eastern time and voted lower the target for the federal funds rate by 50 basis points to 3percent. In its statement the Committee said: The Federal Reserve will continue to supply unusually largevolumes of liquidity to the financial markets, as needed, until more normal market functioning is restored. Asa consequence the FOMC recognizes that the actual federal funds rate may be below its target on occasionin these unusual circumstances.” (Lacker, 2004, p. 949).

16See, for example, “Best Practices to Assure Telecommunications Continuity for Financial In-stitutions and the Payment & Settlements Utilities” published by the Payments Risk Committee(www.newyorkfed.org/prc/telecom.pdf). The Payments Risk Committee is a private sector group of se-nior managers from U.S. banks that is sponsored by the Federal Reserve Bank of New York.

17See Stern and Feldman (2004, chapter 12) for a general discussion of too big to fail policy and thepayment system.

21

6 Settlement Dynamics following the Lehman

Bankruptcy

The increase in uncertainty and erosion of trust that followed the bankruptcy of LehmanBrothers on September 15, 2008 led banks to curtail the provision of intraday credit tocustomers. It also made banks reluctant to submit payments to one another out of fear thatfunds would not return, thereby hindering their ability to make critical payments later in theday. Credit risk is easily incorporated into our model. Following Bech (2008), heightenedcredit risk can be interpreted as an increase in the cost of early processing and a reductionin the cost of delay. The increase in the cost of early processing arises from possibility thatmorning payments may not be returned (due to a default) and hence additional borrowingcosts may be incurred. Let δ denote be the probability a bank is closed by its regulator atnoon and let θ denote the cost given default.18 Then the expected cost of making a morningpayment becomes F + δθ > F . The reduction in the cost of delay is based on the idea thatthere is no reputation cost of delaying to a bank if it defaults. Hence, the expected cost ofdelaying to a single bank is (1 − δ)D < D, and the expected cost of playing the afternoonstrategy is the cost of delay times the expected number of surviving counterparties, namely,(1− δ)(n− 1)D. If we think in terms of the homogeneous bank model, Proposition 3 appliesand the increase in credit risk can move the system from cases (a) or (b) to case (c).

Our model suggests that heightened credit risk can result in widescale deviations to lateprocessing. Empirical evidence that is broadly consistent with our predictions is providedin figures 5 and 6.19 In Figure 5, we show the average time of settlement (value-weighted)of Fedwire payments over the period from April 1, 2008 through March 31, 2009. Wehave removed predictable calendar effects as payment tend to settle later on e.g. largepayment days such as the first, middle and last day of the month. The figure shows thatprior to the Lehman bankruptcy, the average settlement time was around 2:05 pm witha standard deviation of 7 minutes. Over the week following the Lehman bankruptcy theaverage settlement time increased by 25 minutes, or more by than 3 standard deviations,before dropping back quickly close to the prior level. In contrast, from early November, theaverage time of settlement moved noticeable earlier to around 1:30 pm or half an hour earlierthan prior. This subsequent improvement in settlement timing was first and foremost a resultof the Federal Reserves’ effort to restore liquidity and stability to the U.S. financial system.A by-product of these efforts was an unprecedented injection of reserves into the bankingsystem which flooded the payment system in liquidity (see Bech, Martin and McAndrews,forthcoming). This move greatly reduced the cost of payment processing for many banks,lowering F in our model, and returning the system dynamics to case (a) in Proposition 3.

18The costs given default include the bank’s cost of going to the discount window in order to square itsaccount at the end of the day. Missing offsetting payments from defaulted counterparties create a deficit.Moreover, banks may experience principal losses on payments arising from proprietary trading operationsor due to intraday credit extensions to customers that have funded outgoing payments to defaulted banks.

19A model with heterogenous default probabilities and higher-dimensional strategies that allow banks todifferentiate counterparties to whom they delay would be required to match the data exactly.

22

1:00 PM

1:30 PM

2:00 PM

2:30 PM

3:00 PM

Apr-08 May-08 Jun-08 Jul-08 Aug-08 Sep-08 Oct-08 Nov-08 Dec-08 Jan-09 Feb-09 Mar-090

250

500

750

1,000

$ Billions

Average Time of Settlement (Value Weighted) Reserves (Right Axis)

September 15, 2008

Figure 5: Average time of Fedwire settlement.

Payment flows of 16 core participants on Fedwire are depicted as a network in Figure6. Participants are shown as nodes; scaled by the total value of payments that they send.Links illustrate payment flows between participants. One payment between two participantsestablishes a linkage. We do not assign weights to the links depending on the value transferredacross but rather apply a coloring scheme to indicate settlement delay. A gray colored linksignifies that payments were settled within the normal time frame on that particular day(measured relative to a reference period). Black, on the other hand, indicates a significantdegree of delay.20 We show similar networks for each business day in September of 2008 andthe date of the network is indicated by the number preceding it.

As in Soramaki et al (2007), the payments networks, spanned by the largest participantsin Fedwire, form an almost complete network, i.e., a network where all participants areconnected to one another. The network for say September 2, 2008 illustrates a ‘normal’ day

20Specifically, we computed delay as follows. First, we computed the time at which 75 percent of thetotal value sent between any two participants on a given day was settled. Our sample period ran fromApril 1, 2008 (i.e. two weeks after the failure of Bear Stearns) through September 2008. Using days priorto September 15, 2008 as our reference period we constructed an empirical distribution of 75th percentilesettlement times. We then defined settlement on given day to be delayed between two banks if the time atwhich 75 percent of the daily value has settled is later than the 95th percentile from the distribution of suchsettlement times over the reference period. For example, if during the reference period the percentile wasreached no later than say 5:20pm on one in twenty days and on the day of interest, say the percentile wasreached at 5:40pm, then we would label settlement as significantly delayed.

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Monday

Tuesday

Wednesday

Thursday

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in Fedwire with a limited number of links showing delay (less than five percent) and withno discernable pattern. In contrast, the network for September 9, 2008, illustrates the casewhere a single participant (located at 4:30 o’clock) was subject to an idiosyncratic shock andwas late in terms of sending payments to the majority of other participants. This may havecaused a couple of other participants to be late as well as in terms of sending payments toother participants.

During the second week of September 2008, it became increasing likely that LehmanBrothers—the fourth largest investment bank—would not survive as an interdependent in-stitution over the following weekend. Despite high level meetings with relevant stakeholdersat the Federal Reserve Bank of New York, no solution was found and Lehman Brothersfiled for bankruptcy on September 15, 2008. Over the same weekend, Bank of America an-nounced its intention to acquire Merrill Lynch & Co., Inc. and during the following week theFederal Reserve, with the full support of the Treasury, agreed to provide liquidity support toAmerican International Group, Inc. Upon its collapse on September 25, 2008, WashingtonMutual Bank, a large thrift institution, became the largest failure ever of a financial insti-tution in the United States; its stakeholders absorbed significant losses. This tumultuousperiod affected the settlement dynamics over Fedwire.

As shown in Figure 6, settlements over Fedwire among the selected group of participantswas relatively normal through September 12, 2008 with the exception of September 9 whichwas the day of the before mentioned idiosyncratic shock to one participant. However, thischanged significantly the following two weeks. On many days there were significant delaysin the system. On September 17, 19 and 22 more than half of the links between participantswere significantly delayed. The fact that a large number of links are delayed on severaldays highlights that the impact of changing fundamentals on payment system dynamics.There was rapid convergence to late processing that was sustained over an extended periodbefore lower liquidity costs redirected the system back toward a state that approximates the“morning equilibrium.”

7 Conclusion

The containment of systemic risk within the U.S. financial system in the event of a wide-scaledisruption rests on the rapid recovery and resumption of the core clearing and settlementactivities that support the financial markets. Fedwire is a one of the most critical compo-nents of clearing and settlement process and hence the resilience of the system itself andits participants is of paramount importance. As shown by McAndrews and Rajan (2000)and McAndrews and Potter (2002), coordination in the transmission of payments is a vitalpart of the day-to-day operations in Fedwire. The ability to maintain coordination, wherebanks tend to settle promptly and synchronize their payment activity, can potentially beinstrumental in mitigating the impact of a wide-scale disruption to the financial system.

In this paper we have investigated how a wide-scale disruption is likely to impair thesmooth functioning of the interbank payment system. Such a disruption will almost bydefinition - create operational difficulties for the system and its participants in some way

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or another. However, we address a problem that is perhaps less obvious: the operationaldifficulties of some participants may induce other participants to change behavior in termsof how they process payments. In particular, this may lead to a break down of coordination.

This paper argues that the ability of banks in Fedwire to maintain payment coordinationfollowing a wide-scale disruption depends critically on a number of different factors. First ofall, it depends on the size of the disruption. A disruption that affects a large part of the nationor a disruption that hits a key geographical area is more likely to result in the break downof payment coordination as more banks will experience operational difficulties. Secondly, itdepends on the relative cost of liquidity and the cost of postponing payments. The cheaperthe liquidity, the more likely banks will be to maintain coordination by themselves. TheFederal Reserve’s response to the tragic events of September 11, 2001 in terms of providingunprecedented amount of liquidity to the system at virtually zero cost was in part aimedat minimizing the risk of banks holding back processing payments. Thirdly, we argue thatthe banking structure can influence the smooth functioning of the payment system after awide-scale disruption and that a bank can be considered too big to fail in a new interestingway. We showed that the resiliency of a large bank could be important not only in terms ofits share of the payment flow but also in terms of this bank being pivotal in maintaining thecoordination. Moreover, we showed that moral suasion might be an effective policy tool. Ifthe Federal Reserve can persuade the large banks to wait for small banks to resume timelyprocessing following a disruption, then more drastic measures, such as reducing the liquiditycosts, might not be required to restore coordination. Fourthly, we showed that the topologyof the underlying payment flow is key to understand how participants might change behaviorfollowing disruptions and hence for the resiliency of or systemic risk inherent in the interbankpayment system.

In addition to thinking about physical disruptions to payment system participants ouranalysis of payment system dynamics also applies to shocks to fundamentals which affect theparameters of the intraday liquidity management game. We addressed the case of heightenedcredit risk that arose during the time of the Lehman Brothers failure. This episode repre-sents one observation of a disruption and hence cannot be used to formally test our model.However, the observed changes in behavior of system participants in response to increasedcounter party concerns conform with our basic theory.

References

[1] Allen, Franklin, and Douglas Gale. (2000) “Financial Contagion.” Journal of PoliticalEconomy, 108:1, 1–33.

[2] Angelini, Paulo, Giuseppe Maresca and Daniela Russo (1996). “Systemic Risk in theNetting System.” Journal of Banking and Finance, 20:5, 853-868.

[3] Aumann, Robert J. (1974) “Subjectivity and Correlation in Randomized Strategies.”Journal of Mathematical Economics, 1:1, 67–96.

26

[4] Aumann, Robert J. (1987) “Correlated Equilibrium as an Expression of Bayesian Ra-tionality.” Econometrica, 55, 1–18.

[5] Bech, Morten L. (2008) “Intraday Liquidity Management: A tale of Games Banks Play.”Federal Reserve Bank of New York Economic Policy Review, 14:2, 7-23.

[6] Bech, Morten L., and Rodney J. Garratt. (2003) “The Intraday Liquidity ManagementGame.” Journal of Economic Theory, 109:2, 198–210.

[7] Bech, Morten L., Antoine Martin, and James McAndrews. (2012) “Settlement Liquidityand Monetary Policy Implementation – Lessons from the Financial Crisis.” FederalReserve Bank of New York Economic Policy Review, forthcoming.

[8] Bernanke, Ben S. (1990) “Clearing and Settlement during the Crash,” Review of Fi-nancial Studies, 3:1, 133–151.

[9] Coleman, Stacy P. (2002) “The Evolution of the Federal Reserve’s Intraday CreditPolicies.” Federal Reserve Bulletin, February.

[10] Corchon, Luis C. (1994) “Comparative Statics for Aggregative Games: The StrongConcavity case.” Mathematical Social Sciences, 28:3, 151–165.

[11] Devriese, Johan, and Janet Mitchell. (2006) “Liquidity Risk in Securities Settlement.”Journal of Banking and Finance, 30:6, 1807–1834.

[12] Dubey, Pradeep, Abdreu Mas-Colell, and Martin Shubik. (1980) “Efficiency Propertiesof Strategic Market Games: An Axiomatic Approach.” Journal of Economic Theory,22:2, 339–362.

[13] Federal Reserve Bank of New York. (2001) Annual report.

[14] Ferguson, Roger W. (2003) “September 11th, the Federal Reserve, and the FinancialSystem.” Speech at Vanderbilt University, Nashville, Tennessee, February 5.

[15] Fleming, Michael J., and Ken D. Garbade. (2002) “When the Back Office Moved tothe Front Burner: Settlement Fails in the Treasury Market after 9/11.” Federal ReserveBank of New York Economic Policy Review, 8:2, 35–57.

[16] Galbiati, Marco, and Kimmo Soramaki. (2011) “An Agent-Based Model of PaymentSystems.” Journal of Economic Dynamics and Control, 35:6, 859–875.

[17] Goyal, Sanjeev. (2009) Connections: An Introduction to the Economics of Networks.Princeton, NJ: Princeton University Press.

[18] Goyal, Sanjeev, and Fernando Vega-Redondo. (2005) “Network Formation and SocialCoordination.” Games and Economic Behavior, 50, 178–207.

27

[19] Hendricks, Darryll, John Kambhu, and Patrica Mosser. (2007) “Systemic Risk and theFinancial System.” In New Directions for Understanding Systemic Risk: A Report ona Conference Cosponsored by the Federal Reserve Bank of New York and the NationalAcademy of Sciences, edited by John Kambhu, Scott Weidman, and Neel Krishnan, pp.82-106. Washington D.C.: The National Academies Press.

[20] Humphrey, David B. (1986) “Payments Finality and Risk of Settlement Failure.” InTechnology and the Regulation of Financial Markets: Securities, Futures, and Banking,edited by Anthony Saunders and Lawerence J. White, pp. 97-120. Lexington, MA:Lexington Books.

[21] Kahn, Charles M., James McAndrews, and Wlliam Roberds. (2003) “Settlement Riskunder Gross and Net Settlement.” Journal Money Credit and Banking, 35:4, 591–608.

[22] Lacker, Jeffrey M. (2004) “Payment System Disruptions and the Federal Reserve fol-lowing September 11, 2001.” Journal of Monetary Economics, 51:5, 935–965.

[23] McAndrews, James, and Simon Potter. (2002) “Liquidity Effects of the Events ofSeptember 11th.” Federal Reserve Bank of New york Economic Policy Review, 8:2,59–79.

[24] McAndrews, James, and Samira Rajan. (2000) “The Timing and Funding of FedwireFunds Transfers.” Federal Reserve Bank of New york Economic Policy Review, 6:2.,17–32.

[25] Monderer, Dov, and Lloyd S. Shapley. (1996) “Potential games.” Games and EconomicBehavior, 14:1, 124–143.

[26] Rosenthal, Robert W. (1973) “A Class of Games Possessing Pure-Strategy Nash Equi-libria.” International Journal of Game Theory, 2:1, 65–67.

[27] Soramaki, Kimmo, Morten L. Bech, Jeffrey Arnold, Robert J. Glass, and Walt E.Beyeler. (2006) “The Topology of Interbank Payment Flows.” Physica A: StatisticalMechanics and its Applications, 379:1, 317-333.

[28] Stern, Gary H., and Ron J. Feldman. (2004) Too Big To Fail - The Hazards of BankBailouts. Washington D.C.: Brookings Institution Press.

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