rollover risk and endogenous network dynamics

Comput Manag Sci (2013) 10:213–230DOI 10.1007/s10287-013-0172-8

ORIGINAL PAPER

Rollover risk and endogenous network dynamics

Jose Fique · Frank Page

Received: 10 July 2012 / Accepted: 17 April 2013 / Published online: 7 May 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract Using a dynamic network formation model, solved numerically, we studybanks’ rollover decisions. We find that when the existence of linkages between mar-ket participants generates an informational externality, the newly formed networkis conditioned by past architectures. Moreover, this inertia is strongly dependent onmacroeconomic conditions, such as investors’ risk appetite. Simulations show thatfor intermediate values of the risk appetite’s parameter the financial network exhibitstipping points, i.e., the inability to maintain a threshold number of linkages may pushthe market into a gridlock. In this context, we study also how policy instruments, suchas taxes and subsidies, affects debt rollover. Since a reduction in the policy level playsthe same role as an improvement in economic fundamentals, the creation of interbankconnections can be stimulated by it. Thus, in order to restart lending after a majorstress situation in the interbank market a considerable reduction in the policy level isrequired, advising a counter-cyclical policy similar to the ones recently proposed withrespect to capital requirements.

Keywords Interbank networks · Credit crisis · Liquidity freeze ·Endogenous network formation

1 Introduction

One of the most striking phenomena of the 2007–2009 financial crisis was the rapid-ity with which liquidity in the interbank markets dried up, especially in long term

J. Fique (B) · F. PageDepartment of Economics, Indiana University Bloomington, Bloomington, Indiana 47405, USAe-mail: [email protected]

F. PageCentre d’Economie de la Sorbonne, Universite Paris 1, Pantheon-Sorbonne,75647 Paris Cedex 13, France

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214 J. Fique, F. Page

maturities. In network literature terminology, the once dense interbank network thatallowed highly liquid banks to channel liquidity to those banks with investmentopportunities, transited to a sparse architecture. The sudden failure of the once well-functioning interbank loan network during the recent financial crisis has given momen-tum to the movement toward major, worldwide regulatory reform to minimize the pos-sibility of another interbank network failure and to make the financial network morerobust. The shortcomings of the regulatory framework exposed by the crisis lead tothe design and implementation of new instruments aimed at the insuring the stabilityof the financial system. One of these instruments, now in use in several Europeancountries, is a special banking levy/tax. The levy/tax aims not only to raise funds toreduce the cost to taxpayers incurred with past (or future) rescues of the financialinfrastructure but also to provide banks with the correct incentives for risk taking. Bytargeting balance sheet items, this instrument is able to create the proper incentives tomove financial institutions towards safer grounds.

The purpose of this paper is twofold: (1) to analyze within a dynamic network for-mation game how macroeconomic conditions, such as investors’ risk appetite, affectrollover decisions and (2) to determine the effects that policy measures, such as theabove mentioned special banking levy, have on the endogenous dynamics of networkformation. We find that because the existence of linkages between market participantsgenerates an informational externality, the newly formed network is strongly condi-tioned by past architectures. Simulations show that this inertia is strongly dependent onmacroeconomic conditions, such as investors’ risk appetite. The numerical exercisesreveal that for intermediate values of the risk appetite parameter, the inability to main-tain a threshold number of linkages may push the market into a gridlock. Moreover,this tipping point property implies that the recovery from a market freeze situationrequires good conditions of a magnitude considerably greater than the magnitude ofthe bad conditions that precipitated the crisis in the first place leading to a networkinduced inertia. Finally in order to capture in our model the regulatory reform enactedafter the onset of the recent financial crisis we add a policy parameter to our base-line model. Since the dynamic network formation process is strategically driven, wecan account for endogenous effects generated by the policy measures. This strategicelement allows us to contribute to the discussion of what the optimal policy shouldbe, particularly in times of severe stress in the interbank market. Since we model thepolicy measure as a cost to the activation of interbank connections, we find that asubstantial decline in the policy level is required in order to re-start lending activitywhen the market experiences severe stress situations.

We model banks’ rollover decisions within a network formation game. Lenders arefaced with the decision to rollover the debt that a borrower has outstanding, with theborrower’s ability to repay being stochastic. In addition to a common prior distributionwith respect to the borrower’s solvency, each lender receives a private signal. Usingthis updated information and partially observing the network, each lender updates itsbeliefs and decides whether or not to rollover the outstanding debt. Since this lender’sdecision may be observed by another lender prior to its own rollover decision, favorablesignals with respect to the borrower’s creditworthiness are propagated slowly throughthe network. The speed of information diffusion depends on initial conditions and onthe draws of nature.

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Rollover risk and endogenous network dynamics 215

Solving the model numerically, we find the above mentioned tipping point property.One of its implications is that the network formation process exhibits hysteresis—the newly formed network is dependent on past architectures. Moreover, our policyanalysis suggest that when the process is driven into gridlock due to severe conditions,the policy level that allows the system to recover from the market breakdown situationis considerably lower than the one required to precipitate a freeze departing from thecomplete participation scenario. Since a decrease in the policy level plays the same roleas an increase in the ex-post return promised by the lender, the activation of interbanklinkages can be stimulated by it. Thus, in order to restart lending after a major stresssituation in the interbank market a considerable reduction of this parameter is required.

Even though the financial networks literature goes back to the 1970’s (see for exam-ple Thore 1969), it gained a considerable momentum in the recent years. Arguablythis growth can be attributed to the 2007–09 financial crisis. For a substantial literaturereview on this topic see Nagurney (2008) and Allen and Babus (2009), and referencestherein. The increasing popularity of this tool can be attributed to its ability to modelfinancial intermediation, see for example Nagurney and Ke (2001). This has beenproven to be particularly true in the case of interbank markets as Allen and Gale (2000)show. The stress experienced in interbank markets was a distinguishing feature of the2007–2009 financial crisis. One explanation provided to account for the spike in inter-bank market spreads and the decline in loan maturity terms was the rise in counterpartrisk of borrowing banks. From this perspective, which we follow in our model, ratesand volume adjust to the deteriorating borrower’s asset quality. The inability to obtainfinancing in interbank markets can even drive the borrowers to waive profitable invest-ment opportunities and engage in precautionary liquidity hoarding. This can lead to theeffective breakdown of the interbank market. Heider et al. (2009) show how a lemonsproblem can drive the interbank market to breakdown when borrowers have privateinformation with respect to their asset quality. Allen et al. (2012) analyze the welfareimplications of short-term vis-a-vis long-term financing in a network model. They findthat while the financial network is irrelevant when assets are financed in a long-termbasis, short-term rollover decisions depend strongly on the architecture of the system.When investors receive adverse news regarding the solvency of financial institutionsthe pattern of interconnections plays a vital role in assessing the survival perspectivesof each individual borrower. Anand et al. (2012) build a dynamical model in a globalgames framework where rollover decisions are stochastic and assume the form ofbad news with respect to the borrowers’ solvency. Using a random matching process,borrowers and lenders are matched and form the network. They show that it exhibitstipping-points and hysteresis—the breakdown of the interbank market is persistent anda more favorable environment is required to re-start its normal functioning. Our modeldiffers from their model in that we show that even without coordination failure issues,tipping points can occur depending on the importance of the network as an informa-tion propagation mechanism. Given the active role as liquidity insurance providersthat central banks all over the world have had since the onset of the recent financialcrisis we rule out misscoordination as a source of distress in the interbank market.

This paper is also related with the rational herding/information-based contagion lit-erature. When banks’ individual conditions become uncertain, investors may look forexternal sources to update their beliefs with respect to the outcome of their investments.

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When uncertainty increases and signals are noisy, a creditor may become reluctantto refinance an investment opportunity even though the underlying economic funda-mentals are solid. Chen (1999) builds a model where uniformed depositors updatetheir beliefs by observing runs on other banks. Thus, bank runs can become conta-gious since uniformed depositors can infer from a noisy signal that informed depositorshave chosen to withdraw their deposits in anticipation of adverse conditions. Caballeroand Simsek (2009) develop a complexity model, where banks face uncertainty withregards to the likelihood of being hit by a defaulting counterparty. When uncertaintyis severe enough, banks engage in precautionary liquidity hoarding. This in turn leadsto externalities that increase the stress in the interbank market and can drive it tofreeze completely. Our paper is also related to rational herding (see Banerjee 1992and subsequent research). However, we emphasize the role that the network plays inpropagating information. More precisely, we study how the choices made by eachlender interact with their partial knowledge of the network and how that affects theinformation available to the other lenders. Since the network formation process isstrategically driven, we can account for the endogenous effects that policy measureshave on banks’ updating process.

Moreover, since we also study how policy measures affect the structure of thefinancial networks, it is also related with the recent literature on systemic risk taxation.Tarashev et al. (2009), Staum (2009) and Tarashev and Drehmann (2011) proposemeasures to determine the optimal systemic risk charge based on the Shapley Value(Shapley 1952). However, these models take the topology of the network as given. Tothe best of our knowledge our paper is the first to study the effect of a policy parameteron the endogenous formation of the interbank network in a dynamic setting. Finally,the policy implications that derive from endogenous tipping point property have alsosome similarity with the recent interest, of policy makers and academia alike, incountercyclical capital requirements (see Gordy and Howells 2006; Repullo et al.2010; BCBS 2010; Shleifer and Vishny 2010 and Hanson et al. 2011).

2 The model

2.1 The space of directed networks

Let L be a finite set of banks liquidity endowed with typical element l, B a finite setof banks with typical element b that need to rollover debt to avoid bankruptcy, andA (l, b) = {1, 0} be the set of arcs representing the choice of lenders (L) to rollover(or not, respectively) the debt of the borrowers (B).

Definition 1 (Interbank Network) A network G is a subset of A × (L × B) withtypical element (a, (l, b)) such that (i) for all lenders l ∈ L , the section of G at lgiven by

G (l) := {(a, b) ∈ A × B : (a, (l, b)) ∈ G} (1)

is nonempty; and (ii) for all (a, (l, b)) ∈ G, the arc label a ∈ A (l, b) is feasible (e.g.,G ⊂ P (A × (L × B))).

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Fig. 1 Examples of dense and sparse networks

In example, suppose that L = {l1, l2, l3, l4} and B = {b1}. An interbank network isdense if a considerable (to be defined later) fraction of the lenders decide to rollover thedebt of borrower b1, i.e., G1 ={(1, (l1, b1)) , (1, (l2, b1)) , (1, (l3, b1)) , (1, (l4, b1))}.Whereas it is sparse otherwise, i.e., G2 = {(1, (l1, b1)) , (0, (l2, b1)) , (0, (l3, b1)) ,

(0, (l4, b1))}. Figure 1 depicts both networks.

2.2 Players and coalitions

In this game there are two types of nodes: lenders and borrowers. Since only lendersmake an actual decision they belong to the set of players and nodes, while the borrowersthat play a passive role belong solely to the set of nodes. Thus, lenders are price takersand only decide whether to rollover the debt or not1. We will assume throughout thatA-1 (Feasible Coalitions) Given lender set L , the set of all possible coalitions is givenP (L) the collection of all nonempty subsets of L . We will assume that the feasible setof coalitions is given by F1 = {S ∈ P (L) : |S| = 1}, the set of all coalitions havingof one element.

2.3 Stages of the game

Now that some of the primitives are defined, we can proceed to describe the interbanknetwork as a result of a dynamic network formation game. The game plays out in threestages:

Stage 0: Lenders, endowed with a liquidity surplus, enter into a debt contract withborrowers that are endowed with an investment opportunity. This debt contract is ofshort-term nature, though. Before the borrower’s assets mature, lenders are presentedwith an opportunity to rollover or foreclose the obligation. Since after the contract hasbeen agreed upon only lenders decide, the rules of network formation are of the typeunilateral-unilateral, i.e., the rollover decision (the arc) can be modified by the lenderunilaterally. Here, the borrower takes a passive role. The decision to rollover the debtdepends on the perspective of the debt being re-payed by the borrower. A lender is lesslikely to renew the obligation if the perspectives of being re-payed are not particularlyoptimistic.

1 We can think about this setup as a second stage of a broader game where borrowers offer a refinancingcontract based on the rollover decisions of the lenders. However, this extension is left for future research.

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A-2 (normality of the distribution of prior beliefs w.r.t. the health of the borrower)Let v = θ + v be the noisy return to the lender’s investment in the borrower’s project,with fundamental θ . At this stage lenders have a common distribution of prior beliefsof their counterparties’ solvency, i.e., they can only observe a public signal v ∼N

(μ, σ 2

v

). In addition to this public signal, each lender receives a private signal

ε j ∼ N(0, σ 2

ε

), that is assumed to be uncorrelated with θ and E

(εiε j

)=0 ∀ i �= j .Thus, the distribution of prior beliefs that incorporates both signals is given by θ ∼N

(μ, σ 2

), with σ 2

l = σ 2v + σ 2ε.

Stage 1: At the beginning of stage 1, each lender after observing the status quonetwork can update his/her beliefs about the solvency of the borrower and decidewhether to rollover the debt or not. Thus, the network is an information transmissionmechanism.

A-3 (partial observability of the network) Lenders can only observe some networkconnections. Since banks undertake several operations jointly, such as syndicatedloans, each lender may have some information with respect to the interbank operationsof its peers. Thus, we assume that a lender has partial knowledge of how many otherlenders decide to rollover the debt of an interbank borrower. The limited empiricalevidence lends some support this argument (e.g., Akram and Christophersen 2010 andGabrieli 2011 find that interbank interest rates respond to how important a bank is inthe financial network). This knowledge may depend crucially on the structure of thenetwork and initial conditions such as the institutional organization of the bankingsystem. In other words, the number of active arcs in the status quo network can beobserved. However, this status quo network may be only a small part of the networkwhich ultimately prevails in equilibrium. For example, suppose that a bank can onlyobserve the behavior of one of its peers. This can be summarized in a status quo networkwhere only that peer’s arc is active. Then, using this partial information, the status quolender can update its belief distribution with respect to the borrower’s solvency. Notethat since all arcs in the status quo network are inactive, as far as the status quo lenderis concerned that part of the network is not observed. After the updating process iscomplete, the status quo lender makes a proposal. Then, depending in the draw ofnature, this proposal can be used by another lender to update its beliefs based on itsown partial information about the network. Thus, the information propagates slowlythrough the network depending on the strategic moves of each lender and the drawsof nature. In order to capture this asymmetry in the information dynamics, we assumethat unlike the presence of an active arc, the absence of an active arc is uninformative.After observing the network, a coalition can again update its belief distribution usingthe Kalman filter as shown in equation (2).

A-4 (normality of the distribution of the signal w.r.t. the health of the borrower) Atthis stage, lenders can observe the status quo network and infer the beliefs of otherlenders (θ + ul with l ∈ L) with respect to the borrower’s health. Each active arc istaken as a signal ul ∼ N

(0, σ 2

u

)and is uncorrelated with θ, ε j and u j ∀ j ∈ L\ {l}

Then, the lender can use the Kalman filter to update his/her beliefs 2. Now, we get

2 In the special case in which random variables are jointly normally distributed, linear least-squares pro-jections equal conditional expectations.

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θ1l j

= E (θ |θ + ul) = E (θ) + cov (θ.θ + ul)

var (θ + ul)[θ + ul − E (θ + ul)]

= μ + σ 20

σ 20 + σ 2

u(θ + ul − μ) , (2)

var (θ |θ + ul) = var (θ)(

1 − ρ2θ,θ+ul

)= σ 2

u

σ 2u + σ 2

l

σ 2l ≡ K0σ

2l .

Therefore, the updated beliefs w.r.t. the borrower’s health follows a normal distri-bution with mean μ and variance K0σ

2l . Since K0 < 1, the updated distribution has

the same mean but a smaller variance than the one of θl j . If a lender decides to renewthe contract but the asset fails to mature, then the lender faces an opportunity cost cl j .

This stage can be more easily understood with the introduction of the followingdefinition.

Definition 2 (State) The state space is defined as the set Ω = G × F of all feasiblenetwork-coalition duples ω = (G, S) ∈ {ω1, . . . , ωN }. The typical state can be ratio-nalized as the status quo network G when the ability to propose a new network andchoose the succeeding coalition lies with the status quo coalition S.

The subsequent decision to rollover the debt or not can be interpreted as a networkproposal. Let Φl j (ω) ⊆ G denote the subset of network proposals that can be put forthby lender l j for consideration by nature given the state ω. However, not all lendershave the same ability to propose changes to the status quo network.

A-5 (properties of feasible proposals) Only the status quo coalition/lender canchange its decision to rollover the debt or not in this state. All remaining lenders canonly propose the status quo network. Thus, the profile of lender proposals can bedescribed by GL = (

Gl j

)l j ∈L

:= (Gl j , G−l j

).

Since the structure of the network conveys information to lenders, the change in thenetwork structure induced by the rollover decision of a single lender (as expressed bythe addition of an arc representing an active rollover decision) conveys informationabout a borrower’s health to all lenders.

Stage 2: Given the profile of proposed network changes by lenders, the law ofmotion alters the network, leading the subsequent changes brought about by futureproposals by other lenders.

2.4 Payoffs

We will assume thatA-6 (payoff functions) Lenders are mean-variance maximizers and each lender’s

payoff function depends only on the status quo state.

rl (•, •) : Ω × Gm → [−M, M] ,

Thus, for all lenders l ∈ L and for all state-network proposal pairs,

(ω, GL) = ((G,

{l j

}), GL

) ∈ Ω × Gm,

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payoffs are given by

rl (ω, GL) = vl (ω) ,

where vl j (•) is the total payoff to lender l j ∈ L in state ω = (G,

{l j

})with

vl j (ω) = G(l j , b1

) [μ + ε j − λ

2σ 2 (ω)

]+ (

1 − G(l j , b1

))cl j , (3)

where λ is the risk appetite coefficient, i.e., the extent to which investors welcomerisk. Larger values of λ indicate that the investor is less willing to bear risk. Here wedistinguish risk appetite from risk aversion. As Danielsson and Zigrand (2008) show,in the presence of VaR constraints the optimal bank portfolio has a mean-varianceform and the absolute risk aversion coefficient exceeds the innate one in periods ofcrisis3. In their setup, the absolute risk appetite coefficient is given as the inverse ofthe sum of the risk aversion coefficient implied by the investors’ utility function anda term that depends on the Lagrange multiplier associated with the VaR constraint.

2.5 Policy measures

In this section we add a policy parameter to our baseline model. Since the dynamicnetwork formation process is strategically driven, we can account for endogenouseffects generated by the policy measures. We focus on instruments that can potentiallyshift financial institutions’ incentives with respect to the connections established inthe interbank market. This broad definition of policy measures encompasses for exam-ple loan guarantees issued by the government/central bank, levies such as the onesrecently implemented in several European countries or outright government control.These instruments can correspond to positive values of τ (i.e., taxes) as well as negativeones (i.e., loan guarantees). For example, the levy introduced in France is based on theminimum capital requirements as defined in the Basel Accords. Thus, if a financialinstitution decides to lend a higher amount to an interbank counterparty instead ofbuying government bonds, the base of incident increases in the amount of the transac-tion since interbank loans carry a 100 % weight while government bonds carry a 0 %weight for the assessment of risky assets. Since the strategic choice taken by players iswhether to rollover the debt or not, we can see this instrument as a contribution basedon the assets each lender holds in his/her balance sheet. Another example would beloan guarantees where the government provides an explicit (or implicit) guaranteeto interbank creditors, thus increasing the willingness of other lenders to establishinterbank connections. The payoff function can be re-written using equation (3) asfollows:

vl j (ω, τ) = G(l j , b1

) [μ + ε j − λ

2σ 2 (ω) − τ

]+ (

1 − G(l j , b1

))cl j . (4)

3 For a more complete distinction between risk aversion and appetite see Gai and Vause (2006).

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2.6 The law of motion

Given the profile of lender proposals GL and given the current state, ω = (G,

{l j

}),

nature chooses the next state ω′ in accordance with the Markov transition law,q

(ω′|ω, GL

). For each proposal m-tuple GL , let Q (GL) be the resulting N × N

Markov transition matrix with typical entry

qi j (GL) := q(ω j |ωi , GL

)(5)

where qi j (GL) is the probability that nature moves from state ωi to state ω j givenplayer proposals GL . Throughout this paper we will assume that these probabilitiesare given by

q(ω′|ω,

(Gl j

)l j

)=

∑l j ∈L

[e

(vl j (ω

′)−vl j (ω))

IΩ

(Gl j

)(ω′)

]

∑ω′′∈Ω

∑l j ∈L

[e

(vl j (ω

′′)−vl j (ω))

IΩ

(Gl j

) (ω′′)] (6)

where for states ω ∈ Ω .

IΩ

(Gl j

) (ω) ={

1 i f ω ∈ Ω(Gl j

)

0 otherwise.

2.7 Stationary strategies

A stationary strategy(δl j , δl j , . . .

)for lender l j is a constant sequence of conditional

probability measures on Ω . Under stationary strategy(δl j , δl j , . . .

)given the current

(time point n) state ω , the status-quo lender l j , at each and every point in time t ,chooses a network proposal according to the conditional probability measure

δl j

(•|ωt) ∈ P (Φl j

(ωt)) (7)

where P (Φl j

(ωt

))is the set of probability measures with support contained in

Φl j

(ωt

), player l j ’s feasible action correspondences. Moreover, let the set of sta-

tionary strategies for lender l j be given by∑∞

l j:= Π∞

n=1

∑nl j

.

A pure stationary strategy for lender l j is a stationary strategy(δl j , δl j , . . .

)such

that for some function

fl j (•) : Ω → G wi th fl j (ω) ∈ Φl j (ω) ∀ ω ∈ Ω,

δl j

(fl j (ω) |ω) = 1 ∀ ω ∈ Ω. (8)

Thus, under pure stationary strategy in any state ω ∈ Ω , the conditional probabilitymeasure for lender l j assigns probability 1 to the network proposal fl j (ω) ∈ Φl j (ω).

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Given the pure stationary strategy, the discounted expected payoff to player l j

starting at state ω is then given by

El j (δL) (ω) := rl j (δL) (ω) .

3 Pure stationary equilibria

A dynamic network formation game starting at state ω ∈ Ω is given by

Ξω :=(Ω, El j (.) (ω) ,Π∞

l j

)

l j ∈L.

Definition 3 (Nash Equilibrium) A stationary strategy(δ�

l j(.|.)

)

l j ∈Lwith corre-

sponding m-tuple of stationary strategies δ�L (.) =

(δ�

l j(.|.)

)

l j ∈Lis a Nash Equi-

librium of the network formation game Ξ if for all starting states ω and all lenders

l j ∈ L , El j

(δ�

l j, δ�−l j

)(ω) ≥ El j

(δl j , δ

�−l j

)(ω) ∀ δl j ∈ Π∞

l j.

Theorem 1 (Existence of Nash Equilibrium in Stationary Strategies) Under assump-tions [A-1]–[A-7] the dynamic network formation game

Ξ :=(Ω, El j (.) (ω) ,Π∞

l j

)

l j ∈L

has a Nash Equilibrium in stationary strategies.

Proof Theorem 1 is an immediate consequence of Theorem 1 in Federgruen (1978).

Given stationary equilibrium, δ�L =

(δ�

l j(.|.)

)

l j ∈L, let Q� be the result-

ing N × N equilibrium Markov transition matrix, where Q� has typical entryq�

i j

(= q(ω j |ωi , δ

�L (ωi )

)), where q�

i j is the probability that nature moves from stateωi to state ω j given stationary equilibrium proposal strategies δ�

L .Given initial probability measure χ� = (

χ�1 , χ�

2 , . . . , χ�N

) ∈ P (Ω) prescribingthat the process is in state in after n moves is given by

Π{W �

n = ωin

} = (χ�Q�n)

in=

∑

i0

χ�i0

(Q�n)

i0in.

Corresponding to the equilibrium Markov transition matrix Q� there is an uniquedirected network M�—a supernetwork (Page et al. 2005)—where

M� ⊂ [0, 1] × (Ω × Ω) ,

with typical connection(

q�i j

(ωi , ω j

))where q�

i j is the i j th entry in the equilibrium

Markov transition matrix Q�. The connection(

q�i j

(ωi , ω j

)) ∈ M� is active if and

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only if the process of network formation{W �

n

}governed by Q� is such that ∀ n =

1, 2, . . . , N ,

Π{W �

n = ω j |W �n−1 = ωi

} = q�i j > 0.

Often we will be interested in determining the probability with which the formationprocess reaches (hits in finite time) a particular state. To begin, let the integer-valuedrandom variable, T �

ω j(.) : Θ → {0, 1, 2, . . .} , given by

T �ω j

:= min{n ≥ 1 : W �

n = ω j}

(9)

be the hitting time of the formation process{W �

n

}n for state ω j ∈ Ω , and let

ρ�i j := Π

{T �

ω j< ∞|W �

0 = ωi

}

be the probability that the process reaches ω j after leaving ωi at time zero in finitetime.

Note that the concept of supernetwork used in our model differs from Nagurneyand Dong (2002) or Nagurney and Toyasaki (2003). A supernetwork in our setupcorresponds to a network where each node is a state and the directed labeled arcs arethe positive transition probabilities as given by Q�.

Definition 4 (Recurrence and Transience) A state is said to be recurrent if ρ�i i = 1

and transient if ρ�i i < 1.

Definition 5 (Basins of Attraction) A set of states A� ⊆ Ω is a basin of attractionfor the process

{W �

n

}n governed by Markov transition matrix Q� if A� is closed and

irreducible. A� is irreducible if and only if for every pair of states ωi and ω j containedin A�, ρ�

i j > 0 and ρ�j i > 0.

Theorem 2 (Theorem 6 of Page and Resende 2012—Existence of an Unique Decom-position into Basins of Attraction)

Let{W �

n

}∞n=0 be an endogenous Markov process of network and coalition formation

governed by Q� with state space Ω . Then, starting from any state ω ∈ Ω , the process{W �

n

}∞n=0 reaches a state ω′ contained in a basin of attraction in finite time with

probability 1. Thus, for all ω ∈ Ω , there exists an integer nω such that

Π{

W �n ∈

[∪k A�k

]|W �

0 = ω}

= 1 ∀ n ≥ nω.

Proof See Page and Resende (2012).

In the rest of the paper we develop our implications based solely on the concept ofbasins of attraction that arise given a set of parameters. Since when the process entersone of the states contained in these sets it stays there, we can determine by analyzingthe status quo network associated with these states the number of active arcs that wouldprevail under the pre-specified macroeconomic conditions. Then, this number serves

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Fig. 2 Feasible Networks

as a proxy to assess how well the interbank market is functioning. From Theorem 2, wenow for a sufficiently large number of iterations of equilibrium transition matrix Q�

we find the set of basins of attraction. Since this implies a substantial computationaleffort, the model is solved numerically by taking a sufficiently large nω. The next twosections present a simple example and a set of simulations that allows us to retrievesome policy implications from the model.

4 Example—two lenders and one borrower

Suppose that we are in an interbank market where there are two lenders and oneborrower, and there is no tax/levy in place (Fig. 2). The set of all feasible networks isgiven by

G = {G1, G2, G3, G4} ,

with networks described as follows:The set of states is given by

Ω ={

(G1, {l1})︸︷︷︸ω1

,(G1, {l2})︸︷︷︸

ω2,(G2, {l1})︸︷︷︸

ω3,(G2, {l2})︸︷︷︸

ω4,

(G3, {l1})︸︷︷︸ω5

,(G3, {l2})︸︷︷︸

ω6,(G4, {l1})︸︷︷︸

ω7,(G4, {l2})︸︷︷︸

ω8

}

.

Table 1 lists player’s state-contingent network proposal constraint set.Table 2 lists each player’s state-contingent payoffs.Using Table 1 we can calculate under what sufficient conditions tipping points exist:

– G4 is preferred to G3 (ω5, ω7):

c > μ + ε1 − λ

2σ 2

l ⇔ λ >2 (μ + ε1 − c)

σ 2l

; (10)


c > μ + ε2 − λ

2σ 2

l ⇔ λ >2 (μ + ε2 − c)

σ 2l

; (11)

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Table 1 Player’s state-contingent constraint set

States ω1 ω2 States ω3 ω4

G1 Φl1 (•) {G1, G2} {G1, G2} G2 Φl1 (•) {G1, G2} {G2}Φl2 (•) {G1} {G1} Φl2 (•) {G2} {G2, G4}States ω5 ω6 States ω7 ω8

G3 Φl1 (•) {G3, G4} {G3} G4 Φl1 (•) {G3, G4} {G4}Φl2 (•) {G3} {G1, G3} Φl2 (•) {G4} {G2, G4}

Table 2 Player’s state-contingent payoffs


G1 l1 μ + ε1 − λ2 K0σ 2

l μ + ε1 − λ2 K0σ 2

l G2 l1 c c

l2 μ + ε2 − λ2 K0σ 2

l μ + ε2 − λ2 K0σ 2

l l2 μ + ε2 − λ2 σ 2

l μ + ε2 − λ2 σ 2

l


G3 l1 μ + ε1 − λ2 σ 2

l μ + ε1 − λ2 σ 2

l G4 l1 c c

l2 c c l2 c c


μ + ε2 − σ 2u

σ 2l + σ 2

u

λ

2σ 2

l > c ⇔ λ <2 (μ + ε2 − c)

σ 2u

σ 2l +σ 2

uσ 2

l

; (12)


μ + ε1 − σ 2u

σ 2l + σ 2

u

λ

2σ 2

l > c ⇔ λ <2 (μ + ε1 − c)

σ 2u

σ 2l +σ 2

uσ 2

l

. (13)

Table 3 lists the Nash equilibrium proposal strategies for the dynamic game offinancial network formation.

Assuming that the law of motion is such that given any status quo state, only statescontaining networks proposed by one of the players are assigned positive probabilities.Using this assumption we can preview how the supernetwork looks like under the setof sufficient conditions taken previously. Figure 3 displays the resulting supernetwork.

Some preliminary explanation is required to interpret the figure. The regular linesconnecting the states indicate that the transition between them occurs with some pos-itive probability. The thicker lines are the ones where this property only holds condi-tional on the verification of the conditions (10) to (13). For example, if nature assignsthe process to state ω5 and condition (10) does not hold, the transition matrix is0-valued in the qi j entries, with i = 5, 6 and j = 7, 8. When this happens the processstays in states ω5 and ω6 unless condition (12) holds. If that occurs, with some positive

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Table 3 Pure Strategies when2(μ+ε j −c

)

K0σ2l

> λ >2(μ+ε j −c

)

σ2l


G1 f �l1

(•) G1 G1 G2 f �l1

(•) G1 G2

f �l2

(•) G1 G1 f �l2

(•) G2 G4


G3 f �l1

(•) G4 G3 G4 f �l1

(•) G4 G4

f �l2

(•) G3 G1 f �l2

(•) G4 G4

Fig. 3 Supernetwork

probability the process reaches states ω1 and ω2. Thus, when both (10) and (12) hold,the process leaves states ω5 and ω6 and never returns, i.e., these states are transient.

Tracing back the status quo networks from the states, we can see that networks G2and G3 exhibit a tipping point property. Depending on the validity of the sufficientconditions, these states can transit to a completely active network (full participationregime) or to a completely inactive one (no participation regime).

5 Simulations

In this section, we simulate a system comprised of 7 lenders and 1 borrower. As Coccoet al. (2009) show, banks tend to concentrate most of their borrowing activity in a verysmall number of interbank lenders. Thus, we find that considering 7 lenders is notwithout its empirical validation.

5.1 Crises periods

Several empirical studies (see Kumar and Persaud 2002; Scheicher 2003; Coudert andGex 2006 and Coudert and Gex 2008) found that the risk appetite tends to drop intimes of financial turmoil. Thus, it is important to ask what happens to the structureof the financial network when λ increases.

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Fig. 4 Decrease in the risk appetite simulation ran for 7 lenders and 1 borrower with τ = 0, μ = 3, c =1, σ 2

u = 2.1, σ 2l = 1.5, σ 2

ε = 0.09

Figure 4 shows the number of active arcs exhibited by the networks contained inthe set of basins of attraction for each value of the risk appetite parameter.

As we can see when λ is small, the risk appetite is so high that the original networkstructure plays a very modest role in the rollover decision since the final result is fullparticipation. Similarly, when λ is very high the departure network is irrelevant aswell. However, for intermediate levels of λ we find tipping points. In these situations,a completely (in)active market remains (in)active after the drop in risk appetite whileintermediate architectures can transitate to either one of the extreme cases dependingon the law of motion that governs the process. Thus, our simulations suggest an inverseU-shaped relation between risk appetite and original network structure relevance.

5.2 Hysteresis

We now focus on the analysis of the policy parameter. Since a positive policy parameterintroduces a cost to the activation of the arc and since each arc has an externalitiesgiven its importance in the awareness of the solvency of the borrower, it is natural toask what are its effects on the dynamic process of network formation. To answer thisquestion we simulate the outcome for different levels of τ .

Figure 5 displays the number of active arcs for each policy level when the riskappetite is high. As one might expect, low tax values do not have a strong impact onthe interbank network.

However, as Fig. 6 shows, in situations where the risk appetite is very low even smallchanges in incentives can have a measurable effect. Once again, the role played by thenetwork as an information propagation mechanism becomes crucial to the analysis ofexogenous shocks.

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Fig. 5 Effect of the policy parameter in the formation of the network—high risk appetite simulation ran for7 lenders and 1 borrower with λ = [2, 2, 2, 2, 2, 2, 2] , μ = 4, c = 1, σ 2

u = 2.1, σ 2l = 1.5, σ 2

ε = 0.09

Fig. 6 Effect of Taxation in the Formation of the Network - Low Risk Appetite Simulation ran for 7 lendersand 1 borrower with λ = [4.2, 5, 4.2, 5, 5, 6, 6] , μ = 4, c = 1, σ 2

u = 2.1, σ 2l = 1.5, σ 2

ε = 0.09

Similar to risk appetite simulation, we find a region where tipping points occur.In this region, the departing network topology is of pivotal importance. When undersevere market conditions the process is driven into a gridlock, it exhibits hysteresissuch that the policy level that allows the system to recover from the market breakdownis considerably lower than the one required to precipitate a freeze departing from thecomplete participation scenario. Since a decrease in the tax rate plays the same role

123


as an increase in the ex-post return promised by the lender, the activation of interbanklinkages can be stimulated by it.

6 Policy implications

The sudden and profound lack of willingness of banks to lend to each other is with cer-tainty a signature trait of the 2007–2009 financial crisis. Central banks all around theworld were forced to take unconventional measures to re-establish the basic function-ing of interbank markets. However, even with this strong intervention these marketsstill operate far from the pre-crisis conditions. Thus, it is vital to understand how thereforms to the regulatory landscape affect their functioning especially under stressconditions. This paper tries to show that these network may exhibit tipping pointsunder poor macroeconomic conditions and therefore, its endogenous formation maybe strongly conditioned by the past architecture. Thus, in order to restart lending aftera major stress situation in the interbank market a considerable reduction in the specialbanking tax level is required, advising a counter-cyclical policy similar to the onesproposed by BCBS (2010) with respect to capital requirements. That is not to say thatspecial banking taxes/levies should not be a part of the needed reform. These instru-ments can play an important role in aligning private and social incentives. However, ifthe activation of each arcs has an externality that is particularly relevant under stressconditions, the weight of this instruments should be counter-cyclical.

7 Conclusion

Using a dynamic network formation model, solved numerically, we study banks’rollover decisions. We find that when the existence of linkages between market partic-ipants generates an informational externality, the newly formed network is conditionedby past architectures. Moreover, this inertia is strongly dependent on macroeconomicconditions, such as investors’ risk appetite, and market frictions.

A prime example of these frictions is the special banking levy implemented as partof vast regulatory reform that followed the global financial crisis. Given the foundinertia property, the policy level that allows the system to recover from the marketbreakdown situation is considerably lower than the one required to precipitate a freezedeparting from the complete participation scenario. Thus, in order to restart lendingafter a major stress situation in the interbank market a considerable reduction in thepolicy parameter level is required, advising a counter-cyclical policy similar to theones proposed by BCBS (2010) with respect to capital requirements.

Acknowledgments The first author acknwoledges financial support from FCT Ph.D. scholarshipSFRH/BD/62309/2009.

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