Why Are there So Many Banking Crises? The Politics and Policy of Bank Regulation phần 8 potx

The diversified lending structure is always stable when the number N of banks is large enough, whereas N has no impact on the stability of the credit chain structure.. They show that the

Consider now the case of credit chains Still assuming λ = 1, the

balance sheet equations give

D i = 1

2[R i + D i+1 ], i = 1, 2, 3. (7.11)

We can compute the losses experienced by each bank (with respect to

the promised returns R) and it is a simple exercise to check that the only

The results of this section highlight another side of interbank markets

in addition to their role in redistributing liquidity efficiently, as studied

by Bhattacharya and Gale (1987) Interbank connections enhance the

“resiliency” of the system to withstand the insolvency of a particularbank However, this network of cross-liabilities may loosen market dis-cipline and allow an insolvent bank to continue operating through theimplicit subsidy generated by the interbank credit lines This loosening

of market discipline is the rationale for a more active role for monitoringand supervision with the regulatory agency having the right to closedown a bank in spite of the absence of any liquidity crisis at that bank

The effect of a central bank’s guarantee on interbank credit lines would

be that x = (1, , 1) is always an equilibrium, even if one bank is

insolvent The stability of the banking system would be preserved atthe cost of forbearance of inefficient banks

7.4 Closure-Triggered Contagion Risk

7.4.1 Efficiency versus Contagion Risk

We now turn to the other side of the relationship between efficiency andstability of the banking system, and investigate under which conditions

the closure at time t = 1 of an insolvent bank does not trigger the

liquidation of solvent banks in a contagion fashion Suppose indeed that

bank k is closed at t = 1 Assumption 7.2 implies that X k = 0 and

D k = 0 Closing bank k at t = 1 has two consequences First, we have

an unwinding of the positions of bank k since π ki D k assets and π ki D i

liabilities disappear from the balance sheet of bank k In a richer setting

this is equivalent to a situation in which the other banks have reneged

on their credit lines toward bank k, possibly as a result of the arrival of negative signals on its return Second, a proportion π ikof travelers going

to location k will be forced to withdraw early the amount π ik D0and bank

i will have to liquidate the amount π ik D0/α If π ik D0/α is sufficiently large, bank i is closed at t = 1; otherwise the cost at t = 2 of the early liquidation is π ik ((D0/α)R − D i ).

Notice that if π ik D0/α
1, then X i = 0, i.e., bank i is liquidated simply because there are too many depositors going from location i to location

k, the bank is closed at t = 1 The type of contagion that takes place

here is of a purely mechanical nature stemming simply from the directeffect of inefficient liquidation Since this case is straightforward let us

instead concentrate on the other case, namely π ik D0/α < 1 Because

of unwinding and forced early withdrawal, the full general case is more

complex Since x k = 0, we have to suppress all that concerns bank k

from the equations (7.5) We obtain

We now have to check whether x ij ≡ 1 for all i, j ≠ k can correspond

to an equilibrium In this case, X i(k) = max[1 − π ik D0/α, 0] and system

This allows us to establish a result analogous to proposition 7.2

Proposition 7.5 (contagion risk) There is a critical value of the smallest

time t = 2 deposits below which the closure of a bank causes the liquidation of at least another bank This critical value is lower in the credit chain case than in the diversified lending case The diversified lending structure is always stable when the number N of banks is large enough, whereas N has no impact on the stability of the credit chain structure.

Proof This follows the same structure as the proof of proposition 7.2.

Denoting by M k the inverse of the matrix defined by system (7.16),

One can see that all the elements of M kare nonnegative19, thus stability

obtains if and only if D0 /R  ψ k , where ψ kdenotes the minimum of thecomponents

M k

⎛

⎜1 .1

⎞

⎟

⎠

The computation of ψ kis cumbersome in the general case but easy in

our benchmark examples (where, because of symmetry, k does not play

any role) One finds

in the credit chain example and in the diversified lending case,

respec-tively It is immediate from these formulas that Ψcre < Ψdiv (for N
2)

and that Ψdiv tends to 1 when N tends to infinity while Ψcreis independent

of N.

7.4.2 Comparison with Allen and Gale (2000)

It is useful to compare our results with those of Allen and Gale (2000)

Proposition 7.2 establishes that systemic crises may arise for mental reasons, as in Allen and Gale However, the focus of the twopapers is different Allen and Gale are concerned with the stability of thesystem with respect to liquidity shocks arising from the random number

funda-of consumers that need liquidity early in the absence funda-of aggregateuncertainty They show that the system is less stable when the interbankmarket is incomplete (in the sense that banks are allowed to cross-holddeposits only in a credit chain fashion) than when the interbank market

is complete (in the sense that banks are allowed to cross-hold deposits

in a diversified lending fashion)

In our paper interbank links instead arise from consumers’ geographicuncertainty and we focus on the implications of the insolvency of onebank in terms of market discipline and the stability of the system In par-ticular in proposition 7.4 we show how the structure of interbank links

19The fact that the matrix M khas nonnegative elements follows from a property of diagonal dominant matrices (see, for example, Takayama 1985, p 385).

allows the losses of one bank to be spread over other banks We showthat a diversified lending system is more exposed to market discipline(i.e., less resilient) than a credit chain system because in the latter theinsolvent bank is able to transfer a larger fraction of its losses to otherbanks, thus reducing the incentives for its own depositors to withdraw.

In proposition 7.5 we are concerned with the stability of the system withrespect to contagion risk triggered by the efficient liquidation at time

t = 1 of the insolvent bank.

7.5 Too-Big-to-Fail and Money Center Banks

Regulators have often adopted a too-big-to-fail (TBTF) approach in ing with financially distressed money center banks and large financialinstitutions.20 One of the reasons is the fear of the repercussions thatthe liquidation of a money center bank might have on the correspondingbanks that channel payments through it Our general formulation ofthe payments needs, where the flow of depositors going to the variouslocations is asymmetric, offers a simple way to model this case and tocapture some of the features of the TBTF policy We interpret the TBTFpolicy as designed to rescue banks which occupy key positions in theinterbank network, rather than banks simply with large size.21

deal-Consider, for example, the case where there are three locations (N = 3) Locations 2 and 3 are peripheral locations and location 1 is a money

center location All the travelers of locations 2 and 3 must consume

at location 1, and one-half of the travelers of location 1 consume at

location 2 and the other half at location 3 That is, t12 = t13 = 1



; 0



(7.19)and

Suppose now that one of these banks (and only one) is insolvent (this is

known at t = 1) The next proposition illustrates how the closure of a

20 See, for example, the intervention of the monetary authorities in the Continental Illinois debacle in 1984 and, to some extent, in arranging the private-sector rescue of Long Term Capital Management.

21 The failure of Barings in 1996 is an example of the crisis of a large financial institution that did not create systemic risk.

22 Note that we now abandon assumption 7.3 (the symmetry assumption).

bank with a key position in the interbank market may trigger a systemiccrisis.

Our last result concerns the optimal attitude of the central bank when

the money center bank becomes insolvent (R1 = 0) When D0/R is low,

no intervention is needed When D0 /R is large, the central bank has to

inject liquidity More precisely, we have the following proposition

Proposition 7.7 When R1 = 0, x = (1, 1, 1) is an equilibrium if D0/R

is sufficiently low (no central bank intervention is needed) In the other case, the cost of bailout increases with D0/R.

Proof When R1= 0, x = (1, 1, 1) can be an equilibrium if

D > D0

⎛

⎜1 .1

Table 7.1. Summary of central bank interventions.

Type of central Problem bank intervention Costs Results Speculative Coordinating role of Never used in Proposition 7.1 gridlock central bank equilibrium; no

• guarantee credit lines cost apart from

• deposit insurance moral hazard Insolvency in Ex ante monitoring Imperfect monitoring Proposition 7.2

a resilient and supervision leads to forbearance

market Insolvency Orderly closure of No cost, apart from Proposition 7.5;

leading to insolvent bank moral hazard and Proposition 7.6 contagion and arrangement money center banks;

of credit lines in the case of money

to bypass it center banks it may

be too costly or even impossible to organize orderly closure Bailout Transfer of Proposition 7.7

taxpayer money

Solving (7.22) and (7.20) when R1 = 0, R2 = R3 = R yields D1= 4

7R,

D2= D3= 6

7R, which is an equilibrium if and only if D0/R < 47 The

cost of bailout is 0 if and only if D0 /R < 47, it is D0 − 4

7.6 Discussions and Conclusions

We have constructed a model of the banking system where liquidityneeds arise from consumers’ uncertainty about where they need toconsume Our basic insight is that the interbank market allows theminimization of the amount of resources held in low-return liquid assets

However, interbank links expose the system to the possibility that a ber of inefficient outcomes arise: the excessive liquidation of productiveinvestment as a result of coordination failures among depositors; thereduced incentive to liquidate insolvent banks because of the implicitsubsidies offered by the payments networks; the inefficient liquidation

num-of solvent banks because num-of the contagion effect stemming from oneinsolvent bank

7.6.1 Policy Implications

We use this rich setup to derive a set of policy implications (summarized

in table 7.1) with respect to the interventions of the central bank

First, the interbank market may not yield the efficient allocation ofresources because of possible coordination failures that may generate a

“gridlock” equilibrium The central bank thus has a natural coordinationrole to play which consists of implicitly guaranteeing the access toliquidity of individual banks If the banking system as a whole is solvent,the costs of this intervention is negligible and its distortionary effectsmay stem only from moral hazard issues (proposition 7.1)

Second, if one bank is insolvent, the central bank faces a much morecomplex trade-off between efficiency and stability Market forces will notnecessarily force the closure of insolvent banks Indeed, the resiliency ofthe interbank market allows it to cope with liquidity shocks by providingimplicit insurance, which weakens market discipline (proposition 7.2)

The central bank therefore has the responsibility to provide ex ante

monitoring of individual banks However, it is the responsibility of thecentral bank to handle systemic repercussions that may be caused by theclosure of insolvent banks (proposition 7.5) In this case two courses ofaction are available: orderly closure or bailout of insolvent banks Giventhe interbank links, the closure of an insolvent bank must be accompa-nied by the provision of central bank liquidity to the counterparts of theclosed bank.23 This is what we call orderly closure Assuming that this

is possible, theoretically it entails no costs apart from moral hazard

However, the orderly closure might simply not be feasible for moneycenter banks (proposition 7.6) in which case the central bank has nochoice but to bail out the insolvent institution, with the obvious moralhazard implications of the TBTF policy

Our model can be extended in various directions, some of which arediscussed below

7.6.2 Imperfect Information on Banks’ Returns

In reality, both the central bank and the depositors have only imperfectsignals on the solvency of commercial banks (although the central bank’ssignals are hopefully more precise) Therefore, the central bank will have

to act knowing that with some probability it will be lending to ing the credit lines of) insolvent institutions and with some probability

(guarantee-it will be denying cred(guarantee-it to solvent inst(guarantee-itutions Also, depos(guarantee-itors may run

on all the banks which have generated a bad signal

23For instance, in the credit chain case, if bank k is closed the central bank can borrow from bank k − 1 and lend to bank k + 1, thus allowing the interbank arrangements to

function smoothly.

The consequences are different depending on the structure of theinterbank market In the credit chain case, the central bank will have

to intervene to provide credit with a higher probability than in thediversified lending case Therefore in the credit chain case the central

bank has a higher probability of ending up financing insolvent banks Ex ante, therefore, the central bank intervention is much more expensive in

the credit chain case, so that in this case a fully collateralized paymentssystem may be preferred

7.6.3 Payments among Different Countries

Systemic risk is often related to the spreading of a financial crisis fromone country to another Our basic model can be extended to considervarious countries instead of locations within the same country Whendepositors belong to different countries, travel patterns that generate aconsumption need in another location have the natural interpretation

of demand of goods of other countries, i.e., import demand Goods ofthe other country can be purchased through currency (like in autarky inthe basic model) or through a credit line system whereby the imports

of a country are financed by its exports Our results extend to themodel with different countries but the role of the monetary authority

is somewhat different While in our setup the lending ability of thedomestic monetary authority was backed by its taxation power, thelending ability of an international financial organization is ultimatelybacked by its capital Hence the resources at its disposal are limited and

in the case of aggregate uncertainty its ability to guarantee banks’ creditlines is limited.24

7.7 Appendix: Proof of Proposition 7.1

where I is the identity matrix We first need a technical lemma.

Lemma 7.1 All the elements of M(λ) are nonnegative: m ij (λ)
0 for all i, j Moreover, for all i,

j m ij (λ) = 1 As a consequence, if R i > D0

24 See the role of the IMF in the 1997 Asian crises and the 1998 Russian crisis.

for all i, then

Proof M(λ) = (2I − Π  ) −1 Since Π  is a Markov matrix (because of

assumption 7.3), all its eigenvalues are in the unit disk and M(λ) can

be developed into a power series:

⎞

⎟

being an eigenvector of Π (for the eigenvalue 1), it is also an eigenvector

for M(λ).

Proof of proposition 7.1 (i) Because of assumption 7.2, D i = 0 when

x ij = 0 for all j Therefore, x ∗

⎞

⎟

This is an immediate consequence of the above lemma, which implies

that x = (1, , 1) is always an equilibrium when all banks are solvent.

There are no other equilibria when α = D0 Indeed, if xi = 0, then equation (7.5) implies that X i = 0 or D i = R i But X i cannot be zero

(unless all x j are also zero) and D i = R i > D0contradicts the equilibrium

condition Notice, however, that when α < D0 , X i can be zero even if

some of the x j are positive, which implies that other equilibria mayexist

Before establishing proposition 7.3, we have to compute the

expres-sion of matrix M(λ) in the two cases of credit chain and diversified

1− N − 2 (N − 1)2



Recursively, we obtain

T k = β k I + (1 − β k )T  , (7.35)where

β k = 1N

∞



k=0 [θ k β k I + θ k (1 − β k )T  ] (7.37)

D0

1+ θ + · · · + θ N −1 ≡ γcre

It is easy to see that γ Ncre increases in N and in θ (and therefore in γ).

Notice that γcre

Recalling that θ = λ/(1 + λ), we see that γdiv

N increases with λ and N, and that γdiv

∞ = θ.

Proof of proposition 7.4 Comparing γ Ndiv and γ Ncrewe obtain

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PART 4

Solvency Regulations