UNDERSTANDING FINANCIAL CRISES
Section 6:
Contagion and Financial Fragility (Part 1)
April 8, 2002
Franklin Allen
NEW YORK UNIVERSITY
Stern School of Business
Course: B40.3328
( Website: http://finance.wharton.upenn.edu/~allenf/ )
Spring Semester 2002
1. Introduction
- What causes severe financial crises?
- One possibility is that there is a shock somewhere and
this spreads elsewhere. This is the notion of contagion.
- A related notion is that financial crises are started by
small shocks that have large effects. This is the notion
of financial fragility.
- There seem to be a number of historic examples in the
nineteenth century US of both contagion and financial
fragility.
1
Different types of contagion
1. Interlinkages between banks
Rochet and Tirole (1996b): Correlated bank monitoring
Allen and Gale (2000): Interbank markets
Lagunoff and Schreft (2002): Probabilistic model of
linkages
2. Payments systems
Freixas and Parigi (1998): Net versus gross systems
Freixas, Parigi and Rochet (1999): Gridlock
Rochet and Tirole (1996a): Too big to fail
3. Financial markets
King and Wadwhani (1990): Correlated information
Calvo (1999) and Yuan (2000): Correlated liquidity shocks
Kodres and Pritsker (2000): Macroeconomic risk factors
and country-specific asymmetric information
Kyle and Xiong (2000): Wealth effect
2
Theories of financial fragility
Kiyotaki and Moore (1997): Financial multipliers and
credit chains
Chari and Kehoe (2000): Informational cascades
Allen and Gale (2002): Arbitrarily small shocks have a
large effect on prices.
3
2 Contagion with Interlinkages between Banks
2.1 Allen and Gale (2000) model
Three dates t = 0, 1, 2
Single consumption good at each date
t=0
|
1
|
2
|
Liquid asset:
(storage)
1
1
1
Illiquid asset:
1
R = 1.5
r = 0.4
(liquidate)
Four regions A, B, C and D
Each region:
Continuum of investors measure 1
prob.  early consumers - c1
prob. 1- late consumers - c2
EU(c1, c2) = u(c1) + (1-)u(c2)
= ln(c1) + (1-)ln(c2)
Date 0 endowment per investor is 1
4
Table 1: Regional Liquidity Shocks
A
B
C
D
S1
H = 0.75
L = 0.25
H = 0.75
L = 0.25
S2
L = 0.25
H = 0.75
L = 0.25
H = 0.75
No aggregate uncertainty
prob. S1 = prob. S2 = 0.5
Average proportion of early consumers
= Average proportion of late consumers
= 0.5
5
2.2 Optimal Risk Sharing
How would a planner maximize representative ex ante
utility if type (early vs. late) was observable?
y liquid asset; x illiquid asset
max
y, x
0.5 ln(c1) + 0.5 ln(c2)
subject to (i) c1 = y
(ii) c2 = Rx
(iii) y + x = 1
Substituting in
max
y
0.5 ln(y) + 0.5 ln(R(1-y))
First order condition:
1
R

y R (1  y)
Solving:
y = x = 0.5
6
Optimal allocation:
Total:
c1 = 0.5 ; c2 = 0.5  1.5 = 0.75
Per capita:
c1 = 1; c2 = 1.5
This illustrates the first result:
Proposition 1: The first best allocation can be achieved
even if the planner cannot observe
consumers’ types.
To see how the planner would achieve this allocation
consider state S1 (similarly for state S2).
A
B
C
D
Liquidity demand:
0.75
0.25
0.75
0.25
Liquidity supply:
0.5
0.5
0.5
0.5
Date 1:
Transfer
0.25
0.25
Date 2:
Liquidity
demand:
0.251.5
=0.375
0.751.5
=1.125
Liquidity supply: 0.51.5=0.75 0.75
Transfer
0.375
7
0.375
1.125
0.75
0.75
0.375
2.3 Decentralization
Competitive banking sector:
Continuum of representative banks measure 1
Only banks invest in the illiquid asset (because of an
informational advantage)
Representative consumer deposits endowment of 1 in bank
Bank invests endowment in portfolio (y,x)
Consumer obtains deposit contract:
can withdraw c1 at date 1
or
can withdraw c2 at date 2
In order to implement the first best allocation we need to
have some way of implementing transfers between regions.
There is no overall shortage of liquidity, it is just badly
distributed.
8
2.4 The Interbank Deposit Market
Suppose banks are able to exchange deposits at date 0
Crucial issue is the form of market structure.
Complete markets is when each region’s bank can make
deposits in every other region’s bank
A
B
D
C
Figure 1
Complete Markets
9
With this arrangement it can be shown that the first best
allocation can be implemented
Each bank chooses y = 0.5; x = 0.5
Each region is negatively correlated with two other regions
Therefore it is necessary for each region’s bank to deposit
in each other region at date 0
0.25/2 = 0.125
Note that the deposits in the perfectly correlated regions A
and C cancel out so we can ignore them. Similarly for B
and D.
Depositing 0.125 allows a withdrawal of
0.125 at date 1
or
0.1875 at date 2
10
Consider date 1 when S1 has occurred (similarly for S2).
Demand = 0.75
Supply = 0.5
0.125
A
Cross deposits cancel
0.125
Demand = 0.25
Supply = 0.5
Demand = 0.25
Supply = 0.5
B
0.125
D
C
Demand = 0.75
Supply = 0.5
B
Demand = 1.125
Supply = 0.75
0.125
At date 2
0.1875
Demand = 0.375
Supply = 0.75
A
0.1875
Cross deposits cancel
0.1875
Demand = 1.125
Supply = 0.75
D
C
0.1875
Hence the first best is implemented.
The banks can always meet their obligations.
11
Demand = 0.375
Supply = 0.75
Complete markets is not the only market structure that can
implement the first best.
A
B
D
C
Figure 2
Incomplete Market Structure
Again suppose banks choose y = 0.5; x = 0.5
Now each bank can deposit in one negatively correlated
bank so the deposit to implement the first best is
0.25
This allows a withdrawal of
0.25 at date 1
or
0.375 at date 2
12
Consider date 1 when S1 has occurred (similarly for S2).
0.25
Demand = 0.75
Supply = 0.5
A
B
Demand = 0.25
Supply = 0.5
Demand = 0.25
Supply = 0.5
D
C
Demand = 0.75
Supply = 0.5
B
Demand = 1.125
Supply = 0.75
0.25
At date 2
Demand = 0.375
Supply = 0.75
A
0.375
0.375
Demand = 1.125
Supply = 0.75
D
C
Demand = 0.375
Supply = 0.75
Hence the first best is again implemented.
The banks can always again meet their obligations.
13
Yet another market structure can achieve the first best.
A
B
D
C
Figure 3
A Separated Incomplete Market Structure
Each bank can again deposit in one negatively correlated
bank so the deposit to implement the first best is
0.25
This allows a withdrawal of
0.25 at date 1
or
0.375 at date 2
14
Consider date 1 when S1 has occurred (similarly for S2).
0.25
Demand = 0.75
Supply = 0.5
A
B
Demand = 0.25
Supply = 0.5
Demand = 0.25
Supply = 0.5
D
C
Demand = 0.75
Supply = 0.5
0.25
At date 2
0.375
Demand = 0.375
Supply = 0.75
A
B
Demand = 1.125
Supply = 0.75
Demand = 1.125
Supply = 0.75
D
C
Demand = 0.375
Supply = 0.75
0.375
The allocation is again first best.
However, as we shall see the implications of the different
market structures for contagion are very different.
15
2.5 Fragility
Perturb the model by adding a zero probability state S
where there is an excess demand for liquidity in A.
Table 2: Regional Liquidity Shocks with Perturbation
A
B
C
D
S1
H = 0.75
L = 0.25
H = 0.75
L = 0.25
S2
L = 0.25
H = 0.75
L = 0.25
H = 0.75
S
 +  = 0.5 + 
 = 0.5
 = 0.5
 = 0.5
What is the effect of the excess liquidity demand?
Bankruptcy rules:
Banks must meet demands of depositors at date 1
If they can’t they go bankrupt and must liquidate all their
assets
The proceeds of liquidation are split pro rata among
depositors (i.e. we do not assume first come first served –
see Allen and Gale (1998 JF)).
16
The Liquidation “Pecking Order”
A bank is
solvent if it can meet depositor demands from liquid assets
insolvent if it can only meet depositor demands by
liquidating some of the illiquid asset
bankrupt if it can only meet depositor demands by
liquidating all assets.
In the context of the example it can be seen the “pecking
order” for liquidating assets is
1. The liquid asset
2. Deposits in other banks
3. The illiquid asset
Liquidation Values
If late consumers know that the amount they will receive at
date 2 is less than the amount they would receive at date 1
there will be a run on the bank and it will be liquidated
In the liquidation all investors are treated equally
17
Buffers and Bank Runs
A bank can meet a certain excess demand in liquidity at
date 1, which we call the buffer, by liquidating the illiquid
long term asset before a run is precipitated.
Consider the following examples of what happens in S
when there is an incomplete market structure as in Figure
2:
 = 0.04
t=1
0.54
Demands on bank A:
t=2
0.46
At date 1 Bank A goes through its pecking order
- To meet the first 0.5 it liquidates its liquid asset
- It needs more so it calls in its deposit in Bank B
- This precipitates a chain of demands as Bank B calls in
its deposit in Bank C which calls in its deposit in Bank D
which calls in its deposit in Bank A
- Bank A now has an extra 0.25 of the liquid asset but it
also has an extra demand of 0.25 so overall it is no better
off
- It must liquidate 0.04/0.4 = 0.1 of the illiquid asset
- At date 2 it will have 0.41.5 = 0.6 > 0.46 so avoids run
18
 = 0.1
t=1
0.6
Demands on bank A:
t=2
0.4
The sequence of events is the same as before but now
- Bank A must liquidate 0.1/0.4 = 0.25 of the illiquid asset
- At date 2, A has 0.251.5 = 0.375 < 0.4 so there is a run
- In the run on Bank A all depositors including Bank D
withdraw and take a loss
- There is a spillover but is there contagion, i.e. do all
banks go down?
- Assume there is not contagion so Bank A’s claim on B is
worth 0.25 and see if this is consistent
- Pro rata claim on Bank A =
0.5  0.5  0.4  0.25
= 0.76
1.25
- Bank D’s 0.25 claim is worth 0.76  0.25 = 0.19
- To meet their claims by depositors and Bank C they must
start using their buffer and liquidate 0.06/0.4 = 0.15 of
the illiquid
- At date 2, D has 0.351.5 = 0.525 > 0.5 so avoids a run
and there is no contagion
19
 = 0.1, R=1.2
Everything is the same as before except now
- At date 2, D has 0.351.2 = 0.42 < 0.5 so there is a run
on D and there is contagion
- A similar analysis holds for Bank C and Bank B so all
banks go down
- In the contagion equilibrium all banks are liquidated.
Interbank claims are offsetting and so
Pro rata liquidation value =
0.5  0.5  0.4
= 0.7
1
- Critical value of R to avoid contagion =
0.5
= 1.43
0.35
Proposition 2: Provided the shock  on Bank A is big
enough and the return on the illiquid asset R
is low enough a contagion occurs where all
banks are brought down.
Result demonstrated with 4 regions but could have been
done with 1000 or any number. This is the sense in which
a small shock can have a large effect.
20
2.6 Robustness
We have shown contagion can occur. To see that this
depends critically on the market structure consider what
happens if the market structure is complete as in Figure 1.
Now interbank deposits are 0.125 instead of 0.25 and
Banks B and D hold claims on A.
 = 0.1, R=1.2
Analysis is as before except now when considering whether
there is contagion
- Bank B and D’s 0.125 claims are each worth 0.76 
0.125 = 0.095
- Banks B and D each need to liquidate enough to raise
0.125-0.095 = 0.03. The amount liquidated is therefore
0.03/0.4 = 0.075.
- At date 2, Banks B and D have 0.4251.2 = 0.51 > 0.5 so
a run is avoided.
21
2.7 Containment
- If the economy is not fully linked as in Figure 3 the
contagion will be limited
A
B
D
C
Figure 3
A Separated Incomplete Market Structure
- All the allocations discussed above are consistent with
the complete markets structure
- Market structure in Figure 2 is special because the
pattern of holdings that promotes contagion is the only
one that is consistent with this market structure
22
2.8 Alternative Interbank Markets
- The fact that financial interconnectedness takes the form
of ex-ante claims signed at date 0 is crucial.
- It is not important that the contracts are deposit claims:
the same result holds with contingent or discretionary
contracts because the interbank claims net out
- Spillover and contagion occur because of the fall in asset
values in adjacent regions not the form of the contract
- The interbank market operates quite differently from the
retail market in this respect
- If there is an ex post loan market then contagion will not
occur. The interest rate in the ex post market must
compensate lenders for the cost of liquidating assets and
at this rate it will not be worth borrowing.
- But there are the usual difficulties with ex post markets,
such as the lemons problem and “hold-up” problems. If
the asset has a risky return then ex post markets will also
not be optimal.
23
2.9 Discussion
2.9.1 Equilibrium
We have simplified the problem considerably to retain
tractability. In particular, by assuming S occurs with zero
probability we ensure the behavior of the banks is optimal
since the interbank deposits and the resulting allocation
remains efficient.
When S occurs with positive probability the trade-offs will
be more complex. However, provided the benefits of risk
sharing are large enough interbank deposits should be
optimal and this interconnection should lead to the (low
probability) possibility of contagion.
2.9.2 Many States and Regions
We have considered 4 regions. Example of contagion
would hold for an arbitrarily large number of regions.
With a large number of regions the robustness of the
complete market structure will be much greater than that of
(very) incomplete market structures.
24
2.9.3 Sunspot Equilibria
We have focused on financial contagion as an essential
ingredient of equilibrium. There are also self-fulfilling
runs and contagion in sunspot equilibria.
2.9.4 Risky Assets
Both assets are safe in the model above. More realistic to
assume long asset is risky. Similar results should hold.
See Allen and Gale (1998 JF) and (2000 EJ) for analyses
with risky assets and additional issues these raise.
2.9.5 Alternative Market Structures
As we saw in Section 4 many different market structures
allow inter-regional risk sharing. However they have
different implications for robustness.
A
C
D
B
Here Banks A and B have deposits in positively correlated
regions. First best risk sharing can be achieved but only
by having larger interbank deposits.
25
For example, consider date 1 when S1 has occurred
(similarly for S2 and date 2). Deposits are 0.5.
0.25
Demand = 0.75
Supply = 0.5
A
C
Demand = 0.75
Supply = 0.5
0.5
Demand = 0.25
Supply = 0.5
D
B
0.25
Demand = 0.25
Supply = 0.5
Because interbank deposits are higher contagion becomes
more likely.
26
2.10 Concluding Remarks
- A small shock on a single region or bank can bring down
the entire banking system no matter how big it is relative
to the shock, i.e., financial contagion can occur.
- This possibility depends critically on the structure of the
interbank market for deposits as well as the return on
bank assets.
- Central banks and the IMF as a way to complete markets
and prevent contagion.
27
References for Part 1
Allen, F. and D. Gale (2000). “Financial Contagion,” Journal of Political Economy 108,
1-33.
Calvo, G. (1999). “Contagion in Emerging Markets: When Wall Street is a Carrier,”
Unpublished manuscript, University of Maryland.
Chari, V. and P. Kehoe (2000). “Financial Crises as Herds,” working paper, Federal
Reserve Bank of Minneapolis.
Freixas, X. and B. Parigi (1998). “Contagion and Efficiency in Gross and Net Interbank
Payment Systems,” Journal of Financial Intermediation 7, 3-31.
Freixas, X. and B. Parigi and J. Rochet (2000). “Systemic Risk, Interbank Relations and
Liquidity Provision by the Central Bank,” Journal of Money, Credit and Banking 32,
611-38.
King, M. and S. Wadhwani (1990). “Transmission of Volatility Between Stock Markets,”
Review of Financial Studies 3, 5-33.
Kiyotaki, N. and J. Moore (1997). “Credit Chains,” Journal of Political Economy 105,
211-248.
Kodres L. and M. Pritsker (2000), “A Rational Expectations Model of Financial
Contagion,” working paper, International Monetary Fund.
Krugman, P. (1998), “Bubble, Boom, Crash: Theoretical Notes on Asia's Crisis,”
working paper, MIT, Cambridge, Massachussetts.
Kyle, A. and W. Xiong (1999). “Contagion as a Wealth Effect of Financial
Intermediaries,” working paper, Duke University, Durham, N.C.
Lagunoff, R. and S. Schreft (1998). “A Model of Financial Fragility,” forthcoming,
Journal of Economic Theory.
McKinnon, R. and Pill, H. (1997), “Credible Economic Liberalizations and
Overborrowing,” American Economic Review 87, 189-203.
Rochet, J. and J. Tirole (1996a). “Interbank Lending and Systemic Risk,” Journal of
Money, Credit and Banking 28, 733-762.
Rochet, J. and J. Tirole (1996b). “Controlling Risk in Payment Systems,” Journal of
Money, Credit and Banking 28, 832-862.
28
Yuan, K. (2000). “Asymmetric Price Movements and Borrowing Constraints: A
Rational Expectations Equilibrium Model of Crisis, Contagion, and Confusion,”
working paper, Department of Economics, MIT.
29