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MIT LIBRARIES
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3 9080 03317 5842
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Complexity and Financial Panics
Ricardo
J.
Caballero
Alp Simsek
Working Paper 09-17
May 15. 2009
Revised June 17,2009
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Social Science Research
1
1
Complexity and Financial Panics
Ricardo
J.
Caballero and
June
17,
Alp Simsek*
2009
Abstract
During extreme financial
rife
crises, all of a
with profit opportunities
sudden, the financial world that was once
for financial institutions (banks, for short)
becomes
exceedingly complex. Confusion and uncertainty follow, ravaging financial markets
and triggering massive
of this
phenomenon.
In this paper
flight-to-quality episodes.
In our model,
banks normally
collect
we propose a model
information about their
trading partners which assures them of the soundness of these relationships. However,
when acute
enough
to
financial distress emerges in parts of the financial network,
be informed about these partners, as
it
also
becomes important
it is
not
to learn
about the health of their trading partners. As conditions continue to deteriorate,
banks must learn about the health of the trading partners of the trading partners
of the trading partners,
and so on. At some point, the
becomes too unmanageable
cost of information gathering
for banks, uncertainty spikes,
and they have no option
A
flight-to-quality
flight
to quality, cas-
but to withdraw from loan commitments and illiquid positions.
ensues, and the financial crisis spreads.
JEL Codes:
Keywords:
E0. Gl, D8,
E5
Financial network, complexity, uncertainty,
cades, crises, information cost, financial panic, credit crunch.
"MIT and NBER, and MIT,
respectively. Contact information: caball@mit.edu and alpstein@mit.edu.
thank Mathias Dewatripont, Pablo Kurlat, Guido Lorenzoni, Juan Ocampo and seminar participants
Cornell and MIT for their comments, and the NSF for financial support. First draft: April 30, 2009.
We
at
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
Member
Libraries
http://www.archive.org/details/complexityfinanc00caba2
«
1
Introduction
The dramatic
rise in investors'
and banks' perceived uncertainty
All of a sudden, a financial
world that was once
2007-2009 U.S. financial
crisis.
with profit opportunities
for financial institutions (banks, for short),
exceedingly complex.
institutions' capital,
Although the subprime shock was small
banks acted as
if
most of
we have seen
-^r- 6eT
Figure
1:
The
line
.
»5^i55n
Hi,
(3
TED
av
.-sr-
were severely exposed
month Treasury
sir-
spread in basis points (source: Bloomberg),
bills).
reflects a higher perceived risk of default
Great Depression.
5»nr.r
between the interbank loans
banks' perceptions of counterparty
was perceived to be
relative to the financial
their counterparties
in the U.S. since the
corresponds to the
the interest rate difference
government debt
,oK
rife
and uncertainty followed, triggering the worst case
to the shock (see Figure 1). Confusion
of flight-to-quality that
at the core of the
is
An
(3
month LIBOR) and the US
increase in the
TED
on interbank loans, that
is,
spread typically
an increase
in the
risk.
In this paper Ave present a model of the sudden rise in complexity, followed by wide-
spread panic
in the financial sector.
In the model, banks normally collect, information
about their direct trading partners which serves to assure them of the soundness of these
However, when acute financial distress emerges
relationships.
network,
it is
in parts of the financial
not enough to be informed about these partners, but
it
also
tant for the banks to learn about the health of their trading partners.
becomes impor-
And
as conditions
continue to deteriorate, banks must learn about the health of the trading partners of the
trading partners, of their trading partners, and so on. At some point, the cost of information gathering
becomes too
large
and banks, now facing enormous uncertainty, choose to
withdraw from loan commitments and
the financial
crisis
illiquid positions.
spreads.
1
A
flight-to-quality ensues,
and
The
starting point of our
framework
more broadly) have
resenting financial institutions
against local liquidity shocks.
a standard liquidity model where banks (rep-
is
The whole
financial
bilateral linkages in order to insure
system
ages which functions smoothly in the environments that
though no bank knows with certainty
work (that
is,
may
of financial distress arises
is
the
many
it
a complex network of linkis
designed to handle, even
possible connections within the net-
each bank knows the identities of the other banks but not their exposures).
However, these linkages
literature
all
is
we use
that
also
be the source of contagion when an unexpected event
somewhere
in the
this contagion
Our point
network.
mechanism not
as the
with the
of departure
main object of study but
and financial panic. During normal times, banks only need to
as the source of confusion
understand the financial health of their neighbors, which they can learn at low cost. In
contrast,
when
cascades
arises,
that the shock
a significant
the
may
number
network and the possibility of
of nodes to be audited by each
to fully figure out,
uncertainty and they react to
This paper
arises in parts of the
bank
possible
it is
it
which means that banks now face significant
by retrenching into liquidity-conservation mode.
related to several strands of literature. There
is
rises since
spread to the bank's counterparties. Eventually the problem becomes
them
too complex for
problem
is
an extensive literature
that highlights the possibility of network failures and contagion in financial markets.
An
incomplete
list
includes Allen and Gale (2000), Lagunoff and Schreft (2000), Rochet
and Tirole (1996), Freixas, Parigi and Rochet (2000), Leitner (2005), Eisenberg and Noe
(2001), Cifuentes, Ferucci
and Shin (2005)
survey). These papers focus mainly on the
shocks
may
(see Allen
and Babus (2008)
mechanisms by which solvency and
cascade through the financial network. In contrast, we take these
as the reason for the rise in the complexity of the environment in
and focus on the
their decisions,
In this sense, our paper
is
for a recent
effect of this
liquidity
phenomena
which banks make
complexity on banks' prudential actions.
related to the literature on flight-to-quality
and Knightian
uncertainty in financial markets, as in Caballero and Krishnamurthy (2008), Routledge
and Zin (2004) and Easley and O'Hara (2005); and
investigates the effect of
and
Wang
(2005),
to this literature
network
itself.
is
new events and innovations
also to the related literature that
in financial
markets,
e.g.
Liu, Pan,
Brock and Manski (2008) and Simsek (2009). Our contribution
in
More
endogenizing the
rise in
relative
uncertainty from the behavior of the financial
broadly, this paper belongs to an extensive literature on flight-to-
quality and financial crises that highlights the connection between panics and a decline in
the financial system's ability to channel resources to the real economy (see,
and Kurlat
We
e.g.,
Caballero
(2008), for a survey).
build our argument in several steps. In Section 2
we
first
characterize the financial
network and describe a rare event as a perturbation to the structure of banks' shocks.
Specifically,
one bank suffers an unfamiliar liquidity shock
next show that
if
for
which
it
banks can costlessly gather information about the network structure, the
spreading of this shock into precautionary responses by other banks
This scenario with no network uncertainty
is
the
benchmark
similar (although with an interior equilibrium) to Allen
Our main
this context,
for
typically contained.
is
our main results and
is
and Gale (2000).
where we make information gathering
contribution
is
in Section 3,
the cascade
is
small, either because the liquidity shock
if
We
was unprepared.
is
costly. In
limited or because
banks' buffers are significant, banks are able to gather the information they need about
their indirect exposure to the liquidity shock
of Section
2.
and we are back
However, once cascades are large enough, banks are unable to collect the
information they need to rule out a severe indirect
is
to
to the full information results
hit.
Their response to this uncertainty
hoard liquidity and to retrench on their lending, which triggers a credit crunch. In
Section 4
we show that under
certain conditions, the response in Section 3 can
extreme, that the entire financial system can collapse as a result of the
The paper concludes with a
final
flight to quality.
remarks section and several appendices.
The Environment and
2
be so
a Free-Information Bench-
mark
In this section
we
first
introduce the environment and the characteristics of the financial
network along with a shock which was unanticipated at the network formation stage
We
the financial network was not designed to deal with this shock).
the equilibrium for a benchmark case in which information gathering
(i.e.
next characterize
is
free so that the
market participants know the financial network.
2.1
The Environment
There are three dates
which can be kept
{0. 1,2}.
is
a single good (one dollar) that serves as numeraire,
in liquid reserves or
liquid reserves, a unit of the
is
There
loaned to firms at date
before this date. At date
0,
1,
good
it
it
yields one unit in the next date.
then yields
%
throughout this paper.
R >
1
units at date 2
the lender can unload the loan
borrower at a discount) and receive r <
r
can be loaned to production firms.
1
units.
To
(e.g.
Instead,
if it
by
If
is
if
kept
in
a unit
not unloaded
settling
simplify the notation,
it
with the
we assume
Banks and Their Liquidity Needs
The economy has 2n continuums
composed
is
b3
,
which
of identical banks and, for simplicity,
our unit of analysis.
is
of banks denoted by {b3 }~_ 1
1
Each bank b3 has
of liquid reserves set aside for liquidity payments,
aside for
and
—
1
making new loans
y
—
hit
The bank's
A demand
deposit contracts.
shock.
Let
u €
which consist of y units
initial assets
<
fjo
I
—
y units of flexible reserves set
at date 2
>
l}
at date
1
the depositor
if
is
the depositor
if
1
is
not hit by a liquidity
be the measure of liquidity-driven depositors of bank b3
[0, 1]
demand
a measure one of
liabilities consist of
deposit contract pays
by a liquidity shock and ^ > h
3
each continuum b3 as bank
refer to
(but which can also be hoarded as liquid reserves)
at date
yo units of loans.
we
Each of these continuums
.
the
(i.e.
size of the liquidity
shock experienced by the bank), which takes one of the three values
in {<Z\u L ,u H
uj
}
with
h >
y
Note
that,
reserves
j/n
depositors.
if
at
and
lo l
—
u>
=
(u>
H
+ ui L
and
l\(I)
the size of the liquidity shock
central trade-off in this
(which
flexible reserves yo
/2,
(1
—
is Cu,
and suppose
y)
economy
set aside for this
it
R=
1
2 Cj.
the bank that loans
has assets just enough to pay
date
The
)
l\
will
The
of
earl}'
be whether the bank
purpose) or whether
it
will
its
flexible
(resp.
will
late)
loan
its
hoard some of
we describe below.
this liquidity as a precautionary response to a rare event that
The
U) to
(resp.
all
Network
Financial
liquidity needs at date
one of the (man}') reasons
of complexity later
on
a permutation p
:
{1.
..,
among banks, which
not be evenly distributed
highlights
an interlinked financial network. Moreover, the main source
for
will
capture this possibility we
may
1
be confusion about the linkages between different banks. To
let
2n}
E
i
—
>
{1,
{1,
2ra}
..,
..,
denote
slots in a financial
2n} that assigns bank
bp
^
network and consider
to slot
i.
We
consider a
financial network denoted by:
b p
(
where the arc
—
»
)
=
(b
p(1)
-> 6" (2) ->
denotes that the bank
the bank in the subsequent slot
i
+
1
The only reason
for the
continuum
is
for
(3)
in slot
(i.e.,
z
^
?
bank
....
(i.e.,
b
-» b p
W^
bank
p ^ +l)
)
b p{
-
l)
)
6
P(i))
has a
)
demand
deposit in
equal to
= (Q-u L ),
banks
(!)
to take other banks' decisions as given.
(2)
where we use modulo 2n arithmetic
forward neighbor of bank
bank
fr^
!+1 )
).
The
bp
^
for the slot
normal environment, the
bp
in
(i.e.,
Appendix A.l (and
financial
network
l)
^
focus
on the
is
as the
the actual permutation p).
similar to Allen-Gale (2000)), in the
facilitates liquidity
insurance and enables
(uj l
to the
)
even when the financial network b
liq-
banks that
[p) is
unknown
effect of the financial interlinkages in case of
an unan-
experience a high liquidity shock (w w
Our
+1 )
as the backward neighbor of
uidity to flow from banks that experience a low liquidity shock
to the banks.
6 p ('
bank
refer to
on from banks knowing the identity
possibility of confusion arises later
As we formally describe
i
(and similarly, to bank
of other banks but not their particular linkages
We
2
index
).
ticipated shock for which the financial network
not necessarily designed
is
for,
which we
describe next.
A
Rare Event
At date
date
1,
the banks learn that
banks
however, they also learn that one bank, b
,
becomes distressed and
As we formally demonstrate below, the
is
[p),
bank
b3
uj
at
y of
its
might
thus the banks' knowledge of
potentially payoff relevant. In particular, this knowledge influences
whether the banks use the
flexible reserves
to
j/n
make new
b
We
loans or to hoard liquidity.
are thus lead to describe the central feature of our model:
the financial network
<
loses 6
losses in the distressed
over to the other banks via the financial network b
the financial network
average liquidity shock
will experience the
3
liquid assets.
spill
all
the banks
'
uncertainty about
(p).
Network Uncertainty and Auditing Technology
We
let
B = {b
(p)
|
p
:
{1,
..,
2n}
—
>
{1, ,.,2n}
a permutation}
is
denote the set of possible financial networks. Each bank
and the
identities of the
banks
in its neighboring slots
narrows down the potential networks to the
i
b3 observes its slot
—
and
i
+
1.
where
i
=
p
1
(3)
.
i
= p~
l
(j)
This information
set:
p(i-l) = p(i-l)
B (p)= (h{p)<EB
3
p(i)
I
-In particular,
integer
i.
i
p(i
+
l)
represents the slot with index
For example,
i
— 2n+l
=
=
i'
p(i)
p(i
,
+
l
(j)
l)
£ {l,...2n} that
represents the slot with index
1.
is
the modulo In equivalent of
Figure
2:
The
financial
actual financial network.
network and uncertainty. The
Each
circle
corresponds to a slot in the financial network, and in
this realization of the network, each slot
i
the set
B
Bank
l
(p)).
b
l
cannot
(i.e. p (i) = i). The remaining
plausible
finds
after observing its neighbors
contains bank
boxes show the other networks that bank
(i.e.
bottom-left box displays the
tell
6
1
b
1
5
how
4
the banks {b .b ,b
5
}
are ordered in slots
{3.4.5}.
Note that the bank
b3
does not know
how
the remaining banks
assigned to the remaining slots (see Figure
that the bank b
3
is
distressed, but
2).
know how jar removed
Each bank
b3
it
is
key, since
is
from
At date
0,
to audit its forward neighbor 6
neighbor
b
.
it
,
,
{
^
bank b3
„/ i+ x)\
b3
are
knows
slot
l
Q)
means that
a bank b3
^
b3 does not necessarily
the distressed bank.
can acquire more information about the financial network through an
auditing technology.
p(-'^ 2 ^
In particular, each
does not necessarily know the
it
i=P~
of the distressed bank. This
(V)-*,,^^
a bank b3 in slot
p(i+1
in
'
its
a3
+
1
(i.e.
with j
=
p(i)) can exert effort
order to learn the identity of this bank's forward
Continuing this way, a bank
sheets learns the identity of
i
V
that audits a number, a 3
,
of balance
forward neighbors and narrows the set of potential
financial networks to:
p(i-l) = p(i-l)
B> (p
a
|
=
3
)
b
{
B
£
(p)
_
We
B3
where
,
p{i
+
a
+
3
denote the posterior beliefs of bank
=
1)
with f 3
b3
+
p(t
a
3
+
i
=
p
(j)
1)
p,a 3 ) which has support equal to
(.
|
a 3 ) given assumption:
(p.
Assumption (FS). Each bank
In the
example
has a prior belief f 3
illustrated in Figure
3
learn that bank b
2, if
bank
assigned to slot 3 and
is
the two boxes at the
left
hand
it
b
1
B
with
full
support.
audits one balance sheet, then
would narrow down the
bottom row
side of the
over
(.)
in
Figure
would
it
networks to
set of
2.
Bank Preferences and Equilibrium
Consider a bank
by
/
and
q\
and denote the bank's actual payments to early and
b3
(which
qi
may
be different than the contracted values
in principle
and
l^
Because banks are infinitesimal, they make decisions taking the payments of the
2 ).
The bank makes
other banks as given.
{0,1. ,..,277
number
of
some
its
of
—
until
new
the audit and liquidity hoarding decisions, a 3 E
at date
makes
(equivalently,
At date
at date 0).
its
l\
outstanding loans.
its
1,
j/o
The bank makes
maximize the return to the
it
J
y
E
denotes the
<0,j/oi
z3
E
[0, z],
and
these decisions to
liquidity obligations to depositors, that
has no benefit for the bank, thus once
—
withdraw
the bank chooses to
we denote by
deposits in the neighbor bank, which
tries to
We
[0,yo],
loans the bank
can meet
it
beyond
then
and y3 E
3}
unload some of
q\
late depositors
is,
until q\
=
it
may
maximize
l\.
q\
Increasing
satisfies its liquidity obligations, it
late depositors q\.
capture this behavior with the following objective function
«(lW</i}g{' +
where v
:
R+ — K ++
>
is
lW>Zi}^)-d(aO
(4)
a strictly concave and strictly increasing function and d(.)
increasing and convex function which captures the bank's non-monetary disutility
auditing.
When
for (qi^qi)
the bank b
3
is
making a decision that would lead
(which will be the case
expression in (4) given
its
is
able to pay
posterior beliefs f J
its late
then
in Section 3),
Suppose that the depositors' early/late
which
also
(.
|
p,
it
to
l\
an
from
an uncertain outcome
maximizes the expectation of the
a 3 ).
liquidity shocks are observable,
depositors at least
is
at date 2
and a bank
can refuse to pay the late
Banks'
Initial
y liquid reserves
Date
payment
1
depositors: Dl^
-
yo tiexible reserves
-
1
—
y
—
-
Date
,-
y
pavment
2
depositors: Qlz
po loans.
to early
=
Early depodtors demand their deposits
to late
=
— y)R
(1
Late depodtors demand tneir deposits
if
demand depodts in
forward neighbor bank
-
-
j
demand
?
-
and only
if
the
bo.nl;
cannot promise
ql
deposit: neld by
backward neighbor bank.
Insolvent banks (that pay
unload
Date
Banks
- Each
1
Banks moke the depodt withdrawal
dedsion ; e [0,;j.
Liabilities:
-
-
Date
Balance Sheets
Assets:
all
q{
<
/]!
of their outstanding loans
learn that
have shod;
will
u>
frerlected
on above balance sheetcl.
Date 2
Ba.nk b1 becomes distressed and loses 8 liquid reserves
Banks make the audit dedsion
a1
6 {0,1,
3}.
Banks pay
Banks moke the liquidity hoarding decidon y' €
(Equivalent!}- banks extend y — y^ new loans,)
Figure
depositors
bank
V
if
3
they arrive early.
at date
which the bank
1
is
[0,§
3:
With
=
3
li,q2
while the late depositors withdraw at date
insolvent, unloads
is
all
all
is
a no-liquidation equilibrium in
2;
>k,
or there
(5)
is
a liquidation equilibrium in which
outstanding loans, and pays
q{
while
assumption, the continuation equilibrium for
this
and pays
q{
the bank
depodtors.
Timeline of events.
takes one of two forms. Either there
solvent
qi to la,te
]
< u4 =
l
(6)
0,
depositors (including the late depositors) draw their deposits at date
Figure 3 recaps the timeline of events in this economy.
We
1.
formally define the equi-
librium as follows.
Definition
1.
The
equilibrium,
is
withdrawal, and payment decisions
a collection of bank auditing, liquidity hoarding, deposit
J
{a^p),yi(p) ^(p),ql(p),q 2
)
given the realization of the financial network
3
Without
decisions.
this
b
(p)
and
assumption, there could be multiple equilibria
In cases with multiple equilibria, this
depositor withdraws.
assumption
(p)} j
such that,
b(p)£B
the rare event, each bank b3
for late depositors' early/late
selects the equilibrium in
maxi-
withdrawal
which no late
raizes expected utility in (4) according to its prior belief
(with q\ (p)
<
l\)
unload
withdraw deposits early
We
J
(.)
and only
<
if q^ (p)
l\
Eqs. (5)
(cf.
(p)
and
b3
each bank
for
,
1
and
there exists a unique k G {0,
= p(i-
j
which we define as the distance of bank
k)
and
the late depositors
{&)).
Note that
next turn to the characterization of equilibrium.
network b
over B, the insolvent banks
of their outstanding loans at date
all
if
f
..,
for
2n —
from the distressed bank. As we
b3
the distance k will be the payoff relevant information for a bank b
liquidity to
hoard at date
originates at the distressed
bank
respectively with distances
1,
(with distances k € {2.
...
2n
and 2n
—
their forward
..,
how
that decides
it
will
l
-
.
— 1, know
,
their distances,
but the remaining banks
2}) do not have this information a priori
positive probability to each k £ {2,
will see,
determine whether or not the crisis that
cascade to bank b3 The banks b p ^~ \ b p ^ b p( l+1 \
since
b3 will
1} such that
,
3
much
each financial
2n — 2} (they
and they assign a
-
rule out k e {1, 2n
1} by
b3
and backwards neighbors). Note, however, that the bank
observing
can use the
auditing technology to learn about the financial network and, in particular, about
^~ h '
bp
A bank
(if k < a? .+
distance from the distressed bank.
banks either learns
its
distance k
In the remaining half of this section,
case in which auditing
In subsequent sections,
is
free so
we
we
1)
or
(with distance k) that audits a
it
learns that k
>
a
+
3
its
its
>
1
2.
benchmark
characterize the equilibrium in a
each bank learns
3
distance from the distressed bank.
compare
characterize the equilibrium with costly audit and
it
with the free-information benchmark.
Free-Information Benchmark
2.2
We
full
b
first
auditing a
(p)
benchmark case
describe a
p ^~ k}
=
2n —
in
which auditing
is
free so
each bank
bp ^
x
~k
>
chooses
In this context banks learn the whole financial network
3.
and, in particular, their distances.
At date
all
banks anticipate receiving a liquidity shock,
reserves equal to y
liquid reserves y
—
—
9.
CjIi
(plus y
At date
1,
of flexible reserves), except for
the distressed
forward neighbor bank. As we show
in
=
z
bank
bp
^
Appendix A. 2,
until, in equilibrium, all cross deposits are
z3
Co,
at date
bank
withdraws
3
b
its
1
and have liquid
=
6
p(i
Vje{l,..,2n}.
which has
deposits from the
this triggers further
withdrawn. That
'
withdrawals
is
(7)
The bank
drawals.
bp
bank
In particular,
^
tries,
but cannot, obtain any net liquidity through cross with-
also cannot obtain
«
each unit of unloaded loan yields r
additional liquidity at date
by cutting new loans
any liquidity by unloading the loans
0.
Anticipating that
the distressed bank
1,
at date
in order to
meet
In order to promise late depositors at least
the end of date
1
must have
bp
l5
hoards some
liquidity
its
/
^
will
it
at date 1, since
not be able to obtain
reserves y
its flexible
demand
at date
1.
a bank with no liquid reserves
left at
at least
i-y-g.Ozm
The
units of loans.
to deplete
of
all
its
level yj is a natural limit
on a bank's liquidity hoarding (which plans
liquidity at date 1) since
any choice above
amount
necessarily insolvent. If the
can hoard at most
all
(8)
y"o
of flexible reserves
y"o is
this
would make the bank
greater than yj
.
then the bank
of the flexible reserves while remaining solvent; or else
Combining the two
of the flexible reserves yo-
j3
cases, a bank's buffer
= min {j/o,yo}
is
it
can hoard
given by
•
A bank can accommodate losses in liquid reserves up to the buffer 3,
but becomes insolvent
when
6
losses are
beyond
3.
It
follows that the distressed
bank
p(l)
will
be insolvent
whenever
>
that
whenever
is,
the case so bank 6
liquidity as
date
is
can y$
it
Since the bank
1.
Suppose this
liquid reserves are greater than its buffer.
its losses in
/,(l)
(9)
P,
insolvent. Anticipating insolvency, this
=
y"o
is
insolvent,
(since
it
maximizes ql
bank
and unloads
)
all
will
hoard as
is
much
remaining loans at
depositors (including late depositors) arrive early
all
and the bank pays
where
to
/]
if
bank
b
p(l+l)
Partial Cascades.
experience losses
£,p(i-!)-
to
denotes bank 6^' +1 ''s payment to early depositors (which
recall that gf
s
bank
b
equal
solvent).
Since bank b p ^
in its cross
insolvency.
p<~'~ 2
is
is
Once the
\ continuing
its
is
insolvent, its
backward neighbor bank
deposit holdings, which,
crisis
cascades to bank
b
if
severe enough,
p( l ~ l
-
\
it
may
may
^~ l
cause
>
will
bank
then similarly cascade
cascade through the network in this fashion.
10
bp
,
We
K
K
€
conjecture that, under appropriate parametric conditions, there exists a threshold
—
{1, ..,2n
2} such that
banks with distance k
all
cascade through the network but
crisis will partially
banks have
Under
=
p(-'
q
/
and
:
on
distance
its
distressed
also go
b
pl<L
bankrupt
*?i
)
1
Eq.
>
^i
if
+1
condition
cascade size
is
it
can be calculated
(10)
'
_2)
but
—
loses z il\
and only
w hich
K=
if its
K
< 2n —
—
1, is
Consider now the bank
explicitly.
p
q
losses
p
<
on
/]
bank needs to
this
its
deposits from the
pl L ~ l>
Hence, bank
in cross-deposits.
j
solvent. Therefore
b
from cross-deposits are greater than
will
-
its
buffer,
<h--
]
If this
(11)
-
z
is
the original distressed bank
condition holds, then bank b
it
can,
i.e.
=
j/q
and
y
pii
it
~1
^
and the
anticipates insolvency,
will
pay
all
(with k
this point
>
1) is
,(,
{/
~ fc)
given by
k)
this bank's
(12,
onwards, a pattern emerges. The payment by an insolvent bank
qfand
it
depositors
•T" - / (O - !+*£*?..
From
2
can be rewritten as
hoard as much liquidity as
will
be contained after
will
receives only q
it
then the only insolvent bank
1.
remain solvent. In other words, the
from the distressed bank. To remain solvent,
\ so
fails,
are insolvent (there are
1
\ which has a distance 2n
qf
If this
K—
<
as the cascade size.
bp ^
bank
deposits to bank 6 p(
bank
—
'i
in
cfi
K
refer to
this conjecture,
^,'i-j-j
l\
We
failed.
^p(r-i)
K
>
such banks) while the banks with distance k
pay
.
backward neighbor
bp
=
{k -'
f (qf-
^~^ k+l ^
])
)
also insolvent
is
if
and only
p
if
q
'
<
lj
—-
Hence, the payments of the insolvent banks converge to the fixed point of the function
/(.) given by y
-f
y
,
and
if
y
'
J
and
If
it
condition (13)
fails,
h
=
then the sequence lq^'
can be checked that there
is
a full cascade,
—
Q
+ yo >
i.e.
,
!
all
f (q
(13)
p
'
)
)
always remains below
banks are insolvent.
li
—
-,
K
then (under Eq. (11)) there exists a unique
k)
<h--
qf'
p{T-(K-\))
,
and
If
2n
—
2
.
)
each k e
for
shows that
(14)
with distance k 6
(in
2
>
p
from
{1,...A'
cross deposits by hoarding
K)
> h).
least h (i.e.
qf~
{K +
tance k 6
—
=
P
~
h,<J
^2
)
2}
(14)
i.e.
if
(15)
6 p(!)
some
solvent, since
is
liquidity while
Since bank
b
p{J -
In contrast,
is
solvent,
b
p(l
can meet
all
banks
b
do not incur
1} are also solvent since they
~ h)
(that re-
from
its losses
liquidity,
=
j/q
p{T - k)
with
dis-
losses in cross-
and they pay
0,
K
verifying our conjecture for a partial cascade of size
i
banks
promising the late depositors at
still
K)
it
bank
all
)
from cross de-
1} are insolvent since their losses
Hence these banks do not hoard any
deposits.
( <7i
—
l,..,2n
-
addition to the trigger-distressed bank
forward neighbor)
its
K
...
A',
posits are greater than their corresponding buffers.
ceives q
{0.
greater than the solution, A', to this equation,
is
then, Eq.
~k
bp(
such that
•
2n -
L
2
8
,
> m
<7j
>
under
conditions (13) and (15).
Since our goal
we
is
to study the role of network uncertainty in generating a credit crunch,
The next proposition summarizes the
take the partial cascades as the benchmark.
above discussion and also characterizes the aggregate
we
benchmark
use as a
Proposition
in
conditions (9), (13) and (15) hold.
denote the
(i)
(
?2
is
a partial cascade of size
{fr
p(t
~^},
n
For the ex-ante equilibrium
and unload
as they can
K
< 2n —
(at
pi-'~ k
^}
^~ k
date 0): Banks {b p
do not hoard any liquidity or unload any loans.
=
z
(
l\
—
p
q
(with distance k
of their existing loans at date
all
'
)
K
where
2
is
from
= p~
(j)
(14).
—
I) are
which
is
Bank
b
"
just enough to meet
(and does not unload any loans).
12
> K)
<
A'
are solvent.
'
>
}
,
h
_ n hoard as
while banks
1,
p< L ~
the distressed
defined by Eq.
(with distance from the distressed bank k
insolvent while the remaining banks {b
!
let J
l
The banks' equilibrium payments
(at date I):
are (weakly) increasing with respect to their distance k
)
In particular, banks
j/q
and
realized as b(p), auditing is free,
slot of the distressed bank.
bank and there
(ii)
is
For a given financial network b(p),
For the continuation equilibrium
•
9i
which
subsequent sections.
Suppose the financial network
1.
level of liquidity hoarding,
^
its
much
{i>
liquidity
p(t ~ fc)
}
hoards a level of liquidity
losses
from cross deposits
5r
4
-
3
-
2
-
_kr=!L122_
•
1
O
1
0.2
08
l7n=n,llfl
o.e
_i*)=a.,iz3_
fa.
04
0.2
Figure
K
4:
The
04
0.2
1
benchmark. The
free- information
as a function of the losses in the originating
reserves
same
y~o.
The bottom
bank
top figure plots the cascade size
9, for
different levels of the flexible
figure plots the aggregate level of liquidity hoarding, !F, for the
set of {y~o}.
The aggregate
hoarding
level of liquidity
is:
T - 5Z y = K vo + yo
{L ~
A')
(16)
o
Discussion.
Proposition
1
shows
that,
under appropriate parametric conditions, the
equilibrium features a partial cascade and the aggregate level of liquidity hoarding,
is
roughly linear in the size of the cascade
demonstrates this result
The top panel
for particular
K
(and
is
roughly continuous
K
.
Figure 4
as a function of the losses in
the originating bank 9 for different levels of the flexible reserves
flexible reserves y
',
parameterization of the model.
of the figure plots the cascade size
that the cascade size
in 0).
J-
j/o-
This plot shows
increasing in the level of losses 6 and decreasing in the level of
is
Intuitively,
with a higher 9 and a lower y
be contained and the banks have
less
,
there are
more
emergency reserves to counter these
losses to
losses,
thus
increasing the spread of insolvency.
The bottom panel
plots the aggregate level of liquidity hoarding
of the severity of the credit crunch, as a function of
with 9 and
falls
with y
.
This
is
an intuitive
9.
JF which
This plot shows that
result:
is
a
measure
T also increases
In the free-information
benchmark
only the insolvent banks (and one transition bank) hoard liquidity, thus the more banks
are insolvent
(i.e.
the greater
K)
the
more
liquidity
13
is
hoarded
in the aggregate.
Note
.
T increases
also that
These
"smoothly" with
results offer a
benchmark
ing becomes costly, both A' and
can jump with small increases
for the
T
may
next sections. There we show that once auditanch more importantly,
be non-monotonic in y
in 9
Endogenous Complexity and the Credit Crunch
3
We
have now
laid
out the foundation for our main result.
assumption that auditing
realistic
can arise
is
9.
in
is
costly
In this section
and demonstrate that a massive
we add the
crunch
credit
response to an endogenous increase in complexity once a bank in the network
sufficiently distressed. In other words,
bj'
A"
is
large,
it
becomes too
banks
costly for
This means that their perceived uncertainty rises
to figure out their indirect exposure.
and they eventually respond
when
hoarding liquidity as a precautionary measure
(i.e.,
T
spikes).
Note
rium
that, unlike in Section 2.2,
for a particular financial
of the financial network
financial networks
b
(p)
is
we cannot
network b
simplify the analysis by solving the equilib-
(p) in isolation, since,
even when the realization
b(p), each bank also assigns a positive probability to other
6 B.
As
such, for a consistent analysis
equilibrium for any realization of the financial network b (p) £
Solving this problem in
full
generality
B
we must describe the
(cf.
Definition
1).
cumbersome but we make assumptions on
is
the form of the adjustment cost function, the banks' objective function, and on the level
of flexible reserves, that help simplify the exposition.
increasing cost function d
d{l)
First,
we consider a convex and
that satisfies
(.)
=
and
d(2)>v(l +
l2
1
)-v(0).
This means that banks can audit one balance sheet for free but
it is
(17)
very costly to audit
the second balance sheet. In particular, given the bank's preferences in
(4),
the bank will
never choose to audit the second balance sheet and thus each bank audits exactly one
{a3
balance sheet,
(p)
=
1}
L
network b
(p),
a bank
IP
.
J
Given these audit decisions and the actual financial
Jb( P )e/3
has a posterior belief f J
{.\p,l)
with support
B
J
(p,l),
the set of financial networks in which the bank j knows the identities of
and
its
second forward neighbor. In particular, the bank
the distressed bank
b
p(l)
(in
addition to banks 6 pl
information from the outset).
We
'
1)
,
6
6
p(l>
,
6
p( '~ 2)
p(,+1)
learns
its
its
!
is
neighbors
distance
from
which already have this
denote the set of banks that know the
1
which
slot
of the
distressed
bank (and thus
B know
On
is
(
=
p)
{b p
-,c
the other hand, each bank &^'
at least 3
k 6 {3.
...
2n
k
(i.e.
—
2}.
>
We
3),
from
their distance
>
^- 2
this
\b p(I
- l)
bank) by
.b
p{1)
,b
with k 6 {3,..,2n-2} learns that
l
—
1) / (1
—
a) with a
mm
b(p)£BJ(p,l)
(0. 1)
to
distance
its
all
distances
denote the set of banks that are uncertain about their distance by
i
\
_
np(r-3) ^p(T-4)
^p(J-{2n-2))
Second, we assume that the preference function v
(x
.
}
but otherwise assigns a probability in
^uncertain
~a
p{T+l)
—
>
(1 {q{ (p)
in
(.)
oo, so that the bank's objective
< h}
q{ (p)
+
1
{^
(p)
>
\
(4)
is
Leontieff v (x)
=
is:
I,} qi (p))
- d
{a? (p))
(18)
.
This means that banks evaluate their decisions according to the worst possible network
realization, b(p),
The
third
and
which they
find plausible.
assumption
last
is
that
yo
That
is,
<
(19)
PS-
the bank has less flexible reserves than the natural limit on liquidity hoarding
defined in Eq. (8) (which also implies that the buffer
is
ensures that, in the continuation equilibrium at date
liquidity are also solvent (since,
given by 3
1.
j/o)-
This condition
the banks that have
no matter how much of their
they have enough loans to pay the late depositors at least
=
flexible reserves
at date 2).
l\
enough
they hoard,
We
drop this
condition in the next section.
We
next turn to the characterization of the equilibrium under these simplifying as-
sumptions. The banks make their liquidity hoarding decision at date
drawal decision at date
are realized).
At date
1
1
under uncertainty (before their date
the distressed bank b p
neighbor which leads to the withdrawal of
all
^
x)
withdraws
1
its
losses
and deposit withfrom cross-deposits
deposits from the forward
cross deposits (see Eq.
(7)
and Appendix
A. 2) as in the free-information benchmark. Thus, for any distressed bank, the only
to obtain additional liquidity at date
1
is
through hoarding liquidity at date
characterize next.
L5
0,
way
which we
A
Sufficient Statistic for Liquidity Hoarding.
the original distressed bank
the' liquidity
(i.e..
hoarding decision
equilibrium from
certainty that q
is
A
0).
<
(p)
q
^~ k)
sufficient statistic for this
''
bank only needs
For example,
forward neighbor
p
bp
which
li,
bank
amount
the
is
other than
knows with certainty that q
if it
solvent), then
is
p
<
(p)
li
know whether (and how much)
to
hoards no liquidity,
it
— fi/z
its
(i.e.
i.e.
y^
=
forward neighbor
pay so
will
its
(i.e.
li
knows with
If it
0.
flexible
its
=
(p)
'
make
will lose in
it
p
to
receives in
it
forward neighbor. In other words, to decide how much of
its
reserves to hoard, this
cross-deposits.
>
A;
Consider a bank
that this
little
= y
be insolvent), then it hoards as much liquidity as it can, i.e. y^
and its forward
More generally, if the bank b p ^~ k) hoards some y' G [0, j/o] a t date
bank
will also
.
=
neighbor pays x
P
k)
qf-
The
0,
and
can be simplified and
max
(1
{ Ql [y'
,x
its
know x —
its
y'
That
.
Q
shows that
also
q-y
[y'
,
x]
(20)
,
(26)
and (27)
p
q
in
and
(p)
,
x]
and
payment
the bank's
is,
it
x]
are
q2
[y'
is
increasing
,
forward neighbor regardless of the ex-ante liquidity
hoarding decision. Using this observation along with Eq.
in (18)
[y'
q2
hoarding under uncertainty.
Appendix A. 2
from
receives
=
(p)
are characterized in Eqs.
cj2[y' ..x]
level of liquidity
it
qf
the bank does not necessarily
characterization in
amount
~ k}
and
(weakly) increasing in x for any given
in the
then this bank's payment can be written as
1,
q,\y'(y x}
qi[y' ,x]
Appendix A. 2. At date
has to choose the
=
{p)
where the functions
date
(p) &t
'
q
(20), the
bank's objective value
optimization problem can be written as
m >
hjq,
Q ,x
\y'
]
m
+
}
1 { 9l
{y'
.x
m >l
}q2
l
}
\y'
.x
m
})
(21)
,
y e|0.yo]
s.t.
x
m = min fx\z =
In words, a
k^
bp ^
bank
^-
(with k
(
* _1
>
0)
»
(p)
,
b
(p)
G
B
1
(p,l)}
hoards liquidity as
neighbor the lowest possible payment x m
if it will
receive
Next,
we
distance based
if
the bank's equilibrium
Q\,Qi
:
{0.
...
2n
—
1}
(q?-
— R
>
is,
such that
k)
{i
(p),4
~ k)
(p))
16
=
we say
that the equi-
payment can be written only
as a function of its distance k from the distressed bank, that
functions
forward
define two equilibrium al-
location notions that are useful for further characterization. First,
is
its
.
Distance Based and Monotonic Equilibrium.
librium allocation
from
(Qi[k},Q2[k])
if
there exists
payment
b
for all
is
(p)
monotonia
if
..,
—
2n
p
[k]
,
Qi
Then, a bank
=
~
q
(p)
since a
are (weakly) increasing in k. In words,
[k]
—
1,
Q]
which
[k
6
-
p(,
uncertainty about the forward neighbor's
its
equal to one less than
is
bank
b
p(l ' k)
B know
6
we
~^'s uncertainty about the forward neighbor's payment
reduces to
1]
distance based and monotonic (which
is
>
{p) (for k
with x m = Q [k-l].
On the other hand,
a bank b p
distances k G {3, ...,2n
—
own
its
can further be simplified by substituting q
in (21)
that a distance based equilibrium
yield (weakly) higher payments.
l
distance k
we say
Second,
1}.
next conjecture that the equilibrium
verify below).
=
{0,
the payment functions Qi
bank
distressed
x
and k E
based and monotonic equilibrium, the banks that are further away from the
in a distance
We
B
6
=
(p)
knows
0)
distance
p
its
Qi
distance
Hence, the problem
k.
—
[k
In particular,
1].
k, it solves
problem (21)
1
neighbor's payment Qi
^uncertain
We
^
are
sq[wqs pro bl e
now
—
m
B uncertairi
G
^
assigns a positive probability to
Moreover, since the equilibrium
2}.
k
~ k>
^ l
1
minimal
is
(21) With
for the distance
X™ = Q
is
=
fc
monotonic,
3,
forward
its
hence a bank b?^ G
[2].
X
a position to state the main result of this section, which shows that
in
banks that are uncertain about their distances
all
bank hoard liquidity as
to the distressed
all
if
they are closer to the distressed bank than they actually are.
More
specifically, all
—
with distance k
size
3
B uncertam
in
would hoard
sufficiently large
is
banks
A'
(i.e.
hoard the
bank with distance
3) so that the
information benchmark would hoard extensive liquidity,
>
actual distances k
level of liquidity that
the free-information benchmark.
in
>
(p)
K also hoard large amounts of
,
all
banks
in
When
k = 3
B
the
bank
the cascade
in
unceTtain
the free-
(^ with
even though they end up not needing
it.
To
state the result,
we
let
(j^j ree (p)
.
q{j ree (p)
denote the liquidity hoard-
Qojree (p))
,
ing decisions and payments of banks in the free-information benchmark for each financial
network b
(p)
Proposition
B
6
2.
(9), (13), (15),
(characterized in Proposition
Suppose assumptions (FS),
and
(19) hold.
1).
(17),
and
(18) are satisfied
For a given financial network b
(p), let
i
—
and conditions
p~
l
(j)
denote
the slot of the distressed bank.
(i)
For
the continuation equilibrium (at date 1):
based and monotonic.
The cascade
For the ex-ante equilibrium
allocation
size in the continuation equilibrium is the
the free-information benchmark, that
~
banks {b p(l k) }
are solvent where
(li)
The equilibrium
is.
at date 1,
K
is
banks {b p ^~ k%>
}
is
distance
same as
in
are insolvent while
defined in Eq. (14).
(at date 0):
17
Each bank
b>
6
B know
(p)
hoards the
same
:
level of liquidity
b>
e
B
=
(p)
j/q
uncertain
J
as in the
j ree (p)
y
hoards y
(p)
=
J
{p)
Vqj'H
free-information benchmark, while each bank
which
[p),
is
^~ 3
the level of liquidity bank b p
'>
would hoard in the free-information benchmark.
For the aggregate
bank
K
6
<
p(, ~ 3
^uncertain
the
at
which would hoard no
',
^
K=
bank
b
liquidity,
i.e.
y$
hoards no liquidity and the aggregate
\
Tee (p)
=
0-
Thus, each bank
level of liquidity
hoarding
is
bP
£
equal to
(16).
then the crisis in the free-information benchmark would cascade to and stop
3,
pi-'~ 3]
which would hoard an intermediate
.
Thus, each bank
hoarding
the
then the crisis in the free-information benchmark would not cascade to
2.
benchmark Eq.
If
on
K
cascade size
If
there are three cases depending
level of liquidity hoarding,
bP
£
^
^uncertain
is:
^=
level of liquidity y$
\
£
,
(p)
£
[0,yo].
hoards y \~ee (p) and the aggregate level of liquidity
P
i
W
=
+
3y
(2n
- 4)
y^]
(22)
.
3
If
K
>
and would hoard as much
^uncertain
6P
then in the free-information benchmark bank
4,
^^ hoards
as
liquidity as
much
can.
it
liquidity as
i.e.
yQj ree
—
yo-
'
!_3)
would
be insolvent
Thus, each bank
bP
6
can and the aggregate level of liquidity hoarding
it
is:
T = Y,y =
J
o
(°-n-l)yo-
(23)
3
The proof
provided
of this result
in the discussion
verifies that the
relegated to
is
Appendix A. 2
that the cascade size
is
Among
preceding the proposition.
equilibrium allocation at date
1
is
E {l,2,2n-
the payment Qi
same
bank
b
p{T - k)
(that the banks in
B
G
B
know
vncertain
(p)
(ii)
since the
with k >
(p) effectively
is
other features, the proof
the same as in the free-information benchmark.
1} (that a
[2]
most of the intuition
distance based and monotonic,
liquidity hoarding decisions are characterized as in part
for k
since
and
The date
payments Q\
[k
expects to receive)
—
1]
and
expect to receive) are the
as their counterparts in the free-information benchmark.
Discussion.
The
plots in Figure 5 are the equivalent to those in the free-information
case portrayed in Figure
4.
The top
the losses in the originating bank
cascade
size in this case
is
The parameters
K
as a function of
satisfy condition (19) so that
the same as the cascade size in the free-information
characterized in Proposition
The key
8.
panel plots the cascade size
1,
and both
differences are in the
bottom
the
benchmark
figures coincide.
panel,
18
which plots the aggregate
level of liquidity
yv =0.115
0.1
c:
'
Figure
The
5:
04
0.3
C
5
costly-audit equilibrium. The top panel plots the cacade
size
K
as a
function of the losses in the originating bank 6 for different levels of the flexible reserves
yQ
.
The bottom panel
The dashed lines
{y~o}-
T for the same set
bottom panel reproduce the free-information benchmark
plots aggregate level of liquidity hoarding
of
in the
in
Figure 4 for comparison.
hoarding J7 as a function of
characterized in Proposition
benchmark
for
K
<
3),
The
6.
solid lines
while the dashed lines reproduce the free-information
2,
also plotted in Figure
These plots demonstrate
4.
K switches from below 3
above
to
3,
severity of the initial shock) are
that banks are unable to
uncertain banks act as
if
tell
6
(i.e.
same as the
As
the liquidity hoarding in the costly audit equilibrium
That
is,
when
the losses (measuring the
size
becomes so large
whether they are connected to the distressed bank.
All
they are closer to the distressed bank than they actually are,
severe credit crunch episode. This
when
the
is
K
increases roughly continuously with 9.
beyond a threshold, the cascade
hoarding much more liquidity than
crunch)
it
very large and discontinuous jump.
Note
that, for low levels of
the aggregate level of liquidity hoarding with costly-auditing
free-information benchmark, in particular,
make a
correspond to the costly audit equilibrium
is
in the free-information
benchmark and leading to a
our main result.
also that the aggregate level of liquidity hoarding (and the severity of the credit
is
=
not necessarily monotonic in the level of flexible reserves y
0.5,
Figure 5 shows that providing more
flexibility to
f/o
actually increases the level of aggregate liquidity hoarding.
6,
an increase
the shock
is
in flexibility stabilizes the
sufficiently large.
the banks by increasing
That
system but the opposite
Intuitively,
if
l'j
For example,
.
is,
may
the increase in flexibility
at low levels of
take place
is
when
not sufficient
to contain the financial panic (by reducing the cascade size to
flexibility backfires since
it
manageable
levels),
more
enables banks to hoard more liquidity and therefore exacerbate
the credit crunch.
The Collapse
4
of the Financial System
Until now, the uncertainty that arises from endogenous complexity affects the extent of
the credit crunch but not the
show that
if
number
banks have "too much"
K. In
of banks that are insolvent,
we
this section
the sense that condition (19) no longer
flexibility, in
holds and
Vo
(which also implies
=
j3
y£),
then the
e (yl
-
1
(24)
y)
uncertainty
rise in
itself
can increase the number of
insolvent banks.
The
reason
that a large precautionary liquidity hoarding compromises banks' long
is
R
run profitability by giving up high return
for
low return
worst outcome anticipated by a bank does not materialize,
if
bank's large precautionary reaction improves
more benign
the additional
payment x
=
can be a significant
is
q{
~
p
l
some functions
functions
q\ [y' ,x]
date
~ k)
(p)
q^ [y' ,x]
and
choice
its
(p) at
qi
for
number
the
rise in
=
and
qo [y'o,x]
Qi
E
y'
[0,
That
1.
number
critical
to
of insolvencies as a result of
now
there
is
new element
is
and
a t date
yo]
In particular, a bank's
its
forward neighbor's
is:
Wo,*} and Q2
(1
~ k)
(p)
=
<?2
[vo,x]
However, the characterization of the piecewise
q2 [y' ,x].
changes a
little
when
condition (19)
and
(27) in
is
not satisfied.
Appendix A. 2 (as
In
in
an additional insolvency region:
y'
The
In other words, a
banks may find themselves in the
of
particular, these functions are identical to those in (26)
Section 3) but
become insolvent
vulnerability with respect to
its
very similar to that in the previous section.
depends on
still
still
the
flexibility.
analysis
payment
may
if
outcome when very close
liquidation
its
does so at the cost of raising
scenarios. Since ex-post a large
latter situation, there
The
it
it
than K) from the distressed bank.
sufficiently close (but farther
the distressed bank but
In this context, even
1.
the
>
bound
u
y
j/q
[(/,
[(^
—
20
-x)z].
x)
z].
This
is
a function of the losses
from
.
cross-deposits and
and
loans
at least
/j.
is
calculated as the level of liquidity hoarding above which the bank
That
—
is, j/q [(/]
Ril-V-Vl
We
would not
liquid reserves (net of losses)
[ih
z] is
-
x) z\)
>
y'
late depositors
the solution to
x)
where
refer to scenarios
pay the
be sufficient to
's
+
-x)z\-
yl [(h
—
\{l\
y$
x)
-x)z=
(h
h
(1
-
u)
for lack of a better jargon, scenarios
z] as,
of precautionary insolvency.
The
functions
and
qj [y' ,x]
remain (weakly) increasing
q2[y'Q) x]
conjecture as before that the equilibrium
monotonic and distance based (which we
is
verify below), so the banks' liquidity hoarding decisions
be
verified that all
banks (except potentially bank
characterized in part
hoard the
of Proposition
(ii)
level of liquidity that the
equilibrium at date
We
1,
divide the cases by the cascade
limited.
rium
If
K
<
is
size:
B
t
b>
described in part
K
6
transition
^uncertain
bank
6
'~ 31
PI
can be seen that y
^
J
<
hoards yJ
is
j/q
=
£/q
<
—
(p)
(i)
j> ee
=
A'
2,
(i)
<
is
Ho
=
5
Vo
To
>
y~o
f/o
K
0.
in the free-
[(li
—
x)
z],
K
when
Similarly,
.
4.
if
B
unceTtain
J
<
x/q
,
flexibility
equilib-
1
K=
K
=
In this case, each
Since
3,
is
<
ifQ
(p) are solvent.
no
each
i/q,
it
follows
It
again as described in
3.
bank
€
bP
B UTtcertam
an d may experience a precautionary insolvency depending on
see this, first note that y
In the first
4.
(where the inequality follows since the
so the banks in
>
>
The date
solvent in the free-information benchmark).
scenarios arise
i
^
^uncertain
t
particular, there are
2, in
equal to
of Proposition 2, with a cascade size equal to
The new
IP
where banks'
=
hoards y^
of Proposition
K
and
3.
that there are no precautionary insolvencies and the equilibrium
part
level of liquidity
would hoard
3
relative to the case
uncertam
precautionary insolvencies and the cascade size
b3
banks
all
can
of the proposition, which characterizes the
(i)
no additional panic
each bank
2,
in this case is as
bank
hoard the
')
It
(21).
changes once yo exceeds yj.
two of these cases there
is
p(l+1
bank with distance k
However, part
information benchmark.
6
problem
solve
still
In particular,
2.
Moreover, we
in x.
(p)
hoards
its losses
from
which implies
{R-i)vi<RSS-vi = (i-v)R-(i-a)ii-vi,
where the equality follows from Eq. (S). Combining this inequality with the inequality y JQ >
uriceTtaln
(since A" < 3. the banks in B
(p) have sufficient liquid reserves at date 1) leads to
:
v
Vo
Note also that condition
the above steps,
_
< (l-i/)fl-(l-u»/i-«i-r)= S
^_ j
(19) implies y
]
<
yo
<
Vo
21
2/o
I'.'i
an<^ thus rules
~ x
)
(/j
—
x)
z
~l-
out precautionary insolvencies by
.
To analyze
cross-deposits.
in
—
(ly
and thus increasing
x) z,
Second, note the inequality,
<
/2//1
R).
y
tjq
<
[0]
1
—
it
more a bank
the
is,
will
[(/]
j/q
—
x)
receives
decreasing
z] is
from
its
forward
experience a precautionary insolvency.
y (which follows from
some algebra and using
Then, there are two subcases to consider depending on whether or not the
level of flexible reserves y
v
That
in x.
bound above which
neighbor, the higher the
bound
this case, first note that the
greater than y^
is
[0]
(which
the highest value of the
is
bound
[(h-x)z}).
Subcase
upper bound
less of the
If
1.
—
[(/]
j/q
amount x
£)P('-( 2 "- 2 ))
{bp ^
in the interval
is
j/o
j
and a bank
x)
z]
it
receives from
are insolvent.
2.
€ [yo^Vo
If yo
It
p ,t
insolvency by hoarding some y
Subcase
(j/q [0]
[0])>
,
1
—
then y
y],
b7 experiences a
always greater than the
is
precautionary insolvency regard-
forward neighbor.
In particular,
all
can be verified that the informed bank
6 p(?
its
<
l
y Q (see
banks
'
''
in
averts
Appendix A. 2).
tnen there exists a unique x
[yo]
6
(/j
—
(3/zJi) that
solves
yt[(h-x[y
In this case, a
from
its
bank b3 that hoards
forward neighbor x
<
x
[yo]
])z]
yo of liquidity
(so that y%
K
are insolvent while the banks
is
a partial cascade which
is
<
E [K,2n b
p ^~
^
'
..,
(25)
.
insolvent
is
x)
we
z]
<
if
and only
By
yo).
carry out in
1- ' 2 " -1
))
I
if it
receives
a similar analysis to
Appendix A. 2),
such that the banks ib
1]
fr^
y
—
[(/1
that in Section 2.2 for the partial cascades (which
be checked that there exists
=
p(J
\
^(M^"
...
can
it
1
))
j
are solvent. In other words, there
at least as large as (and potentially greater than) the partial
cascade in the free-information benchmark.
We
summarize our findings
Proposition
3.
6
Suppose assumptions (FS), (17) and (18) are
Suppose also that condition (24) (which
(9), (13), (15) hold.
6
in the following proposition.
Given the possibility of precautionary insolvencies, one may
also
is
satisfied
the opposite of condition
wonder whether there could be
multiple equilibria due to banks' coordination failures. Suppose, for example,
contained after 3 banks
level of liquidity,
hoard the
and
maximum
fail.
Could there
also be a
bad equilibrium
in
and conditions
which
all
K=
3,
so that the crisis
banks hoard the
is
maximum
their liquidity hoarding decisions are justified since their forward neighbors also
level of liquidity
This kind of coordination failure
is
and experience a precautionary insolvency (thus paying a small
not possible
in
our setup, precisely because of conditions (13)
These conditions ensure that bank b pi + 1) is always solvent, even if all other banks hoard the
maximum level of liquidity and experience precautionary insolvencies. To see this, note that the losses
from cross-deposits decrease as we move away from the distressed bank and eventually qP('^ 2 > l x — /3/z.
pil+l>
expects to receive at least l± — 3/z from its forward neighbor, it can avoid insolvency
Since bank b
'~
by hoarding an intermediate level of liquidity. Hence, it is never optimal for bank b pl
to undergo a
p(l +
= l lt the rest of the equilibrium is uniquely determined
precautionary insolvency. But once we fix <y
and
-'
(15).
)
1
^
1
'
as described above, that
is,
there
is
no coordination failure
22
among
banks.
For
(19)^ holds.
a given financial
network b(p),
let
—
i
}
p
denote the
(j)
slot
of the
distressed bank.
For the ex-ante equilibrium
(i)
Each bank
(at date 0):
^
gknow
p{T
£ {b \
b>
hoards the same level of liquidity Uq (p) = y ree (p)
unceTta n
free-information benchmark, while each bank b> 6 B
(p)
J
,
that
'
which
the level of liquidity the bank b
is
The bank
mark.
p ^ +l
6
bP
hoards
^
p ^'~ 3
pl T -
l
\
bp
(T_2)
}
C
would hoard in the
it
=
hoards yfQ (p)
f
yQ
rj
(p),
would hoard in the free-information bench-
'
<
(p)
j/q
b
yfi
which
is
enough
just
to avert insol-
vency.
(ii)
For
tance based and monotonic.
h
p{1
\...b
The equilibrium allocation
the continuation equilibrium (at date I):
p{ J '^
-
K ' ^\
1
K
cascade size
is
There
K
a unique
exists
are insolvent while banks {&
..,
K
such that banks
1]
^(M2n-i))
)
potentially larger than the cascade size
—
[K,2n
£
p (*- if )
dis-
is
The
j a re solvent.
in the free-information bench-
mark. In particular, there are two cases:
If
K
<
then each bank
3,
bP
B uncerta,n (p)
£
precautionary insolvency. The cascade size in
benchmark,
If
to a
K
>
K
i.e.
4,
this case is identical to the
If Vo
€
(l/S [°]
K=
payment
is
below x
Discussion.
1
-
2n
€ (yo'^o
(25) such that bank
£
bP
- V)>
—1 >
B uncertain
(p)
£
[yo\-
aU banks b e
'
hoards
=
yf (p)
>
y
yfi,
which
may
lead
B
B unceTtam
exists a
unceTtain
(p)
The cascade
unique x
is
free-information
case in which y
yfi,
are insolvent
£ ih — P/z,
if
and only
an intermediate
size is
K
sufficiently large 9
when condition
y£
.
K
as a function of
forward neighbor's
£ [K. 2n
0, for
—
1].
different levels of
the
size A' in
(19) holds.
By
Proposition
is
the
same
2,
in this
as the
In this case, precautionary insolvencies are possible,
more banks
are insolvent in the costly audit
i.e.
a sufficiently large shock 9
-
if its
case
cascade
benchmark. The second panel corresponds to a higher level
free-information benchmark,
2n
characterized by Eq.
same parameters. The top panel corresponds to the
for the
i.e.
size in the free-information
>
li)
level
there are no precautionary insolvencies and the cascade size
that satisfies y
and the cascade
For comparison, the dashed lines plot the cascade
benchmark
<
[j/o]
insolvent
Figure 6 plots the cascade size
.
(p)
K > 4.
^ ere
[0]]-
bP
the flexible reserves y
K=
free-information
= K.
then each bank
size is given by
j/o,
and avoids a
y~Q.
precautionary insolvency. There are two sub-cases:
If Ho
of y
<
hoards som.e y Q3 {p)
K
may
> K. The
benchmark than
for
in the
shows that, as we increase
third panel
trigger a collapse of the
and
whole financial system
(i.e..
1).
The bottom panel
in
Figure 6 shows that as
23
y~Q
continues to
rise,
then at some point the
A
r
Figure
6:
The
costly-audit equilibrium with precautionary insolvencies.
K
one of the four panels plots the cascade size
(in increasing
K
amplification disappears and again
on the
size of the
an intermediate
is
as a function of 6 for a different level of
order of y from top to bottom).
in the free-information benchmark.
the flexible reserves y
plot the cascade size
cascade
K
is
K
=
K
.
That
is.
.
The
restored (and, in fact,
T
on
.
is
not sufficiently large, A" does not
As long
and. in the current case,
However,
if
it
as there
is
to
manageable
levels
a financial panic, the increase in y
also amplifies the cascade
the increase in y
fall
is
is
Increasing the level of
reduces the cascade size A' in the free-information benchmark.
financial panic remains.
insolvencies.
is
intuition for this non-monotonicity
the same as the intuition for the non-monotonic effect of y
increase in flexibility
lines
non-monotonic: The whole financial system collapses with
stronger) with sufficiently high levels of y
flexible reserves yo
The dashed
the effect of the flexible reserves yo
but the health of the financial system
level of yo,
Each
If this
and the
backfires
by generating more precautionary
sufficiently large,
it
may end
the financial
panic and restore the health of the financial system.
5
Final
Remarks
Our model captures what appears
to be a central feature of financial panics:
During
severe financial crises the complexity of the environment rises dramatically, and this in
itself
arises
causes confusion and financial retrenchment.
even
in transactions
among apparently sound
24
The perception
of counterparty risk
financial institutions
engaged in long
term relationships. All of a sudden, economic agents are faced with massive uncertainty
as things are no longer business-as-usual.
current financial crisis
is
money market
In the
cascades.
collapse of
Lehman Brothers during
one such instance, which froze essentially
and triggered massive run downs of
kets
The
credit lines
all
the
private credit mar-
and withdrawals even from the safest
funds.
model we capture the complexity
When
of the
environment with the
size of the partial
these cascades are small, banks only need to understand the financial
health of their immediate neighbors to
make
their decisions. In contrast,
when
financial
conditions worsen and cascades grow, banks need to understand and be informed about a
larger share of the network.
At some point,
from intermediation rather than
risk
this
is
simply too costly and banks withdraw
exposure to enormous uncertainty, which triggers a
flight to quality.
We
also
showed that banks'
new
not extending
flexibility,
defined as their ability to hoard liquidity by
loans or by selling illiquid assets while in distress,
for large cascades to develop,
but
if
makes
they do develop they can trigger more severe credit
crunches and even a collapse in the financial system. Intuitively, a gain
very useful
not as
An
it
if it
succeeds in containing panic, but
facilitates
aspect
in a related
it
can be counterproductive
is
in this
With
full
that of secondary markets for loans at date
the) cease to
r
does
information, the distant banks
(i.e.,
> K)
are the
However, once distant banks
close to the distressed
would rather hoard
as they
highlight in this
the banks with k
and become worried that they may be too
buy loans from these banks
Our preliminary
0.
mechanism we
natural buyers of the loans sold by the distressed banks.
face uncertainty
if it
paper but one which we are currently pursuing
findings point to yet another amplification aspect of the
paper:
in flexibility is
banks' withdrawal from intermediation.
we did not explore
work,
harder
it
their liquidity,
bank,
which
exacerbates the network's distress.
There are some obvious policy conclusions that emerge from our framework. For example, there
is
clearly scope for having
banks hold a larger buffer than
the}'
would be
privately inclined to do. Also, transparency measures, by reducing the cost of gathering
information, increase the resilience of the system to a lengthening in potential cascades.
There
ever,
is
we
even an argument to limit banks'
are interested in going
flexibility to cut
loan commitments.
beyond these observations, and
in particular in
the impact of policies that modify the structure of the network.
an emerging consensus that the prevalence of bilateral
tions
it
is
compounded the confusion and complexity
imperative to organize these transactions
25
OTC
exploring
For example, there
markets
for
CDS
of the current financial crisis,
in well capitalized
How-
is
transac-
and that
exchanges to prevent
a recurrence.
Our framework can
We
considerations.
A
help with the formal analysis of this type of policy
leave this analysis for future research.
Appendix
The Normal Environment
A.l
The
analysis in the text characterizes the equilibrium following a rare event for
network
financial
is
which the
not prepared. In this Appendix, we describe the functioning of the
financial network in the
normal environment. In particular, we show that the financial
network enables the banks to insure against heterogenous liquidity shocks and facilitates
the flow of liquidity across banks, even
the banks are uncertain about the network
if
structure.
normal environment, there are three aggregate states of the world, denoted by
In the
and
s (0), s (r)
s (g),
revealed at date
the same liquidity shock
and the
ui.
The
0.
In state s (0)
states s (r)
and
all
banks expect to receive
Uff,
date
1
with equal probability
s {g) are realized
liquidity shocks in these states are heterogeneous across banks.
More
specifically,
the banks are divided half and half between two types: red and green. In state
s (g)),
at
s (r)
(resp.
the banks with red type (resp. green type) expect to receive a high liquidity shock,
and the other banks expect
states s (r)
and
s (g)
there
is
to receive a low liquidity shock,
enough aggregate
uj l
liquidity but there
.
This means that
in
a need to transfer
is
liquidity across banks.
To
transfer liquidity in states s (r)
bilateral deposits in (1).
(resp.
all
even
slots)
We
and
s {g),
the banks form the financial network of
say that the financial network
is
consistent
if all
odd
slots
contain banks of the same type, which means that red and green
type banks alternate around the financial
circle.
We
restrict the set of feasible
networks
to consistent ones (as opposed to, for example, any circular network in which banks
be arbitrarily ordered around the
circle), since
these networks ensure that each
may
bank that
needs liquidity has deposits on a bank with excess liquidity, facilitating bilateral liquidity
insurance (see below) with the minimally required level of cross-deposits
Banks' types and the financial network are realized as follows:
banks are realized at random
(half of the
b
(p)
(with respect to these types)
Banks' types are their private information, thus the set
set of consistent financial
First the types of
banks become red type and the other half green
type); then a particular consistent financial network
realized.
z in (2).
B
networks from an ex-ante point of view
in (3) represents
(i.e.
is
the
before the types
of the banks are realized). This shock structure ensures that the actual realization of the
26
Banks'
y liquid reserves.
-
ya flexible reserves.
-
1
-
Measure
or
1/
—
of
1
at date
(-,
decision
demand
deposits that pay
/
3
at date
r'
€
[0, r],
1
Early depositors
2.
yo loans.
Lo.te depositors
and only
if
Date
Banks* types axe realised
random
a.t
the audit decision a'
€
demand
demand
their deposits.
their deposits
the bank cannot promise
if
insolvent banks (that pay
A consistent hnanciai network is realised.
NORMAL EN V: State in {:{0),:(r),5{g)}
RARE E\ ENT: State r'(0> is reoteed.
Banks make
1
make the deposit withdrawal
Ba.nks
Liabilities::
-
—
Dote
Balance Sheets
Initio]
Assets
{0,1,
unload
is
all of their
<j,
<
q-_
>
ij
'])
outstanding, loans
realised
Date
2
.
Banks mo.ke the liquidity hoarding, decision y^ 6
(Hquivalently banks extend yo — yi nw' loans.
|0,"/
Banks pay
]
qi to late
depositors.
|
Figure
7:
Timeline of events, for both the normal environment and the rare event.
network
financial
network
always consistent while the banks' uncertainty about the financial
is
described as in the rare event case. In particular, a bank b3 observes the
is still
(and since the network
slots of its neighbors
types of
banks
its
neighbors), but
it
(b3 j£{p(i-l),p(i)Xi+l)}
further narrow
down
is
consistent,
also indirectly learns the
it
does not know the slots (or the types) of the remaining
(see
Figure
By
2).
auditing a 3 balance sheets, the bank can
B3
the set of possible networks to
{p
a3
j
).
Figure 7 recaps the timeline of events in this economy both for the normal environment
and the rare event analyzed
main
text
is
that one bank, b3
[a
3
(p)
,
financial
(p)
if
,
z
3
(p)
network b
decisions that
early
Note that the rare event analyzed
normal environment
the
in
,
(p)
q[ (p)
,
such
maximize the preferences
and only
if
qi
< h
(cf.
The Normal Functioning
emdronment, the
financial
pay the contracted values
that,
state in {s (0)
in (4),
Eqs. (5) and
and the
(6)).
We
.
[q\
=
=
very similar
s (r)
in
h)
'J
7
,
s
(<"")},
decisions
of
realization
each bank
late depositors
bank
of
payment
each
for
b3
the
makes
withdraw deposits
next characterize this equilibrium.
facilitates liquidity flow
l\,q\
is
collection
and
of the Financial Network.
network
a
is
withdrawal,
,
and the aggregate
which
(0)
in the
the state s (0)) except for the fact
environment
deposit
cf2 (p)\
(i.e.
s
3
loses 9 of its liquid reserves.
normal
hoarding,
liquidity
j/q
text.
becomes distressed and
,
equilibrium
auditing,
main
characterized by an unanticipated aggregate state
to one of the states in the
The
in the
We
claim that, in the
normal
and enables each bank b3 to
each state of the world.
Suppose that a
b
consistent financial network,
and state
(p)
of generality that red type banks are assigned to
banks are assigned to even
slots
case since the case in which s (g)
in
which
We
realized
s (0) is
symmetric).
is
is
realized
odd
prove the statement for this
symmetric to the
is
the equilibrium at date
zP( 2l
chooses
2t)
in ,B p (
(/?),
>
A
1.
and hoards no
0)
~1
^
=
liquidity,
red type bank, 6
draws
it
/5
2
drawing z^ ^ £
U > h and the
i.e.
(which
,
a3
is
—
and y J
(.l-y)R
b
p{2 '\
p(2l)
=
deposits regardless of their beliefs f p
t
-
LU L
p), i.e.
(,
|
through the network at date
1
they choose c
p(2! '
even though each bank
is
=
0.
(and similarly
in states s (g)
choose not to audit and not to hoard any
at a cost of
liquidity
U/h
1 is
able to obtain
by hoarding
liquidity hoarding at date
a
P(i)
and
R >
l2
its
parity),
liquidity. First
0.
I).
and
its
Thus the bank does not
—
(whenever d{.) >
0),
l^,
is
p ^ l)
The bank could
Therefore, each bank
it
=
cf2
—
l-i)
banks
need of
in
deposits in the forward neighbor
Second note that a bank's,
in particular,
(q{
note that a bank b
but this would cost
date
and deposit withdrawal
(p).
q
> h >
follows that
verify our conjecture that the
by withdrawing
flexible reserves at
not to hoard any liquidity at date
(and only on
it
It
their
s (0)).
units at date 2 for each unit of liquidity.
each unit of liquidity (since
B ^
and
next consider the equilibrium at date
liquidity at date
draw
uncertain about the
network structure. In particular, each bank b3 pays the contracted values
in state s (r)
and
l\
+ Z** \ + (z-Z* 2i>)l 2
2lS
>
i.e.
network
regardless of the financial
preferences are given by (4), the green type banks do not
^
Consider
0.
assigned to an odd slot by
deposits leads to the payments q
[0, z]
=
deposits from the forward neighbor bank,
its
1
liquidity flows
'
2t-1)
For each green type bank,
z.
„ (2i)
p
and the case
s (r) case,
trivial.
is
assumption) needs liquidity so
We
which red type
slots (the case in
It suffices to
loss
conjecture (and verify below) that each bank b3 chooses not to audit (for any
positive audit costs d{.)
Since
and suppose without
s (r) is realized,
bp
p{
b ~'\
at date 1) only
also
obtain
R > U/h units for
^ optimally chooses
optimal actions (for
depend on
its slot
i
independent of the financial network in
benefit from auditing
and optimally chooses not to audit,
thus verifying our conjecture. This completes the proof of
our claim that, in the normal environment, the financial network facilitates liquidity flow
across banks and enables each
bank b
3
to
pay the contracted values
each aggregate state.
2-
[<j[
=
l\.qi
—
h) in
Proof of Eq.
Eq.
i.e.
original distressed bank, b
bank
b
that, for
We
p(-'~ k)
.
proves Eq.
some k 6
we
claim,
As the
cases in turn.
is
{0,
pays q
it
first
first
further implies that the
_
solvent,
benchmark analyzed
bank
1},
~'^
p{l
it
also
withdraws
b
p ^'~^ k+:
all
of
By
3.
its
deposits,
^ withdraws
withdraws deposits,
condition (9), the
of
all
z
i.e.
:p
i.e.
its
pl-'~ k>
=
which
z,
< ^ and qf~
—
(k ' l))
=
0).
Recall that bank b
forward neighbor as given (see footnote
its
its
forward neighbor by withdrawing
bank withdraws
from
claim that z p ^" k]
'
h- In this case,
=
withdraws
small
than
This
z.
of its deposits
its
^
bank
units per unit of liquidity.
bank W*
is
>
is
>
in particular,
1),
less
pi '- k
^~ k%
forward neighbor, i.e.
~
suppose that the forward neighbor bank, bp l ~^ k 1 ''., is
all
bp
^'~ k ^
needs liquidity
can obtain this liquidity either by withdrawing
it
bp
suppose that the forward neighbor of bank
i
q
R > U/h
liquidity,
z.
deposits in
z (to
pay
its
backward
deposits, which costs
its
units at date 2 per unit of liquidity, or by hoarding flexible reserves at date
costs
^ =
consider the free-information benchmark and analyze two
seconc C ase,
pi
i.e.
neighbor) and
U/h
^e
^g
;>
2n —
..,
case,
cannot potentially bail out
P(i-k)
insolvent thus
p{, - [k - l))
and takes the payment of
z
\
Section
in
cross-deposits are
all
by induction.
(7)
(i.e.
p^l
claim that
holds, in both the free-information
(7)
claim that bank b
To prove the
insolvent
We
3.
and the costly audit model analyzed
in Section 2.2
Suppose
and
for Sections 2.2
(7)
withdrawn,
fully
it
Main Text
Proofs Omitted in the
A. 2
all
of
its
Since the former
deposits from
is
which
0,
a cheaper way to obtain
forward neighbor, proving our
its
z.
Next consider the costly audit model of Section
Recall that bank b p{
3.
'~ k)
makes
the deposit withdrawal decision before the resolution of uncertainty for cross-deposits
(see Figure 3).
As the
first case,
to a network structure
b
(p)
suppose that bank
such that q
assigns a positive probability that
payment
takes the
its
z
(that
_
is,
z
]\T
(p)
of its forward neighbor as given
ext SU ppose bank
the bank
p
knows that
^~ k^
bp
its
>
<
li
(that
and
its
is,
suppose the bank
preferences are given
bank necessarily withdraws
believes that q
forward neighbor
information benchmark, the bank withdraws z
to its
^~ k assigns
a positive probability
forward neighbor will be insolvent). Since the bank
Leontieff form in (18), in this case the
p(i-k)
bp
is
p ^'~ k ^
P
(p)
all
=
li
of
its
by the
deposits,
with probability
=
z
to
meet
its
liquidity obligations
(7)
induction.
1.
Contained
in
the discussion preceding the proposition.
29
1
solvent). In this case, as in the free-
backward neighbor. This completes the proof of the claim and proves Eq.
Proof of Proposition
i.e.
by
Characterization of Banks' Payment Functions
3.
bank
If
pays x
is
=
b
p(
l
-
~ k)
hoards a level of liquidity
p
given by functions
Case
date
(p) at
q
If
1.
x e
[l\
qx
and
qi [y' .x]
-
=
/3/z,
,
/j
q2
and
lj]
,
and
(and
1
q2
[0,
>
-
(/]
If
2.
<
x
lj
—
/3/z or y'
=
<?i
The
<
1
y
its
buffer
characterizes the
0,
and
-
+ (l-y-y' )R
—
Section
forward neighbor
its
payment
follows:
>
(26)
l\.
to
x) z, then
7—+ zx
y'
<
:
=
and q2
'1
0.
(27)
payment when the bank's
and the bank has hoarded enough
these losses. In this case, the bank
when
+
case characterizes the
first
do not exceed
or
(lj
date
in
x) z, then
1
Case
yo] at
qi \y' .x}
which are characterized as
-(l -x)z
y'
=
€
and
condition (19) holds), then this bank's
if
[y'o-.x}
y'
y'
q^ [y' ,x]
losses
from cross-deposits
flexible reserves to
solvent and pays according to (26).
is
payment when the bank's
losses
counter
The second case
from cross-deposits exceed
its
buffer,
the losses do not exceed the buffer but the bank has not hoarded enough flexible
reserves to counter these losses. In this case, the
bank
insolvent
is
and pays according to
(27).
Proof of Proposition
First consider part
2.
of the liquidity hoarding decisions in part
of each bank depends only on
its
taking as given the characterization
(i)
Note that the
(ii).
liquidity
hoarding decision
distance from the distressed bank, which implies that
the payments of banks in the continuation equilibrium can be written as a function of
that the equilibrium is distance based. The characterization in part
~
shows that each bank b p{J k) G {b p{T\ b p{1 l \ ... £>MMA'-i))} that would be insolvent in
their distances,
(ii)
i.e.
the free-information benchmark hoards y^
would pay
Qi
in the
=
[*]
The bank
b
p{
-
l
^>
benchmark, thus
y
.
and thus
it
pays the same allocation
it
free-information benchmark:
lij'rel (P)
~K
=
<
'1
and ^2
[k]
=
fi
P
(
hoards at least as much liquidity as
it is
)
=
it
for k
€
{0,
would hoard
...
K
in the
-
1}
.
free-information
solvent given condition (19) (which ensures that hoarding too
liquidity does not cause insolvency)
and pays
Qi[K\
=
h
(cf.
andQ 2
30
(28)
much
Eq. (26)):
[^]
>
h-
(29)
The banks
piT
b
~V
|^-(^+i)) ^-(^+2))
<=
j
>
^
foP
(r-(2 n -i))|
and thus pay
are so i vent
(
efi
Eq. (26)):
p (T - k)
Q
l
=h
\k)
and
Q2
=
[k]
k)
+ (i-y- yf- ) R
yo
—>
-
—
1
'
+
h, for k E {A'
1,
..,
-
2n
1}
.
CO
(30)
In particular, the size of the cascade
Since q
p
as
also implies that the payments,
distance based equilibrium
Qi
[k]
and
1)
the characterization in (28) through (30)
(ii)),
and Q2
and proves that the
are increasing in k
[k],
monotonic.
is
are optimal. Consider
(ii)
decreasing in k (given
is
j/q
and prove that the choices
next turn to the liquidity hoarding decisions at date
prescribed in part
benchmark.
in the free-information
is
it
increasing in k (see Proposition
l~ee (p) is
the liquidity hoarding decisions in part
We
K
is
the banks in
first
B know
Comparing the
(p).
characterization of the continuation equilibrium in (28) through (30) to the characterization in Proposition
each bank
1,
IP
forward neighbor compared to what
its
mark
(i.e.
each bank
hoards the same
B know{p)
6
b>
We
B
g
bank
either case
K
Q\
<
[2]
which shows
K—
(21) with x
would hoard
uncertavn
[p)
bench-
in the free-information
m =
P
q
l))
j~^~
Thus
)-
which solves problem (21)
2.
=
Q
same
Note that
in this case
Q\
level of liquidity
[2] is
Note that by Proposition
l\.
benchmark.
with ,r m = Q [2].
l
1 [2]
=
q[
p !~
yQ
Ze l
K
>
(p),
3
^ (!_3)
that
ee
(the
=
q Jree
and note that
=
l\
when
in this case
completing the proof of part
K
Qi
<
[2]
2.
is
which
would
p
q j ree
given by Eq. (29) or Eq. (30)
p
1,
also
it
in the free-information
benchmark. To prove the claim that Qi[2]
claim in this case. Next suppose
The
would receive
problem
b 3 hoards the
in the free-information
suppose that
(28)
expects to receive the same payment from
[2]
in turn proves that the
hoard
V
it
it
(p)
characterized in Eqs.
(28) through (30) is equal to q\* ?
~
3(!
3)
in the free-information benchmark),
of the forward neighbor of bank 6'
claim that Qi
payment
solves
level of liquidity that
Next we consider a bank
2.
B know
€
,
first
and
piwing the
given by Eq.
(ii).
characterization for the aggregate level of liquidity hoarding for the cases A"
3
and
K
>
4 then trivially follow
from part
(ii)
in
and Proposition
1,
<
thus completing
the proof.
Proof of Proposition
the proposition. Here,
3.
Most
we consider
verify the claims in the
main
text.
of the proof
in
is
contained
turn the subcases
We
1
in the discussion
and
2 (for case
A
>
preceding
4)
and we
also verify the conjecture that the equilibrium
distance based and monotonic.
31
is
.
for
Subcase
1.
any x 6
[0, /]].
If
e
y
(j/q [0]
,
—
1
y]
then
,
This implies that
banks
all
in [b p(
l
-
1
\
..., fr/H
-' 2 1 - 2
'
are insolvent since
))}
they hoard a level of liquidity greater than their corresponding upper limits. These banks'
payments are characterized by the system of equations
q
where /
By
p(X-(k-\))
f^-^»j
(
,
>
foreach/ce{l,...2n-2},
defined in Eq. (12) and the initial condition q
(.) is
m^
plugging
o{l-k)
K _
^=
+
the fixed point y
<
f/o
< M>
1
P
=
q
\
we have
(15),
Then, since bank
Tee
£>
K
p( '~ A
K
—
—
2n
which
1,
— ^o^
< y^
ized in Eq.
if
and only
2.
If j/o
receives
the equilibrium
h
pir
K...b
p
is
^-^'-
The payments
(j/o>
i
from
we
its
there exists a unique x
in
t/
of the
banks
fc
>
=
in
we
<
b p(
l
-
p ^ ~^
..,
b
{qf-^~
for
^
an increasing sequence (by condition
are back to subcase
~ 1 J)
l
\
1 (i.e.
benchmark
h^
K—
2n —
l),
(13)).
6
(/]
B
K
(under
—
character-
yfi/z, l\),
unceTtam
(p) is
insolvent
Using the conjecture that
k
)
in
equilibrium in this case.
t
].
^
is
follows that the cascade
banks
further conjecture that the
p
...
j/>(M2n-i))j are so i vent
.
are characterized by
I
each
]
b3
forward neighbor x < x[y
l))
/
It
.
1
[y
such that a bank
,
y)\ are insolvent while the banks
"
is
[0])'
distance based and monotonic,
qf
which
Vo
and increasing
(25)
if
£
>
greater than the free-information cascade size
is
distance
is
p
the informed bank b p ^ +l
condition (15)), completing the characterization of the date
Subcase
implies q
in the free-information
'
this case can also avert insolvency by hoarding y$
is
increasing (and converges to
is
< In — 2, which
able to avert insolvency by hoarding the level J/o/ree
size
given by Eq. (10) (after
verifying our conjecture that the equilibrium
based and monotonia By condition
q
is
=/i).
condition (13), the solution to the above system
p
P
(31)
6
jfc
{l,
..,
A'
- l}
Then, either q P
or there exists a unique
l
(32)
,
'
K
< x
E [K, 2n
—
and
[j/o]
1]
such
that
^-^"•czlSo]^-'*-'".
In the latter case, the banks in
(since they receive less
than x
h
[yo]
pt J
-
-K
K
..,
b'M*-
1
))
}
C B uncertain
(33)
(/»)
are insolvent
from their forward neighbor) but the bank
32
b
p ^'
is
solvent since
\y>{i-{K+i))
it
receives at least x
^y,(i-(2n-2))
I
its
analysis implies that the equilibrium
1
b
p ^ +1
is
^
is.
The banks
forward neighbor.
are a | so so i ven t since the}' receive ^
ward neighbors. The informed bank
characterization of the date
from
[j/o]
> x[y
also solvent as in subcase
]
from their
1.
in
for-
Finally, this
distance based and monotonic, completing the
equilibrium in this case.
33
References
[lj
Allen, F.
and A. Babus (2008). "Networks
in Finance,"
working paper, Wharton
Financial Institutions Center, University of Pennsylvania.
[2]
Allen. F.,
and D. Gale (2000). "Financial Contagion," Journal of
Political
Economy,
108, p. 1-33.
[3]
Brock W. A. and C. F. Manski.(2008). "Competitive Lending with Partial Knowledge
of
[4]
Loan Repayment," working paper, University of Wisconsin.
Caballero, R.
and A. Krishnamurthy (2008). "Collective Risk Management
J.
in a
Flight to Quality Episode," Journal of Finance, 63(5), p. 2195-2230.
[5]
Caballero, R.
and
J.
P.
Kurlat (2008). "Flight to Quality and Bailouts: Policy Re-
marks and a Literature Review," <http://econ-www.mit.edu/files/3390>, October
2008.
[6]
Cifuentes,
R.,
G.
gion," Journal of
[7]
Diamond
D.,
Ferrucci,
and H.S. Shin (2005).
European Economic Association,
and
P.
Easley, D.,
3, p. 556-566.
Dybvig (1983). "Bank Runs, Deposit Insurance and Liquidity,"
Journal of Political Economy, 91,
[8]
and Conta-
"Liquidity Risk
p. 401-419.
and M. O'Hara (2005). "Regulation and Return: The Role of Ambiguity,"
working paper, Cornell University.
[9]
Eisenberg L. and T.
Noe
(2001). "Systemic Risk in Financial Systems,"
Management
Science, 47, p.236-249
[10]
Freixas, X.. B. Parigi,
and J.C. Rochet (2000). "Systemic Risk, Interbank Relations
and Liquidity Provision by the Central Bank," Journal of Money, Credit and Banking. 32, p. 611-638.
[11]
Lagunoff, R., and L. Schreft (2001).
Economic Theory,
[12]
"A Model
of Financial Fragility," Journal of
99, p. 220-264.
Leitner, Y. (2005): "Financial Networks: Contagion,
Commitment, and Private Sec-
tor Bailouts," Journal of Finance, 60(6), p. 2925-2953.
[13]
Liu
Its
J..
Pan
J.
and T.
Wang
(2005)
"An Equilibrium Model
of Rare-Event
Implication for Option Smirks," Review of Financial Studies, 18(1),
34
Premia and
p.
131-164.
[14]
[15]
C,
Rochet,
J.
Money.
Credit,
J.
Tirole (1996). "Interbank Lending and Systemic Risk," Journal of
and Banking,
Routledge, B. and
S.
28, p. 733-762.
Zin (2004). "Model Uncertainty and Liquidity," working paper,
Carnegie Mellon University.
[16]
Simsek A. (2009) "Speculation and Risk Sharing with
paper,
MIT.
35
New Financial Assets," working