MIT LIBRARIES
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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Fire Sales in a
Model of Complexity
Ricardo
J.
Caballero
Alp Simek
Working Paper 09-28
October 28, 2009
RoomE52-251
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Cambridge,
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496592
o
:
Model
Fire Sales in a
Ricardo
J.
of Complexity
Caballero and Alp Simsek*
October
2009
28,
Abstract
Financial assets provide return and liquidity services to their holders. However,
during severe financial crises
many
asset prices
plummet, destroying their
provision function at the worst possible time. In this paper
fire sales
follows
and market breakdowns, and of the
we present a model
financial amplification
from them. The distinctive feature of our model
is
liquidity
of
mechanism that
the central role played by
endogenous complexity: As asset prices implode, more "banks" within the financial
network become distressed, which increases each (non-distressed) bank's likelihood
of being hit by an indirect shock.
As
this
happens, banks face an increasingly
complex environment since they need to understand more and more interlinkages
in
making
their financial decisions.
uncertainty, which
reluctant to
buy
makes
crisis
ensues.
relatively healthy banks,
since they
control or understand.
This complexity brings about confusion and
The
now
fear
becoming embroiled
liquidity of the
The model exhibits
and hence potential
in
asset buyers,
a cascade they do not
market quickly vanishes and a financial
a powerful "complexity-externality." As a potential
asset buyer chooses to pull back, the size of the cascade grows,
which increases the
degree of complexity of the environment. This rise in perceived complexity induces
other healthy banks to pull back, which exacerbates the
JEL Codes:
Keywords:
crises,
E0, Gl, D8,
fire sale
and the cascade.
E5
Fire sales, complexity, financial network, cascades, markets freeze,
information cost, financial panic, credit crunch, pecuniary externality.
"MIT and XBER, and MIT,
respectively. Contact information: caball@mit.edu
We thank David Laibson, Frederic Malherbe. Juan Ocampo and
MIT for their comments, and the NSF for financial support. First
and alpstein@mit.edu.
at Harvard and
seminar participants
draft.:
July 28. 2009.
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium Member Libraries
http://www.archive.org/details/firesalesinmodelOOcaba
Introduction
1
Financial assets provide return and liquidity services to their holders.
severe financial crises
many
asset prices
plummet, destroying
function at the worst possible time. These
mechanism observed
However, during
their liquidity provision
are at the core of the amplification
fire sales
Large amounts of distressed asset sales
in severe financial crises:
depress asset prices, which exacerbates financial distress, leading to further asset sales,
and the downward
spiral goes on.
There are many instances of these dramatic
recent financial history.
Bernanke
floor
TARP
and the chaos they
trigger in
As explained by Treasury Secretary Paulson and Fed Chairman
to Congress in
goal of the
fire sales
an emergency meeting soon after Lehman's collapse, the main
during the subprime
on the price of the assets held by
crisis as initially
proposed was,
precisely, to
put a
financial firms in order to contain the sharp contrac-
tionary feedback loop triggered by the confusion and panic caused by Lehman's demise.
And
a few years back, after the
in his congressional
LTCM intervention,
testimony of October
1,
then Fed Chairman Greenspan wrote
1998:
"Quickly unwinding a complicated portfolio that contains exposure to
manner
of risks, such as that of
to conducting a fire sale.
flect
The
LTCM,
in
such market conditions amounts
prices received in a time of stress
longer-run potential, adding to the losses incurred
be sufficiently intense and widespread that
elevates uncertainty
enough
all
it
a
do not
fire sale
seriously distorts markets
re-
may
and
to impair the overall functioning of the economy.
Sophisticated economic systems cannot thrive in such an atmosphere."
In this paper
we propose a model
of
fire sales
that builds on the idea that complexity,
a feature strongly disliked by investors during downturns for the uncertainty
rises
endogenously during
crises:
As
asset prices implode,
(banks, for short) within the financial network
may go
more
it
generates,
financial institutions
under, which increases each bank's
likelihood of being hit by an indirect shock from counterparty risk.
This means that
banks face an increasingly complex environment since they need to understand more and
more
interlinkages in
making
their financial decisions.
Thus, perceived uncertainty
rises
with complexity and makes relatively healthy banks reluctant to buy since they now fear
becoming embroiled
in the cascade themselves,
rule out this option in the time available.
and a
financial crisis ensues.
and no reasonable amount of research can
The
liquidity of the
market quickly vanishes
In Caballero and Simsek (2009)
we show
the basic interaction between the size of the
equilibrium cascade in a financial network and the degree of complexity of the portfolio
problem facing banks. In that model banks have
bilateral linkages in order to insure each
The whole
other against local liquidity shocks.
system
financial
a complex network
is
of linkages which functions smoothly in the environments for which
though no bank knows with certainty
(that
is,
all
the
many
possible connections within the network
each bank knows the identities of the other banks but not their exposures).
However, these linkages
financial distress arises
may
also
be the source of contagion when an unexpected event of
somewhere
During normal times, banks only need
in the network.
to understand the financial health of their neighbors,
contrast,
designed, even
it is
when a
significant
problem
which they can learn
at low cost. In
network and the possibility of
arises in parts of the
cascades arises, the number of nodes to be audited by each bank rises since
that the shock
may
uncertainty and they react to
is
it
means that banks now
Our
the starting point of this paper.
focus here
is
fire sales
on secondary
that arise in
In particular, this paper introduces a secondary market in which banks
these markets.
can
face significant
by retrenching into a liquidity-conservation mode.
markets and on the feedback between that environment and the
in distress
possible
spread to the bank's counterparties. Eventually, the problem becomes
too complex for them to fully figure out, which
This structure
it is
sell
their legacy assets to
meet the surprise
liquidity shock.
The natural
buyers of the legacy assets are other banks in the financial network, which
receive an indirect hit.
When
the surprise shock
is
small, cascades are short
can inspect their neighbors to rule out an indirect
may
also
and buyers
In this case, buyers purchase
hit.
the distressed banks' legacy assets at their "fair" prices (which reflect the fundamental
value of the assets). In contrast,
when the
surprise shock
is
large, longer cascades
possible which increases the complexity of the environment,
an indirect
hit.
As a precautionary measure, they hoard
trades in (now illiquid) legacy assets and
assets
A
plummets
shock.
When
and buyers cannot rule out
liquidity
even turn into
and disengage from
sellers.
model
is
price of legacy
the dependence of the cascade size on the price
which leads to multiple equilibria
for intermediate levels of the surprise
legacy assets fetch a fair price in the secondary market, the banks in distress
have access to more liquidity and thus the surprise shock
bankrupt, leading to a relatively simple environment.
the natural buyers rule out an indirect hit and
demand
is
contained after fewer banks are
When
the environment
fire-sale equilibrium
is
simple,
legacy assets, which ensures that
these assets trade at their fair prices. Set against this benign scenario
a
The
to "fire-sale" levels.
central aspect of our
of legacy assets,
may
become
where the price of legacy assets collapses to
is
the possibility of
fire-sale levels,
which
.
As the
leads to a longer cascade and a greater level of complexity.
become worried about an
increases, natural buyers
and/or
sell their
own
and
complexity
hoard liquidity
the)'
legacy assets, which reinforces the collapse of asset prices.
This amplification mechanism
tial asset
indirect hit
level of
is
exacerbated by a complexity- externality. As a poten-
buyer chooses to pull back, the
size of the potential
cascade grows and with
it
the degree of complexity of the environment (each bank needs to explore larger segments
of the network to understand the risk
it is
exposed
to)
This
.
complexity induces
rise in
other healthy banks to pull back as a precautionary measure, which further exacerbates
the
fire sale
and cascade.
1
Our framework has two
liquidity externality that
bank chooses
it
additional (and
more standard)
externalities:
A
network-
stems from the interlinkages of the financial system; when a
to raise external liquidity rather
spreads the distress to other banks.
And
than to generate
a
it
from
fire-sale externality
its
own
resources,
that arises from the
negative effects that a bank's distressed asset sales have on other banks' balance sheets.
These two externalities are present
in
many network and
liquidity models.
They
interact
but are distinct from the complexity externality that we highlight, which stems from the
feature that any decision that lengthens the potential cascade, increases the complexity
of the environments that other banks need to consider.
This paper
sales, these
is
related to several strands of literature.
happen because during
(cf.
Shleifer
may
if
some
choose not to arbitrage the
run because they anticipate a better deal
in the future.
potential buyers are
fire sale in
Our model
between these two views: Most potential buyers are unconstrained,
and Pedersen
fire
(2008), but they are confused
and hence
More
and Vishny (1992,1997)).
Brunnermeier and Pedersen (2008) show that even
not distressed or constrained, these
model of
financial crises the natural buyers of the assets (other
banks) also experience financial distress
recently,
In the canonical
fearful of going
lies
as in
the short
somewhere
Brunnermeier
about their normal
arbitrage role (and in this sense they are distressed as in Shleifer and Vishny (1992)).
is
in
It
the complexity of the environment that sidelines potential buyers.
There
is
an extensive literature that highlights the possibility of network failures and
contagion in financial markets.
An
incomplete
list
includes Allen and Gale (2000), La-
gunoff and Schreft (2000), Rochet and Tirole (1996), Freixas, Parigi and Rochet (2000),
Leitner (2005), Eisenberg and
Although we do not pursue
this
Noe
(2001),
avenue
in
and Cifuentes, Ferucci and Shin (2005)
the paper, complexity probably plays a role
the incompleteness needed for a pecuniary externality to arise.
If
in
(see
creating
agents could sign contracts contingent
on the events that follow a cascade, then the externality would be greatly reduced, However, as potential
cascades lengthen, the number of contingencies that need to be written into contracts grow exponentially.
Allen and Babus (2008) for a recent survey). These papers focus mainly on the mecha-
nisms by which solvency and liquidity shocks
In contrast,
we take
environment
these
phenomena
plexity on banks' prudential actions.
mechanism we emphasize
lieve that
That
is,
in this
The contagion
many
even
cascade through the financial network.
as the reason for the rise in the complexity of the
which banks make their decisions, and focus on the
in
of contagion.
may
paper
literature
financial institutions
there
if
is
a cascade,
also
It is
is
is
com-
worth pointing out that the complexity
operational even for relatively small
sometimes
criticized
would be caught up
it is
effect of this
because
it
is
amounts
hard to be-
in a cascade of bankruptcies.
reasonable to expect that
2
would eventually be
it
contained (especially since banks take precautionary actions to fight the cascade). But
as this paper illustrates, even partial cascades can have large aggregate effects, since they
3
greatly increase the complexity of the environment.
Our paper
is
also related to the literature
tainty in financial markets, as in Caballero
on flight-to-quality and Knight ian uncer-
and Krishnamurthy (2008), Routledge and
Zin (2004), and Easley and O'Hara (2005); and to the related literature that investigates
the effect of
(2005),
new
events and innovations in financial markets,
e.g.
Liu, Pan,
Brock and Manski (2008), and Simsek (2009). Our contribution
literatures
network
is
in endogenizing the rise in uncertainty
itself.
More
broadly, this paper belongs to an extensive literature on flight-to-
the financial system's ability to channel resources to the real
for
The organization
economy
and a decline
(see, e.g.,
in
Caballero
a survey).
of this paper
network and the secondary market
event) in the network.
We
is
as follows.
we
describe the financial
and we introduce a surprise shock
(a rare
benchmark case without complexity
effects
for assets,
also discuss a
In Section 2
(because banks can understand the network at no cost).
results.
relative to these
from the behavior of the financial
quality and financial crises that highlights the connection between panics
and Kurlat (2008),
and Wang
Section 3 contains our main
There, banks have only local knowledge about the financial network, and a
sufficiently large surprise
shock increases the complexity of the environment and leads to
2
See. e.g..Brunnermeier, Crockett, Goodhart, Persaud, an Shin (2009) which argue that the domino
model of financial contagion is not useful for understanding financial contagion in a modern financial
system since ".... It is only with implausibly large shocks that the simulations (of their model) generate
any meaningful contagion. The reason is that the domino model paints a picture of passive financial
institutions who stand by and do nothing as the sequence of defaults unfolds. In practice, however, they
will take actions in reaction to unfolding events, and in anticipation of impending defaults...''
3
The
role of cascades in elevating
party's risk
complexity was also highlighted
in
Haldane's (2009) speech,
who
mechanism when he wrote that at times of stress "knowing your ultimate counterbecomes like solving a high-dimension Sudoku puzzle."
nicely captures the
.
a breakdown in secondary markets.
level of
This section also highlights the dependence of the
complexity on asset prices and demonstrates the possibility of multiple equilibria.
In Section 4
we
describe the three externalities in our setup and analyze their role in our
The paper concludes with a
main
results.
2
Equilibrium without Complexity
In this section,
remarks section and several appendices.
final
we describe the basic environment and characterize the equilibrium
a benchmark case in which financial institutions (banks, for short) have
of the interlinkages between
we show
that
the network
if
the banks in the financial network.
all
deep
is
(i.e.,
there
markets do not break down and the financial network
the size of financial cascades
is,
These
number
a large
is
is
resilient to
knowledge
full
benchmark
In this
of banks) secondary
a perturbation. That
contained and aggregate loan contraction
is
we obtain
relatively benign results contrast with those
for
is
limited.
in the next section
once we
introduce complexity.
The Environment
2.1
We
consider an
can be kept
economy with three dates {0,1,2}
in liquid reserves or
reserves, a unit of the
to firms,
it
then yields
it
1
units at date
The economy has n continuums
is
tP,
composed
which
is
in the next date.
2.
units of legacy loans, z units of
reserves set aside for
Each bank has
demand
making new loans
measure of demand deposits held by
R
units at date 2)
driven depositors.
pay
'The only reason
for
loaned
is
illiquid.
continuums
of these
each continuum
initial assets
retail depositors
as long as the
economy
Each
.
1
refer to
and
b>
as bank
which consist of
liabilities consist of
units of
z
in this
bank
is
solvent,
2.
At date
all
demand
1
—
y
economy pay
1
a unit
deposits held
unit at date
will
(only) there
be whether the bank uses
the continuum
is
for
all retail
1
depositors would rather
of its flexible reserves has just
legacy loans and the (equilibrium) price for these loans
trade-off in this
a unit
the depositor arrives early (resp. late). There are no liquidity
depositors at date
its
•
The bank's
at date 0.
Note that a bank that loans
arrive at date 2.
assets to
I.e.
if
if
deposits in one other bank, and y units of flexible
by one other bank. The demand deposit contracts
(resp.
by {V }
we
of identical banks and, for simplicity,
our unit of analysis.
Instead,
kept in liquid
If
These new loans are completely
of banks denoted
4
which a single good (one dollar)
can be loaned to production firms.
good yields one unit
R>
in
is
r
<
is
1
enough
a secondary market for
(see below).
its flexible
The
reserves to
banks to take other banks' decisions as given.
central
make new
loans and to purchase legacy loans in the secondary market, or whether
we describe below).
of this liquidity in response to a rare event (that
The banks'
cross
demand
b (P
w.here p
{l,..,n}
:
demand
now,
all
network
A
make
These
all
bank
but we
-* 6" (1)
in the
subsequent
will shut
slot
model
down
is
i
(i.e.,
2
+
and
1,
the bank
9 dollars of
losses
of
might
may
3
(to
an outsider)
1.
To prepare
{HS, H, B}, which
and Simsek (2009)
for
reserves y as liquidity
choose
its
A =
J
H,
slot 1 as a
remain solvent
in order to
at date
for
date
1,
For
will
,
bank
this
1.
may
bring
the banks take one
at date
are restricted to a discrete choice set for
a related model with unrestricted action
and
sell all of its
its
its flexible
keep or withdraw
to hoard
its flexible
legacy loans
its
reserves either to
is
more
make new
profitable).
bank which
to pay the retail depositors
equilibrium for bank
In Caballero
b°
y in the
and to keep
A =
3
1,
if
is
it
able to pay
they arrive
its
early.
at date 1 takes one of
its
this
W}.
it
ends up paying
are no liquidity driven retail
q\ at
date 2 can refuse
assumption, the continuation
two forms. Either the bank
and Simsek (2009), we motivate the formation of the
and Gale (2000).
facilitating bilateral liquidity insurance, as in Allen
in the
depositors the contracted
depositors at least
With
is,
the bank chooses whether to
takes), but instead
Banks know that there
its
B, to keep
buy legacy loans
deposits on the forward neighbor bank, A\ 6 {A',
to all depositors.
depositors, thus a
choose
loans or to
At date
R) (despite the precautionary measures
(01,132)
—
balance sheet and to be a potential buyer of loans. That
Given the rare event, a bank may not be able to pay
(1,
may
1
to
extreme measure,
less
reserves as liquidity
balance sheet. Alternatively, the bank
secondary market (whichever
J
n has
slot
over to other banks via the financial network and
A —
legacy loans on
bank uses
some
has z units
this ingredient until the next section.
and Gale (2000),
payment
spill
its flexible
legacy loans on
amount
p W)
the banks' uncertainty about the
secondary market, keeping a completely liquid balance sheet. As a
this
fc
in the
As the most extreme precautionary measure, the bank may choose Aq — HS,
all
own
bank
i
the financial linkages (cross-deposits) in the network.
simplicity (see Caballero
its
(l)
)
the banks learn that a rare event has happened and one bank, b3
of the following actions
hoard
bW
b?w ->...;-»
denotes that the bank in slot
>
into financial distress at date
space).
-*
a permutation that assigns bank b p ^ to sZof
is
distressed. Similar to Allen
needs to
them
0,
w^
-»
central feature of the
in (1),
banks know
At date
become
—
arc
deposits in the
forward neighbor.
financial
{l,..,n}
»
The
financial network.
of
—
deposit holdings form a financial network denoted by:
= (ww
)
hoards some
it
financial
is
solvent,
network
pays
for its role in
5i
=
1, <fi
q{
<
1,
>
=
t/o
1)
and the
0,
and
J
actions (A ,A{) to maximize q\ until
ations to depositors, that
bank, thus once
it
at date 2; or the
depositors draw their deposits at date
all
The bank chooses
withdraw
retail depositors
is,
until q[
—
1.
Increasing
satisfies its liquidity obligations,
to late depositors (f2
Note that the banks make
-
it
q[
it
is
insolvent, pays
its
liquidity oblig-
1.
can meet
beyond
then
bank
1
has no benefit for the
tries to
maximize the return
their decisions while facing uncertainty
about the network structure. For expositional simplicity we assume that banks are
infi-
nitely risk averse (rather than just risk averse) with respect to the financial network,
i.e.
they evaluate their decisions according to the worst possible network realization which
they find plausible.
Secondary Market
for
Legacy loans are traded
price
r,
Legacy Loans
in
a centralized exchange that opens at date
the banks that choose
3
bank) while the banks that choose
spend up to y
HS
A =
sell all
AJ = B
—y
units for each
are potential buyers of legacy loans
There
(their flexible reserves).
of their legacy loans (1
Given the loan
0.
is
no demand or supply
and may
for legacy loans
outside of the network.
A
legacy loan traded in the secondary market
and
originator,
it
yields returns
R—
<
5
R
is
at date 2.
held by a bank different than
We
assume
6
>
which simplifies
the subsequent analysis, but the economic results generalize also to the case 5
that there
is
an upper bound on endogenous loan
of legacy loans give
up
units at date
2,
R
If
while
r
if
2.
(1992).
Note
buyers
Since secondary loans return
= (R~5)
R—5
/R.
freely disposed of at the scrap value rf ire
6 (0,ry a7T ), so
a lower bound on prices.
<
r
Tj aiTl potential buyers
=
rj air
,
spend
all
of their flexible reserves y on legacy loans,
they are indifferent between buying legacy loans and making
In other words, even though banks get returns slightly lower than
in the
6
no potential buyer would demand legacy loans at a price greater than
Legacy and new loans can be
is
0.
prices. In particular, potential
units at date
r fajr
rf ire
=
which they could also use to make new loans, thus
flexible reserves
each loan has an opportunity cost of
its
secondary market, they are the natural buyers of these loans
R
new
loans.
from legacy loans purchased
and Vishny
in the sense of Shleifer
1
Banks
-
-
Liabilities:
1
—
demand
z
-
a legacy loans.
forward
-
A\
-• A':
-
A\
=
11':
Keep deposits
in
forward neighbor.
Withdraw deposits
in
forward ireaghbor.
demaud
deposits held by
backward neighbor batik.
-
deposits in
iH'itfhlxir
1
Banks choose an action A[ e {AMI'):
Dare 2 puyment to retail depo -itors
of measure due: 11 units total.
a flexible reserves,
-
Date
Balance Sheets
Initial
Assets:
bank.
.:
Retail depositors dei.ri.aud
Date
and only
if
Banks learn li.it bank b becomes distressed
and Iihh to make a liquidity payment of at date
if
t
t
licit
deposits
bank cannot promise
lie
fii
>
.1.
t
1.
Date 2
Banks choose an action
€ \ IIS, II. IS}:
= IIS: Hoard liquid reserves arid sell legacy loans.
•= //: Hoard liquid reserves, keep legacy loans
,4(| = 11: Use reserves to buy legacy loans or to make new loans
.1;',
- A(,
.'1,'',
Legacy loans are traded
in a centralized
exchange
at price V.
Figure
1:
Thus, the market clearing condition
Banks pay
to late depositors.
<fj
Timeline of events.
can be written as
for legacy loans
>
;i-y)X>{^ = tfS}-^l{^ = B]
=
if
<
The
term on the
first
maximum
term denotes the
for
each r £
hand
left
[rfire
,
rfair],
potential
demand.
If
enough legacy loans to
the
[rf^e, rfair],
r
£
[rfire,
value
rfire
rf a
j
r ],
then r
the
left
then legacy loans trade at their
and
£
clear the market.
is
If
selling
left
is
hand
fair
if
r flTe
(r fire, r aiT )
r
=
(2)
r fair
left
side of Eq. (2)
if
negative
,
loans,
and they buy just
some
side of Eq.
(2) is
hand
positive for each
side
excess supply of loans and the price
(note that this happens only
is
value rf air potential buyers
new
hand
the equilibrium price. If the
then there
£
r
=
side denotes the total supply of loans while the second
are indifferent between buying legacy loans
r
r
if
is
is
for
given by the scrap
there are no potential buyers).
We
refer to this
situation as a breakdown in the legacy loan market.
Equilibrium
Figure
1
recaps the timeline of events in this economy.
Definition
{AJ
(p)
,
A\
1.
An
(p) }
J
equilibrium
{q\(p) ,qi (p)
}J
is
a
collection
and a price
Jb(p)
We
of
can now define equilibrium:
bank
level r
£
and
actions
[r fire
,
r /a7>
]
payments
for legacy loans
such
W
given the realization of the financial network b (p) and the rare event, each bank
that,
chooses
its
actions according to the worst case financial network that
the legacy loan
and only
early if
We
market
if q\ (p)
<
Eq.
(cf.
(2)
retail depositors
tP
,
finds plausible,
withdraw deposits
1.
bank bJ=p ^K Note
there exists a unique k 6 {0,
network b
that, for each financial
..,
j
—
n
=
which we define as the distance of bank
p
b3
the payoff relevant information for a bank
...
n} denote the
and
(p)
for
each
1} such that
-
(i
k)
,
from the distressed bank.' The distance k
V and
will play a
key role in the analysis.
is
8
The Benchmark without Complexity
2.2
We
and the
J,
next turn to the characterization of this equilibrium. Let T € {1,
slot of the distressed
bank
clears
it
We
characterize the equilibrium in two steps:
and payoffs
for
a given price
start
by describing the banks' actions
and then solve
level of legacy loans, r;
for the
equilibrium
price using the legacy loan market clearing condition (2).
Suppose the loan prices are
fixed at
some
r
£
[rfi re ,Tf a ir]
and consider the banks'
Consider a distressed bank that needs to
optimal actions and payments
in this setting.
obtain liquidity at date
the original distressed bank b p ^). This bank could try to
1 (e.g.
obtain the required liquidity either by withdrawing
choosing A\
A
3
= W)
£ {H. HS}).
It
cross deposits at date
its
and/or by taking a precautionary action at date
(i.e.
1
by
by choosing
(i.e.
can be checked that the bank always weakly prefers ex-post liquidity
withdrawal to the ex-ante precautionary actions, thus we
fix
banks' liquidity pecking order
as follows:
Assumption (LPO). The
liquidity pecking order is such that a
at least z units of liquidity at date
1
first
chooses A\
— W, and
bank that
then
(if
will
there
is
need
need)
resorts to ex-ante precautionary measures.
Under assumption (LPO) and the parametric condition
bank
6
p(l)
,
which has to make a liquidity payment of
from the forward neighbor bank
liquidity,
'We
8
use
which
also
withdraws
modulo n arithmetic
b
p(l+l
its
^ .
>
6
6 at date
This puts bank
b
p ^ +l
z,
1,
the original distressed
withdraws
1
index
i,
e.g. T
—
k
= —1
deposits
also in need of z units of
deposits on the forward neighbor bank.
for the slot
its
'
represents the slot
As
n—
in Allen
1.
Note that the definition of equilibrium has built in a conservative feature which characterizes financial
markets in distress: Banks make their precautionary decisions based on the worst outcome they find
plausible. This feature plays no role in the benchmark since banks know the actual financial network.
and Gale (2000),
are withdrawn,
this triggers further
A\
i.e.
=
W
withdrawals
until, in equilibrium, all cross
bank
for all j. In particular, a
the original distressed bank b p ^)
withdrawals. Anticipating that
tries,
will
it
in
deposits
need of liquidity (including
but cannot, obtain any net liquidity through cross
not be able to obtain additional liquidity at date
1,
the distressed banks try to obtain some liquidity by taking precautionary actions at date
0.
To
simplify the characterization of the banks' date
y
The
hand
left
bank that
sells its
<K(l~y)R.
+ (l-y)r fair
side of this condition
legacy loans
is
not new, as
it
bank that keeps
promise
to
make
A =
J
it
its
depositors at least
a net liquidity
H.
The
1.
legacy loans on
its
1
payment
its
hand
right
book
unit at date
(i.e.
chooses A\
—
H
believes that
and averts insolvency.
it
liquidation outcome.
of losses the
unit to
If
1
less
than
even
3
€ {H, B}) can always
(3),
a
bank that expects
its flexible
it
bank can sustain while remaining
i.e.
the action
reserves y, then
is
greater than y,
A = H. Hence, this
chooses A = HS to improve its
since this is the maximum level
if it
refer to y as the bank's buffer,
We next conjecture that,
A
the bank's expected loss
1
= HS)
depositors at date 2, and
its
that chooses
is
Aq
side of this condition ensures that a
cannot avoid insolvency and thus
We
ensures that a
It
0.
at date 1 first considers hoarding liquidity,
then the bank expects to be insolvent at date
bank
1
Under condition
2.
the bank's expected loss at date
If
>
that chooses the most precautionary action
(i.e.
be insolvent at date
will
(3)
follows from 5
it
does not have enough resources to promise at least
thus
we assume
actions,
J
chooses
J
solvent.
under appropriate parametric conditions (including condition
K (r) £ {l,..,n — 2}, which depends on loan prices, such
that
banks with distance k < K (r) —
are insolvent (there are K (r) such banks)
while the banks with distance k > K (r) remain solvent. That
the crisis will partially
cascade through the network but will be contained after K (r) < n — 2 banks have failed.
We refer to K (r) as the cascade size.
(3)),
there exists a threshold
all
1
is,
The
than
original distressed bank, b p{ \
its
l
-
is
insolvent as long as
buffer y (and greater than z so that assumption
its
required payment
(LPO)
applies),
i.e.
if
is
greater
and only
if
6>max(y,z).
Suppose
this
is
the case so bank
f^
is
insolvent
Id
(4)
Anticipating insolvency, this bank
— HS,
chooses Aq
all retail
depositors will arrive early at date
{r)
Ql
_
~
+ r(l-y)-e +
l + z
y
z
and the bank will pay
1,
<l
Note that the
to each depositor (where the inequality holds in view of condition (3)).
bank
it
have y
will
+
r (1
—
Consider next the bank
its
6
P '' -1
bank needs to pay
1
on
its
Hence, bank
6 p(
'
^
are greater than
will also
become
buffer, z
its
(
1
—
pay
p(l)
insolvent
p
qx
>
j
so
,
1
it
fr
6
P * !+1
'
and
of 8,
(which
is
unit for each unit of deposit).
p( '~ 2)
loses z
(and only
if
make a payment
from the distressed bank. To remain
1
deposits to bank
deposits from the distressed bank 6
-1
will
with distance
^
will
1, it
forward neighbor bank
its
and thus
solvent in our conjectured equilibrium
on
from
will receive z units in cross-deposits
solvent, this
date
y) units of liquidity at
but
it
—
P
1
(
1
in cross-deposits.
q
J
losses
if) its
<
receives only q[
from cross-deposits
which can be rewritten as
y,
qf <\-\.
]
condition
If this
cascade size
is
fails,
the only insolvent bank
= 1. If this condition holds, then bank
= HS, and it will pay its depositors
^
,*-„ .
j
(with k
this point
>
1) is
this bank's
i.
=
+
6
p ( t_1
>
and the
anticipates insolvency
rd-,)-^
(5)
onwards, a pattern emerges. The payment by an insolvent bank
b
pi-'~
given by
p{%— k)
9i
and
the original distressed bank
K (r)
and chooses Aq
From
is
backward neighbor
6P
'
=
I
r
p(
/ (Si
!_ ' A:+1
"
is
also insolvent
if
and only
if
q{
<
1
— -.
Hence, the payments by insolvent banks converge to the fixed point of the function /
and
if
y
then there exists a unique
K (r)
>
k)
qf~
and g**-C*(r)-U>
If
(.),
n —
2
is
+ r(l-y)>l-Z,
2
such that
<l>
greater than the solution,
(6)
!
_
V
-
for
each k E
{0.
...
K (r)
V__
K (r),
to this equation,
11
i.e.
if
-
2}
(7)
n >
K (r) + 2,
(8)
then, Eq. (7) shows that (in addition to the trigger-distressed
with distance k 6
..,K
{1,
(r)
—
bank
1} are insolvent since their losses
b p ^) all
banks
fe^(
t_fc '
from cross deposits are
to improve their
greater than their corresponding buffers. These banks choose Aq =
p(l ~ h( r
^ is solvent, since it can meet its losses by
liquidation outcome. In contrast, bank b
= H. Since bank £^( l ~ A M) is solvent,
hoarding liquidity, i.e. this bank chooses Aq
HS
-
all
6 p(
banks
-/c
'
)
{K
with distance k £
(r)
+
1,
...
n
—
they do not
1} are also solvent since
incur losses in cross-deposits. These banks are potential buyers of legacy loans,
choose
AQ —
J
conditions
Note
B.
This
(3), (4), (6)
verifies
,
also that Eqs.
our conjecture for a partial cascade of size
and
(8).
(5)
and
imply that
(7)
loan price, the liquidation value of each bank
is
K (r)
is
decreasing in
greater, thus the crisis
i.e.
K (r)
they
under
with a higher
r:
contained after
is
a smaller number of insolvencies. For future reference, we strengthen condition
(6) to
+ r fire (l-y)>l-^,
y
so that there exists a partial cascade for any r e \vj lTe rj a i T
,
(9)
]
The next lemma summarizes
.
the above discussion.
Lemma
1.
Suppose information
is
that the loan prices are exogenously fixed
and that conditions
of size K(r), where
bank k
<
distance
k
K (r)
K (r)
K (r) +
>
Figure
1
—
(3), (4), (8)
K (r)
1
is
,
and
atr£
2 displays
illustrating that
K (r)
is
[rfire,
Then, there exists a partial cascade
A =H
AQ = B
r.
.
the distressed
The transition bank with
The remaining banks with distance
3
r fair] for
decreasing in
A — HS.
J
J
the equilibrium cascade size
ogenously fixed at some r G
assumption (LPO) holds,
Banks with distance from
and they choose
and they choose
network b(p),
the financial
ry ajr ], that
,
(9) are satisfied.
averts insolvency by choosing
are solvent
[ry, re
defined by Eq. (7).
are insolvent
know
free so that banks
K (r)
when the
loan prices are ex-
a particular parameterization of the model,
The negative dependence
of the cascade size
the price of loans will play an important role in the next section in which
on
we consider the
equilibrium with endogenous complexity
We
rium
next consider the legacy loan market clearing condition and solve for the equilib-
level of prices in the free-information
large, the
To
endogenous loan price
benchmark.
in the free-information
see this, note that the insolvent banks (there are
12
We
claim that
benchmark
K (r)
of
is
if
n
is
sufficiently
= Tf air
A J — HS
given by r
them) choose
.
K-(r)
05
15
1
25
2
45
4
35
3
5
Cascade size in the free-information benchmark. The plot
cascade size K (r) when the loan prices are exogenously fixed at r £ [0, rf air
Figure
displays the
2:
]
a given
(for
level of 9).
and
hence the aggregate supply of loans is given by
^~
K T )) chooses to hoard liquidity while the remain(r) (1 — y). The transition bank bp
^~ k
J
— B, i.e.
ing solvent banks {b p
(there are n —
(r
1 of them) choose
sell
all
of their existing loans,
K
^
^
the}' are potential
(n
Under
'fair
'
K
,
buyers of loans. Suppose n
-
is
level r
<
demand from
rf mr
,
if
the cascade
is
only partial and banks
When
there are sufficiently
to absorb the supply
r fair
(10)
)
assumption.
The next
1.
that much.
flexible reserves to
many
banks, the
the financial network, then
These banks do not
purchase loans from distressed
demand from
these solvent banks
is
.
We refer
to condition (10) as the deep secondary
market
proposition summarizes the above discussion and characterizes the
symmetric equilibrium
Proposition
know
know
from the distressed banks, ensuring that the secondary loans
are traded at their fair price rj air
and
[r fire ,
thus the loan market clearing condition (2) implies that
hoard liquidity and are ready to use their
b(p).
G
the potential buyers exceed the supply of loans
there exist banks which will remain solvent and
enough
for all r
•
Intuitively,
banks.
A
sufficiently large so that
K (r) - 1) y > r (1 - y) K (r)
this condition, the
any price
for
,
for the free-information
Suppose information
is
benchmark.
free so that banks
Suppose assumption (LPO) holds, conditions
the deep secondary
market assumption
(3),
(10) holds.
13
know
(4).
Then:
(8),
the financial
and
(9)
network
are satisfied,
r(8)
=
r, a ir
0.5
i-
0.6
0.7
0.9
0.8
1.2
1.1
1
1.3
1.4
1.5
6r
4
-
2
-
—
/
'
oL
0.6
0.7
0.9
0.8
1
1.1
1.2
1.3
1.4
1.5
11
12
13
1.4
1.5
~~\
Y(B)
11
>-
10.5
10
0.6
08
07
09
1
t
Figure 3: Equilibrium in the free-information benchmark. The top, the middle,
and the bottom panels respectively plot the loan prices, the cascade size, and the aggregate
new
level of
(i)
loans as a function of the losses in the originating bank.
The unique equilibrium price
(ii)
given by r
is
For the continuation equilibrium
with distance k
< K{rf au —
.)
1
=
rj mT
.
All banks choose A\
(at date I):
=
are insolvent while the banks with distance k
W
.
Banks
> K(rj air
)
are solvent.
(Hi)
For the ex-ante equilibrium
the transition bank with distance
banks with distance k
(i.e.
(iv)
>
K (rj
K {rf
air
The insolvent banks choose
(at date 0):
)
ai r )
+
chooses
1) choose
For the loan market: The insolvent banks
market, while the solvent banks with distance k
buying legacy loans and making
new
loans.
>
A Q = H,
A = B.
J
The aggregate
all
J
other solvent banks
J
of their loans in the secondary
sell all
K (r/ a
,>)
+
are indifferent between
1
These banks spend a portion of their flexible
reserves to buy the legacy loans sold by the insolvent banks
the rest of their reserves.
and
A — HS
level of
new
and they make new loans with
made
loans
in this
economy
is
given
by:
y=
Discussion.
(
n -
K [rfair) -
1)
V
~
Figure 3 displays the equilibrium
K (r }air
in the
)
(1
-
11)
y) r fc
benchmark economy
for particular
parameterization of the model. The figure demonstrates that the loan prices are fixed at
rf ai r
,
the cascade size
is
increasing in
6,
and the aggregate
14
level of
new
loans
is
decreasing
more
Intuitively, as 9 increases, there are
in 9.
As the insolvency
the spread of the insolvency.
reserves as liquidity instead of
decreases "smoothly" with
we show
making new
These
9.
losses to
with small increases
more banks hoard
spreads,
results offer a
benchmark
K
There
for the next section.
y may
and
3-'
experience large changes
in 9.
Endogenous Complexity and Fire Sales
3
Let us repeat the steps of the previous section but
now with
only have local knowledge about the financial network
Each bank observes
its
two forward neighbors but
b
is
we assume
this uncertainty takes a simple form:
our key assumption: Banks
(p).
otherwise uncertain about
remaining banks are allocated to the remaining financial
slots (cf. Eq.
>
their distances
and assign a
In this context,
when
positive probability to
the shock
mark. But when the shock
is
large,
is
small, the
all
complex linkages
2
know
distances k £ {3.4,
—
..,n
system behaves exactly as
their
about
3 are uncertain
1}.
in the
bench-
banks need to understand distant (complex) linkages
order to assess the amount of counterparty risk they are facing.
figure out these
how the
(1)). In particular,
Banks with distance k <
distances from the distressed bank, and banks with distance k
in
their flexible
which lowers y. Note, however, that
loans,
that once auditing becomes costly, both
be contained, which increases
Their inability to
which overturns
triggers a set of precautionary actions
the relatively benign implications of the benchmark environment.
As
in the
benchmark, the original distressed bank
bp
^
withdraws
its
deposits from
the forward neighbor bank, which triggers further withdrawals until, in equilibrium,
cross deposits are withdrawn,
it
i.e.
A\
=
W
cannot obtain any net liquidity at date
condition
(4), it also
knows that
Next consider a bank
bank
to choose action
from
its
b>
it
for all j. Since the distressed
1, it tries
with distance k >
Aq £ {HS,H, B)
is
0,
the
and note that a
amount
it
it
the bank chooses
A
J
Formally,
at date
then this bank's paj'ment can be written as a function
However, the bank chooses
consequently about
choice of action.
x.
That
AQ
Note
is,
J
its
it
its
precautionary
will lose in cross-
forward neighbor pays x
deposits.
1,
— HS.
expects to receive in equilibrium
and
at date
Under
sufficient statistic for this
measure, this bank only needs to know whether (and how much)
if
chooses Aq
In other words, to decide on the level of
forward neighbor.
bank knows that
to obtain liquidity at date 0.
cannot avoid insolvency, thus
all
[q\ [-4q,.t]
.
<£>
[^cb
x ])-
while facing uncertainty about the financial network, and
3
also that qi \A ,x]
the bank's payment
15
is
and
q2
[Ag,x] are increasing in x for any
increasing in the
amount
it
receives
from
its
forward neighbor regardless of the ex-ante precautionary measure
the bank
from
its
infinitely risk averse,
is
will
it
choose
precautionary action as
its
forward neighbor the lowest possible payment
To characterize the bank's optimal action
We
librium.
further,
say that the equilibrium allocation
Thus, since
takes.
it
if it will
receive
x.
we
define a useful notion of equi-
distance based and monotonic
is
the
if
banks' equilibrium payments can be written as an increasing function of their distance
That
from the distressed bank.
Qi,Q 2
{0, ...n
-1}-*R
p {i
b
and
(p)
We
k.
payment functions
there exists weakly increasing
such that
- k)
{i
- k)
(p)^2
(q1
for all
is,
=
(p))
(Qi[k]y
conjecture (and verify in Appendix A.l) that the equilibrium
is
distance based and monotonic.
Under
this conjecture, consider again
V
a bank
with distance k
the payment of the bank's forward neighbor can be written as x
the bank's uncertainty about the forward neighbor's payment x
its
than
own
its
distance
k E {1-2}, then
=
receive x
Q\
uncertain about
it
[k
—
its
\k
—
1
is
k.
the bank knows
If
chooses
its
from
1]
distance,
distances k E {3, ...,n
Qi
—
uncertainty about the forward neighbor's distance k
—
minimal
chooses Aq E {HS, H,
1}.
i.e.
if
k
>
3,
then
Moreover, since Qi
for the distance k
B}
as
A
J
if it
=
x
Q\
Q\
[k
is
and note that
—
[k
—
1]
Then,
1],
reduces to
equal to one less
On
B} knowing
the other hand,
that
the
if
i.e.
if
will
it
bank
is
it
assigns a positive probability to all
is
an increasing function, the payment
[.]
Hence a bank
3.
will receive
=
which
£ {HS, H,
forward neighbor.
its
0,
distance to the distressed bank,
its
optimal action
1,
—
>
=
Qi
[2]
from
with distance k
b>
its
>
forward neighbor.
3
In
words, the banks that are uncertain about their distances to the distressed bank choose
their precautionary action as if they are closer to the distressed bank than they actually
are.
Formally, our next
lemma
establishes that
action that the bank with distance k
available (characterized in
K (r)
=
so that the
3,
Lemma
1).
=
If
= H to avert insolvency, then all banks
A\ = H even though ex-post they end up
is
even greater
(i.e.
if
K (r)
>
banks with distance k >
would choose
if
-
A
—
with k
3
>
3 fold
the information was freely
>
is
given by
3 also take the precautionary action
not needing liquidity.
4) so that the
by taking the most precautionary action
16
choose the
would take the precautionary action
— HS,
A Q — HS,
J
If
the free-information
bank with distance k —
be insolvent and choose the most precautionary action Aq
k
3
the free-information cascade size
bank with distance
Aq
cascade size
3
all
then
all
3
would
banks with
which ensures that their
insolvency (even though some of these banks would be solvent in the free-information
economy). The proof of the following lemma
is
Lemma
Consider the setup of
free, so that
banks
know
Lemma
the equilibrium,
b
Let
(p).
K (r)
of
(characterized in
is
than
costly rather
denote the cascade size that would obtain
Lemma
Lemma
that
it
Each bank
IP
with distance
would choose in the free-information economy
Each bank V with distance k £
1).
if
1).
distance based and monotonic.
is
same action
2} chooses the
1,
now information
but
1,
information was freely available (characterized in
k £ {0,
most
only their two forward neighbors and they are otherwise uncertain
about the financial network
Then
since
provided by the above discussion.
the intuition
2.
Appendix A.l
relegated to
is
{3,
...
n
—
1} chooses the
same
action that the bank with distance 3 would choose in the free-information economy.
In
particular, there are three cases to consider, depending on the cascade size:
If
K (r)
<
then the crisis in the free-information economy would not cascade to
2.
bank with distance
k £ {3.
...
n —
3,
1} chooses
to the free-information
given by
If
A =
J
still
=
same as
3.
which would choose
K (r)
>
£
{3,
..,77
insolvent,
i.e.
—
1} chooses
In particular, the cascade size
is
A — HS.
J
the cascade size
free-information cascade size
in the loan
=
3) or
A =H
J
A =
J
Thus each
H. The equilibrium cascade
K (r)
i.e.
to avert insolvency.
=
K (r).
size is
but the banks' actions
then bank with distance 3 would be insolvent in the free-information
4,
is
K (r).
market, while
Thus,
given by
Note that the loan trade decisions
K (r)
1.
different.
economy and would choose
(if
Lemma
in
the free-information economy,
and payments are
buyers
each bank b3 with distance
Thus,
then the crisis in the free-information economy would cascade to and
3.
b3 with distance k
If
B.
K (r) = K (r).
K (r)
the
J
B, and the equilibrium actions and payments are identical
economy described
stop at bank with distance
bank
A =
which would choose
if
all
K (r) =
of the
they are potential
>
and they are
all
n.
if
K (r)
<
2,
>
3
depend on the
these banks are potential
they are either neutral in the loan market
3,
sellers (if
3
banks with distance k
In particular,
K (r)
A = HS
banks choose
K (r)
loan trades on the free-information cascade size A"
price) plays a key role in subsequent analysis,
>
(7')
4).
This dependence of the banks'
(which
itself
depends on the loan
where we endogenize the loan prices and
complete the characterization of the equilibrium.
We
next solve the equilibrium level of loan prices in the costly information setting and
present our
K (r)
main
results.
There are three cases to consider depending on the cascade
over the price range r £
\rj ire
,
rjfair]
a
17
size
Case
K (r/
If
(i).
we
rium, that
jre
<
)
then we conjecture that there
2,
and loans trade
K (r)
is
J
K (r)
a decreasing function so that
r$ aiT
<
Lemma
r,
To
.
K (rf
ire
verify this conjecture, recall that
<
)
€ [r^ re
2 for all r
the equilibrium price
Case
If
(ii).
which we
r
Lemma
A = HS
J
K (rf mr
)
(10), the
unique equilibrium price
Lemma
in
>
3,
2,
3 choose
(if
=
r flTe
To
.
we conjecture
that there
see this, note that 3
2 implies that
K (r)
>
is
all
<
is
which there
K (r/
azr
)
<
is
there
is
Case
Once
.
a breakdown in the loan
K (r)
banks with k > 3 choose either
G
for all r
H
A —
3
(if
[rj lTe
,
rj air ],
K (r) = 3),
or
words, these banks are either neutral in the legacy loan market
4). In
note that the banks with distance k
sellers in the legacy
least 3 sellers
rf aiT
a unique symmetric equilibrium,
or they are sellers, in particular, they are not potential buyers. Using
and become
=
given by r
proving our conjecture for this case.
refer to as the fire-sale equilibrium, in
market and
<
Lemma 2
once more,
AJ = HS
2 are necessarily insolvent, so they choose
loan market. Thus, the legacy loan market features at
but no buyers. The market clearing condition
(2) implies
that r
=
r
ire
,
i.e.
a breakdown in the secondary loan market.
(iii).
equilibria:
If
K{rj air
<
)
Then, the analysis
fair-price equilibrium.
3
< K{rj ire ), we
loans,
given by r
(i)
=
ire
)
>
3.
To
see this, first sup-
all
K (r)
price of loans
is
given by r
for case
(ii)
—
rj ire
,
2.
—
is
a
rj ire so that
applies unchanged,
3 are either neutral in the loan
prices collapse 'to r
<
banks with distance
rj mT clears the market, verifying that there
Then, the analysis
banks with distance k >
and loan
fire-sale equilibrium.
applies unchanged, in particular,
and the price
K (rf
conjecture that there are two stable
rj aiT so that the cascade size satisfies
Next suppose that the
the cascade size satisfies
in particular, all
is
for case
3 are potential buyers
sell their
<
2
one fair-price equilibrium and one
pose that the price of loans
>
>
banks with k
2 implies that all
Hence,
determined, the remaining equilibrium allocations are uniquely
is
determined as described
k
rf a i r ].
,
B, and as such, these banks are potential buyers of loans. Thus, under the deep sec-
ondary market assumption
thus
=
at their fair price r
regardless of the endogenous price
A =
which loan markets endogenously
refer to as the fair-price equilibrium, in
clear
a unique symmetric equilib-
is
market or they
verifying that there
is
a fire-sale
equilibrium.
We
summarize these
Proposition
2.
results in the following,
and main, proposition.
Consider the setup of Proposition
1
with the difference here being that
banks only have local understanding of the network, so that banks know only their two
forward neighbors and they are otherwise uncertain about the financial network h(p).
18
K (r)
Let
[r fi re ,
denote the cascade size in the free-information economy with price level r £
r a i r ] (characterized in
Lemma
loan market clearing condition (2)
=
price r
r fair
same
size is the
make new
using their reserves to
and
r
level of
new
loans
Fire sale equilibrium.: If
=
r f ire
K (rfi
=
re )
K {rfire)
(u.2) If
K{rf aiT <
)
A = HS and sell their loans,
> A" + l choose Aq = B and are indifferent between
J
1) choose
equal to the benchmark Eq. (11).
K (rf
air
)
>
then there
3,
i.e.
>
4,
banks with distance k
K = K (rf
size is
all.
is
a unique equilibrium, in which
there
In either sub-case,
>
an excess supply of loans
is
3 choose
A =H
3
These banks
.
iTe ).
banks with distance k
banks go under and the cascade size in
this case is
K=
>
A = HS.
}
3 choose
These
n.
new
reserves are hoarded as liquidity and there are no
all flexible
0.
Multiple equilibria: If
(ivi)
K—
i.e.
loans or to buy legacy loans in the secondary market.
is
3, all
remain solvent and the cascade
y=
at their fair
.
(u.l) If
loans,
K
<
—
a breakdown in the secondary loan market,
is
and loans trade
as the free-information benchmark,
while the solvent banks with distance k
(ii)
a unique equilibrium in which the
satisfied with equality
is
The insolvent banks (with distance k
The aggregate
2, there is
.
The cascade
there
<
Fair-price equilibrium: If K{rj., re )
(i)
2.
1).
K (r/ a
jr
)
<
<
2
3
< K{rj ire ),
there are two stable equilib-
ria.
In the fair-price equilibrium, loans trade at their fair price r
size is
K = K (rf a
ir)
<
2.
the solvent banks are indifferent between
purchasing loans in the secondary market, and the aggregate
level of
—
rj mr
the cascade
,
making new loans or
new
loans
y
is
given
by the benchmark Eq. (11).
In the fire sale equilibrium, r
K (rf
dump
ire
)
—
3
)
or
K
=
n
K (rf
(if
ire
)
=
>
rf ire
4), all
their loans in the secondary market,
Discussion.
the cascade size
,
is
given by either
banks hoard liquidity and
and there are no new
loans,
(if
y
K=3
K (r/,re
)
(if
>
4)
—0.
Figure 4 displays the equilibria with network uncertainty for a particular
parameterization of the model. The top panel reproduces the cascade size
free-information
benchmark
loan prices are fixed at r
=
as a function of the losses in the originating
rf ire
and
7'
=
r/
,>.
The remaining
K (r) in
bank
6,
the
when
three panels display
the equilibria with network uncertainty, illustrating the characterization in Proposition
Note that there
is
a unique equilibrium for small and large levels of
multiple equilibria for intermediate levels of
0.
19
6,
2.
however, there are
5
^
/
0.7
0.6
08
0.9
1.1
1
1
1.2
13
4
1.5
14
1.5
1
r(fl)
Fair Price Equilibrium
i-
5
Fire Sale Eq.uilibriuin
|
0.6
0.7
08
09
1.1
1
.JL-:
1,2
1.3
40
^
__--.
«-
*2)
20
06
0.7
8
0.9
1
15
1.1
1.2
11
1.2
T
1.3
,
,
1.4
1.5
14
15
Y(B)
5
j
.1.
06
07
08
09
1
1.3
8
Figure
4:
Equilibria with network uncertainty. The top panel plots the cascade
size
K (r) in the free-information benchmark as a function of the losses in the originating bank
6,
when
loan prices are fixed at r £ {0,r/ ajr }.
size
The remaining
three panels display the
network uncertainty: They respectively plot the loan prices, the cascade
and the aggregate level of new loans as a function of 0, for both the fair-price and
equilibria with
the fire-sale equilibria.
L'O
When
K(rj ire
8 is sufficiently small so that
benchmark features a short cascade even
fair price
)
<
2,
when
i.e.
=
for the price level r
the free-information
rf lTe
there
,
is
a unique
equilibrium. This equilibrium features a low level of complexity thus the banks
that are uncertain about their distance to the distressed bank can rule out an indirect hit.
Consequently, these banks use their reserves to
The aggregate
new
level of
loans
is
make new
loans and to
demand
assets.
the same as in the free-information benchmark, and
assets trade at their fair prices.
In contrast,
when
8
is
sufficiently large so that
size in the free-information
is
and discontinuous drop to
zero.
That
is,
when
there
new
when
3, i.e.
a unique fire-sale
is
loans
the cascade
makes a very
large
the losses (measuring the severity of the
shock) are beyond a threshold, the cascade size becomes so large that banks are
initial
unable to
whether they are connected to the distressed bank or not. All uncertain
tell
banks act as
much more
if
they are closer to the distressed bank than they actually are, hoarding
liquidity
than
benchmark, become
in the free-information
prices.
in the free-information
sellers
to a severe
and
this leads to a collapse in asset
This result provides a rationale for the collapse of asset prices in an environment
which complexity suddenly (and endogenously)
re
benchmark and leading
Moreover, these banks, some of which would be potential buyers
credit crunch episode.
K (rfi
>
)
sufficiently large,
In this equilibrium, the aggregate level of
equilibrium.
in
benchmark
K{rf mr
>
)
rises.
that the uncertain banks panic so
4)
much
Furthermore,
it
is
possible
(if
that they take an extremely
precautionary action to increase their liquidation outcome. However, ex-post, this action
ensures that the uncertain banks become insolvent.
When
is
high
8
(i.e.
sale price
is
in
an intermediate range, the cascade
K (rj <
K (r/,>
air
(i.e.
endogenous
)
e
level of
)
however
2),
>
3).
it
size
is
manageable
becomes unmanageable
price of loans
loans trade at the
In this case, the interaction between asset prices
fire-
and the
complexity generates multiple equilibria.
In the fair-price equilibrium, loans trade at a higher price
relatively small,
if
if
which reduces the
level of complexity.
and the cascade
With the lower
size
is
level of complexity,
the banks that are uncertain about their distance to the distressed bank become potential
buyers of loans, which ensures that loans trade at the higher price and that the cascade
is
shorter.
Set against this benign scenario
is
the price of loans collapses and there
complexity.
As the
distances panic and
Note
also that,
level of
sell
the possibility of a fire-sale equilibrium, in which
is
a longer cascade, which increases the level of
complexity increases, banks that are uncertain about their
their loans,
which reinforces the collapse of loan
whenever there are multiple
21
prices.
equilibria, the fair-price equilibrium
Pareto
dominates the
equilibrium for
fire-sale
sale equilibrium entails
In the next section,
insolvent.
account
two distinct
we
all
banks and their depositors.
social costs: fewer
new
loans
Intuitively,
the
fire-
and more banks become
identify the negative externalities in our setup that
for these social costs.
Three Externalities
4
In this section,
we
discuss the various externalities present in our setup
the role they play in our main results.
which we
call
features three distinct externalities,
the network-liquidity externality, the fire-sale externality and the complex-
The
ity externality.
The second
Our model
and we highlight
first
of these emerges directly from the interlinkages
externality stems from the interaction of loan prices
between banks.
and the endogenous
determination of bank runs. These two externalities are relatively standard. Models of
interconnected banks typically feature externalities akin to our network-liquidity externality (see, for
example, Rotemberg (2009)), while models based on the Diamond and Dybvig
(1983) setup typically lead to fire-sale externalities whenever prices affect the likelihood of
bank runs
(see, for
example, Allen and Gale (2005)). The complexity externality
and
to our analysis,
it
is
novel
emerges from the interaction of loan prices and the complexity of
the financial network.
4.1
Network-liquidity Externality
In our setup, a distressed
either
withdraw
serves (or
a
it
can
its
bank can obtain
sell its
it
can hoard
its
liquidity externality
on
it.
bank scrambles
(cf.
A
is
scarce,
forward neighbor bank also in distress,
In particular,
unable to meet this liquidity demand.
withdraw deposits
its
if
the bank decides to withdraw
for liquidity
distressed
bank
in
and
faces insolvency
it
can.
'
its
own
Our
The bank's
liquid reserves,
its
if it
our setup always (weakly)
assumption (LBO)). 9 In Appendix A. 2, we study the
polar opposite case, enforced by the government, in which a bank that needs liquidity
hoards
can
It
liquid re-
legacy loans, as a last measure). Since aggregate liquidity
cross deposits, the forward neighbor
prefers to
through two distinct sources:
deposits on the forward neighbor bank, or
bank that chooses the former option puts
imposing a
is
liquidity
and thus avoids
inflicting
first
a liquidity externality whenever
analysis shows that internalizing the liquidity externalities in this fashion
preference for liquidity withdrawal
is
strict
if
the forward neighbor pays q%
<
R.
In
the more general model analyzed in Caballero and Simsek (2009). the bank always strictly prefers to
withdraw deposits and thus always inflicts a liquidity externality.
22
leads to shorter cascades in aggregate (see Proposition 3 in the appendix).
Intuitively, the
scramble for liquidity
a hot potato which the banks can either
like
is
pass straight to their neighbors, or which they can cool
before passing
it
on.
When
reaches a vulnerable bank,
all
banks pass
it
without cooling
a bank which
i.e.
is
This bank cannot pass the potato to
bank.
down
it,
a bit using their resources
the hot potato eventually
sufficiently close to the original distressed
neighbor, which
its
already bankrupt.
is
Moreover, the resources of this bank alone are not sufficient to cool
hence the bank burns
it
(i.e.
down
the potato
cannot meet the liquidity demand and becomes insolvent),
which lengthens the cascade. In contrast, when each bank cools down the potato before
passing
much
it
on, then the potato
cold before
is
it
reaches the vulnerable bank, leading to a
shorter cascade.
Fire Sale Externality
4.2
Consider a bank that decides to
sell
This action has a small positive
loans, while
loans.
it
effect
has a small negative
Absent further
"net out," which
is
effects,
some
on the net budgets of the banks that buy legacy
effect
on the net budgets of the banks that
sell
legacy
the welfare impacts of these budget changes would typically
the content of the
increases the likelihood of a
loans, leading to a small decline in loan prices.
first
bank run
in
welfare theorem. However, a small decline also
our setup, leading to a further decline in the
welfare of these banks and their depositors. Consequently, the effects of price changes do
not necessarily "net out," and loan prices feature a negative pecuniary externality.
To formalize
this point, let us consider the
This setting shuts
down
benchmark economy analyzed
the complexity externality (since the banks
in Section 2.2.
know the
financial
network) which we will analyze in the next subsection.
K (r)
Recall that the cascade size
liquidation
payment
number
after a smaller
f £
[rfire, ffair]
of each
bank
is
for
any
e
>
decreasing in
greater
of insolvencies
such that,
is
Eq.
(cf.
Eq.
(cf.
(7)).
(5)),
is,
thus the
crisis is
contained
In particular, there exists
some
we have
K(f-e/2) = K(f + e/2) +
That
with a lower loan price, the
r:
l.
the cascade size increases by one in response to an arbitrarily small decrease
in loan prices.
one more bank,
The small
inflicting
backward neighbor, while
price
drop from f
+
a discrete negative
it
has a continuous
23
e/2 to f
effect
effect
—
e/2 leads to the insolvency of
on the welfare of
this
bank and
its
on the welfare of other banks. Hence,
for sufficiently small e, the net welfare effect of the price
is
negative, demonstrating the
The
fire sale externality.
general intuition behind this externality
change from r
+ e/2
to f
—
e/2
10
that a drop in asset prices lowers the
is
liquidation value of an insolvent bank, which increases the probability of a
bank run
in
the counterparties of this bank.
Complexity Externality
4.3
Consider next the model analyzed
in Section 3 in
which banks only have
about the financial network. Similar to the above analysis
bank that
sells
local information
for the fire sale externality, a
a loan has a small negative impact on loan prices, which in turn increases
the length of the cascade. However, in this case, longer cascades also increase the comSince banks are averse to complexity (which
plexity of the financial network.
we model
as infinite risk aversion with respect to the financial network), an increase in complexity
leads to a welfare reduction for banks that are uncertain about the financial network.
To formalize
this point, consider the set of
K (rj
in Proposition 2,
that, for
any
e
>
air
<
)
2
<
3
<
is,
ire ).
Then, there exists f £
[r fire,
1"
fair]
such
we have
K (f - e/2)
That
K (rf
parameters that lead to multiple equilibria
=
3
>
K (f + e/2) =
2.
small decline in loan prices makes the cascade size in the free-information
benchmark exceed the
critical
threshold of
Recall that the banks with distance
2.
k G {3, ..,n— 1} mimic the bank with distance 3 in the free-information benchmark.
Then, when the loans trade at price f
+
e/2, these
banks know that they
will
not lose
anything in cross deposits, thus they do not take any precaution and they pay out at least
(<^
= 1,^ =
R)
to their depositors (and perhaps more,
if
they can acquire legacy loans
at a discount).
In contrast,
{3, ..,n
—
when
the loans trade at price f
1} are worried that they might suffer
—
e/2, the
an indirect
these banks are infinitely risk averse, their welfare
is
banks with distance k G
hit
from the cascade. Since
greatly reduced.
Moreover, these
banks hoard liquidity to precaution against the worst case scenario, and they end up
paying
[q\
=
1,
ql,
< R)
to their depositors. Hence, a small drop in prices inflicts a discrete
10
The discrete nature of the fire sale externality is due to our modeling assumptions, in particular, our
assumption that the good equilibrium is selected whenever there are multiple equilibria for bank runs.
However, the externality would also be present in other variants of the model, as long as the probability
of a bank run is increasing in the date 1 losses (i.e. the severity of the liquidity shock) of the bank.
24
negative effect on
all
of the banks with distance k
£
—
{3, ...n
and
1}
their depositors,
demonstrating the complexity externality.
The complexity
setup, as
externality
we have already seen
complexity due to a reduction
banks and
may
also lead to multiple Pareto-ranked equilibria in our
in Proposition 2. In particular,
in the price of assets not
their depositors, but also induces these
in panic
mode reduces
asset prices further,
many
banks to take extreme precautionary
sales.
The
sale of assets
by
which leads to a vicious cycle culminating
an increase
in asset prices reduces the
complexity
which may mitigate the precautionary measures and turn more
sellers into
in the fire-sale equilibrium. In contrast,
externality,
in the level of
only lowers the welfare of
measures, which, in some instances, includes further asset
banks
an increase
buyers, leading to a virtuous spiral towards the fair price equilibrium.
Finally, note that the complexity externality
the
fire sale
erature.
is
potentially
externality analyzed in the previous subsection
The reason
is
and highlighted
in
the
lit-
that the fire-sale externality affects banks that are on the verge
of insolvency. In contrast, the complexity externality affects
about the financial network, which,
The
much more potent than
all
banks that are uncertain
in practice, includes virtually all financial institutions.
greater scope of the complexity externality also leads to widespread (precautionary)
actions,
which has the potential to create aggregate
effects, price
changes, and multiple
equilibria.
Conclusion
5
In this paper
we provide
a model that illustrates
how
fire sales
and complexity can
ger a very powerful negative feedback loop within a financial network.
sales lengthen the potential cascades,
ponentially.
This triggers confusion
exacerbate the
fire sale.
and
raise the
among
trig-
More severe
fire
complexity of the environment ex-
potential asset buyers, which pull back
and
In extreme scenarios these potential buyers can turn into sellers,
leading to a complete collapse in secondary markets.
In our model, the distressed institutions' motive for
outcome
in the
worst case scenario of insolvency. This
only to simplify the exposition. In
reality,
is
fire
sales
is
to improve their
an extreme assumption
the distressed institutions
may
multitude of reasons that do not imply anticipating insolvency. E.g. they
to
do so by regulatory requirements, or they may hope
liquidity.
main
to avoid a
sell
made
assets for a
may be
forced
run by obtaining more
These reasons can be incorporated into our framework while preserving our
conclusions.
25
Having said
what would happen
of
we
the particular insolvency motive
this,
consider also raises the question
the distressed institutions chose to gamble for resurrection by
if
not selling their assets, which would improve their outcome in good states at the cost of
a greater bankruptcy
mixed blessing
Our model suggests that gambling
risk.
Gambling by potential buyers, that
for the aggregate.
know
are far from the cascade but that do not
downward
On
spiral of prices.
would increase the cascade
We
ates
size
and supporting
institutions that
and the
limit the fire sales
it
from counterparty
apparent that our environment cre-
Supporting secondary markets, insulating
and
risk,
is
stress testing (increasing
transparency
needed) systemically important financial institutions, are
if
be a
and trigger the complexity mechanism.
policy opportunities during crises.
financial institutions
would
this,
is,
may
the other hand, gambling by institutions near the cascade
did not explore policy questions, but
many
for resurrection
policies
all
practiced during the current financial crisis and supported by our framework.
A
Appendix
Proofs Omitted in the
A.l
Proof of
Case
(i):
Lemma
K (r)
an equilibrium,
2
<
first
To prove that the conjectured actions and payments constitute
2.
it
level
Q\
[0]
amount
receives the full
in the conjectured equilibrium).
z
that
it
from
would pay
its
forward neighbor, which
it
is
it
equal to what
it
its
date
action as
if it
it
will receive
would receive
optimally chooses the same action
chooses
+l
bp ^
\ which
is
solvent
Next consider a bank with distance k £ {1,2}. Under
>
Consider next a bank with distance k
thus
economy
in the free-information
forward neighbor,
the conjectured equilibrium, this bank knows that
Hence,
= HS
note that the original distressed bank optimally chooses Aq
and pays out the same
(since
Main Text
it
3.
—
Q\
[k
—
1]
in the free-information
would choose when information
This bank
will receive
Under the conjectured equilibrium, the cascade
x
x
size is
=
is
Qi
K (r)
uncertain about
[2]
<
from
2,
its
thus
from
economy.
is free.
its
distance,
forward neighbor.
we have
Q
x [2]
Since the bank makes no losses in cross deposits (even in the worst case scenario of k
it
optimally chooses
A =
J
its
=
—
1.
3),
B, verifying the optimality of the conjectured action.
Since the banks' actions are the
same
as the free-information
are also identical to that case, which verifies (by
26
Lemma
1)
economy, their payments
that the equilibrium
is
distance
based and monotonia
Case
K (r)
(ii):
=
the banks with distance k £ {1. 2} optimally choose the same action and
payments
it
chooses
K (r) =
Since
its
date
action as
>
with distance k
chooses
it
This bank
3.
will receive
if it
x
=
uncertain about
is
Q\
[2]
from
under the conjectured equilibrium, we have Q\
3,
benchmark)
level in the free-information
thus
make the same
as in the free-information economy.
Consider next a bank with distance k
thus
bank and
Similar to the previous case, the original distressed
3.
>
some
3 expects (with
H
A —
J
lies in
the interval (1
probability) to
make
—
its
forward neighbor.
(which
[2]
y/z.
distance,
its
1).
is
equal to
its
Hence, the bank
losses less
than
its buffer,
to counter these losses, verifying the optimality of its conjectured
action.
These banks with distance k >
free-information economy.
amount (Qi
with distance k
>
[k]
>
1
it
would pay
pay potentially
bank with distance k
=
than the
less
3 pays the
in the free-information case, while the
same
banks
4 end up paying
Q
where Q%
In particular, the
that
[3] ,(5-2 [3])
3 are solvent, but they
x
[k]
follows
=
1
and
from Eq.
Q-2
[k]
(3).
= y + (1 - y) R
It
G
(1,
R)
,
follows also that the equilibrium
is
distance
based and monotonia
Case
(iii):
A' (r)
>
4.
In this case, note that the banks' payments, given the conjectured
actions, are the solutions to the following
Q,
[0]
ft
[k]
=
^(l-0)r-^^[n-l]
=
V+SL^jQlVtzl,
(12)
+z
1
Note
that,
system of equations:
by condition
(3),
tor k
€
{1, ..,*
also verifies that the equilibrium allocation
{Q\
is
.
[k]
<
The system
1
in Eq. (12)
verify the optimality of banks' actions, first note that Eq. (12) implies the
i^]}fc=o
are l°wer than
what they would be
would be solvent and would pay
1
for all k,
distance based and monotonia
in the free-information
the forward neighbor of the original distressed bank pays Q\ [n
it
1}
the solution to this system satisfies Qi
verifying that the banks are indeed insolvent given their actions.
To
-
—
1]
<
1
payments
economy, because
in this case (while
unit per deposit in the free-information economy).
Consider a bank with distance k G {1-2}, and recall that this bank chooses
27
its
date
action knowing that
bank would be insolvent
4, this
Q\
—
[k
will receive
it
1] is
[k
—
it
would be
chooses
it
neighbor.
date
its
K (r)
Since
from
its
>
>
action as
if it
economy.
we
optimally chooses
This bank
is
uncertain about
necessarily have
3 believes that
go bankrupt. Consequently,
it
Qi
[2]
=
x
Q\
r
<
1
—
[2]
A = HS.
J
its
distance
its
forward
from
are even lower than
[k]} k:i
might experience
it
economy. Thus the
it
will receive
>
In the present case,
and
and since the payments {Qi
4,
the free-information economy,
bank with distance k >
3.
K (r)
forward neighbor. Since
in the free-information
{1, 2} is necessarily insolvent,
Consider next a bank with distance k
and thus
1]
in the free-information
even lower than what
bank with distance k 6
Q\
y/z. In other words, the
losses
beyond
buffer
its
chooses the most extreme precautionary action
and
A = HS
J
to improve their liquidation outcome, verifying the optimality of the conjectured action.
Equilibrium without Liquidity Externalities
A. 2
In the main text, the bank always considers deposit withdrawal as the
liquidity
(cf.
first
assumption (LBO)), thus imposing a liquidity externality on
source of
forward
its
neighbor bank. To clarify the role of liquidity externalities, in this appendix we analyze
the equilibrium under a government policy that prevents banks from withdrawing cross
deposits unless they have used
To
facilitate the analysis,
it
all
of their liquid reserves.
helps to consider a slightly greater action space for banks:
(Al A\) e {H
where
A = H (y)
and A{
3
=
W
(z)
(y)
,
HS, B} x
denotes that the bank hoards y e
,
(0, y]
denotes that the bank withdraws z 6
the forward neighbor bank. In other words,
flexible reserves
{W (z) K}
and to
main assumption of
partially
this
withdraw
(13)
,
units of
(0, z]
we allow the banks
their cross deposits.
its flexible
units of
reserves,
deposits on
its
to partially hoard their
Next, we introduce the
appendix: the government enforces the reverse of the liquidity
pecking order in assumption (LPO) so that banks do not impose a liquidity externality
on
their forward neighbors,
whenever they can avoid doing
Assumption (LPO-R).
forward neighbor bank
bank
is
Consider a bank that needs liquidity at date
solvent.
such that the bank
first
The government imposed
considers hoarding
J
28
1
and whose
liquidity pecking order for this
its flexible reserves, i.e. it
A = H (y). The bank withdraws some of its cross deposits
if Aq — H (y) is not sufficient to meet its liquidity demand.
action
only
is
so.
considers the
(chooses A\
=
W
(z))
Finally,
we strengthen
the right hand side of condition (3) to
l
own
so that a bank's
+ z<(l-y)R,
legacy loans are enough to promise
backward neighbor bank)
at least 1 unit at date
Under these assumptions and when n
an equilibrium
in
(14)
which
is
2.
we conjecture
sufficiently large,
>
banks with distance k
all
depositors (including the
all
1
all
(i.e.
bank
In the conjectured equilibrium, the original
and
W
withdraws
also
for the liquidity
its flexible
and
=
(Af
b
2,
p(-^:
in
'
need of
pecking order, bank
reserves.
<
z
If
bp ^
H {z),Af
+1)
if z — y <
y,
W
=
z
(z
—y
(with distance n
'
- y)\
Otherwise
= H(z),A
>
hoarding
=A
p
its flexible
j
reserves,
y and this bank chooses
bank
b
p{T+2 ^
is
with distance
similar to
payment purely by hoarding
withdraws an even smaller amount from
it
in
—
= HS and A P
1) first resorts to
In this case, consider
liquidity
1,
2.
units of liquidity. This bank's response
its
most
Under the new assumption
—
z
at
of its liquid reserves
chooses Aq
z units of liquidity.
+1
all
is
its
bank
its flexible
forward neighbor.
follows that a pattern emerges for the banks' cross withdrawal decisions. In partic-
n >
ular, let
1
denote the unique integer such that
yn >
and suppose n >
chooses
n.
Then,
for
,
—
its
need any
H (z
—
(n
z
each j £ {1,
(Af+J) = H (y) Af+j) =
chooses [Aq
keeps
it
hoards
payments purely by hoarding
then the bank meets
and otherwise
i.e.
®
and
1
then this bank chooses [Aq
y,
liquidity
its
which needs to find
reserves,
It
P ' I+1
\
does not withdraw any cross deposits.
+1]
n—
6
the bank meets
is,
it
deposits on bank b
which puts bank
(z),
that
its
pi-' +1
bp
is
banks except potentially
the original distressed bank) are solvent. In other words, the cascade size
stark contrast with the equilibria characterized in Propositions
that there
—
W
\)y)
{z
-
> y(n-
..,
n
—
1)
1},
the bank b p(,+j) with distance n
jy)) while the bank
,
,A P
= A
j
.
Since the
cross deposits, the remaining banks with distance k
liquidity,
and these banks withdraw
their deposits
,(T+ri)
with distance
n-n
bank with distance n — n
6
if
^
—j
{1, ...,n
and only
—n—
if
1} do not
their forward
neighbors are insolvent.
Going back
sections, this
to the original distressed
bank obtains a
units from cross deposits
total of y
(if it
+
z
bank
b p ^\
note that, unlike in the previous
units of liquidity: y units from its buffer
remains solvent). Hence, this bank
29
is
insolvent
if
and
z
and only
if
9>y + z,
which
bp
^
a stronger condition than
is
is
W
this condition
and consider the backward neighbor bank
insolvent
=
Suppose
(4).
(15)
6
satisfied so that
is
p (' _1
bank
This bank chooses
).
and receives some gf' 2 < z from the original distressed bank. Despite
~1
is solvent and it can promise
incurring some losses, in view of condition (14), bank b p<Kl
Ai
(~)
^
its late
depositors (including the backward neighbor bank) at least
=
?
it
pays
(<2i
liquidity,
it
1
1, q^'
chooses to keep
with distances k G
{2, ..,n
>
'
—n—
1
Since the backward neighbor bank 6
•
deposits in
its
{1, ..,n
)
—n—
bank
Repeating
2} are solvent, and thus
= B,A^
1} choose (Aq
^~ l
bp
\
1
K),
==
all
unit at date 2,
1
p ('~ 2
i.e.
does not need
)
this reasoning, all
banks
banks with distances k G
in particular,
keeping their deposits
in their forward neighbor banks.
Note
also that there
distressed
6 P '^)
bank
is
at
most one
the conjectured equilibrium (the original
seller in
while the banks with distance k G {2, ...n
—h—
buyers of loans. Hence, under the deep secondary market assumption
1} are potential
the analogue
(i.e.
of condition (10) for this setup),
in
the unique equilibrium price
is
2)y>l-y,
=
given by r
regardless of whether the banks
have local knowledge
—n
know
r_/
air
(16)
Note
.
also that this analysis applies
the financial network
b
(p) or
whether they only
they know only their two forward neighbors).
(i.e.
We
summarize
this result in the following proposition.
Proposition
Suppose the banks' action space
3.
for the liquidity pecking order
the government)
know
,
is
b
(p)
(LPO-R) (which
In particular, the original distressed bank
the banks with distance k
is
—
rj aiT
and
—
1,
..,
is
the cascade size
insolvent while
G {1,0, n
is
(LPO)
imposed by
Then, regardless of whether the banks
or just their two forward neighbors, there
equilibrium in which loan prices are given by r
1,
extended to (13), assumption
replaced by assumption
conditions (15) and (16) hold.
the financial network
At date
is
n—n—
all
a symmetric
is
equal to
other banks are solvent.
1} withdraw all or
their cross deposits, while the banks with distance k G {2,3, ..,n
1.
— n}
some of
keep their deposits
in the forward neighbor bank.
Proposition 3 establishes our main result in this appendix:
a liquidity externality, then the cascade size
in Propositions
1
and
2.
To
is
much
If
the banks avoid inflicting
shorter relative to the cases analyzed
see the intuition, consider the equilibrium in Proposition 3
30
and consider what the banks would do
if
we removed the government imposed
pecking order in assumption (LPO-R). Consider bank
demand from
its
backward neighbor purely by hoarding
bank, hoarding reserves delivers
2),
(as
1
some
•
Given the characterization
of the losses in the original
bank
not restricted by government policy, bank
cross deposits to hoarding liquidity, which
absent government policy, bank
1+n
\ which meets
its
its flexible reserves.
R
unit of liquidity at an opportunity cost of
while withdrawing cross deposits would deliver
cost of ^2
bp ^
+n+l
bp ^
in 3,
1
it
liquidity
For
(in
this
period
unit of liquidity at an opportunity
< R
can be checked that q^
will spillover to
p ('+")
£>
liquidity
would
would put bank
bank
+n+l
bp ^
strictly prefer to
'
)
).
Hence,
withdraw
if
its
£/('+ n+1 ) in distress. Similarly,
'
>
would prefer to withdraw
its
cross deposits
and
the scramble for liquidity would continue to cascade in similar fashion. Eventually, a
~
vulnerable bank, b p(
\ which is sufficiently close to the original distressed bank (and
l
1
-
thus has incurred some losses) would become distressed. This bank might be unable to
find the required liquidity
and might become
insolvent. Hence, the liquidity externalities
have the potential to make vulnerable banks go insolvent.
liquidity order in
The government imposed
assumption (LPO-R) internalizes these liquidity externalities, which in
turn ensures that the cascade size
is
shorter.
31
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