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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Financial System Risk
and
Flight to Quality
Ricardo J. Caballero
Arvind Krishnamurthy
Working Paper 05-31
November 21, 2005
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MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
DEC
1
2 2005
LIBRARIES
Financial System Risk and Flight to Quality
Ricardo
J.
Arvind Krishnamurthy*
Caballero
This draft: November 21, 2005
Abstract
We
present a model of flight to quality episodes that emphasizes financial system risk and the Knigh-
tian uncertainty surrounding these episodes.
In the model, agents are uncertain about the probability
distribution of shocks in markets different from theirs, treating such uncertainty as Knightian. Aversion
demand
to this uncertainty generates
for safe financial claims.
own
intermediaries to lock-up capital to cover their
certainty over other markets.
accelerator.
A
agents to require financial
markets' shocks in a manner that
These actions are wasteful
lender of last resort
It also leads
in the aggregate
is
robust to un-
and can trigger a
financial
can unlock private capital markets to stabilize the economy during
these episodes by committing to intervene should conditions worsen.
JEL Codes:
E30, E44, E5, F34, Gl, G21, G22, G28.
Keywords: Locked
collateral, flight to quality, insurance, risk premia, financial intermediaries, lender
of last resort, private sector multiplier, collateral shocks, robust control.
•Respectively:
We
MIT and NBER;
Northwestern University. E-mails: caball@mit.edu, a-krishnamurthy@northwestern.edu.
are grateful to Marios Angeletos, Olivier Blanchard, Phil Bond, Jon Faust, Xavier Gabaix, Jordi Gali, Michael Golosov,
William Hawkins, Burton
Hollifield,
Bengt Holmstrom, David Lucca, Dimitri Vayanos, and seminar participants
Imperial College, LBS, Northwestern, MIT, Wharton,
Conference,
Bank
of England,
and the Central Bank
assistance. Caballero thanks the
NSF
NY
Fed Liquidity Conference,
of Chile for their
for financial support.
comments.
NBER MIDM
Conference,
at
Columbia,
UBC Summer
David Lucca provided excellent research
Policy practitioners operating under a risk-management paradigm may, at times, be led to
"...
When
undertake actions intended to provide insurance against especially adverse outcomes
confronted with uncertainty, especially Knightian uncertainty,
medium
to disengage from
commitments
to long-term
human
beings invariably attempt
in favor of safety
and
The
liquidity....
immediate response on the part of the central bank to such financial implosions must be to
inject
large quantities of liquidity..." Alan Greenspan (2004).
Flight-to-quality episodes are an important source of financial instability.
US
episodes in the
include the
Penn Central
and ending with the bailout
the events that followed the attacks of 9/11. Behind each of these episodes
may
slowdown as
lead to a prolonged
Depression.
in
of these
default of 1970; the stock market crash of 1987; the events
of the Fall of 1998 beginning with the Russian default
that
Modern examples
lies
of
LTCM;
as well as
the specter of a meltdown
Japan during the 1990s, or even a catastrophe
like the
Great
1
From a broad-brush
standpoint, flight-to-quality episodes involve an increase in perceived
risk.
However,
the risk does not circumscribe a purely fundamental shock, and instead centers around the financial system.
For example, the Russian default in the
of
As
US
summer
of 1998 eliminated a small fraction of the trillions of dollars
wealth. Although small, the default created circumstances that severely strained the financial sector.
prices of illiquid assets
fell,
losses
grew
in
commercial banks, investment banks, and hedge funds, leading
investors to question the safety of the financial sector.
Investors withdrew risk-capital from the affected
markets and institutions and moved into short-term and liquid
capital
emerged as sophisticated parts of the
economy were
relatively unaffected.
In this paper
financial
The primary
we develop a model
risk
claims;
and
effects that arise
during this episode was financial system
of a flight-to-quality episode.
and demand
flight to quality into
multipliers.
risk.
for safety are
of
risk.
In practice, the occurrence of an unex-
We take
this
during flight-to-quality episodes: agents
financial bottlenecks develop that hinder capital
increase in risk
movement
system were compromised while other sectors of the
pected event triggers an increase in agents' perception of
two important
Bottlenecks in the
assets.
shock as exogenous, and focus on
demand
safe
movement and segment
a hallmark of flight-to-quality episodes.
and certain
financial
financial markets.
The
Bottlenecks transform
a macroeconomic problem by triggering and amplifying collateral shortages and financial
It is this
systemic dimension that motivates the Fed's reactions, as indicated by Greenspan's
comments.
Our model
on promises of
focuses on agent's perceptions regarding the ability of financial intermediaries to deliver
credit.
arise in their markets.
'See Table
century.
1
Agents contract with financial intermediaries to cover liquidity shocks that
may
However, agents are uncertain about the probability model describing shocks in
(part A) in Barro (2005) for a comprehensive
list
of
extreme events
in
developed economies during the 20th
markets different from
collateral, agents
intermediaries
theirs, treating
grow concerned that shocks may
and compromise
Knightian uncertainty or a
conservatism and a
risk
such uncertainty as Knightian.
demand
their promises of credit.
fall in
for safety.
financial intermediaries have limited
markets that
Such riskiness
intermediation collateral.
The
will deplete the resources of
rises either
through an increase
We show
by formulating plans that are robust to their uncertainty regarding other markets.
what happens
in other
some
in
increase in perceived riskiness generates
We study the case in which agents shield themselves from the
require financial intermediaries to lock-up
in
arise in other
If
capital to devote to their
markets. Once locked-up, intermediaries' capital
own markets'
is
that agents
shocks, regardless of
move
not free to
increased
across markets
response to shocks, resulting in bottlenecks and market segmentation.
The
interaction between Knightian uncertainty and limited intermediation collateral
our analysis. The following example
some
liquidity needs,
which are provided
There are two states that can occur,
reversed in S2-
Thus
it is
A
si
A and
Suppose that two agents,
our example), that
for
and
S2
it
B, have
must allocate between these two agents.
with probabilities n(si) and
has a higher marginal valuation of liquidity in
"efficient" to
at the heart of
by contracting with an intermediary. However, the intermediary
for
has a limited supply of liquidity (one unit
value liquidity in each state,
our main ideas.
illustrates
is
provide more liquidity to
A
in s\
7r(s2).
s\
While both
A and B
than B, with the situation
and more to
B
in
S2'-
s2
s\
X
1
-X
X>±
B
Now
fashion.
suppose that
A
and
1
B
-x
view the uncertainty over the realization of states
In particular, suppose that
A
and
B
may
S\ or S2 in
are unsure over the probabilities 7r(si)
potential problem with achieving this efficient solution
in S2, he
X
is
that since
A
is
and
a Knightian
7r(s2).
Then a
receiving a lower liquidity allocation
overweight the chances that S2 will occur when making decisions.
B may
overweight
s\.
Consider an alternative "non-contingent" hoarding solution, where each agent hoards | unit of liquidity
in
each state, independent of their relative economic liquidity needs.
s\
B
s2
1
1
2
2
1
1
2
2
A
and
B may
reject the efficient solution in favor of the non-contingent
unknown
robust to the
hoarding solution because
probabilities. In equilibrium, the desire for robustness
may
it is
lead to a non-contingent
allocation of liquidity.
we develop
In our model,
we mean by
"efficient"
this
example
in
and "non-contingent."
a competitive, general equilibrium setting.
We
show that
robustness concern vanishes. In terms of our example above,
does not need to ration the liquidity between
of si or S2,
We
One
and the robustness concern
is
A
if
explain what
intermediaries have sufficient collateral, the
if
the intermediary has sufficient liquidity,
and B. Then A's allocation depends
less
it
on the realization
reduced.
present examples to connect our model to the behavior of agents during flight to quality episodes.
of our examples, as alluded to earlier,
flight to quality,
risk capital in
the allocation of risk capital by a financial intermediary. In a
is
the intermediary locks-up risk capital to devote to individual markets, instead of allocating
an
efficient
shock-contingent fashion across markets.
In equilibrium, robustness concerns magnify an aggregate liquidity shortage
consequences.
We
show how a
central bank's
commitment
and potentially improve outcomes. In our model, the
away from the hoarding solution and toward the
Our paper
Moore (1997)
highlight
how
and have macroeconomic
as a lender of last resort can affect equilibrium
central
bank can
shift the private sector's allocation
efficient solution.
relates to several strands of literature.
frictions in business cycle models.
how
We
It
helps to
fill
a gap
in the literature
on financial
Papers such as Bernanke, Gertler, and Gilchrist (1998) and Kiyotaki and
financial frictions in firms amplify aggregate shocks.
Instead,
uncertainty in response to shocks, and
financial frictions lead to greater (Knightian)
we emphasize
how
this rise in
uncertainty feeds back into the financial accelerator.
Our paper
sense, our
also studies the
paper
is
closer to
macroeconomic consequences of
frictions in financial intermediaries.
In this
Holmstrom and Tirole (1998) and Krishnamurthy (2003) that emphasize that
with complete financial contracts, the aggregate collateral of intermediaries
is
ultimately behind financial
amplification mechanisms.
In terms of the policy implications,
Holmstrom and Tirole (1998) study how a shortage
collateral limits private liquidity provision (see also
government can
The key
issue
difference
for liquidity provision.
between our paper and those of Holmstrom and Tirole, and Woodford,
limited. In our model, the
provision
is
but
and more
also,
Woodford, 1990). Their analysis suggests that a credible
government bonds which can then be used by the private sector
even a large amount of collateral in the aggregate
centrally,
The model we develop
it
may
is
government intervention not only adds to the private
how a
that
we show how
be inefficiently used, so that private sector liquidity
improves the use of private
illustrates
of aggregate
sector's collateral
collateral.
lender of last resort
commitment
to insure a low probability
extreme event stabilizes the economy. 2 This result resembles Diamond and Dybvig (1983), where the benefit
of a lender of last resort in
equilibrium; the bad event
bank runs
is
is
to rule out a "bad" equilibrium. However, our
just a low probability
by depositors' concern that they
will
be
extreme event. In Diamond and Dybvig, a run
last in line to
the bank run
is
is
triggered
withdraw deposits. In our model, agents exaggerate
the odds that intermediaries will have exhausted their capital, and
Diamond and Dybvig,
model has a unique
fail
to deliver on promises of credit. In
In our model,
a coordination failure.
on the other hand,
flight to
quality results from agents overweighting the chance of an extreme event.
Flight-to-quality episodes have financial and real effects.
On
the financial side, papers such as Krish-
namurthy (2002) and Longstaff (2004) document that spreads between
assets
widen during
away from
firms
flight to quality.
risky assets such as
draw down
credit lines
Gatev and Strahan (2005) document that investors
Commercial Paper and toward
safe
and liquid/safe
illiquid/risky assets
shift their portfolios
bank deposits during these
episodes, while
from these banks. These actions are consistent with our model, as we predict
that agents shift toward well capitalized financial intermediaries that are less sensitive to the robustness
concerns of agents.
A
number
of papers have also
Commercial Paper and Treasury
Bills
is
documented that the widening of the spread between
a powerful leading indicator for cyclical downturns
and Watson, 1989, Friedman and Kuttner, 1993, Kashyap,
There
is
Stein,
(see, e.g.,
and Wilcox, 1993).
a growing economics literature that aims to formalize Knightian uncertainty (a partial
Dow and Werlang
contributions includes, Gilboa and Schmeidler (1989),
(1992), Epstein
and
Hansen and Sargent (1995, 2003), Skiadas (2003), Epstein and Schneider (2004), and Hansen,
As
in
much
of this literature,
Knightian uncertainty
a class of
applies
These firms typically
in
in practice.
mind.
most similar to Gilboa and Schmeidler,
max-min expected
stress-test their
The widespread
policies.
to describe agents expected utility.
Wang
of
(1994),
et al. (2004)).
Our treatment
in that agents choose a worst case
We
utility
theory to agents
who we
of
among
interpret as running financial firms.
models to various extreme scenarios before formulating investment
use of Value-at-Risk as a decision
Corporate liquidity management
is
making
also
criterion
is
an example of robust decision
done with a worst case scenario
for cash-flows
view the max-min preferences of the agents we study as descriptive of decision rules that
overweight a worst-case rather than as stemming from a deeper psychological foundation. In
paper we
list
priors.
Our paper
making
is
we use a max-min device
Stock
refer to agents as robust decision
making process
much
of the
makers. This terminology most closely corresponds to the decision
of the financial institutions that concern us in this paper.
Routledge and Zin (2004) also begin from the observation that financial institutions follow decision rules
2
Easley and O'Hara (2005) study a model where ambiguity averse traders focus on a worst case scenario
investment decision. Like us, Easley and O'Hara point out that government intervention
effects.
in a
when making an
worst-case scenario can have large
They develop a model
to protect against a worst case scenario.
maker
averse market
and asks
sets bids
to facilitate trade of
of market liquidity in which an uncertainty
an
asset.
Their model captures an important
aspect of flight to quality: uncertainty aversion can lead to a sudden widening of the bid-ask spread, causing
agents to halt trading and reducing market liquidity.
Both our paper and Routledge and Zin share the
emphasis on financial intermediation and uncertainty aversion as central ingredients in
But each paper captures
episodes.
different aspects of flight to quality.
In our model, agents are only Knightian with respect to other markets
stein (2001) explores the
home
flight to quality
and not
bias in international portfolios in a similar setup.
modeling also highlights the difference between high
risk aversion
their
own market. Ep-
As Epstein
points out, this
and aversion to Knightian uncertainty.
Moreover, our modeling shows that max-min preferences interact with macroeconomic conditions in ways
that are not present in models with an invariant
mediary
collateral
is
plentiful,
amount
of risk aversion.
We show
that
1
we
Knightian model.
describe the environment and characterizes decisions and equilibrium. Section 2 explains
the implications of our model for flight-to-quality episodes.
resort in our
inter-
Knightian and standard agents behave identically. However when aggregate
collateral falls, the actions of these agents differ, leading to flight to quality in the
In Section
when aggregate
economy. Sections 4 and 5
Section 3 derives the value of a lender of last
illustrate the interaction
between aggregate collateral and robustness.
Section 6 concludes.
1
We
The Environment
consider a continuous time
economy with a
a phase where liquidity shocks
may
exponentially with hazard rate
8.
Liquidity
take place, and
We
consumption good. At date
single
it
exits this
0,
the economy enters
phase at some random date
r, distributed
study the economy conditional on entering this liquidity phase.
demand
There are two groups of agents, each with unit measure, whose members we label as agent-type
B. Each individual derives utility from consumption at date
17*=u(4)
where the
log-utility
assumption
is
primarily
notation u(.) where convenient). There
Each agent
is
endowed
at date
is
= ln4
made
A
and
r:
ie{A,B}.
for expositional reasons
(1)
(we retain the more general
no discounting.
with wo in consumption goods. These goods are perishable, and are
used to purchase financial claims - see below.
In the normal course of business, the agents can expect to earn a profit of
phase, so that cv
=
II.
However, the agents
may
II
when
exiting the liquidity
receive a "liquidity shock" that reduces profits to
unless
they have some liquidity available to them immediately.
the
We
fall in profits.
such liquid funds, agents can insure against
normalize the return on the liquidity investment to one, so that investing
funds results in a consumption level of cT (l)
The
With
liquidity shock
is
common
to
all
=
I
of liquid
I.
agents of a given type.
We
will think of all agents of a
type as
belonging to a particular market, so that the shock has the interpretation of a market-specific shock. The
shock occurs to each market at most once between
A.
The shocks
and
The
t.
arrival of the
shock
Poisson, with intensity
is
are uncorrelated across markets.
Agents purchase financial claims that deliver some liquidity to them
and thereby insure against the
in the event of the liquidity
shock
fall in profits.
Liquidity supply and aggregate collateral constraint
The
financial sector consists of intermediaries
who
liquidity insurance to
sell
think of these intermediaries as having some capital available to allocate across
A
A
and
B
at date 0.
and B's markets
We
in the
event of shocks.
Financial intermediaries are required to collateralize
insurers have ready access to
liquidity insurance
in the
by their
consumption goods
collateral.
all sales
at all dates,
of liquidity insurance.
and are only
For now, we assume that collateral
is
We
assume that
restricted in their supply of
riskless
and worth Z, but
later
paper we relax this assumption and study situations where the collateral value varies over time and
and
across states (see Sections 5
The
intermediaries' objective
6).
is
to
maximize the date
of the insurance resources disbursed at time s of
/3
revenue from the sale of the insurance
a cost
per unit:
U 1 = 4 + 001,
and subject
less
/3>0
(2)
to the collateral constraint that, in any state, the insurer does not write insurance exceeding Z.
Financial claims
Each agent
is
uncertain about whether he will be hit by a shock or not and about the order in which
shocks will take place. There
of both types of agents.
s
being the
date
s
<
first
We
one. y s (t)
is
a complete
set of
Arrow-Debreu claims contingent on the shock
define x(s) as a claim that pays one at date
is
a claim paying one at date
t,
s,
realizations
conditional on the shock at date
conditional on the
first
shock having occurred at
t.
Robustness
Agents purchase financial claims from the intermediaries to insure against their liquidity shocks.
making the insurance
decisions, agents have a probability
model
In
of the liquidity shocks in mind. But, while
each agent knows the intensity of the shock in his
the other agent-types' market.
We
own market, he
unsure of the intensity of the shock in
is
explore the case in which agents formulate decisions that are robust to
the intensity of the other agents' shock.
We write
all
of our problems from the standpoint of agent(s) A, referring to the other agent-type as B,
and denoting A's assessment
More
formally,
we
of B's intensity as Xb-
follow Gilboa and Schmeidler (1989),
max
and express agent
min Eo[u(c6)
A
is
a class of alternative probability models that agent
decision problem as:
As.A^l
I
{x(s)},{y s (i)}A B 6A
where
j4's
A deems
as possible.
3
Discussion of setup
Our model
studies an
episode. In practice,
economy conditional on entering a
we think that
liquidity episode. It
is
silent
the occurrence of an unexpected event, such as the
We focus
causes agents to re-evaluate their models and triggers robustness concerns.
play out during the liquidity episode:
How
do agents' robustness concerns
do these robustness concerns interact with aggregate constraints?
What
LTCM or Enron
crises,
on the mechanisms that
affect prices
is
on what triggers the
and quantities?
How
the role of an outside liquidity
provider such as the central bank?
We
think of the agents in our model as financial specialists
For example, the agents
may be
Alternatively, the agents
firms in a single industry that
may be banks
we model the agents who
We
actively participate in a single market.
susceptible to a temporary drop in cash-flow.
that are susceptible to an outflow of deposits.
may be hedge-funds/market-makers who
cases,
is
who
are susceptible to a shock to their trading capital. In
case scenario analysis in decision
affected
goal
is
all
of these
are responsible for insuring against a temporary liquidity shortfall.
view agents' max-min preferences as descriptive of their decision
Our main
Finally, the agents
making example
why
to investigate
by aggregate conditions. In
rules.
The widespread
use of worst-
of the robustness preferences of such agents.
these robustness concerns are important in the aggregate
principle, the
demand
for safety
the Russian default, can be easily
accommodated by the
on asset prices or quantities. Yet,
in practice, capital does not
and are
by a sector of hedge funds, say during
rest of the capital
move
market, with negligible effects
freely across
markets to provide the
necessary liquidity. There are bottlenecks, suggesting that there are constraints on the supply of capital.
We model
3
We
these constraints by assuming that financial intermediaries,
who
specialize in capital reallocation,
study arrangements where the financial intermediary provides insurance, ruling out direct consumption insurance
arrangements between
A
and B. In keeping with our
collateral restrictions
on the intermediary, we may think
having no collateral and therefore being unable to support such insurance arrangements.
of the agents to support
If
we assigned Z/2
of
A and B
as
collateral to each
mutual insurance, agents' robustness concerns would reduce such insurance. Intermediaries maximize
the use of collateral.
4
See Routledge and Zin (2004) for a more extensive discussion of examples of max-min behavior
among
financial specialists.
are limited by a collateral constraint.
The
collateral of the intermediary
measures the
maximum amount
of
capital they can supply to a particular market. Since the intermediaries are risk-neutral, they will supply
how much
insurance to our agents. But, in aggregate, the collateral constraints place a limit on
the supply-
side can respond to the robustness-concerns triggered during a liquidity episode.
Our model assumes
that there are complete liquidity-shock-contingent financial claims. Moreover, agents
prearrange insurance against liquidity shocks at date
in asset
markets -
for
0. In practice, liquidity
shocks trigger dynamic trading
example, by shedding risky assets and moving to Treasury
responses are ex-post and not
all
ex-ante.
We
have chosen our strategy despite
- so that agents'
Bills
its
interpretability costs
because, as usual, with complete Arrow-Debreu markets a dynamic problem can be more easily stated as a
static one.
More
substantively, our objective
is
why
to provide an endogenous explanation for
capital does
not move across markets, as opposed to making an exogenous segmentation/incomplete markets assumption.
The assumption
As a matter
of complete liquidity-shock-contingent claims isolates the
mechanism we
highlight.
of interpretation, the ex-ante insurance contracts can be thought of as collateralized contin-
gent credit-lines.
Decisions and Equilibrium
2
In this section
we
where robustness
describe agents' decisions and equilibrium,
is
Flight-to-quality, which
not a concern.
and contrast them with a benchmark case
we
discuss
more
fully in the next section, is
described in terms of the difference in agents' equilibrium decisions across these two cases.
The
2.1
The problem
cost of locking collateral
of a financial intermediary
order to maximize revenue
Denote the price of the
<?!>i(s)
less
to
sell
x(s) first-shock claims and y s (t) second-shock claims in
the cost of providing liquidity insurance, subject to the collateral constraint.
first-shock claim p(s)
as the (true) probability of the
We
is
first
and the
event,
and
price of the second-shock claim q s (t).
5
(i)
as the (true) probability of the second event. 5
assume that the insurer has no robustness concerns and knows these true
out that for most of the conclusions we derive, this
is
Also, denote
without
probabilities, but
loss of generality.
it
turns
Then the intermediary
solves,
max
{x(s)},{y(s}}
Jf
(
\
(p(s)
- /?<Ms))x(s) + /
Js
(<7.(0
-
0</> s
(t))y s (t)dt) ds
(3)
J
subject to,
x{s)
5
With
4>i(s)
= Xe-V+W 3
and
4>s(t)
=
2
X e-
Xs
e-^+ x ^.
< Z
(4)
=
and
x(s)+ys {t)<Z.
(5)
Constraints (4) and (5) are standard collateral constraints that agents impose on the intermediary.
We
Consider an agent looking to purchase first-shock insurance.
balance sheet of the intermediary,
i.e.,
assume that
This gives the constraint
collateral, Z, to cover all of his liabilities, x(s).
first-
if
the intermediary has sufficient
(4).
the other hand, an agent looking to purchase second-shock insurance
and second-shock insurance that the insurer has
sold. Since
concerned about both the
we
is
paid only after
up second-shock insurance
claim buyers require that the effective collateral of the insurer covers
insurance sold. Thus,
all
Z — x{s).
is
second-shock
arrive at (5).
Because of the linearity of the objective function,
problem
is
second-shock insurance
first-shock claims have been settled, the effective collateral that backs
The second-shock
can observe the
the collateral levels of the intermediary and the claims the intermediary
has sold to other agents. The agent only purchases first-shock insurance
On
this agent
(3) yields that x{s)
and ys (t) are
at
an
the collateral constraints do not bind the solution to
if
interior only
if,
p(s)=0Ms)
Qs{t)
For example, when
We
(3
=
1,
when insurance
the second constraint binds in equilibrium for
values, the
first
P4>s(t)-
prices are actuarially fair at an interior solution.
are interested in the case
shadow
=
is
limited by the insurer's collateral. In this case, assuming
all (s,t) pairs,
and denoting by
(J-
S {t)
>
the corresponding
order conditions yield:
q s (t)=l3<f> s (t)+»s(t),
/oo
H s (t)dt.
Prices
now
rise to reflect
the scarcity of collateral
price of a second-shock claims for all
>
t
s,
(/j,
>
0).
Integrating the expression for q s (t) to obtain the
gives:
/oo
q a (t)dt
=
0<h{s) +(*{*),
(6)
with
2
The term
is
</>2(s) is
(
s
)=
r°°
/
x
&(t)<ft=0i(s)—
X
—6
the probability of the economy receiving two shocks before the liquidity event
equal to the probability of receiving a single shock,
10
<j>i(s),
(7)
+
is
over. It
multiplied by the conditional probability of
receving one
more shock
£$. Thus
prior to the end of the liquidity episode,
q(s)
is
made
sum
of the
of the
expected cost of a two-shocks policy for the insurer, and the multiplier on the collateral constraint, m( s )-
An
important point to note about the expressions
for q(s)
and p(s)
is
that the multiplier
both and does not depend on the probability of each of these events. That
is,
is
common
a tight collateral constraint
pushes up the price of contingent claims regardless of the probability of the insurance-event.
this
is
because the collateral constraint requires that
curtailed for
>
all t
Thus the opportunity
s.
if
x(s)
cost of writing
is
first-shock claims
is
claims can be offered. Both claim prices have to be in line because they both reflect a
collateral.
We
The common
in
cost of locking
terms of
q(s):
= q(8)+/3(4>i(*)-M*))-
(8)
cost of locking collateral arises because of the sequential shock structure of the economy.
The second shock only occurs
and
after the first shock has occurred. If the x(s)
two mutually exclusive events, then the intermediary could use the same
in each event.
is
that less second-shock
common
can make this point more precise by solving out p(s) and writing p(s)
p(a)
Intuitively,
on the second shock
high, then insurance
more
to
y(s) claims corresponded to
collateral to
back liquidity insurance
In such case, selling more y(s) claims does not impinge on the intermediary's ability to
sell
x (s) claims, so that the prices of claims need not be aligned.
We
are
most
interested in the opportunity cost of using collateral, as
To
writing liquidity insurance.
simplify
some
of our expressions,
drop the second term on the right-hand-side of
is
the
main
result
we assume
for
now
that
=
/?
and thereby
(8) to yield:
p(s)
This pricing expression
opposed to the actuarial cost of
=
q(s).
(9)
from modelling the supply side of the economy.
It
captures the
equilibrium cost of "locking" collateral.
A
2.2
benchmark
Before turning to the agent's decision problem, we pause to establish a "benchmark" for the model
there are no robustness concerns. In this case,
max
{x(s)},{y 3 (t)}5
where
r)(t
-
s)
=
(A
—+
+ <5)e _
/•oo
5
-
u(U)
,
\
(
+
A+l5 )( t_s )
shock, while y s (t)
term
is
in the objective
the
is
is
the
amount
/*oo
0i(s)u(i(s))
+ Ms)
/
V(t
-
s)u(y s (t)) dt
ds
Js
I
represents the probability that a second shock takes place at time
conditional on a second shock taking place at
In this objective, x(s)
directly at the planner's problem:
r
/
JO
we can look
when
amount
some date greater than
s.
of liquidity insurance delivered to the
of insurance delivered to the second agent
who
first
agent
who
receives a
receives a shock.
the expected utility in the event that no shocks arrive and agents consume
11
t,
The
first
II.
This
term
invariant to the x(s)
is
expected utility from the
date
(exit)
The
and y s (t)
first
and where convenient we omit
decisions,
shock arriving at date
6
it.
The second term
and the second shock occurring
s,
at a date
is
the
prior to
t
r.
constraints in the optimization are:
<
< Z
x(s)
and
0<x{s)+ys {t)<Z.
Since
,5=0, the only
limitation on liquidity insurance delivered to the agents
financial intermediaries.
The second
Thus the planner simply
is
the supply of collateral of
allocates the limited collateral across time
collateral constraint has to bind since
subsumes the
it
first
constraint,
and agents.
and there
is
no other
cost of providing insurance than the collateral limitation.
y s (t)=y(s)
Thus we
t.
The
find that y s
(t)
= Z-x(s).
allocations are not a function of the time interval between the
expression also reflects the cost of locking collateral that
we
first
referred to earlier.
Z
to back x(s) insurance, or to back y s (t) insurance, but not both. In this sense, offering
amount
insurance restricts the
We
,
V?
8
?, m
are,
for each time
with y(s)
the
(
)
for
benchmark economy,
[Ms)u(x(s))
/
each
For
s.
u'(x(s))
_ Ms) =
u'(y(s))
4>i(s)
A
A
+
[W>
5
the log utility case, the quantity of insurance claims are:
v
(
y»<
*
<57
A
V2A +
<5
satisfy:
x^_
y
We
+ fo(s)u(y(s))] ds,
the first order conditions characterizing insurance decisions
^-(^A)z
2A +
3
of one kind of
s,
= Z — x(s).
and hence
more
as:
rrT u n +
which can be solved by maximizing pointwise
Proposition 1 In
can either be used
of the other kind.
can then rewrite the planner's problem
,
and second shock,
be
_ \+5
~ A
also omit a 2 that should multiply the objective function throughout to weight the
12
two groups.
an agent
In words, once
is
hit
by a liquidity shock, the planner has an incentive to allocate more than
This
half of the resources to the agent in distress.
is
because as long as 5
>
there
a chance that the
is
second agent never gets hit and scarce liquidity goes wasted.
We
note that x(s)
episodes where S
is
is
constant and increasing with respect to
large (temporary phenomena). In these cases, the
to inject funds aggressively to the
In the
We
5.
first
group of agents
hit
are interested in liquidity crisis
benchmark
reveals a strong incentive
by a shock.
benchmark competitive equilibrium without robustness considerations, the
more x than y claims
to purchase
intermediary that we derived in the
claims reflect a
common
follows directly from the pricing schedule of a collateral constrained
As
part of this section.
first
probable than the two-shock event.
incentive for agents
5 rises, the single-shock event
However, because the prices of both
cost of locking collateral, the prices of
first
becomes more
and second shock
financial
both claims remain equal, favoring the
purchase of the first-shock claim over the second-shock claim.
The
2.3
An
agents' decision
agent type
A
problem
(henceforth agent A, for short) solves a decision problem where he chooses the paths of
x(s) and y s {t), attempting to be robust to values of As.
choice of x(s) and y s
We
{t)
In this subsection,
we
characterize the optimal
given a particular value of A^.
define the expected utility function for
/•oo
V({x(s)},{y s (t)},X B )
=
J
A
given choices of {z(s)} and {y s {t)} and
r
some value
of Ag:
/>oo
V (t-s)u(y s (t))dt
4>?(s)u(x(s))+4>£(s)J
ds
where, from the exponential distribution function, the probabilities (as perceived by A) are:
4>f{s)
=
P(First shock
<p£(s)
=
P(Second shock
is
=
A's, at s)
is
A e (<5+A+As)s
A's, at t>s)
=
—+
A
The agent maximizes
s.t.
f
r
{x(s)>0},{y,(t)>0}
The budget
endowment
u>o-
The
first
(p(s)x(s)+
Jo
constraint reflects purchase of x(s) and y s
of
^-r</>f (s).
o
V(-) subject to his budget constraint:
™« ^^(M 5 )}.^*)}.^)
,
,
V
f
Js
q s (t)y s (t)dt) ds
<w
(11)
J
given prices of p(s) and q s (t) and the initial
(t),
order conditions for the optimization are:
4>?(s)u'(x(s))=Ms)
(12)
and
^{s)-q{t
where
ip is
-
fl)u'(y.(t))
= iMO.
the Lagrange multiplier on the budget constraint.
13
(13)
We
can simplify the characterization of the agent's decision using the result from our
the intermediary's problem. There
we found
earlier analysis of
that p(s) and q s (t) are related to each other by a
of locking collateral. Integrating both sides of (13) with respect to
t
and using equation
(9),
common
we
cost
find that,
/oo
T){t-s)u'(y s (t))dt
Then, combining this relation with
we can eliminate
(12),
'
U {X{S))
=
each
is
only a function of s and not of
the agent's insurance decisions
s,
(14)
simplify this relation considerably by using the knowledge that in
so that y s {t)
for
p(s) to find:
vit-sWiy^dt
W{S)J
We can
= l>p{s)
t.
sum
Since
to
Z
all
all equilibria,
= Z — x(s),
y s (t)
insurance has to be backed by the collateral of Z,
in equilibrium.
7
Since the probability of receiving a second shock prior to the end of the liquidity episode conditional on
having received a
first
shock at date
Proposition 2 The solution
s
equal to
is
we can
^r^-,
to the agent's decision
simplify the right-hand side of (14), to find:
problem, (11), given a value of
Xb
is
characterized by
the relation:
which for
u'[x(a))
= 44 {s) =
u'(y(s))
cpf(s)
_
X
As one would
benchmark,
(10),
expect, the solution to the agent's decision problem coincides with the solution in the
when Xb
than the second-shock.
likely
=
A.
8
More
generally, for
As <
X+S
is
Note that we are not imposing the requirement that y s (t)
in equilibrium, ys{t)
=Z—
= Z—
c(s).
FVom the
first
8
that, y s {t)
Given equilibrium
=
A
x(s) in agents' decisions.
reallocates
Instead,
we
demand
to
are using the
Xb,
all
feasible
t),
A 5
c(s)e-< + >«-*>,
order condition, (13), and using:
*£(«)
we conclude
=
first-shock
decreasing with respect to Xb-
x(s), only certain price functionals are possible. For a given
q s (t)
some
is
more against the
expected to be hit by a shock, the more agent
equilibrium price functionals, q s (t), must take the form (with respect to
for
agents insure
Note, however, that the ratio of x to y claims
the other agent
knowledge that since
+5
XB
y(s)
7
[l0>
+S
log utility implies:
x(s)
The more
AB
X
y(s), regardless of
prices, p(s)
and
q(s),
=
rrr^W.
+X
Xbfrom (12) and
/
Js
n(t
-«)* =
(13), as well as the
x (s)=^-(X
i,
intertemporal budget constraint we find that:
+ 6)e-( 6+x B-l-»°
(16)
p(s)
y{ s)
= ^Lx B e-(^ B +»°-.
(17)
g(«)
(Note that the multiplier on the budget constraint
is
V = ^~ JTX
14
f° r
tne case °f '°S
utility.)
A
second-shock claims.
thereby ensures that he
still
payments
receives
if
he
is
hit
by a liquidity shock as
well.
Robustness and equilibrium
2.4
We now
in
turn to the robustness step.
In equation (15)
we
characterized the decision problem for agent
A
choosing x(s) relative to y s (t), given a particular value of Xb- These choices defined an expected utility
function V({x(s)}, {y s
Xb)-
{t)},
to alternative values of
Xb
The robustness
chooses x(s) and y s (t), then "nature" chooses
problem
full
for agent
is
make
these choices while being robust
A
Xb £ A
max-min expected
minimize the
to
utility theory,
utility of
agent
A
^/X ™*M m n ^W*)}.
{»•(*)},
given equilibrium prices p(s) and q s (t) and the budget constraint.
over which agents would like to be robust
is
an
agent A, given his choices.
We
Ab)
assume the
(18)
set of alternative
models
interval defined by:
max{0, X -
K} <
XB
<
preferences of agents, with a larger
X
+ K.
K corresponding to a more extreme worst-case
scenario.
The
following proposition presents the
Proposition 3 The following insurance
• For
•
K < 8,
agents' decisions are as
main
result of this section:
decisions constitute an equilibrium in the robust
if
Xb
=
X
+ K:
x pr
X
yP r
X
We
refer to this case as the "partially robust" case.
For
K>
5,
agents' decisions are as
if
Xb
=
X
+
+5
+K
6:
x fT
yfr
We
1.
refer to this equal claims case as the "fully robust" case.
In both cases,
first
is:
/ I
i
iA
{x(s)>0}
,{y 3 (t)>0} A B SA
K indexes the robustness
to
•
In the game-theoretic language often used to describe
The
A
step for agent
agents' decisions are time invariant.
15
economy:
When
are as
if
K=
Xb =
0,
A.
the model collapses to the benchmark model without the robustness concern, and decisions
As
K
agents become concerned that they will receive a shock second, in which case
rises,
their larger purchases of first-shock insurance will
and increase
first-shock insurance
be wasted.
first
they reduce their purchases of
At the extreme, when
reflects the intuition
K
is
and second-shock claims, and thereby insulate themselves
against their uncertainty over the likelihood of market
make
result,
their purchases of second-shock insurance.
large enough, they equate their purchases of
The formal proof
As a
B
receiving a shock.
we have provided, but
is
their insurance decisions anticipating that the worst case
complicated by two
Xb
issues. First,
depend on those
will
agents
decisions;
and,
second, equilibrium prices are a function of the anticipated worst case of agents (but in an individual agent's
decision problem, prices are taken as given).
Proof.
we
First,
The market
decisions in equilibrium as A.
clearing conditions imply,
x A (s)+yf(t)
x
where we have used the
Since the relative
B
(s)
+
first
y
A (t)
= Z,
=
fact that the collateral availability of
demand
Z,
Z determines
the supply of liquidity insurance. 9
satisfies
x(s)
we can
Denote the Xb on which agents base
derive the prices in the proposed equilibrium.
To
substitute to solve out for x(s) and y(s).
_ X+5
find prices,
we
substitute the derived quantities into the
order conditions for agent A, (12) and (13), to yield:
p{s )
^
=
{
X
+ S + X)e-^ +5+ ^ s
(19)
and,
qs{t)
Small K:
=
^z
(X
+ 5 + A)(A +
,5) e
-(A+*+A),-(A+*)(t-,)
(20)
Consider the robustness step next. For small K, we wish to show that, at equilibrium prices, the
highest utility attainable
when
solving,
mmV({x{s)},{y
max
V= {i(s)>0},{y
(t)>0} AbEA
s
(t)},X B )
(21)
a
occurs at x pr and y pr and given these choices by the agent, "nature" always chooses X B
,
no other choice of
{:r(s)}
and {y s (t)} can induce nature to choose a X B that
=
X
+ K.
Moreover,
results in a higher utility for
the agent.
9
If
agent
economizes
A
receives a shock
first,
his limited collateral
then his second-shock insurance claim disappears. The same applies to B. Thus, the insurer
by only backing the two possible shock sequences of
16
A
then B, and
B
then A.
Given an assumed worst case of A
max
agents
problem, (11),
W = X + +S + X
X
=
6
_
e -(A*-*)«z
and
pr
y
,
denote probability of a
<j>2
and
(20), the
unique optimum to the
and,
=
y(s)
ys
(t)
A«
_
+5+
X
=
X
Using
e
-(A a -*)«
Z
knowledge, we can derive the
this
as,
V{x p \
4>?
prices (19)
Xg, x{s) and y(s) are constant functions of time.
expected utility function for the agent
where
and equilibrium
is:
x f a)
At A
+ K,
= tfu{x pr + <i>$u(ypr ),
XB )
(22)
)
and second shock at any time
first
(as perceived
by agent A),
respectively:
/•OO
Using the envelope theorem, we
which
is
values of
pr
'
2/
apparent that the agent
it is
,
As, since such a choice makes first-shock claims
(i.e.
=
is
X)
and as long as x pr
particularly unprotected
Next, define
V
=
X
+
K
= V{x
pr
utility attainable in (21).
pT
(i.e.
,y
pr
,Xb
=
Suppose there
X
+
set in
choosing A £
We now
K).
.
Thus, since
A
is
for small
nature chooses to increase
is
over- weight in
linear, for small
optimum.
If
is
utility if
enough
Xb = X + K, then
choose
Xb = A +
K given
the induced utility
is
result in
they induce nature to choose
x
pT
and y
strictly lower
pr
than
and nature continues
V
pr
is
K=
5.
At
is
the highest
V > V pr
pr
and y
.
this point, the agent
becomes
a unique
to choose a worst
Xb
further lowers the utility of
pr
.
robust with respect to any Xb, since
fully
It
Xb < X + K. Note
For large values of K, the gap between first-shock and second-shock claims narrows until
disappears once
K
Moreover, since nature can always
.
the agent's deviation, nature's optimal choice of
the agent. Thus, the agent can do no better than to choose x
Large K:
and {y s {t)} which
V pr
strictly concave, the first order conditions define
the agent makes any choice other than
case of
argue, by contradiction, that
exist choices of {x(s)}
that since the optimization problem in (11)
it
when
y,
Vx B
for
pr
highest value possible).
should be clear that such choices can only increase
=
if
y
pr
pr
the agent considers only small deviations in the model), the utility of the agent at choices (x ,y )
minimized when Xb
x
>
valuable to the agent, and the agent
less
Moreover, since the constraint
holding these claims.
\
find that,
around the benchmark (Xb
strictly negative
K, x 91 >
/>oo
\
any Xb- At
K=
5,
the agent attains utility of
=0
V^ r =
V(f-,
f As = X+5).
,
Further increases in
K expand
the constraint set for nature, and thereby weakly decrease the highest utility attainable in (21). However,
17
the agent can guarantee himself
V fr
by choosing x
=
K>
5.
attain the highest possible utility in (21) for
all
= f
y
.
Thus, the fully robust choices continue to
m
Flight-to- Quality Episodes
3
We now
discuss the connection between our
model and
shows how, given a limited capacity of the economy
may
for liquidity provision (Z),
We
further reduce the liquidity available to the economy.
robust actions and
how they reduce aggregate
At a broad
flight-to-quality episodes.
liquidity.
We
level
our model
the actions of robust agents
begin this section with examples of these
then discuss the macroeconomic consequences
of flight to quality episodes.
We
generate a flight to quality in the model as a comparitive static in which
accounts of
flight to quality
As
rises.
In practice,
episodes begin with an unexpected event that leads agents to reassess their
models. In 1970, the unexpected event was the Penn Central default on Commercial Paper. In 1987,
thesharp drop
K
rise in
5),
which then leads
more
terms of financial intermediary
3.1
is
clear enough.
our model
A
fall
in
Z
grow concerned that intermediaries may not have enough
Agents worry about being second and
in
flight to quality in
detail in Section 5, but the intuition
risk as agents
was
to a rise in \b-
For a fixed K, an alternative way to generate
case in
it
terms of our model, a reassessment of models can be thought of as a
in the stock market. In
K<
(for
many
raise \b-
Some
is
We explain this
to reduce Z.
increases financial intermediary
collateral to cover their shocks.
episodes, such as the Fall of 1998, can be thought of
risk.
Example: Intermediary capital allocation
In equilibrium, markets
A
and
B
are guaranteed liquidity of
y
In addition to the
minimum
level of insurance,
=
min{z,;y}.
x - y >
of liquidity
is
delivered to the
first
market that
receives a shock.
We can
intepret this behavior in terms of intermediaries' capital allocation decision, y of the capital of an
intermediary
capital") to
is
locked-up to service each market ("committed capital"), and x
move
to the
first
market with the shock. As Xb
to allocate flexibly across markets. In the fully robust case,
In
many
flight to quality episodes, bottlenecks arise in
mented. For example,
in the Fall of
rises,
x—y
x—y
=
the
0,
falls
—
y capital
and there
is
is less
free ("trading
trading capital
and markets are segmented.
movement
of capital
and markets appear
seg-
1998 episode the markets where hedge funds specialized were particularly
affected by the crisis, as capital did not flow into these markets.
18
Notice that segmentation
is
the market's response to agents' robustness concerns. Since agents require
financial claims that are uncontingent on shocks outside their
by committing intermediaries to segment their
can interpret the financial claims of
that intermediaries have a limited
is
x
the market provides such claims
capital.
Example: Run on banks' credit facility/deposits
3.2
We
own markets,
A
that
= y,
A
amount
and
B
of capital to back
up these
The
credit lines.
preemptively drawing
down a
credit line at
complete in our model, the future action by
Although the actions of agents
in our
a date when the fear of large
A
is
prearranged at date
0,
B
shocks
may
=
first.
By
As markets
A
are
10
y.
model are most naturally interpreted as a run on banks'
at a deeper level, agents' actions are also related to the
setting
be reflected by
arises.
by choosing x
recognize
robustness concern
agents are worried that B's shocks will deplete the bank's limited credit facility
they insulate themselves from this concern. In practice, the robustness action
facility,
The agents
as credit lines from intermediaries.
credit
more commonly analyzed run on banks'
As Diamond and Dybvig (1983) emphasize, a deposit contract implements an optimal shock-
deposits.
contingent allocation of liquidity. In the fully robust equilibrium of our model, agents choose an allocation
that
non-contingent on each other's shocks instead of the contingent liquidity allocation of the benchmark
is
economy. Agents each hoard Z/2 units of liquidity to cover their own shocks, independent of other markets'
shocks.
In this sense, robust agents' preference for shock independent liquidity allocations
is
related the
behavior of panicked depositors in a bank run.
The
the
essence of a flight-to-quality episode
demand
uncontingent insurance as \b
for
well collateralized financial institutions.
financial claim
demand
for
is
is
agents'
that agents
rises is
With a
demand
for certainty.
may
shift their
empirical counterpart of
demands toward extremely
less well collateralized institution,
contingent on the other markets' shocks. Thus our model
bank deposits and bank
The
accompanies
credit lines that
is
an agent fears that
his
consistent with the increased
flight to quality (see
Gatev and Strahan,
2005).
In Section 6
of their
own
we show that
collateral.
these actions by the agents trigger a reaction by insurers to ensure the safety
Insurers increase their valuation of riskless assets relative to risky assets, which
consistent with the widening of the spread between
Commercial Paper and Treasury
bills
is
that accompanies
flight to quality.
10
We
can also interpret robustness
bank lobbies
for
more
risk capital,
in
terms of the internal
because of fear that desk
risk
B
management
of an investment bank: Desk
will suffer losses
19
A
of an investment
soon, reducing the bank's total risk capital.
Example: Liquidity suppliers become demanders
3.3
In the crises of the Fall of 1998, hedge funds were retrenching from
many markets and
thereby reducing the overall supply of liquidity (see Kyle and Xiong, 2001).
became
to markets
We may
The
liquidating assets,
natural liquidity providers
liquidity demanders.
think of the robustness concern in our model as corresponding in practice to hedge funds' (or
their investors') fears that
market liquidity
outcomes, further reducing market
investors) requires
enough other
causing the hedge funds to overprotect against adverse
will fall,
liquidity.
A
liquidity providers to participate in
B
robust hedge fund grows concerned that shocks in market
from market A. As a
result, the
A
For example, suppose a hedge fund in market
make
to
(x) used currently for liquidity
provision in market A, and instead increases the buffer (y) against the possible
A
The
the business profitable.
could cause other liquidity providers to pull back
hedge fund reduces the amount of capital
behavior reduces the liquidity of both market
(or the fund's
B
shock.
The
conservative
and B.
In any flight to quality episode, the capacity of financial intermediaries - commercial and investment
banks, hedge funds, trading desks, market makers,
At the abstract
level of
liquidity to markets.
our model,
Z
Our model does not
distinguish
economy and compounds an aggregate
(13), to find the prices of first
modes
the various
provide
we
of liquidity provision
the behavior of robust agents reduces the effective
Z
of
liquidity shortage.
and second-shock
P(«)
As Xb
rises,
claims:
=
?(*)
linear
<5
(24)
.
Z
in
K
s,
the exponential term in (24)
increases the prices of both
=
g(0)
= ^(A + 5 +
increase in prices of the first-shock insurance contract
second-shock insurance.
As we have shown
Because the model has only two shocks,
on surviving until date
fall.
is
liquidity insurance:
first
is
and second-shock
11
The
which
order conditions for agent A, (12)
= ^(A + + X B )e-^ +6+x ^ s
term so that the increase
p(0)
claims
first
equilibrium prices of insurance change as well. For small
dominated by the
11
how
among
can substitute the equilibrium values of x(s) and y{s) into the
and
constrained.
Price of liquidity provision
3.4
We
is
reflects the capacity of liquidity providers, in aggregate, to
observe in practice. Instead our model shows
the
- to provide liquidity to markets
etc.
the
s
when As
is
indirect, driven
rises,
large s occurs at
the recessionary shock.
20
s
=
by increased demand
robust agents decrease their
for s sufficiently large, the increase in
without a shock. In these cases, the exponential term
The boundary between the small and
half-life of
earlier,
A B ).
A^
in (24)
for
demand
lessens the odds that the agent places
dominates and the prices of financial
j^Vj; for 5 large relative to A, the
boundary
is
at j
for first-shock insurance.
collateral for y
They
increase
demand
for
and decrease the supply available to
second-shock insurance, causing insurers to lock-up
first-shock insurance, thereby indirectly driving
up the
price of first-shock insurance.
The
prices,
p and
q,
represent the marginal cost of liquidity provision in the economy. Since intermediaries
have a limited amount of collateral to back liquidity provision, at the aggregate
capacity of the economy to provide liquidity to markets. As robustness
rises
the
parameterizes the
the cost of liquidity provision
rises,
Macroeconomic
active.
cost
Suppose we introduce a small number of rational Savage agents into the economy. These agents,
benchmark economy,
will calculate that
it is
prices that the robust agents face.
of insurance they purchase,
economy-wide
At these high
is
same
inflated liquidity insurance
non-robust agents will also reduce the amount
to liquidity shocks (for low s).
12
In this sense,
not just a problem affecting the decision rules of robust agents.
It
has an
cost.
robust agents lock-up resources in their desire for certainty, the financial system becomes inflexible
in its response to shocks.
promise to the
first
In the
benchmark economy, x be > y be
market that receives a shock. The freedom
good chance that the second shock never takes
An
prices, the
and leave themselves exposed
our model
flight to quality in
as in the
optimal to insure more against the first-shock as opposed to the
second-shock. However, notice that these agents will have to trade at the
When
structure,
the cost of liquidity provision will also be reflected as higher liquidity premia in the asset markets
where the intermediaries are
3.5
Z
Z of the economy. With a richer asset
because the actions of robust agents reduce the effective
rise in
level,
alternative
way
is
,
so that
x be
—
y
be
of collateral
valuable since as long as 5
>
0,
is left
there
is
to
a
place.
to see the collateral waste associated to a flight-to-quality episode
is
to re-write the
constraint
+y < Z
x
in
terms of the risky and
riskless
claims
Z
can support:
(x
The
riskless
claim consumes twice as
much
Recall that the change in prices depends on
s.
y)
+ 2y <
Z.
collateral as a risky claim.
claims, as in the fully robust equilibrium, locks
12
-
up
For large
Having
collateral that goes
s,
all
agents holding only riskless
unused when 5
prices fall since the rise in
Xb
>
0.
leads to a decline in the perceived
probability of shocks taking place. Hence rational Savage agents lower early insurance (both of x(s) and y(s)) as a result of the
presence of robust agents (the point we have highlighted) but increase late insurance.
21
In practice, systemic risk
financial
system becomes
is
the important macroeconomic concern in flight to quality episodes.
inflexible, bottlenecks arise that
When
the
can trigger accelerators and create system-wide
problems.
Locked Collateral and Extreme Event Intervention
4
In this section
we study
the impact of a lender of last resort on the economy.
that obtains resources ex-post, at
event. In practice, this pledge
may be
guaranteeing, against default, the
markets.
We
some
cost,
which
it
We
bank
consider a central
can credibly pledge to agents in the two-shock extreme
supported by costly ex-post inflation or taxation and carried out by
liabilities of financial
intermediaries
who have
sold financial claims to both
analyze the impact and marginal benefit of such a guarantee.
Formally, we assume that the government credibly expands the collateral of the financial sector in the
two-shock event by an amount G. Thus, the collateral constraints on intermediaries are altered to
x
Since
we
are interested in computing the marginal benefit of intervention,
vention of G.
4.1
A
The
infinitesimal inter-
Welfare criterion
when agents have non-Savage
also adopt the agents'
preferences'?
concerned with agents' ex-post average
As noted
earlier in the paper,
Or, should
utility
we think
decision rules of financial specialists
why
we study an
intervention has no effect on the constraint for first-shock insurance.
delicate issue arises
bank
+ y< Z + G.
(e.g.,
of robust control to central
it
In evaluating policy, should the central
take a more paternalistic approach in which
of the robustness preferences as a realistic depiction of the
its
From
objective function that
Sims (2001) has made
may
this perspective,
it is
this point in questioning the application
bank policymaking. He argues that max-min preferences
we begin by analyzing a
to alternative,
4.2
more
are
simpy shortcuts
first
In
what
welfare function based on agents' expected utility functions, and then turn
paternalistic, welfare criteria.
Robust expected
Consider
not clear
lead to obvious average losses,
to generate observed behavior of economic agents, but should not be seen as deeper preferences.
follows,
it is
from consumption?
value-at-risk constraints).
a central bank should build biases into
just because agents exhibit these biases.
preferences:
utility
the case where the central bank uses the agents' expected utility functions to evaluate welfare.
In this case, the envelope theorem
tells
us that the indirect effect due to the change of insurance decisions
22
is
second-order, and that the only benefit
In our model,
we have
and
(see equations (22)
is
the direct
is
direct,
(23))
is,
y,
=
XB )
(j>iu{x)
more insurance against the second-shock
whereas the
effect
on x
is
an indirect
+ <t>£u{y)
leads agents to increase both x
the
effect
on y
Z
can be computed assuming
all
of the
Z
is
(25)
J^^,
j^^Z.
=
While there
tially
The
y.
y:
VG = d4Av) =
since y
and
reoptimization of agents' portfolios. However, using
effect of
the envelope theorem, the utility gain from an extra unit of
spent increasing
and y
derived that the expected utility for an agent at the optimal choices of x
V(x,
Offering
effect.
same
is
a welfare benefit of providing a government guarantee, the value of the benefit
in the
benchmark and the robust economies. 13 In both
fully insure agents against shocks.
The
benefit of the extra unit of
agents more insurance against an adverse outcome.
It
Z
is
cases, the
economy
(1998).
As we show
substan-
lacks collateral to
limited to the direct effect of providing
stems from our assumption that the government can
create liquidity to insure a shock from sources that the private sector cannot, as in
Holmstrom and Tirole
is
next, this logic does not apply
if
Woodford (1990) and
we consider
alternative welfare
criteria.
4.3
bank
Fully informed central
Consider a welfare criterion based on agent's decisions, but evaluated at the reference
envelope theorem argument breaks
down
for the robust
from second-shock to first-shock insurance has
first
order
economy and the
A's.
In this case the
reallocation of private insurance
Since in equilibrium the agent "exagerates"
effects.
the likelihood of a two-shock event relative to a single-shock event, the reallocation of resources toward
shock insurance has a
The expected
first
order benefit.
value of the direct effect of intervention for the central bank
is:
(26)
fcu'(y).
7r
13
Recall that a liquidity shock eliminates
>
0.
We find that in
economy, even
may eschew
if
first-
all profits.
We
have explored a setup where the shock reduces profits to a
this case the benefit of the lender of last resort policy
agents' expected utility
is
the welfare criterion.
When
may be
7r is
greater
in
the robust than in the
sufficient large, agents in the
level of
benchmark
benchmark economy
insurance agains the extreme two-shock event in favor of increasing insurance against the single-shock event. In
contrast, agents in the robust
economy always insure against the two-shock
against an extreme event, while agents in the
event.
Thus robust agents further value insurance
benchmark economy have limited (expected) valuation
for
such insurance. Notice
that here the benefit of the policy comes from the agent's valuation of the extra insurance rather than from the reorganization
of its portfolio in the presence of the policy.
23
where
(date
<j>
2 is
the (true) probability the central bank assigns to a two-shock event at the time of commitment
0).
The
total (direct plus indirect) expected value of the
—
government intervention
dec
is:
dii
VG = (t>W{x)—+4> 2 u\y)^.
(27)
where Vq denotes the expected value of intervention evaluated using the central banks's probability
distri-
bution.
The
first
order condition for agents' optimization
u'(x)
Using
VG =
Finally,
we divide
= ^T^'(y)
^
this relation, as well as the constraint that
is:
-I-
4^
—
1,
we
find,
^y)(l + {^-l)%)
by the direct
this total effect of intervention
(28)
effect of intervention to
construct a private
sector multiplier of the lender of last resort commitment:
M-(l + (£-l)§).
Proposition 4 When X B
benefit
=X
(benchmark economy), the
and
M
For Xb
total benefit of intervention is limited to the direct
>
X, there is
an indirect
benefit
be
=
stemming from
M
r
>
When
agents
make
the reoptimization of agents' portfolios:
1.
economy stems from the
Intuitively, the benefit of intervention in the robust
to the central bank's guarantee.
1.
reaction of the private sector
their insurance decisions, they are overly concerned
about
receiving the shock second, and use too high an assessment of the likelihood of the two-shock event. This
breaks the envelope theorem argument under the central banks' objective function.
By
offering to insure
the two-shock event directly, the central bank leads the agent to free up resources to insure the more likely
one-shock event
4.4
(^ >
0).
Partially informed central
Our assumption
bank
that the central bank knows the true A's of the economy, despite the fact that agents
be unsure about these
A's,
is
a strong requirement.
requirement on the central bank.
24
In this section,
we
may
consider a weaker informational
Agent
central
A
bank
knows that X B £ [max{A — K,
some non-degenerate symmetric
robust economy).
+ K]
joint distribution
We
(and likewise for agent B).
about the values of the
also uncertain
is
0}, A
and only knows that the
A's,
cb
F(X A A^).
,
We study
the case where
suppose that the
drawn from
A's are
K
>
5
(i.e.
the fully
14
Because of symmetry
symmetric interventions,
in F(-),
i.e.
it is
straightforward to show that the central bank restricts attention to
interventions that are equal regardless of whether
it is
A
or
B
that
is
hit first,
or second.
we
In this case
define the multiplier as:
[Vg (\a,Xb)]
m= EEXaXb[DE(X
a ,X b
)}'
XaXb
where Vc{X A ,X B ) and
DE(X A Xb)
and
respectively stand for the total
,
direct expected effect of the inter-
vention conditional on X A and A^.
Then,
for the fully robust
economy,
EXaXb [Vg (X a ,Xb)} =
where
denotes the probability that agent
Pr(z; Aj)
economy, the promised resources
event.
^u'(Z/2)EXaXb [Pv(A;X a
Thus the marginal value
by the probability that shocks
hit
i is
of resources in the shock
may
)}
equally across the
Proposition 5
is
u'(Z/2) times one-half. This value
,
i
is
hit
first
is
weighted
+ Pr(B\A; X a ,X b )},
second given intensities X A and Asi is
effect reflects that resources are all
hit
by a shock while j
'
is
For
not, given
spent in the event of the second shock.
knowledge of the markets probability models
If the central bank
and second shock
split
Px(i\Noj;X A ,X B ) denote the probability that agent
X A andAs. The direct
intensity A*. In the fully robust
is
u'(Z/2)E XaXb [Pt{A\B; X a X b )
where Pr(i\j\X A ,X B ) denotes the probability that agent
intensities
its
occur, to either agent.
EXaXb [DE(X a ,X b =
later use, let
+ Pt(B-X b )},
by a shock given
two-shock event
in the
)
is
limited to a non-degenerate
symmetric joint distribution F(X cAb Ag) with some mass for finite and positive X A and X cg, the multiplier in
,
the fully robust
economy
is,
M
T
>
1.
Proof. From the symmetry of F(X Ab X CB ), we have that the multiplier can be written as
,
M
14 In
T
=
EXaXb [Pt(A;X a
)}
2E XaXb {Pv(A\B;X a ,X b
)
the fully robust economy, the x and y decisions of agents are independent of A, which simplifies our analysis because
avoid computing the central bank's expectations over agents insurance decisions.
25
we
Thus, the proposition holds
if:
E XaXb [Pt(A- X A
Expanding the
left
hand
)}
side of this expression,
> 2E XaXb [Pt(A\B;X a ,X b )).
we have
EXaXb [Pr(i4; X A = E XaXb [Pv(A\B; \ a
)}
,
Xb )
that:
+ Pv(A\NoB; X A ,X B + Pr(B\A; X A
)
= EXaXb {2 Pr(A\B; X A ,X B +
)
,
X B )}
Pr(A\NoB; X A) X B )}-
Replacing this expression back into the inequality, we have that the latter holds
and only
if
X B and positive X Ab since 5
>
if
E XaXb {Pt(A\NoB;X a ,X b )}>0,
which
is
satisfied as long as
F(X A
,
X'g
)
has some mass for
Intuitively, the proof exploits the central bank's
exaggerates the probability of being second. Agent
finite
knowledge of symmetry
B
the distribution). Agent
A
does the same. Thus the total benefit of the lender
of last resort reflects exaggerated probabilities of agents being second.
the central bank knows that they are drawn from similar distributions.
A,
(in
0.
Despite not knowing the true
With a symmetric
A's,
distribution over
the probabilities the central bank uses reflects that the agents cannot both be second. This information
affects the central
thereby sufficient to conclude that the multiplier on intervention
4.5
The symmetry information
bank's calculation of the direct benefit of intervention.
is
is
greater than one.
Moral hazard critique
Under the
decisions.
paternalistic view, the public provision of insurance leads to improved private sector insurance
The complementarity between
public and private provision of insurance in our model cuts against
the usual moral hazard critique of central bank interventions. This critique
to the provision of public insurance
by cutting back on their own insurance
is
predicated on agents responding
activities. In
our model, in keeping
with the moral hazard critique, agents reallocate insurance away from the publicly insured shock. However,
when
flight to quality is
the concern, the reallocation improves (ex-post) outcomes on average. 15
Note that the central bank can achieve the same distribution
intervene in the
first
shock.
(likely) single-shock event.
15 Note
that
if
is
instead
it
commits to
larger than the extreme event
bank rather than the private sector bears the
cost of insurance against the
cost of this policy
is
Agents would reallocate the expected resources from the central bank to the
the direct effect of intervention
inconsistent. This result
the policy
if
much
However the expected
intervention, since the central
of insurance
is
insufficient to justify intervention,
then the lender of last resort policy
is
time
not surprising as the benefit of the policy comes precisely from the private sector reaction, not from
itself.
26
two-shock event, which
is
what the
exactly the opposite of
central
bank wants
to achieve.
In this sense,
interventions in intermediate events are subject to the moral hazard critique.
To see
this, let us
return to the case of the fully informed central bank.
in the event of the first
shock
is
4>\u'{x), while the total effect
,'
From
this equation,
dx
A
The
direct effect of the intervention
is,
dx
/
follows that,
it
Vr
<piu'{x)
since
AB
That
>
is,
<
A and
^g-
<
1.
the lender of last resort
facility, to
be effective and improve private financial markets, has to be
a last not an intermediate resort.
Financial Intermediary Risk and Aggregate Collateral
5
Throughout the paper we have made assumptions
aggregate collateral
a
fall in
limited. In this section
is
we
so that the
some
relax
economy
we
and show that
of these assumptions
derive a comparative static to contrast the response of the
the robust economy to a tightening of collateral.
We
in this case
The only
that insurers
substantive modification
now
we make
some
Abundant
that
our baseline model
Z
We
results characterizing the equilibrium.
solutions illustrating the equilibria for both
first
in this section to
we have discussed
face a "consumption" cost of using insurance resources.
solution for this case but present
benchmark economy and
find that the response to a collateral tightening
qualitatively similar to the increase in preference for robustness that
Suppose
always in a situation where
intermediation collateral can trigger a flight to quality.
In this context,
5.1
is
is
in earlier sections.
is
to set
cannot derive a
full
P >
so
analytical
The section ends with numerical
benchmark and robust economy.
collateral
is
large
enough so
that, at equilibrium levels of
demand, the
financial sector
is
uncon-
strained in selling insurance. In this case, the prices of the two financial claims just reflect probabilities:
In the case of robust agents, using the
first
p(s)
= 0Ms)
q(s)
=
Pfa(a)
order conditions (12) and (13)
x[s)
y[)
0* M*)
27
MM*)'
we
find that:
It is
easy to verify that the solution for robust agents
for all
s.
16
The argument resembles
At equal
(22)
levels of
x and
y,
the proof of Proposition
the agent
is
equal to zero. Picking the point
is
=
we
X,
y(s)
= ya (t),
and equal
to.
a constant
3.
robust to any value of
Xb
=
to set x(s)
is
Xb
that nature chooses and V\ B in equation
see that choosing the constant levels for
x and y
is
the
unique optimum for agents.
If
agents are fully informed about Xb, the same
first
the solution. Intuitively, since prices are actuarially
likely
order conditions, evaluated at
Xb
=
X,
determine
agents choose to insure equally against both the
fair,
one-shock event as well as the unlikely two-shock event.
The
benchmark case
solution in the
is
identical to the solution in the robust economy.
Both economies
respond equally to the typical (single shock) event, as well as to the extreme two-shock event.
5.2
Collateral tightening
Let
That
is,
when
prices are "fair"
Note
Z_ represents
the level of intermediation collateral which exactly satisfies agents insurance
(ji
=
at this point that
demand
0).
when
=
/?
in the collateral-constrained region.
0,
Z —
as in the previous sections,
Henceforth
let
>
oo so that the economy
us specialize to the case with
(3
=
1
so that
is
always
when Z >
Z_
prices correspond to the standard actuarially fair prices.
Suppose now that
Z
falls
constraint of the insurers
where
fj,(s)
The
= / n s (t) >
first
below
and
Z_.
Then, at actuarially
fair prices, agents'
demand
saturate the collateral
prices rise to reflect the collateral tightening:
p(s)
=
<t>i(s)
q s (t)
=
4>2{s)ri{t
+n(s)
-
s)
+ fis (t),
0.
order conditions for agents remain as in (12) and (13). For the x decision:
±-4>A(s)u'{x(s))=Ms)
while for the y 3
{t)
+ n(s)
(30)
decision:
4^(s)r?(s -
t)u'(y s (t))
=
<t>
2
(s)v(s
~t)
+ n,(t).
V
As
before,
we can
use the knowledge that y s (t)
=
y(s) to integrate this equation
^(s)u'(y(s)) = 4>2(s)+^s)In addition, at this solution, the multiplier on the budget constraint
28
is
i/i
=
,
find:
(31)
ip
16
and
-
x+s )wo
We
consider the effect of a collateral tightening by inspecting (30) and (31).
multiplier
More
and
/j.(s)
raises prices, leading to
interestingly, for
a
an equal change
both x(s) and
fall in
tightening increases the
y(s).
in /i(s), the relative fall in x(s)
and
y(s)
depends on the ratio
Beginning from the case where x and y are equal, from (31) we can see that y
-rfr-
<
response to the tight collateral constraint since ^ftthe decrease in y
To prove
yu(s)
The
=
smaller in the robust
is
and
these statements, for each of (30)
and
/i(s)
>
for the
benchmark case
the right-hand side across these two values of n(s)
sides across the
two values of
in the
x*(s)
1
A).
For each of (30) and (31), the only difference in
is
+
X
and
y°{s). Since
-A^
<
Using this knowledge, we find that, 17
1
when the
constraint does bind.
the percentage
1,
(31).
\y°(s)
(s) are
=
Thus, the difference in the left-hand
0.
/V(a)
S
and y l
bind, x°(s)
>
due to n(s)
x°(s) and y°{s) are the equilibrium values of x and y
when the
benchmark economy.
=
Where
equilibrium values
A,
construct the difference between the case where
= _A_
x°{s)
>
we
are equal, for each of (30)
fi(s)
in
(31),
Xb
(i.e
more than x
Moreover, since in the robust economy Xb
1.
economy than
falls
fall in
x
is
As
constraint does not bind, and
before,
when the
x 1 (s)
constraint does not
smaller than the percentage
fall in y.
Then
it
follows that
1
la:
In the
^) -
benchmark economy, the tightening
a:
(*)|
1
<
|y (a)
- v°(*)|.
of the collateral constraint leads to a smaller fall in
x than in
y.
Let us compare the robust economy next to the benchmark economy, where both economies are constrained. Subtracting (31) from (30)
Xb
In the robust case,
right-hand side to
rises
beyond
rise (recall
we
A.
find:
From
that 0f(s)
is
the above expression
decreasing in
Xb
we can
for all s)
see that increasing
and x(s) to
fall.
Xb
causes the
Thus, x(s)
is
lower
economy than the benchmark economy.
in the robust
To summarize:
Proposition 6 Comparing
levels of aggregate collateral,
Z>
we
economy and
the
benchmark economy across two
//
•
When Z < Z, x
find:
is
lower in the robust economy than in the benchmark economy.
robustness concerns only arise
difference between
"risk-aversion"
when
the aggregate constraint binds.
and robustness
in
our setting.
Indeed, this
^ = ZT
vjq XT?
x+6
*i
ant^ s ecI ua l ' n these two cases.
'
29
is
an important
Comparing across two economies with
different degrees of risk aversion yields differences independently of aggregate constraints.
11
different
Z, the equilibrium in the robust economy and the benchmark economy are identical.
•
The
equilibria in the robust
Numerical example
5.3
Figure
The
Notes:
A B and Collateral Squeeze
1:
Xb
figure illustrate the equilibrium selection of
A (0.25) and X
4- 6
(0.75).
Z
is
on the horizontal
for
As the
axis.
Z <
Z_.
Xb
is
where
in the case
last
on average
/?
we
=
We
2 years.
first
Z
and always
falls,
lies
between
robust agents become
and therefore increase Xb above
A.
present a numerical example to quantitatively illustrate the workings of our model
We
1.
vertical axis
financial sector's collateral of
increasingly focused on the chance that the other market will receive a shock
In this subsection,
on the
study the fully robust economy.
choose A
=
We
choose 5
=
1/2, to imply that recessions
1/4. If recessions occur once every 8 years, this choice of
A implies
that the two-shock extreme event happens about every 48 years.
We
normalize
choose wq
Figure
\b begins
II
=
1.
In order to prevent
insurance by the agents,
we assume
=
0.3.
1
illustrates the effect of squeezing the financial sector's collateral
With
these parameters,
to rise above A until
Figure 2 illustrates the
K
effect of
that
is
that
wo^p <
1,
and
1.8.
maximum
we discussed
at A
+ 5. We
on As. As
Z
note that this behavior
falls
is
below Z,
qualitatively
earlier.
squeezing the financial sector's collateral on the agents' choice of x/y. For
the robust economy, x/y varies over
This average behavior of x/y
Z_=
reaches a
it
similar to the effect of increasing
As the
full
s.
We
present the probability-weighted average value of x/y over
representative of the behavior of x/y for moderate values of s
financial sector's collateral of
Z
falls,
(s
<
all s.
3 years).
robust agents become increasingly focused on the chance that
30
Figure
2:
x/y and Collateral Squeeze
Benchmark
Robust
The
Notes:
Z <
Z_.
x/y
figure contrasts the behavior of
is
on the vertical
Z
axis.
is
x/y between the robust economy and the benchmark economy, over the range
on the horizontal
probability-weighted average value of x/y over
moderate values of
s (s
<
As the
3 years).
on the chance that the other market
second. Agents in the benchmark
the
more
likely single-shock event.
For the robust economy, x/y varies over
The average behavior
financial sector's collateral of
will receive
a shock
economy account
Thus x/y
the other market will receive a shock
second. Agents in the
all s.
axis.
rises as
first,
first,
of
Z
recession.
When
for the fact that collateral
Z
falls for
the
earlier,
in the robust
x
is
representative of the behavior of x/y for
robust agents
become
increasingly focused
is
scarce,
first
or
and direct insurance purchases
to
and insure roughly equally against receiving the shock
benchmark economy account
for the fact that collateral
Z
falls for
is
scarce,
and
n.
When
there
lower in the robust
economy.
When
constraint imposes that x
is
the benchmark economy.
output responses to a typical single shock
one shock during the liquidity episode, output
economy than
in the
benchmark economy. As a
there are two shocks, output
+y= Z
in
is
or
first
direct insurance
there are no shocks during a liquidity episode, per-capita output in both the
is
present the
benchmark economy.
in insurance-portfolios translate into different
and robust economy
noted
falls,
is
We
and insure roughly equally against receiving the shock
purchases to the more likely single-shock event. Thus x/y rises as
These difference
x/y
s.
simply ^p-.
falls
result,
benchmark
to
^j5
output
is
-
As
lower
However, since the collateral
both benchmark and robust economy, the output upon two shocks
identical across both economies.
31
is
Thus, in terms of output, the benchmark economy does
robust
economy
suffers deeper "typical" recessions, because robust agents
two-shock extreme event. This "over-reaction"
example, when
of
GDP
An
6
Z
is
high and close to Z, at
more than the benchmark economy.
in the robust
than
in the
is
1.7,
than the robust economy. The
become overly concerned with the
more severe when aggregate
the robust
Instead,
economy
when Z
=
1.2,
sees
collateral
is
more binding. For
output decline by only one percent
the decline
is five
percent of
GDP
Extension to Risky Collateral
when
collateral
are amplified as financing conditions tighten, reducing
collateral values,
is
both an input into production as well as security
feedbacks emerge between the financial and real side. Negative shocks
for financial claims, potentially large
and reinforcing the
demand
for collateral as
a productive input, reducing
financial tightening.
can consider how Kiyotaki and Moore's insight applies within our model by assuming that the
eral of the financial sector is (exogenously) risky, but related to the shocks in
some understanding
gives us
more
benchmark economy.
Kiyotaki and Moore (1997) observe that
We
strictly better
market
A
collat-
and B. This exercise
of the effect of collateral risk, without working out a full-blown production
model.
6.1
Modeling assumptions
The only
uncertainty in the economy
is
number
in the
of shocks.
zero or one, then the total collateral of the financial sector
falls
to z
<
Z. This could happen, for example,
for collateral assets (such as
The
If
Z, but
assume that
if
if
the
number
of shocks
is
there are two shocks, then collateral
the second shock results in a
fall in
investment and
demand
real estate).
introduction of risky collateral requires us to rewrite the collateral constraint of insurers:
x
< Z
y
< max{z —
(32)
x, 0}.
(33)
only one shock hits the economy, insurers have enough collateral to cover up to x financial claims.
new
constraint
is
that
if
The
the second shock affects the economy, not only has some of the collateral of insurers
been pledged to cover the
7T
land or
if
is
We
first
shock, but also there
We also modify our model so that in
= 0. When it > 0, agents effectively are
is less
collateral in total.
the event of a liquidity shock, profits drop to
"endowed" with
7r
7r
>
rather than
units of first-shock and second-shock claims, so
that their decisions are taken net of this n. In particular their decisions over x(s) and y s (t) are as described
thus
far, less
n
(or zero,
if
the net
is
negative).
32
•
We
assume that
X'
Z<n< X + tmn{K,5} Z
6-mm{K,5}
^
Then
straightforward to show that for the benchmark case,
it is
the (low probability) second-shock that agents use
probability)
first
case of the fully robust economy,
=Z
be
K
>
we
5,
still
x-fr
y
to
revert to the case
where
(3
=
be
=
0.
demand second-shock
insurance,
and
in the particular
have that,
Z
=—
_, r
y
2
6.2
provides sufficient insurance against
of their wealth to purchase insurance against the (high
all
economy continue
In contrast, agents in the robust
we
tt
shock:
x
Finally,
-
Z
=—
2
0.
Amplification
Since in the benchmark economy, x be
=Z
agents recognize that the risk in collateral
and y be
is
=
0, collateral risk
has no
not relevant for their payoffs.
The
effect
events are sufficiently rare that forgoing insurance
In the robust economy, both
in the robust
x and y are
economy because when agents
if
there
is
amount
relevant
constraining the financial sector continues to be Z. Of course, by not insuring
experiences large ex-post (distributional) utility costs
on equilibrium because
y,
the
of collateral
benchmark economy
a two-shocks event.
Nonetheless, these
optimal.
is
The
positive.
risk in collateral values
has a significant impact
insure against two shocks, the relevant collateral constraint for
the single-shock shift to (33). Offering more y claims requires intermediaries to lock-up collateral. But this
reduces the
The
event
amount
of liquidity insurance available for x.
effective collateral for the robust
(z).
economy
is
the
amount
of collateral in the worst-case, two-shock
Relative to the benchmark, insurance against the typical single-shock event
falls
by at
least
Z — z,
amplifying the flight-to-quality problem we have identified.
6.3
Demand
Because
in the
for Assets
and Collateral Premia
robust economy agents evaluate insurers based on their collateral levels in the worst-case
scenario of two shocks, insurers have an incentive to shore
up
their worst-case collateral so as to increase
their capacity.
In a sense, the agents' robustness preferences induces "risk-aversion" in the insurers.
contrast, in the
benchmark economy the
event.
While
still
credibility of the insurer
averse to losses, the insurer
is
is
based on his collateral
more concerned about moderate
33
losses.
in the
In
one-shock
We
parameterize this effect by calculating the risk-premia that an insurer
amount
of the following
the occurrence of the
of the second shock.
We
two Arrow-Debreu claims:
first
The
A
shock; and (2)
A
(1)
will assign to
an infinitesimal
claim that delivers an extra unit of collateral on
claim that delivers an extra unit of collateral on the occurrence
objective probabilities of each of these events are
<t>\
and
<p2-
note that the claims are not typical Arrow-Debreu securities, since the events are not mutually
exclusive.
The
The second shock can occur only once
the
shock has occurred,
first
valuations of the two claims are denoted as £i0i and
£,24>2,
4>\
>
where the
intermediaries are willing to pay for claims on first-and second-shock events
(i.e.
4>2-
£'s are the price
premia
they are the premia on
these claims after accounting for the probability of the events).
In the
is
benchmark economy
we have
(recall that
set
P=
binding), the extra collateral in the second-shock event
On
the other hand, the extra collateral in the
selling
x claims.
claims.
Thus the
We
note in the equilibrium
price of
an x claim
In contrast, in the robust
first
&=
o.
shock
is
we have
and that
is
in this case the single
shock constraint
useless to the insurer so that
helpful since
it
loosens the collateral constraint on
constructed, agents pay
wq
in total to
buy
Z
insurance
u/q/Z. Then,
is
economy the extra
collateral in the second-shock event
is
valuable. Following
the same logic to calculate prices in the worst case scenario, the value of an extra unit of collateral
Then
the collateral-premium assigned to this claim by the insurer
/r
_ w
Z
Collateral in the first-shock event
subsumed by the
is
first-shock event
However, the probability of the
shock
is
The
insurer in the robust
1
economy, because the second-shock event
is
higher than the probability of the second-shock event, thus,
^^<e
Z
2
01
are ordered as follows:
&>
Proof. Follows directly from
is,
the extra collateral loosens the constraint in the worst-case scenario).
er =
Proposition 7 Collateral-premia
wq/ z.
4>2
also valuable in the robust
(i.e.
first
is
Z>
z
economy
and
is
(f>\
r
t(
>
e
>
£r
>
&=
0.
(j>2-
most concerned about the worst-case event and
the highest collateral-premia for claims that help him shore up the worst-case collateral
34
is
willing to
level.
pay
Moreover,
> wo/Z,
wq/z
so that the robust insurer always pays a higher collateral-premium than the
- even against
insurer
This pattern of prices
first-shock events.
is
in
keeping with an agent
benchmark
who
more
is
"risk-averse" against losses.
An
empirical regularity during flight-to-quality episodes
in risk
premia.
Demand
for riskless assets
between Commercial Paper and Treasury
is
money put options on
the
S&P500
and the
rise
The spread
index, or the
VIX
phenomena
are
the financial intermediaries
we
too stylized to replicate these observed patterns precisely, these
have identified are the marginal investors
plausibly
for safety
widens during these episodes. Volatility sensitive
Bills typically
consistent with increased aversion to the worst-case scenario.
broadly
demand
rise.
Although our model
If
markets, then their risk-valuations affect prices, and
in these asset
match the observed phenomena.
Our model
also provides a
new
perspective on the cyclical indicator property of the Commercial-Paper
to Treasury-Bills spread (see e.g., Stock
and Wilcox, 1993). In our model the
and Watson, 1989, Friedman and Kuttner, 1993, Kashyap,
rise in
the spread
is
a
symptom
Stein,
of agents' robustness concerns, as
But the robustness concerns trigger actions that lead to
opposed to a contributing factor to a recession.
underinsurance against the single-shock event, and
7
the increase in
such as Treasury Bills or bank deposits increases.
financial measures, such as the prices of out-of-the
implied volatility index,
is
may
thereby lead to a recession.
Final remarks
Flight to quality
is
a pervasive phenomenon that exacerbates the impact of recessionary shocks.
In this
paper we present a model of this phenomenon based on robust decision making by financial specialists.
show that when aggregate intermediation
collateral
is
plentiful, robustness considerations
with the functioning of private insurance markets (credit
intermediation collateral
is
scarce, they take
a
lines).
do not
We
interfere
However, when agents think that aggregate
set of protective actions to
guarantee themselves safety, but
which leave the aggregate economy overexposed to recessionary shocks.
In this context, a Lender of Last Resort policy
events.
The main
events,
which are very
benefit of this policy
rare,
but from
is
useful
if
used to support extreme rather than intermediate
comes not so much from the direct
its
effect of the policy
ability to unlock private sector collateral
during extreme
during milder, and far
more frequent, contractions.
The
implications of the framework extend beyond the particular interpretation
and policymakers. For example,
the policymaker as the
IMF
in the international context
or other IFFs. Then, our
we have given
one could think of our agents as countries and
model prescribes that the
IMF
not support the
country hit by a sudden stop, but to commit to intervene once contagion takes place.
35
to agents
The
first
benefit of this
policy comes primarily from the additional availability of private resources to limit the impact of the initial
pullback of capital flows.
36
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38
Financial System Risk and Flight to Quality
ABSTRACT
We present a model
of flight to quality episodes that emphasizes financial system risk
and the Knightian uncertainty surrounding these episodes. In the model, agents are
uncertain about the probability distribution of shocks in markets different from theirs,
treating such uncertainty as Knightian. Aversion to this uncertainty generates demand for
safe financial claims.
capital to cover their
It
also leads agents to require financial intermediaries to lock-up
own markets'
shocks in a manner that
is
robust to uncertainty over
other markets. These actions are wasteful in the aggregate and can trigger a financial
accelerator.
A lender of last resort can unlock private capital markets to
economy during
JEL
these episodes
by committing
to intervene
stabilize the
should conditions worsen.
Codes: E30, E44, E5, F34, Gl, G21, G22, G28.
Keywords: Locked
collateral, flight to quality, insurance, risk premia, financial
intermediaries, lender of last resort, private sector multiplier, collateral shocks, robust
control.
'.284
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