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Liquidity And Risk Management

Search-and-Matching Financial MarketS†
Liquidity and Risk Management
By Nicolae Gˆarleanu and Lasse Heje Pedersen*
This paper provides a model of the interaction
Not only does risk management affect liquid-
between risk-management practices and market
ity; liquidity can also affect risk-management
liquidity. Our main finding is that a feedback
practices. For instance, the Bank for International
effect can arise. Tighter risk management leads
Settlements (2001, 15) states, “For the internal
to market illiquidity, and this illiquidity further
risk management, a number of institutions are
tightens risk management.
exploring the use of liquidity adjusted-VaR, in
Risk management plays a central role in insti-
which the holding periods in the risk assessment
tutional investors’ allocation of capital to trad-
are adjusted to account for market liquidity,
ing. For instance, a risk manager may limit a
in particular by the length of time required to
trading desk’s one-day 99 percent value at risk
unwind positions.” For instance, if liquidation is
(VaR) to $1 million. This means that the trad-
expected to take two days, a two-day VaR might
ing desk must choose a position such that, over
be used instead of a one-day VaR. Since a secu-
the following day, its value drops no more than
rity’s risk over two days is greater than over one
$1 million with 99 percent probability. Risk
day, this means a trader must choose a smaller
management helps control an institution’s use
position to satisfy his liquidity-adjusted value at
of capital while limiting default risk, and helps
risk (LVaR) constraint. One motivation for this
mitigate agency problems. Phillipe Jorion (2000,
constraint is that, if an institution needs to sell,
xxiii) states that VaR “is now increasingly used
its maximum loss before the completion of the
to allocate capital across traders, business units,
sale is limited by the LVaR.
products, and even to the whole institution.”
The main result of the paper is that subjecting
We do not focus on the benefits of risk man-
traders to an LVaR gives rise to a multiplier effect.
agement within an institution adopting such con-
Tighter risk management leads to more restricted
trols, but, rather, on the aggregate effects of such
positions, hence longer expected selling times,
practices on liquidity and asset prices. An institu-
implying higher risk over the expected selling
tion may benefit from tightening its risk manage-
period, which further tightens the risk manage-
ment and restricting its security position, but as a
ment, and so on. This feedback between liquidity
consequence it cannot provide as much liquidity
and risk management can help explain why liquid-
to others. We show that, if everyone uses a tight
ity can suddenly drop. We show that this “snow-
risk management, then market liquidity is low-
balling” illiquidity can arise if volatility rises, or
ered in that it takes longer to find a buyer with
if more agents face reduced risk-bearing capac-
unused risk-bearing capacity, and, since liquidity
ity—for instance, because of investor redemp-
is priced, prices fall.
tions, losses, or increased risk aversion.
Our link between liquidity and risk manage-
ment is a testable prediction. While no formal
†Discussants: Dimitri Vayanos, London School of
empirical evidence is available, to our knowl-
Economics; Neil Wallace, Pennsylvania State University;
Manuel Amador, Stanford University.
edge, our prediction is consistent with anecdotal
evidence on financial market crises. For exam-
* Gˆarleanu: Wharton School, University of Pennsylva-
ple, in August 1998 several traders lost money
nia, 3620 Locust Walk, Philadelphia, PA 19104-6367, and
National Bureau of Economic Research, and Centre for
due to a default of Russian bonds and, simulta-
Economic Policy Research (e-mail: garleanu@wharton.
neously, market volatility increased. As a result,
upenn.edu); Pedersen: Stern School of Business, New York
the (L)VaR of many investment banks and other
University, 44 West Fourth Street, Suite 9-190, New York,
institutions increased. To bring risk back in line,
NY 10012-1126, NBER, and CEPR (e-mail: lpederse@
stern.nyu.edu). We are grateful for helpful conversations
many investment banks reportedly asked traders
with Franklin Allen, Dimitri Vayanos, and Jeff Wurgler.
to reduce positions, leading to falling prices and
193
194
AEA PAPERS AND PROCEEDINGS
MAY 2007
lower liquidity. These market moves exacerbated
holding ut shares of the asset incurs a holding
the risk-management problems, fueling the crisis
cost of d . 0 per share and per unit of time if he
in a similar manner to the one modeled here.
violates his risk-management constraint
We capture these effects by extending the
search model for financial securities of Darrell
(2) vart (ut [P(Xt )
1t 2 P(Xt)]) # (si)2,
Duffie, Gârleanu, and Pedersen (2005, forthcom-
ing, henceforth DGP). This framework of time-
where si is the risk-bearing capacity, defined by
consuming search is well suited for modeling
sh 5 s
¯ . 0 and sl 5 0. The low risk-bearing
liquidity-based risk management as it provides a
capacity of the low-type agents can be inter-
natural framework for studying endogenous sell-
preted as a need for more stable earnings, hedg-
ing times. While DGP relied on exogenous posi-
ing reasons to reduce a position, high financing
tion limits, we endogenize positions based on a
costs, or a need for cash (e.g., an asset manager
risk-management constraint, and consider both a
whose investors redeem capital).1
simple and a liquidity-adjusted VaR. Hence, we
We use this constraint as a parsimonious
solve the fixed-point problem of jointly calculat-
way of capturing risk constraints, such as the
ing endogenous positions given the risk-manage-
very popular VaR constraint,2 which are used
ment constraint and computing the equilibrium
by most financial institutions. Our results are
(L)VaR given the endogenous positions that deter-
robust in that they rely on two natural proper-
mine selling times and price volatility. Pierre-
ties of the measure of risk: the risk measure
Olivier Weill (forthcoming) considers another
increases with the size of the security position,
extension of DGP in which market maker liquid-
and the length of the time period t over which
ity provision is limited by capital constraints.
the risk is assessed. While the constraint is not
Our multiplier effect is similar to that of Markus
endogenized in the model, we note that its wide
K. Brunnermeier and Pedersen (2006) who show
use in the financial world is probably due to
that liquidity and traders’ margin requirements
agency problems, default risk, and the need to
can be mutually reinforcing.
allocate scarce capital.
We consider two types of risk management:
I.  Model
(a) “simple risk management,” in which the vari-
ance of the position in (2) is computed over a
The economy has two securities: a “liquid”
fixed time horizon t; and (b) “liquidity-adjusted
security with risk-free return r (i.e., a “money-
risk management,” in which the variance is
market account”), and a risky illiquid security.
computed over the time required for selling the
The risky security has a dividend-rate process X
asset to an unconstrained buyer, which will be a
and a price P(X), which is determined in equi-
random equilibrium quantity.
librium. The dividend rate is Lévy with finite
Because agents are risk neutral and we are
variance. It has a constant drift normalized to
interested in a steady-state equilibrium, we
zero, Et (X(t 1 T) − X(t)) 5 0, and a volatility
restrict attention to equilibria in which, at any
sX . 0, i.e.,
given time and state of the world, an agent holds
either 0 or u¯ units of the asset, where u¯ is the largest
(1) vart (X(t 1 T) − X(t)) 5 s2X T.
Examples include Brownian motions, (com-
1 An interesting extension of our model would consider
pound) Poisson processes, and sums of these.
the direct benefit of tighter risk management, which could
The economy is populated by a continuum of
be captured by a lower ld.
2
agents who are risk neutral and infinitely lived,
A VaR constraint stipulates that Pr(−u[P(Xt )
1t
2
P(X
have a time-preference rate equal to the risk-free
t )] $ VaR) # p for some risk limit VaR and some con-
fidence level p. If X is a Brownian motion, this is the same
interest rate r . 0, and must keep their wealth
as (2). We note that rather than considering only price risk,
bounded from below. Each agent is characterized
we could alternatively consider the risk of the gains process
by an intrinsic type i [ {h, l}, which is a Markov
(i.e., including dividend risk) Gt, t5 P(X(t 1 t)) − P(X(t))
T
chain, independent across agents, and switching
1 et X(s) ds. This yields qualitatively similar results (and
quantitatively similar for many reasonable parameters since
from l (“low”) to h (“high”) with intensity lu,
dividend risk is orders of magnitude smaller than price risk
and back with intensity ld. An agent of type i
over a small time period).
VOL. 97 NO. 2
LIquIDITY AND RISk MANAGEMENT
195
position that satisfies (2) with si 5 s¯, taking the
holding u¯, and the price P. Naturally, low-type
prices and search times as given.3 Hence, the set
owners lo want to sell and high-type non-owners
of agent types is T 5 {ho, hn, lo, ln}, with the
hn want to buy, which leads to
letters “h” and “l” designating the agent’s current
intrinsic risk-bearing state as high or low, respec-
(5) 0 5 22lmhn(t)mlo(t) 2lumlo(t) 1 ldmho(t)
tively, and with “o” or “n” indicating whether the
agent currently owns u¯ shares or none, respec-
and three more such steady-state equations.
tively. We let m (t) denote the fraction at time t of
Equation (5) states that the change in the fraction
z
agents of type z [ T . These fractions add up to
of lo agents has three components, correspond-
1 and markets must clear:
ing to the three terms on the right-hand-side of
the equation. First, whenever a lo agent meets a
(3) 1 5 mho 1 mhn 1 mlo 1 mln ,
hn investor, he sells his asset and is no longer a
lo agent. Second, whenever the intrinsic type of a
(4)
Q 5 u¯ (mho 1 mlo),
lo agent switches to high, he becomes a ho agent.
Third, ho agents can switch type and become lo.
where Q . 0 is the total supply of shares per
Duffie, Gârleanu, and Pedersen (2005) show that,
investor.
taking u¯ as fixed, there is a unique stable steady-
Central to our analysis is the notion that the
state mass distribution as long as u¯ $ Q. Here,
risky security is not perfectly liquid, in the sense
agents’ positions u¯ are endogenous and depend
that an agent can trade it only when she finds
on m, so that we must calculate a fixed point.
a counterparty. Every agent finds a potential
Agents take the steady-state distribution m as
counterparty, selected randomly from the set of
fixed when they derive their optimal strategies
all agents, with intensity l, where l . 0 is an
and utilities for remaining lifetime consumption,
exogenous parameter characterizing the mar-
as well as the bargained price P. The utility of an
ket liquidity for the asset. Hence, the intensity
agent depends on his current type z(t) [ T (i.e.,
of finding a type-z investor is lm , that is, the
whether he is a high or a low type and whether he
z
search intensity multiplied by the fraction of
owns zero or u¯ shares), the current dividend X(t),
investors of that type. When two agents meet,
and the wealth W(t) in his bank account:
where 1the ty 2
they bargain over the price, with the seller hav-
1 2
ing bargaining power q [ [0, 1].
(6) V X(t), W
,
z
t 5 Wt 1 1 z[{ho, lo} u
¯ X(t)/r1u¯ vz
This model of illiquidity directly captures
the search that characterizes over-the-counter
pe-dependent utility coefficients
(OTC) markets. In these markets, traders must
v are to be determined. With q the bargaining
z
find an appropriate counterparty, which can be
power of the seller, bilateral Nash bargaining
time consuming. Trading delays also arise due
yields the price
to time spent gathering information, reach-
ing trading decisions, mobilizing capital, etc.
(7) Pu¯ 5 (Vlo 2 Vln) (1 2 q) 1 (Vho 2 Vhn) q.
Hence, trading delays are commonplace, and,
therefore, the model can also capture features of
We conjecture, and later confirm, that the equi-
other markets such as specialist and electronic
librium asset price per share is of the form
limit-order-book markets, although these mar-
kets are, of course, distinct from OTC markets.
(8)
P(X(t)) 5 X(t)/r 1 p,
II.  Equilibrium Risk Management, Liquidity, 
for a constant p to be determined. The value-
and Prices
function coefficients v and p are given by a set of
z
Hamilton-Jacobi-Bellman equations, stated and
We now proceed to derive the steady-state
solved in the Appendix available at www.e-aer.
equilibrium agent fractions m, the maximum-
org/data/may07/p07048_app.pdf. The Appendix
contains all the proofs.
3 Note that the existence of such an equilibrium requires
that the risk limit s¯ not be too small relative to the total sup-
PROPOSITION 1: If the risk-limit s¯ is suffi-
ply Q, a condition that we assume throughout.
ciently large, there exists an equilibrium with
196
AEA PAPERS AND PROCEEDINGS
MAY 2007
holdings 0 and u¯ that satisfy the risk manage-
the equilibrium adjusting so that the constraint
ment constraint (2) with equality for low- and
is not violated. If an equilibrium exists, then a
high-type agents, respectively. With simple risk
stable equilibrium exists. Indeed, the equilib-
management, the equilibrium is unique and
rium with the largest u¯ is stable and has the high-
est welfare among all equilibria.
rs 1
!
The main result of the paper characterizes the
(9)
u¯ 5

.
equilibrium connection between liquidity and
sX
t
risk management.
With liquidity-adjusted risk management, u¯
PROPOSITION 2: Suppose that s¯ is large
depends on the equilibrium fraction of potential
enough for the existence of an equilibrium.
buyers mhn and satisfies !
Consider a stable equilibrium with liquidity-
rs
adjusted risk management and let t 5 1/(2lmhn),
(10)
u¯ 5
2lm
which means that the equilibrium allocations
s
hn.
X
and price are the same with simple risk man-
agement. Consider any combination of the
conditions (a) higher dividend volatility sX, (b)
In both cases, the equilibrium price is given by
lower risk limit s¯, (c) lower meeting intensity l,
(d) lower switching intensity lu to the high risk-
X
bearing state, and (e) higher switching intensity
(11) P(X
t
t) 5 r
1 2
ld to the low risk-bearing state. Then, (i) the
equilibrium position u¯ decreases, (ii) expected
search times for selling increase, and (iii) prices
decrease. All three effects are larger with liquid-
r
ity-adjusted risk management.
1 1ld 1 2 2lmlo 1 2 q 1 1lu 1 2 2
d
r 1 2 q 1 ld 1 2lmlo 1 2 q
2 r
lmhnq,

To see the intuition for these results, consider
the impact of a higher dividend volatility. This
where the fractions of agents m depend on the
makes the risk-management constraint tighter,
type of risk management.
inducing agents to reduce their positions and
spreading securities among more agents, thus
These results are intuitive. The “position limit”
leaving a smaller fraction of agents with unused
u¯ increases in the risk limit s
¯ and decreases in
risk-bearing capacity. Hence, sellers’ search time
the asset volatility and in the square root of the
increases and their bargaining position worsens,
VaR period length, which is t under simple
leading to lower prices. This price drop is due to
risk management and (2lmhn)21 under liquid-
illiquidity, as agents are risk neutral.4
ity-adjusted risk management. In the latter case,
With liquidity-adjusted risk management, the
position limits increase in the search intensity
increased search time for sellers means that the
and in the fraction of eligible buyers mhn.
risk over the expected liquidation period rises,
The price equals the present value of divi-
thus further tightening the risk-management
dends, Xt /r, minus a discount for illiquidity.
constraint, reducing positions, increasing search
Naturally, the liquidity discount is larger if there
times, and so on.
are more low-type owners in equilibrium (mlo is
This multiplier also increases the sensitiv-
larger) and fewer high-type nonowners ready to
ity of the economy with liquidity-adjusted risk
buy (mhn is smaller).
management to the other shocks (b)–(e). Indeed,
Of the equilibria with liquidity-adjusted risk
a lower risk limit (b) is equivalent to a higher
management, we concentrate on the ones that are
stable, in the sense that increasing u¯ marginally
would result in equilibrium quantities violating
4 In a Walrasian market with immediate trade, the price
the VaR constraint (2). Conversely, an equilib-
is the present value of dividends X/r when (Q/ u¯ ) , lu /
(l
rium is unstable if a marginal change in hold-
u 1 ld ), a condition that is satisfied in our examples
below. (When Q/ u¯ . lu /(lu 1 ld ), the Walrasian price is
ings that violates the constraint would result in
(X2d) /r .)
VOL. 97 NO. 2
LIquIDITY AND RISk MANAGEMENT
197
Figure 1
Note: The effects of dividend volatility on equilibrium seller search times (left panel) and prices (right panel) with simple
(dashed line) and liquidity-adjusted (solid line) risk management, respectively.
dividend risk. The “liquidity shocks” (c)–(e) do
with t 5 0.0086, which is chosen so that the
not affect the equilibrium position u¯ with simple
risk management schemes are identical for sX 5
risk management, but they do increase the sell-
0.3. Search times increase and prices decrease
ers’ search times and reduce prices. With liquid-
with volatility. These sensitivities are stron-
ity-adjusted risk management, these liquidity
ger (i.e., the curves are steeper) with liquidity-
shocks reduce security positions, too, because of
adjusted risk management due to the interaction
increased search times and, as explained above,
between market liquidity (i.e., search times) and
a multiplier effect arises.
risk management.
The multiplier arising from the feedback
between trading liquidity and risk manage-
ment clearly magnifies the effects of changes
REFERENCES
in the economic environment on liquidity and
prices. Our steady-state model illustrates this
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point using comparative static analyses that
Pedersen. 2006. “Market Liquidity and Fund-
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Duffie, Darrell, Nicolae Gârleanu, and Lasse Heje 
economy if there were random variation in the
Pedersen. 2005. “Over-the-Counter Markets.”
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Duffie, Darrell, Nicolae Gârleanu, and Lasse Heje 
Pedersen (forthcoming). In the context of such
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Studies.
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Bank for International Settlements.  2001. “Final
We illustrate our model with a numerical
Report of the Multidisciplinary Working
example in which l 5 100, r 5 0.1, X0 5 1,
Group on Enhanced Disclosure.” http://www.
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bis.org/publ/joint01.htm.
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¯ 5 1. Figure 1 shows how prices (right panel)
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and sellers’ expected search times (left panel)
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