Download the fragile capital structure of hedge funds and the

THE FRAGILE CAPITAL STRUCTURE OF
HEDGE FUNDS AND THE LIMITS TO
ARBITRAGE
X. LIU, A. S. MELLO
Centre for Hedge Fund Research, Risk Management Laboratory
Imperial College Business School
Working Paper 06
Date: 2009
The Fragile Capital Structure of Hedge Funds and the Limits to
Arbitrage
Xuewen Liu
Antonio S. Mello
Imperial College Business School
Imperial College Business School
Imperial College London
Imperial College London
February, 2009
Abstract
During a …nancial crisis, when markets most need liquidity and arbitrage trading to correct
prices, hedge funds reduce their exposures and positions. The paper explains this phenomenon in
light of coordination risk. We argue that the fragile nature of the capital structure of hedge funds,
combined with low market liquidity, introduces coordination risk to hedge fund’s investors.
Coordination risk e¤ectively limits hedge funds’ arbitrage capabilities. We present a model
of hedge funds’ optimal asset allocation with coordination risk. We show that hedge fund
managers behave conservatively and even give up participating in the market when they factor
coordination risk into their investment decisions. The model gives a new explanation to the
limits to arbitrage. We also discuss other implications of the model.
Contact: [email protected]; [email protected].
1
1
Introduction
“One fund-of-funds manager says he rushes to be the …rst out if he suspects that others may
desert a hedge fund...If managers worry that clients will bail out, they may try to raise cash in
anticipation.”
The Economist, October 25, 2008, page 87-88.
In the …nancial crisis of 2007-2008, hedge funds reduced their exposures to risky investments
and increased their cash holdings very quickly. The Economist estimates that between July and
August 2008 alone, the hedge fund industry’s cash holdings rose from $156 billion to a record
$184 billion, approximately 11% of the assets under management.1 The reallocation towards liquid
securities happened simultaneously with a drastic reduction in assets under management by hedge
funds, which declined by nearly 30% during 2008.2 The decline, the largest on record in a year, was
caused by a combination of negative performance and a surge in redemptions. Ironically, the poor
performance was self-in‡icted, as many hedge funds under pressure had to sell and close positions
to meet margin requirements and redemption calls, and the …re sale caused fall in fund value, in
turn generating a cycle of further redemptions and losses. Some hedge funds were forced to suspend
redemptions towards the end of 2008 to break down the vicious cycle of redemptions and losses.
It is possible that the evidence above - asset recomposition towards liquid securities - re‡ects
hedge funds’ astute grasp of investors’ sentiment and prevailing market conditions. However, we
could argue that precisely during times of high ambiguity and high volatility the opportunities for
arbitrage are more evident. Why then did hedge funds become much less aggressive in exploiting
price distortions when the opportunities to pro…t became all the more interesting?
In their quest for arbitrage gains, hedge funds also perform the important social role of guaranteeing that markets operate e¢ ciently. But such a role can only be achieved if hedge funds decide
to trade risky assets. It is therefore also socially ine¢ cient that hedge funds allocate an important
1
Banks, on the other hand, only started to switch systematically to riskless securities later, after the collapse of
Lehmann Brothers in September 2008. (Financial Times, March 31 2009, page 39. The Economist, October 25, 2008,
page 87.)
2
See "Hedge Funds 2009", International Financial Services London Research.
2
fraction of their portfolios to riskless securities, instead of pro-actively pursuing more arbitrage
opportunities in risky assets. Holding cash/riskless assets is synonymous to imposing limits on arbitrage activities. Cutting short on exposures during a crisis means that hedge funds underperform
the social role of liquidity provision and price formation precisely when markets most need correct
price signals for allocative purposes. In the absence of smart investors, …nancial markets can wander o¤ erratically, and risk falling into a vicious cycle that goes from lower price informativeness
to less liquidity and back, a process of continued deterioration that can prolong a …nancial crisis.
Thus, to analyze a modern …nancial crises it is necessary to understand the reasons that make
hedge funds limit their own arbitrage activities.
In this paper we ask why hedge funds, with a reputation for being aggressive investors seeking
high returns, hold non insigni…cant amounts of riskless assets in their balance sheet?3 And if
cash can be interpreted as negative debt, why then do hedge funds carry low leverage ratios in
comparison to investment banks, which are essentially like large hedge funds?4 Also, what are the
consequences to capital markets of hedge funds limiting arbitrage?
Until now, the …nance literature has attributed the existence of limits to arbitrage to problems
of asymmetrical information between investors placing their money with arbitrageurs and arbitrageurs managing investors’money. According to Shleifer and Vishny (1997), when investors do
not understand or observe perfectly the trades and positions taken by arbitrageurs, they can react
with their feet upon witnessing poor performance. Knowing that bad performance can lead to
redemptions, fund managers choose to abstain from taking positions that might lose in the short
term, even if a pro…t could be realized in the long run.5 This explanation is quite plausible. But
is it necessary to rely on asymmetrical information to support the conclusion that hedge funds
face limits to arbitrage? What happens if the assumption of asymmetrical information between
investors and hedge fund managers is removed? And what if the arbitrage positions in risky assets
3
Bates, Kahle and Stulz (2008) provide the evidence that U.S. non-…nancial …rms increase cash holdings in recent
years due to precautionary motives.
4
The hedge fund industry as a whole had ratios of assets-to-equity at the beginning of the crisis of 1.8, compared
to more than 20 in investment banks (The Economist, October 25, 2008, page 87).
5
Gromb and Vayanos (2002) build on the intuitions of Shleifer and Vishny (1997) and show that margin constraints
have a similar e¤ect in limiting the ability of arbitrageurs to exploit price di¤erences. Abreu and Brunnermeier (2002,
2003) provide the explanation of synchronization risk to market ine¢ ciency.
3
were virtually riskless?
In this paper we show that hedge funds - as representative arbitrageurs - still limit their actions
even when investors know perfectly well that the long-term performance of hedge funds’positions
is good. The reason is that although investors might feel comfortable about performance, they
might be concerned about the stability of a hedge fund’s equity when they do not know what other
investors will do. A coordination problem among investors in a hedge fund may lead investors to
withdraw when each investor suspects that other investors might redeem their holdings and as a
result of the hedge fund is forced to liquidate positions. This problem is particularly acute when
markets are illiquid and positions are closed at a discount. The possibility that the hedge fund is left
with insu¢ cient equity after liquidating positions to satisfy withdrawals makes investors in a fund
unequal, because it gives a clear advantage to …rst movers. Then, even investors with long-horizons
decide to withdraw, and the vicious cycle of redemptions and asset sales can trigger a disorderly
collapse of the hedge fund. The following quote by The Economist illustrates the problem at hand:
“One fund-of-funds manager say he rushes to be the …rst out if he suspects that others may desert
a hedge fund...Since then the vicious cycle of forced sales to fund anticipated or actual redemptions
has only intensi…ed.”This risk of coordination and market illiquidity must be taken into account by
prudent hedge fund managers, who then limit their arbitrage activities, reduce leverage and hold
liquid assets, which serve to both honour redemptions at little cost and reassure investors that the
fund is robust.
We use global game techniques to model the asset allocation decision of a hedge fund that is
subject to the possibility of a run by its investors. In the benchmark equilibrium, we assume that
there is no coordination problem. A random number of investors need to redeem early for exogenous
reasons. The random size of the early redemptions requires the hedge fund to decide how much cash
it needs to hold ex ante. The trade o¤ faced by the hedge fund is between lower liquidation costs
if it holds more cash, on the one hand, and achieving a higher return if it holds risky assets, on the
other hand. This trade o¤ gives an optimal level of cash holdings. In the presence of coordination
risk, however, we can show that hedge funds choose to hold a non insigni…cant fraction of liquid
securities. Coordination risk a¤ects the optimal cash position of a fund in two ways: First, fear
makes more investors redeem even if they do not have to redeem. To satisfy redemptions at a
4
minimum cost, hedge funds optimally choose to hold more cash.6 Second, the cash holding by
itself in‡uences investors’decisions and a¤ects the degree of coordination risk and the probability
of panic run. In fact, the probability of run is a decreasing function of cash holding. Intuitively,
cash holding can assure investors and this reduces the probability of run. The fund managers take
investors’response into account and hold more cash.
Our model provides a new explanation to the limits of arbitrage. We do not emphasize the
ignorance of investors about the performance of the hedge fund, as Shleifer and Vishny (1997).
Instead, we choose to focus on frictions that result from market illiquidity and the unstable nature of
the equity capital of hedge funds. The combination of fragile equity and low market liquidity implies
that an investor in a hedge fund cares not only about the fundamental value of the assets under
management, but also about the actions of other investors. Both strategic risk (i.e. uncertainty
about what other investors will do) and fundamental risk have an impact on investors’likelihood of
redemption. We show in this paper that strategic risk makes hedge funds behave conservatively and
limit arbitrage, while Shleifer and Vishny (1997) explain the limits of arbitrage with fundamental
risk. Alternatively, in Schleifer and Vishny (1997) the limits to arbitrage occur because investors
are concerned about the asset side of the balance sheet of a hedge fund, i.e. the asset quality.7
In contrast, in our model, investors worry about the liability side of the balance sheet. Limits
to arbitrage due to strategic risk are particularly relevant during a …nancial crisis, when investors
become in general nervous and even behave irrational, causing severe coordination problems. We
believe that the full explanation for the limits to arbitrage requires the two motives to be taken
simultaneously.
Brunnermeier (2009) discusses the funding-liquidity reason for a run and the fundamental reason. In our model, the realization of early large withdrawals that force a hedge fund to incur in
large …re sales that depress the fund value end up by triggering a run. In contract, the run can be
driven by fundamentals as in Goldstein and Pauzner (2005). Bad fundamentals lead to insu¢ cient
asset values necessary to pay investors who withdraw late and this causes an immediate run. In
the recent …nancial crisis, capital markets experienced many runs triggered by funding-liquidity
reasons, rather than fundamental reasons, given that investors ran on many …nancial institutions
6
7
In the model part, we address this subtle issue. The hedge fund may reduce cash holding in this case.
In fact, for the mutual fund industry, Johnson (2006) …nds that existing investors do not redeem more following
bad performance (the indicator of fundamental).
5
that were fundamentally sound, at least with asset fundamentals and capital requirements that appeared adequate before collapse. Even so, investors still decided to run, worried that the liquidity
of these …nancial institutions was insu¢ cient to meet massive withdrawals and assets would have
to be liquidated at a deep discounts.
It is interesting to contrast the behavior of hedge funds with that of investment banks before
and during the …nancial crisis of 2007-08. Before the crisis, on average hedge funds operated with
much less leverage than investment banks. Hedge funds also tended to hold higher levels of cash
balances and more liquid securities in their portfolios than investment banks. During the crisis,
hedge funds reduced their exposures drastically and quickly. We argue that the di¤erences in
investment and …nancing policies of hedge funds, relative to banks, are partly due to the di¤erent
nature of the capital structure of the two institutions. The equity of hedge funds is fragile since
it can be redeemed at investors’discretion, while that of banks is locked permanently and cannot
be tendered back to the bank for redemption. The fragile capital structure of hedge funds can
work as an e¤ective market disciplining mechanism, since it indicates that hedge funds should run
less risky portfolios and operate with lower leverage ratios than banks. Needless to say that the
conservative balance sheet of hedge funds, in turn, make them more resilient than banks to liquidity
shocks. This helps to understand why banks rather than hedge funds are at the center of the recent
…nancial crisis. Many observers seem to have been surprised that hedge funds, typically regarded
as the riskiest segment of institutional investors, are in much better shape than banks. Since banks
were regulated and hedge funds not, it is quite possible that market discipline worked better than
government regulation in limiting exposures. In sum, while the fragile capital structure in hedge
funds encourages micro-prudence, since it imposes limits to arbitrage, it creates the side-e¤ect of
curtailing the social role these investors perform in keeping capital markets e¢ cient, and therefore
it leads to macro-unsoundness.
Our paper relates to a large body of literature on bank runs. Gorton and Winton (2003) provide
an excellent survey in this …eld. Gorton (2008) discusses in detail the panic runs in the recent crisis.
The setup of our model is similar to that in Diamond and Dybvig (1983). The di¤erence is that we
assume that the size of early withdrawals is ex-ante random, and this allows us to study the optimal
cash holding problem of hedge funds. In general, there are multiple equilibria in a run, but we
resort to global game techniques to solve for the unique equilibrium solution. In this respect, our
6
paper is close to the work of Goldstein and Pauzner (2005),
8
but the purpose and the aim of the
two papers di¤er signi…cantly. We wish to study the optimal cash holding in hedge funds’assets,
while Goldstein and Pauzner study the optimal liquidity provision of banks (short-term interest
rate).
Our work also relates to that by Chen, Goldstein and Jiang (2007). These authors …nd that
investors’‡ows out of mutual funds are signi…cantly more sensitive to poor performance in the case
of funds that invest in illiquid securities than of funds that invest in liquid securities. This insight
is similar to our result that market illiquidity can potentially generate a panic run. Compared
to their work, we explicitly model how hedge fund managers factor the ex-post panic run into
their ex-ante investment and …nancing decisions. Stein (2005), in the same vein as Schleifer and
Vishny (1997), argues that open-end type of funds may be suboptimal as a result of competition,
because the early withdrawals in response to bad short-term performance limits their discretion to
invest. In this paper, the emphasis is in the mechanisms that limit arbitrage. Our argument that
limited arbitrage can be due to a coordination problem and a panic run di¤ers from the arguments
presented in this set of papers. In a banking context, Shin and Morris (2008) also use the argument
that cash holding by a debtor bank can lower the threshold for coordination of non-withdrawing
creditor banks. However, they do not formalize the argument, and we develop the argument further
by studying the optimal asset allocation problem of hedge funds and the limits to arbitrage.
The paper is organized as follows. We …rst describe the model setup. Next we solve for the
optimal asset allocation of a hedge fund with and without a coordination problem between investors
in the fund. After that we analyze the model’s implication - limits to arbitrage. In the discussion
part, we generalize the model and discuss some broader issues. In the …nal section we conclude
and add some additional remarks.
8
The global game methodology has been used in various contexts in literature. Rochet and Vives (2004) also
apply global game to study bank run. Other applications include currency crises (Morris and Shin (1998), Corsetti,
Dasgupta, Morris, and Shin (2004)), contagion of …nancial crises (Dasgupta (2004), Goldstein and Pauzner (2004)),
and market liquidity (Morris and Shin (2004), Plantin (2009)).
7
2
The model
2.1
The setup
This is a four-date model: T0 , T 1 , T1 and T2 . All agents are risk-neutral. There is no discount
2
factor between T1 and T2 :
Hedge fund
Consider a simple setup of a hedge fund which depends on its investors for funding. These
investors are the limited partners of the fund. The fund begins at T0 and ends at T2 . Investors
have the right to redeem their investment at an intermediate date, T1 . We assume that the total
amount of assets under management of the fund at T0 is 1, that there is a continuum of investors
with measure 1, and each investor contributes 1 unit of capital. We model the liquidity risk of
the hedge fund on its liability side as following: Each investor in the hedge fund is either be an
‘early investor’ or a ‘late investor’. Early investors are investors who, although initially have the
intention of staying for a long time, end up by redeeming their shares at T1 , for reasons that we
leave unspeci…ed, but could be related to consumption as well as portfolio allocation reasons, or due
to other …nancial emergencies. The probability for an investors to be an ‘early investor’, depending
on the states of economy, is ex ante a random variable uniformly distributed with support
8
2 [0; ]
where 0 <
< 1. That is, in aggregate, the proportion of investors that end up being early investors
ex ante is given by the random variable .
At T0 , the hedge fund needs to make the asset allocation decision - how much capital X is
invested in cash (C). The remaining capital, 1
X, is invested in illiquid assets (AL ). In order to
distinguish our argument from that in Schleifer and Vishny (1997), we assume that illiquid assets
have a (gross) return of R at T2 , without any uncertainty, where R > 1. However, if illiquid assets
are liquidated early, at T1 , it is sold at a discount, where the discounted price at …re sale is , with
0<
< 1. Figure 1 shows the balance sheet of the hedge fund. Notice that there is no leverage,
only equity. Later, we will discuss the situation of a levered hedge fund.
Investors
At T 1 the uncertainty about investors’status is resolved. Speci…cally, at that time, each investor
2
knows whether he is an ‘early’ investor or not, but the investor does not know the status of the
other investors. Nevertheless, ‘late’ investors receive signals about the state of the economy or
i
=
is independent from
j
the aggregate number of ‘early’ investors ( ). More speci…cally, late investor i0 s signal is
+ i , where
i
is uniformly distributed variable with support [
; ].
i
for i 6= j. Further, at T 1 , investors are perfectly informed of the cash position X and market
2
depth . The position of the fund in liquid assets can be known either because the fund discloses
it in a regular letter to investors, or it is easily obtained since many investors in hedge funds
are themselves professional investors and have access to the management of the fund (the general
partners). Usually, hedge funds are very careful about disclosing trading strategies and about
positions in speci…c assets, but do not raise obstacles about disclosing their allocation by general
asset classes.
At T 1 , all investors need to make a decision about whether they stay with the fund until T2 ,
2
or redeem their shares at T1 , in which case they inform the fund about the withdrawal. All early
investors have to redeem. A late investor might decide to redeem if he thinks that too many other
investors will redeem, because too much withdrawal will force the hedge funds to liquidate illiquid
investments at a cost. It is this coordination problem among late investors that we want to focus
in this paper.
9
Let us consider the payo¤ structure of an investor, be that an early or a late investor, between
redemption and non-redemption. Suppose that the aggregate number of investors that redeem at
T1 is given by s, where 0
s
1.
The payo¤ function for an investor to redeem is
8
< 1
wR (s) =
: X+(1
if 0
X)
s
s
if X + (1
X + (1
X)
X) < s
(1)
1
In the expression (1), note that investors who withdraw their capital at T1 receive a share of the
hedge fund’s net asset value. At T 1 , when investors give notice of their redemptions to the fund,
2
the hedge fund’s mark-to-market net asset value is 1, rather than the amount X + (1
X) shown
in expression (1). This is because the hedge fund has a cash cushion on balance sheet, and as long
as it has not started to liquidate illiquid assets, the marking-to-market value of fund is 1. Investors
who withdraw are paid based on a proportion of total value 1. However, in case that too many
withdrawals occur, once the hedge fund exhausts its cash balances, X, it starts to liquidate its
illiquid assets. The realized sale price of illiquid asset is only , less than 1.9 When this happens,
9
We can assume that when the hedge fund sells its illiquid assets, the fund faces a special ‘downward-sloping’
demand curve. Speci…cally, the market can absorb only a tiny amount of the asset selling at a price of 1. After
10
the realized value of an investor’s share of the assets is less than his claim. In order to ful…ll early
investors’claims, the fund has to liquidate part of late investors’shares of the assets. It is possible
that the fund exhausts all assets at T1 , in which case late investors are left with nothing. This
…rst-mover advantage of early withdrawal is the fundamental reason for a run on hedge funds and
on mutual funds [see Brunnermeier (2009) for details].
The payo¤ function for an investor that does not redeem at time T1 (i.e. stays) is
wS (s) =
8
>
>
>
<
>
>
>
:
(X s)+(1 X)R
1 s
(1 X) s X
R
1 s
0
if 0
s
X
if X < s
X + (1
X)
if X + (1
X) < s
.
(2)
1
In expression (2), when the aggregate number of investors redeeming at T1 is less than X, the
fund does not need to liquidate any long-term illiquid asset. So the total realized value of fund at
T2 is (X
s) + (1
X)R. However, if the aggregate number of investors withdrawing is higher
than X, the fund has to sell some illiquid assets at T1 . The number of units of the illiquid asset
needed to sell is
s X
, where s
X is the amount of cash short to meet redemptions. If the number
of withdrawing investors is higher than the total liquidation value of the fund, X + (1
X) , the
hedge fund is fully liquidated and there is nothing left after T1 - the third line in expression (2).
As in Diamond and Dybvig (1984) and Shleifer and Vishny (1997), in our model we assume
that the hedge fund is not able to replace the redeeming investors with new investors.10
We de…ne the di¤erence between the payo¤s of not withdrawing at time T1 and withdrawing
as:
w(s)
wS (s)
wR (s).
Figure (2) shows the payo¤s wS (s) and wR (s), and Figure (3) shows
absorbing that amount, the demand price immediately drops to
with constant price
w(s).
and the demand curve becomes perfectly elastic
. Therefore, before the sale, the mark-to-market value of the fund is 1, but at liquidation it
becomes X + (1 X) .
10
In Shleifer and Vishny (1997) new investors are reluctant because they do not observe the quality of the assets of
the hedge fund. In our model, we can rationalize this assumption in the following way: the actual number of investors
withdrawing and thus the actual amount of re…nancing required are unobservable by outsiders. The fund managers
can always claim that they need money and in fact they can use the money for other purposes and can even misuse
the funding. The Ponzi scheme that Mado¤ put in place for mnay years is an example of how this could happen.
11
Figure (4) describes the timeline.
2.2
The equilibrium
In this subsection we solve for the equilibrium and determine the optimal asset allocation of the
hedge fund.
12
2.2.1
The benchmark case: no coordination problem among hedge fund investors
In the benchmark equibrium, we assume there is no coordination problem for late investors. All
investors that inform the fund at T 1 that they will redeem at T1 are genuinely early investors. Late
2
investors do not withdraw. We work out the optimal amount of cash holding for the hedge fund
by backward induction from T 1 to T0 .
2
Like the setup in Shleifer and Vishny (1997), the hedge fund’s objective is to maximize the
fund’s total equity value at T2 . This objective is consistent with a hedge fund manager who tries
to maximize the value of the assets under management. The …nal value of the fund is an indicator
of the fund’s performance and a¤ects its future capacity to raise funds. If the redemption shock
is small and the cash holdings X are su¢ cient to cover the withdrawals by investors at T1 , the
total equity value of the fund at T2 is (X
) + (1
X
is bigger than X, the fund has to liquidate minf
value at T2 is [(1
> X + (1
W I (X)
X)
X
]R if 1
X
X
X)R. By contrast, if the redemption shock
;1
(or
Xg units of illiquid assets and the equity
X + (1
X)), and 0 if 1
X<
X
(or
X)). Therefore, the expectation of fund’s equity value at T2 can be expressed as
Z
= f
X
1
[(X
) + (1
X)R]d +
Z
min[ ;X+(1 X) )]
[(1
X
X)
X
0
]Rd g.
(3)
The problem of the hedge fund manager is to choose an X at T0 that maximizes the expected
value of the equity in the fund at T2 : M axW I (X). In the maximization problem, the trade-o¤ is
X
between return on investment and the cost of liquidating illiquid assets. More X reduces a costly
sale of illiquid positions in case redemptions are unexpectedly large. However, more X wastes
investment opportunities and foregoes the high return R in case the redemption shock is small .
The trade-o¤ leads to a unique optimal cash holding, X
Theorem 1 If 0 <
R
R(2
) 1,
survives any redemption shock
the optimal amount of cash holding is X =
2 [0; ], where
optimal amount of cash holding is X =
within
2 [0; X + (1
X ) ]
R(1
)
R(2
) 1.
X + (1
13
R
R(2
) 1
. The fund
<
< 1, the
The fund survives if the redemption shock falls
[0; ], where X + (1
Proof: see the Appendix.
X ) . If
R R
R
X ) < .
2.2.2
Equilibrium when there is a coordination problem among hedge fund investors
When the cash holdings X are lower than the realized redemption shock, , a costly sale of illiquid
assets is unavoidable. The …re sale depresses the fund’s net asset value and potentially causes a
run by investors. The fund manager anticipates the possibility of the run and therefore takes the
run into account in deciding the fund’s cash balances.
Again, we solve the equilibrium by backward induction, from T 1 to T0 . In the …rst step, we
2
work out investors’decisions at T 1 for a given cash balance X. In the second step, we go back to
2
T0 and solve the fund’s optimal cash holding X by taking investors’response into consideration.
Investors’decision at T 1 given the cash holding X
2
At T 1 , late investor i’s decision rule can be written as a map: ( i , X) 7 ! (Withdraw, Stay),
2
where ( i , X) is the information set for investor i and (Withdraw, Stay) is the choice.
We need to work out investors’decision rules that form an equilibrium.
We …rst examine late investors’ strategies when the state of economy
is extremely low or
extremely high. We have following two extreme cases.
1) For the lower dominance region of
2 [0;
L
), staying is the dominant strategy for late
investors. Following Goldstein and Pauzner (2005), we modify the technology for a very low .
Speci…cally, we assume that when the proportion of early investors (the state of economy)
within [0;
i.e.
L
) (where
L
is close to 0), there is no discount in the sale price of the illiquid asset,
= 1. With this modi…cation, we can conclude that if a late investor knows
region [0;
L
falls
is in the
), she doesn’t run whatever he believes other investors will do. This is because no
discounting in selling price means that early withdrawals will not erode late investors’shares and
late investors don’t have disadvantage in withdrawing at T2 . In most of our analysis, " is taken to
be arbitrarily close to 0, so late investors are sure that
is in [0;
L
). Therefore, staying is their
dominant strategy.
2) Considering that s = X + (1
U
X)
(R 1)
R
solves the equation wS (s) = wR (s), we de…ne
s . Then, for the upper dominance region of
14
2 (
U
; ], withdrawing is the dominant
strategy for late investors. The reason is as follows: when
>
U
, it means that there are a
su¢ cient number of redemptions by (early) investors, and even if all late investors choose to stay,
the payo¤ a late investor gets if he decides to redeem early is higher than the payo¤ if he decides to
stay. If a late investor is sure that
>
U
, he would redeem whatever his belief about the actions
of other late investors. As " is taken to be arbitrarily close to 0, late investors are informed when
>
U
. Therefore, if
is in in the interval (
U
; ], withdrawing is the dominant strategy for late
investors. In our model, we will show that it is not optimal for the hedge fund to choose cash
holdings X such that X + (1
X)
(R 1)
R
. Therefore, the condition
our model. That is, the upper dominance region (
For the intermediate region of
2 [
L
;
U
U
>
U
is always true in
; ] exists.
], late investors’ decisions depend on their beliefs
about other late investors’actions. The signals regarding
form their beliefs. Our main interest
is in the threshold strategy equililirium where a late investor’ strategy depends on his signal.
Speci…cally, the threshold strategy equilibrium is the equilibrium
where every late investor sets a
8
< Withdraw i > (X)
i
threshold (X) and uses the threshold strategy ( , X)7 !
, such that,
i
: Stay
(X)
given that all other late investors set the threshold as (X), it is optimal for a late investor himself
to do that. We prove that there exists a unique threshold equilibrium for our model.
In the following, we use global game techniques to solve for the unique threshold equilibrium. It
is important to note that in this paper we do not consider equilibria other than threshold strategy.
But with the assumptions of existence of two extreme dominance regions of
made above, it is
possible to give a tight proof as Goldstein and Pauzner (2005) that the threshold equilibrium is the
only equilibrium.
The equilibrium result is summarized in the following theorem.
Theorem 2 The model has a unique threshold equilibrium. In the equilibrium, every late investor
redeems if and only if his signal is above the threshold
(X).
We proceed to prove the theorem in several steps.
First, suppose that every late investor uses the threshold strategy and sets the threshold as
(X). We want to determine the total number of investors (both early and late ones) who decide
15
to redeem at T1 for a given realization of
+
proportion of late investors running is
and the threshold
, the
. Hence, the total number of investors redeeming,
2
denoted by s( ,
s( ,
(X)),
is
8
>
>
>
<
(X)) =
+ (1
>
>
>
:
. Given the realized
When
if
)
+
(X)
if
2
1
(X)
(X)
if
+ , s( ,
(X) +
.
(4)
(X) +
(X)) has two components. The …rst component is the
redemption by early investors and the second one is by late investors that change their minds and
decide to redeem early.
Second, for the late investor i, after he receives the signal
in his eyes is uniform in the interval [
i
i
;
i
, the posterior distribution of
+ ]. Therefore, his expected net payo¤ of staying
versus redeeming is
i
( ;
)=
1
2
Z
i
+
w(s( ;
))d .
i
Third, for the marginal late investor who receives the threshold signal
with redeeming and staying. Therefore, (
Z +
1
w(s( ; ))d = 0.
2
In our model, we focus on the case of
;
, he should be indi¤erent
) = 0. That is,
(5)
being arbitrarily small:
! 0. This assumption is to
make the model tractable and to highlight the main insight of paper. When
! 0, fundamental
uncertainty disappears while the strategic uncertainty is unchanged (see Morris and Shin (1998)).
In our context, from the marginal late investor’s perspective,
! 0, the uncertainty of the …rst element of s( ;
As a whole, s( ;
falls within [
idea. When ! 0, the blue curve, which depicts function s( ;
16
+ ]. When
) diappears while the second element does not.
) is uniformly distributed within the interval [
points A and B.
;
; 1]. Figure 5 illustrates the
), becomes a straight line between
Based on the uniform distribution of s( ;
Z 1
w(s)ds = 0.
lim
), we can rewrite expression (5) as (6):
(6)
!0
From (6), we can solve for the unique
. Geometrically,
6 that equates the shaded area below and above the x-axis.
17
corresponds to the point in Figure
Finally, to show that
(X) indeed forms a threshold equilibrium, we need to show that the
late investor i who receives a signal higher (lower) than
( ( i;
[
i
) > 0). In fact, for the late investor i’s perspective,
;
i
+ ]. If
i
>
, as ! 0, s( ;
( i;
prefers to run (stay):
) < 0
is distributed within the interval
) is uniform in the interval [
i
+ (1
i
)
i
; 1] and has
2
positive probability measure at 1. This is equivalent to take away a positive part from the shaded
area above the x-axis in Figure 6 and add a negative part of the shaded area. Hence ( i ;
A similar argument applies to the case when
i
<
.
With the proof of Theorem 2 we have also shown how to …nd out the threshold
guarantee the existence of
always to the right of X (i.e.
unique
(X) and
)+
(X). To
(X), we need to …gure out the parameter conditions. Theorem 3
establishes su¢ cient conditions for the existence of
Theorem 3 If (R
) < 0.
R
(1
(X), and also makes sure that
(X) is
(X) > X).
) log(1
)>1
and
> X + (1
X)
(R 1)
,
R
there exists a
(X) > X.
Proof: See Appendix.
Further, we want to make sure that
(X) is increasing in X, so that the higher the balance
of cash X, the higher is the threshold for redemption by late investors (running). However, in our
18
model, 4w(s) is not strictly monotonic decreasing in s. The proof is not trivial. We …nd out the
necessary and su¢ cient conditions to guarantee that
(X) is an increasing function with respect to X when
Theorem 4
solves ( R
(X) is increasing in X.
1)(
b) +
R
(1
) log( 11
b)
=
R
log( 11
b)
< log
(where b
log ).
Proof: See Appendix.
As we have obtained the unique threshold and because
investors run if
>
(X) and none runs if
! 0 , we conclude that all later
(X). Therefore, the total number of investors
redeeming their investments from the hedge fund can be expressed as
8
<
if
(X)
s( ) =
.
(7)
: 1 if > (X)
The fund’s optimal cash holding decision at T0
Now we go back to T0 and look at the hedge fund’s optimal cash holding decision by taking the
run into consideration. To save space and simplify the analysis, we focus on the case
For the case
R
R(2
) 1,
X)
(R 1)
R
R
R(2
) 1.
the result and the main insight do not change.
We analyze two regions of X, in order: i) X 2 fX jX + (1
jX + (1
>
X)
(R 1)
R
< g; and ii) X 2 fX
g. In the …rst region, X is low and the only equilibrium is the threshold
equilibrium. In the second region of X, we make a weak assumption and assume the investors
coordinate and choose the more e¢ cient equilibrium - the staying-strategy equilibrium. That
is, late investors ignores their signals and set the strategy as ‘staying’ for whatever signals: ( i ,
X)7 !Stay.11
We begin the analysis from the low range of X, i.e. X 2 [0; X C ), where X C +(1 X C )
(R 1)
R
=
. In the region of X, investors use the threshold strategy in equilibrium. Because investors use the
threshold strategy and set the threshold at
11
(X), the fund knows that if the realized redemption
If investors still use the threshod equilibrium for the second region of X, it only strengthens our result.
19
shock is larger than
(X), all late investors will redeem and desert the fund, driving it net asset
value to zero. That is, the surviving window of fund is given by
2 [0;
(X)].12
The expectation value of the fund is
W II (X)
1
= f
Z
X
[(X
) + (1
X)R]d +
Z
(X)
[(1
X
X)
X
0
]Rd g for X 2 [0; X C ) (8)
Since W II (X) is a continuously and bounded function with respect to X in the support [0; X C ),
the optimization problem M axW II (X) has a solution. Speci…cally, the …rst order condition (FOC)
of expression (8) is
[(1
R
)X +
1
(X)R] + [(X + (1
X)
(X)]
R
d
(X)
dX
= 0.
(9)
The tradeo¤ mechanisms in (9) is as follows: The …rst term on the left hand side of (9) is the
e¤ect of increasing the amount X on fund value, given the surviving range [0;
(X)]. This term
captures the tradeo¤ without the e¤ect of a run. This tradeo¤ is between having a high return (R)
and incurring large …re sale costs. This tradeo¤ has been examined before when we looked at the
benchmark equilibrium. As X increases, the …rst term is negative, because
(X) increases at a
lower rate than X, and a very high X negatively impacts the optimal tradeo¤ as in the benchmark
case. The second term of (9) is new: it represents the e¤ect of increasing the surviving window for
on the value of fund. This is always positive: increasing the amount of cash X in the portfolio
of the hedge fund increases the surviving range [0;
(X)]. Overall, for some parameter values, the
sum of the two terms equals zero for some X. Thus, we have a unique interior solution of X. We
denote this unique interior solution as X .
Theorem 5 There exists a unique optimal amount of cash holding in the hedge fund’s balance
sheet, X , in the interval (0; X C ) that maximizes the hedge fund value W II (X).
We aim to prove that the optimal cash holdings when there is the possibility of run are higher
than the the optimal amount of cash holdings when a run does not exist, that is, X
12
By the properties of
(X), the inequality
limit of the surviving window is min[
(X) < X + (1
(X); X + (1
X) ; ] =
20
X)
(X).
(R 1)
R
<
>X .
is valid and, therefore, the upper
We look at what happen if the fund still keeps the cash holdings as X . Since
know from Theorem 1 that X + (1
we have X + (1
X )
(R 1)
R
X ) < . Because X + (1
X )
(R 1)
R
>
R
R(2
< X + (1
) 1,
we
X ) ,
< . Therefore, we conclude that X indeed lies in the …rst region.
Furthermore, by the property of
(X), we have
(X ) < X + (1
the run e¤ect, the surviving range of the fund shrinks to [0;
X ) . That is, because of
(X )] from [0; X + (1
X ) ].
.
Because an increase in X can enlarge the surviving range (i.e.
(X) increases with X), the
hedge fund may need to distort the previous optimal tradeo¤ X as in the benchmark to achieve
a larger surviving range and maximize the fund value. We evaluate the …rst derivative of W II (X)
at X = X and check the change of W II (X) when X is slightly changed:
dW II (X)
jX=X
dX
= [(1
R
)X + 1
(X)R]jX=X + f[(X + (1
X)
(X)]
R
d
(X)
dx gjX=X
.
(10)
An increase in X at X = X has two e¤ects on W II (X). The second term in (10) is positive,
which is the dominant e¤ect. It captures the e¤ect of gaining from increasing the surviving range.
Nevertheless, interestingly, the …rst term of (10) is negative. The economic intuition for the …rst
term is the following: Given the e¤ ective maximum shock being X + (1
holdings of cash is X . Since the e¤ective maximum shock decreases to
X ) , the optimal
(X ) , the optimal cash
holdings should be lower. But this e¤ect is of second order compared to the …rst e¤ect under some
parameter values. Therefore, overall,
dW II (X)
jX=X
dX
> 0, meaning that the fund can increase its
value at T2 by increasing its holdings of cash.
Theorem 6 The optimal cash holdings X
(when a run is possible) is higher than X (when a
run is not taken into account).
Proof: See Appendix.
Theorem 6 is a key result of the paper. Due to the fact that the equity capital of the hedge
fund is a fragile commitment from its limited partners, because these can ask for redemption if
they believe that the fund may not have enough assets to pay them back, the optimal amount of
cash the hedge fund holds should be always higher than without run. It is important to emphasize
that there are two e¤ects by which ex-post runs impact on optimal cash holding ex ante. The …rst
21
one is the response e¤ect, which is the …rst term on LHS of (9). Because of the run e¤ect, the
total redemptions not only include the early investors but also late investors. That is, the size of
total redemption is changed because of a run. Hedge fund manager, in response to the changed
redemption size, has to adjust the optimal cash holding. The second channel is the signalling e¤ect.
The probability of run and hence the size of total redemptions is a function of the hedge fund’s
cash holding itself. The fund manager may need to hold more cash to make a ‘signal’and assure
investors in order to reduce the probability of a run. This is captured by the second term on LHS
of (9).
Next, we analyze the second region of X, that is, when X 2 [X C ; 1]. For this region of X, since
cash holdings are very high, all late investors decide to stay with the fund. The expected value of
the fund is therefore equal to that in the benchmark case:
Z X
Z
1
II
I
W (X) = W (X) = f
[(X
)+(1 X)R]d +
[(1 X)
X
0
Because
>
R
R(2
) 1,
X
]Rd g for X 2 [X C ; 1]13 .
based on the proof of Theorem 1, it is easy to show that W II (X) is
decreasing in the interval X 2 [X C ; 1]. Therefore, W II (X) achieves the maximum at X = X C .
We need to prove that this maximum value is less than that in the region X 2 [0; X C ], that is,
W II (X C ) < W II (X ). This is true under some parameter values. Therefore, it is not optimal for
the hedge fund to choose cash holdings in the second region. It is too costly.
Theorem 7 It is not optimal for the hedge fund to hold cash in the interval X 2 [X C ; 1]. Globally,
the optimal cash holding is X .
Proof: See Appendix.
The intuition for theorem 7 is straightforward. If the fund holds a large amount of cash, it will be
able to ‘convince’late investors that might redeem at T1 to stay, hence eliminating the coordination
problem and increase survival chance of fund. However, this policy is too costly because it seriously
distorts the tradeo¤, as in the benchmark case. That is, the fund holds too much cash and foregoes
valuable investment opportunities. Therefore, overall, it is better that the fund holds less cash.
Globally, the optimal liquid holdings is still X , which lies in the …rst region.
13
Since X + (1
X)
(R 1)
R
> , the upper limit of fund’s surviving window is min[ ; X + (1
22
X) )] = .
Before closing this subsection, we illustrate the numerical results derived above with the help
of a numerical example. We set the parameter values as R = 3;
= 0:8 and
= 0:9. With
these values, the optimal amount of cash holdings without considering a run is X = 0:23. The
hedge fund survives as long as
2 [0; X + (1
X ) ] = [0; 0:85]
[0; 0:9]. However, if the
fund sets the cash holdings at X = 0:23, the e¤ect of a run reduces the fund’s survival range to
2 [0;
(X )] = [0; 0:59]. The fund value is W II (X ) = 1:27. Therefore, when a run is possible,
it is optimal to increase the amount of cash holding since this increases the range of values of
for
which the fund survives. In fact, at X = 0:23, the survival range of the fund changes at a rate of
d
(X)
dX jX=X
= 0:52. The fund value rate of change is
dW I (X)
dX jX=X
= 0:31. Therefore, it is possible
to increase the value of the fund by an increase in the cash holdings, X.
If, indeed, the cash holding is increased to X
[0;
(X )] = [0; 0:64]. We have
d
(X)
dX jX=X
= 0:34, the surviving range widens to
= 0:53 and
dW II (X)
dX
jX=X
2
= 0. Therefore, the fund
achieves the highest expected value at X = X . This expected value is W II (X ) = 1:30. Note
that X + (1
X)
(R 1)
jX=X
R
is true.
= 0:82 <
If the fund increases the cash holdings to X C = 0:63, there is no coordination problem. The
survival range of the fund is the largest,
2 [0; ] = [0; 0:9]. But this policy is too costly and
therefore it is not optimal. In fact, the maximum value of the fund within the region X 2 [X C ; 1]
is W II (X C ) = 1:18, which is less than W II (X ) = 1:30.
Figure 7 depicts how W I (X) and W II (X) change when X changes. W I (X) is above W II (X),
meaning that the run e¤ect leads to a lower fund value. W I (X) achieves its highest value at
X = 0:23, while W II (X) has a positive slope at X = 0:23 and achieves a maximum at X = 0:34.
For values of X above 0:63, there is no coordination problem anymore, and therefore W I (X) and
W II (X) coincide. The optimal cash holding without considering a run is X = 0:23 and the optimal
cash holding with considering run is X
= 0:34.
23
2.3
The model implication - limits to arbitrage
From Theorem 6, the panic run forces the hedge fund to hold more cash and invest less in illiquidity
assets. This is a formal proof that coordination problem induces the hedge funds to reduce their
exposures.
Alternatively, we can formalize the argument in the following way: Suppose that the hedge fund
manager’s reservation utility (opportunity cost) is C. That is, only if the expected value of the
fund at T2 is higher than C, the fund manager is willing to participate in the market at T0 . We can
prove that W I (X ; R) and W II (X ; R) are both increasing functions in R. Because W I (X ; R)
> W II (X ; R), there exists an R and an R
such that W I (X ; R ) = C, W II (X ; R ) = C
and R < R . Therefore, for moderate returns R : R < R < R , coordination problem prevents
the hedge fund manager from taking these sound opportunities of illiquid positions.
Theorem 8 The coordination risk leads to hedge funds giving up participating in market when the
return R is moderately high (i.e. R < R < R ).
Proof: See Appendix.
24
In theorem 8, the limited arbitrage opportunities (R < R < R ) occur not because of the
reason presented in Shleifer and Vishny (1997), but because of the coordination risk and the
potential panic run on the hedge fund.
3
Discussion
In the last section of paper, we model hedge funds and assume that all the capital of hedge funds
comes from their investor-clients (equity-holders), that is, the hedge fund is completely …nanced
by equity and carries no debt. Obviously, this is a gross simpli…cation, since hedge funds use debt,
most of it provided by their prime-brokers. But by assuming no debt, we wish to emphasize a key
distinctive characteristic of hedge funds: the equity of hedge funds is fragile. That is, hedge funds’
equity is subject to panic run, and the equity holders have the right to redeem their investments.
This feature in the capital structure of hedge funds distinguishes these arbitrageurs from other
main …nancial institutions, namely investment banks and commercial banks, where the panic run
can only happen to the debt component of the capital structure, rather than the equity portion.
Equity in banks cannot be redeemed at investors’discretion, and because of that equity once issued
is locked permanently. Investors can only trade ownership in the secondary market, but cannot
tender the shares back to the bank for redemption.
In addition to equity, hedge funds also use debt. Debt is certainly apt to run, be that margin
calls in …nancial contracts, loan covenants or an increase in hair-cuts charged by prime brokers.
Therefore, the whole liability side of the balance sheet of hedge funds, including both debt and
equity, is fragile and is subject to runs. In particular, the run by debtholders and by equityholders
can interact and reinforce each other. A call on the debt made by prime brokers of a fund can scare
the hedge fund’s clients and trigger redemptions. The opposite is also true: the shrinkage in the
assets under management due to a large scale of redemptions will certainly induce prime brokers
to impose strict limits on leverage and a reduction in the available credit.
Our argument is that the fragile nature of both equity and debt of hedge funds has important
implications on individual hedge funds’behavior and the whole …nancial market. We discuss these
implications at both the micro and the macro levels next.
25
3.1
Capital fragility encourages micro-prudence
The works of Calomiris and Kahn (1991) and Diamond and Rajan (2001) argue that the coordination problem inherent in a bank run arises as a commitment device in that the bank’s managers
can commit not to engage in actions that dissipate the value of the assets. In particular, Diamond
and Rajan (2000) developed this argument further, and argue that bank capital (requirements)
sometimes can be harmful and reduce welfare, since by making the bank less prone to a run, it
weakens the management’s commitment. In a world of certainty, the bank maximizes the amount
of credit it can o¤er by …nancing with a rigid and fragile all deposit capital structure.
Compared with commercial banks or investment banks, hedge funds have little or no capital in
a strict sense, when capital is de…ned as ‘a long-term claim without a …rst-come-…rst-served right
to cash ‡ows’ (see Diamond and Rajan (2000)). In other words, hedge funds do not have a real
‘capital cushion’ as part of their balance sheet to mitigate a panic run. Therefore, the market
disciplining mechanism of fragile capital structure is a lot stronger in the case of hedge funds than
in the case of banks. Hedge fund managers need to be a lot more careful in their investment and
…nancing decisions than banks.14
With respect to investment decisions, our paper presents in a simple model how the asset
allocation of hedge funds is a¤ected by the fragile nature of capital structure. Due to potential
panic runs, the funds optimally choose to hold more cash in their balance sheets. Furthermore, we
show that if the fund relies on some locked capital (i.e. some clients’investments are locked until
T2 and cannot be redeemed at an earlier date, T1 ), the fund would hold less cash and invest more
in illiquid assets ex ante. In order to prove this formally, we adjust the original setup and assume
that there are a tiny proportion ( ) of investors that are locked in. The proportion of investors
that can run is 1
. We can prove that the threshold that investors redeem early increases
and hence the optimal cash holdings decrease:
Theorem 9 The presence of locked capital leads to the hedge fund holding less cash and investing
more in illiquid assets.
Proof: see the Appendix.
14
This conclusion is even stronger when banks are protected by deposit insurance and hedge funds are not.
26
The intuition for the result in Theorem 9 is that the commitment by some investors not to run
(i.e. locked) can reassure other investors and make them less prone to run.15
The result in Theorem 9 sheds light on why hedge funds must be less aggressive in their trading
positions involving illiquid assets than major investment banks, and even many commercial bank,
which it was recently discovered had a signi…cant mismatch in their asset-liability management.
Banks rely on equity that is locked instead of being redeemable. Thus, the equity of banks provides
a cushion to debtholders’withdrawals. This, in turn, ensures debtholders and makes them be less
prone to run. The lower risk of a panic run induces banks to be more aggressive and hold less cash
ex ante when compared to hedge funds. Thus it is not surprising that capital requirements lead to
more aggressive investments on the part of banks.
Not only before the crisis but also during the crisis hedge fund behaves quite di¤erently with
banks. It is reported that hedge funds have in the recent crisis rebalanced their portfolios much
more aggressively and much more swiftly into liquid assets than banks. Accordingly, not like hedge
funds, banks only started to seriously consider increasing their holdings in liquid assets after the
fall of Lehman. This is a further piece of evidence that di¤erent degree of fragility leads to di¤erent
response to market liquidity change between hedge funds and banks. In our model, when the
crisis hits and the market liquidity
deteriorates, it naturally requires hedge funds to voluntarily
liquidate some illiquid positions at T0 and hold more cash.
With respect to …nancing decisions, cash can be seen as a substitute for negative debt, that
is, more cash is equivalent to less debt. Equity fragility forces hedge funds to be less aggressive
in using leverage than banks. Indeed, any roll-over problem of hedge funds’ short-term debt can
ignite a run not only by other debtholders but also by the equity holders. In contrast, equity
holders of banks are locked and cannot run. If there is a rollover problem by some lenders of a
bank at most it can trigger a run by the other lenders, not by the equity holders. Furthermore, the
fact that equityholders cannot run also ensures unconstrained lenders and makes them less prone
to run. Formally, consider that a hedge fund and a bank have the same debt-to-equity ratio and
that the probability of rolling-over problem of the debt is the same for both institutions (i.e. the
15
It is worth noting that this mechanism is di¤erent with the explanation of duration match. The locked investors
may have long-term horizon anyway and don’t need to withdraw early at all if no panic. So locked capital doesn’t
increase duration, but increases commitment.
27
only di¤erence is the fragility in equity). Then, we can show that the ex-post probability of a run,
triggered by the inability to rollover part of lenders, is higher for the hedge fund than for the bank.
As a consequence, ex-ante, this encourages banks to operate with higher leverage ratios than hedge
funds. It is possible to formalize the above idea with global game techniques. To do that, we need
to explicitly model the payo¤ structure of both debtholders and equityholder, rather than that of
equityholders only, as in our model16 .
In summary, the di¤erence in investment and …nancing policies, arising from the di¤erent degrees
of fragility in capital structure claims, helps to understand why banks rather than hedge funds are
at the center of the current …nancial crisis. Most people seemed to have been surprised that hedge
funds, which are typically believed to be risky …nancial institutions, are in much better shape than
banks. This article explains that hedge funds have incentives to run less risky portfolio of assets
and operate with lower leverage ratio. These conservative behaviors of hedge funds help them
survive better, when the economy is hit by an unexpected large negative shock (e.g. the large scale
of mortgage loans default).
3.2
Capital fragility limits arbitrage
The arguments related to micro-prudence discussed before do not necessarily imply macro-soundness.
A fragile capital structure dramatically increases the liquidity risk to hedge funds. Thus, hedge
funds, although equipped with the most professional expertise and perceived as the most sophisticated investors, are not able to freely and fully exploit their trading capabilities. Liquidity risk
e¤ectively limits arbitrage activity. Constraints on arbitrage have important implications to …nancial markets. Presumably, during the recent crisis, the hedge fund industry must have spotted many
interesting investment opportunities, and had hedge funds taken these opportunities, these would
contribute to a greater market e¢ ciency. In reality, the opposite happened. Hedge funds drastically
reduced their exposures to risky and illiquid positions and reduced their role of liquidity providers
precisely when the markets were in greater need of liquidity. The absence of professional investors
is particularly harmful to markets because arbitrageurs contribute to reduce price ambiguity and
16
Within such framework, it is possible to model the optimal leverage ratio for …nancial institutions.
28
price misalignments, both needed for prices to provide correct signals for investment decision regarding the allocation of resources. Unsured about the reliability of prices, this, in particular,
prevents other investors (e.g. individual investors with spare liquidity, sovereign funds) with …nancial resources but without expertise from participating in the markets. The result is a breakdown
of the smooth movement of liquidity within the …nancial markets. The situation deteriorates and
the …nancial crisis prolongs.
4
Concluding remarks
In this paper, we present a model that shows how the fragile nature of the capital structure of
hedge funds combined with imperfect market liquidity can hamper the ability of these investors
to arbitrage the markets. This has negative consequences to the e¢ ciency of markets and also
imposes negative externalities to market participation by other investors. During the recent crisis
of 2007-08, the hedge fund industry was presumed to have many good investment opportunities.
However, faced with the risk of massive redemptions from nervous investors, many hedge funds
decided to pull back taking advantage of the frequent and signi…cant mispricings. In the absence of
arbitrageurs, markets were not able to correct themselves, which prevented many other investors
from participating in the markets. The result was a breakdown in market e¢ ciency and a shortage
of liquidity ‡owing into the markets, a form of vicious cycle that prolongs a …nancial crisis.
Our explanation for the limits of arbitrage di¤ers from that in Shleifer and Vishny (1997),
which is based on asymmetries of information. In their work, investors’ worry about the quality
of the assets of a money manager and bad performance leads to early redemptions, which, in turn
limits arbitrage. We assume that investors do not worry about the assets of the money manager,
and instead face a coordination problem in short-term redemptions. The mere apprehension that
other investors might pull out, makes an investor consider pulling out as well. During the crisis,
the market liquidity is low, a …re sale is costly and investors are nervous. All these contribute to a
panic run. The ex-post potential panic run curtails hedge funds’willingness to arbitrage markets
ex-ante.
This model may also help to understand two facts that surprised most observers of the recent
29
…nancial crisis: that hedge funds showed greater resilience in crisis and operate with lower leverage
ratios than investment banks (and many commercial banks). We claim that the reason behind
these facts partially lies with the di¤erent nature of the …nancial claims in hedge funds relative to
claims in banks. Both the equity and the debt of hedge funds are fragile in the sense that they
can be withdrawn by investors on a …rst-come-…rst-served basis, while in the case of banks only
the debt can be called, since the equity is a non-redeemable long-term claim. The fragility on
the whole liability side of balance sheet of hedge funds implies they are much more prone to run.
Hedge fund managers factor the possible run into their investment as well as …nancing decisions.
In the …nancing decision, equity fragility forces hedge funds to be less aggressive than banks in
using leverage, because any roll-over problem of hedge funds’short-term debt can ignite a run not
only by other debtholders but also by the equity holders. In turn, in the absence of a commitment
by limited partners not to rush to redeem, hedge funds are not able to assure other claimholders
and this strengthens a possible run. In sum, more prone to run institutions like hedge funds, forces
them to use less leverage. An important implication from our paper is that an unregulated sector
of the …nancial industry can be safer than a regulated sector because of the nature of the …nancial
contracts that exist in each case.
30
5
Appendix
Proof of theorem 1:
We can rewrite (3) as
8
Z X
>
1
>
<
[(X
f
0
W I (X) =
Z X
>
>
: 1f
[(X
) + (1
) + (1
R
2
X)R]d +
Z
X)R]d +
[(1
X
X)
X
X+(1 X)
[(1
]Rd g
)X 2 +
X)
X)
.
]Rd g if X + (1
X) <
2
1
R
2
2 )X
R+
if X + (1
X
X
0
Hence, we 8
have
< 1 [( 1
2
W I (X) =
: 1 [( 1
2
Z
RX + (
+ R(1
2
)X +
)R]
if X
R
2 ]
if X <
1
(A1)
1
(A2)
.
The …rst order conditions of A1 and A2 are
X1 =
) R
(1
R
Because X 1
and X 2 =
1
(R
) [R(2
) 1]
[R(2
) 1](1
)
=
(1
R(1
)
R(2
) 1,
respectively.
) R
=
R
1
, the relation (X 1
1
(R
(R
)(X 2
are on the same side of the vertical line X =
(R
R(2
If
)
) 1,
If
) R
(1
R
, where X 2 [ 1
(R
R(2
)
) 1,
1
)
and X 2
=
1
R(1
)
R(2
) 1
1
=
0 is true. It means that X 1 and X 2
.
1
], while it increases
; 1]. Overall, W I (X) achieves the maximum value at
; 1).
W I (X) increases …rst and then decreases in the interval [0;
decreases in the interval [ 1
where X 2 (0;
1
1
) 1]
)=
W I (X) is an increasing function in the interval [0;
…rst and then decreases in the interval [ 1
X =
) [R(2
)(1
1
; 1]. Overall, W I (X) achieves the maximum value at X =
].
Proof of theorem 2:
31
], while it
R(1
)
R(2
) 1,
As shown in Figure 8, to guarantee that
(X) is on the right to X , we need to make sure
that the area of ADE (denoted as SABC ) is greater than the area of EFGH (denoted as SBDGF )
for whatever X.
It is easy to show that SABC
) ln(1
) +(R
)] and SDEF G =
SBDE =
Z 1
X+(1 X)
(1
X)(1
)(
t ln t
1 t jt=X+(1 X)
).
Z
X+(1 X)
[
X
X+(1 X)
s
(1 X) s
1 s
ds =
X
R
1]ds = (1
X)[ R (1
[X +(1 X) ] ln[X +(1 X) ] =
Therefore, we can work out the area di¤erence
4S = SABC
= SABC
= (1
Note that
SBDGF
(SBDE + SDEF G )
X)f[ R (1
t ln t
1 t
) ln(1
) + (R
)]
(1
is increasing in the interval [0; 1),
)(
t ln t
1 t jt=X+(1 X)
t ln t
1 t jt=0
)g:
= 0, and lim
t!1
guarantee the area di¤erence 4S is positive for whatever X is to make sure that
) + (R
) > (1
Also, we need
)(
t ln t
1 t jt=1 ),
that is
R
(1
to be big enough so that
su¢ ces for this because X + (1
X)
(R 1)
R
>
) ln(1
>
) + (R
)>1
(X). The condition
t ln t
1 t
R
= 1. To
(1
) ln(1
.
> X + (1
X)
(X) is de…nitely true by the property of
Proof of theorem 4:
32
(R 1)
R
(X).
We still use Figure 8 to illustrate the proof.
We express
(X) as
(X + 4X) with
(X) = X +b(1 X). Suppose X changes to X +4X, we want to compare
(X). The idea of comparison is as follows. After X changes to X + 4X, the
shape of payo¤ curve will of course change because the parameter becomes X + 4X instead of
X. We want to know under the new shape of curve whether the area di¤erence SBKH
becomes negative or positive if we still keep the threshold as
SBDGF
(X). If 4S becomes positive, it
means that the above shaded area is bigger than the below one, we need to remove part of the
above shaded area to make the two shaded areas equal, and
We check how the area di¤erence SBKH
(X + 4X) >
(X).
SBDGF changes in two steps.
In the …rst step, we wonder what the area di¤erence SBKH
SBDGF under the new curve is if
we keep b unchanged. That is, we want to …nd out the area di¤erence SBKH
new curve if the threshold is (X + 4X) + b[1
It is easy to show SHBK
SEF GD =
[X + (1
Because SHBK
R
[(
) + (1
(X + 4X)].
X)f R [(
SBED = (1
X) ] ln[X + (1
SBDGF under the
X) ] = (1
) ln 11
) + (1
X)(1
b]
t ln t
1 t jt=X+(1 X)
)(
(
b)g and
).
SBED = SEF GD , we have
) ln 11
b]
(
Also, we
@(SHBK SBED )
= R [(
@X
GD )
can work out @(SEF
=
@X
Overall,
@(SBKH SBDGF )
@X
b) = (1
(1
(1
)[
t ln t
1 t
t ln t
1 t jt=X+(1 X)
) ln 11
)+(1
Therefore,
=
)(
b ]+(
)(1 + ln t)jt=X+(1
).
b) =
X)
(1 + ln t)]jt=X+(1
(A3)
(1
)(
t ln t
1 t jt=X+(1 X)
).
.
X)
.
In the second step, we work out under the new curve how much the above shaded area SHBK
increases if the threshold reduces from (X + 4X) + b[1
to show that the increase in area is (4X)(1
b)( 1
b
b
R
(X + 4X)] to X + b(1
X). It is easy
1).
Aggregating the results in the above two steps, if the curve shape is changed to new parameter
X + 4X while the threshold is still
(X), the area di¤erence SBKH
33
SBDGF becomes
(4X)f(1
b
b
b)( 1
Because
(1
R
1)
t ln t
1 t
)[
(1
t ln t
1 t
)[
(1 + ln t)]jt=X+(1
g.
X)
(A4)
(1 + ln t)] is positive for any t 2 (0; 1), we conclude that b must be
decreasing with X. Hence the …rst term of (A4) is increasing with X. Also, it is easy to check
that the second term (1
t ln t
1 t
)[
(1 + ln t)]jt=X+(1
is decreasing function with respect to
X)
X. Therefore, the whole term of (A4) is increasing with X. To guarantee that (A4) is positive for
any X 2 (0; 1), we need to have its value at X = 0 be positive. That is, (
ln
)(1 + ln ) where b solves ( R
(1
R
the above inequation can be simpli…ed as
In sum,
b) + R (1
1)(
ln( 11
b)
) ln( 11
b)
=
1)(
b) +
R
(1
) ln( 11
b)
(1
b) >
ln . Alternatively,
< ln .
(X) is an increasing function with respect to X if parameters satisfy
(where b solves ( R
R
b)
=
R
ln( 11
b)
< ln
ln ).
Proof of theorem 5:
Again, we express
By A3, we have
(X) as
R
[(
(X) = X + b(1
) ln 11
) + (1
X). We have
b]
(
b) = (1
@
(X)
@X
)(
)2
(1
Thus,
@b
@X
)2
= (1
R
(
@b
@X
+
@b
X) @X
.
).
1
@b
1 b @X )
+
@b
@X
=
.
(ln t) 1+t
(1 t)2
+1+ R 11 b
jt=X+(1
X)(1
)2
R
b) + (1
t ln t
1 t jt=X+(1 X)
Di¤erentiating on both sides of the above equation, we obtain
(ln t) 1+t
jt=X+(1 X)
(1 t)2
= (1
.
X)
Therefore,
@
(X)
@X
= (1
b) + (1
R
(ln t) 1+t
(1 t)2
+1+ R 11 b
After we work out the explicit expression of
@
jt=X+(1
(X)
@X ,
X)
.
we can also explicitly express
dW II (X)
,
dX
that
is,
dW II (X)
dX
)2
= [(1
(ln t) 1+t
(1 t)2
R
+1+ R 11 b
R
)X +
jt=X+(1
X)
1
].
(X)R] + [(X + (1
X)
(X)]
R
[(1
b) + (1
(A5)
Based on A5, we can directly evaluate
dW II (X)
dX
for any X.
To guarantee the interior solution of X , we need parameters satisfying two conditions:
0 and
dW II (X)
jX=X C
dX
X)(1
< 0.
34
dW II (X)
jX=0
dX
>
Proof of theorem 6:
Based on A5, we can directly evaluate
the ones such that
dW II (X)
jX=X
dX
dW II (X)
jX=X
dX
. Therefore, the parameter conditions are
> 0.
Proof of theorem 7:
Because we know the expression of W II (X), the parameter conditions are the ones such that
W II (X C ) < W II (X ). These do exist such parameters. The numerical example in the text is one
case.
Proof of theorem 8:
We only need to prove that W I (X ; R) and W II (X ; R) are both increasing functions in R. For
8
Z X
Z
> 1
X
>
<
f
[(X
) + (1 X)R]d +
[(1 X)
]Rd g
if X + (1 X)
I
0
X
W (X) =
Z X
Z X+(1 X)
>
>
X
: 1f
[(X
) + (1 X)R]d +
[(1 X)
]Rd g if X + (1 X) <
X
X
0
both (X
) + (1
X)R and (1
X)
]R are increasing with R. Therefore, when R
increases, W I (X ; R) increases even if we don’t adjust X . But in fact, when R increases, X
changes as well in maximizing W I (X). Therefore, W I (X ; R) certainly increases with R.
For
that
W II (X)
1
= f
Z
X
[(X
) + (1
X)R]d +
Z
(X)
[(1
X)
X
X
0
]Rd g , we need to prove
(X) is increasing with R …rst. In Figure 8, as R increases, SEF GD is unchanged while SBED
decreases. So if keep
(X) unchanged, SBKH
(X) has to increase. After we prove that
SBDGF > 0. Therefore, to restore equilibrium,
(X) is increasing with R, we use the similar argument
as for the case of W I (X), we can prove W II (X ; R) is increasing in R.
Proof of theorem 9:
In Figure 6, the existence of a tiny proportion ( ) of locked investors is equivalent to cutting
a -width block at the right end of below shaded area. So the below shaded area is reduced. In
order to make two shared area be equal, the above shaded area should be reduced as well. So the
threshold
(X) has to increase. That is, for the same level of X, the surviving window of fund
is bigger if there is locked capital than that if there is not. Therefore, in the …rst order condition
of (8), if we use the new increased threshold
(X) substitute for the original
35
(X) while keep
,
X as the same, the …rst order derivative becomes negative with wide parameters. That is, as the
threshold
(X) improves, the optimal cash holding becomes less.
36
6
References
Abreu, Dilip and Markus K. Brunnermeier (2002) “Synchronization Risk and Delayed Arbitrage”,
Journal of Financial Economics (2002) 66(2–3): 341–600.
Abreu, Dilip and Markus K. Brunnermeier ( 2003) “Bubbles and Crashes”, Econometrica,
Econometric Society, vol. 71(1), pages 173-204.
Allen, Franklin and Douglas Gale (2000) “Financial Contagion”, Journal of Political Economy,
108, 1-33.
Brunnermeier, Markus (2009) “De-Ciphering the Credit Crisis of 2007”, Journal of Economic
Perspectives, forthcoming.
Calomiris, Charles and Charles Kahn (1991) “The Role of Demandable Debt in Structuring
Optimal Banking Arrangements”, American Economic Review, 81, 497-513.
Chen, Qi, Goldstein, Itay and Jiang, Wei (2007) “Payo¤ Complementarities and Financial
Fragility: Evidence from Mutual Fund Out‡ows”, working paper.
Dasgupta, A. (2004) “Financial contagion through capital connections: a model of the origin
and spread of bank panics,” Journal of the European Economic Association, 2, 1049— 1084.
Diamond Douglas and Philip Dybvig (1983) “Bank Runs, Deposit Insurance, and Liquidity”
Journal of Political Economy, 91, 401-19
Diamond, Douglas and Raghuram Rajan (2000) “A Theory Of Bank Capital”, Journal of
Finance, v55 (6, Dec), 2431-2465.
Diamond Douglas and Raghuram Rajan (2001) “Liquidity Risk, Liquidity Creation, and Financial Fragility: A Theory of Banking”, Journal of Political Economy, 109, 287-327.
Goldstein, Itay, and Ady Pauzner (2004) “Contagion of self-ful…lling …nancial crises due to
diversi…cation of investment portfolios,” Journal of Economic Theory, 119, 151— 183.
Goldstein, Itay, and Ady Pauzner (2005) “Demand Deposit Contracts and the Probability of
Bank. Runs,” Journal of Finance 60, 1293-1328.
Gorton, Gary and Andrew Winton (2003) “Financial Intermediation,”in The Handbook of the
Economics of Finance: Corporate Finance, edited by George Constantinides, Milton Harris, and
Rene Stulz (Elsevier Science; 2003).
37
Gorton, Gary (2008) “The Panic of 2007”, Federal Reserve Bank of Kansas City Symposium
2008, http://www.kc.frb.org/publicat/sympos/2008/Gorton.10.04.08.pdf.
Gromb, Denis and Dimitri Vayanos (2002) “Equilibrium and Welfare in Markets with Financially Constrained Arbitrageurs” Journal of Financial Economics, 66, 361-407.
Johnson, Woodrow T. (2006) “Who monitors the mutual fund manager, new or old shareholders?” AFA 2006 Boston Meetings Paper Available at SSRN: http://ssrn.com/abstract=687547.
Morris, Stephen and Hyun Song Shin (1998) “Unique Equilibrium in a Model of Self-Ful…lling
Currency Attacks” American Economic Review, 88, 587-597.
Morris, Stephen and Hyun Song Shin (2004) “Liquidity Black Holes” Review of Finance, 8,
1-18.
Morris, Stephen and Hyun Song Shin (2008) “Financial Regulation in a System Context”paper
prepared for the Brookings Papers conference, Fall 2008.
Plantin, G. (2009) “Learning by Holding and Liquidity,” Review of Economic Studies, 76, 395412.
Rochet, J.-C., and X. Vives (2004) “Coordination failures and the lender of last resort: was
Bagehot right after all?,” Journal of the European Economic Association, 2, 1116— 1147.
Shleifer, Andrei, and Vishny, Robert (1990) “The Limits of Arbitrage”, Journal of Finance 52,
35-55.
Stein, J. (2005) “Why are most funds open-end? competition and the limits of arbitrage,”
Quarterly Journal of Economics, 120, 247— 272.
38