The Fragile Capital Structure of Hedge Funds and the Limits to Arbitrage

Xuewen Liu Antonio S. Mello

Imperial College Business School

February 2009

Facts and evidence

• During a financial crisis, when markets most need liquidity and

arbitrage tradings, hedge funds often reduce their exposures and

positions.

Between July and August in 2008 alone, the hedge fund industry's cash holdings

rose from $156 billion to a record $184 billion, equivalent to 11% of assets under

management (The Economist, October 25, 2008)

Why it is puzzle?

• Micro perspective

Hedge funds’ expertise would present them with greater opportunity from the more

frequent and more evident misalignment in prices during a crisis. Why did they do the

opposite?

---“Be fearful when others are greedy and greedy when others are fearful”

(Warren Buffett)

• Social perspective

Hedge funds' behavior had a negative impact on the functioning of financial

markets.

Without arbitrage, no correct price signals, the vicious cycle of lower price

informativeness and less liquidity

The research questions

Research questions:

• Why hedge funds, aggressive investors seeking high returns, hold riskless assets?

• Why hedge funds quickly increase their cash holdings during a crisis?

• What factors determine the cash holdings in hedge funds' balance sheets?

• Cash is negative debt. Alternately, why hedge funds have lower leverage than investment banks?

Why these questions matter?

To help understand an important phenomenon – limits to arbitrage

• Holding cash/riskless assets is synonymous to hedge funds limiting their arbitrage activities.

•The limits-to-arbitrage is one of three key elements to understand financial crisis (Allen (2009))

Existing explanation

Shleifer and Vishny (1997):

• Asymmetrical information: investors do not understand or perfectly observe the trades and positions taken by arbitrageurs

• Investors might react with their feet to short-term bad performance.

• Ex ante, professional fund managers decide to abstain from taking positions that might lose money in the short term, which limits arbitrage.

Our explanation

Strategic risk rather than fundamental risk causes the limits to arbitrage

• No information asymmetry: investors know perfectly the long-term performance of hedge funds' position

• Frictions result from market illiquidity and the fragile nature of the equity

• The frictions mean that investors care about both fundamentals and the actions of other investors

• Both strategic risk (i.e. uncertain about what other investors will do) and fundamental risk cause redeem.

• Hedge fund managers , in response to ex-post possible run, decide to limit arbitrage ex ante.

Investors worry about both asset side and liability side of balance sheets of hedge funds

Our Framework

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.

• Solve the unique optimal cash holding with and without the run effect

The coordination problem and the panic run induce hedge funds to hold more

cash ex ante

• Run is due to funding-liquidity reasons rather than fundamental reasons

Brunnermeier (2009) discusses two reasons

The run can be fundamental-driven as in Goldstein and Pauzner (2005).

Related Literature

― Limits to arbitrage: Schleifer and Vishny (1997), Stein (2005)

― Coordination problem: Diamond and Dybvig (1983), Goldstein and Pauzner (2005),

Chen, Goldstein and Jiang(2007)

― Applciaitons of global game: Bank run (Rochet and Vives (2004), Goldstein and Pauzner

(2005)); currency crises (Morris and Shin (1998), Corsetti, Dasgupta, Morris, and Shin (2004));

contagion of financial crises (Dasgupta (2004), Goldstein and Pauzner (2004)); market liquidity

(Morris and Shin (2004), Plantin (2009)).

The model – setup

Four dates:

• T0 : X in cash, 1-X in long-term illiquid asset

• T1 : λ proportion are ‘early’ investors needing to redeem, λ uniformly

Discounted price at fire sale of illiquid asset: α, where α<1.

• T2 : the illiquid asset realizes gross return R, where R>1.

• : each investor knows whether he is ‘early’ or not. Late investor i receives signal

; εi uniform in [- ε ,+ ε].

All investors make notices to fund about redemption.

The model – payoff structure

Redeem:

Stay:

The model – payoff structure

The equilibrium - benchmark

• The trade-off

In case λ≤X: liquidate 0 units In case λ>X: liquidate

• The expectation of fund’s equity value at T2 :

• The unique optimal cash holding:

The full equilibrium

• Idea

Coordination problem among late investors causes ex-post potential run Fund mangers factor the ex-post run into ex-ante cash holding decision

• Procedure: backward induction

Investors’ redemption decisions at T1/2 given the cash holding X The fund's optimal cash holding decision at T₀

The full equilibrium: decisions at T1/2

• Late investor i’s decision rule:

• Two dominance regions of λ

The lower dominance region : dominate strategy (λi ,X) → stay Assumption: no discounting at fire sale (i.e. α=1)

The upper dominance region : dominant strategy(λi ,X)→ Withdraw

where solves WS(s)=WR(s) and To verify: , so that the upper region exists

The full equilibrium: decisions at T1/2

• The threshold equilibrium

The equilibrium where every late investor sets a threshold and uses the threshold strategy

such that, given that all other late investors set the threshold as , it is optimal for a late investor himself to do that.

• The unique threshold equilibrium

The full equilibrium: decisions at T1/2

• Step 1: Given the realized λ and the threshold , the proportion of late investors running:

• Step 2: For the late investor i, his expected net payoff of staying versus redeeming:

• Step 3: For the marginal late investor who receives the threshold signal :

The full equilibrium: decisions at T1/2

• When ε→0, s(λ, λ*) is uniformly distributed within the interval [λ*,1].

• Geometrically,

The full equilibrium: decisions at T1/2

• The existence of

• The threshold increasing in X

The full equilibrium: decisions at T0

• The total number of investors redeeming

• Only consider the case of

• Equilibrium selection

: the unique threshold equilibrium

: staying-strategy equilibrium

• Consider two regions of X, in order

i) ii) where

The full equilibrium: decisions at T0

• The expectation of fund’s equity value at T2 :

• First order condition

• The trade-off First term: high return and costly fire sale; the tradeoff in benchmark; being negative when X is high Second term: the effect of increasing surviving window λ on fund value

• The unique optimal cash holding X:

The full equilibrium: decisions at T0

• We check the change of WII(X) when X is slightly changed at X=X*

• The first term is negative and the second term is positive. Overall,

• Run causes more cash holding:

• Two effects of run on cash holing

Response effect : the size of redemption is changed Signalling effect: cash can assure inventors and reduces the probability of run

The full equilibrium: decisions at T0

• The expected fund value at the second region of X

• Not optimal to hold cash in this region

• Intuitively, too much cash convinces investors not to run, but it is too costly

A numerical example

•

The model implication

• Suppose that the hedge fund manager's reservation utility (opportunity cost) is C.

WI(X*,R)≥C and WII(X*,R)≥C

• The limited arbitrage opportunity R such that WI(X*,R)>C> WII(X*,R)

• Define

Discussions

• The difference between hedge funds and banks

Both liability and equity of hedge funds are fragile while only debt is fragile for banks

• Capital fragility encourages micro-prudence

Investment decision: less risky investment

Financing decision: lower leverage

These conservative behaviors of hedge funds help them survive better

• Capital fragility limits arbitrage and implies macro-unsoundness

Conclusions

• A new explanation of limits to arbitrage

• Coordination risk in addition to fundamental risk matters

• Fragile equity structure plus low market liquidity leads to coordination risk

• Fragility encourages micro-prudence but leads to macro-unsoundness

• An unregulated sector of the financial industry can be safer than a regulated sector because of the nature of the financial contracts that exist in each case.