Liquidating Illiquid Collateral Martin Oehmke Columbia University

February 20, 2009

Motivation Many traders, especially hedge funds, rely on collateralized borrowing When a hedge fund defaults, lenders need to liquidate collateral These liquidations are often big (potential) shocks for the financial system Examples: I

LTCM

I

Amaranth

I

Current credit crunch

⇒ What determines dynamics of these liquidations? ⇒ Will they be orderly?

This paper Model liquidation conditional on the default of a hedge fund. Two realistic features: I

Collateral asset is illiquid

I

Prime brokers face balance sheet constraints

Liquidation takes form of a continuous-time trading game: I

Prime brokers take into account price impact when liquidating

I

Maximize expected recovery given their balance sheet constraint

Equilibrium determines price dynamics, liquidation values, etc.

Talk Outline I

Liquidation game setup and equilibrium trading strategies

I

Main results: 1. Price overshooting: When prime brokers are sufficiently balance-sheet constrained 2. Creditor structure: Tradeoff between risk sharing and strategic racing 3. Margin setting: Should reflect balance sheet constraints and strategic interaction 4. Block trades: Can determine block price and expected profit at which a vulture buyer can purchase entire collateral position

I

Related literature

I

Conclusion

Model Setup (1): Players and Timing

Continuous-time setup, t ∈ [0, ∞) HF borrows from PB, risky asset position as collateral At t = 0 HF defaults, n PBs liquidate collateral X i each (symmetric) PBs are risk neutral, but balance-sheet constrained No discounting

Model Setup (2): Illiquidity

P(t) =

F (t) + γ[X (t) − X (0)] + λY (t)

I

F fundamental, follows a Brownian Motion: dF (t) = σF dW (t)

I

permanent price effect of trading: γ[X (t) − X (0)] temporary price pressure: λY (t), Y (t) = X˙ (t)

I

I

Empirical motivation: Kraus and Stoll (1972), Holthausen, Leftwich and Mayers (1990), Cheng and Madhavan (1997), Shleifer (1986) etc.

I

Theoretically: Downward sloping demand curve, where trader does not have full instant access to whole demand curve

Model Setup (3): PB Variance Constraint

I I I

PB would like to liquidate slowly, but slow liquidation is risky ˜ ∼ N(Π, V ) Selling illiquid asset is like choosing Π Assumption: During the liquidation the PB can only incur a limited amount of variance: V ≤V

Interpretation: I

Internal risk management constraint

I

Regulatory constraint

I

Balance sheet strength

Liquidation Game I

Each PB optimally liquidates his collateral position X i

I

Choice variable: trading rate Yi (t) ∈ L2 on [0, ∞)

I

PB i’s optimization problem: Z max E Yi (t)

∞

−Yi (t)P(t)dt

0

s.t. Xi (0) = X i

(trading constraints)

lim Xi (T ) = X i + Yi (t)dt = 0 Z ∞ 0 σF2 [Xi (t)]2 dt ≤ V

(variance constraint)

Z

∞

T →∞

0

Equilibrium

Definition Equilibrium. An equilibrium in time-dependent strategies is a set of admissible trading strategies {Yi (t)}, i = 1, .., n, such that each PB maximizes liquidation payoff subject to the variance constraint, taking the trading strategies of the other PBs, {Y−i (t)}, as given.

Equilibrium Derivation I

Write Lagrangian: Z L

"

# X X Yi (t) F (0) + γ [Xi (t) − X i ] + λ Yi (t) dt

∞

= 0

·Z

−

φ 0

I

i ∞

σF2 [Xi (t)]2 dt − V

We can write the Hamiltonian: " H

=

# X X Yi (t) F (0) + γ [Xi (t) − X i ] + λ Yi (t)

−

φσF2 [Xi (t)]2

i

I

i

¸

i

+ q(t)Yi (t)

FOCs imply a second-order ODE: (n − 1)γYi (t) + λ(n + 1)Y˙ i (t) − 2φσF2 Xi (t) = 0.

Equilibrium Liquidation Strategies Optimal trading rate is exponential: Yi (t) = −ae −at X i q

1 (n − 1)γ + Γ 2 (n + 1)λ

a=

Xi (t) = e −at X i

[(n − 1)γ]2 + 8(n + 1)φσF2 λ

Γ=

t 0

1

2

3

4

5

0 10 -2 8 -4 6 Y -6

X 4

-8

-10

-12

2

0 0

1

2

3

4

5

t

Figure: Trading rate (left) and remaining collateral position (right).

Equilibrium Liquidation Strategies Optimal trading rate is exponential: Yi (t) = −ae −at X i

a=

Xi (t) = e −at X i

q

1 (n − 1)γ + Γ 2 (n + 1)λ

Γ=

[(n − 1)γ]2 + 8(n + 1)φσF2 λ

Variance constraint not binding: a=

(n − 1)γ (n + 1)λ

Variance constraint binding: 2

σ2 X a= F i 2V

Expected Unwind Value

Proposition The equilibrium aggregate expected unwind value when n traders liquidate a total of X units of collateral is given by nΠ(X i , V ) = F (0)X −

γ 2 λ n 2 X − a (X i , V )X 2 2

⇒ Trading speed an (X i , V ) determines illiquidity inefficiency.

Expected Price Dynamics: Overshooting I

When φ >

γ2 , σF2 λ

the expected price overshoots.

100

100

90

90

80

80

P

P 70

70

60

60

50 -2

0

2

4 t

6

8

10

50 -2

0

2

4

6

8

10

t

Figure: No overshooting φ < γ 2 /σF2 λ (left) and overshooting φ > γ 2 /σF2 λ (right)

Creditor Structure: A Stylized Financial System Consider a setup with two HF and two PB:

PB

PB

PB

PB

HF

HF

HF

HF

Figure: Monopolistic Finance (left) and Distributed Finance (right)

More Creditors Reduce Overshooting When one HF defaults: I

Monopolistic case: s X >

I

2

γ V λ σF2

Duopolistic Case: s X >2

2

γ V . λ σF2

Conditional on overshooting, amount larger in monopolistic case When both HFs default: Overshooting condition and amount is the same in both setups

What Happens to Expected Liquidation Payoffs? I

HF invest in same risky asset, each position X

I

Individual HF defaults with probability p

I

Both HFs default with probability q

Monopolistic Finance

Distributed Finance

M

HF1 defaults HF2 defaults Both HF default

Π (X , V ) ΠM (X , V ) 2ΠD (X , V )

Just need to check 2ΠD (

X , V ) > ΠM (X , V ) 2

2ΠD ( X2 , V ) 2ΠD ( X2 , V ) 2ΠD (X , V )

Compare trading rates...

Remember: inefficiency determined by an (X i , V ) Monopolist always constrained: aM (X , V ) =

σF2 X 2 2V

Duopolists may be constrained or unconstrained  2 2  σF X when constrained X 8V aD ( , V ) = γ 2 when unconstrained 3λ

Tradeoff: Risk-bearing vs. Strategic Racing

Proposition Given variance constraint V , splitting the financing between two prime brokers raises expected recovery when s 2γ V X > , 3 λ σF2

Intuition: Relaxing Constraints vs. Racing Consider a monopolist liquidating a position: 7

6

5

4 Y 3

2

1

0 0

1

2

3

4

5

t

Monopolist only has risk-bearing capacity V

Intuition: Relaxing Constraints vs. Racing Spreading across multiple PB relaxes constraint... 7

6

5

4 Y 3

2

1

0 0

1

2

3

4

5

t

...as duopolists have joint risk bearing capacity of 2V

Intuition: Relaxing Constraints vs. Racing But need to consider strategic effect... 7

6

5

4 Y 3

2

1

0 0

1

2

3

4

5

t

...for low γ/λ also duopolists constrained.

Intuition: Relaxing Constraints vs. Racing But need to consider strategic effect... 7

6

5

4 Y 3

2

1

0 0

1

2

3

4

5

t

...intermediate γ/λ, balance sheets not fully used, but still beneficial.

Intuition: Relaxing Constraints vs. Racing But need to consider strategic effect... 7

6

5

4 Y 3

2

1

0 0

1

2

3

4

5

t

...high γ/λ, strategic effect outweighs benefit from risk-bearing capacity!

Monopolistic finance preferable when balance sheets are strong...

460

440

420

400

380

360 10

20

30

40

50

60

V

Figure: Expected aggregate liquidation payoff Π as a function of the variance constraint in the monopolistic case (left panel) and the duopoly case (right panel). F = 50, X = 10, γ = 1, λ = 0.2, σF2 = 1.

Monopolistic finance preferable when balance sheets are strong...

460

440

420

400

380

360 10

20

30

40

50

60

V

Figure: Expected aggregate liquidation payoff Π as a function of the variance constraint in the monopolistic case (left panel) and the duopoly case (right panel). F = 50, X = 10, γ = 1, λ = 0.2, σF2 = 1.

Default Correlation

Proposition The expected difference of liquidation proceeds under distributed and monopolistic finance is decreasing in the default correlation of the two funds. Intuition: I

When both HF defaults, two financing schemes equivalent

I

Less default correlation ⇒ more probability weight on scenario in which regimes differ

Margin Setting What are ex-ante implications for risk management? Assume: I

Prime brokers set margin to cover expected loss given default

I

Loss given default entirely driven by illiquidity costs

Proposition Per share margin in the monopolistic setup: γ λ M M = (p + 2q) X + [paM (X , V ) + 2qaD (X , V )] X . 2 2 Per share margin in duopolistic setup: γ X λ M D = (p + 2q) X + [paD ( , V ) + 2qaD (X , V )] X . 2 2 2

Margin Setting: Implications

Creditor Structure matters: I

When many creditors, margin should anticipate ‘crowded trade’

Balance sheet correlation matters: I

Prime broker may have own positions that lose money exactly when hedge fund defaults

I

Correlation between V and hedge fund default state

Block Trades Strong balance sheet buyer, V = ∞ I

Can purchase entire collateral position and unwind orderly

I

Has to pay fixed cost c

Example: Citadel’s acquisition of Amaranth and Sowood

Proposition Block trade results in positive gains from trade when c<

λ 2 aX , 2

Vulture buyer’s expected profit ΠV =

λ 2 aX − c. 2

Block Trades

950

940

930

Pi 920 910

900

890 40

60

80

100

120

140

V

Figure: Gains from block trade. F = 100, γ = 1, λ = 1, σF2 = 1, c = 30.

Related Literature Strategic Trading I

Brunnermeier and Pedersen (2005)

I

Carlin, Lobo and Viswanathan (2007)

Optimal Liquidation I

Almgren and Chriss (1999)

I

Bertsimas and Lo (1998)

I

Huberman and Stanzl (2005)

I

Engle and Ferstenberg (2006)

Optimal Debt Structure I

Bolton and Scharfstein (1996)

Conclusion Paper provides continuous-time model for strategic collateral liquidations Price can overshoot during large collateral liquidations Illiquidity implies tradeoff between spreading risk and strategic inefficiencies Margins should take into account creditor structure and PB balance sheet constraints Model provides a framework to analyze and price balance-sheet driven block trades

Solving for the Multiplier

For n traders: h φ = λ(n + 1)

i 2 2

σF X i 8V

2

2

− γ(n − 1)

For a monopolist: i 2 2

h φ=λ

σF X i 4V

2

σF X i 4V