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February 16, 2010
Incentives in Hedge Funds Hitoshi Matsushima Faculty of Economics, University of Tokyo
February 4, 2010
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Hedge Fund as Delegated Portfolio Management
Investor (Unsophisticated) 1 Unit of Fund, No Withdrawal
Manager
M Units of Personal Fund: Manage Investor’s and Personal Funds ‘Separate Management’ or ‘Equity Stake’ Weak Regulation, Low Transparency Generate Alpha Skilled Type
Select Alpha (Action) a ∈ [0, ∞ ) with Non-Pecuniary Cost C (a )
Unskilled Type
Alpha 0
Manager
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Incentive Problem
Hidden Type Hidden Activity
Investor Cannot Identify whether Manager is skilled or not Investor Cannot Observe Manager’s Activity
Q: Can We Solve Incentive Problem? Skilled Entry
HF Survives:
Skilled and Investor
Pareto Improvement
Positive Capital Gain Tax (Fulcrum Scheme or Equity Stake) Investor Entry
No HF No CG Tax Unskilled
A: Yes, but We Need Capital Gain Tax!
Unskilled Exit
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Manager’s Incentive Fee Scheme
y :[0, ∞ ) → [− M , ∞ ] , y( x ) ∈ [− M , ∞ ) Return-Contingency, Penalty, Escrow for Solvency
Manager
Transfer 1 Unit
Investor
Fee Scheme y :[0, ∞ ) → [− M , ∞ ] Give Back Return
Maximal Penalty
w ( y ) ≡ max[− min y( x ),0] x∈[0,∞ )
Pay Fee Pay Penalty
Escrow Account Unmanageable, Alpha 0
Generate Return
x ∈ [0, ∞ ) Unit (Alpha x − 1 )
x
to Investor
y( x ) to Manager if y( x ) is Positive
− y( x )
from Escrow if Non-Positive
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Real Fee Scheme
‘2:20’ Scheme Asymmetry, No Penalty, Convexity, High-Powered
y= ( x ) 0.2 x + 0.02
Criticisms (Warren Buffet): ‘2:20’ Makes Manager More Risk-Taking by Side Contracting with Third Party. We Should Change ‘2:20’ Scheme to
‘Fulcrum’ Scheme Symmetric, Positive Penalty, Linear, Low-Powered
y(= x ) k ( x − 1)
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Side Contracting: Performance Mimicry Randomize Return Cumulative Distribution F :[0, ∞ ) → [0,1] E[ z | F ] = x
Side Contract F F :[0, ∞ ) → [0,1]
HF Return x
E[ z | F ] = x
Give x to Third Party (Arbitrageur) Return z is Randomly Determined According to F
Give z to Investor Receive Fee y( z )
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Example (Lo (2001)) Capital Decimation Partners (CDP) Unskilled Can Generate Alpha
Unskilled Manager Safe Asset 1 Unit (HF)
Escrow
p > 0 with Prob. 1 − p 1− p Covered Option Sale: Transfer Safe Asset to Arbitrageur if S&P500 Index Decline 20% (Prob. Price
p
Safe Asset p Unit
Price
p2
Safe Asset
Price
p3
p2
Unit
p)
Arbitrageur
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Previous Works: Hedge Fund Never Survives
Foster + Young (08/09)
With No CG Tax, No Scheme Can Solve Incentive Problem
Media:
FT (18/3/08), NYT (3/8/08) “HF Never Survives. We Need More Transparency!”
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Results of This Paper ・CG Tax Functions ・With No CG Tax, We Cannot Solve Incentive Problem ( a la Foster + Young) ・With Positive CGT Rate t > 0 , We Can Solve Incentive Problem ・Constrained Optimal Scheme ・Fulcrum After Taxation: Low-Powered ・Income Tax on Fee Functions ・Income Tax Rate Should be Greater than CG Tax Rate, τ > t ・Manager Selects Constrained Optimal Scheme Voluntarily ・Equity Stake Functions ・We Can Solve Incentive Problem without Fulcrum
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Assumption: Separate Management
Unskilled Manager
Skilled Manager HF 1
Personal Fund M − w( y )
Action a Cost c(a )
Escrow w( y )
Action a = 0
Action a ′ { M − w ( y )}c(a ′ )
HF 1
Personal Fund M − w( y ) Action a ′ = 0
Action 0
Action 0 HF Return
HF Return
1
a+1
Alpha 0 Return
Return Side Contract F
{ M − w ( y )}(a′ + 1)
Random Return
z
Escrow w( y )
Side Contract F Return
w( y ) Alpha 0
Random Return
z
M − w( y ) Alpha 0 Return
w( y ) Alpha 0
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Incentive Problem: Five Constraints
①
Skilled Entry
②
Unskilled Exit
③
Investor Entry
④
Welfare Improvement
⑤
Skilled Non-mimicry: Skilled Needs No Third-Party Side Contract
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Skilled Entry: V ( y , t ,τ ) ≥ V ( t ) HF Industry
Outside Opportunity Manage Entire Personal Fund M Payoff V ( t ) ≡ M {(1 − t )a (1 − t ) − c(a (1 − t ))}
CG Tax tMa (1 − t )
Put w ( y ) in Escrow, Unmanageable Payoff V ( y ,τ , t ) ≡ min[(1 − τ ) y(a ( y ,τ ) + 1), y( a * ( y ,τ ) + 1)] − c( a * ( y ,τ )) *
+{ M − w ( y )}{(1 − t )a (1 − t ) − c(a (1 − t ))} CG Tax t { M − w ( y )}a (1 − t ) Income Tax max[τ y(a * ( y ,τ ) + 1), 0]
Skilled
a (1 − t ) Maximize (1 − t )a − c(a ) a * ( y ,τ ) Maximize (1 − τ ) y(a + 1) − c( a )
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Unskilled Exit: max E[min[(1 − τ ) y( z ), y( z )]| F ] ≤ 0 F ∈Φ
HF Industry No Skill but Side Contracting
Payoff max E[min[(1 − τ ) y( z ), y( z )] | F ] F ∈Φ
Outside Opportunity CG Tax 0 Income Tax E[max[τ y( z ), 0)] | F ]
Payoff 0
UnSkilled
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Investor Entry: U ( y , t ,τ ) ≥ 0 , i.e., a * ( y ,τ ) ≥ y(a * ( y ,τ ) + 1)
HF Industry Payoff U ( y , t ,τ ) ≡ min[(1 − t ){a * ( y ,τ ) − y(a * ( y ,τ ) + 1)}, a * ( y ,τ ) − y(a * ( y ,τ ) + 1)]
Outside Opportunity CG Tax max[ t {a ( y ,τ ) − y( a ( y ,τ ) + 1)},0] *
Payoff 0
Investor
*
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Welfare Improvement: S ( y , t ,τ ) > S
HF Industry
No HF (Status Quo) t= τ= 0
Surplus S ≡ M {a (1) − c(a (1))}
Surplus Increases
Surplus S ( y , t ,τ ) ≡ a * ( y ,τ ) − c(a * ( y ,τ )) +{ M − w ( y )}{a (1 − t ) − c(a (1 − t ))}
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No Capital Gain Tax: Impossibility Theorem: Suppose CGT Rate t = 0 . Then, There Exists No Fee Scheme that Satisfies Skilled Entry, Unskilled Exit, and Welfare Improvement. Outline of Proof: Assume a > 0 is only available, y(0) = − w ( y ) Unskilled (CDP)
Alpha a Pro.
1 a +1
Alpha −1 Pro.
y(a + 1)
y(0)
Skilled
a a +1
Put w ( y ) in Escrow
Alpha a
y(a + 1) − C (a )
− w ( y ){a − c(a )}
Skilled Entry Unskilled Exit 1 a y(a + 1) ≤ w( y ) a+1 a+1
y(a + 1) ≥ w ( y )a + {1 − w ( y )}c(a ) > w ( y )a
Contradiction!
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Positive Capital Gain Tax: Possibility Theorem: There exist Tax Rates ( t ,τ ) ∈ [0,1]2 and Fee Scheme y ∈ Y * (τ ) that satisfy All Constraints.
Outline of Proof: Assume a > 0 is only available Skilled’s Outside Opportunity Manage Entire Personal Fund
Pay CG Tax tMa
Skilled’s HF
Save CG Tax
Put w ( y ) in Escrow
tw ( y )a
Pay CG Tax t { M − w ( y )}a
“Larger Fund + Less Active” is Better Than “Smaller Fund + More Active”
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Constrained Optimization: ( y* , t * ,τ * )
(1) Fulcrum Scheme after Taxation
y( x )= x − 1
for all x ∈ [1, ∞ )
y( x ) = (1 − τ )( x − 1)
for all x ∈ [0,1)
V ( y , t ,τ ) = V ( t )
(2) Skilled Entry Binding
We Specify ( y , t ,τ ) = ( y , t ,τ ) As Maximizing Surplus S ( y , t ,τ ) Subject to (1) and (2) *
*
*
Theorem: ( y* , t * ,τ * ) Satisfies All Constraints. There exists No ( y , t ,τ ) that Satisfies All Constraints and S ( y , t ,τ ) > S ( y * , t * ,τ * ) .
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Constrained Optimization: Properties
・Manager is Willing to Select y* Voluntarily: y* is the Only Scheme that Satisfies Skilled Entry, Unskilled Exit, Investor Entry, and Skilled Non-mimicry.
・Manager Prefers to Put Personal Fund in Escrow as Large as Possible, Distorting Welfare.
・Income Tax Rate τ * is Greater than CG Tax Rate t * : High Income Tax Rate
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Another Assumption: Equity Stake
Skilled Manager HF 1 + M − w( y )
Unskilled Manager
Escrow w( y )
HF 1 + M − w( y )
Action a {1 + M − w ( y )}c(a )
Escrow w( y )
Action 0 Action 0
Action 0 HF Return
1 + M − w( y )
HF Return
{1 + M − w ( y )}(a + 1)
Zero Alpha
Side Contract F
Random Return
z,
{ M − w ( y )}z
Side Contract F Return
w( y ) Zero Alpha
Return Random Return
z,
{ M − w ( y )}z
w( y ) Zero Alpha
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We Don’t Need Penalty, But CG Tax and Big Stake
Theorem: Suppose CGT Rate t = 0 . Then, There Exists No Fee Scheme that Satisfies Skilled Entry, Unskilled Exit, and Welfare Improvement. Additional Assumption: a > 0 is only available, τ = 0 Theorem: For Sufficiently Large Personal Fund M , There exist ( t , y ) that Levy No Penalty but Satisfy All Constraints. Outline of Proof: CDP Must be Covered by Not only Investor’s Fund But also Personal Fund Return M ( a + 1) CG Tax tMa
Equity Stake M (Sufficiently Large)
Unskilled’s CDP (Prob.
Covered by Equity Stake
1 ) a +1
Expected Return M − tMa < M
Return 0 (Prob.
a ) a +1
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Further Comments
Investor’s Optimization ・Investor Prefers higher-Powered and More Penalty than Constrained Optimal Scheme. ・By Transferring Total Tax Revenue to Investor, Government Can Incentivize Investor to Select Constrained Optimal Scheme Voluntarily. ・Investor’s Payoff May be Greater than Manager’s Payoff per Unit: Manager May Fold HF Business.
Entry Cost Entry Cost Functions, if, and Only if, It is Non-Pecuniary!
