Incentives in Hedge Funds

Transcription

1 1 February 4, 2010 Incentives in Hedge Funds Hitoshi Matsushima Faculty of Economics, University of Tokyo February 3, 2010

2 2 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 Manager Skilled Type Select Alpha (Action) a [0, ) with Non-Pecuniary Cost Ca ( ) Unskilled Type Alpha 0

3 3 Incentive Problem Hidden Type Investor Cannot Identify whether Manager is skilled or not Hidden Activity Investor Cannot Observe Manager s Activity Q: Can We Solve Incentive Problem? HF Survives: Skilled Entry Skilled and Investor Pareto Improvement Positive Capital Gain Tax (Fulcrum Scheme or Equity Stake) Unskilled Exit No HF No CG Tax Investor Entry Unskilled A: Yes, but We Need Capital Gain Tax!

4 4 Manager s Incentive Fee Scheme y:[0, ) [ M, ], yx ( ) [ M, ) Return-Contingency, Penalty, Escrow for Solvency Maximal Penalty x [0, ) Escrow Account Unmanageable, Alpha 0 Manager w( y) max[ min y( x),0] Generate Return x [0, ) Unit (Alpha x 1) Transfer 1 Unit Fee Scheme y :[0, ) [ M, ] Investor Give Back Return x to Investor Pay Fee yx ( ) to Manager if yx ( ) is Positive Pay Penalty yx ( ) from Escrow if Non-Positive

5 5 Real Fee Scheme 2:20 Scheme Asymmetry, No Penalty, Convexity, High-Powered yx ( ) = 0.2x 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 yx ( ) = kx ( 1)

6 6 Side Contracting: Performance Mimicry Randomize Return Cumulative Distribution F :[0, ) [0,1] E[ z F] = x Side Contract F HF Return x F :[0, ) [0,1] E[ z F] = x Give x to Third Party (Arbitrageur) Give z to Investor Receive Fee y( z ) Return z is Randomly Determined According to F

7 7 Example (Lo (2001)) Capital Decimation Partners (CDP) p Unskilled Can Generate Alpha 0 1 p > with Prob. 1 p Unskilled Manager Safe Asset 1 Unit (HF) Covered Option Sale: Transfer Safe Asset to Arbitrageur if S&P500 Index Decline 20% (Prob. p ) Price p Escrow Safe Asset p Unit Price 2 p Arbitrageur Safe Asset 2 p Unit Price 3 p

8 8 Previous Works: Hedge Fund Never Survives Foster + Young (08/09) With No CG Tax, No Scheme Can Solve Incentive Problem Medias: FT (18/3/08), NYT (3/8/08) HF Never Survives. We Need More Transparency!

9 9 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

10 10 Assumption: Separate Management Skilled Manager Unskilled Manager HF 1 Personal Fund M w( y) Escrow w( y ) HF 1 Personal Fund M w( y) Escrow w( y ) Action a Cost ca ( ) Action a { M w( y)} c( a ) Action 0 Action a = 0 Action a = 0 Action 0 HF Return a + 1 Side Contract F Return { M w( y)}( a + 1) HF Return 1 Alpha 0 Side Contract F Return M w( y) Alpha 0 Random Return z Return w( y ) Alpha 0 Random Return z Return w( y ) Alpha 0

11 11 Incentive Problem: Five Constraints Skilled Entry Unskilled Exit Investor Entry Welfare Improvement Skilled Non-mimicry: Skilled Needs No Third-Party Side Contract

12 12 Skilled Entry: V( y, t, τ ) V( t) Outside Opportunity Manage Entire Personal Fund M Payoff V() t M{(1 t)(1 a t) c( a (1 t))} HF Industry 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 tmta (1 t) CG Tax tm { wy ( )} ta (1 t) Income Tax * max[ τ ya ( ( y, τ ) + 1),0] Skilled a (1 t) Maximize (1 ta ) ca ( ) a * ( y, τ ) Maximize (1 τ ) ya ( + 1) ca ( )

13 13 Unskilled Exit: max E[min[(1 τ ) y( z), y( z)] F] 0 F Φ HF Industry No Skill but Side Contracting Outside Opportunity Payoff 0 Payoff max E[min[(1 τ ) yz ( ), yz ( )] F] F Φ CG Tax 0 Income Tax E[max[ τ yz ( ),0)] F] UnSkilled

14 14 Investor Entry: U( y, t, τ ) 0, i.e., * * a y τ y a y (, ) ( (, τ) + 1) HF Industry Outside Opportunity Payoff 0 Payoff * * τ τ τ U( y, t, ) min[(1 t){ a ( y, ) y( a ( y, ) + 1)}, CG Tax * * a y τ y a y (, ) ( (, τ) + 1)] * * max[ ta { ( y, τ) ya ( ( y, τ) + 1)},0] Investor

15 15 Welfare Improvement: S( yt,, τ ) > S No HF (Status Quo) t = τ = 0 Surplus S M{ a (1) c( a (1))} Surplus Increases HF Industry Surplus * * S( y, t, τ ) a ( y, τ) c( a ( y, τ)) + { M w( y)}{ a (1 t) c( a (1 t))}

16 16 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) Skilled Alpha a Pro. ya+ ( 1) 1 a + 1 Alpha 1 Pro. y (0) a a + 1 Alpha a ya ( + 1) Ca ( ) Put w( y ) in Escrow wy ( ){ a ca ( )} Unskilled Exit 1 a y( a+ 1) w( y) a+ 1 a+ 1 Skilled Entry y( a+ 1) w( y) a+ {1 w( y)} ca ( ) > wya ( ) Contradiction!

17 17 Positive Capital Gain Tax: Possibility Theorem: There exist Tax Rates satisfy All Constraints. 2 (, t τ ) [0,1] and Fee Scheme y Y * ( τ ) that Outline of Proof: Assume a > 0 is only available Skilled s Outside Opportunity Manage Entire Personal Fund Save CG Tax tw( y) a Skilled s HF Put w( y ) in Escrow Pay CG Tax tma Pay CG Tax t{ M w( y)} a

18 18 Constrained Optimization: * * * ( y, t, τ ) (1) Fulcrum Scheme after Taxation yx ( ) = x 1 for all x [1, ) yx ( ) = (1 τ )( x 1) for all x [0,1) (2) Skilled Entry Binding V( y, t, τ ) = V( t) We Specify = As Maximizing Surplus S( ytτ,, ) Subject to (1) and (2) * * * ( yt,, τ ) ( y, t, τ ) Theorem: * * * ( y, t, τ ) Satisfies All Constraints. There exists No ( ytτ,, ) that Satisfies All Constraints and * * * S( y, t, τ ) > S( y, t, τ ).

19 19 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

20 20 Another Assumption: Equity Stake Skilled Manager Unskilled Manager HF 1 + M w( y) Escrow w( y ) HF 1 + M w( y) Escrow ( ) w y Action a {1 + M w( y)} c( a) HF Return {1 + M w( y)}( a + 1) Action 0 Action 0 HF Return 1 + M w( y) Zero Alpha Action 0 Side Contract F Side Contract F Random Return z Return w( y ) Zero Alpha Random Return z Return w( y ) Zero Alpha

21 21 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: 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 Equity Stake M Unskilled s CDP Covered by Equity Stake Return Ma+ ( 1) CG Tax tma (Prob. 1 a + 1 ) Return 0 Expected Return M tma< M (Prob. a a + 1 )

22 22 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!