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- 1. Problem Model Solution Welfare Implications The Incentives of Hedge Fund Fees and High-Water Marks Paolo Guasoni (Joint work with Jan Obłoj) Boston University and Dublin City University Workshop on Foundations of Mathematical Finance January 12th , 2010
- 2. Problem Model Solution Welfare Implications Background Paul Krugman, How Did Economists Get It So Wrong? NY Times Magazine, September 2, 2009 “...the economics profession went astray because economists, as a group, mistook beauty, clad in impressive-looking mathematics, for truth.” “Economics, as a ﬁeld, got in trouble because economists were seduced by the vision of a perfect, frictionless market system.”
- 3. Problem Model Solution Welfare Implications Background Paul Krugman, How Did Economists Get It So Wrong? NY Times Magazine, September 2, 2009 “...the economics profession went astray because economists, as a group, mistook beauty, clad in impressive-looking mathematics, for truth.” “Economics, as a ﬁeld, got in trouble because economists were seduced by the vision of a perfect, frictionless market system.” John Cochrane, How Did Krugman Get It So Wrong? “No, the problem is that we don’t have enough math.” “Frictions are just bloody hard with the mathematical tools we have now.”
- 4. Problem Model Solution Welfare Implications Background Paul Krugman, How Did Economists Get It So Wrong? NY Times Magazine, September 2, 2009 “...the economics profession went astray because economists, as a group, mistook beauty, clad in impressive-looking mathematics, for truth.” “Economics, as a ﬁeld, got in trouble because economists were seduced by the vision of a perfect, frictionless market system.” John Cochrane, How Did Krugman Get It So Wrong? “No, the problem is that we don’t have enough math.” “Frictions are just bloody hard with the mathematical tools we have now.” Make Frictions Tractable.
- 5. Problem Model Solution Welfare Implications Background Paul Krugman, How Did Economists Get It So Wrong? NY Times Magazine, September 2, 2009 “...the economics profession went astray because economists, as a group, mistook beauty, clad in impressive-looking mathematics, for truth.” “Economics, as a ﬁeld, got in trouble because economists were seduced by the vision of a perfect, frictionless market system.” John Cochrane, How Did Krugman Get It So Wrong? “No, the problem is that we don’t have enough math.” “Frictions are just bloody hard with the mathematical tools we have now.” Make Frictions Tractable. One Step at a Time.
- 6. Problem Model Solution Welfare Implications Outline High-Water Marks: Performance Fees for Hedge Funds Managers.
- 7. Problem Model Solution Welfare Implications Outline High-Water Marks: Performance Fees for Hedge Funds Managers. Model: Power Utility with Long Horizon.
- 8. Problem Model Solution Welfare Implications Outline High-Water Marks: Performance Fees for Hedge Funds Managers. Model: Power Utility with Long Horizon. Solution: Effective Risk Aversion and Drawdown Constraints.
- 9. Problem Model Solution Welfare Implications Outline High-Water Marks: Performance Fees for Hedge Funds Managers. Model: Power Utility with Long Horizon. Solution: Effective Risk Aversion and Drawdown Constraints. Fees and Welfare: Stackelberg Equilibrium between Investor and Manager
- 10. Problem Model Solution Welfare Implications Two and Twenty Hedge Funds Managers receive two types of fees.
- 11. Problem Model Solution Welfare Implications Two and Twenty Hedge Funds Managers receive two types of fees. Regular fees, like Mutual Funds.
- 12. Problem Model Solution Welfare Implications Two and Twenty Hedge Funds Managers receive two types of fees. Regular fees, like Mutual Funds. Unlike Mutual Funds, Performance Fees.
- 13. Problem Model Solution Welfare Implications Two and Twenty Hedge Funds Managers receive two types of fees. Regular fees, like Mutual Funds. Unlike Mutual Funds, Performance Fees. Regular fees: a fraction ϕ of assets under management. 2% typical.
- 14. Problem Model Solution Welfare Implications Two and Twenty Hedge Funds Managers receive two types of fees. Regular fees, like Mutual Funds. Unlike Mutual Funds, Performance Fees. Regular fees: a fraction ϕ of assets under management. 2% typical. Performance fees: a fraction α of trading proﬁts. 20% typical.
- 15. Problem Model Solution Welfare Implications Two and Twenty Hedge Funds Managers receive two types of fees. Regular fees, like Mutual Funds. Unlike Mutual Funds, Performance Fees. Regular fees: a fraction ϕ of assets under management. 2% typical. Performance fees: a fraction α of trading proﬁts. 20% typical. High-Water Marks: Performance fees paid after losses recovered.
- 16. Problem Model Solution Welfare Implications High-Water Marks Time Gross Net High-Water Mark Fees 0 100 100 100 0 1 110 108 108 2 2 100 100 108 2 3 118 116 116 4 Fund assets grow from 100 to 110. The manager is paid 2, leaving 108 to the fund.
- 17. Problem Model Solution Welfare Implications High-Water Marks Time Gross Net High-Water Mark Fees 0 100 100 100 0 1 110 108 108 2 2 100 100 108 2 3 118 116 116 4 Fund assets grow from 100 to 110. The manager is paid 2, leaving 108 to the fund. Fund drops from 108 to 100. No fees paid, nor past fees reimbursed.
- 18. Problem Model Solution Welfare Implications High-Water Marks Time Gross Net High-Water Mark Fees 0 100 100 100 0 1 110 108 108 2 2 100 100 108 2 3 118 116 116 4 Fund assets grow from 100 to 110. The manager is paid 2, leaving 108 to the fund. Fund drops from 108 to 100. No fees paid, nor past fees reimbursed. Fund recovers from 100 to 118. Fees paid only on increase from 108 to 118. Manager receives 2.
- 19. Problem Model Solution Welfare Implications High-Water Marks 2.5 2.0 1.5 1.0 0.5 20 40 60 80 100
- 20. Problem Model Solution Welfare Implications Risk Shifting? Manager shares investors’ proﬁts, not losses. Does manager take more risk to increase proﬁts?
- 21. Problem Model Solution Welfare Implications Risk Shifting? Manager shares investors’ proﬁts, not losses. Does manager take more risk to increase proﬁts? Option Pricing Intuition: Manager has a call option on the fund value. Option value increases with volatility. More risk is better.
- 22. Problem Model Solution Welfare Implications Risk Shifting? Manager shares investors’ proﬁts, not losses. Does manager take more risk to increase proﬁts? Option Pricing Intuition: Manager has a call option on the fund value. Option value increases with volatility. More risk is better. Static, Complete Market Fallacy: Manager has multiple call options.
- 23. Problem Model Solution Welfare Implications Risk Shifting? Manager shares investors’ proﬁts, not losses. Does manager take more risk to increase proﬁts? Option Pricing Intuition: Manager has a call option on the fund value. Option value increases with volatility. More risk is better. Static, Complete Market Fallacy: Manager has multiple call options. High-Water Mark: future strikes depend on past actions.
- 24. Problem Model Solution Welfare Implications Risk Shifting? Manager shares investors’ proﬁts, not losses. Does manager take more risk to increase proﬁts? Option Pricing Intuition: Manager has a call option on the fund value. Option value increases with volatility. More risk is better. Static, Complete Market Fallacy: Manager has multiple call options. High-Water Mark: future strikes depend on past actions. Option unhedgeable: cannot short (your!) hedge fund.
- 25. Problem Model Solution Welfare Implications Questions Portfolio: Effect of fees and risk-aversion?
- 26. Problem Model Solution Welfare Implications Questions Portfolio: Effect of fees and risk-aversion? Welfare: Effect on investors and managers?
- 27. Problem Model Solution Welfare Implications Questions Portfolio: Effect of fees and risk-aversion? Welfare: Effect on investors and managers? High-Water Mark Contracts: consistent with any investor’s objective?
- 28. Problem Model Solution Welfare Implications Answers Goetzmann, Ingersoll and Ross (2003): Risk-neutral value of management contract (future fees). Exogenous portfolio and fund ﬂows.
- 29. Problem Model Solution Welfare Implications Answers Goetzmann, Ingersoll and Ross (2003): Risk-neutral value of management contract (future fees). Exogenous portfolio and fund ﬂows. High-Water Mark contract worth 10% to 20% of fund.
- 30. Problem Model Solution Welfare Implications Answers Goetzmann, Ingersoll and Ross (2003): Risk-neutral value of management contract (future fees). Exogenous portfolio and fund ﬂows. High-Water Mark contract worth 10% to 20% of fund. Panageas and Westerﬁeld (2009): Exogenous risky and risk-free asset. Optimal portfolio for a risk-neutral manager. Fees cannot be invested in fund.
- 31. Problem Model Solution Welfare Implications Answers Goetzmann, Ingersoll and Ross (2003): Risk-neutral value of management contract (future fees). Exogenous portfolio and fund ﬂows. High-Water Mark contract worth 10% to 20% of fund. Panageas and Westerﬁeld (2009): Exogenous risky and risk-free asset. Optimal portfolio for a risk-neutral manager. Fees cannot be invested in fund. Constant risky/risk-free ratio optimal. Merton proportion does not depend on fee size. Same solution for manager with Hindy-Huang utility.
- 32. Problem Model Solution Welfare Implications This Paper Manager with Power Utility and Long Horizon. Exogenous risky and risk-free asset. Fees cannot be invested in fund.
- 33. Problem Model Solution Welfare Implications This Paper Manager with Power Utility and Long Horizon. Exogenous risky and risk-free asset. Fees cannot be invested in fund. Optimal Portfolio: 1 µ π= γ ∗ σ2 ∗ γ =(1 − α)γ + α γ =Manager’s Risk Aversion α =Performance Fee (e.g. 20%)
- 34. Problem Model Solution Welfare Implications This Paper Manager with Power Utility and Long Horizon. Exogenous risky and risk-free asset. Fees cannot be invested in fund. Optimal Portfolio: 1 µ π= γ ∗ σ2 ∗ γ =(1 − α)γ + α γ =Manager’s Risk Aversion α =Performance Fee (e.g. 20%) Manager behaves as if owned fund, but were more myopic (γ ∗ weighted average of γ and 1).
- 35. Problem Model Solution Welfare Implications This Paper Manager with Power Utility and Long Horizon. Exogenous risky and risk-free asset. Fees cannot be invested in fund. Optimal Portfolio: 1 µ π= γ ∗ σ2 ∗ γ =(1 − α)γ + α γ =Manager’s Risk Aversion α =Performance Fee (e.g. 20%) Manager behaves as if owned fund, but were more myopic (γ ∗ weighted average of γ and 1). Performance fees α matter. Regular fees ϕ don’t.
- 36. Problem Model Solution Welfare Implications Three Problems, One Solution Power utility, long horizon. No regular fees.
- 37. Problem Model Solution Welfare Implications Three Problems, One Solution Power utility, long horizon. No regular fees. 1 Manager maximizes utility of performance fees. Risk Aversion γ.
- 38. Problem Model Solution Welfare Implications Three Problems, One Solution Power utility, long horizon. No regular fees. 1 Manager maximizes utility of performance fees. Risk Aversion γ. 2 Investor maximizes utility of wealth. Pays no fees. Risk Aversion γ ∗ = (1 − α)γ + α.
- 39. Problem Model Solution Welfare Implications Three Problems, One Solution Power utility, long horizon. No regular fees. 1 Manager maximizes utility of performance fees. Risk Aversion γ. 2 Investor maximizes utility of wealth. Pays no fees. Risk Aversion γ ∗ = (1 − α)γ + α. 3 Investor maximizes utility of wealth. Pays no fees. Risk Aversion γ. Maximum Drawdown 1 − α.
- 40. Problem Model Solution Welfare Implications Three Problems, One Solution Power utility, long horizon. No regular fees. 1 Manager maximizes utility of performance fees. Risk Aversion γ. 2 Investor maximizes utility of wealth. Pays no fees. Risk Aversion γ ∗ = (1 − α)γ + α. 3 Investor maximizes utility of wealth. Pays no fees. Risk Aversion γ. Maximum Drawdown 1 − α. Same optimal portfolio: 1 µ π= γ ∗ σ2
- 41. Problem Model Solution Welfare Implications Price Dynamics dSt = (r + µ)dt + σdWt (Risky Asset) St dSt α ∗ dXt = (r − ϕ)Xt dt + Xt πt St − rdt − 1−α dXt (Fund) α dFt = rFt dt + ϕXt dt + dX ∗ (Fees) 1−α t Xt∗ = max Xs (High-Water Mark) 0≤s≤t One safe and one risky asset.
- 42. Problem Model Solution Welfare Implications Price Dynamics dSt = (r + µ)dt + σdWt (Risky Asset) St dSt α ∗ dXt = (r − ϕ)Xt dt + Xt πt St − rdt − 1−α dXt (Fund) α dFt = rFt dt + ϕXt dt + dX ∗ (Fees) 1−α t Xt∗ = max Xs (High-Water Mark) 0≤s≤t One safe and one risky asset. Gain split into α for the manager and 1 − α for the fund.
- 43. Problem Model Solution Welfare Implications Price Dynamics dSt = (r + µ)dt + σdWt (Risky Asset) St dSt α ∗ dXt = (r − ϕ)Xt dt + Xt πt St − rdt − 1−α dXt (Fund) α dFt = rFt dt + ϕXt dt + dX ∗ (Fees) 1−α t Xt∗ = max Xs (High-Water Mark) 0≤s≤t One safe and one risky asset. Gain split into α for the manager and 1 − α for the fund. Performance fee is α/(1 − α) of fund increase.
- 44. Problem Model Solution Welfare Implications Dynamics Well Posed? Problem: fund value implicit. Find solution Xt for dSt α dXt = Xt πt − ϕXt dt − dX ∗ St 1−α t
- 45. Problem Model Solution Welfare Implications Dynamics Well Posed? Problem: fund value implicit. Find solution Xt for dSt α dXt = Xt πt − ϕXt dt − dX ∗ St 1−α t Yes. Pathwise construction.
- 46. Problem Model Solution Welfare Implications Dynamics Well Posed? Problem: fund value implicit. Find solution Xt for dSt α dXt = Xt πt − ϕXt dt − dX ∗ St 1−α t Yes. Pathwise construction. Proposition ∗ The unique solution is Xt = eRt −αRt , where: t t σ2 2 Rt = µπs − π − ϕ ds + σ πs dWs 0 2 s 0 is the cumulative log return.
- 47. Problem Model Solution Welfare Implications Fund Value Explicit Lemma Let Y be a continuous process, and α > 0. Then Yt + 1−α Yt∗ = Rt if and only if Yt = Rt − αRt∗ . α
- 48. Problem Model Solution Welfare Implications Fund Value Explicit Lemma Let Y be a continuous process, and α > 0. Then Yt + 1−α Yt∗ = Rt if and only if Yt = Rt − αRt∗ . α Proof. Follows from: α α 1 Rt∗ = sup Ys + sup Yu = Yt∗ + Yt∗ = Y∗ s≤t 1 − α u≤s 1−α 1−α t
- 49. Problem Model Solution Welfare Implications Fund Value Explicit Lemma Let Y be a continuous process, and α > 0. Then Yt + 1−α Yt∗ = Rt if and only if Yt = Rt − αRt∗ . α Proof. Follows from: α α 1 Rt∗ = sup Ys + sup Yu = Yt∗ + Yt∗ = Y∗ s≤t 1 − α u≤s 1−α 1−α t Apply Lemma to cumulative log return.
- 50. Problem Model Solution Welfare Implications Long Horizon The manager chooses the portfolio π which maximizes expected power utility from fees at a long horizon.
- 51. Problem Model Solution Welfare Implications Long Horizon The manager chooses the portfolio π which maximizes expected power utility from fees at a long horizon. Maximizes the long-run objective: 1 p max lim log E[FT ] = λ π T →∞ pT
- 52. Problem Model Solution Welfare Implications Long Horizon The manager chooses the portfolio π which maximizes expected power utility from fees at a long horizon. Maximizes the long-run objective: 1 p max lim log E[FT ] = λ π T →∞ pT Dumas and Luciano (1991), Grossman and Vila (1992), Grossman and Zhou (1993). Risk-Sensitive Control: Bielecki and Pliska (1999) and many others.
- 53. Problem Model Solution Welfare Implications Long Horizon The manager chooses the portfolio π which maximizes expected power utility from fees at a long horizon. Maximizes the long-run objective: 1 p max lim log E[FT ] = λ π T →∞ pT Dumas and Luciano (1991), Grossman and Vila (1992), Grossman and Zhou (1993). Risk-Sensitive Control: Bielecki and Pliska (1999) and many others. Certainty Equivalent Rate: λ as risk-free rate above which the manager would prefer to retire and invest at such a rate, and below which would rather keep his job.
- 54. Problem Model Solution Welfare Implications Long Horizon The manager chooses the portfolio π which maximizes expected power utility from fees at a long horizon. Maximizes the long-run objective: 1 p max lim log E[FT ] = λ π T →∞ pT Dumas and Luciano (1991), Grossman and Vila (1992), Grossman and Zhou (1993). Risk-Sensitive Control: Bielecki and Pliska (1999) and many others. Certainty Equivalent Rate: λ as risk-free rate above which the manager would prefer to retire and invest at such a rate, and below which would rather keep his job. 2 1 µ λ = r + γ 2σ2 for Merton problem with risk-aversion γ = 1 − p.
- 55. Problem Model Solution Welfare Implications Solving It Set r = 0 and ϕ = 0 to simplify notation.
- 56. Problem Model Solution Welfare Implications Solving It Set r = 0 and ϕ = 0 to simplify notation. Cumulative fees are a fraction of the increase in the fund: α Ft = (X ∗ − X0 ) ∗ 1−α t
- 57. Problem Model Solution Welfare Implications Solving It Set r = 0 and ϕ = 0 to simplify notation. Cumulative fees are a fraction of the increase in the fund: α Ft = (X ∗ − X0 ) ∗ 1−α t Thus, the manager’s objective is equivalent to: 1 ∗ max lim log E[(XT )p ] π T →∞ pT
- 58. Problem Model Solution Welfare Implications Solving It Set r = 0 and ϕ = 0 to simplify notation. Cumulative fees are a fraction of the increase in the fund: α Ft = (X ∗ − X0 ) ∗ 1−α t Thus, the manager’s objective is equivalent to: 1 ∗ max lim log E[(XT )p ] π T →∞ pT Finite-horizon value function: 1 V (x, z, t) = sup E[XT p |Xt = x, Xt∗ = z] ∗ π p 1 dV (Xt , Xt∗ , t) = Vt dt + Vx dXt + Vxx d X t + Vz dXt∗ 2 2 = Vt dt + Vz − α 1−α Vx dXt∗ + Vx Xt (πt µ − ϕ)dt + Vxx σ πt2 Xt2 2
- 59. Problem Model Solution Welfare Implications Dynamic Programming Hamilton-Jacobi-Bellman equation: 2 Vt + supπ xVx (πµ − ϕ) + Vxx σ π 2 x 2 ) 2 x <z α Vz = 1−α Vx x =z V = z p /p x =0 V = z p /p t =T
- 60. Problem Model Solution Welfare Implications Dynamic Programming Hamilton-Jacobi-Bellman equation: 2 Vt + supπ xVx (πµ − ϕ) + Vxx σ π 2 x 2 ) 2 x <z α Vz = 1−α Vx x =z V = z p /p x =0 V = z p /p t =T Maximize in π, and use homogeneity V (x, z, t) = z p /pV (x/z, 1, t) = z p /pu(x/z, 1, t). 2 ut − ϕxux − µ22 ux = 0 x ∈ (0, 1) 2σ uxx ux (1, t) = p(1 − α)u(1, t) t ∈ (0, T ) u(x, T ) = 1 x ∈ (0, 1) u(0, t) = 1 t ∈ (0, T )
- 61. Problem Model Solution Welfare Implications Long-Run Heuristics Long-run limit. Guess a solution of the form u(t, x) = ce−pβt w(x), forgetting the terminal condition: 2 2 µ wx −pβw − ϕxwx − 2σ2 wxx = 0 for x < 1 wx (1) = p(1 − α)w(1)
- 62. Problem Model Solution Welfare Implications Long-Run Heuristics Long-run limit. Guess a solution of the form u(t, x) = ce−pβt w(x), forgetting the terminal condition: 2 2 µ wx −pβw − ϕxwx − 2σ2 wxx = 0 for x < 1 wx (1) = p(1 − α)w(1) This equation is time-homogeneous, but β is unknown.
- 63. Problem Model Solution Welfare Implications Long-Run Heuristics Long-run limit. Guess a solution of the form u(t, x) = ce−pβt w(x), forgetting the terminal condition: 2 2 µ wx −pβw − ϕxwx − 2σ2 wxx = 0 for x < 1 wx (1) = p(1 − α)w(1) This equation is time-homogeneous, but β is unknown. Any β with a solution w is an upper bound on the rate λ.
- 64. Problem Model Solution Welfare Implications Long-Run Heuristics Long-run limit. Guess a solution of the form u(t, x) = ce−pβt w(x), forgetting the terminal condition: 2 2 µ wx −pβw − ϕxwx − 2σ2 wxx = 0 for x < 1 wx (1) = p(1 − α)w(1) This equation is time-homogeneous, but β is unknown. Any β with a solution w is an upper bound on the rate λ. Candidate long-run value function: the solution w with the lowest β.
- 65. Problem Model Solution Welfare Implications Long-Run Heuristics Long-run limit. Guess a solution of the form u(t, x) = ce−pβt w(x), forgetting the terminal condition: 2 2 µ wx −pβw − ϕxwx − 2σ2 wxx = 0 for x < 1 wx (1) = p(1 − α)w(1) This equation is time-homogeneous, but β is unknown. Any β with a solution w is an upper bound on the rate λ. Candidate long-run value function: the solution w with the lowest β. 1−α µ2 w(x) = x p(1−α) , for β = (1−α)γ+α 2σ 2 − ϕ(1 − α).
- 66. Problem Model Solution Welfare Implications Veriﬁcation Theorem µ2 1 1 If ϕ − r < 2σ 2 min γ∗ , γ∗ 2 , then for any portfolio π: 1 1 µ2 lim log E (FT )p ≤ max (1 − α) π + r − ϕ ,r T →∞ pT γ ∗ 2σ 2 α 1 µ2 Under the nondegeneracy condition ϕ + 1−α r < γ∗ 2σ 2 , the µ unique optimal solution is π = γ1∗ σ2 . ˆ
- 67. Problem Model Solution Welfare Implications Veriﬁcation Theorem µ2 1 1 If ϕ − r < 2σ 2 min γ∗ , γ∗ 2 , then for any portfolio π: 1 1 µ2 lim log E (FT )p ≤ max (1 − α) π + r − ϕ ,r T →∞ pT γ ∗ 2σ 2 α 1 µ2 Under the nondegeneracy condition ϕ + 1−α r < γ∗ 2σ 2 , the µ unique optimal solution is π = γ1∗ σ2 . ˆ Martingale argument. No HJB equation needed.
- 68. Problem Model Solution Welfare Implications Veriﬁcation Theorem µ2 1 1 If ϕ − r < 2σ 2 min γ∗ , γ∗ 2 , then for any portfolio π: 1 1 µ2 lim log E (FT )p ≤ max (1 − α) π + r − ϕ ,r T →∞ pT γ ∗ 2σ 2 α 1 µ2 Under the nondegeneracy condition ϕ + 1−α r < γ∗ 2σ 2 , the µ unique optimal solution is π = γ1∗ σ2 . ˆ Martingale argument. No HJB equation needed. Show upper bound for any portfolio π (delicate).
- 69. Problem Model Solution Welfare Implications Veriﬁcation Theorem µ2 1 1 If ϕ − r < 2σ 2 min γ∗ , γ∗ 2 , then for any portfolio π: 1 1 µ2 lim log E (FT )p ≤ max (1 − α) π + r − ϕ ,r T →∞ pT γ ∗ 2σ 2 α 1 µ2 Under the nondegeneracy condition ϕ + 1−α r < γ∗ 2σ 2 , the µ unique optimal solution is π = γ1∗ σ2 . ˆ Martingale argument. No HJB equation needed. Show upper bound for any portfolio π (delicate). Check equality for guessed solution (easy).
- 70. Problem Model Solution Welfare Implications Upper Bound (1) Take p > 0 (p < 0 symmetric).
- 71. Problem Model Solution Welfare Implications Upper Bound (1) Take p > 0 (p < 0 symmetric). For any portfolio π: T σ2 2 T ˜ RT = − 0 2 πt dt + 0 σπt d Wt
- 72. Problem Model Solution Welfare Implications Upper Bound (1) Take p > 0 (p < 0 symmetric). For any portfolio π: T σ2 2 T ˜ RT = − 0 2 πt dt + 0 σπt d Wt ˜ Wt = Wt + µ/σt risk-neutral Brownian Motion
- 73. Problem Model Solution Welfare Implications Upper Bound (1) Take p > 0 (p < 0 symmetric). For any portfolio π: T σ2 2 T ˜ RT = − 0 2 πt dt + 0 σπt d Wt ˜ Wt = Wt + µ/σt risk-neutral Brownian Motion Explicit representation: ∗ ∗ µ ˜ µ2 WT − T E [(XT )p ] = E[ep(1−α)RT ] = EQ ep(1−α)RT e σ π 2σ 2
- 74. Problem Model Solution Welfare Implications Upper Bound (1) Take p > 0 (p < 0 symmetric). For any portfolio π: T σ2 2 T ˜ RT = − 0 2 πt dt + 0 σπt d Wt ˜ Wt = Wt + µ/σt risk-neutral Brownian Motion Explicit representation: ∗ ∗ µ ˜ µ2 WT − T E [(XT )p ] = E[ep(1−α)RT ] = EQ ep(1−α)RT e σ π 2σ 2 For δ > 1, Hölder’s inequality: δ−1 1 µ ˜ 2 δ ∗ µ ˜ 2 W − µ2T ∗ δ δ δ−1 W − µ2T σ T p(1−α)RT σ T δp(1−α)RT 2σ EQ e e 2σ ≤ EQ e EQ e
- 75. Problem Model Solution Welfare Implications Upper Bound (1) Take p > 0 (p < 0 symmetric). For any portfolio π: T σ2 2 T ˜ RT = − 0 2 πt dt + 0 σπt d Wt ˜ Wt = Wt + µ/σt risk-neutral Brownian Motion Explicit representation: ∗ ∗ µ ˜ µ2 WT − T E [(XT )p ] = E[ep(1−α)RT ] = EQ ep(1−α)RT e σ π 2σ 2 For δ > 1, Hölder’s inequality: δ−1 1 µ ˜ 2 δ ∗ µ ˜ 2 W − µ2T ∗ δ δ δ−1 W − µ2T σ T p(1−α)RT σ T δp(1−α)RT 2σ EQ e e 2σ ≤ EQ e EQ e 1 µ2 T Second term exponential normal moment. Just e δ−1 2σ2 .
- 76. Problem Model Solution Welfare Implications Upper Bound (2) ∗ δp(1−α)RT Estimate EQ e .
- 77. Problem Model Solution Welfare Implications Upper Bound (2) ∗ δp(1−α)RT Estimate EQ e . Mt = eRt strictly positive continuous local martingale. Converges to zero as t ↑ ∞.
- 78. Problem Model Solution Welfare Implications Upper Bound (2) ∗ δp(1−α)RT Estimate EQ e . Mt = eRt strictly positive continuous local martingale. Converges to zero as t ↑ ∞. Fact: inverse of lifetime supremum (M∞ )−1 uniform on [0, 1]. ∗
- 79. Problem Model Solution Welfare Implications Upper Bound (2) ∗ δp(1−α)RT Estimate EQ e . Mt = eRt strictly positive continuous local martingale. Converges to zero as t ↑ ∞. Fact: inverse of lifetime supremum (M∞ )−1 uniform on [0, 1]. ∗ Thus, for δp(1 − α) < 1: ∗ ∗ 1 EQ eδp(1−α)RT ≤ EQ eδp(1−α)R∞ = 1 − δp(1 − α)
- 80. Problem Model Solution Welfare Implications Upper Bound (2) ∗ δp(1−α)RT Estimate EQ e . Mt = eRt strictly positive continuous local martingale. Converges to zero as t ↑ ∞. Fact: inverse of lifetime supremum (M∞ )−1 uniform on [0, 1]. ∗ Thus, for δp(1 − α) < 1: ∗ ∗ 1 EQ eδp(1−α)RT ≤ EQ eδp(1−α)R∞ = 1 − δp(1 − α) 1 In summary, for 1 < δ < p(1−α) : 1 1 µ2 limlog E (FT )p ≤ π T →∞ pT p(δ − 1) 2σ 2
- 81. Problem Model Solution Welfare Implications Upper Bound (2) ∗ δp(1−α)RT Estimate EQ e . Mt = eRt strictly positive continuous local martingale. Converges to zero as t ↑ ∞. Fact: inverse of lifetime supremum (M∞ )−1 uniform on [0, 1]. ∗ Thus, for δp(1 − α) < 1: ∗ ∗ 1 EQ eδp(1−α)RT ≤ EQ eδp(1−α)R∞ = 1 − δp(1 − α) 1 In summary, for 1 < δ < p(1−α) : 1 1 µ2 limlog E (FT )p ≤ π T →∞ pT p(δ − 1) 2σ 2 1 Thesis follows as δ → p(1−α) .
- 82. Problem Model Solution Welfare Implications High-Water Marks and Drawdowns Imagine fund’s assets Xt and manager’s fees Ft in the same account Ct = Xt + Ft . dSt dCt = (Ct − Ft )πt St
- 83. Problem Model Solution Welfare Implications High-Water Marks and Drawdowns Imagine fund’s assets Xt and manager’s fees Ft in the same account Ct = Xt + Ft . dSt dCt = (Ct − Ft )πt St Fees Ft proportional to high-water mark Xt∗ : α Ft = (X ∗ − X0 ) ∗ 1−α t
- 84. Problem Model Solution Welfare Implications High-Water Marks and Drawdowns Imagine fund’s assets Xt and manager’s fees Ft in the same account Ct = Xt + Ft . dSt dCt = (Ct − Ft )πt St Fees Ft proportional to high-water mark Xt∗ : α Ft = (X ∗ − X0 ) ∗ 1−α t Account increase dCt∗ as fund increase plus fees increase: t t α 1 Ct∗ −C0 = ∗ ∗ (dXs +dFs ) = ∗ + 1 dXs = (X ∗ −X0 ) 0 0 1−α 1−α t
- 85. Problem Model Solution Welfare Implications High-Water Marks and Drawdowns Imagine fund’s assets Xt and manager’s fees Ft in the same account Ct = Xt + Ft . dSt dCt = (Ct − Ft )πt St Fees Ft proportional to high-water mark Xt∗ : α Ft = (X ∗ − X0 ) ∗ 1−α t Account increase dCt∗ as fund increase plus fees increase: t t α 1 Ct∗ −C0 = ∗ ∗ (dXs +dFs ) = ∗ + 1 dXs = (X ∗ −X0 ) 0 0 1−α 1−α t Obvious bound Ct ≥ Ft yields: Ct ≥ α(Ct∗ − X0 )
- 86. Problem Model Solution Welfare Implications High-Water Marks and Drawdowns Imagine fund’s assets Xt and manager’s fees Ft in the same account Ct = Xt + Ft . dSt dCt = (Ct − Ft )πt St Fees Ft proportional to high-water mark Xt∗ : α Ft = (X ∗ − X0 ) ∗ 1−α t Account increase dCt∗ as fund increase plus fees increase: t t α 1 Ct∗ −C0 = ∗ ∗ (dXs +dFs ) = ∗ + 1 dXs = (X ∗ −X0 ) 0 0 1−α 1−α t Obvious bound Ct ≥ Ft yields: Ct ≥ α(Ct∗ − X0 ) X0 negligible as t ↑ ∞. Approximate drawdown constraint. Ct ≥ αCt∗
- 87. Problem Model Solution Welfare Implications Certainty equivalent rates Certainty equivalent rates under parametric restrictions
- 88. Problem Model Solution Welfare Implications Certainty equivalent rates Certainty equivalent rates under parametric restrictions Manager: 1 − α µ2 − (1 − α)(ϕ − r ) γ∗ 2σ 2
- 89. Problem Model Solution Welfare Implications Certainty equivalent rates Certainty equivalent rates under parametric restrictions Manager: 1 − α µ2 − (1 − α)(ϕ − r ) γ∗ 2σ 2 Investor: 1 − α µ2 γI − γM 1 − (1 − α) − (1 − α)(ϕ − r ) γ∗ 2σ 2 γ∗
- 90. Problem Model Solution Welfare Implications Certainty equivalent rates Certainty equivalent rates under parametric restrictions Manager: 1 − α µ2 − (1 − α)(ϕ − r ) γ∗ 2σ 2 Investor: 1 − α µ2 γI − γM 1 − (1 − α) − (1 − α)(ϕ − r ) γ∗ 2σ 2 γ∗ Dependence on fees?
- 91. Problem Model Solution Welfare Implications Manager Performance fees affect the manager in two ways.
- 92. Problem Model Solution Welfare Implications Manager Performance fees affect the manager in two ways. Income effect. Accrued to manager’s account, but only at safe rate. Positive impact.
- 93. Problem Model Solution Welfare Implications Manager Performance fees affect the manager in two ways. Income effect. Accrued to manager’s account, but only at safe rate. Positive impact. Drag effect. Reduce fund growth, hence future fees. Negative impact.
- 94. Problem Model Solution Welfare Implications Manager Performance fees affect the manager in two ways. Income effect. Accrued to manager’s account, but only at safe rate. Positive impact. Drag effect. Reduce fund growth, hence future fees. Negative impact. Because horizon is long, and no participation is allowed, second effect prevails.
- 95. Problem Model Solution Welfare Implications Manager Performance fees affect the manager in two ways. Income effect. Accrued to manager’s account, but only at safe rate. Positive impact. Drag effect. Reduce fund growth, hence future fees. Negative impact. Because horizon is long, and no participation is allowed, second effect prevails. Manager’s certainty equivalent rate decreases with α. Manager prefers 10% in rapidly growing fund, than 20% in slowly growing fund.
- 96. Problem Model Solution Welfare Implications Investor Performance fees affect the investor in two ways.
- 97. Problem Model Solution Welfare Implications Investor Performance fees affect the investor in two ways. Cost effect. Reduce fund growth. Negative impact.
- 98. Problem Model Solution Welfare Implications Investor Performance fees affect the investor in two ways. Cost effect. Reduce fund growth. Negative impact. Agency effect. Shrink manager’s risk aversion towards one. Ambiguous impact.
- 99. Problem Model Solution Welfare Implications Investor Performance fees affect the investor in two ways. Cost effect. Reduce fund growth. Negative impact. Agency effect. Shrink manager’s risk aversion towards one. Ambiguous impact. Do observed levels of performance fees serve investors?
- 100. Problem Model Solution Welfare Implications Investor Performance fees affect the investor in two ways. Cost effect. Reduce fund growth. Negative impact. Agency effect. Shrink manager’s risk aversion towards one. Ambiguous impact. Do observed levels of performance fees serve investors? If investors could choose performance fees themselves, at which levels would they set them?
- 101. Problem Model Solution Welfare Implications Equilibrium Fees 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Pairs of risk aversions for the manager (x) and the investor (y) such that investors’s optimal α∗ is within 0 and 1, and certainty equivalent rate greater than r . ϕ = r = 2% (left panel) and ϕ = r = 3% (right panel). Optimal fees 20% on solid line.
- 102. Problem Model Solution Welfare Implications Agency Effect Limited Equilibrium fees require very low risk aversion both for the investor and for the manager.
- 103. Problem Model Solution Welfare Implications Agency Effect Limited Equilibrium fees require very low risk aversion both for the investor and for the manager. Investor Risk aversion must be lower than 2.
- 104. Problem Model Solution Welfare Implications Agency Effect Limited Equilibrium fees require very low risk aversion both for the investor and for the manager. Investor Risk aversion must be lower than 2. Manager’s risk aversion must be lower than 1.
- 105. Problem Model Solution Welfare Implications Agency Effect Limited Equilibrium fees require very low risk aversion both for the investor and for the manager. Investor Risk aversion must be lower than 2. Manager’s risk aversion must be lower than 1. Otherwise no equilibrium exists.
- 106. Problem Model Solution Welfare Implications Parameter Restrictions ϕ = 1% ϕ=1%, r = 1% α µ/σ 10% 15% 20% 25% 30% 0.25 3.0 2.9 2.9 2.8 2.7 0.5 12.4 12.3 12.3 12.2 12.1 1.0 49.9 49.8 49.8 49.7 49.6 1.5 112.4 112.3 112.3 112.2 112.1 α 1 µ2 Maximum risk-aversion γ for which ϕ + 1−α r < γ∗ 2σ 2 , and µ hence the optimal portfolio is π = γ1∗ σ2 .
- 107. Problem Model Solution Welfare Implications Parameter Restrictions ϕ = 2% ϕ=2%, r = 1% α µ/σ 10% 15% 20% 25% 30% 0.25 1.5 1.5 1.5 1.5 1.4 0.5 6.5 6.6 6.7 6.8 6.9 1.0 26.2 26.9 27.5 28.2 29.0 1.5 59.1 60.6 62.3 64.0 65.7 α 1 µ2 Maximum risk-aversion γ for which ϕ + 1−α r < γ∗ 2σ 2 , and µ hence the optimal portfolio is π = γ1∗ σ2 .
- 108. Problem Model Solution Welfare Implications Testable Implications The model predicts that: Funds with higher fees should have higher leverage, (for γ > 1, and viceversa for γ < 1).
- 109. Problem Model Solution Welfare Implications Testable Implications The model predicts that: Funds with higher fees should have higher leverage, (for γ > 1, and viceversa for γ < 1). Funds with higher fees should have smaller drawdowns.
- 110. Problem Model Solution Welfare Implications Testable Implications The model predicts that: Funds with higher fees should have higher leverage, (for γ > 1, and viceversa for γ < 1). Funds with higher fees should have smaller drawdowns. Leverage may differ across funds, but for a given fund it should remain constant over time.
- 111. Problem Model Solution Welfare Implications Conclusion Performance fees with High-Water Marks: Make managers more myopic. Higher fees: manager’s preferences matter less.
- 112. Problem Model Solution Welfare Implications Conclusion Performance fees with High-Water Marks: Make managers more myopic. Higher fees: manager’s preferences matter less. Akin to Drawdown constraints, for long horizons.
- 113. Problem Model Solution Welfare Implications Conclusion Performance fees with High-Water Marks: Make managers more myopic. Higher fees: manager’s preferences matter less. Akin to Drawdown constraints, for long horizons. Manager’s nonparticipation important assumption.

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