The Incentives of Hedge Fund Fees and High-Water Marks

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


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.”


Background

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.”


Background

have now.”
Make Frictions Tractable.


Background

have now.”
Make Frictions Tractable.
One Step at a Time.


Outline

High-Water Marks:
Performance Fees for Hedge Funds Managers.


Outline

High-Water Marks:
Model:
Power Utility with Long Horizon.


Outline

High-Water Marks:
Model:
Solution:
Effective Risk Aversion and Drawdown Constraints.


Outline

High-Water Marks:
Model:
Solution:
Effective Risk Aversion and Drawdown Constraints.
Fees and Welfare:
Stackelberg Equilibrium between Investor and Manager


Two and Twenty

Hedge Funds Managers receive two types of fees.


Two and Twenty

Regular fees, like Mutual Funds.


Two and Twenty

Unlike Mutual Funds, Performance Fees.


Two and Twenty

Regular fees:
a fraction ϕ of assets under management. 2% typical.


Two and Twenty

Regular fees:
Performance fees:
a fraction α of trading proﬁts. 20% typical.


Two and Twenty

Regular fees:
Performance fees:
a fraction α of trading proﬁts. 20% typical.
High-Water Marks:
Performance fees paid after losses recovered.


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.


High-Water Marks

0 100 100 100 0
1 110 108 108 2
2 100 100 108 2
3 118 116 116 4

Fund drops from 108 to 100.
No fees paid, nor past fees reimbursed.


High-Water Marks

0 100 100 100 0
1 110 108 108 2
2 100 100 108 2
3 118 116 116 4

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.


High-Water Marks

2.5

2.0

1.5

1.0

0.5

20 40 60 80 100


Risk Shifting?

Manager shares investors’ proﬁts, not losses.
Does manager take more risk to increase proﬁts?


Risk Shifting?

Option Pricing Intuition:
Manager has a call option on the fund value.
Option value increases with volatility. More risk is better.


Risk Shifting?

Static, Complete Market Fallacy:
Manager has multiple call options.


Risk Shifting?

High-Water Mark: future strikes depend on past actions.


Risk Shifting?

High-Water Mark: future strikes depend on past actions.
Option unhedgeable: cannot short (your!) hedge fund.


Questions

Portfolio:
Effect of fees and risk-aversion?


Questions

Portfolio:
Welfare:
Effect on investors and managers?


Questions

Portfolio:
Welfare:
Effect on investors and managers?
High-Water Mark Contracts:
consistent with any investor’s objective?


Answers

Goetzmann, Ingersoll and Ross (2003):
Risk-neutral value of management contract (future fees).
Exogenous portfolio and fund ﬂows.


Answers

High-Water Mark contract worth 10% to 20% of fund.


Answers

Panageas and Westerﬁeld (2009):
Exogenous risky and risk-free asset.
Optimal portfolio for a risk-neutral manager.
Fees cannot be invested in fund.


Answers

Panageas and Westerﬁeld (2009):
Optimal portfolio for a risk-neutral manager.
Constant risky/risk-free ratio optimal.
Merton proportion does not depend on fee size.
Same solution for manager with Hindy-Huang utility.


This Paper
Manager with Power Utility and Long Horizon.


This Paper
Optimal Portfolio:

1 µ
π=
γ ∗ σ2
∗
γ =(1 − α)γ + α
γ =Manager’s Risk Aversion
α =Performance Fee (e.g. 20%)


This Paper
Optimal Portfolio:

1 µ
π=
γ ∗ σ2
∗
γ =(1 − α)γ + α

Manager behaves as if owned fund, but were more myopic
(γ ∗ weighted average of γ and 1).


This Paper
Optimal Portfolio:

1 µ
π=
γ ∗ σ2
∗
γ =(1 − α)γ + α

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.


Three Problems, One Solution

Power utility, long horizon. No regular fees.



1 Manager maximizes utility of performance fees.
Risk Aversion γ.



Risk Aversion γ.
2 Investor maximizes utility of wealth. Pays no fees.
Risk Aversion γ ∗ = (1 − α)γ + α.



Risk Aversion γ.
Risk Aversion γ. Maximum Drawdown 1 − α.



Risk Aversion γ.
Risk Aversion γ. Maximum Drawdown 1 − α.
Same optimal portfolio:

1 µ
π=
γ ∗ σ2


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.


Price Dynamics

dSt
St
dSt α ∗
α
1−α t
0≤s≤t

Gain split into α for the manager and 1 − α for the fund.


Price Dynamics

dSt
St
dSt α ∗
α
1−α t
0≤s≤t

Gain split into α for the manager and 1 − α for the fund.
Performance fee is α/(1 − α) of fund increase.


Dynamics Well Posed?

Problem: fund value implicit.
Find solution Xt for
dSt α
dXt = Xt πt − ϕXt dt − dX ∗
St 1−α t



dSt α
St 1−α t
Yes. Pathwise construction.



dSt α
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.


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


Fund Value Explicit

Lemma
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


Fund Value Explicit

Lemma
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.


Long Horizon
The manager chooses the portfolio π which maximizes
expected power utility from fees at a long horizon.


Long Horizon
Maximizes the long-run objective:
1 p
max lim log E[FT ] = λ
π T →∞ pT


Long Horizon
1 p
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.


Long Horizon
1 p
π T →∞ pT

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.


Long Horizon
1 p
π T →∞ pT

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.


Solving It
Set r = 0 and ϕ = 0 to simplify notation.


Solving It
Cumulative fees are a fraction of the increase in the fund:
α
Ft = (X ∗ − X0 )
∗
1−α t


Solving It
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


Solving It
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


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



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 )


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)


Long-Run Heuristics

Long-run limit.
2 2
µ wx
wx (1) = p(1 − α)w(1)

This equation is time-homogeneous, but β is unknown.


Long-Run Heuristics

Long-run limit.
2 2
µ wx
wx (1) = p(1 − α)w(1)

Any β with a solution w is an upper bound on the rate λ.


Long-Run Heuristics

Long-run limit.
2 2
µ wx
wx (1) = p(1 − α)w(1)

Candidate long-run value function:
the solution w with the lowest β.


Long-Run Heuristics

Long-run limit.
2 2
µ wx
wx (1) = p(1 − α)w(1)

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).


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


Veriﬁcation

Theorem
µ2 1 1
If ϕ − r < 2σ 2
min γ∗ , γ∗

1 1 µ2
π
+ r − ϕ ,r
T →∞ pT γ ∗ 2σ 2

α 1 µ2
µ
ˆ

Martingale argument. No HJB equation needed.


Veriﬁcation

Theorem
µ2 1 1
If ϕ − r < 2σ 2
min γ∗ , γ∗

1 1 µ2
π
+ r − ϕ ,r
T →∞ pT γ ∗ 2σ 2

α 1 µ2
µ
ˆ

Show upper bound for any portfolio π (delicate).


Veriﬁcation

Theorem
µ2 1 1
If ϕ − r < 2σ 2
min γ∗ , γ∗

1 1 µ2
π
+ r − ϕ ,r
T →∞ pT γ ∗ 2σ 2

α 1 µ2
µ
ˆ

Show upper bound for any portfolio π (delicate).
Check equality for guessed solution (easy).


Upper Bound (1)
Take p > 0 (p < 0 symmetric).


Upper Bound (1)
For any portfolio π:
T σ2 2 T ˜
RT = − 0 2 πt dt + 0 σπt d Wt


Upper Bound (1)
T σ2 2 T ˜
˜
Wt = Wt + µ/σt risk-neutral Brownian Motion


Upper Bound (1)
T σ2 2 T ˜
˜
Explicit representation:
∗ ∗ µ ˜ µ2
WT − T
E [(XT )p ] = E[ep(1−α)RT ] = EQ ep(1−α)RT e σ
π 2σ 2


Upper Bound (1)
T σ2 2 T ˜
˜
∗ ∗ µ ˜ µ2
WT − T
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


Upper Bound (1)
T σ2 2 T ˜
˜
∗ ∗ µ ˜ µ2
WT − T
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 .


Upper Bound (2)
∗
δp(1−α)RT
Estimate EQ e .


Upper Bound (2)
∗
δp(1−α)RT
Estimate EQ e .
Mt = eRt strictly positive continuous local martingale.
Converges to zero as t ↑ ∞.


Upper Bound (2)
∗
δp(1−α)RT
Estimate EQ e .
Fact:
inverse of lifetime supremum (M∞ )−1 uniform on [0, 1].
∗


Upper Bound (2)
∗
δp(1−α)RT
Estimate EQ e .
Fact:
∗

Thus, for δp(1 − α) < 1:
∗ ∗ 1
EQ eδp(1−α)RT ≤ EQ eδp(1−α)R∞ =
1 − δp(1 − α)


Upper Bound (2)
∗
δp(1−α)RT
Estimate EQ e .
Fact:
∗

∗ ∗ 1
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


Upper Bound (2)
∗
δp(1−α)RT
Estimate EQ e .
Fact:
∗

∗ ∗ 1
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) .


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


dSt
St
Fees Ft proportional to high-water mark Xt∗ :
α
Ft = (X ∗ − X0 )
∗
1−α t


dSt
St
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


dSt
St
α
Ft = (X ∗ − X0 )
∗
1−α t
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 )


dSt
St
α
Ft = (X ∗ − X0 )
∗
1−α t
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∗


Certainty equivalent rates

Certainty equivalent rates under parametric restrictions



Manager:
1 − α µ2
− (1 − α)(ϕ − r )
γ∗ 2σ 2



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∗



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?


Manager

Performance fees affect the manager in two ways.


Manager

Income effect.
Accrued to manager’s account, but only at safe rate.
Positive impact.


Manager

Income effect.
Positive impact.
Drag effect.
Reduce fund growth, hence future fees.
Negative impact.


Manager

Income effect.
Positive impact.
Drag effect.
Negative impact.
Because horizon is long, and no participation is allowed,
second effect prevails.


Manager

Income effect.
Positive impact.
Drag effect.
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.


Investor

Performance fees affect the investor in two ways.


Investor

Cost effect.
Reduce fund growth.
Negative impact.


Investor

Cost effect.
Reduce fund growth.
Negative impact.
Agency effect.
Shrink manager’s risk aversion towards one.
Ambiguous impact.


Investor

Cost effect.
Reduce fund growth.
Negative impact.
Agency effect.
Ambiguous impact.
Do observed levels of performance fees serve investors?


Investor

Cost effect.
Reduce fund growth.
Negative impact.
Agency effect.
Ambiguous impact.
Do observed levels of performance fees serve investors?
If investors could choose performance fees themselves, at
which levels would they set them?


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.


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.



Manager’s risk aversion must be lower than 1.
Otherwise no equilibrium exists.


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 .


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 .


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.



Funds with higher fees should have smaller drawdowns.
Leverage may differ across funds, but
for a given fund it should remain constant over time.


Conclusion

Performance fees with High-Water Marks:
Make managers more myopic.
Higher fees: manager’s preferences matter less.


Conclusion

Akin to Drawdown constraints, for long horizons.


Conclusion

Akin to Drawdown constraints, for long horizons.
Manager’s nonparticipation important assumption.

The Incentives of Hedge Fund Fees and High-Water Marks

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