The Incentives of Hedge Fund Fees and High-Water Marks

Problem Model Solution Welfare Implications

The Incentives ofHedge Fund Fees and High-Water Marks

Paolo Guasoni(Joint work with Jan Obłoj)

Boston University and Dublin City University

Workshop on Foundations of Mathematical FinanceJanuary 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-lookingmathematics, for truth.”“Economics, as a field, got in trouble because economists wereseduced 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 wehave now.”

Make Frictions Tractable.One Step at a Time.


Background



Make Frictions Tractable.

One Step at a Time.


Background



Make Frictions Tractable.One Step at a Time.


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


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


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


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


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.


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.


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.


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.


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.


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.


High-Water Marks

Time Gross Net High-Water Mark Fees0 100 100 100 01 110 108 108 22 100 100 108 23 118 116 116 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.


High-Water Marks


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.


High-Water Marks


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.


High-Water Marks

20 40 60 80 100

0.5

1.0

1.5

2.0

2.5


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.


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.


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.


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.


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.


Questions

Portfolio:Effect of fees and risk-aversion?

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


Questions

Portfolio:Effect of fees and risk-aversion?Welfare:Effect on investors and managers?

High-Water Mark Contracts:consistent with any investor’s objective?


Questions

Portfolio:Effect of fees and risk-aversion?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.

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.


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.


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.


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.


This Paper

Manager with Power Utility and Long Horizon.Exogenous risky and risk-free asset.Fees cannot be invested in fund.

Optimal Portfolio:

π =1γ∗

µ

σ2

γ∗ =(1− α)γ + α

γ =Manager’s Risk Aversionα =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.


This Paper

Manager with Power Utility and Long Horizon.Exogenous risky and risk-free asset.Fees cannot be invested in fund.Optimal Portfolio:

π =1γ∗

µ

σ2

γ∗ =(1− α)γ + α




This Paper


π =1γ∗

µ

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.


This Paper


π =1γ∗

µ

σ2

γ∗ =(1− α)γ + α




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:

π =1γ∗

µ

σ2



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.



π =1γ∗

µ

σ2




Risk Aversion γ.2 Investor maximizes utility of wealth. Pays no fees.

Risk Aversion γ∗ = (1− α)γ + α.



π =1γ∗

µ

σ2




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.


π =1γ∗

µ

σ2


Price Dynamics

dSt

St= (r + µ)dt + σdWt (Risky Asset)

dXt = (r − ϕ)Xtdt + Xtπt

(dStSt− rdt

)− α

1−αdX ∗t (Fund)

dFt = rFtdt + ϕXtdt +α

1− αdX ∗t (Fees)

X ∗t = max0≤s≤t

Xs (High-Water Mark)

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.


Price Dynamics

dSt



(dStSt− rdt

)− α






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.


Price Dynamics

dSt



(dStSt− rdt

)− α






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.


Dynamics Well Posed?

Problem: fund value implicit.Find solution Xt for

dXt = XtπtdSt

St− ϕXtdt − α

1− αdX ∗t

Yes. Pathwise construction.

Proposition

The unique solution is Xt = eRt−αR∗t , where:

Rt =

∫ t

0

(µπs −

σ2

2π2

s − ϕ)

ds + σ

∫ t

0πsdWs

is the cumulative log return.


Fund Value Explicit

Lemma

Let Y be a continuous process, and α > 0.Then Yt +

α1−αY ∗t = Rt if and only if Yt = Rt − αR∗t .

Proof.Follows from:

R∗t = sups≤t

(Ys +

α

1− αsupu≤s

Yu

)= Y ∗t +

α

1− αY ∗t =

11− α

Y ∗t

Apply Lemma to cumulative log return.


Long HorizonThe manager chooses the portfolio π which maximizesexpected power utility from fees at a long horizon.

Maximizes the long-run objective:

maxπ

limT→∞

1pT

log E [F pT ] = λ

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 preferto retire and invest at such a rate, and below which wouldrather keep his job.

λ = r + 1γ

µ2

2σ2 for Merton problem with risk-aversionγ = 1− p.


Long HorizonThe manager chooses the portfolio π which maximizesexpected power utility from fees at a long horizon.Maximizes the long-run objective:

maxπ

limT→∞

1pT

log E [F pT ] = λ


λ = r + 1γ

µ2




maxπ

limT→∞

1pT

log E [F pT ] = λ

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 preferto retire and invest at such a rate, and below which wouldrather keep his job.

λ = r + 1γ

µ2




maxπ

limT→∞

1pT

log E [F pT ] = λ


λ = r + 1γ

µ2



Solving ItSet r = 0 and ϕ = 0 to simplify notation.

Cumulative fees are a fraction of the increase in the fund:

Ft =α

1− α(X ∗t − X ∗0 )

Thus, the manager’s objective is equivalent to:

maxπ

limT→∞

1pT

log E [(X ∗T )p]

Finite-horizon value function:

V (x , z, t) = supπ

1p

E [X ∗Tp|Xt = x ,X ∗t = z]

dV (Xt ,X ∗t , t) = Vtdt + VxdXt +12

Vxxd〈X 〉t + VzdX ∗t

= Vtdt +(

Vz − α1−αVx

)dX ∗t +

(VxXt(πtµ− ϕ)dt + Vxx

σ2

2 π2t X 2

t

)dt


Solving ItSet r = 0 and ϕ = 0 to simplify notation.Cumulative fees are a fraction of the increase in the fund:

Ft =α

1− α(X ∗t − X ∗0 )

Thus, the manager’s objective is equivalent to:

maxπ

limT→∞

1pT

log E [(X ∗T )p]

Finite-horizon value function:

V (x , z, t) = supπ

1p

E [X ∗Tp|Xt = x ,X ∗t = z]

dV (Xt ,X ∗t , t) = Vtdt + VxdXt +12

Vxxd〈X 〉t + VzdX ∗t

= Vtdt +(

Vz − α1−αVx

)dX ∗t +

(VxXt(πtµ− ϕ)dt + Vxx

σ2

2 π2t X 2

t

)dt


Dynamic ProgrammingHamilton-Jacobi-Bellman equation:

Vt + supπ

(xVx(πµ− ϕ) + Vxx

σ2

2 π2x2)

)x < z

Vz = α1−αVx x = z

V = zp/p x = 0V = zp/p t = T

Maximize in π, and use homogeneityV (x , z, t) = zp/pV (x/z,1, t) = zp/pu(x/z,1, t).

ut − ϕxux − µ2

2σ2u2

xuxx

= 0 x ∈ (0,1)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−pβtw(x),forgetting the terminal condition:{

−pβw − ϕxwx − µ2

2σ2w2

xwxx

= 0 for x < 1wx(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.

w(x) = xp(1−α), for β = 1−α(1−α)γ+α

µ2

2σ2 − ϕ(1− α).


Long-Run Heuristics



2σ2w2

xwxx

= 0 for x < 1wx(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.

w(x) = xp(1−α), for β = 1−α(1−α)γ+α

µ2

2σ2 − ϕ(1− α).


Long-Run Heuristics



2σ2w2

xwxx

= 0 for x < 1wx(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.

w(x) = xp(1−α), for β = 1−α(1−α)γ+α

µ2

2σ2 − ϕ(1− α).


Long-Run Heuristics



2σ2w2

xwxx

= 0 for x < 1wx(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.

w(x) = xp(1−α), for β = 1−α(1−α)γ+α

µ2

2σ2 − ϕ(1− α).


Verification

Theorem

If ϕ− r < µ2

2σ2 min{

1γ∗, 1γ2∗

}, then for any portfolio π:

limT→∞

1pT

log E[(Fπ

T )p] ≤ max

{(1− α)

(1γ∗

µ2

2σ2 + r − ϕ), r}

Under the nondegeneracy condition ϕ+ α1−α r < 1

γ∗µ2

2σ2 , theunique optimal solution is π̂ = 1

γ∗µσ2 .

Martingale argument. No HJB equation needed.Show upper bound for any portfolio π (delicate).Check equality for guessed solution (easy).


Verification

Theorem

If ϕ− r < µ2

2σ2 min{

1γ∗, 1γ2∗


limT→∞

1pT

log E[(Fπ

T )p] ≤ max

{(1− α)

(1γ∗

µ2

2σ2 + r − ϕ), r}


γ∗µ2


γ∗µσ2 .

Martingale argument. No HJB equation needed.

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


Verification

Theorem

If ϕ− r < µ2

2σ2 min{

1γ∗, 1γ2∗


limT→∞

1pT

log E[(Fπ

T )p] ≤ max

{(1− α)

(1γ∗

µ2

2σ2 + r − ϕ), r}


γ∗µ2


γ∗µσ2 .

Martingale argument. No HJB equation needed.Show upper bound for any portfolio π (delicate).

Check equality for guessed solution (easy).


Verification

Theorem

If ϕ− r < µ2

2σ2 min{

1γ∗, 1γ2∗


limT→∞

1pT

log E[(Fπ

T )p] ≤ max

{(1− α)

(1γ∗

µ2

2σ2 + r − ϕ), r}


γ∗µ2


γ∗µσ2 .

Martingale argument. No HJB equation needed.Show upper bound for any portfolio π (delicate).Check equality for guessed solution (easy).


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

For any portfolio π:

RT = −∫ T

0σ2

2 π2t dt +

∫ T0 σπtdW̃t

W̃t = Wt + µ/σt risk-neutral Brownian MotionExplicit representation:

E [(XπT )

p] = E [ep(1−α)R∗T ] = EQ

[ep(1−α)R∗T e

µσ

W̃T− µ2

2σ2 T]

For δ > 1, Hölder’s inequality:

EQ

[ep(1−α)R∗T e

µσ

W̃T− µ2

2σ2 T]≤ EQ

[eδp(1−α)R∗T

] 1δ EQ

[e

δδ−1

(µσ

W̃T− µ2

2σ2 T)] δ−1

δ

Second term exponential normal moment. Just e1

δ−1µ2

2σ2 T .


Upper Bound (1)Take p > 0 (p < 0 symmetric).For any portfolio π:

RT = −∫ T

0σ2

2 π2t dt +

∫ T0 σπtdW̃t


E [(XπT )

p] = E [ep(1−α)R∗T ] = EQ

[ep(1−α)R∗T e

µσ

W̃T− µ2

2σ2 T]


EQ

[ep(1−α)R∗T e

µσ

W̃T− µ2

2σ2 T]≤ EQ

[eδp(1−α)R∗T

] 1δ EQ

[e

δδ−1

(µσ

W̃T− µ2

2σ2 T)] δ−1

δ


δ−1µ2

2σ2 T .



RT = −∫ T

0σ2

2 π2t dt +

∫ T0 σπtdW̃t

W̃t = Wt + µ/σt risk-neutral Brownian Motion

Explicit representation:

E [(XπT )

p] = E [ep(1−α)R∗T ] = EQ

[ep(1−α)R∗T e

µσ

W̃T− µ2

2σ2 T]


EQ

[ep(1−α)R∗T e

µσ

W̃T− µ2

2σ2 T]≤ EQ

[eδp(1−α)R∗T

] 1δ EQ

[e

δδ−1

(µσ

W̃T− µ2

2σ2 T)] δ−1

δ


δ−1µ2

2σ2 T .



RT = −∫ T

0σ2

2 π2t dt +

∫ T0 σπtdW̃t


E [(XπT )

p] = E [ep(1−α)R∗T ] = EQ

[ep(1−α)R∗T e

µσ

W̃T− µ2

2σ2 T]


EQ

[ep(1−α)R∗T e

µσ

W̃T− µ2

2σ2 T]≤ EQ

[eδp(1−α)R∗T

] 1δ EQ

[e

δδ−1

(µσ

W̃T− µ2

2σ2 T)] δ−1

δ


δ−1µ2

2σ2 T .


Upper Bound (2)Estimate EQ

[eδp(1−α)R∗T

].

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:

EQ

[eδp(1−α)R∗T

]≤ EQ

[eδp(1−α)R∗∞

]=

11− δp(1− α)

In summary, for 1 < δ < 1p(1−α) :

limT→∞

1pT

log E[(Fπ

T )p] ≤ 1

p(δ − 1)µ2

2σ2

Thesis follows as δ → 1p(1−α) .



[eδp(1−α)R∗T

].

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:

EQ

[eδp(1−α)R∗T

]≤ EQ


]=

11− δp(1− α)


limT→∞

1pT

log E[(Fπ

T )p] ≤ 1

p(δ − 1)µ2

2σ2




[eδp(1−α)R∗T

].

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:

EQ

[eδp(1−α)R∗T

]≤ EQ


]=

11− δp(1− α)


limT→∞

1pT

log E[(Fπ

T )p] ≤ 1

p(δ − 1)µ2

2σ2




[eδp(1−α)R∗T

].

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:

EQ

[eδp(1−α)R∗T

]≤ EQ


]=

11− δp(1− α)


limT→∞

1pT

log E[(Fπ

T )p] ≤ 1

p(δ − 1)µ2

2σ2



High-Water Marks and DrawdownsImagine fund’s assets Xt and manager’s fees Ft in thesame account Ct = Xt + Ft .

dCt = (Ct − Ft)πtdSt

St

Fees Ft proportional to high-water mark X ∗t :

Ft =α

1− α(X ∗t − X ∗0 )

Account increase dC∗t as fund increase plus fees increase:

C∗t −C∗0 =

∫ t

0(dX ∗s +dFs) =

∫ t

0

(α

1− α+ 1)

dX ∗s =1

1− α(X ∗t −X0)

Obvious bound Ct ≥ Ft yields:

Ct ≥ α(C∗t − X0)

X0 negligible as t ↑ ∞. Approximate drawdown constraint.

Ct ≥ αC∗t


Certainty equivalent rates

Certainty equivalent rates under parametric restrictions

Manager:1− αγ∗

µ2

2σ2 − (1− α)(ϕ− r)

Investor:

1− αγ∗

µ2

2σ2

(1− (1− α)γI − γM

γ∗

)− (1− α)(ϕ− r)

Dependence on fees?


Certainty equivalent rates

Certainty equivalent rates under parametric restrictionsManager:

1− αγ∗

µ2

2σ2 − (1− α)(ϕ− r)

Investor:

1− αγ∗

µ2

2σ2

(1− (1− α)γI − γM

γ∗

)− (1− α)(ϕ− r)

Dependence on fees?


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% inslowly growing fund.


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% inslowly growing fund.


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% inslowly growing fund.


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% inslowly growing fund.


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% inslowly growing fund.


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, atwhich levels would they set them?


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, atwhich levels would they set them?


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, atwhich levels would they set them?


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, atwhich levels would they set them?


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, atwhich levels would they set them?


Equilibrium Fees

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

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 certaintyequivalent rate greater than r . ϕ = r = 2% (left panel) andϕ = r = 3% (right panel). Optimal fees 20% on solid line.


Agency Effect Limited

Equilibrium fees require very low risk aversion both for theinvestor 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.



Equilibrium fees require very low risk aversion both for theinvestor 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.



Equilibrium fees require very low risk aversion both for theinvestor 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.



Equilibrium fees require very low risk aversion both for theinvestor 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.


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 2.70.5 12.4 12.3 12.3 12.2 12.11.0 49.9 49.8 49.8 49.7 49.61.5 112.4 112.3 112.3 112.2 112.1

Maximum risk-aversion γ for which ϕ+ α1−α r < 1

γ∗µ2

2σ2 , andhence the optimal portfolio is π = 1

γ∗µσ2 .


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 1.40.5 6.5 6.6 6.7 6.8 6.91.0 26.2 26.9 27.5 28.2 29.01.5 59.1 60.6 62.3 64.0 65.7

Maximum risk-aversion γ for which ϕ+ α1−α r < 1

γ∗µ2

2σ2 , andhence the optimal portfolio is π = 1

γ∗µσ2 .


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, butfor a given fund it should remain constant over time.



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, butfor a given fund it should remain constant over time.



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, butfor 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.

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


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.


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.

The Incentives of Hedge Fund Fees and High-Water Marks

Economy & Finance

Transcript of The Incentives of Hedge Fund Fees and High-Water Marks