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Incentive Contracts and Hedge Fund Management

James E. Hodder Jens Carsten Jackwerth

jhodder@bus.wisc.eduhttp://www.wiwi.uni-konstanz.de/jackwerth/

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The Basic Model

• A manager dynamically controls the stochastic process for a hedge fund’s value by altering the proportion of risky vs. riskfree assets in its portfolio.

• The manager’s compensation depends on the fund’s value at a future evaluation date T.

• Poor performance results in the fund being shutdown and the manager terminated at a lower boundary.

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Setup

• Risky investment (technology) grows at rate and has volatility

• Riskfree investment grows at the rate r

• Hedge fund value process is optimally controlled at each time step by the manager investing into the risky strategy and (1 - ) into the riskfree security.

• The manager chooses to maximize her expected utility of wealth at time T.

1

( )1T

TWU W

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• There is an indexed high-water mark which starts at H0 and grows at the riskless rate.

• Managerial compensation at time T if the fund is not liquidated:

– Manager owns a fraction (a) of the fund but has no outside wealth.

– Earns a management fee at the rate of b % annually on (1-a) of fund assets.

– Earns an incentive fee at the rate of c % of the amount fund value exceeds the high-water mark at time T on (1-a) of fund assets.

0(1 ) (1 ) ( )rTT T T TW aX a bTX a c X H e

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• Liquidation Barrier t in the basic model is set at half the high-water mark (t = 0.5 H0ert).

• If that barrier is hit at some time ≤ T, the manager receives:

– the value of her share ownership

– the prorated management fee

– these payments are reinvested at the riskless rate until time T:

( )0[ (1 ) (0.5 )]r r T

TW aX a b H e e

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• We use a grid in discrete time and log asset value.

• Given κ the change in log X is distributed normally with:

• Probabilities are calculated for +/- 60 moves on the grid.

• Recursive indirect utility:

2 21, ,2[ (1 ) ] and t tr t t

, , , ,; max

, 2 ,..., ,0X T X T X t X t tJ U J E J

where t T t T t t

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Standard Parameters

Time to maturity T 0.25 Interest rate r 0.05

Log value steps below/above X0 600/600 Initial fund value X0 1.00

Risk aversion coefficient 4 Mean 0.07

Number of time steps n 60 Volatility 0.05

High water mark H0 1.00 Incentive fee c 0.20

Exit boundary at t=0 0.50 Mgt. fee rate b 0.02

Manager’s share ownership a 0.10

Offset steps for the Normal approx. 1+2×60 = 121

Log X step (log (1/0.5))/600≈0.001155

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Optimal Kappa Surface with No Incentive Option and No Managerial Share Ownership

0.50 0.52 0.53 0.55 0.56 0.58 0.60 0.61 0.63012345678

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10

Kappa

Fund Value

Hill of Anticipation

Gambler's Ridge

Merton Flats

0.25Time

0

Valley of Prudence

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Optimal Kappa Surface with a Managerial Incentive Option but No Share Ownership

0.5 0.6 0.7 0.8 0.9 1.1 1.3 1.5 1.7012345678910

Kappa

Fund Value

Hill of Anticipation

Gambler's Ridge

Option Ridge

Ramp-up toMerton Flats

0.25Time

0

Valley of Prudence

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Optimal Kappa Surface with an Incentive Option and Managerial Share Ownership

0.5 0.6 0.7 0.8 0.9 1.1 1.3 1.5 1.7012345678910

Kappa

Fund Value

Gambler's Ridge

Option Ridge

Ramp-up toMerton Flats

0.25Time

0

Valley of Prudence

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Comparison of Kappa Choices in Related Models I

0

2

4

6

8

10

12

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

Fund Value, discounted at the riskless rate

Kappa

Hodder and Jackwerth [0.5-1.2] Merton kappa = 2 Carpenter [0.95-1.2]

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Comparison of Kappa Choices in Related Models II

0

2

4

6

8

10

12

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

Fund Value, discounted at the riskless rate

Kappa

Hodder and Jackwerth [0.5-1.2] Merton kappa = 2 GIR [0.5-0.55] BPS [0.5-1.2]

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Endogenous Shutdown

• The manager may decide to voluntarily shut down the fund in order to pursue other employment opportunities (e.g. investment bank, another fund, or start a new hedge fund).

• Outside opportunities become more attractive when the incentive option is unlikely to finish in-the-money.

• We let L represent a known annual compensation rate for the manager’s best outside opportunity.

• The manager’s wealth at time T if she chooses to shut down is:

( - ) ( - )(1- ) ( ) 0r T r TTW aX e b a X e L T for T

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Optimal Kappa Surface with an Endogenous Shutdown Option

0.5 0.6 0.7 0.8 0.9 1.1 1.3 1.5 1.701234

56

7

8

9

10

Kappa

Fund Value

Option Ridge

Ramp-up toMerton Flats

0.25Time

0

Fund Closure Region

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Managerial Effort

• Suppose the manager can enhance the drift of the risky return process by expending extra effort.

• Let ψ denote the effort level. This is scaled so that ψ = 0 denotes normal effort, ψ = 0.01 increases the drift of the risky investment by 1%, and ψ = 0.02 increases that drift by 2%.

• The manager’s indirect utility function now has a modified form with g being a parameter that scales the aversion to effort:

2,ψ, ,ψ,[ ] 0.5 ψX t X t tG E G g

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Optimal Kappa Surface with Managerial Effort and

Standard Compensation Package

0.5 0.6 0.7 0.8 0.9 1.1 1.3 1.5 1.701234567

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9

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Kappa

Fund Value

Gambler's Ridge,maximum effort

Top of Option Ridge,

maximum effort

Ramp-up toMerton Flats,normal effort

0.25Time

0

Valley of Prudence,maximum effort

Ramp-up toOption Ridge,

high effort

Merton Flats,maximum effort

Merton Flats,high effort

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Concluding Comments

• With hindsight, the manager’s behavior makes sense. However, that behavior can vary dramatically both with location in the state space and with the compensation structure. Some previous papers have identified parts of that behavior but missed seeing the “whole elephant”.

• Behavior approaching the lower boundary is particularly complex and depends strongly on the structure of severance compensation (including penalties), managerial shareholding, and outside opportunities (including the possibility of voluntary shutdown).

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• Managerial control with non-linear incentive compensation leads to fund value distributions which are far from lognormal (with our basic parameters, it’s bimodal). This can result in derivative values which differ substantially from those based on standard lognormal assumptions.

• Allowing the manager to use extra effort to improve the Sharpe Ratio for the risky investment typically leads to greater risk taking in regions of the state space where she is expending greater effort. Effort and risk taking act like complements rather than substitutes in this model.