1. 1 Incentive Contracts and Hedge Fund Management James E. Hodder Jens Carsten Jackwerth jhodder@bus.wisc.edu http://www.wiwi.uni-konstanz.de/jackwerth/

2. 2 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.

3. 3 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.

4. 4 There is an indexed high-water mark which starts at H 0 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.

5. 5 Liquidation Barrier  t in the basic model is set at half the high- water mark (  t = 0.5 H 0 e rt ). 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:

6. 6 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:

7. 7 Standard Parameters Time to maturity T0.25 Interest rater0.05 Log value steps below/above X 0 600/600Initial fund valueX 0 1.00 Risk aversion coefficient  4Mean  0.07 Number of time stepsn60Volatility  0.05 High water markH 0 1.00Incentive feec0.20 Exit boundary at t=0   0.50 Mgt. fee rateb0.02 Manager’s share ownershipa0.10 Offset steps for the Normal approx.1+2×60 = 121 Log X step(log (1/0.5))/600≈0.001155

8. 8 Optimal Kappa Surface with No Incentive Option and No Managerial Share Ownership

9. 9 Optimal Kappa Surface with a Managerial Incentive Option but No Share Ownership

10. 10 Optimal Kappa Surface with an Incentive Option and Managerial Share Ownership

11. 11 Comparison of Kappa Choices in Related Models I

12. 12 Comparison of Kappa Choices in Related Models II

13. 13 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:

14. 14 Optimal Kappa Surface with an Endogenous Shutdown Option

15. 15 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:

16. 16 Optimal Kappa Surface with Managerial Effort and Standard Compensation Package

17. 17 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).

18. 18 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.