Inspired by GIR, to focus on our purposes, we also assume that the impacts of all above factors on the fund's AUM are fixed and consider a moderate-skill manager. Let αt be the extra return, or the risk-adjusted return in excess of r, and σ be the volatility of the fund's AUM, Pt.6 To analyze the optimal effort made by the fund manager, here the only varying variable (the control variable) is the extra return αt, which can be chosen by the manager to maximize her present value from all future fees. The assumption means that in our model, except for the manager's endogenous decision, other conditions make no difference to the fund's AUM. Given that the effects derived from other conditions are invariant, here the extra return αt can be regarded as a measure to represent the effort chosen by the hedge fund manager. By this means, we can evaluate the effort explicitly.
where i is a positive constant parameter measuring the degree of the effort cost. The convex function of the effort cost relates to both the realized effort αt and the fund value Pt, and agrees with decreasing returns to scale concerning the increased marginal effort expenditure. To choose her optimal effort, the manager needs to make a trade-off between return benefits against the effort cost while considering impacts of liquidation and the high-water mark contracts
2.1. Compensation structure
Since the 1990s, the global hedge fund industry has developed substantially and quickly, becoming increasingly important to the modern portfolio management. According to HFR inflows and performance gains through the volatile macroeconomic environment in 2015Q1 increased total hedge fund capital to a new record of .94 trillion. Although the hedge fund capital posted a decline in the first quarter of 2016, it still remained above .87 trillion. One major feature of hedge funds is the special compensation contracts. Highwater mark contracts can be regarded as the combination of option-like compensation contracts and the high-water mark (HWM), which is known as a loss carry-forward provision. Besides the management fees that are typical for mutual funds and are usually collected as 2% of the fund assets under management (AUM), i.e., the fund value, as long as the fund survives, hedge fund managers also charge performance fees. The performance fee relies on the HWM, which for each investor is the maximum value ever reached by the past fund's AUM since her investment (in some contracts, the HWM is also subject to certain adjustments). When the fund's AUM exceeds the HWM, the HWM is reset as the current fund's AUM and the manager usually receives 20% of this excess profit as a reward for good performance. In addition, the compensation contracts vary with different funds.Optimal effort under high-water mark contracts（上）
Throughout the paper, time is continuous and the hedge fund does not possess a pre-specified expiration date. The manager is risk-neutral and discounts her wealth at a constant rate β. Since the manager has the ability to pursue risk-adjusted extra returns, the time value of the manager's wealth may be more expensive and her subjective discount rate may be larger than the risk-free rate r. Following GIR, we assume that β is equal to r. Extending our model to allow for differences between the manager's subjective discount rate and the risk-free rate is straightforward, but it is not the main concern of our model and no additional insight would be obtained for our main issue. The manager's objective is to maximize her expected present value of total wealth, by choosing the optimal dynamic effort α. Let F (P, H; α) be the manager's value function, and it satisfies，
In the absence of payouts and the probability of liquidation, the AUM P is assumed to follow a log-normal diffusion process in GIR. In the model, the expected rate of extra return and the volatility of the fund value are both constant, indicating that the impacts of stochastic market conditions and various leverage choices on the fund value are excluded from the modeling. If not, either one of these could have significant effects on the fund's returns and risks. Besides, GIR do not consider the effects of a manager's performance due to either skill or luck.
The relationship between the extra return and the manager's effort is natural. If the manager spends more time and money on searching for valuable information, finishing in-depth market analysis report, doing due diligence on products, and performing more sophisticated models to select better products and time the market within her capacity, she would construct better investment strategies and thus obtain higher extra returns on the fund. A similar relationship also exists between students' effort and their test scores. For mediocre students, their test scores are positively related to the effort they make. If they work harder, it is reasonable that the gained scores will be better.
where Bt is a standard Brownian motion. The first and second terms together represent the changes of the fund value relative to the fund manager's effort. The third and forth terms reflect investors' withdrawals and management fees, respectively, and the fifth corresponds to the performance fees paid if the fund value exceeds the HWM. The last term refers to a Poisson jump process accounting for the exogenous liquidation, where Jt is a jump process with intensity λ.黄瓜自卫器女性视频
Because the manager's optimal problem is homogeneous in the fund's AUM P and HWM H of degree one, we can apply the ratio of P and H to rewrite the above problem. Let the lowercase letters denote the corresponding variables or functions divided by the HWM H, so the variable p=P/H ranges from the liquidation boundary b to the upper boundary 1. The manager's value function can also be written as F (P, H) = Hf (P/H) = Hf (p), and in this way, the problem can be simplified from solving PDE equations to ODE ones.
Fund liquidation may occur for two reasons. The first one is exogenous where the liquidation is assumed to be a Poisson process with probability λ per unit of time, and such liquidation can be derived from investors' liquidity need. Let τ1 denote this exogenous liquidation time. When the exogenous liquidation occurs, the AUM Pτ1 jumps to zero, and the manager can do nothing about it. The second one depends on the fund manager's performance and is hence endogenous. If the AUM Pt drops to a substantially low level, investors would lose faith in the manager and therefore liquidate the fund. Let the level leading to liquidation be the product of a constant fraction b and the HWM H, and τ2 be this endogenous liquidation time, so that we have
subject to the dynamics of Pt (5), the dynamics of Ht (1), the liquidation boundary condition (4) and the effort constraint (2). The manager's value function contains her payoffs from management fees and performance fees, and the loss of wealth from the cost of the effort. Upon fund liquidation, the manager loses all future payoffs, and the value function equals to 0.
Since the degree of the effort cost i and the AUM P in the duration of the fund are both positive, the second-order condition is naturally met.