韩信点兵（idobzooki.com）
编辑：jh-epOK
时间：2017-08-17 09:48:23

Investors's discount rate is the risk-free rate r. They receive money outflows at a fixed fraction before liquidation, and gain the whole fund value once liquidation occurs.

本篇编辑：牛淑雅

Optimal effort under high-water mark contracts（上）韩信点兵央广网西安3月13日消息（记者雷恺）外表呆萌、内心体贴、穿着白大褂穿梭于实验室、用科技守护健康——西安交大的医学生就像《超能陆战队》中的“大白”一样，他们的创意项目——“手术语音图文记录系统”实现了外科关键手术步骤的图像采集和图文手术记录的生成，获得第二届中国“互联网+”大学生创新创业大赛创意组铜奖。

Our boundary conditions follow GIR and PW. However, in our model, the fund manager influences the likelihood that the fund value hits the lower or upper boundary through the optimal effort.

We mainly develop a model measuring the optimal effort of a risk-neutral hedge fund manager in a continuoustime framework. The fund manager chooses the optimal effort to maximize the present value of total fees and reduce liquidation risks, trading off extra return benefits against the cost of the effort. We find that the manager's effort depends on the ratio between the fund's assets under management (AUM) and the high-water mark (HWM), and endogenous fund liquidation has key influence on the dynamics of the effort. Our calibration suggests that when the fund is close to liquidation, the manager exerts greatest effort. The more distant the fund value is from the liquidation boundary, the less effort the manager chooses to make; when the fund value is approaching the HWM, the manager's optimal effort still decreases, but the rate of decline becomes far slower. The optimal effort contributes to both increasing the likelihood of survival for the fund and preserving the fund's going-concern value. A growth of degree of the effort cost, volatility of the AUM, exogenous liquidation probability or endogenous liquidation boundary decreases the optimal effort. We also find empirical evidence that may support our theoretical conclusion.

To our knowledge, our model is the first to discuss the optimal effort of the hedge fund manager under a continuous-time framework and more practical assumptions. We extend the log-normal diffusion process setting of the AUM in Goetzmann et al. (2003) (GIR henceforth), to constant volatility σ and undetermined extra return α that the manager chooses to maximize her value function (present value of total fees). Just as GIR, throughout the paper, we assume that market opportunities, the manager's skill and the leverage are fixed, and also rule out the influence of luck on the extra return. Under these assumptions, the undecided extra return can be regarded as a measure to judge the fund manager's effort, that is, α is equivalent to the manager's effort in our model. It is a natural approach, since if the manager devotes herself to information acquisition within her capacity, she would employ a better investment strategy that results in higher excess returns on the fund. In this way, we associate the fund's extra return with the effort, and measure the manager's effort explicitly.

However, both Zhan (2011) and Ray and Chakraborty (2008) only apply a single-period discrete-time model, assuming that hedge funds have a determined termination, and do not consider the possibility of fund liquidation during the limited period. In contrast, we study the dynamics of the managerial effort in a continuous-time framework of hedge fund valuation. Our setting is more reasonable in that the fund has a infinite horizon and can be liquidated once the exogenous or endogenous liquidation condition is triggered.We find that liquidation is one crucial factor that determines the effort dynamics.

医学是关系人类生命与健康的学科，学习医学需要大量积累实践经验。尤其是外科学——操作性很强的一门课程，只有依靠大量的操作练习和现场观摩才能提高教学效果。囿于课堂教学时间有限，传统书本教育模式的局限以及国内相关资料的空白，学员只能学会基本动作要领，无法形成应有的潜意识操作习惯。西安交大的医学生们便萌生了做一个外科手术记录系统的创意。

据介绍，项目成员大都来自外科梦工厂实验室。他们的想法得到了吕毅教授的肯定和支持。吕毅教授认为“这个项目符合寻证医学的发展潮流”，经常询问项目的进展情况，给团队提供了大力支持，并与医院方面联系沟通与协调。刚开始做项目时，资金还没到账，都是吕毅教授先垫付的。队长徐向华告诉记者，团队除了软件设计师，其他都是纯粹的医学生。而参加比赛需要准备大量的图片素材、制作视频以及完成详尽的项目策划说明书。为此，队员杨桓特地跑去广东、深圳等地学习摄像机使用，考察哪种设备更适合记录手术过程，以使手术室里的拍摄所产生的辐射不影响手术。为了项目的推进，团队通宵达旦地改善实验方案，整理资料，完善策划书，在校园寂静的夜里挥洒的，满满的都是心血和努力。

While the manager has incentives to seek for higher extra returns so as to receive higher compensation fees, she needs to consider the accompanying cost for making certain effort, since the effort is not costless. Without loss of generality, we assume that the effort cost can be measured by a loss of wealth to the manager, and the function to measure the loss of wealth, G(αt, Pt), has a standard quadratic form, similar to the adjustment cost function in Hayashi (1982) and Bolton et al. (2011):

韩信点兵

Zhan (2011) investigates and compares five compensation schemes that are commonly employed in the mutual fund or hedge fund industry, likewise under the principal-agent framework. Specially, the paper finds that the HWM provision induces more effort when the fund's AUM is slightly under the HWM, but it dampens the manager's effort when the fund's AUM is far away from the HWM and the manager's skill is poor. Ray and Chakraborty (2008) construct a simple optimization problem assuming that the portfolio follows uniform distribution, and find that as the distance between the fund's AUM and the HWM increases, the manager's effort falls.

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.

韩信点兵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.

文章来源：Economic Modelling 2017年4月10日（本文仅代表作者个人观点）

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韩信点兵The manager's payoffs come from management fees as well as performance fees. Management fees are not sensitive to the fund performance and paid continuously at a constant percentage of the AUM (denoted as c) provided that the fund survives. The payment of performance fees relies on the HWM H. In the simplest case, H is the highest historical value of the fund's AUM P. Besides that, H is also subject to certain adjustments in some compensation contracts. As in GIR, H is adjusted up at a contractually growth rate g, and down at the withdrawal rate δ, as well as the charged percentage of management fees c. The growth rate g represents the rate required by investors to compensate for their opportunity costs and the value of g can be equal to the risk-free rate r, 0, or other values depending on the contracts. When the fund's AUM P exceeds H, H is reset as the new value of P and the performance fee is paid as a fraction k of this excess profit.

The main feature of our model is the incorporation of the manager's effort as an endogenous variable to model the dynamics of hedge funds. At the same time, we also consider the adjusted HWM, management fees and performance fees, the exogenous liquidation, and performance- induced endogenous liquidation.

In our dynamic framework, the manager's effort depends on the moneyness of the fund p, i.e., the ratio of the fund's AUM and its HWM, ranging from the lower liquidation boundary to the upper boundary, 1. The higher the moneyness p is, the lower the effort α and the decline rate. When the fund approaches liquidation, the manager tends to exert greatest effort, in spite of resulted expensive cost. The costly liquidation (downside risk) in our model damages both the manager's future payoffs and reputation. So to reduce the risk, the riskneutral manager behaves as if she was risk-averse. Also, the degree of effort decreases with the distance of the fund from liquidation. Besides, as the fund's AUM gets closer to the HWM, the manager's optimal effort still decreases, although with a slower rate, which indicates that the manager's motivation to make the fund value exceed the HWM is not strong. The payoff of performance fees shrinks the fund's AUM, increasing the probability of liquidation in the future. Our result covers the fact that a risk-neutral manager is averse to collect performance fees too soon. However, the rate of decline is becoming slower when p approaches 1, due to the coming performance fees. The optimal effort induced in our model contributes to both increasing the likelihood of fund survival and preserving the fund's going-concern value. It can be concluded that incorporating both the HWM contracts and fund liquidation is beneficial for the survival of funds and fund investors. We also find empirical evidence that may support our theoretical conclusion.

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.

韩信点兵在团队的努力下，“手术语音图文记录系统”项目通过运用语音识别、互联网远程控制等技术，实现外科术中关键手术步骤的图像采集和图文手术记录的生成，同时运用互联网技术实现手术记录的存取和调阅。这不仅为医疗工作者及科研工作者提供一个外科手术相关研究的大数据平台，而且为医学教育提供了丰富的外科手术资料。该项目在2015年获得中华医学会外科学分会手术学组专家的一致认可。

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，

The remainder of the paper is constructed as follows. Section 2 presents the valuation model containing the manager's effort to be chosen optimally. In Section 3 we deduce the ODEs for value functions of both the manager and investors and determine the optimal effort. Section 4 shows some comparative statics, economic implications and empirical results. Section 5 concludes.

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