Risk Choice under High-Water Marks

The following post comes to us from Itamar Drechsler of the Department of Finance at New York University Stern School of Business.

High-water mark (HWM) contracts are the predominant compensation structure for managers in the hedge fund industry. In the paper, Risk Choice under High-Water Marks, forthcoming in the Review of Financial Studies, I seek to understand the optimal dynamic risk-taking strategy of a hedge fund manager who is compensated under such a contract. This is both an interesting portfolio-choice question, and one with potentially important ramifications for the willingness of hedge funds to bear risk in their role as arbitrageurs and liquidity providers, especially in times of crises. High-water mark mechanisms are also implicit in other types of compensation structures, so insights from this question extend beyond hedge funds. An example is a corporate manager who is paid performance bonuses based on record earnings or stock price and whose choice of projects influences the firm’s level of risk.

This paper provides a closed-form solution to this problem. The solution shows that the optimal risk choice is a function of the ratio of the fund’s assets under management to its high-water mark. Even though the manager is risk neutral, the incentives induce him to act as if he is risk-averse, with his effective risk-aversion a function of the ratio of the fund’s value to its HWM. The resulting optimal-risk taking dynamics take one of two possible forms: “derisking,” whereby the manager continuously reduces risk as the fund falls further below its HWM, or “gambling,” where the manager instead increases risk as the fund falls further below its HWM. Which optimal dynamic arises depends on important parameters of the manager’s environment, including investors’ termination policy, the manager’s outside option, the Sharpe ratio of the manager’s risky strategy, investors rate of withdrawals, and the management fee. In either case the dynamic is highly nonlinear, so that risk-taking becomes much more sensitive following a large negative return. This result may help explain patterns of derisking displayed by hedge funds in the “quant crisis” in 2007 and at the onset of the financial crisis in 2008.

The paper further examines the manager’s optimal walk-away decision and its impact on risk taking. I show that if the manager finds it optimal at some point to exercise his walk-away option, then having the right to do so increases his risk taking. However, for a class of reasonable outside option assumptions, including outside options that are linear in the fund’s size, walk-away is actually never optimal and hence risk-choice is unaffected. This result can be reversed if there are binding limits on risk taking, which can make walk-away optimal.

Though the HWM contract is very prevalent, whether it is optimal is an open question. Even though this is not the question addressed by this paper, it is worth mentioning several characteristics of the HWM which seem quite reasonable. First, the inclusion of a performance fee may be important in inducing the manager to share proprietary knowledge (or make a high “effort”), because it ties his compensation closely to his performance. In contrast, a pure management-fee compensation is not very sensitive to the fund’s performance. Second, the HWM, together with an indefinite horizon, produces concavity in the manager’s value function. As Panageas and Westerfield (2009) first showed, this mitigates the manager’s incentive to risk shift, so that even a risk-neutral investor avoids unlimited risk taking.

Third, the HWM implies that the investor can never pay more performance fees than the maximum cumulative gain of the fund, a possibility under alternative performance fee arrangements. For example, suppose the manager is instead paid a tiny fraction of every positive return made by the fund. Then no matter how small the fraction, the investor would soon have nothing left, since the fund return process has unbounded positive variation. Finally, under a HWM contract investors only pay the manager performance fees when they are doing particularly “well” (i.e., increasing cumulative gains) and do not pay at all when realized returns are negative. In this sense the fund’s performance hedges the payment to the manager, making the payment smaller on a marginal-utility-weighted basis.

The full paper is available for download here.

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