Extension: Model Without Margins

In the previous section, I obtained that the margin requirements drive the hedge fund manager behavior. This section investigates managerial behavior in case the margin requirements are relaxed. In this case the only difference between this model and the model considered in Panageas and Westerfield (2009) is the timing of fee payment. In Panageas and Westerfield (2009) the fees are paid continuously in infinitesimal increments, while in this paper the fund manager is paid at discrete time moments. This minor change produces a huge change in optimal leverage.

Consider one possible managerial strategy. Assume the manager chooses a constant leverage level πt= π for t ∈ [0, 1]. Then from equation (1) we have

dAt = At(r + π(µ − r))dt + AtπσdBt. (10)

Consequently, A1 = A0+er+(µ−r)π− σ2 2 π 2 +σπξ_{, (11)}

where ξ is a standard normal variable. As a result,

E[A1] = A0+er+(µ−r)π. (12)

Therefore, the expected value of the AUM at time t= 1 prior to the payment of fees goes to

infinity as π goes to infinity.

The dollar value of the management fee is equal to fmA1, so the expected value of the

management fee paid at time t = 1 is equal to fmA0+er+(µ−r)π, which increases to infinity

The dollar value of the performance fee is equal to fp((1 − fm)A1− (1 + hr)H0)+. The

expected value of the performance fee satisfies

E[fp((1 − fm)A1− (1 + hr)H0)+] ≥ fpE[(1 − fm)A1− (1 + hr)H0] = fp((1 − fm)E[A1] − (1 + hr)H0).

(13)

The right hand side of equation (13) goes to infinity as π goes to infinity. Consequently, the expected value of the performance fee goes to infinity as π goes to infinity.

The reasoning used in inequality (13) with respect to the indexed high-water mark can be used with respect to any level that needs to be outperformed in order for the manager to obtain the performance fee. Therefore, the expected value of the performance based portion of the managerial compensation goes to infinity as leverage π increases to infinity provided

fp >0 and 1 − fm >0.

All the fees the hedge fund charges across time are nonnegative, so the total expected utility for the risk-neutral manager goes to infinity since the first fee already goes to infinity. As a result, the manager cannot have an optimal strategy that has a bounded leverage level and delivers a finite utility, since he can always choose a higher leverage level that provides a higher utility than the given strategy.

Consider a case where the manager is restricted to rebalancing portfolio only at t = 1

after the fees are paid. In this situation denoting π the weight of the risky asset we obtain

A1 = A0(πeµ−

σ2

2 +σξ+ (1 − π)er),

so

E[A1] = A0(πeµ+ (1 − π)er) = A0(er+ π(eµ− er)).

E[A1] goes to infinity as leverage goes to infinity, since eµ_{− e}r _{> 0. Considering the man-}

agement and the performance fees as before, we obtain that the expected value of each of the fees goes to infinity as the leverage π goes to infinity. This shows that the result that the risk-neutral manager does not have an optimal bounded leverage does not depend on the ability to continuously rebalance the fund’s portfolio.

Therefore, unlike in the models where the fees are charged continuously we obtain that when the fees are charged at discrete time moments the model needs some additional limi- tations in order for the manager to use a limited leverage. For example, Panageas and West- erfield (2009) find that a risk neutral manager who has a continuous flow of fees chooses a limited leverage, while I find that a small change to their model where the continuous flow of fees is substituted with payment of fees at discrete time moments, the optimal managerial behavior changes. When the manager faces a continuous flow of fees, he has less incentive to choose high leverage, since the closest payment option is infinitesimal, while a loss would lead to a drop in value of all the future options that add up to a substantial sum. In a discrete case, however, the closest option value is already substantial and infinitely high leverage delivers infinitely high expected utility from this option alone, and therefore the manager chooses to behave differently in this case.

There are a number of arguments for limitations on hedge fund leverage provided in the academic literature: future career concerns, managerial investments in the fund, and liquidation in the case of a poor performance. I consider these and other arguments in terms of the model with the risk neutral hedge fund manager in Appendix C and find that they do not lead to limitations on the optimal leverage level. If the hedge fund manager is risk averse, he chooses a finite leverage, but this result comes from risk aversion rather than these listed reasons. The question of an average risk aversion level of hedge fund managers is an interesting one, but unfortunately, according to my knowledge, there is no empirical study which reports it. There are many books about individual hedge fund manager stories, for example, Richard (2010), who points out that if anything, at least some hedge fund manager behavior is risk-seeking rather than risk-averse. Hedge fund flows are convex (see Chevalier and Ellison, 1997 and Sirri and Tufano, 1998), and this also effectively results in risk-seeking behavior because it increases utility gains from positive results and decreases utility losses from negative results.

Lan, Wang, and Yang (2012) propose that the presence of a liquidation boundary in- duces limited leverage levels without an exogenous margin requirement. Dai and Sundaresan (2010) provide two possible types of hedge fund liquidation that are relevant to the model:

liquidation by the investor and liquidation by a prime broker. In a continuous fee payment framework with a liquidation boundary, as in Lan, Wang, and Yang (2012), it is impossible to distinguish between these two cases. Below I consider two extensions of the basic model which allow me to study managerial behavior given each of the two possibilities separately.