Investment Constraints and Delegated Portfolio Management Wei-Lin Liu* August 14, 015 Abstract: This paper examines the optimal use of investment constraints in delegated portfolio management. We show that investment constraints, which limit managers uses of margin purchase and short-sales, can benefit investors by enhancing managers incentives to acquire longterm investment information information that identifies significant asset mispricing, which may take a long time to be corrected. Our analysis helps explain why mutual fund managers are typically subject to much tighter investment constraints and receive less high-powered compensation than do hedge fund managers. JEL Classification: G10, G0, D80. Keywords: Fund managers, Investment constraints, Long-term investment information, Interim compensation risk * Contact Address: Department of Finance, School of Management, Fudan University, Shanghai, 00433. Tel.: 86 1 5011085, E-mail: wl liu@fudan.edu.cn. I would like to thank Suman Banerjee, Steve Dimmock, David Hirshleifer, Clive Lennox, Jiang Luo, Jay Ritter, and Hanjiang Zhang, for helpful comments. All remaining errors are my own.
I. Introduction The optimal design of employment contracts for portfolio managers is an important issue that has attracted significant research interests. Previous studies on this issue have mostly focused on analyzing optimal managerial compensation (see Cuoco and Kaniel, 011, and references therein). In practice, however, employment contracts for portfolio managers generally go beyond just specifying the compensation schemes. Studies show that in the case of mutual funds, managers are typically subject to investment constraints that prohibit short positions and limit long positions in individual assets. For example, examining a broad sample of actively managed U.S. domestic equity funds over the period between 1994 to 000, Almazan et al. (004) find that in about 70 percent of the cases managers are expressly prohibited by fund charters to engage in shortsales, and about 90 percent of the funds do not allow managers to engage in margin purchase. Such constraints represent binding commitments as funds must publicly disclose them through SEC filings (Form N-SAR). 1 Moreover, as observed by Almazan et al., the funds adoptions of investment constraints appear to be more than mere regulatory prohibitions. What is the rationale behind imposing investment constraints on fund managers? Standard models of security markets under asymmetric information indicate that managers with highly positive or negative private information about assets future returns can benefit fund investors by using margin purchase or short-sales (e.g., Grossman and Stiglitz, 1980). It thus seems puzzling why investors desire to subject managers to investment constraints. Moreover, since constraints prevent managers from fully utilizing their information, would they discourage managers from engaging in the information acquisition activities that are central to the rationale for using portfolio managers? Finally, in circumstances where investors desire to constrain their managers, how do these same circumstances affect managerial compensation? One possible answer to the first of the above questions might be that investment constraints are a response to the agency problem arising from excessive managerial risk taking. Since managers are rewarded for high investment returns but only bear limited losses should investments go sour, they may have incentives to invest excessively in risky assets for large personal gains. Investment constraints limit managerial risk taking. The above explanation to the use of investment constraints, however, appears at odds with the practice in the hedge fund industry. Prior studies show that in comparison to mutual fund man- 1 Other than margin purchase and short-sales, mutual funds may also prohibit managers from taking positions in options, futures, or restricted stocks. However, Almazan et al. (004) find that restrictions on margin purchase and short-sales are far more prevalent than the other types of constraints. 1
agers, hedge fund managers typically receive more high-powered compensation. Consequently, the problem of excessive managerial risk taking is likely to be more acute for hedge funds than for mutual funds. Yet, casual observations reveal that in comparison to mutual fund managers, hedge fund managers generally face less constraints they routinely use significant leverage and take short positions in assets. While the two types of funds differ in the respective regulatory environments and investor clienteles, these other differences are unlikely to result in sharply more excessive managerial risk taking for mutual funds (for a detailed discussion of this point, see Section IV). 3 It appears, therefore, that there might be reasons beyond limiting managerial risk taking that motivate mutual funds adoptions of investment constraints. This paper proposes some answers to the above questions. Our answers are based on the insight that investment constraints can benefit fund investors by enhancing risk averse managers incentives to acquire long-term investment information information that identifies significant asset mispricing, which may take a long time to be corrected. We show that by moderating the adverse selection problem in the trading of the mispriced assets, constraints reduce the compensation risk to managers when they make investments based on long-term information, thereby encouraging managerial efforts to acquire such information. Since mutual funds typically follow buy-and-hold strategies, they are likely to mostly focus on profiting from long-term information. Thus, subjecting managers to investment constraints can be especially beneficial to mutual fund investors. To see the intuition behind the above explanation, suppose that after expending efforts to acquire long-term investment information, some managers perfectly identify a mispriced asset. The informed managers, however, lack sufficient capital to immediately bring the price of the asset to its fundamental value (Shleifer and Vishny, 1997). As such, after the managers take positions in the asset, the market values of the positions may experience considerable fluctuation at the interim due to uninformed noise trade an effect that has been emphasized in the seminal work Hedge fund managers are typically paid through both a base fee of 1 to percent of the assets under management, and an incentive fee of about 15 to 0 percent of the profit above the high water mark (Goetzmann, Ingersoll, and Ross, 003). In comparison, most of the mutual fund managers are exclusively paid through a base fee, which used to be around 0.5 percent of the assets under management, but has risen steadily to 0.75 percent of the assets under management by 1995 (Golec, 003). Incentive fees that are tired to fund performance relative to a pre-specified benchmark are of rare occurrences in the mutual fund industry (Coles, Suay, and Woodbury, 000; Elton, Gruber, and Blake, 003; Golec, 003). 3 The difference in the uses of investment constraints between hedge funds and mutual funds can not simply be due to possible differences in the risk preferences between investors of the two types of funds. For, if managers incentives are perfectly aligned with those of investors, managers will diligently collet investment information and choose the appropriate risk level of funds portfolios in the absence of investment constraints.
of DeLong et al. (1990). 4 If mispricing is not widespread and the managers tilt fund investments toward the mispriced asset, noise-induced variation in the interim value of the asset can surface at the fund levels, even if noise trades are uncorrelated across assets. Unpredictable changes in the interim fund values present a risk to the managers. Since the managers receive periodic compensation that is tied to the amounts of assets under management (see footnote ), variations in the interim fund values subject the risk averse managers to a compensation risk. Because of this risk, the managers will trade conservatively on long-term information, thereby reducing the value of and managers incentives to acquire such information. Noise traders influence on the interim asset price is, however, mitigated by rational competitive market makers, who collectively possess a large amount of capital and observe the net order flow at the interim (Kyle, 1985; Glosten and Milgrom, 1985). If the order flow is known to be from uninformed noise traders, market makers will clear the flow at the prevailing market price of the asset, thereby removing noise-induced price variation. However, since market makers generally do not know the timing and the content of the managers information, they view the interim order flow as potentially comprising in part of trades by the informed managers. This adverse selection problem tempers market makers incentive to meet the order flow at the prevailing price, thereby limiting their mitigating effect on noise-induced interim price variation. Investment constraints on managers abate the above adverse selection problem. As the managers are expected to trade less intensely on their information, the market makers will be more willing to trade against the interim order flow, thereby dampening noise-induced interim price variation. As a result, the managers face a lower compensation risk when they trade on long-term information, and thus have enhanced incentives to obtain such information. Since the interim compensation risk presents a cost to the managers, it might appear that the managers will voluntarily limit their trading intensity, rendering the contractual constraints unnecessary. This voluntary behavior is, however, not time-consistent: while, ex ante, the managers may have an incentive to limit their trading intensity so as to reduce the compensation risk when they trade on long-term information, such an incentive dissipates when they do get significant information at the interim. To be effective, the constraints must be contractually binding. In addition to explaining the benefit of investment constraints, the above argument has three implications. First, high-powered compensation schemes may be of limited values to mutual 4 Reasons motivating noise trade include liquidity and hedging needs, changes in investor sentiment (Black, 1986; DeLong et al., 1990), or agency conflict (Trueman, 1988; Dow and Gorton, 1997). For a recent survey on noise trading, see Dow and Gorton (008). 3
funds: since such schemes tie compensation more closely to fund performance, they exacerbate the noise-induced compensation risk to managers. When the cost of increased compensation risk outweighs the benefit of high-powered compensation schemes in motivating managerial effort, mutual funds may be better off avoiding such schemes. Second, the benefit of investment constraints tends to be small when, on average, managers have limited exposure to noise-induced compensation risk. We argue in Section V of the paper that this is likely to be the case for hedge funds, due to hedge funds market specialization and investment strategies. As a result, investment constraints are less valuable for hedge funds than for mutual funds. On the other hand, the limited exposure to noise-induced compensation risk renders high-powered compensation schemes especially beneficial to hedge fund investors: such schemes motivate managerial effort to collect investment information, without creating an excessive compensation risk to managers. Third, our argument for the benefit of investment constraints applies well to large mutual funds. Since each such fund tends to take large positions in assets and create significant adverse selection problem in the trading of the assets, it is individually optimal for each fund to adopt constraints. However, a small mutual fund generally trades relatively modest positions that have little price impact. It might appear that small funds will not use constrains, so that the managers can free-ride on the benefit of the constraints on the large-fund managers and trade fully on valuable information. Yet, the evidence in Almazan et al. (004) reveals that investment constraints are more prevalent among small funds than among large funds. To provide a more complete explanation to the use of constraints in the mutual fund industry, in the later part of the paper (see Section IV) we extend our basic analysis by considering competition between large and small funds. Realism suggests that in comparison to managers of large funds, those of small funds tend to be less well established and have more uncertain abilities in collecting long-term information. As such, investors view better return performance by small-fund managers relative to that of large-fund managers as indicative of high ability by the former managers. In the absence of constraints, small-fund managers that are incapable of acquiring long-term information may nevertheless be able to exploit the advantage of conducting unconstrained trading to obtain return performance superior to that of constrained large-fund managers. To enable speedy resolution in the uncertainties about the managers competence, we show that investors of small funds may optimally adopt the same constraints used by the large funds, so as to ensure that performance can be compared on equal footings. 4
This paper complements Almazan et al. (004) and He and Xiong (013). Almazan et al. (004) argue that investment constraints are optimal devices in mitigating agency problems in fund investor-manager relationship when other forms of monitoring are weak or absent. Though not on investment constraints per se, He and Xiong (013) provides insights on funds uses of investment mandates, which restrict the segment(s) of markets that managers can invest in. They argue that mandates can enhance managers incentive to acquire information. In contrast, this paper follows the noise-trader approach pioneered by DeLong et al (1990), and identifies noiseinduced risk as the distinctive element behind the differences in the managerial contracts for mutual funds and hedge funds. We show that investment constraints are effective contractual devices to motive acquisition of long-term information by mutual fund managers. Our findings also help illustrate how contractually stipulated limit of arbitrage may arise as funds seek to mitigate the negative incentive effect of noise-induced risk. This paper is also closely related to the previous studies on the link between investment constraints and asset prices. Diamond and Verrecchia (1987), among others, show that constraints on traders materially affect asset prices. Focusing on equity mutual funds, Chen, Hong, and Stein (00) provide evidence that constraints on mutual fund managers significantly influence stock returns. The present analysis builds on this link to provide an explanation to why constraints on mutual funds managers should exist in the first place. The rest of the paper proceeds as follows. In Section II, we outline the model. In Section III, we present the analysis of the model and our main findings. Section IV considers several extensions of the model, including competition between large and small funds. In Section V, we discuss the differences between hedge funds and mutual funds, and argue that the benefit of investment constraints is likely to be limited for hedge fund investors. In Section VI, we conclude the paper. II. The Model The model has four dates, 0, 1,, 3. The key date is date, which was referred to as the interim date in Introduction section. A. The Assets There is one riskless and one risky asset. The riskless asset provides zero return over time with certainty. Each unit of the risky asset pays a random liquidating dividend v at date 3, where 5
(1) v = v + θ. In Eq.(1), v > 0 is the prior mean of v, and θ is the random component of v. Later in Section IV we extend our key results to a setting with multiple risky assets. B. Managerial Employment Contract For simplicity, we consider an enviroment with one manager. Generalizing our results to the symmetric equilibrium with multiple identical managers is straightforward but notationally cumbersome. The manager we focus on operates a fund of size w 0. At date 0, investors of the fund offer the manager an employment contract, denoted as (A 0, λ, X). The contract pays the manager an upfront fee of A 0 0, which covers the manager s cost of setting up the fund. The contract also stipulates periodic compensation λŵ t at dates t=1,, and 3, where 1 > λ > 0 and ŵ t is the date-t fund value w t normalized by the fund s initial value w 0, i.e., ŵ t = w t w 0. We refer to ŵ t as the date-t fund performance, and λ as the performance sensitivity of the compensation. Clearly, if λ = 0 the manager does not have any incentives to collect information to improve fund performance. We assume, therefore, that there is a minimum feasible value λ 0 so that performance sensitivity λ λ 0. 5 Finally, the contract imposes investment constraints. For simplicity, we assume that the constraints require that the manager s position in the risky asset to be in the interval [ X, +X], where X > 0. These constraint are not exactly the same as prohibitions of margin purchase and short-sales; nevertheless, they capture the main spirit behind the prohibitions, which limit the manager s position in an asset. The employment contract is publicly observable. The manager s compensation does not include a benchmark-adjusted incentive fee or implicit compensation associated with fund flows. This simplification is in part motivated by the fact that in the mutual fund industry, benchmark-adjusted incentive fees are rarely used (Coles, Suay, and Woodbury, 000; Elton, Gruber, and Blake, 003; Golec, 003). Nevertheless, in Section IV we show that including these two types of compensation does not alter our key results. C. The Information Structure According to the date-0 common prior belief, θ is uniformly distributed over the interval [ Θ, +Θ]. 5 Note that our analysis does not derive λ 0. Our primary interest relating to the compensation scheme concerns the issue as to why mutual fund investors may choose not to use high-powered compensation even if they are permitted to do so. 6
Thus, the date-0 price of the risky asset p 0 = v. At date 3, the realized θ becomes publicly observable, so p 3 = v + θ. For our analysis, we will be most interested in the case where Θ is very large such that information about θ before date 3 potentially reveals significant mispricing. There is a signal φ, which is perfectly informative of θ, i.e. φ = θ. If the manager expends effort e between dates 0 and 1, where e [0, 1], he gets to observe the signal before date 1 with probability e ; otherwise, he observes the signal after date 1 but before date with probability κ < 1/. With probability (1 κ e ), the manager remains uninformed throughout. Information about the signal before date 1 represents long-term information. Thus, a higher effort increases the manager s chance to obtain long-term information. On the other hand, the possibility of observing the signal between dates 1 and captures the uncertainty in the timing of the manager s information. We assume that both the signal the manager observes and the effort expended are his private information. Exerting effort e costs the manager c(e) = e. Figure 1 illustrates the time line of the key events. t = 0 T he manager expends e, and observes φ with probability e. Date 1 trading t = 1 T he manager observes φ with probability κ. Date trading t = θ realized t = 3 F igure1. T ime line of events D. The Trading Process Participants in trading at dates 1 and include noise traders, a large number of perfectly competitive and risk-neutral market makers, and the manager if he choose to trade at that date. The market makers set the market clearing price of the risky asset at each round of trading. They, however, do not observe the informative signal. Suppose that the manager observes signal φ after date 1 but before date. Since the manager is risk averse and before date 1 he views the risky asset as providing zero return, the manager does not take a position in the risky asset at date 1 but does so at date. Date- trading proceeds according to the Kyle (1985) model. Specifically, the manager submits order x (φ), which is the number of units of the risky asset he wishes to trade. Simultaneously, the noise traders submit their orders. The aggregate noise traders order is denoted as ɛ, and is uniformly distributed over [ Ω, +Ω]. The market makers do not separately observe the manager s and noise traders orders, but they observe the total net order flow y = x (φ) + ɛ. They then clear flow y by setting the 7
date- price, p (y ), equal to the expected date-3 value of the risky asset. Back to date-1 trading, we adopt two simplifying assumptions. First, when the manager observes signal φ before date 1, he follows an all-in strategy: he trades fully on the long-term information at date 1 without further trading at date. This assumption is not as restrictive as it may seem: by slightly generalizing our model, the all-in strategy can indeed be shown as the manager s optimal strategy. 6 Second, denote the manager s date-1 trade as x 1 (φ), we assume that x 1 (φ) has a negligible impact on the price, so that the date-1 price of the risky asset p 1 = v. There are at least two ways to justify this assumption. First, since information about θ before date 1 corresponds to long-term information, date 1 is way before date 3. As such, date 1 should be more realistically construed as representing a period of time, during which physical trading can take place at a number of calender dates. Thus, to take a large position in the risky asset, the manager can spread out his trades over the different calender dates, making the size of the trade on each date very small. Second, the key issue that concerns us is the effect of interim (date-) compensation risk on the manager s incentive to acquire long-term information. Explicitly accounting for possible price impact of the manager s trade at date 1 significantly complicates our analysis without adding insights to the key issue. Finally, if the manager does not observe the signal either before or after date 1, the expected return of the risky asset remains at zero throughout, so the risk averse manager never takes any positions in the risky asset. E. The Manager s and Investors Preferences The manager has time-additive expected utility over compensation with no discounting. Since the fund s performance is zero at date 1 (recall p 1 = v), the manager receives compensation only on dates 0,, and 3. The manager s expected utility of a compensation stream (A 0, λŵ, λŵ 3 ) is () E[U(A 0, λŵ, λŵ 3 )] = A 0 + E[U(λŵ )] + E(λŵ 3 ), where (3) E[U(λŵ )] = E(λŵ ) a V ar(λŵ ). 6 Specifically, following the argument in Shleifer and Vishny (1997), suppose that at date, there is a chance that φ may become publicly observable, and the price of the risky asset converges to its fundamental value. When the manager observes the signal before date 1, by delaying taking some of the positions till date the manager loses the opportunity to profit fully from his information when convergence occurs at date. Thus, if the probability of early convergence is beyond a certain threshold level, the manager will optimally follow the all-in strategy. 8
In Eq.(), E[U(λŵ )] is the manager s expected utility of the date- compensation, while E(λŵ 3 ) is his expected date-3 compensation. In Eq.(3), E(λŵ ) and V ar(λŵ ) are the expected and variance of date- compensation, and a > 0 is the manager s coefficient of risk aversion. Together, Eqs.() and (3) suggest that the manager is only risk averse with respect to the date- compensation. We assume risk neutrality with respect to date-3 compensation because we are most interested in the compensation risk to the manager from acting upon long-term information, but when the manager trades on such information at date 1, there is no uncertainty about the fund s performance at date 3. Fund investors are risk neutral and only care about their payoff at date 3. Thus, investors are unconcerned about variation in the interim fund value. In Section IV, we show that if investors are also risk averse with respect to the interim fund value, their benefit from imposing constraints on the manager increases. III. Analysis of the Model and Main Results Our analysis proceeds by first specifying the optimization problem that the optimal employment contract solves. We then analyze the manager s equilibrium trading strategies at date and date 1, followed by the derivation of the manager s optimal choice of information acquisition effort. Finally, we characterize the optimal employment contract for the manager. A. Investors Optimization Problem Given a stream of compensation to the fund manager, (A 0, λŵ, λŵ 3 ), the date-3 payoff to the fund s investors is w 0 +(ŵ 3 λŵ λŵ 3 ) A 0. Thus, the optimal employment contract (A 0, λ, X) solves the following optimization problem: (4) Max A0,λ,X(w 0 + E[ŵ 3 λŵ λŵ 3 e] A 0 ), (5) e = Argmax e {E[U(A 0, λŵ, λŵ 3 ) e ] e /}, (6) E[U(A 0, λŵ, λŵ 3 ) e] e / U, (7) x 1 (φ) [ X, +X], x (φ) [ X, +X] (8) λ λ 0. Eqs.(5) and (6) are standard in a moral hazard problem: Eq.(5) is the incentive compatibility condition for effort provision, while Eq.(6) is the manager s participation constraint. In Eq.(6), U is the manager s reservation utility, which is the highest utility the manager can fetch from outside employment opportunities. Eq.(7) states the investment constraints on the manager s 9
date-1 and date- trades, which we solve for in the next two subsections. Finally, Eq.(8) indicates the lower bound on λ. B. The Manager s Equilibrium Trading Strategy at Date When the manager observes signal φ after date 1 he trades x (φ) at date. We follow Kyle (1985) to consider a linear trading strategy, where: (9) x (φ) = β φ. Variable β > 0, and measures the manager s trading intensity at date. Given that φ [ Θ, +Θ], the manager s maximum buy order is Γ = β Θ, while the maximum sell order is Γ. The investment constraints require Γ X. As will be shown, the investment constraints only bind the manager s trading decision at date but not at date 1. Thus, without loss of generality, we set Γ = X. 7 Moreover, as can be shown, when Γ > Ω (the maximum noise trade) no equilibrium exists in which the manager follows a linear trading strategy as in Eq.(9) (notes are available upon request). Thus, we require Γ Ω. These conditions are summarized as follows: (10) Γ = X Ω. After both the manager and noise traders submit their orders, the market makers observe net order flow y. In contrast to the Kyle model (1985) where the market makers follow a linear pricing strategy, i.e. the price is linear in y, in the present setting the manager trades under constrains, so we permit the price to be nonlinear in y. The market makers, based on the observed y, revise their belief about the distribution of signal φ, and then set the price of the risky asset equal to its expected date-3 value. Since the market makers do not know whether the manager observes the signal before or after date 1, in principle they must try to infer from y both the likelihood that the manager is trading at date and the signal he has observed. However, purely for exposition ease we adopt the simplifying assumption that at date, the market markers invariably believe that part of the observed order flow y comes from the informed manager, and only try to infer the signal value from y (see footnote 8 for a description of the market makers inference when they seek to infer 7 This is without loss of generality because if in equilibrium, β Θ = Γ < X, the same equilibrium applies when X is reduced such that Γ = X. 10
both the likelihood that the manager is trading at and the manager s signal from the order flow). Though the market makers do not directly observe the manager s trade, they know about the manager s investment constraints and can rationally conjecture the manager s trading strategy in Eq.(9). Denoting the market makers updated belief about φ conditional on observed order flow y as f M (φ y ). Since both φ and ɛ are uniformly distributed, f M (φ y ) also follows a uniform distribution. To find f M (φ y ), we define function φ + (y ) as follows: (11) (1) β φ + (y ) = y + Ω if y < Γ Ω, φ + (y ) = Θ if y Γ Ω. Function φ + (y ) is the maximum possible value of signal φ that the market makers view as compatible with order flow y. A value φ of φ is compatible with y if it can lead to order flow y. That is, when the manager observes φ = φ and trades according to the strategy in (9), there is a noise trade ɛ such that the total order flow β φ + ɛ = y. Since the minimum value of ɛ is Ω, compatible value φ reaches the maximum when β φ = y + Ω, which occurs if y < Γ Ω, or when it is equal to its maximum feasible value Θ. Similarly, we define φ (y ), the minimum possible value of φ that is compatible with y, as follows: (13) β φ (y ) = y Ω ify > Γ + Ω (14) φ (y ) = Θ if y Γ + Ω. Figure illustrates functions φ + (y ) and φ (y ). +Θ φ φ + (y ) Γ Ω Γ Ω φ (y ) Θ Γ + Ω Γ + Ω y F igure. Graphic representations of φ + (y ) and φ (y ). T he solid line upper contour of the parallelogram represents φ + (y ), and the dashed line lower contour represents φ (y ). 11
The updated belief f M (φ y ) is uniformly distributed over [φ (y ), φ + (y )]. From Fig., if y assumes large positive values such that y > Γ + Ω, φ (y ) > Θ and φ + (y ) = Θ. In this case, since φ (y ) is increasing in y, a larger order flow is indicative that the manager has observed a more positive signal. In contrast, if y assumes large negative values such that y < Γ Ω, φ (y ) = Θ and φ + (y ) < +Θ. In this case, since φ + (y ) is increasing in y, a more negative order flow is indicative that the manager has observed a poorer signal. For the intermediate values of y, i.e. Γ Ω y Γ + Ω, φ (y ) = Θ and φ + (y ) = +Θ. Since, by the prior belief signal φ is uniformly distributed over [ Θ, +Θ], in this case compatible values of the manager s signal average to zero. Therefore, order flow y is completely uninformative about the manager s signal, and it does not make a difference to the market makers if y comprises in part of the manager s informed order or entirely of noise trade. This discussion provides the following lemma about f M (φ y ). 8 Lemma 1: The market makers updated belief I(φ (y ) < φ < Θ) Θ φ (y ) (15) f M (φ y ) = 1 Θ I( Θ < φ < φ + (y )) Θ + φ + (y ) if y > Γ + Ω if Γ Ω y Γ + Ω if y < Γ Ω, where I( ) is the indicator function that takes the value of one if φ satisfies the condition in the parenthesis, and zero otherwise. Based on the revised belief f M (φ y ), the market makers set the date- price, p (y ), equal to the expected value of the risky asset conditional on order flow y : Θ (16) p (y ) = v + φf M (φ y )dφ. Θ The following lemma describes p (y ). The proof of the lemma is in the Appendix. 8 If the market makers try to infer both the likelihood the manager trades at date and the manager s signal, the straight line in the upward sloping part of φ + (y ) or φ (y ) in Fig. are replaced by two connected line segments with differing slopes. However, φ + (y ) or φ (y ) over the intermediate values of y remain unchanged. 1
Lemma : The date- price is v + φ (y ) + Θ (17) p (y ) = v v + φ+ (y ) Θ if y > Γ + Ω if Γ Ω y Γ + Ω if y < Γ Ω. Unlike in the Kyle model, Lemma shows that when the manager trades under constraints, the price is nonlinear in y. In particular, when y is outside the intermediate-value range, p (y ) is increasing in order flow y, as both φ + (y ) and φ (y ) are increasing functions of y. In the intermediate-value range, y is uninformative of the manager s signal, and the market makers view y as if it arises entirely from noise trade. In this case, therefore, p (y ) equals the prior mean value of the risky asset v, independent of y. A key implication of Lemma is that tighter investment constraints (smaller X) reduces the variability of date- price p (y ). Specifically, as X declines, Γ declines (see Eq.(10)), expanding the intermediate-value range of order flow y. Since the date- price invariably assumes v over the intermediate-value range, as this range expands the variability of p (y ) declines. To complete the discussion on date- trading, we solve for the condition under which the investment constraints are strictly binding. To do so, note that with binding constraints, β Θ = Γ = X, so β = X Θ. In the absence of the constraints, the manager chooses β to maximize the expected utility of compensation subject only to the constraint Γ Ω. Since when the manager trades at date, the date- performance ŵ = 0, by Eq.() maximizing the expected utility of compensation is equivalent to maximizing the expected date-3 fund performance. Thus, the unconstrained optimal trading intensity solves Max β (β φ){(φ + v) E[p (y ) x (φ)]}, (18) s.t. β Θ Ω. In the above optimization problem, E[p (y ) x (φ)] is the price at which the manager expects to trade x (φ). The following lemma provides the expression for this expected price and the condition under which the constraints are strictly binding. 13
Lemma 3: Upon trading x (φ), the manager expects the date- price to be (19) E y [p (y ) x (φ)] = v + Θ Ω x (φ). The first-order-condition for the unconstrained optimal intensity applies when β Θ = Ω, so the unconstrained optimal intensity β = Ω Θ. In the equilibrium with investment constraints, therefore, the manager s date- trading intensity (0) β = min{ X Θ, Ω Θ }. If X < Ω, the investment constraints are strictly binding. Lemma 3 indicates that when the investment constraints are binding, i.e. X < Ω, tighter constraints reduce the manager s date- equilibrium trading intensity β. In turn, the reduced trading intensity mitigates the variability of the date- price. C. The Manager s Equilibrium Trading Strategy at Date 1 If the manager observes the signal before date 1, he trades x 1 (φ) units of the risky asset at date 1. The date- and date-3 fund performance are ŵ = x 1 (φ)(p (y ) v) and ŵ 3 = x 1 (φ)φ, respectively. The only element of uncertainty in fund performance is the date- price p (y ). The manager chooses x 1 (φ) to maximize the expected utility of compensation, subject to the investment constraints. It follows from Eqs.() and (3) that optimal x 1 (φ) solves the following problem: Max x1 (φ){λe y [x 1 (φ)(p (y ) v)] a (λx 1(φ)) V ar y (p (y )) + λx 1 (φ)φ}, (1) s.t. x 1 (φ) [ X, +X] The first two terms in the manager s objective function represent the expected utility of the date- compensation, while the last term is the date-3 compensation. Since following the all-in strategy the manager does not trade again at date, he knows that the date- order flow y will be due entirely to noise trade, i.e., y = ɛ. Setting x (φ) = 0 in Eq.(19) shows that the expected date- price E y (p (y )) = v. Thus, the objective function in Eq.(1) can be rewritten as () λx 1 (φ)φ a (λx 1(φ)) V ar y (p (y )). 14
In this form, the manager s objective function resembles a mean-variance utility function, with the variance of the date- price, V ar y (p (y )), representing the interim compensation risk to the manager when he trades on long-term information. To clearly indicate that V ar y (p (y )) represents a noise-induced compensation risk, in the following we write it as V ar ɛ (p (ɛ)), instead. We solve for the manager s optimal date-1 trading strategy in the appendix. The results are stated in the following lemma. Lemma 4: The noise-induced interim compensation risk, (3) V ar ɛ (p (ɛ)) = β Θ3 1Ω. For large Θ, the investment constraints are not binding at date 1, and the manager optimally follows a linear trading strategy x 1 (φ) = β1 φ, where the date-1 equilibrium trading intensity (4) β 1 = 1 aλv ar ɛ (p (ɛ)). Eq.(3) reveals the determinants of the interim compensation risk. First, consistent with our previous discussion, this risk worsens as the adverse selection problem in date- trading becomes more severe, that is, as the intensity with which the informed manager trades at date, β, increases. Second, this risk is increasing in Θ, suggesting that the risk is most pronounced when the information that the manager trades on may reveal significant mispricing. Eq.(4) shows that the effect of the interim risk on the manager s incentive to act on long-term information: β 1 is decreasing in V ar ɛ(p (ɛ)). Thus, a higher interim compensation risk causes the manager to be more conservative when trading on long-term information. Eq.(4) also shows the cost of high-powered compensation: β 1 is inversely proportional to λ. Thus, more high-powered compensation leads the manager to act more conservatively on long-term information. This effect arises because more high-powered compensation increases the manager s exposure to the interim compensation risk (see the second term in Eq.()), and thus reduces the manager s incentive to act upon long-term information. D. The Manager s Optimal Effort Choice A higher effort increases the manager s chance of obtaining long-term information (observing the signal before date 1), and reduces the manager s chance of remaining uninformed throughout. In the latter case, the manager never takes a position in the risky asset, so, other than the upfront fee, he receives zero compensation. Thus, by Eq.(), the effort-dependent portion of the 15
manager s expected utility is (5) e +Θ dφ Θ Θ {λx 1(φ)φ a (λx 1(φ)) V ar ɛ (p (ɛ))} e, where x 1 (φ) and V ar ɛ (p (ɛ)) are from Lemma 4. The first term in Eq.(5) is the manager s expected benefit from trading on long-term information, while the second term is the cost of expending effort e. The manager s optimal effort, denoted as e, follows from taking the first-order condition of the expected utility in Eq. (5): (6) e = 1 +Θ Θ dφ Θ {λx 1(φ)φ a (λx 1(φ)) V ar ɛ (p (ɛ))}. Substituting β1 from Eq.(4) into x 1(φ) = β1 φ, and calculating the integration in the above equation provides the following lemma. Lemma 5: The manager s optimal effort (7) e = Θ 1aV ar ɛ (p (ɛ)) Since the interim compensation risk causes the manager to be more conservative when trading on long-term information, it reduces the value of such information. As Eq.(7) shows, the manager s optimal effort decreases as the interim compensation risk increases. Eq.(7) also reveals that the performance sensitivity of the compensation, λ, has no effect on the manager s optimal effort choice. This follows because high-powered compensation creates two opposing effects on the manager s incentive to acquire long-term information. First, more high-powered compensation encourages information acquisition effort by increasing the manager s share of the fund s profit. However, more high-powered compensation also increases the manager s exposure to the interim risk and causes the manager to be more conservative in trading on long-term information (see Eq.(4)), thereby reducing the value of such information. The two opposing effects tend to balance out one another, leaving high-powered compensation ineffective in motivating managerial effort. A related result appeared previously in Stoughton (1993). Unlike in this analysis, however, in Stoughton (1993) compensation risk originates from the imprecision of the manager s information about the fundamental value of the risky asset. E. The Optimal Employment Contract The optimal employment contract maximizes investors expected payoff, i.e., the expected date-3 16
fund value net of the compensation to the manager, subject to the constraints in Eqs.(5)-(8). Since the upfront fee A 0 is a pure transfer, fund investors can always adjust A 0 to ensure that the manager s participation condition is binding. Thus, the optimal employment contract maximizes the total surplus the sum of the manager s expected utility of compensation and investors expected payoff. Up to the initial fund value w 0, the total surplus is (8) e + (1 λ)e Θ Θ dφ Θ Θ x dφ 1(φ)φ + κ Θ Θ x (φ)(φ Θ Ω x (φ)), where e is from Lemma 5, and the manager s equilibrium trading strategies at dates 1 and, x 1 (φ) and x (φ), are respectively from Lemmas 3 and 4. The first term in Eq.(8) is the manager s expected utility from expending effort e and then obtaining and trading on long-term information. This follows from combining Eqs.(5) and (6). The second term represents investors proportional share of the fund s expected date-3 profit from the manager trading on long-term information. The last term is the expected date-3 fund profit, which is shared between the manager and investors, from the manager observing the signal after date 1 and trading on the information at date. This follows from the objective function in the optimization problem in (18) and from (19). The following proposition characterizes the investment constraints in the optimal employment contract. Proposition 1: The optimal employment contract specifies investment constraints. These constraints strictly bind the manager s date- trading strategy. To see the optimality of binding investment constraints, suppose initially that the constraints are not binding. Then, upon observing the signal after date 1 the manager chooses date- trading intensity to maximize the expected date-3 fund profit, and by Lemma 3 the unconstrained optimal intensity β = Ω Θ. Suppose now that investors set X slightly below Ω, so that the investment constraints become strictly binding and, by Lemma 3, β reduces to X Θ. Because of the firstorder-condition on the unconstrained optimal intensity (see Lemma 3), the marginal effect of the reduction in the trading intensity on the expected date-3 fund profit, the third term in Eq.(8), is zero. On the other hand, the binding investment constraints mitigate the adverse selection problem in date- trading and thereby reduce the noise-induced interim compensation risk to the manager when he trades on long-term information. Thus, binding constraints induce the manager to trade more intensely on long-term information, i.e., x 1 (φ)φ increases, and expend 17
more information acquisition effort e, leading to increases in the first two terms in Eq.(8). The above argument makes clear the benefit and cost to fund investors from imposing investment constraints on the manager. On one hand, binding constraints limit the fund s profit when the manager gets valuable information after date 1. On the other hand, binding constraints improve the fund s profit from the manager diligently collecting and trading on long-term information. On balance, the benefit of investment constraints dominates, so it is optimal for a fund to impose investment constraints on the manager. Turning to the optimal performance sensitivity of compensation, we have the following. Proposition : In the optimal employment contract, the performance sensitivity of compensation λ assumes the minimum feasible value λ 0. To see the intuition behind Proposition, recall that higher performance sensitivity does not incentivize the manager to work more diligent to collect long-term information, i.e. e is independent of λ. Moreover, the performance sensitivity does not influence the manager s date- trading strategy and thus the third term in Eq.(8). However, by increasing the manager s exposure to interim compensation risk, higher performance sensitivity causes the manager to be more conservative when trading on long-term information, so x 1 (φ)φ declines, thereby reducing the second term in Eq.(8). Consequently, to encourage active trading on long-term information, in the optimal employment contract fund investors set the performance sensitivity of compensation at the minimum feasible value. IV. Discussion A. Fund Flows For open-end mutual funds, prior studies show a positive relation between fund flows and fund performance (e.g., Chevalier and Ellison, 1997; Sirri and Tufano, 1998). Since the manager s date-t compensation is positively related to the date-t fund value, fund flows may significantly influence the manager s compensation. To consider such an influence, suppose that performance ŵ t at date t induces a flow sŵ t, where s > 0 is the flow-performance sensitivity. Then, the manager s date-t compensation becomes λ(1 + s)ŵ t. Clearly, fund flows magnify the effect of variation in fund performance on the manager s compensation. Thus, when the manager trades on long-term information, the anticipated date- fund flow exacerbates the noise-induced interim compensation risk, making the case for imposing investment constraints stronger. The empirically documented flow-performance curve is convex when performance is partic- 18
ularly good. That is, s is an increasing function of ŵ t when ŵ t is very large. This convexity suggests a possible managerial preference for high variability in interim performance, when ŵ is likely to be exceedingly good (Chevalier and Ellison, 1997). In our model, such a preference is likely to be fairly weak. This is because when the manager trades on long-term information, order flow y is due entirely to noise trade, so the manager expects ŵ to be on average zero, away from the range where the convexity of the flow-performance curve is important. B. Benchmark-adjusted Compensation Previous literature suggests that properly designed benchmark-adjusted compensation, by filtering out components of a manager s performance that are beyond the manager s control, promotes managerial incentive to diligently collect investment information (Stoughton, 1993). In our model, however, useful benchmark-adjusted compensation is likely to be hard to implement because of the difficulty in finding a suitable benchmark. Specifically, when the manager trades on long-term information the date- fund performance is determined entirely by noise trade, which is beyond the manager s control; but, the date-3 performance is attributable solely to the manager s information acquisition effort. A suitable benchmark, therefore, needs be a portfolio whose interim performance is publicly known to be closely correlated with the noise trade in the asset that the manager invests in, but whose date-3 performance is known to be highly predicable. Such a benchmark, however, can not exist: given that the portfolio s date-3 value is predicable, investors know precisely the date- value of the portfolio, violating the requirement that the interim performance of the benchmark portfolio needs be correlated with that of the asset in which the manager invests. C. Risk-averse Investors Suppose that similar to the manager, investors are risk averse with respect to their interim wealth the date- fund value net of the compensation to the manager. Since given date- fund performance ŵ the manager receives compensation λŵ, the net date- fund value is w 0 +(1 λ)ŵ. We assume that risk-averse investors have the following expected utility function: (9) [w 0 + E(ŵ 3 λŵ λŵ 3 ) A 0 ] b V ar((1 λ)ŵ ), where b is investors coefficient of risk aversion. 9 The first term in (9) is the same as in the 9 The utility function in (9) can be formally derived by slightly modifying our model. In the modified model, investors care about the date- fund value because they may need sell the fund portfolio at date to meet liquidity or portfolio rebalancing needs. 19
previous section, but the second term introduces investors aversion to variation in the interim fund value. When investors are risk averse, in addition to providing incentives to the manager the employment contract also offers risk sharing between the manager and investors. To see how the need for risk sharing might modify the optimal employment contract, note that given the manager s and investors mean-variance utility functions, the compensation scheme that offers efficient sharing of the interim risk is a linear scheme with λ = b a+b (e.g. Stoughton, 1993; Admati and Pfleiderer, 1997). Since investors are typically much better diversified than the manager (b is much smaller than a), optimal risk sharing requires λ be small, constrained only by the need to provide the manager with the incentive to work hard. However, since the manager s effort choice is independent of λ (see Lemma 5), the usual friction between the risk sharing and incentive provision roles of the compensation scheme (e.g. Holmstrom, 1979) does not apply here. Thus, optimal λ continues to assume a small value as in Proposition. On the other hand, since the interim risk now directly creates a disutility to investors, the optimal employment contract sets tighter investment constraints than those when investors are risk neutral. D. Multiple Risky Assets Suppose that there are N risky assets. For simplicity, we assume that the payoff from, the process of trading, and the timing and content of the manager s information about, each of the assets are identical to those in the single-asset case in the previous section. We use the same variables from the previous section, but add subscript i, where i = 1,..., N, when the variables apply to asset i. We assume that date- noise trade {ω i } N i=1 across the N assets is independent of one another. Moreover, we assume that date- trading is not fragmented, so that there is a common pool of market makers who observe and clear the net order flow in the trading of each of the N assets. To illustrate the applicability of the optimal employment contract in Propositions 1 and to multi-risky-asset environment, we consider two cases. In the first case, θ i s and φ i s are independently distributed across the N assets, and the times at which the manager observes φ i s are also independent across the assets. In addition, expending information acquisition efforts {e i } N i=1 costs the manager N i=1 e i. Since φ i s are independently distributed, in revising the date- belief about the value of asset i the market makers rely exclusively on the net order flow in the trading of asset i, y i. Thus, the date- price of asset i, p i (y i ), assumes the same form as that in Lemma, and when the manager observes φ i after date 1, his optimal date- trading decision 0