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Article Excerpt
Summary. In this paper we study delegated portfolio management when the manager's ability to short-sell is restricted. Contrary to previous results, we show that under moral hazard, linear performance-adjusted contracts do provide portfolio managers with incentives to gather information. We find that the risk-averse manager's effort is an increasing function of her share in the portfolio's return. This result affects the risk-averse investor's choice of contracts. Unlike previous results, the purely risk-sharing contract is now shown to be suboptimal. Using numerical methods we show that under the optimal linear contract, the manager's share in the portfolio return is higher than what it is under a purely risk sharing contract. Additionally, this deviation is shown to be: (i) increasing in the manager's risk aversion and (ii) larger for tighter short-selling restrictions. As the constraint is relaxed the deviation converges to zero.
Keywords and Phrases: Third best effort, Linear performance-adjusted contracts, Short-selling constraints.
JEL Classification Numbers: D81, D82, J33.
1 Introduction
Investors delegate portfolio decisions to managers because of their alleged skill in gathering superior information on movements in security prices. When the manager's research activity is not observed, the investor could face problems associated with moral hazard. Then, it could be in the investor's interest to provide the manager with incentives to gather better information. In studying the nature of such incentive contracts, past literature has assumed the manager's portfolio choice to be unbounded. Yet, we seldom observe environments where the manager's portfolio choice is totally "unrestricted." Practices like borrowing money, margin purchases, short-selling or investment in derivative securities are usually restricted. Our purpose is to study the effect of such constraints on incentive provision.
We assume that the manager's ability to short-sell is restricted and that investors have to cope with moral hazard. Our primary interest is in the impact of short selling restrictions on the power of incentives provided by linear symmetric contracts. We report three main results. First (Corollary 2), linear performance-adjusted contracts do provide managers with incentives for gathering better information. Second (Proposition 4), we show that the manager's share in the portfolio return is different from that under the purely risk sharing contract, (we shall refer to the purely risk sharing contract as the first best contract). (3) Third, using numerical methods, we show that the manager's share in the optimal portfolio is higher than that under the first best and decreases as we relax the leverage constraint. We also present some additional results. In a scenario without moral hazard, but with short selling restrictions: (i) under the optimal linear contract, the manager's share in the portfolio is equal to the one under the first best contract (Proposition 4); (ii) linear contracts dominate quadratic contracts (Proposition 6, in Appendix A). With moral hazard and short selling restrictions, numerical methods show that, quadratic contracts dominate linear contracts only for certain parameter values (Table 2 in Appendix A).
We take restrictions on short selling as given. Almazan et al. (2004) report that 70% of mutual funds explicitly state (in Form N-SAR handed to the SEC) that short selling is not permitted. The authors, however, assert that these restrictions are more than regulatory prohibitions. Hence, endogenizing short selling constraints may be a valuable line for future research.
Our main focus is on the incentives provided by linear symmetric contracts. Such contracts need not be optimal in the domain of all contracts and quadratic contracts are known to perform better than linear contracts in certain environments. We compare linear and quadratic contracts in Appendix A. (4) There are two reasons for focusing on linear contracts in the main text of the paper. First, from an institutional point of view, the Security Exchange Commission (SEC) restricts compensation contracts in the mutual fund industry to only linear symmetric contracts. Second, restricting our domain to symmetric linear contracts provides us with the very well known "no-incentive" benchmark. When no restrictions on short-selling exist, Stoughton (1993) and Admati and Pfleiderer (1997) have shown that linear (fulcrum) contracts fail to affect the manager's decision to gather better information. In other words, the manager's optimal effort choice is independent of the contract she receives from the investor. As a consequence, the only role for the linear contract is to split risk efficiently between the manager and the investor: a higher risk aversion of the former relative to the latter would then imply no performance adjustment component in managers fees.
In contrast to the "no-incentive" result, our first result asserts that under moral hazard and finite short-selling bounds, linear contracts do provide the manager with incentives to gather better information. Both assumptions are necessary for this result. With moral hazard but no short-selling bounds, the no-incentive result prevails. With short-selling constraints but no moral hazard, incentives for performance are not required. Hence, as we show in Proposition 4, the first best split is optimal.
The intuition behind our first result is as follows. With no short selling constraints the manager is able to undo the effects of incentives by appropriate modifications of the portfolio. Hence, we get the "no incentive" result. With finite short selling bounds, no matter how large they are, the manager anticipates that with positive probability she shall not be able to form the portfolio of her choice. This leads her to reduce effort in gathering better information. Under such circumstances, by increasing the incentive fee the investor expands the manager's portfolio set, thereby partially undoing the effects imposed by short-selling bounds. This in turn, provides her with incentives for spending more effort.
Given the investor's utility function, the cost of increasing effort through linear contracts may be too high. As a result, the investor may simply desire to share risk through the first best sharing rule and ignore effort inducement. Our second result rules out such behavior: the first best sharing rule is never optimal.
We are not able to derive closed form solutions for the optimal linear contract. (5) Using numerical methods, we show that the manager's share in the portfolio is higher than in the first best. Importantly, this share converges to the first best level as the bounds on short selling get relaxed. Thus, the "no-incentive" result is a special case. This final result can be interpreted as follows. In the constrained scenario, the performance adjustment fee plays an additional role beyond risk sharing, namely effort inducement. When the short-selling bounds shrink (making the restriction tighter) the volatility of the portfolio decreases as well since fewer "extreme" portfolios are feasible. If the investor does not increase the performance adjustment fee the manager will be under-exposed to management risk. As a consequence, effort will also decrease. The risk sharing and the effort inducement arguments are aligned in the same direction: the optimal incentive fee increases above the first best value. This effect is enhanced by the manager's risk-aversion: given a certain level of short-selling, the (percentage) deviation from the first best share increases as the manager's risk-aversion augments.
The rest of the paper is organized as follows. Section 2 introduces the basics of the model. We distinguish four possible scenarios, depending on the restrictions on portfolio choice (constrained/unconstrained) and the observability of effort (public-information/moral hazard). The optimal linear unconstrained contract under public-information is termed the first best. The second best scenario is reserved for one where there are no constraints on short selling but where the manager's effort is not observable. The third best scenario pertains to the one where constraints on short selling are exogenously imposed and the manager's effort is unobservable. Section 3 studies linear contracts. Here we study linear contracts without restrictions on portfolio choice, both in the first best and second best scenarios. The same analysis is repeated for constrained portfolio problems in Section 4. Section 4.1 presents numerical results on the optimal linear contract under limited leverage, i.e. on the third best contract. Linear and quadratic contracts are compared in Appendix A. All proofs are provided in Appendix B.
2 The model
A typical fund will inform the customer that managers (who are involved in investment research) are responsible for choosing each fund's investments. Customers may also be informed about how the managers are compensated. Given the information, the customer decides how much to invest in the fund. In this paper we shall abstract from the decision problem of the consumer. Instead, assuming that the interests of the customer and the fund owner are the same, we shall focus on the determination of the manager's compensation scheme by the owner of the fund. Slightly abusing terminology, we call the owner of the firm the investor.
Let the manager and the investor have preferences represented by exponential utility functions. Throughout the paper we will use a > (b > 0) to denote the manager (investor) as well as her (his) absolute risk aversion coefficient.
The manager's investment opportunity set consists of two assets: a risky asset with net return [~.x] and a riskless bond. Assume that [~.x] is distributed as a standard normal variable. The distribution of the risky asset return and the return on the bond are public information. As in Heinkel and Stoughton (1994), the bond is taken as the benchmark portfolio against which the returns on the manager's portfolio are measured. The investment horizon is one period. At the beginning of the period, the investor transfers one unit of wealth to the manager who also receives a compensation contract from the investor. This contract sets the management fee as a percentage of the wealth under management and consists of two components: a fixed flat fee, denoted by F, and a performance adjustment fee. The performance adjustment rate is calculated as a percentage [alpha] of the portfolio's excess return over the net...
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