Incomplete Contracts in Investment Models∗ Diego Garc´ıa† January 24, 2001
∗
I would like to thank Peter DeMarzo for his advice and suggestions on the notes upon which this pa-
per is based.
I would also like to thank Sandro Brusco, Ben Hermalin, Hayne Leland, Stefan Reichel-
stein, Rafael Repullo, Mark Rubinstein, Ilya Segal, Matthew Spiegel, and Nicholas Wonder for their comments, as well as seminar participants in UC Berkeley, Universidad Carlos III, CEMFI, Dartmouth College, UT Austin, Wisconsin University, University of Washington, XV Jornadas de Econom´ıa Industrial, and the 1999 FMA PhD Student Symposium. The latest version of this paper can be downloaded from http://mba.tuck.dartmouth.edu/pages/faculty/Diego.garcia/. † Correspondence information: Diego Garc´ıa, Tuck School of Business Administration, Hanover NH 037559000, tel: (603) 646-3615, fax: (603) 646-1308, email:
[email protected].
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Abstract This paper studies optimal contracts in a standard investment model. A risk-averse principal hires a risk-averse agent to manage her wealth by deciding the amounts to be invested in a set of risky securities. I focus on the frictions caused by the inability of the parties to write contracts that are functions of the state of nature: payments to the agent are allowed to depend only on final output. It is known (Ross (1973)) that arbitrary contracts written on final wealth fail to achieve the first-best outcome in general, even in a pure risk-sharing environment. I show that driven by this inefficiency, the principal may find it in her best interest to forgo investment in a subset of the assets. This provides a theoretical foundation for the existence of constraints on the portfolios that mutual fund managers choose. In this framework, fixed wages can be optimal even when the principal and the agent can write contingent contracts. The model can be used to explain the allocation of securities among different funds, allowing for a characterization of this optimal allocation on the basis of the assets’ risk-return attributes. JEL classification: D20, G11, L20. Keywords: incomplete contracts, portfolio choice, risk-sharing.
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1
Introduction
Financial markets are dominated by institutional investors. Pension funds, mutual funds, investment advisors and money managers in general comprise a larger proportion of the global holdings of assets in the 1990’s than ever before. Several empirical regularities characterize the industry.1 The Investment Advisers Act of 1940 restricts performance-based compensation to being symmetric around a chosen index, thereby prohibiting asymmetric fees. Perhaps due to this restriction, the typical fund manager is compensated through a fixed wage.2 Another pervasive feature of the industry is the specialization within asset classes of different funds. Even though the investment abilities that the managers claim to have should allow them to gain excess returns on most asset classes, managers seem to find it in their best interest to restrict themselves to analyzing a particular set of securities. The standard labels of growth, value, yield, . . . , as well as the existence of funds specialized in stocks, high-yield bonds, . . . , are some of the most prominent features of the mutual fund industry. This specialization could be due to some economies of scale in the information processing of money managers. But note that often funds commit to not trade on derivative securities (e.g. future contracts, see Almazan, Brown, Carlson, and Chapman (2000)). An argument based on economies of scale would suggest that the mutual fund manager should be allowed to trade the underlying assets and derivatives written on them, since all are subject to the same uncertainty. The most natural way to study the relationship of the fund manager with the investors is as an agency problem. In the finance literature, there are a large number of research projects that study the optimal contracting problem faced by an investor and a mutual fund manager. In most models, private information, and/or superior abilities of the agent plays a central role (Dybvig and Ross (1985), Admati and Ross (1985), Bhattacharya and Pfleiderer (1985), Starks (1987), Stoughton (1993), Heinkel and Stoughton (1994), Huddart (1994), Admati and 1
The following discussion does not attempt to be comprehensive. See Grinblatt and Titman (1989b), Lakon-
ishok, Shleifer, and Vishny (1992), Golec (1992), Ippolito (1992), and Elton, Gruber, Das, and Hlavka (1993) for a discussion of empirical regularities in the invesment industry. 2 Fund managers usually receive most of their compensation in terms of “fraction-of-funds” fees, which are a percentage of the total money under management. This is probably due to incentive issues related to gathering new clientele. The focus of this paper is on the incentives that are related to the actual investment decision of the manager, so the relevant fees are those related to the performance of the fund.
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Pfleiderer (1997), Ou-Yang (1998), Das and Sundaram (1999), Palomino and Prat (1999)). With few exceptions, these papers do not study multi-dimensional action problems and rather focus on one-dimensional models in which the driving forces are either costly effort and/or adverse selection. Therefore, the above empirical regularity with respect to the specialization of the mutual fund managers cannot be endogeneously derived from these models. The theoretical framework introduced in this paper matches the above empirical facts. Symmetric contracts (linear payment functions) and fixed wages are shown to be the optimal contracts in the model introduced below. Moreover, it may be optimal for the principal to restrict access to a subset of the securities, giving away their return and hedging potential in order to improve the outcome of the contracting game. When the principal is allowed to contract with several agents, it is shown that as the risk-aversion of the agent (relative to the principal’s) becomes large, the optimal contracts and security allocation among managers is one of specialization: each manager should hold assets that are “similar” to each other (see section 4.2 for the precise statements). These results provide a foundation for the existence of specialization in the mutual fund industry, as well as a rationale for having restrictions in the set of trading strategies that managers can take (see Almazan, Brown, Carlson, and Chapman (2000) for an empirical study of some types of restrictions on the portfolios of US mutual funds). Carpenter, Dybvig, and Farnsworth (2000) and Garcia (2000) provide other theoretical explanations to the existence of portfolio constraints. Both models’ results are driven by having a privately informed agent. In these two papers the portfolio constraints appear after a signal about the excess returns of the assets has been observed, rather than at the time of contracting, as is the case for the restrictions studied by Almazan, Brown, Carlson, and Chapman (2000) and in this paper. Risk-sharing between the principal and the agent must be an important ingredient in a theoretical model of delegated portfolio management. Most of the principal-agent literature focuses on the case where the principal is risk-neutral, in order to abstract from pure risksharing issues. In a standard portfolio selection problem without trading constraints, a riskneutral individual’s optimization problem would not be well-defined, so risk-aversion for both parties is a crucial part of the problem. The analysis of this paper first focuses on characterizing optimal contracts in a pure risk-sharing environment, i.e. without adverse selection or costly 4
effort. Even though private information is probably an important part of the problem, it is crucial to have a thorough understanding of the model when risk-sharing is the only ingredient. Moreover, contracting inefficiencies present in a risk-sharing environment should prevail in a more complicated model, as I argue in section 5. This paper studies the problem faced by a risk-averse principal who can write (arbitrary) contracts on the total value of the assets under management in order to induce an investment decision from a risk-averse agent.3 I focus on the standard parametric case for which closed form solutions can be obtained: the agent and the principal both have CARA preferences, and the returns of the assets under management are normally distributed. The early literature (Wilson (1968), Ross (1973), Ross (1974), Leland (1978), Haugen and Taylor (1987), Dybvig and Spatt (1986)) finds conditions under which the first-best is achieved when the action is unobservable. When the principal and the agent have the same subjective beliefs and the same endowment, the result of the contracting game with unobservable actions is the same as the one in a first-best world (where the action is observable and verifiable). This result relies on the fact that the optimal risk-sharing rule is linear, which is sufficient to yield first-best outcomes under the above conditions. The point of departure in this paper is to study a model in which these conditions are not satisfied, i.e. when the principal and the agent have different endowments and/or different beliefs about the risk-return trade-off of the assets under management.4 It is shown that with the above assumption on preferences and returns, linear contracts are still optimal. The outcome of the game is nevertheless inefficient, and I investigate different ways for the principal to overcome these contracting frictions. The paper proves that restricting access to a subset of the assets may be optimal when the actions are unobservable. This result generates interesting implications regarding the allocation of assets among funds. The inefficiencies caused by the incompleteness of the contract space create an incentive for the principal to set up different smaller funds, where the frictions would partially disappear. This 3
It is in this sense that the paper refers to incomplete contracts as the forces generating the inefficiencies
in this model, since if the parties would be able to write state-contingent payment functions, the inefficiencies would disappear. 4 See Heaton and Lucas (1999) for an empirical investigation of the importance of non-tradable endowment risk for entrepreneurs, and its effects on their portfolio choice.
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is similar in flavor to the “task allocation” results of Holmstrom and Milgrom (1991). The trade-off in my model is between the cost of creating a new fund, and the benefits generated through better contracting possibilities. The theoretical framework can be used to describe several economic problems. The most natural interpretation of the role of the agent is that of an investment manager in charge of a set of assets in which to allocate the wealth of the principal. The contracts written for mutual fund managers are cast in terms of the total return of the assets under management, so the setup of this paper seems to fit well with this application. But the model can also be used to discuss contracting issues in a corporation: the agent could be thought to be the manager of a firm that has access to some production technologies and whose sole decision is how much money to allocate to each of them. In contrast to standard contracting models, I use the Arrow-Debreu paradigm to describe an asset allocation decision: choice of securities characterized by different payoffs across states of nature.5 The key ingredient of the model, as compared to standard moral hazard models,6 is that the agent can control the risk of the value of the assets under management. The formal model is closely related to that of Holmstrom and Milgrom (1991), who present a standard moral hazard problem with multiple tasks for the agent. They show that if some aspects of performance are not measurable, fixed wages can be optimal. The driving force in their paper is the costly effort that the agent incurs. Using contingent contracts has the negative effect of diverting attention from tasks that are not easily measurable to those that are. Multidimensionality of the action space is a key ingredient of the model to be introduced in the next 5
The distributional approach (see Holmstrom and Milgrom (1987)) that is commonly seen in most papers in
the contracting literature loses some of its bite when considering multi-dimensional action problems. Namely, it is impossible to build the notion of contracting incompleteness by using the standard (reduced-form) model as laid out in Grossman and Hart (1983), since the output of the firm is a sufficient statistic for the state of the world. 6 In the moral hazard literature, the actions of the agent map into probability distributions that can be ranked by first-order stochastic dominance. In a Gaussian environment (see Holmstrom and Milgrom (1987)), the agent’s action only affects the expected return of the wealth, not its variance. Analyzing contracting problems with more general dependencies between output and actions is an open question to which this paper brings a partial answer (see Hermalin and Katz (1994) and Sung (1995) for other models that relax the distributional assumptions of standard contracting models).
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section. But note that the results in this paper are not driven by the relative “costs of effort” of the different actions of the agents, but rather simply due to contractual incompleteness. Ou-Yang (1998) solves for optimal contracts in a continuous-time economy, where the agent has CARA preferences, and the assets under management follow geometric Brownian motion.7 He extends Ross’s optimality results to this dynamic setting by showing that the first-best outcome can be achieved by simply writing contracts that are a function of the path of total wealth, even when the preferences of the principal are outside the CARA class. His results crucially depend on the fact that the principal and the agent would like to invest in the same mean-variance efficient mutual fund. The one-period model in this paper with CARA preferences yields the same type of model as in Ou-Yang (1998) with constant market coefficients. I extend his results by considering the effects of endowment risk or different beliefs about the expected returns of the assets. Admati and Pfleiderer (1997) study the optimality of linear contracts written on total output and (possibly) a benchmark in the case where the agent and the principal have CARA preferences and returns are Gaussian. A crucial aspect of their analysis is that the first-best is always achieved when the principal can use a particular benchmark to compensate the agent. But note that this benchmark depends on the particular endowment risk that the agent faces. I depart from the Admati and Pfleiderer paper by ruling out benchmarks in the compensation of the manager and by studying the possibility of introducing constraints on the managers’ portfolios. This paper is also related to the contracting literature that studies project selection. Different versions of the choice of projects within a firm have appeared in the literature (see for example Hermalin (1993), Hermalin and Katz (1994), Hirshleifer and Suh (1992), Hirshleifer and Thakor (1992), Sung (1995)). The driving force in these models is the informativeness of output, as this is used in the compensation scheme. Reduction or amplification of risk may be optimal if the risk-sharing inefficiencies can be controlled through either of these actions. 7
Another strand of the literature that is related in terms of the actual model of this paper studies the
trading strategy induced by a particular incentive contract, without analysing what the optimal contracts may be (Grinblatt and Titman (1989a), Goetzmann, Ingersoll, and Ross (1998), Carpenter (1998), Cuoco and Kaniel (2000)).
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The results of my model should be taken as complementary to the ones existing in this literature, since the frictions are generated by a different set of primitive assumptions and I address other issues. Another set of related papers are those considering aggregation of signals for compensation purposes. Feltham and Xie (1994) extend the Holmstrom and Milgrom (1991) model to study the congruence of “incomplete performance measures.” Paul (1991) analyzes a rational expectations equilibrium for the stock of a firm to assess the use of the stock price as a contracting variable.8 The results of the paper are also related to the literature that attempts to explain noncontingent contracts (fixed wages) as the result of an optimal contracting problem (see Allen and Gale (1992), Holmstrom and Milgrom (1991), Spier (1992), Bernheim and Whinston (1998)). The optimal contracting solution in this paper is a fixed wage for an open set of the model’s parameters, even though in a first-best world (with observable actions) optimal risk-sharing calls for contingent contracts. Section 2 presents a general model of contracting in a standard optimal investment framework. Section 3 solves for the optimal contracts in the case of Gaussian returns and CARA preferences, fixing the number of assets under management. Section 4 considers the optimal composition of a fund, and studies how the principal should allocate assets across two different managers. Section 5 argues that the main qualitative features of the model do not depend on the above parametric assumptions. This section also discusses the effects of introducing costly effort. Section 6 concludes.
2
A model of contracts in an investment setting
2.1
Some general considerations
Consider an abstract contracting environment between an agent and a principal. Let W denote a real-valued random outcome, which depends both on the action that the agent takes and on 8
Other rational expectations models with similar flavor include Bushman and Indjejikian (1993), and Kim
and Suh (1993). In these models the informativeness of each of the signals is the driving force in the outcome of the contracting game.
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the state of nature. Assume that the agent and the principal have von Neumann-Morgenstern utility functions uA (·) and uP (·). Moreover, suppose that the principal and the agent have some non-tradable endowment, given by the random variables eA and eP . If the action of the manager were observable, the Pareto-optimal risk-sharing rules would be the solution to max E [uP (W − f (W ) + eP )|a] ;
a,f (·)
such that
E [uA (f (W ) + eA )|a] ≥ u ¯;
where f (W ) is the (possibly nonlinear) sharing rule, or payment function, u ¯ is the reservation utility of the agent, and a denotes the action of interest. Note that the action does not enter directly into the utility function of the agent, so we are ruling out standard moral hazard models with costly effort in the above formulation. Note that, for any action, efficient risk sharing implies that E u0P (W − f (W ) + eP )|W = λE u0A (f (W ) + eA )|W ;
(1)
for some positive constant λ, where E [X|W ] denotes the conditional expectation of the random variable X given W . The first-order condition that characterizes the optimal action is dW dW 0 0 0 0 + λE uA (f (W ) + eA )f (W ) = 0. E uP (W − f (W ) + eP )(1 − f (W )) da da
(2)
This first-best solution has the property that on the margin the principal and the agent value increases in the action equally. The contracting problem under observable actions reduces to one of risk-sharing: the contract f (·) plays no role in the determination of the optimal action besides its role in allocating risk. The problem has a simple separation in terms of the decision making, given by (2), and risk-sharing, characterized by (1). Note that if there were no endowment risk, equation (1) becomes u0P (·) = λu0A (·), which using (2) implies that the optimal action is the solution to dW 0 E uA (f (W )) = 0. da
9
(3)
Consider now the case of unobservable action. The above problem has the added constraint that a ∈ arg maxa E [uA (f (W ))|a]. This incentive compatibility constraint has the effect of restricting the set of actions that are implementable. In general, we would expect this constraint to be binding.9 In order to see this, note that given the sharing rule in equation (1) the firstorder condition to the agent’s problem is dW 0 0 = 0. E uA (f (W ) + eA )f (W ) da
(4)
In general (2) and (4) will yield different optimal actions. Even in the case with no endowment risk, if f 0 (W ) is not a constant, equations (2) and (4) will not characterize the same action, so there will be a loss in efficiency from the first-best outcome. In this case the contract f (·) plays a double role: it is used both for risk-sharing reasons and to induce an action choice. Thus, even in a pure risk-sharing environment the Pareto-optimal allocation will not be achievable. Of course, if f 0 (W ) is a constant and there is no endowment risk, then (3) and (4) are equivalent, and the outcome of the contracting game with unobservable actions coincides with the first-best. One case in which this occurs is if the principal and the agent have preferences described by utility functions in the HARA class with the same cautiousness parameter. Linear payments under this preference restriction achieve optimal risk sharing. Therefore, we obtain the separation of the optimal action and risk sharing discussed above, and the first-best outcome will be obtained (see Ross (1974)). Note that this last argument required the non-existence of endowment risk. Even within the HARA class, inefficiencies will arise if the agent and the principal do not share the same endowment, since equation (2) does not reduce to (3). This is due to the fact that aggregate wealth is no longer a sufficiently rich contracting variable. The state dependency created by the presence of endowments makes the contracting game with unobservable actions fall short of the first-best outcome. 9
Take for example the standard moral hazard problem (Holmstrom (1979)). With a risk-neutral principal,
optimal risk sharing implies that no risk should be born by the agent. But since under a flat wage the agent would choose the least costly action, contracting inefficiencies arise. By choosing f (·) that is not constant the principal is able to induce some effort level from the agent, but this occurs at the cost of exposing him to risk.
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This paper explores the contracting problem that arises when (4) binds, specializing the setup to one in which the action available to the agent is one of asset allocation: the manager’s sole decision is how much to invest in a set of assets. Inefficient risk sharing due to endowment risk creates a conflict of interest between the principal and the agent. I also study the case in which the principal and the agent have different beliefs about the relative likelihood of the states of nature, and show that the same type of conflict of interest arises. The objective of the paper is to study the effects of these inefficiencies in the outcome of the contracting game.
2.2
Setup and contracts
All underlying uncertainty in the economy is summarized by the probability space (Ω, F, P). The state of the economy is an element ω ∈ Ω. The investment universe is composed of m + 1 different securities. Asset zero has a riskless gross return of Rf . For each dollar invested in security i, for i = 1, . . . , m, the fund receives a gross return Ri (ω). The gross payoffs Ri (ω) are assumed, without loss of generality, to be linearly independent, i.e. there are no redundant investment opportunities. Moreover, these assets do not admit arbitrage opportunities. The agent has control over the investment decisions in these assets. The fund has an amount W0 available for investment. The agent’s action is the amount to be invested in each of the m+1 assets. Let θ ∈ Re (ω), where Re (ω) = R(ω) − Rf 1 ∈ 1 . The agent’s and the principal’s preferences are defined over consumption plans, random variables c(ω). I assume that preferences can be described by von Neumann-Morgenstern utility functions, but allow for the possibility of different beliefs for the agent and the principal. Given a consumption plan x, agent i receives utility Ui (x) = Ei [ui (x(ω))], where Ei [x(ω)] denotes the expectation of the random variable x(ω) under the probability measure Pi , for i = A, P . These probability measures are assumed to be absolutely continuous with respect to P. The functions ui (x) are assumed to be strictly increasing, differentiable and concave. The agent and the principal have some endowments outside the fund, with payoffs of eA (ω) and eP (ω)
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respectively. These can be thought of as investments in some other assets (for the principal) or the value of human capital (for the agent). One possible way to specify a contracting environment would be to allow the two parties to agree to a payment function defined on the whole probability space, which specifies the payoff to the agent in each state of nature. I will only consider contracts defined on the total value of the assets under management.10 Formally, the contract specifies a payment f (W ), where f : R → R belongs to some nice functional space C. I will assume that C includes fixed wages (constant functions) and is restricted to functions f (·) that are non-decreasing.11 Moreover, I assume that f (·) is such that θ∗ ∈ arg max E [uA (f (W ))] is finite. Instead of considering the set of assets to be included in the fund as exogenous, I let the principal choose what securities the agent can invest in. The set of all possible combinations of the assets in the fund will be denoted by D, and D∗ denotes the set that includes all m + 1 assets. In what follows it will be assumed that the principal always includes the risk-free security as part of the contracting aggregates.12
2.3
First-best and second-best problems
The first-best contract is the result of the parties contracting under observable and verifiable actions. This seems to be the natural case to take as a benchmark. Definition 1. The first-best outcome is the solution to: max
θ,f ∈C,D∈D
EP [uP (W − f (W ) + eP )] ;
(5)
such that EA [uA (f (W ) + eA )] ≥ u ¯; where W = W0 Rf + θ> Re , and θi = 0 if i ∈ DC . 10
More general contracts could be written, which could potentially achieve the first-best outcome (for example
allowing the payment to depend on the return of each of the assets under management). It is in this sense that the paper refers to the inefficiencies that appeared in the model as generated due to “incomplete contracts.” 11 This last assumption can be motivated by assuming that the agent has access to a free disposal technology (so he can destroy output), which forces the optimal contract to be non-decreasing. 12 The reason for this assumption is for simplicity in the exposition. As it will become clear in what follows, it is never optimal to restrict access to this risk-free security.
12
In a first-best world, the generic case has the property that the parties would invest nontrivial amounts in all assets, so the solution to the above problem entails D = D∗ , i.e. the principal will allow the agent to control all assets. This follows from a simple revealed preference argument, since when θ is contractible, the parties can always specify θi = 0 in the contract. Note that if the parties cannot write contracts that yield this first-best outcome, it is not obvious that the principal would allow the agent to invest in all possible assets. As the paper shows, this is indeed the case in many situations. The second-best contract is taken to be the solution to the contracting problem where the action of the manager is unobservable. Definition 2. The second-best contract is defined to be the solution to: max EP [uP (W − f (W ) + eP )] ;
f ∈C,D∈D
(6)
such that EA [uA (f (W ) + eA )] ≥ u ¯; θ ∈ arg max EA [uA (f (W ) + eA )] ; θ
where W = W0 Rf + θ> Re , and θi = 0 if i ∈ DC . If the contract specifies a fixed wage f (W ) = α, I assume the agent will take on the preferred trading strategy for the principal:13 θ ∈ arg maxθ E [uP (W − α + eP )]. This is the standard approach in the literature: the agent, when indifferent about what actions to take, will choose the one that is optimal for the principal. This assumption has an important implication: the agent’s action will not be continuous with respect to the contract offered by the principal. Solving the above contracting problem for arbitrary functions f ∈ C seems to be a difficult task. For example, just as in the standard costly-effort model, there is no assurance that the first-order condition to the agent’s optimization problem will be sufficient. In order to see this, note that the second-order condition to the agent’s optimization problem requires the matrix EA u00A (·)f 0 (·)2 + u0A (·)f 00 (·) Re (Re )> to be negative semidefinite. If f is sufficiently convex, 13
This would be the Nash equilibrium of a bargaining game between the two parties in which the bargaining
power is on the principal’s side.
13
this second order condition may be violated. But f is determined endogenously, and it seems hard to get bounds on its derivatives based only on the model’s primitives. The next proposition restates the result from Ross (1974) on the conditions under which linear contracts achieve the first-best outcome in this model, and extends his results by pointing out that linear-sharing rules and a single risky asset are sufficient even in the presence of endowment risk. Proposition 1. The first-best outcome is achieved under unobservable actions if either of the following conditions holds: 1. There is no endowment risk, the beliefs of the agent and the principal coincide, and their preferences are in the HARA-class with the same cautiousness parameter. 2. There is a single risky asset and equation (1) defines a linear sharing rule. The next section solves the above contracting problem when the returns are jointly Gaussian and the principal and the agent have CARA preferences. Within this parametric framework different implications of the model are studied. Section 4 discusses the robustness of the results by relaxing these parametric assumptions.
3
Optimal contracts
This section considers a special parametric case of the model presented previously. I assume CARA preferences both for the agent and the principal, and take the random payoffs to be Gaussian. It is shown that linear contracts are actually optimal, which makes the model tractable, since the optimal contract can be solved in closed form. Two properties of this model are worth recalling. If the principal and the agent had the same beliefs and the same endowment, then a linear payment function would be optimal, and it would achieve the first-best outcome under unobservable actions. The bonus coefficient would equal the relative risk aversion of the agent with respect to the principal. Moreover, as shown in Proposition 1, the first-best is always attained with a single risky asset. With multiple assets and endowment risk (or different beliefs) the agent will generically want to 14
diversify across assets in a different way from what the principal would want. This drives the inefficiencies to be presented in this section.
3.1
Optimal contingent contracts
The gross returns of the m risky securities are jointly Gaussian. These assets have mean µ and variance-covariance matrix Σ. I will denote by h the vector of excess returns of the risky assets, h = µ − r1. A trading strategy is a vector θ ∈ h + W0 Rf and variance θ> Σθ. The agent has CARA preferences with absolute risk-aversion parameter bA , i.e. uA (x) = − exp(−bA x). The principal also has CARA preferences with risk-aversion parameter bP . The principal and the agent have some risky endowment ej . Each endowment j (for j = A, P ) is assumed to have an expected gross return of e¯j , variance σj2 , and correlation ρji with each asset i in the fund. All random variables are jointly normal. The following proposition characterizes the solution to the contracting problem with these parametric assumptions in the case of unobservable actions, excluding the case of a fixed wage, which will be dealt with in the next section. Proposition 2. The optimal contingent contract is linear, i.e. f (W ) = α + βW , with β= with γ =
κM κA ,
1 γbA /bP + 1
−1 > −1 κA = h> A Σ hA , κM = hA Σ hP , hA = h − bA νA , hP = h − bP νP ; where the ith
element of νA is of the form σA σi ρiA ; and νP is a vector whose ith element is σP σi ρiP . The optimal trading strategy is θ=
1 −1 Σ (h − bA νA ). bA β
(7)
The optimality of linear contracts should not be surprising given the preference and distributional assumptions made in this section. The standard linear risk sharing rule with coefficient β = bP /(bA + bP ) is recovered if there is no endowment risk. It is interesting to note that the agent will always invest in the mutual fund Σ−1 (h − bA νA ), and that the contract only changes the amount that he allocates to this risky fund. Moreover, the higher the sensitivity to output 15
(higher β), the smaller the risk that the agent would take: he undoes the leveraging effects of the sharing rule by adjusting the trading strategy accordingly. This appears to be a typical result in these types of portfolio choice problems.14 It is immediate from the expression for the trading strategy in the proposition that the first-best is not attained: simply note that the principal’s endowment risk does not appear in the agent’s optimal trading strategy. The loss in efficiency is created by the unobservability of the agent’s trading strategy, and the fact that only contracts written on final wealth are allowed. The exposure to each of the risky assets for the principal in this second-best solution is (γ/bP )Σ−1 (h − bA νA ). If the principal could invest on her own, her exposure would be (1/bP )Σ−1 (h − bP νP ). As argued above, the composition of the mutual fund in the secondbest solution does not depend on the principal’s preferences. The variable γ measures the relative endowment risk that the agent and the principal have. When the agent faces no endowment risk and the principal’s endowment is positively correlated with the assets in the fund, under the optimal contract the agent keeps a larger portion of the risky payoff with respect to the first-best outcome. In the general case, the variable γ can be larger or smaller than 1, depending on the relative sizes of the endowment risks of the principal and the agent.
3.2
An alternative formulation
In order to gain some further intuition about these contracting frictions, this section of the paper considers a variant of the model in which the agent and the principal have different subjective beliefs about the expected return of the assets under management.15 The agent believes the assets have mean µA and the principal believes the mean is µP . I will denote by hi the vector of excess returns of the risky assets, hi = µi − r1, for i = A, P . Neither party faces any endowment risk. 14
Carpenter (1998, 1999) finds the optimal trading strategy for a mutual fund manager facing a call option
as payment for his services. She shows that the amount invested in the risky assets is a decreasing function of the number of options that the agent holds. The models of Admati and Pfleiderer (1997), Ou-Yang (1999) and Carpenter et al. (2000) also satisfy this property. 15 See Morris (1995) for a discussion of the merits of the heterogeneous prior assumption in Economics.
16
With CARA preferences, it is easy to see that the model with endowment risk and the model with different beliefs should be equivalent. Namely, given some endowment e(ω), we can con struct a probability measure Qi equivalent to P, such that EP e−b(x(ω)+e(ω)) = EQi e−b(x(ω)) . The Radon-Nikodym derivative for this measure change is simply given by
dQi dP
= ae−be(ω) , for
some constant a (which normalizes Qi to integrate to 1). Given the solution presented in the previous section, if we pick the endowments such that hA = h − bA νA and hP = h − bP νP , the solution to the contracting problem with different beliefs is obtained. Note that, as in the model with different endowments, the agent invests in the mutual fund of risky assets that he perceives as being mean-variance efficient, Σ−1 hA , and the bonus coefficient β simply scales his position in this fund. This formulation brings in a very natural interpretation of the variables κi . It is easy to see that κA is simply the Sharpe ratio of the agent’s optimal portfolio squared. At the same time, κ2M /κA is the Sharpe ratio squared of the portfolio picked by the agent, but using the principal’s beliefs. Equivalently, these could be interpreted as a measure of the benefit to the agent and the principal of the portfolio in (7), i.e. as the Sharpe ratio of the holdings of the agent and the principal (taking into consideration the endowment risk that both face). The principal, under the optimal contract, keeps a fraction (1 − β)/β = γbA /bP of the risky payoff. The variable γ can be further interpreted as a measure of the discrepancies between the agent and the principal. When their beliefs coincide, then γ = 1 and the first-best outcome is achieved. Depending on the exact parameters of the model, the optimal (second-best) contract will give the principal a higher or lower share of the risky payoff. A sufficient condition for γ < 1 (so the agent keeps a higher fraction of the fund’s risky payoff than in the first-best outcome) is that the agent’s beliefs about these expected returns are higher than those of the principal. The amount of the risk-free asset that is split between the principal and the agent does not depend on the initial wealth W0 of the fund (since the constant term of α + βW is − log(−¯ u)−0.5κA ). bA
Note that it does depend on the risk-return attributes of the assets through
κA : the fixed wage that must be paid to the agent decreases as κA increases, since he is getting increased utility from being able to invest in the risky assets.
17
The crucial assumption in this model is that there is some state-dependency in the preferences of the agent and the principal, and this can be achieved (as a modeler) by introducing endowment risk or disagreement about the relative likelihood of the different states.
3.3
Optimality of fixed wages
As pointed out in the introduction, nothing prevents the principal from offering the agent a fixed-wage contract. With this payment function, the agent is indifferent with respect to the trading strategy that he will take, since his payoff is independent of the return of the portfolio. As customary in the contracting literature, I will assume that the agent, when indifferent, will invest in the principal’s optimal portfolio. An important consequence of the loss in efficiency that occurs when using a contingent contract is the following proposition. Proposition 3. Fixed wages will be optimal if and only if κ2M κA 1 κP − ≥ bP κA bA
(8)
−1 where κP = h> P Σ hP . Fixed wages are optimal for a non-degenerate set of the model’s prim-
itives.16 The proposition shows that in this model, even though the parties would like to share risk, and are allowed to do so by writing contracts on final wealth, they may optimally choose not to contract on the final payoff at all. The outcome of the contracting game when the principal uses a fixed wage is suboptimal from a risk-sharing perspective, but it induces a better portfolio choice by the agent (given the principal’s endowment risk or beliefs). The condition for fixed wages to be optimal is a complicated function of the model parameters. First note that when hA = hP the left-hand side of (8) is zero, so efficient risk-sharing makes contingent contracts optimal. The term κP /bP measures the benefits of using a fixed wage contract, since the principal’s certainty equivalent (under the fixed-wage solution) is an increasing function of κP .17 The term κA /bA can be interpreted as the direct gain in risksharing from using a contingent contract. But using a contingent contract, inefficiencies arise 16 17
By a non-degenerate set I mean an open set of the model’s primitives. Note that κP is the Sharpe ratio squared of the optimal portfolio chosen by the principal.
18
due to the different beliefs of the agent and the principal. The term κ2M /κA captures the benefits to the principal of the risky portfolio chosen by the agent. Note that κ2M /κA ≤ κP , so the term in the left-hand side of the above expression is always positive. When κ2M /κA < κP , fixed wages may be optimal, so the size of κ2M /κA versus κP is a measure of the costs of contingent contracts. The following example illustrates the optimality of fixed-wage contracts. Example 1. There are three assets, all with the same attributes with respect to their expected return, 20%, standard deviation, 40%, and correlation among them, 0.5. The risk-free rate is 0.10. The risk aversion of the agent is bA = 10, and that for the principal is bP = 1. The principal has no endowment outside this fund, whereas the agent has some risky endowment with volatility σA = 0.10. The agent’s risky endowment is uncorrelated with assets 2 and 3, but has a correlation ρ with asset 1. Figure 1 plots the optimal bonus coefficient β for different values of the correlation coefficient between the agent’s risky endowment and the first asset. The first thing to note is that the bonus coefficient is increased as the correlation becomes further away from zero. The reason for the increase in the bonus coefficient is that the agent is benefitting relatively more from the assets than the principal is. Therefore, under the optimal contract, he must bear a larger portion of the risky payoff of the fund. Some further intuition can be gained from looking at the agent’s investment strategy as the correlation coefficient changes. The optimal investment in the risky assets is given by θ∗ =
1 −1 bA β Σ (h − ba νA ),
where νA1 = ρσA σ1 , and νAj = 0 (for j = 2, 3). The agent will always
invest in the same mutual fund, proportional to Σ−1 (h − ba νA ). As one of the components of νA changes, this mutual fund becomes more and more “extreme.” In this example, as ρA1 rises, the agent shifts his investment from asset 1 to assets two and three. The further this mutual fund is away from that which is optimal for the principal (Σ−1 hP ), the more likely it is that fixed wages become optimal. As it is seen from figure 1, this happens for a (large) open set of the model’s parameters. The following proposition considers some cases under which fixed wages are optimal. Proposition 4. In the generic case:
19
1. there exists M < ∞ such that for all bA /bP > M fixed wages become optimal; 2. there exists σ ¯A < ∞, such that for all σA > σ ¯A contigent wages are optimal; 3. if the agent’s beliefs with respect to the risky assets can be written as µA = sµ, for some scaling constant s ∈ R1 , then there exists s¯ such that for all |s| > s¯ fixed wages are not optimal. The intuition for the first result is fairly straightforward: as the agent becomes highly riskaverse (compared to the principal) the private benefits that he obtains from investing in the assets become negligible with respect to the private benefits that the principal will get under his optimal trading strategy. Lastly, note that the generic qualification simply rules out the case where κP κA = κ2M , i.e. the principal’s preferred mutual fund happens to be the same as that for the agent. The second and third statements in the proposition highlight one of the potential benefits of incentive compensation. Contingent contracts offer the agent the risk-return trade-off from the assets under management. Even if the agent is more risk-averse than the principal (the statement holds for any bA and bP ), the optimal contract may entail non-trivial sensitivities to output due to the fact that it is cheaper to compensate the agent this way than by using fixed wages. Note that one can interpret this as being the result of an agent being very optimistic about the assets under management, or equivalently as the result of having large endowment risk outside of the relationship that is highly correlated (either positively or negatively) with the assets in the fund.
4
A theory of mutual fund specialization
The first part of this section characterizes the optimal set of securities that a single fund should hold as a function of the model’s primitives. Then I allow the principal to contract with two agents, deriving both the optimal contracts and the optimal allocation of securities among them. These results yield a theory that can explain the observed mutual-fund specialization.
20
4.1
Optimal fund composition
In the previous section I found the optimal contract given a set of securities D ∈ D. The issue of what assets to include in the fund is tackled next. The next proposition characterizes the optimal fund composition, and shows that, in general, the principal may find it in her best interest to limit the agent’s investment decisions. Proposition 5. Let D denote the set of securities included in the contract. Define κi (D) (i = A, M, P ) as in the previous propositions with all vectors and matrices restricted to the assets in D. The optimal fund composition using contingent contracts solves max D∈D
κ2M (D) κA (D) + κA (D)bP bA
If the value from the above optimization problem is smaller than
κP (D∗ ) bP
then a fixed wage
is optimal. Otherwise the optimal contract is a linear contract on total wealth, including the set of securities that solve the above maximization problem. The set of model primitives for which it is optimal not to include all assets in the fund is non-degenerate. The main result is that restricting the agent to invest in a subset of the assets may be optimal for the principal. The next two examples illustrate this idea. Example 2. Consider the above setup with the following parameter values. There are three assets, all with a standard deviation of 40% and correlation among them of 0.5. The risk-free rate is 0.10. The risk aversion of the agent is bA = 10, and that for the principal is bP = 1. The expected return vector is µ = (0.30, 0.25, 0.20). The agent faces endowment risk σA = 0.10, which is uncorrelated with assets 1 and 2, but has a correlation of ρA3 = 0.125 with asset 3. Suppose now that the principal can ban access to the agent to one of the assets if she considers it optimal to do so. The following table presents the value function for the principal, as well as the optimal bonus coefficient, under different fund arrangements. The third column indicates the increase in the value of the contract offered to the agent with respect to the situation in which he has access to three assets (i.e. (κA (D∗ ) − κA (D))/bA ). The last column is the Sharpe ratio of the risky portfolio chosen by the agent.
21
∝ E [uP (·)]
β
∆α
Sharpe ratio
3 assets
0.2844
0.10
0.00
0.5031
Assets 1 and 2
0.2979
0.0909
0.0021
0.5204
Assets 1 and 3
0.2578
0.0977
0.0021
0.4804
Assets 2 and 3
0.1403
0.0972
0.0083
0.3546
In this example it is optimal for the principal to ban access to asset 3, even though both parties would like to invest non-trivial amounts in this asset. Restricting access to this asset improves the risk-sharing possibilities that the two individuals face by reducing contracting frictions. With assets one and two, the portfolio that the agent chooses has a higher Sharpe ratio than the portfolio that he chooses when allowed to trade the three assets. This is due to the fact that the agent uses asset 3 to hedge some of its endowment risk by investing relatively less in this asset than what the principal would like him to. Even though the salary that the principal pays is increased across these two options (due to the loss in the agent’s utility caused by restriction on investment in the third asset), the principal finds it optimal to pay this extra amount to be able to invest in the portfolio with the higher Sharpe ratio. Example 3. There are two risky assets, with standard deviation of 30%, and correlation of 0.5. The risk-aversion parameters are bA = 10 and bP = 1. Assume there is no endowment risk, but the agent and the principal have different beliefs about the risky assets’ expected A returns. The beliefs of the agent are assumed to be µA 1 = µ2 = 0.20. Figure 2 presents the
optimal asset restrictions for different values of the beliefs of the principal. The graph shows that for extreme values of the relative quality of the different assets (proxied by the expected return for each asset perceived by the principal), it is optimal for the principal to restrict access to the asset with the highest expected return. The intuition is that the agent is always going to invest in an equally-weighted portfolio of the two risky assets. From the principal’s perspective, she may be better off giving away the expected return on asset 2 when the expected return in asset 1 is sufficiently high, since she does not find it in her best interest to diversify the portfolio equally among both assets.
22
4.2
Allocation of securities among funds
In this section I allow the principal to contract with two agents. I show that the incompleteness of the contract space, in the sense introduced in this paper, can be used as a theoretical foundation for the allocation of securities into different funds. Suppose that there are two managers with CARA preferences with the same risk-aversion parameter and who face the same endowment risk.18 The principal and the agents have at their disposal m different securities, characterized by their expected return µ and variance-covariance matrix Σ. The principal can structure the allocation of assets and wealth by creating two funds, one for each agent, each with a total initial wealth of Wi (i = 1, 2), and allocate a set D1 of the securities the first fund, and a set D2 to the second agent. I’ll assume in what follows that D1 ∩ D2 = ∅.19 The principal can only write contracts for each manager i that are functions of the total wealth in fund i. The problem that she faces is max Di ,Wi ,fi (·)
E uP (W 1 − f1 (W 1 ) + W 2 − f2 (W 2 )
such that W i = Wi Rf + θi∗> Rie (ω), θi∗ ∈ arg max E uA (fi (W i )) , θ
i = 1, 2; i = 1, 2;
W1 + W2 ≤ W0 ; plus the obvious participation constraints, where the subscripts refer to the fund, and each fund i is composed of those assets in the set Di . The next proposition characterizes the solution to the above contracting problem in closed form. Proposition 6. The optimal contingent contract for each manager is linear. Fixing the set of assets in each fund, the optimal bonus coefficient for each agent is given by βi = 18 19
1 ; γi bA /bP + 1
i = 1, 2;
Heterogeneity of the agents could be analyzed in the same lines as below. This restriction is with loss of generality, although it seems a natural assumption in the case of real pro-
duction decisions. At the end of this section I discuss what happens when this assumption is relaxed.
23
where γi =
κiM κjA − κL κjM κiA κjA − κ2L
i 6= j;
;
i = 1, 2;
−1 κiA = h> Ai Σii hAi ;
i = 1, 2;
−1 κiM = h> Ai Σii hP i ;
i = 1, 2;
−1 κL = h> A1 Σ12 hA2 .
The optimal arrangement of securities among the two funds solves max
D1 ,D2
κ1A + κ2A γ1 κ1M + γ2 κ2M − γ12 κ1A − 2γ1 γ2 κL − γ22 κ2A + ; bA bP
(9)
where for notational convenience the explicit dependence of each of the variables γi and κik on the sets Di has been dropped. In principle there is no easy characterization of the solution to (9). This is not surprising, since both the risk-aversion of the manager and principal should play a role, as well as the characteristics of the assets themselves. The next proposition states some properties of the optimal solution. Proposition 7. For an open set of parameters: 1. It may be optimal to have D1 ∪ D2 ⊂ D∗ , i.e. the principal may forgo investment in some securities even when there are two different managers. 2. The solution with one single fund may dominate the solution with two funds, even for low values of the reservation utility of the agents. From the results in the previous section it is straightforward to see the forces behind the first point of the proposition: if it is optimal to exclude one asset when there is a single fund, it is not surprising that this would be the case with two managers (although it will happen less often). The principal’s asset span increases with the number of funds that she sets up. Nevertheless, in order to conclude that the solution with two funds will dominate the solution with one single fund, it is necessary to check that the total compensation to the agents is not increased 24
when using two managers. The proposition shows it may be optimal to just have one single contracting unit, and that this is not driven by the fixed cost of hiring an agent (his reservation utility). The intuition is that having one single agent manage all assets may be the most economical compensation scheme (this would be the case, for example, when the first-best can be achieved). The next proposition puts more structure into the distributions of the assets under management in order to get a sharper characterization of the optimal fund composition. It is shown that if there are two factors that generate the returns of the assets, it is optimal for the principal to separate these assets into two different contracting units. Proposition 8. Suppose that the assets belong to two different classes, and that the returns are generated by a factor model of the form: Rie = f1 + i
for i ∈ C1
Rie = f2 + i
for i ∈ C2
where i (for i = 1, . . . , m) is a normal random variable with mean 0 and variance σ 2 ; and the factors fj (j = 1, 2) are also normal with mean µF j and variance σF2 j . The idiosyncratic shocks i are uncorrelated with all other random variables in the model, whereas the factors are correlated with the endowments of the agent and the principal. Then, in the generic case, if the risk tolerance and the reservation utility of the agent are low enough, it will be optimal to set up two funds, each with the same type of securities in it, i.e. D1 = C1 and D2 = C2 . The proposition provides conditions under which investment funds should control assets that belong to the same “type,” i.e. they are in similar assets classes or industries. This implication is similar in flavor to that of Holmstrom and Milgrom (1991), who show in a costly-effort model that it is never optimal to allocate two different tasks to the same agent.20 This result provides a theoretical foundation for “division of responsibility” across agents based 20
As explained before, the model in this paper can explain grouping assets into funds due to the benefits
in terms of cheaper compensation for management, which is an implication that cannot be generated by the Holmstrom and Milgrom model.
25
on the risk-return attributes of the assets under management, rather than on the “costly-effort” variables of Holmstrom and Milgrom (1991). Under the conditions of the proposition, both the principal and the agent will invest in equally-weighted portfolios within each asset class. If the principal combines all assets into one single fund, the agent will invest in them in a proportional way which will be different from the one that is optimal for the principal. If she separates each asset class in a different contracting unit, she can decide the relative exposure to each of the factors on her own. Note that in order to conclude that the solution with two funds dominates that with one fund it is necessary to assume that bA is large relative to bP , so the actual holdings of the risky assets by the agent are unimportant relative to the exposure that the principal obtains under the optimal contract (see the second result of Proposition 7).
4.3
Other implications and caveats
The model can also generate conditions under which a new security should be included in an existing fund or not. If we think of the model as being about “real” production decisions, then it also has implications for the existence of spin-offs and/or mergers. These conditions will be related to the value of the project itself, the cost of creating the firm and its attributes with respect to the other projects in a firm. The next example illustrates this application of the model. Example 4. Consider a firm that is originally formed by two types of projects, with expected returns µ1 = 0.20, µ2 = 0.25, and standard deviations σ1 = 0.20, σ2 = 0.30. A new project arrives with risk of σ3 = 0.20. All assets are uncorrelated. The principal and the agent agree on these parameter values, but disagree with respect to the expected return of the new project. The agent believes that it is µ3 = 0.20. I will use the beliefs for the principal as a measure of the “quality” of the project. Unless the principal believes that the project has µ3 = 0.20, there will be a non-trivial conflict of interest if the project is kept within the firm, due to the restriction of the contracting space. If the principal spins off the project into a new firm, this conflict of interest will be eliminated. But creating a new firm in our model has a fixed cost k that is a monotonic
26
function of the agent’s reservation utility. Figure 3 shows the set of values for µ3 and for k for which spinning-off the project is optimal. I close this section discussing the assumption of separation of the assets under management across the two different contracting units. If the principal is allowed to arrange assets such that D1 ∩ D2 6= ∅, then she will always find it in her best interest to do so. In order to see this consider the factor model of proposition 8. By having one fund with all assets, and another fund with assets affected by only one of the factors, the principal faces the same span as in the case where each fund has the assets from one class. But this arrangement allows the principal to pay a lower wage to the agent who has access to all assets. Therefore this arrangement dominates the one with D1 ∩ D2 = ∅. This suggests that, if it is reasonable to assume that the principal can arrange assets arbitrarily, she should have one fund that contains all possible securities (i.e. a broad market index), and other funds in which she adjusts her exposure to the factors for which her endowment is different from that of the agent (i.e. industry funds).
5
Extensions
The results in the previous section were developed under special parametric assumptions. The issue remains as to whether the same results could be generated in a more general class of problems. First I consider the contracting problem when the principal is restricted to write linear contracts. I show that for arbitrary preferences and distributions, the same mutual-fund interpretation of the problem holds. Then I argue that the main qualitative features of the model are preserved with arbitrary contracts if the distribution of the assets belongs to the separating class of Ross (1978) (with some mild restrictions on the payment functions). I consider next the effect of including a costly-effort component into the model.
5.1
The case of linear contracts
The payment to the agent under this contracting agreement is f (W ) = α + βW . I will drop the endowments and different beliefs in the following discussion to ease the notation. 27
The first thing to note is that the payoff to the agent under this contract is given by f (W ) = α + β(W0 Rf + θ> Re (ω)) ≡ WA Rf + φ> Re (ω) where I make the change of variables WA = α/Rf + βW0 , and φ = βθ. For β 6= 0 the optimization problem that the agent faces is equivalent to a standard investment problem in which he has an initial wealth of WA and has to decide how much to invest in each of the assets, i.e. the solution to the agent’s problem θ∗ is equal to φ∗ (WA )/β, where h i φ∗ (WA ) ∈ arg max E uA (WA Rf + φ> Re (ω)) . φ
It is worth remarking that there is a discontinuity in the agent’s optimal action as we move from a contract with β 6= 0 to a fixed wage contract. As long as the agent is offered some compensation for his action, the solution to the investment problem is unique, and may generally be different from the one that the principal would ask him to take if β = 0. The next proposition states the implications of the above argument for the principal. Proposition 9. For β 6= 0, the principal’s problem is equivalent to (1 − β) max E uP [(W0 − WA )Rf + φ(WA )> Re (ω)] ; WA ,β β WA ≥ w; ¯
such that
where I have made the previous change of variables from α to WA . This is equivalent to an optimal investment problem with a risk-free asset and a single risky asset. Fixing the number of assets that the agent can control, the principal’s problem reduces to that of investing in the risk-free asset and a single risky asset, characterized by the excess returns φ(WA )> Re (ω). We can view this risky asset as the agent’s optimal mutual fund, which depends both on his preferences and his reservation utility. This proposition establishes that the principal’s control variables, WA and β, allow her to choose the share of the risky portion of the fund’s payoff that she receives (which is (1 − β)/β), and the composition mutual fund that the agent chooses in his auxiliary problem (through φ(WA )). If the participation constraint is binding, then the agent’s problem will always involve investing in the same mutual fund
28
φ∗ (WA ), so the principal’s problem is further reduced to the allocation of her wealth among the risk-free asset and this particular mutual fund. Few other general statements can be made in this model. It is interesting to note that the participation constraint may not be binding. To see this, consider raising WA above w. ¯ This would be optimal if (1 − β) dφ(WA ) e 0 R (ω) > 0. E uP (·) −Rf + β dWA h i dφ e 6= 0 it is straightforward to see that it may be optimal, for some β, to offer If E u0P dW R A the agent a contract that gives him a utility level above his reservation utility. Namely, the wealth effect associated with a larger reservation utility may make the agent tilt his portfolio choice closer to the one that is optimal for the principal. Consider now the principal’s decision of what assets to include in the fund. She can choose to include all of them, pay the agent a fixed wage (which we will denote by w ¯0 ), and instruct him to take on her preferred trading strategy. She may also choose to let the agent control only one security. Between these two extremes, the principal has the option of banning access to different subsets of the assets. Taking away an asset from the fund gives away the expected return and hedging potential of the securities. Nevertheless, giving away exposure to this asset may be optimal, since the inefficiencies due to contracting are reduced with a smaller number of securities. Proposition 10. The principal’s problem reduces to that of investing freely in all assets with initial wealth W0 − w ¯0 , or investing in the risk-free asset and a risky portfolio whose excess returns are given by φ(WA , Di )> Re (ω), with initial wealth W0 −WA (Di ), subject to the constraint WA (Di ) ≥ w(D ¯ i ). For two contracting sets D1 ⊂ D2 we have w ¯0 ≥ w(D ¯ 1 ) ≥ w(D ¯ 2 ). The proposition highlights the trade-off that the principal is facing when deciding the number of assets to allow the agent to control. With a fixed wage, the principal can simply instruct the agent what trading strategy to follow. When the principal offers a linear contract with β 6= 0 and allows access to n assets, the agent will optimally allocate the fund’s resources across these securities according to his own preferences (so he chooses his own mutual fund).
29
The principal, under this option, can choose investment among this mutual fund of n-assets chosen by the agent and the risk-free security. The last statement of the problem refers to the fact that the agent’s participation constraint is harder to meet when we take away from him the control of a subset of assets. Banning access to assets reduces the utility of the agent from his investment in these securities, so the principal needs to give him a higher certainty equivalent in order to meet his participation constraint. In general restricting access to a subset of the assets has two negative effects for the principal: on one hand she needs to increase w, ¯ on the other her investment opportunities are reduced. The advantage is that the conflict of interest about the mutual fund chosen by the agent may be reduced with a smaller number of assets.
5.2
Distributional restrictions
The driving force for the results in the previous section is that the agent’s optimal portfolio of risky assets is independent of the contract that he is offered. The next proposition shows that the same result holds more generally. Proposition 11. Suppose that the asset returns belong to the 2-fund separating class of Ross (1978). Let RA (x) denote the absolute risk aversion of the agent. Then, if we restrict the contracts so that f 00 (x)/f 0 (x)2 < RA (f (x)), the portfolio of risky assets chosen by the agent is independent of the contract being offered. The intuition for this result is simple: if assets belong to one of the distributions from the separating class, then the agent’s choice of his optimal mutual fund does not depend on the contract that he is offered. Therefore we obtain the separation discussed in section 2.1 between the risk-sharing problem and the optimal action to be chosen. The crucial aspect of this class of economies is that no matter what payment function the principal designs, the agent will choose the same mutual fund of risky assets. To see this simply note that under a nonlinear contract, the agent is solving maxθ E [u(f (W ))]. The effective preferences of the agent under this nonlinear payment function become u ◦ f . The conditions
30
on f in the proposition simply require that these effective preferences are concave.21 With concavity of the objective function, the results of Ross (1978) imply that when facing assets that belong to the separating class the agent’s mutual fund of risky assets does not change with different f (·), since this mutual fund is independent of the agent’s effective preferences. Even though a nonlinear contract may be optimal, this argument suffices to generate the discontinuities in the payoff function caused by introducing or taking away assets from the agent (take for example a case in which θA 6= aθP if all n assets are in the fund, but θA = aθP if n − 1 assets are in the fund, for some constant a). Therefore the previous results seem to be robust to the particular parametric assumptions that I use.
5.3
Costly effort
Up to now, the model has simply considered a pure risk-sharing environment. One possible solution to the contracting problem, as pointed out throughout the paper, is to give the agent a fixed wage and have him follow the instructions of the principal. This would eliminate the investment inefficiencies that I presented in previous sections. Note that the outcome of the game with fixed wages is still inefficient, since both parties are risk-averse. Nevertheless, the investment choices will be optimal, since all assets will be included in the fund. Intuitively, extending the model to include costly effort on the side of the agent would change the optimality of fixed wages, since under this scheme the agent would simply shirk and not take on the principal’s preferred strategy. I develop this idea in this section. One reduced-form variant would be to introduce a fixed cost of effort c that the agent incurs when making the investment decisions (where this cost is independent of the actual amounts invested). If the agent is given a fixed-wage contract, then he would optimally choose to shirk, since he does not internalize the benefits of his effort. For c sufficiently low it will 21
It is not true that these conditions are necessary. In principle it may be optimal to create convex payoff
functions that do not satisfy the above restriction. But note that the convexity cannot be too large, since otherwise the agent’s problem may not have a solution. I conjecture that if we consider problems for which the solution of the agent’s maximization exists, then the above restriction will be without loss of generality. A concavification argument such as the one in Carpenter (1998) could prove useful in pursuing an answer to this question.
31
always pay the principal to give the agent a contingent contract, so that he is induced to take on a positive effort (invest non-negative amounts in the assets). The only change to the optimal contingent contract would be to raise the fixed wage that the agent receives by an amount c. The investment inefficiencies would remain on this version of the model, since the multi-dimensionality of the action space of the agent may make it optimal for the principal to restrict access to some assets, just as in the previous section. A more general model would make the agent’s effort depend on the particular portfolio choice he makes. To illustrate the trade-offs in a stylized manner, suppose that the agent incurs P 22 Assuming that the parameters of a costly effort equal to m i=1 ci |θi |, where ci is a constant. the model are such that the solution to the agent’s problem is interior and that θ∗ > 0, the first-best outcome would involve investing as if the expected return of each asset was reduced by the factor ci . Note that with unobservable actions and linear contracts, we can write the agent’s problem as c > e . max E uA α + β W0 Rf + θ R − θ β
(10)
The above equation shows that the agent would act as if the expected returns of the assets were reduced by a factor ci /β (which is greater than ci ), so he will underinvest in the risky assets. He cannot internalize his cost of effort, since he only receives a fraction β of the output. This is the typical result in costly effort models. The next proposition shows that as the cost of exerting effort vanishes, the optimal linear contract converges to the linear contract presented in the previous section (in a pure risksharing environment). This is not surprising since both the agent’s trading strategy and the principal’s objective function are continuous in the costly effort parameter c. Proposition 12. As c → 0, the optimal linear contract under costly effort converges to the linear contract characterized in proposition 2. The results of the previous sections are unchanged. Introducing a costly effort component simply changes the relative costs of the different securities. The investment inefficiencies will 22
The same convergence result as stated in proposition 12 follows for the standard quadratic cost formulation,
in which costly effort amounts to θ> Cθ, for some m × m matrix C.
32
now be driven by both the inability of the parties to write state-contingent contracts and the usual failure to internalize the agent’s costly effort. The qualitative features of the model do not change for the solutions that involve contingent contracts.
6
Conclusion
This paper has studied contracting frictions within the standard investment model. It extends the work on optimal contracts between risk-averse agents by considering a pure risk-sharing model in which the first-best is not achieved (for arbitrary contracts on final wealth). It fully characterizes in closed-form the optimal contracts for the case of Gaussian returns and CARA preferences. The model has interesting implications with respect to investment inefficiencies. It has been shown that in the presence of contractual incompleteness, the agent and the principal may forgo investment in some assets, even though they would both like to take advantage of the risk-return attributes of those securities. Moreover, it may be in their best interest to have fixed wages as the result of the trade-off between inefficient risk-sharing and the conflict of interest about the unobservable investment choice. This paper contributes to the literature on compensation by pointing out one more reason why “powered” incentive schemes may not be optimal: the inability for the two parties to write state-contingent contracts. The framework studied brings out an alternative theoretical explanation for the “principle of unity of responsibility” (see Holmstrom and Milgrom (1991)) based on pure risk-sharing motives and the inability of parties to write state-contingent contracts. Moreover, the paper builds an endogenous theory of fund composition, in terms of the allocation of assets among different contracting units. In contrast to alternative theories, this explanation is solely based on the risk-return characteristics of the securities.
33
Appendix
Proof of proposition 1. If there is no endowment risk the statement has been proved in Ross (1974). For the case with endowment risk, we need to show that with one single risky asset the first-best is still achieved. The first-best outcome is characterized by EP u0P (W − f (W ) + eP )|W = λEA u0A (f (W ) + eA )|W ; EP u0P (W − f (W ) + eP )(1 − f 0 (W ))R + λEA u0A (f (W ) + eA )f 0 (W )R = 0.
(11) (12)
In the case of unobservable actions, the first-order condition with respect to the portfolio choice is given by EA u0A (f (W ) + eA )f 0 (W )R = 0
(13)
The equivalence between (12) and (13) when there is a single asset and (11) defines a linear sharing rule follows from EP u0P |W = EP u0P |R
⇒
EP u0P (1 − f 0 )R|R = λEA u0A (1 − f 0 )R|R
and substituting for EP [u0P (1 − f 0 )R|R] equation (12) reduces to (13), i.e. the solutions under observable and unobservable actions coincide.
Proof of proposition 2. The first step in the proof is to show that the agent’s relative investment in the risky assets is invariant to the contract offered. Let g(x, e) denote the joint density of the excess returns and endowment for the agent, and let p(x) denote the marginal density of the excess returns. For an arbitrary contract f , the agent’s preferences are given by Z h i > e E uA (f (W0 Rf + θ R ) + eA ) = e−bA (f (x)+e) g(x, e)dxde m+1 R Z e−bA f (x) e−bA e g(x, e)dxde = m+1 ZR Z g(x, e) −bA f (x) = e p(x) e−bA e dedx p(x) m R R 34
Z
h i e−bA f (x) p(x)E e−bA e |x dx
= m R Z
= c
e−bA f (x) q(x)dx
Rm
where q(x) is the density of a Gaussian random vector with mean h − bA νA . The last line in the above derivation follows by “completing the square,” noting that conditional on x the > Σ−1 x + endowment of the agent is distributed as a normal random variable with mean νA
c, where νA is the vector of covariances between the returns of the assets and the agent’s endowment. Given any contract f (·), the agent is then solving maxθ E u(W0 Rf + θ> Re ) , where u = uA ◦f are the effective preferences of the agent. But since the above change of measure preserves the normality of the excess returns, the standard mutual fund theorem with normal returns implies that the agent’s optimal investment in the risky assets will be proportional to Σ−1 hA , irrespective of the contract f (·). Note that the mutual-fund theorem holds irrespective of the potential convexity of f (·), since with Gaussian returns the mean-variance efficient mutual fund first-order stochastically dominates any other portfolio of risky assets. I next show that linear contracts achieve optimal risk-sharing in the presence of endowment risk, which together with the previous statement about the set of implementable actions implies that linear contracts are optimal. The first-order condition with respect to f (·) follows from the pointwise maximization of E [uP (·)] + λE [uA (·)], which using the above result about the agent’s trading strategy reduces to h i h i bP e−bP (W −f (W )) E e−bP eP |W = λbA e−bA f (W ) E e−bA eA |W . Due to the joint normality of ei and W , E e−bi ei |W = eαi W +βi , for some αi , βi . The solution to the above equation yields f (W ) = α + βW , i.e. the optimal contract must be linear. Next I derive the optimal linear contract. Fix a contract characterized by (α, β), where β 6= 0. The agent’s maximization problem is maxθ E −e−bA (α+βW +eA ) . It turns out more convenient to work with the normalized certainty equivalent of the agent’s expected utility, UA = − b1A log (E [uA (α + βW + eA )]). Note that the agent’s wealth is distributed as a Gaussian random variable with mean 35
2 , where the ith element of ν α + e¯A + β(θ> h + W0 Rf ) and variance β 2 θ> Σθ + 2βθ> νA + σA A
is of the form σA σi ρiA . The agent’s problem reduces to solving max θ
1 βθ> h − bA β 2 θ> Σθ + 2βθ> νA . 2
From the first-order conditions it follows that θ∗ =
1 −1 bA β (Σ )hA ,
where hA = h − bA νA .
Equating the certainty equivalent for the agent under this optimal action to U¯A we can solve for α as a function of β: α(β) = U¯A − βW0 Rf −
1 > −1 2bA hA Σ hA
2 −e + 12 bA σA ¯A .
Now consider the principal’s problem. For a given β, the total wealth for the principal is (1−β) > −1 bA β hA Σ h + a, where a 2 (1−β) > −1 > −1 2 h Σ hA + 2 (1−β) bA β hA Σ νP + σP where b2 β 2 A
distributed as a normal with mean
is some constant independent of
β; and variance
νP is a vector whose ith element
A
is σP σi ρiP . The principal is then solving max β
(1 − β) 1 (1 − β)2 bP κM − κA β 2 β2 bA
(14)
−1 > −1 where κM = h> A Σ hP , κA = hA Σ hA , and hP = h−bP νP . The expressions in the proposition
follow from the first-order conditions to the above maximization problem. For further reference, the certainty equivalent for the principal equals 1 UP = W0 Rf − U¯A + 2
κ2 κA + M bA b P κA
.
(15)
Proof of proposition 3. If the principal offers the agent a fixed wage without incentives, then he will choose the trading strategy θ =
1 bP
Σ−1 P hP . The fixed wage that meets the agent’s reservation utility is
α = U¯A . The certainty equivalent for the principal from this contracting arrangement is UP = W0 Rf − U¯A +
1 κP 2 bP
(16)
where UA is the certainty equivalent for the agent when offered a fixed wage (which is a monotonic function of his reservation utility). 36
The principal’s certainty equivalent under the optimal contingent contract is given by (15). Comparing the value of these two expressions the following inequality needs to be satisfied for fixed wages to be optimal 1 bP
κ2 κP − M κA
≥
κA . bA
(17)
In order to see that the inequality (17) holds for a non-degenerate set of the model’s primitives note that the left hand side of that equation is positive for an open set of the model’s parameters (the set for which κP κA − κ2M > 0). Therefore letting bA → ∞ it is easy to see that the inequality will be satisfied for all bA larger than some cutoff value bA .
Proof of proposition 4. If the left hand side of the inequality (17) is negative then it is immediate that contingent contracts are optimal. Otherwise, if it is positive, note that it does not depend on the scale parameter s. Therefore by letting s → ∞ the right-hand side of the inequality becomes arbitrarily large, so a fixed wage is not optimal.
Proof of proposition 5. From proposition 2, the principal’s certainty equivalent when he includes the set of assets D in the fund and uses contingent contracts is given by equation (15), with each vector and matrix restricted to the securities in D. Therefore the maximum certainty equivalent for the principal using contingent contracts is max
β,D∈D
κM (D)2 κA (D) + . κA (D)bP bA
Using a fixed wage the principal gets certainty equivalent κP (D∗ )/bP . The results in the proposition follow by comparing these two quantities. To show non-degeneracy, consider first the case with two securities, with the assets being uncorrelated. The principal payoff if he includes two securities in a fund becomes P A P 2 2 A 2 (hA (hA 1 h1 /σ11 + h2 h2 /σ22 ) 1 ) /σ11 + (h2 ) /σ22 + ; A 2 2 bA bP ((hA 1 ) /σ11 + (h2 ) /σ22 ) −1 where ai is the ith element of the vector h> A ΣA .
37
(18)
The payoff to the principal if he includes just project one is (hP1 )2 (hA )2 + 1 . bP σ11 bA σ11
(19)
In order to show that (19) can be larger than (18), consider a set of parameter values for P A P P A A which hP1 > hA 1 h1 /σ11 + h2 h2 /σ22 (take h2 and h1 low enough). For h2 large enough the
denominator of the first term in (18) will be larger than the corresponding term in (19). By this argument there exists an open set of parameters for which the first-term is larger in (19). Letting bA ↑ ∞ it follows that it will be in the principal’s best interest to exclude the second project from the agent’s investment decision. The general case follows in the same way, since given n assets, we can recycle the above proof to find an open set of parameter values so that a mutual fund of n − 1 assets dominates the payoff with n assets.
Proof of proposition 6. Given a contract βi , the optimal trading strategy for agent i is θi =
−1 1 bA βi Σii hAi .
Linear
contracts are optimal for each agent, since the same set of implementable actions is achievable under non-linear contracts, and linear contracts maximize joint surplus conditional on any action. Let ψi =
1−βi βi
denote the fraction of the risky payoff that the principal keeps in the fund
−1 > −1 under the linear contract with bonus coefficient βi . Define κAi = h> Ai Σii hAi , κM i = hAi Σii hP −1 −1 > and κL = h> A2 Σ22 Σ12 hA1 Σ11 .
The total payoff for the principal is distributed as a Gaussian random variable with mean (ψ1 κM 1 + ψ2 κM 2 )/bA and variance (ψ12 κA1 + 2ψ1 ψ2 κL + ψ22 κA2 )/b2A . The first-order conditions with respect to the choice variables ψi are κM 1 −
bP (ψ1 κA1 − ψ2 κL ) = 0 bA
κM 2 −
bP (ψ2 κA2 − ψ1 κL ) = 0 bA
which yield ψ1 =
bA (κM 1 κA2 − κL κM 2 ) ; bP (κA1 κA2 − κ2L )
38
ψ2 =
bA (κM 2 κA1 − κL κM 1 ) . bP (κA1 κA2 − κ2L )
The expressions in the propositions follow immediately from the above.
Proof of proposition 7. In order to see the optimality of placing restrictions on the investment on some securities let’s consider the case of uncorrelated assets. The principal’s certainty equivalent under this scenario becomes 1 1 (κA1 + κA2 ) + bA bP
κ2M 1 κ2M 2 + κA1 κA2
.
In the proof of proposition 5 it was shown that there existed an open set of parameters such that
1 bA κA
+ b1A
κ2M κA
was maximized using one project instead of two. The same argument
applied pairwise to a set of three assets yields the existence of one group of securities that will not be included in either fund. In order to see that the solution with one single fund may dominate that with two funds, consider the case in which all assets are the same. In this case the first-best outcome can be obtained setting up one single fund. It is straightforward to check that small perturbations of the parameters still keep the one-fund solution as optimal.
Proof of proposition 8. Under the assumptions of the factor model the principal’s first-best portfolio is of the form θ∗ = aφ1 + bφ2 , where φi puts equal weights on each of the assets in class i, zero in all assets in class j, and a and b are real-valued constants. For large values of the risk-aversion of the agent, his exposure to the risky assets does not play a role in the principal’s optimization problem: the first two terms in (9) can be ignored for bA large enough. Finally note that the agent will invest equal amounts in each of the assets if he can only control assets within a similar class. The principal can achieve the exposure θ∗ by writing contracts with coefficients β1 = 1/(1 + a) and β2 = 1/(1 + b).
Proof of proposition 9. 39
Following the argument in the text, the principal’s payoff is W (1 − β) − α = (W0 − WA )Rf + (1 − β)θ(WA )> Re (ω). where we have used the relation WA = α/Rf + βW0 . Moreover, θ(WA ) = φ(WA )/β, where φ(WA ) solves h i V (WA ) ≡ max EA uA (WA Rf + φ> Re ) φ
(20)
The principal maximizes her expected utility with respect to the variables WA , β, with the constraint that V (WA ) ≥ u ¯. It is obvious that V (·) is strictly increasing, so the previous constraint is equivalent to WA ≥ w, ¯ for some w. ¯ From the above discussion, the principal’s problem reduces to one in which he chooses how much to invest in the risk-free asset (W0 −WA ) and a portfolio of risky assets with excess return φ(WA )> Re (ω). When the participation constraint is binding (WA = w) ¯ then the principal’s sole decision is how to allocate his wealth into the riskfree asset and the risky portfolio with excess returns φ(w) ¯ > Re (ω), where φ(w) ¯ solves (20).
Proof of proposition 10. The first part of the proposition follows by simply noting that the problem of the agent in (20) depends on the set of assets that he has under control, D. Consider the agent’s optimization problem (20) under two sets of assets D1 ⊂ D2 . A simple revealed preferences argument shows that V (WA , D1 ) ≤ V (WA , D2 ). The participation constraint and the monotonicity of V (·, D) imply that w ¯0 ≥ w(D ¯ 1 ) ≥ w(D ¯ 2 ).
Proof of proposition 11. Under a contract f (·) the agent is maximizing E [uA (f (W ))]. Note that the effective preferences of the agent, represented by the function uA (f (·)) are increasing (since f (·) was assumed increasing). Moreover d2 uA (f (x)) = u00A (f (x))f 0 (x)2 + u0A (f (x))f 00 (x) dx2 40
which is negative under the assumptions of the proposition, i.e. when f 0 (x)2 /f 00 (x) > RA (f (x)). Therefore the agent’s problem becomes a standard optimal investment problem with concave preferences. The results of Ross (1978) yield the invariance of the agent’s choice of mutual fund to the contract offered.
Proof of proposition 12. It is immediate from equation (10) that the optimal trading strategy for the agent is 1 −1 c θ= Σ h− . βbA β The principal’s problem in certainty equivalent terms then becomes (1 − β) c > −1 c > −1 bP (1 − β)2 c max hA − hA − Σ hP − Σ hA − . β bA β β β β b2A β 2 The optimal solution β ∗ to the above equation will be a continuous function of c, and as c ↓ 0 the bonus coefficient converges to the solution to (14), i.e. to the optimal contract without costly effort.
41
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45
0.20 0.15 0.10 0.05 0.0
Sharing rule (beta)
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.2
0.3
1.0 0.9 0.8 0.7 0.6 0.5
Principal’s exposure to risky assets (% of first best)
Correlation between asset 1 and agent’s endowment
-0.3
-0.2
-0.1
0.0
0.1
Correlation between asset 1 and agent’s endowment
Figure 1: The top graph presents the optimal bonus coefficient β for different values of the correlation between asset 1 and the endowment of the agent. The dotted line is the optimal linear contract if fixed wages were not allowed. The bottom graph presents the scaled share of the risky payoff kept by the principal (relative to the first best). 46
0.40 Expected return on second asset 0.25 0.30 0.35
Only asset 2 optimal
2 assets optimal
0.20
Only asset 1 optimal
0.20
0.25
0.30 0.35 Expected return on first asset
0.40
Figure 2: The graph presents the optimal composition of the fund as a function of the beliefs
Expected return of project 0.4 0.5
0.6
of the principal about the return of the two different assets.
Spin-off region
0.3
Keep in firm
0.0
0.1
0.2 Cost of new firm
0.3
0.4
Figure 3: The graph presents the optimal allocation of the new risky project as a function of its expected return and cost of the new firm. 47
