Environment: We consider a setting with two risk neutral players: a repre- sentative equity holder of a fund (principal) and a fund manager (agent). The equity holder delegates an investment decision, denoted by d, to the fund manager. The investment decision consists of two options: buying a given asset (“going long”, d = L) or short-selling a given asset (“going short”,d =S).
The expected return of each position depends on the expected price movement of the asset,Pθ
1 −P0, which in turn, depends on the ex-ante un-
known quality (state of the world) of the asset, denoted byθ. The manager and the equity holder share common prior beliefs about the quality of the asset. In particular, they expect the asset to be of good quality (θ =G), with probabilityp, and of bad quality (θ = B), with probability 1−p. Following the binary nature of the quality, the value of p incorporates any publicly available information regarding the asset’s quality.4
We assume that if the asset is of good quality, its price is expected to increase to P1G = P0+e, whereas if the asset is of bad quality, its price is expected to decrease to P1B = P0−e. This assumption captures the idea that the asset price moves towards its fundamental value.
Information Acquisition Technology: Before the investment decision is taken, the manager can acquire a signal about the quality of the asset. The manager’s private information consists of the signal realization and signal 4The publicly available information, which is captured by p, can include analyst’s
forecasts, decisions of other managers, and credit ratings. We do not impose any structure on how these beliefs are formed.
acquisition, the latter of which incurs a costc.5 This cost can be interpreted a utility loss due to effort of acquiring a signal. Conditional on acquiring information, the manager receives a noisy signals ∈ [0, 1]about the quality of the asset. The signal s is distributed according to a continuous density function fθ(.), with a distribution function Fθ(.), whereθ={B,G}. Hence, for a signal realizations0, the following hold:
fθ(s 0) ≡ Pr(s=s0|θ) Fθ(s 0) ≡ Pr(s≤s0|θ)
After observing the signal realizations0, the manager updates his beliefs as follows:
Pr(θ =G|s =s0) = fG(s
0)p
fG(s0)p+ fB(s0)(1−p)
=1−Pr(θ =B|s=s0)
Assumption 1: Monotone Likelihood Ratio Property (MLRP)
For any signal realization s0 ∈ [0, 1], the ratio fG(s0)
fB(s0) is increasing in s
0.
A direct consequence of Assumption 1 is that the probability that the asset is of good quality is increasing in the signal realization. In addition, given that fG(s)and fB(s)represent probability density functions, Assumption 1 implies that fG(s) and fB(s)satisfy the single-crossing condition.
5This characteristic relates to the nature of the delegation problem – the manager, as
opposed to the equity holder – has expertise in inferring the unknown state of the world. Thus, the equity holder is unaware of the evidence the agent needs to collect to infer the unknown state of the world, and how costly it is for the manager to collect these evidence.
Preferences and Actions: The equity holder designs the manager’s com- pensation contract, which is denoted by ˆWand specified below, in order to maximize his expected profits:
EΠ=E[R|Wˆ ]−E[C|Wˆ ] (1.1)
where E[R|Wˆ ] stands for the expected revenue of the portfolio, whereas E[C|Wˆ ] stands for the manager’s expected compensation, given a contract
ˆ
W. Note that the compensation contract affects the expected portfolio re- turn,ER, through the investment decision of the manager.
The manager faces two kinds of decisions: the information acquisi- tion decision, and the decision to go short or long. The objective of the manager is to maximize his expected utility:
EV =E[C|Wˆ ]−1c (1.2)
where1equals 1 if information is acquired, and zero otherwise.
Set of available contracts: We allow for contracts contingent on the im- plemented position, d, and the state realization, θ, i.e., ˆW : d×θ 7→ R+, where d = {S,L} and θ = {B,G}. Thus, the contract is characterized by the following quadruple:
ˆ
W ={wSG,wSB,wLG,wLB}
wherewSG (wSB) denotes the payment when the manager goes short and the asset quality is revealed to be good (bad). Likewise,wLG (wLB) denotes
the payment when the manager goes long, and the quality is revealed to be good (bad). For the sake of tractability, we assume that the agent is protected by limited liability, i.e., paymentwd,θ is non-negative. In section 1.4.2, however, we show that the limited liability assumption can be relaxed without affecting the main findings qualitatively.
In Appendix A.1, we explore the case where it is not feasible for the principal to offer contracts contingent on the realized state of the world. We show that, as long as there is a public signal, which is revealed af- ter the decision is taken and it is informative about the actual state of the world, the main findings go through. Besides, in Appendix A.2, we allow for contracts contingent on: i) messages sent by the manager to the equity holder, and ii) the realized state of the world. We show that the optimal contract is effectively the same, independently of whether it is contingent on the implemented position or the agent’s messages.
Timing: The sequence of events is the following:
1. The equity holder offers a compensation contract ˆW. 2. The manager decides whether to acquire information.
3. If information is obtained, the manager observes the signal realiza- tion.
4. The manager chooses the implemented position. 5. The state is realized, and the contract is executed.