Passive Manager Equilibrium

In this section, we consider a benchmark case when the manager is given a linear compensation contract. In this case, as shown in Lemma III.2 above, the optimal

action ataken by the manager does not depend on µ. Moreover, as shown in Propo- sition III.3 a linear compensation contract does not provide any incentives to learn about the fundamentals. We label the manager to be “Passive”.

The passive manager’s action do not add any additional uncertainty in the value of the firm beyond the uncertainty due to fundamentalsθ. Thus the price formation process through trading in the secondary market follows Kyle (1985). The trading intensity, price impact and price informativeness obtained in the equilibrium with a passive manager serve as a benchmark to illustrate the effects of manager’s learning on the financial markets with an active manager in section 3.5 below.

We solve the equilibrium strategies of all agents as follows. Given the manager’s decision, we first solve for informed trader’s trading intensity. At t = 0 the informed trader places his orderx(st) to maximize his expected profitsE[(V −P)x|st] given the pricing rule of the market maker. The informed trader chooses the trading intensity as: γ₁P = 1 2αP 1 σ2 θ σ2 θ+σt2 . (3.4)

The informed trader trades with a greater intensity if his information is precise (σ_t2

low) or if the price impact αP₁ is low. Since the manager is passive, her actions do not add any additional uncertainty about the firm value. Thus, the informed trader’s information about the fundamental θ is useful to predict the overall firm value. Put differently, in the case of a passive manager, her actions do not affect the trading intensity of the informed trader.

Given the trading intensity, the market maker chooses the price impact to make zero expected profits. The market maker upon observing the total order flow q can- not distinguish between the orders coming from informed trader or liquidity traders. Thus, in order to minimize the adverse selection risk, he sets the price P = E[V|q]

such that his expected losses E[(P −V)q] equal 0. The price impact αP

1 set by the

market maker is:

αP₁ = γ P 1σθ2 (γP 1 )2(σt2+σ2θ) +σ2y . (3.5)

The price impact is non-linear in trading intensity γP

1 and increasing in informed

trader’s precision 1/σ2

t, while it is decreasing in the amount of noise tradingσy2. The price impact also does not depend on the manager’s action, since a passive man- ager’s action do not add any uncertainty to the firm value (beyond the fundamental uncertainty aboutθ).

Next, we solve for the equilibrium trading intensity and the price impact given the manager is passive. We find the trading intensity and the price impact as

γ₁P = _σ2 y σ2 θ +σ 2 t 1/2 , (3.6) αP₁ = σ 2 θ 2σy(σt2+σ2θ)1/2 . (3.7)

We plot the informed trader’s trading intensity as a function of price impact and the market maker’s price impact as a function of trading intensity in figure 3.2. When the manager is passive, the equilibrium values of (γP

1,αP1) are shown at point A. Note,

point A corresponds to the maximum αP

1 for all possible values of γ1P, implying that

the market maker chooses the highest possible price impact given trading intensity

γ₁P.

Next, we calculate the price informativeness, price volatility and the expected profits of the informed trader for this benchmark case. In our setting the market price, not only provides information about the value of the firm, but also serves as a signal about θ, which the manager may find useful to learn about the fundamental. Thus, we define price informativeness about fundamental as the amount of reduction

in uncertainty about θ.

P I = V[θ]−V[θ|P]

V[θ] .

We obtain the equilibrium price informativeness and price volatility as:

P I= σ 2 θ 2(σ2 θ +σ 2 t) , (3.8) V(P) = (σ 2 θ)2 2(σ2 θ +σ2t) . (3.9)

Note in this equilibrium, since the manager is passive, her actions do not affect the price impact, price informativeness and price volatility. The reason is that the passive manager’s action does not increase uncertainty about the firm value. This also implies that the price informativeness about the firm value is equal to the price informativeness about the fundamental. We show later that, in the case of an active manager, the price informativeness about the firm value is different from the price informativeness about the fundamental.

We also calculate the equilibrium expected profits of the informed trader. It can be easily shown that the expected profits of the informed trader are:

E[π(st)] =

σyσ2θ 2(σ2

θ+σt2)1/2

. (3.10)

We summarize the results of this section in the following proposition.

Proposition III.4. There exists a unique linear equilibrium when the manager is passive. In the equilibrium the market maker sets the price impact αP

1 as: αP₁ = σ 2 θ 2σy(σt2+σ2θ)1/2 .

The informed trader’s trading intensity γP 1 is: γ₁P = _σ2 y σ2 θ +σ2t 1/2 .

The price volatility is:

V(P) = (σ 2 θ)2 2(σ2 θ +σ2t) .

When the manager is passive, the trading intensity, price impact, price informa- tiveness and price volatility are the same as in a Kyle setting. In the next section, we modify the model such that the manager has incentives to use her information

I to take optimal actions which affect the price formation process in the secondary market.