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Premium for going against the flow & bias in the in-

1.5 Implications of the Optimal Contract

1.5.1 Premium for going against the flow & bias in the in-

Proposition 3 explores the relation between the optimal payment when the manager “goes against the flow” and the optimal payment when the manager “follows the flow”. By “following the flow” we define the case where the manager takes the same position as the position that the equity holder would take, had he not hired the manager. In this setting, “going against the flow” means going long when the asset is overvalued (p<0.5) and going short when the asset it undervalued (p >0.5).

Proposition 3: Premium for going against the flow.

Under the optimal contract:

(i) If p>0.5, then wSB(sˆ∗) >wLG(sˆ∗).

(ii) If p<0.5, then wLG(sˆ∗) >wSB(sˆ∗).

Proof. See Appendix A.3.

Hence, under the optimal contract, there is a premium for going against the flow. Proposition 4 explores how ˆs∗ relates to: i) the optimal value of ˆs

if there is no agency problem (first best), ˆsFB, and ii) the value of ˆs which minimizes the cost of incentivizing information acquisition, ˆsmin.

Proposition 4: Bias in the investment decision.

Under the optimal contract:

(i) If p≥0.5, thensˆFB <sˆ∗ ≤sˆmin.

(ii) If p≤0.5, thensˆFB >sˆ∗ ≥sˆmin.

The proof of Proposition 3 and Proposition 4 can be captured in the follow- ing figures, which illustrates Lemma 3 and Lemma 4. Figure 1.4 presents the case where p > 0.5. Note that for these values of p, an investment threshold ˆs0 ∈ [0, ˆsFB] cannot be optimal, because switching to ˆs00 = sˆ0+η, whereη is a small positive number, decreases the expected compensation cost and increases the expected revenue. Likewise, an investment threshold ˆ

s0 ∈ [sˆmin, 1] cannot be optimal, because switching to ˆs00 =sˆ0−η decreases the expected compensation cost and increases the expected revenue. Thus, for p > 0.5, ˆs∗ ∈ (sˆFB, ˆsmin]. Similar intuition applies for the case where

p<0.5, which is illustrated in Figure 1.5.

0.2 0.4 0.6 0.8 1 0.5 1 1.5 ˆ sFB sˆ∗ sˆmin ER(sˆ,p>0.5) EC(sˆ)

Figure 1.4. Bias in the investment decision - Case where p>0.5, Flow: long

Proposition 4 captures the key feature of the optimal contract: the man- ager goes against the flow more often than the first best. Proposition 4 is a consequence of Proposition 3 and the MLRP property. The intuition behind the premium and the bias for going against the flow relies on the interaction of unobservable information acquisition and the multi-tasking nature of the problem. In order to incentivize the manager to acquire in-

0.2 0.4 0.6 0.8 1 0.5 1 1.5 ˆ sFB ˆ smin sˆ∗ ER(sˆ,p<0.5) EC(sˆ)

Figure 1.5. Bias in the investment decision - Case wherep<0.5, Flow: short

formation, the equity holder should promise a positive payment only if the ex-post right position is taken. However, the compensation contract – apart from affecting the incentive of the manager to acquire information – also affects his investment decision, which in turn, determines the portfolio’s expected revenue.

In this environment, the portfolio’s revenue is maximized when the payments which correspond to a short and long position are equal. A critical property is that under the optimal contract, the manager’s utility should coincide with his outside option. For equal payments, the outside option of the manager is to follow the flow without acquiring information. The equity holder can thus worsen outside option of the manager by low- ering the payment when the flow is followed. Worsening outside option of the manager enables the principal to incentivize information acquisition at a lower cost– however, changing the ratio of payments comes at the cost of decreasing the portfolio’s expected revenue; for a ratio of payments differ- ent than one, the first best is no longer implemented. The optimal contract thus solves the inherent trade-off between the cost of incentivizing learning

and benefit of implementing an investment decision as close as possible to the revenue-maximizing one. We show that for small deviations from the first best, the decrease in the expected cost is greater than the decrease in the expected revenue.

Corollary 2 explores the main implications of the bias in the invest- ment decision on: i) the probability of going short or long, ii) the prob- ability that the implemented position is revenue-maximizing, and iii) the beliefs about the state of the world, after the position is implemented.

Corollary 2: Implications of the Optimal Contract compared to First Best.

If p> 0.5 (p < 0.5) and compared to the case where there is no agency problem, under the optimal contract:

(i) The manager is more (less) likely to go short.

(ii) Given that a short position is implemented, it is less (more) likely to be revenue-maximizing.

(iii) Given that a long position is implemented, it is more (less) likely to be revenue-maximizing.

(iv) Given the implemented position, the beliefs about the asset being good are higher (lower).

Proof. See Appendix A.3.

Part one implies that compared to the first best, it is less likely that the manager will invest in an asset where prior beliefs indicate that it is undervalued. This is a direct consequence of Proposition 4. The intu- ition behind part two is that under the optimal contract and for p > 0.5,

the manager is tempted by the premium to go short, even for signal re- alizations where he believes that it is more likely that going long is the revenue-maximizing position. Thus, for p >0.5 and conditional on going short, the probability of going short being the right position is lower than in the case where the first best threshold is implemented. The intuition behind part three is similar: the manager is willing to go long and forgo the premium of the short position, only if he holds strong evidence that the price will increase. Part four is a direct consequence of part two and three.