Informed External Investors

The common prior about the Board’s type isθ ∼N(y,1/τθ). While sophisticated

shareholders do not have additional information, there is a continuum of external investors, e.g. risk-arbitrageurs and hedge fund activists, of mass one where each investorj receives aniidsignalxj =θ+εj withεj ∼N(0,1/τε) and updates beliefs

accordingly. The interim stock priceM is set by market makers with zero expected profits. External investors acquire stock at the interim if their value is higher than

M, i.e. if they receive a sufficiently low signal of the Board’s type.

The equilibrium analysis is analogous to that inSection 2.3. For clarity, we denote our endogenous variables in this section with the subindex I.

Suppose that investors follow anx∗_I-threshold strategy: buy in the offer pe- riod if and only if they observex < x∗_I. Then stock sales equal the mass of investors that received a signal below the threshold relative to the number of sophisticated shareholders,ρI =δΦ

√

τε(x∗−θ)

. As a result, the Board has θ∗_I-threshold equi- librium behavior. We do not require a uniqueness condition, in contrast to our benchmark setting, asθ∗_I is unambiguously increasing in x∗_I.

tion. Thus, both the probability of a takeover βI and the interim stock price MI

are characterised by expressions equivalent to (2.6) and (2.7) respectively.

Consider now the strategy of an investor j observing signal xj and hence

with an updated belief θ|xj ∼ N

_τ θy+τεxj τθ+τε , 1 τθ+τε

. If other investors follow an

x∗_I-threshold strategy, her expected benefit to buying over not buying is

u(xj, x∗I) = (P −V) Φ _√ τθ+τε θ_I∗− τθy+τεxj τθ+τε +V −M = (P −V) Φ _√ τθ+τε θ∗_I−τθy+τεxj τθ+τε −Φ (√τθ(θ∗I−y)) (2.31)

It is possible to see that the investor’s utility of buying is monotonically decreasing in the signal received xj. Thus, the game of external investors is opposite to the

game of sophisticated shareholders (characterised in the benchmark model), whose utility of selling is increasing as a function of the signal they receive. With an argument analogous to the one developed in Proof of Lemma16, we find that there exists a unique solution tou(xj, x∗I) = 0 and hence external investors have a unique

x∗_I-threshold equilibrium. The equilibrium tuple{θ_I∗, x∗_I}is characterised in the next proposition:

Proposition 29 There exists a unique Bayesian Nash Equilibrium in which all external investors buy shares if and only if they observe a signal below x∗_I. The takeover succeeds if, and only if, the takeover resistance is below the threshold θ∗_I. The thresholds are characterised implicitly by the following equations:

θ∗_I =P −V +κδ[1−Φ (∆ (θ_I∗−y))] (2.32)

x∗_I =θ∗_I−√∆

τε

(θ∗_I−y) (2.33)

Proof. Analogous to Proof of Proposition17.

As a result, stock sales at the interim areρI =δΦ

√

τε(θ_I∗−θ)−∆ (θ∗_I−y)

. Comparing θ_I∗ with the threshold in (2.9) it is possible to appreciate that revers-

ing the information structure switches the mass of strategic decisions influencing the takeover, i.e. 1−Φ (∆ (θ∗_I−y)). In particular, the proportion of sophisticated shareholders selling now corresponds to the probability that an investor receives a signal x < x∗_I. This is in contrast to the benchmark model, where the proportion of sophisticated shareholders selling equals the probability that x > x∗. As a con- sequence, while the effect of the fundamental variables on both the probability of a takeover and the interim stock price has the same sign as in the original setting, their effect on the volume of sales is different. The following can be shown emulating the method of proof used in the prior analysis:

Proposition 30 Responses of the main outcomes to marginal increases of the model fundamentals are as follows

Fundamental Effect on outcomes

Pr. Takeover β Interim Price M Stock Sales ρ

Bid premium: P −V + + +

New-shareholder pressure: κ + + +

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