Comments on the Signaling Theory of IPO Underpricing and Investor Protection Laws

3.1 Complete Information Case

The equilibrium can be found by backward induction. Since this is a complete information game, it suffices to find the subgame perfect equilibrium in this dynamic game.

It is clear that given any IPO price p, the investors accept the price offer (buy the shares at the price) if p ≤ π.

Now, consider the firm’s pricing decision. Let p*(π) be the equilibrium price of type π. Taking the investors’ decisions into account, the firm will choose the IPO price p which is the maximal price that the investors will accept. Therefore, the equilibrium prices under full information must be p*(H) = H and p*(L) = L. It is also clear that the H type prefers this outcome to no investment outcome which is obtained when it offers p > H, because αH + (1 − α)H = H > L, while the L type is indifferent between investing and not investing because αL + (1 − α)L = L.

3.2 Incomplete Information Game

Our main interest in this section is whether a separating equilibrium in which an IPO price signals the value of the firm is possible. We denote the separating equilibrium price of the high type and the low type by p _H and p _L respectively, and the investors’ posterior belief updated after observing p by θ ̂ ( p ) . Instead of θ ̂ , we may denote by π 1 e the perceptions of investors about π in the first period. That is, π 1 e ≡ θ ̂ H + ( 1 − θ ̂ ) L , so π 1 e = H if θ ̂ = 1 and π ^e  = L if θ ̂ = 0 .

If the true value of the firm is not known to the investors, a low type wants to pretend to be a high type because he could sell his shares at a higher price by doing so. However, if the true value of the firm is revealed in the second period and is fully reflected in π, i.e.,  π 2 e = π where π 2 e is the second-period perception of incestors about π, regardless of the IPO price p, a low type gains nothing in the second period by pretending a high type. If p _H  > p _L , a low type can successfully imitate the high type by offering p _H which will be always accepted. So, it cannot be an equilibrium. If p _H  < p _L , a low type loses by pretending to be a high type. In this case, he has no incentive to deviate from p _L . So, a necessary condition for a separating equilibrium is that p _H  < p _L .

Suppose p _H  < p _L , i.e., a high type underprices in equilibrium. It is easy to see that the low-type firm’s separating equilibrium price is not distorted, i.e., p _L  = p*(L), because the low type would deviate to p*(L) if p _L  ≠ p*(L), under the most pessimistic belief, because p*(L) is the best price for the low-type firm among the prices such that θ ̂ ( p ) = 0 . The equilibrium price of the high-type firm must satisfy the following incentive compatibility condition of the type:

Inequality (1) is the incentive compatibility condition of the high type which will be called [ICH1]. It requires that a high-type firm has no incentive to deviate to any other price than its equilibrium price p _H . The right hand side of (1) is the high type’s payoff when it deviates from p _H . If it deviates to p > L, investors do not buy the shares, so its payoff is just L. If it deviates to p ≤ L, investors invest and thus the firm’s payoff is αp + (1 − α)H.^[8]

It is not difficult to see that this incentive compatibility condition cannot be compatible with the optimal decision of the investors. Since p _H  < L, the high-type firm would deviate to p ∈ (p _H , L) because such a price would always be accepted by the investors. The investors would infer from this price that the firm must be a low type, but finds the price still attractive enough to buy the shares because the price is too low (p < L);^[9] hence, no IPO underpricing in equilibrium. To summarize, we have

Some may suspect that this result is an artifact of the assumption that only pure strategies are available. What if we allow mixed strategies of the investors? Unfortunately, it turns out that separation (semi-separating equilibrium) is not possible even with mixed strategies.

Let r(p) ∈ [0, 1] be the probability that investors accept the IPO price p. It is true that the incentive of a high-type firm to deviate to p ∈ (p _H , L) could be deterred if we allow mixed strategies of the investors so as to make the probability that the investors accept p strictly less than one. However, the mixed strategy r(p) < 1 for any p ∈ (p _H , L) cannot be optimal for the investors because they always strictly prefer investing at the price p to not investing, because α(π ^e  − p) > 0 for any p ∈ (p _H , L) and for any belief π ^e . As long as the investors always buy the shares at p ∈ (p _H , L) with probability one, the high type always deviates to p from the equilibrium price p _H (<L).

5.1 Investor Protection Laws

We examine how investor protection laws can affect IPO underpricing. Since many rules and amendments to enhance investor protection in IPO consider information asymmetries as one of the main causes in IPO underpricing, we may identify a stronger investor protection law by stronger a disclosure requirement.

Suppose that the firm releases some required report to the public before IPO pricing. Then, the investors receive some signal s = G or B from the report. We assume that investors receive G with probability q ( > 1 2 ) or B with the remaining probability 1 − q if π = H and they receive G with probability 1 − q or B with q if π = L. It means that a report from a high-type firm is not always interpreted as a good news (G). It may be interpreted as a bad news (B) by some investors. If the disclosure gets more reliable, q gets higher, that is, q = 1 if disclosure is perfect, and q = 1 2 if the report is completely unreliable. We assume that q is higher as the investor protection law is stronger.

In the first period, the firm sells a fraction α of its equity by offering the IPO price p. Then, investors decide whether to purchase the shares, based on the observations of s and p.

In this modified game with investor protection laws, the investors’ purchasing decision is slightly modified. Their posterior belief is updated based on p and s. So, we will denote the resulting posterior belief by θ ̂ ( p , s ) . Then, their perceptions about π in the first period is π e = θ ̂ ( p , s ) H + ( 1 − θ ̂ ( p , s ) ) L . It is clear that the investors buy the shares if p ≤ π ̂ 1 ( p , s ) .

Since IPO underpricing can occur only in a separating equilibrium, we will restrict our attention to separating equilibria. Then, it is not difficult to see that the analysis for a separating equilibrium is not affected very much. This is mainly because in a separating equilibrium, the true value of the firm is revealed by its strategy (price or the equity shares), regardless of the signal s. This implies that there is no separating equilibrium in the model with one-dimensional signal. The same intuition applies. In a separating equilibrium, if p _H  < p _L , the high-type firm would profitably deviate to any p ∈ (p _H , p _L ) because investors will always buy the shares under the most pessimistic belief. If θ ̂ ( p , s ) = 0 , investors’ net benefit from buying the shares is π 1 e − p = L − p > 0 for any p ∈ (p _H , p _L ). This is true even for the most optimistic belief, because the investors’ benefit is then π 1 e − p = H − p > L − p > 0 for any p ∈ (p _H , p _L ). The result that we obtained in the model with two-dimensional signals is not affected, either. It is clear that p _H  = p _L  = L for the same reason. Now, suppose that α _H  ≠ α _L . The definition of a separating equilibrium requires us to infer that θ ̂ ( α H , L ; s ) = 1 and θ ̂ ( α L , L ; s ) = 0 for any s = G, B. Then, under the most pessimistic belief, the incentive compatibility conditions given by [ICH2] and [ICL2] are not affected. Since [FC] condition also remains unaffected, Proposition 2 saying that a high-type firm can signal its type only by selling a lower fraction of equity is still valid, regardless of q. This implies that investor protection laws have no effect on IPO underpricing at all.

5.2 Partial Revelation of Information

In this paper, we assumed that all uncertainty regarding the profitability of the firm is resolved at the end of the first period, but readers may be interested in whether this assumption is crucial to our result. So, in this section, we will examine whether the result would be affected if the true value of the firm is not completely realized in the final (second) period so the second-period value depends partially on the first-period belief of the investors, as in Allen and Faulhaber (1989).

Let us assume that some investors become fully informed of π in the second period and others are not. Let μ ∈ (0, 1) be the proportion of investors who are uninformed of π in the second period. Then, we can assume that the firm’s perceived valuation in the second period is π 2 e = μ π 1 e + ( 1 − μ ) π .

Under this alternative assumption of partial realization, let us check the incentive compatibility conditions. Now, [ICH1] and [ICL1] are modified as the following [ICH3] and [ICL3]:

Note that the most profitable deviation of an H-type firm is to p = L which will be accepted by investors. The firm’s profit in this case, which is the right hand side of [ICH3], is αL + (1 − α)(μL + (1 − μ)H) = L + (1 − α)(1 − μ)(H − L). So, [ICH3] is satisfied if μ ≥ α 1 − α L − p H H − L , or equivalently,

The inequality [ICH3] can be more easily satisfied if the punishment for a deviation from p _H by the worst belief is more severe (μ is larger) or the price distortion is less severe (p _H is higher).

On the other hand, a low-type firm will have no incentive to imitate a high type by choosing p _H if p _H satisfies [ICL3] which boils down to μ ≤ α 1 − α L − p H H − L , or equivalently,

Again, note that it is easier to satisfy [ICL3] if the punishment for a deviation to p _H is more severe (i.e., μ is smaller) or the price that he will imitate is lower (p _H is lower). Since (4) and (5) are contradictory to each other except for the knife-edge case that both types are just indifferent between choosing its own equilibrium price and the other type’s equilibrium price, we can say that there is essentially no meaningful separating equilibrium in which a high type signals its type by underpricing.

The intuitive reason for the nonexistence of a separating equilibrium is quite clear. Since the low-type firm’s IPO price cannot be distorted from p _L  = L in any equilibrium, it remains to check whether it is possible that p _H  > L or p _H  < L in equilibrium. If p _H  > L, a low type always has an incentive to deviate to p _H to imitate a high type. If p _H  < L, a high type always has an incentive to deviate to p′ ∈ (p _H , L), as we argued, because such a price p′ is always accepted insofar as the price is lower than the worst perception on the firm value; hence, no separating equilibrium. Note that those two intuitions are both so robust that this result cannot be affected by any assumption on the second-period perceived value of the firm.^[10]

5.3 Model with Issuing Costs

To examine the possibility that the robust intuition for nonexistence of a separating equilibrium (in the case of a single-dimensional signal) may break down, we may consider a more enriched model by adding more realistic features. As suggested by a commentator, we will incorporate the issuing cost into the model, although the paper by Allen and Faulhaber did not. The motivation for considering the issuing cost is that a low type may not want to imitate p _H which is higher than L if imitation incurs additional costs.

Let c > 0 be the issuing cost. Then, the firm first chooses whether to go public (enter the IPO market) with incurring additional costs or stay private and forgo the investment opportunity. After the decision, the game assumed in Section 2 proceeds.

It is easy to see that there is no separating equilibrium in which the high type enters the IPO market and the low type does not enter. The reason is obvious. If there is such an equilibrium, the type of the firm is fully revealed by the entering decision, implying that the high type must choose p = H after it enters. Unfortunately, however, this cannot be an equilibrium. If c is small enough, i.e., c < H − L, a low type will imitate the strategy by choosing p = H after entering the IPO market. If c > H − L, a high type will not enter because H − c < L. Anyway, there will be no separating equilibrium involving underpricing in this game.

If the firm chooses the issuing decision and the pricing decision simultaneously,^[11] underpricing may occur in a separating equilibrium. Consider an outcome in which a high type enters the IPO market and chooses p _H while a low type does not enter. The incentive compatibility conditions in this case require the following inequalities;

First, to prevent a low type from imitating p _H , p _H must be very low. More specifically, the gain from imitation α(p _H  − L) must not be higher than the entry cost c, i.e.,

Second, a high type firm’s incentive to deviate from choosing p _H should be prevented. The most profitable deviation of a high type may be to enter or not to enter, depending on c. If c < (1 − α)(H − L), the high type prefers entering and deviating to L to not entering, so max{L, αp + (1 − α)H − c} = αL + (1 − α)H − c. Then, [ICH4] simply implies that p _H  ≥ L because the equilibrium price p _H must be better (higher) than or equal to the highest price (except for p _H ) that can be accepted which is L.

If c > (1 − α)(H − L), not entering is better than entering to the high type. So, max{L, αp + (1 − α)H − c} = L. In this case, [ICH4] implies that αp _H  + (1 − α)H − c ≥ L, i.e.,

Note that L α − 1 − α α H + c α > L , since c > (1 − α)(H − L). This implies that p _H  > L. Therefore, in both cases, i.e., whether the entering cost c is greater or less than the gain from deception, (1 − α)(H − L), a high type can signal its type by p H ∈ ( L , L + c α ] . This is consistent with our discussion in Section 3 that p _H  < L cannot be a separating IPO price of a high type.

Intuition goes as follows. With a positive issuing cost, a low type cannot imitate the price of a high type even if p _H  > L so that imitation gives him some gain by p _H  − L. Also, with a positive issuing cost, a high type would always deviate if p _H  < L. To prevent this incentive, either c must be large or p _H  > L. In fact, even if c is large, p _H must be greater than L. Otherwise, the high-type firm’s payoff when he enters the market with incurring a significant entering cost would be lower than his payoff when he does not enter. To see this, suppose that p _H < L when c > (1 − α)(H − L). Then, the high-type firm’s equilibrium payoff is lower than his payoff when he does not enter the IPO market, since αp _H  + (1 − α)H − c < αL + (1 − α)H − c < αL + (1 − α)H − (1 − α)(H − L) = L. Thus, a high type can separate himself by underpricing. Note that there is a discontinuity at c = 0. Although there is no separating equilibrium when c = 0, there does exist a separating equilibrium for any c > 0. This is mainly because the low-type firm cannot imitate a high-type’s equilibrium price p _H which is higher than L if it must bear a positive cost c.

The upshot of this section is that the AF result of signaling by underpricing can be recovered with considering the issuing cost only if the firm can choose its IPO price at the same time as it enters the IPO market.