Divergence of opinion and initial public offerings

Abstract

This article analyzes several IPO patterns in the framework of divergence of opinion. Considering a new industry with few publicly traded companies, the investors in this IPO market do not initially have complete knowledge about the industry, but may learn from other IPOs in the sector. Our model shows that the equilibrium is consistent with empirical evidence documented for IPO underpricing and hot issue markets. We also characterize the association between share overhang, trading volume, and IPO prices. Furthermore, we discuss the decision of going public, analyst coverage, and IPO lockup expiration in the presence of divergent opinions.

Appendix

Lemma 1

The function

$$ G(x) = \frac{\phi (x|N,k)}{{\phi (x|N,k) + (1 - \phi (x|N,k))\left( {\tfrac{1 - q}{q}} \right)^{2j - 1} }}V $$

is increasing in x.

Proof of Lemma 1

Because $\frac{\partial G(x)}{\partial x} = \frac{\partial G}{\partial \phi }\frac{\partial \phi }{\partial x},$ the lemma is proved by noting that

$$ \frac{\partial \phi }{\partial x} > 0 $$

is obtained from both (1) and (2), and $\frac{\partial G}{\partial \phi } = \frac{{\left( {\tfrac{1 - q}{q}} \right)^{2j - 1} }}{{\left[ {\phi + (1 - \phi )\left( {\tfrac{1 - q}{q}} \right)^{2j - 1} } \right]^{2} }}V > 0.$

Proof of Proposition 1

We prove the uniqueness of equilibrium first. From the conditions (6) and (7), it follows that the solution satisfies:

$$ \frac{(1 - x)e}{m} = \frac{\phi (x|N,k)}{{\phi (x|N,k) + (1 - \phi (x|N,k))\left( {\tfrac{1 - q}{q}} \right)^{2j - 1} }}V $$

The left term is a monotonically decreasing function of x, while the right term increases in x based on Lemma 1. Then we have a unique solution based on the intermediate value theorem. Next, we prove the result of underpricing by contradiction. Let p* ≤ p ₀. From (6), it follows

$$ x^{*} \ge 1 - m\frac{{p_{0} }}{e} > 1 - m = x_{0} . $$

Then it leads to p* > p ₀ because the function in (4) increases in x. Thus, we prove p* > p ₀ and x* > x ₀.

Proof of Proposition 2

From (1), it is obvious that B(x|N,k) increases in 2k − N. Then from (2), it follows that ϕ(x|N,k) is increasing in 2k − N. Therefore, from (5) and (7), we obtain that p ₀ and p* increase in 2k − N.

Proof of Proposition 3

When N approaches infinity, B(x|N,k) converges to 1 (successful industry) or 0 (unsuccessful industry), which implies that investors have full knowledge about whether the industry is successful, as in the case of rational expectation.

Proof of Proposition 4

From x ₀ = 1 − m, (5–7) it can be proved. Because B(x|N,k) is positively associated with x, a decrease in m will increase x ₀, and thus increase ϕ(x ₀|N, k). From Eq. 5 it follows that p ₀ will decrease with m. The effect of m on p* can be proved by contradiction. Assume that p* does not increase with a decrease in m. From (6) it follows that x* will increases, and therefore from Eq. 7 p* will increase. This leads to a contradiction.

Proof of Proposition 5

If $\tilde{e} > e,$ because $\tilde{x}_{0} = x_{0} = 1 - m,$ by Eqs. 2 and 4 we have $\tilde{p}_{0} = p_{0}.$ Next we prove $\tilde{x}^{*} > x^{*}$ and $\tilde{p}^{*} > p^{*}$ by contradiction. If $\tilde{x}^{*} \le x^{*},$ then from Eq. 6 and $\tilde{e} > e,$ it leads to $\tilde{p}^{*} > p^{*}.$ However, from (7) and $\tilde{x}^{*} \le x^{*},$ it leads to$\tilde{p}^{*} \le p^{*},$ which is a contradiction.

Proof of Proposition 6

When investors have received a good signal, j = 1, from (5–7) it can be shown that p ₀ and p* increase in q by a similar argument to Proposition 4 because

$$ \frac{\phi (x|N,k)}{{\phi (x|N,k) + (1 - \phi (x|N,k))\left( {\tfrac{1 - q}{q}} \right)^{2j - 1} }} $$

increases (decreases) in q if j = 1 (j = 0).

Proof of Proposition 7

If $\tilde{B}(x) > B(x),$ because $\tilde{x}_{0} = x_{0} = 1 - m,$ by Eq. 2 we have $\tilde{\phi }(\tilde{x}_{0} |N,k) > \phi (x_{0} |N,k)$, and therefore $\tilde{p}_{0} > p_{0}.$ Next we prove $\tilde{x}^{*} < x^{*}$ and $\tilde{p}^{*} > p^{*}$ by contradiction. If $\tilde{x}^{*} \ge x^{*},$ then from Eq. 6 it leads to $\tilde{p}^{*} \le p^{*}.$ However, from (7) and $\tilde{x}^{*} \ge x^{*}$ it leads to $\tilde{p}^{*} > p^{*},$ which is a contradiction.

Proof of Proposition 8

From (6), x* can be stated as x* = 1 − mp*/e. Thus, trading volume would be: $(x^{*} + m - 1)I = I(1 - mp^{*} /e + m - 1) = Im(e - p^{*} )/e.$

By the proof of Proposition 4, p* is decreasing in m. Differentiating the right hand side of the above identity with respect to m, we conclude that trading volume is increasing in m. Because share overhang is decreasing in m, the chain rule would establish that trading volume is negatively associated with share overhang. In the proof of Proposition 5, we know that $\tilde{x}^{*} > x^{*}$ if $\tilde{e} > e.$ Similarly, by the proof of Proposition 7, we have observed that $\tilde{x}^{*} > x^{*}$ if $\tilde{B}(x) > B(x).$ Thus, trading volume in the aftermarket is positively related to the investment amount e and the extent of investor optimism.

Proof of Proposition 9

From (9), the going public condition becomes mp ₀ + A ≥ mλV when the entrepreneur believes that the industry is a successful one, and mp ₀ + A ≥ m(1 − λ)V in an unsuccessful industry. Because p ₀ is positively associated with the degree of optimism of investors, the results are obviously obtained. From Eqs. 1, 2, and 5, we have:

$$ H(\lambda ,B_{0} ,2k - N) = 1 + \frac{{\lambda + (1 - \lambda )\left( {\tfrac{{1 - B_{0} (x_{0} )}}{{B_{0} (x_{0} )}}} \right)\left( {\tfrac{1 - \lambda }{\lambda }} \right)^{2k - N} }}{{1 - \lambda + \lambda \left( {\tfrac{{1 - B_{0} (x_{0} )}}{{B_{0} (x_{0} )}}} \right)\left( {\tfrac{1 - \lambda }{\lambda }} \right)^{2k - N} }}\left( {\tfrac{1 - q}{q}} \right)^{2j - 1} $$

$$ p_{0} = \left[ {\frac{1}{{1 + \frac{{\lambda + (1 - \lambda )\left( {\tfrac{{1 - B_{0} (x_{0} )}}{{B_{0} (x_{0} )}}} \right)\left( {\tfrac{1 - \lambda }{\lambda }} \right)^{2k - N} }}{{1 - \lambda + \lambda \left( {\tfrac{{1 - B_{0} (x_{0} )}}{{B_{0} (x_{0} )}}} \right)\left( {\tfrac{1 - \lambda }{\lambda }} \right)^{2k - N} }}\left( {\tfrac{1 - q}{q}} \right)^{2j - 1} }}} \right]V = \left[ {\frac{1}{{H(\lambda ,B_{0} ,2k - N)}}} \right]V. $$

The two inequalities in Proposition 9 are obtained by substituting the above equation into (9). In addition, ${{(H(\lambda ,B_{0} ,2k - N) - 1)} \mathord{\left/ {\vphantom {{(H(\lambda ,B_{0} ,2k - N) - 1)} {\left( {\tfrac{1 - q}{q}} \right)^{2j - 1} }}} \right. \kern-\nulldelimiterspace} {\left( {\tfrac{1 - q}{q}} \right)^{2j - 1} }} = \frac{{\lambda + (1 - \lambda )\left( {\tfrac{{1 - B_{0} (x_{0} )}}{{B_{0} (x_{0} )}}} \right)\left( {\tfrac{1 - \lambda }{\lambda }} \right)^{2k - N} }}{{1 - \lambda + \lambda \left( {\tfrac{{1 - B_{0} (x_{0} )}}{{B_{0} (x_{0} )}}} \right)\left( {\tfrac{1 - \lambda }{\lambda }} \right)^{2k - N} }} = \frac{\lambda + (1 - \lambda )h}{1 - \lambda + \lambda h},$ where $h = \left( {\tfrac{{1 - B_{0} (x_{0} )}}{{B_{0} (x_{0} )}}} \right)\left( {\tfrac{1 - \lambda }{\lambda }} \right)^{2k - N}.$  Note that λ > 0.5 and h is strictly decreasing in B ₀ and 2k − N. The following derivative shows that:

$$ \frac{{\partial \{ (\lambda + (1 - \lambda )h)/(1 - \lambda + \lambda h)\} }}{\partial h} = (1 - 2\lambda )/(1 - \lambda + \lambda h)^{2} < 0. $$

Therefore, H(λ, B ₀, 2k − N) is increasing in 2k − N and B ₀. Eq. 11 holds if B ₀ and 2k − N is relatively large. Moreover, Eq. 11 also tends to hold if A is relatively large and m is relatively small.

Proof of Proposition 10

It can be shown that

$$ \frac{k(x)f}{k(x)f + (1 - k(x))(1 - f)} > k(x)\quad {\text{if}}\;f > 0.5, $$

because

$$ \frac{k(x)f}{k(x)f + (1 - k(x))(1 - f)} - k(x) = \frac{k(x)(1 - k(x))(2f - 1)}{k(x)f + (1 - k(x))(1 - f)} $$

and k(x) < 1. Therefore, from (6a) and (12), it follows to that $p_{a}^{*} (s,N,k) > p^{*} (s,N,k)$ by a contradiction. From (6a), the representation $p_{a}^{*} \left( {s,N,k} \right) = \left( {1 - x_{a}^{*} } \right)e/m$ is a decreasing function in $x_{a}^{*},$  and from (12), $p_{a}^{*} (s,N,k) = \frac{{k\left( {x_{a}^{*} } \right)f}}{{k(x_{a}^{*} )f + \left( {1 - k\left( {x_{a}^{*} } \right)} \right)(1 - f)}}V$  is an increasing function of $x_{a}^{*}.$ By the fact that

$$ \frac{k(x)f}{k(x)f + (1 - k(x))(1 - f)} > k(x) $$

if $p_{a}^{*} (s,N,k) \le p^{*} (s,N,k)$ holds, by $p_{a}^{*} (s,N,k) = (1 - x_{a}^{*} )e/m$ it follows that $x_{a}^{*} \ge x^{*},$ which contradicts to $p_{a}^{*} (s,N,k) = \frac{{k\left( {x_{a}^{*} } \right)f}}{{k\left( {x_{a}^{*} } \right)f + \left( {1 - k\left( {x_{a}^{*} } \right)} \right)(1 - f)}}V.$ Note that

$$ \frac{k(x)f}{k(x)f + (1 - k(x))(1 - f)} = k(x) $$

when f = 0.5, that is, the analyst coverage does not provide additional information to investors. In this degenerated case, it leads to $p_{a}^{*} (s,N,k) = p^{*} (s,N,k).$ Moreover,

$$ \frac{{\partial p_{a}^{*} (s,N,k)}}{\partial f} > 0 $$

can be proved because $\frac{{\partial \left[ {\frac{k(x)f}{k(x)f + (1 - k(x))(1 - f)}} \right]}}{\partial f} = \frac{k(x)(1 - k(x)}{{[k(x)f + (1 - k(x))(1 - f)]^{2} }} > 0.$

Proof of Proposition 11

From (13), $p_{{m^{\prime}}}^{*} (s,N,k) = (1 - x_{{m^{\prime}}}^{*} )e/(m + m^{\prime})$ is decreasing in m′ and x. From (12a), p is increasing in x. We prove it by contradiction similarly to the proof of Proposition 10. If $p_{{m^{\prime}}}^{*}$ is not decreasing in m′, then from (12a) an increase in m′ will be related to an increase in x, which contradicts to (13). Therefore, the proposition is proved.

Proof of Proposition 12

First, note that p ₀ is unchanged and solved by (5). The aftermarket price and marginal investor (p ^*,x ^*) will satisfy the equilibrium conditions (15) and (16). Because B(x) is increasing in (2k − N) from (1), it follows that e(x, B(x)) is increasing in (2k−N), which means that the endogenous investment amount will be positively related to the number of positive outcomes minus the total number of IPOs in history. Next, we prove the impact of investment function e(x, B(x)) on the aftermarket price by contradiction. For any (N,k) and given $\tilde{e}(x, B(x)) > e(x, B(x)),$ if $\tilde{p}^{*} \le p^{*},$ from (16) it follows that $\tilde{x}^{*} \le x^{*}.$ From (15), this suggests that

$$ p_{{}}^{*} = \int_{{x^{*} }}^{1} {e(y, B(y)){\text{d}}y} /m $$

is decreasing in x* but increasing in e. Therefore, by $\tilde{x}^{*} \le x^{*}$ and the inequality $\tilde{e}(x, B(x)) > e(x, B(x))$ for all x, it leads to $\tilde{p}^{*} > p^{*},$ which is a contradiction.