IPO activity and information in secondary market prices

Abstract

This paper explores the link between IPO underpricing and financial markets. In my model the IPO is a mean for a capital constrained initial investor to exit and thereby to raise funds for a new investment opportunity. This investor is privately informed vis-a-vis outside investors about the profitability of the new opportunity and the quality of the firm to be offered in the IPO. He can then use the offer price and the fraction of shares sold as signals of his private information. The model shows that underpricing is not only linked to firm’s characteristics, i.e. firm value, but to elements external to the firm, i.e. new investment profitability and financial markets characteristics. In particular higher market efficiency reduces the cost of listing. This results in lower underpricing and the listing of more valuable firm. Similarly, a higher lower bound of the new investment’s profitability reduces the information asymmetry and hence reduces underpricing and widens the range of firms listed.

Appendix

1.1 Proof of Proposition 1

Proof

The proof proceeds in three steps: $(a)\quad \pi \equiv \pi _L;\;(b)\quad \pi \ge \pi _L;\;(c)$ The verification of the conditions under which a unique Pareto dominant reactive equilibrium exists

Part 1: $\pi =\pi _{L}$. The initial owner has no reason to incur any underpricing since the investors cannot form a belief over $\pi$ worse than the true $\pi =\pi _{L}$. The problem reduces then to a one-dimensional signalling problem. Hence, the initial owner signals the firm value by limiting the amount of shares sold. Define with $V_{\pi _{L}}\left(\beta \right)$ the signalling schedule, that is the investors’ beliefs on firm value, observing $\beta$ and knowing that $\pi =\pi _{L}$.

The owner chooses $\beta$ to maximize his objective function (1) taking into consideration how $\beta$ affects outside investors’ beliefs. The first order condition is

$$\begin{aligned} -\tau V+(\pi _{L}-1+\tau )V_{\pi _{L}}\left(\beta \right) +(\beta (\pi _{L}-1)+1-\tau (1-\beta ))V^{\prime }_{\pi _{L}}\left(\beta \right)=0 \end{aligned}$$

(15)

Imposing the self-fulfilling belief condition $V_{\pi =\pi _{L}}\left(\beta \right) =V$ gives a family of informationally consistent signalling sets. Imposing the condition that $V\left[1\left|\pi =\pi _{L}\right.\right]=L$ gives the Pareto optimal set. This can be seen noticing that schedules where $V_{\pi =\pi _{L}}\left(1\right) >L$ allow for arbitrage for the owner: the owner who owns a firm whose value is $L$, could sell all his shares for $V_{\pi =\pi _{L}}\left(1\right)$ and make a profit for $V_{\pi =\pi _{L}}\left(1\right)-L$. Those where $V_{\pi =\pi _{L}}\left(1\right)<L$ are Pareto dominated by $V_{\pi =\pi _{L}}\left(1\right)=L$ as in this case the investor receives higher proceeds to reinvest. This gives Eq. () which is the informationally consistent Pareto-dominant signalling set.

Part 2: $\pi \ge \pi _{L}$. The owner chooses the underpricing such that his wealth is maximized taking into account that $\pi$ is not known to outside investors. So he chooses $P$ taking into account the effect on the beliefs about firm value, $V[P,\beta ]$:

$$\begin{aligned} \max _{P}\beta \pi P+\left(1-\beta \right)\left(\tau V+\left(1-\tau \right)V\left[P,\beta \right]\right) \end{aligned}$$

(16)

again $V\left[V,\beta \right]=V_{\pi _{L}}\left(\beta \right)$ is required in order to have the Pareto dominant schedule. As in Part 1 if $V\left[V,\beta \right]>V_{\pi _{L}}\left(\beta \right)$ there are arbitrage opportunities for the owner, while the schedule $V\left[V,\beta \right]<V_{\pi _{L}}\left(\beta \right)$ is Pareto dominated. This gives:

$$\begin{aligned} V[P,\beta ]=\frac{1}{1-A}V_{\pi _{L}}\left(\beta \right)+ \frac{A}{1-A}P \end{aligned}$$

(17)

where

$$\begin{aligned} A=\frac{\beta \pi }{\left(1-\tau \right)\left(1-\beta \right)+\beta \pi } \le 1 \end{aligned}$$

Second, $\beta$ is chosen such that wealth is maximized given Eq. () and such that the resulting schedule is both informationally consistent and Pareto efficient, i.e. $V[P,\beta ]=V$ and $P=V$ when $\pi =\pi _{L}$. This gives:

$$\begin{aligned} \pi =\pi _{L}+\tau \left(\frac{V}{V_{\pi _{L}}(\beta )}-1\right) \end{aligned}$$

(18)

Solving the system of Eqs. (17) and (18) for $\beta$ and $P$, yields (5) and (6).

The second order condition is satisfied as it is given by:

$$\begin{aligned} 2(\pi _{L}-1)V^{\prime }_{\pi _{L}}(\beta )+(1-r-\beta +\beta \pi _{L}) V^{\prime \prime }_{\pi _{L}}(\beta ) \end{aligned}$$

Part 3. Engers (1987) has shown that a sufficient condition of uniqueness is that the marginal cost of the signal is decreasing and convex in the value of the firm. Since this is the case here, we have that the above set is the only separating equilibrium set.

The signalling schedule (5) and (6) is Pareto dominant among all the separating equilibria. When $\pi =\pi _{L}$, there is no underpricing and hence $P=V$ and the optimal schedule is given by (9). When $\pi >\pi _{L}$ the price, and hence the underpricing, is chosen such that it maximizes the objective function of the investor.

As $\beta >0$ this equilibrium exists only when condition (7) is satisfied. When $\beta \le 0$, the owner does not take the firm public. When $P=L$ a pooling equilibrium exists. Note that given the signalling schedule $\beta \le 1$ and $P\le V(\beta )$. $\square$

1.2 Proof of Corollary 1

Proof

The derivatives of Eqs. (5) and (6) with respect to $V$ and $\pi$, shows how $P$ and $\beta$ change with $V$ and $\pi$. $\square$

1.3 Proof of Proposition 2

Proof

As $\delta =\frac{V-P}{P}$

$$\begin{aligned} \frac{d\delta }{d\tau }=-\frac{V}{P^2}\frac{dP}{d\tau } \end{aligned}$$

where

$$\begin{aligned} \frac{dP}{d\tau }=\frac{dV_{\pi _{L}}(\beta )}{d\tau }\left(1+\frac{(1-\tau )(1-\beta )}{\beta \tau }\right)+ \left(V-V_{\pi _{L}}(\beta )\right)\frac{1-\beta }{\beta \pi } \end{aligned}$$

Hence, the derivative is negative if $\frac{\partial V_{\pi _{L}}(\beta )}{\partial \tau }>0$. This occurs when:

$$\begin{aligned} \pi _{L}+(\tau (\beta -1)+\beta (\pi _{L}-1)+1)\left(\ln \left(\frac{\tau (\beta -1)+\beta (\pi _{L}-1)+1}{\pi _{L}}\right)-1\right)>0 \end{aligned}$$

Define $B=(1-\tau )(1-\beta )+\pi _{L}\beta$. Hence the above inequality can be rewritten as:

$$\begin{aligned} \pi _{L}+B\ln B-B\ln \pi _{L}-B>0 \end{aligned}$$

Rearranging:

$$\begin{aligned} -B\left(\ln \frac{\pi _{L}}{B}\right)+\left(\pi _{L}-B\right)>0 \end{aligned}$$

Because $B=(1-\tau )(1-\beta )+\pi _{L}\beta <\pi _{L}$, the above inequality is always true. $\square$

1.4 Proof of Corollary 2

Proof

The derivative of the right hand side of condition (7) in respect to $\tau$ is positive. $\square$

1.5 Proof of Corollary 3

Proof

Direct consequence of the study of the first derivative of conditions (7) and (8). $\square$

1.6 Proof of Proposition 3

Proof

The proof is structured in the following way. I show (1)$\frac{d \beta }{d \pi _L}>0$; (2) $\frac{d V_{\pi _{L}}(\beta )}{d \pi _L}>0$; (3) $\frac{d \delta }{d \pi _L}>0$; (4) I analyze the effect of $\pi _l$ on conditions (7) and (8).

Part 1: $\frac{d \beta }{d \pi _L}<0$.

$$\begin{aligned} \frac{d \beta }{d \pi _L}=\frac{-1+\tau -\pi _L (\pi _L-\tau +1) z^{\prime }(\pi _L)+(\tau -1) z(\pi _L)}{(\pi _L-\tau +1)^2}<0 \end{aligned}$$

where

$$\begin{aligned} z(\pi _L)=h(\pi _{L})^k(\pi _{L}), \quad h(\pi _{L})=\frac{\tau }{\pi -\pi _{L}+\tau }\frac{V}{L} \quad k(\pi _{L})\frac{\pi _{L}-1+\tau }{\pi _{L}-1}{1-\tau +\pi _{L}} \end{aligned}$$

It can be easily seen that because $z^{\prime }(\pi _L)>0$, $\frac{d \beta }{d \pi _L}<0$.

Part 2: $\frac{d V_{\pi _{L}}(\beta )}{d \pi _L}>0$.

$$\begin{aligned} \frac{d V_{\pi _{L}}(\beta )}{d \pi _L}=\frac{\delta V_{\pi _{L}}(\beta )}{\delta \pi _L}+\frac{\delta V_{\pi _{L}}(\beta )}{\delta \beta }\frac{\delta \beta }{\delta \pi _L} \end{aligned}$$

From above I know that $\frac{\delta V_{\pi _{L}}(\beta )}{\delta \beta }\frac{\delta \beta }{<}0$ and $\frac{\delta \beta }{\delta \pi _L}>0$. I now look at $\frac{\delta V_{\pi _{L}}(\beta )}{\delta \pi _L}$:

$$\begin{aligned} \frac{\delta V_{\pi =\pi _{L}}(\beta )}{\delta \pi _L}=V_{\pi =\pi _{L}}(\beta )\left[\frac{g(\pi _L) }{f(\pi _L)}f^{\prime }(\pi _L)+\log (f(\pi _L)) g^{\prime }(\pi _L)\right] \end{aligned}$$

where

$$\begin{aligned} f(\pi _L)=\frac{\pi _{L}}{\left(1-\beta \right)\left(1-\tau \right)+\beta \pi _{L}}L\quad \text{ and} \quad g(\pi _L)=\frac{\pi _{L}-1}{\pi _{L}-1+\tau } \end{aligned}$$

It can be easily be seen that $V_{\pi =\pi _{L}}(\beta )>0$, $g(\pi _L)>0$, $f(\pi _L)>1$ and $g^{\prime }(\pi _L)>0$.

Part 3: $\frac{d \delta }{d \pi _L}<0$.

$$\begin{aligned} \frac{d \beta }{d \pi _L}=-\frac{V}{P^2} \frac{d P}{d \pi _L} \end{aligned}$$

where $P$ is given by Eq. (6) and

$$\begin{aligned} \frac{d P}{d \pi _L}= \frac{d P}{d V_{\pi _{L}}(\beta )}\frac{d V_{\pi _{L}}(\beta )}{d \pi _L} \end{aligned}$$

From Part 1 $\frac{d V_{\pi _{L}}(\beta )}{d \pi _L}$ and it can be easily seen that $\frac{d P}{d V_{\pi _{L}}(\beta )}>0$.

Part 4: The boundary conditions.

I first study condition (7). The right hand side of this condition can be written as:

$$\begin{aligned} z(\pi _L)f(\pi _l)^{g(\pi _L)} \end{aligned}$$

where

$$\begin{aligned} z(\pi _L)=\frac{\pi -\pi _L+\tau }{\tau }, \quad f(\pi _l)=\frac{1-\tau }{\pi _L} \quad \text{ and} \quad g(\pi _L)=\frac{\pi _L-1}{\pi _L-1+\tau }. \end{aligned}$$

$$\begin{aligned} \frac{dz(\pi _L)f(\pi _l)^{g(\pi _L)}}{d \pi _L}&= z(\pi _L) f(\pi _L)^{g(\pi _L)} \left(\frac{g(\pi _L) f^{\prime }(\pi _L)}{f(\pi _L)}+\log (f(\pi _L)) g^{\prime }(\pi _L)\right)\\&+f(\pi _L)^{g(\pi _L)} z^{\prime }(\pi _L) \end{aligned}$$

As $z(\pi _L)>0$, $0<f(\pi _L)<1$, $g(\pi _L)>0$, $z^{\prime }(\pi _L)<0$, $f^{\prime }(\pi _L)<0$ and $g^{\prime }(\pi _L)>0$, $\frac{dz(\pi _L)f(\pi _l)^{g(\pi _L)}}{d \pi _L}<0$.

I know study condition (8). Defining $\iota (\pi _L)$ the right hand side of the condition we have that:

$$\begin{aligned} \frac{d\iota (\pi _L)}{d \pi _L}=\frac{d\iota (\pi _L)}{d V_{\pi _{L}}(\beta )}\frac{V_{\pi _{L}}(\beta )}{d \pi _L}= -\frac{(1-\tau )(1-\beta )}{\beta }\frac{V-L}{(V_{\pi _{L}} (\beta )-L)^2}\frac{V_{\pi _{L}}(\beta )}{d \pi _L}<0 \end{aligned}$$

$\square$

1.7 Proof of Proposition 4

Proof

The proof follows the same steps as the proof of Proposition 1. $\square$