A+Theory+of+IPO+Underpricing%2c+Issue+Activity%2c+and+Long Run+

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A Theory of IPO Underpricing, Issue Activity, and

Long-Run Underperformance

Ralph Bachmann

Nanyang Business School

Division of Banking and Finance

Nanyang Avenue, S3-B1A-14

Singapore 639798

[email protected]

Phone: +65 6790 5000

Fax: +65 6794 6321

Comments welcome!

March 11, 2004

I wish to thank Vik Nanda, Jay Ritter, Masako Ueda, seminar participants at Nanyang Tech-

nological University, and participants of the 2003 Australasian Banking and Finance Conference

in Sydney for valuable comments.


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A Theory of IPO Underpricing, Issue Activity, and

Long-Run Underperformance

Abstract

We present a dynamic model of an IPO market in which firms go public to raise capital

for investment. The original shareholders have inside information with respect to the quality of

their firms investment opportunities, and they decide whether to go public, how much capital

to raise and invest, and how to price the IPO. Outside investors are of one of two types: someinvestors know and understand even the not directly observable economic incentives of the original

shareholders, whereas all other investors learn about the IPO market only from the publicly

observable IPO market data. Market participation by the latter investors can upset the stationary

rational equilibrium and give rise to a dynamic equilibrium that is characterized by: (1) IPO

underpricing, (2) underperformance of IPO shares in the long run, and (3) cyclical variations

in IPO volume. Learning investors upset the stationary equilibrium not because of ex ante

unreasonable beliefs, but because they fail to condition their beliefs on variables that convey only

redundant information in the stationary equilibrium. The model provides a joint explanation for

the three empirically documented IPO market anomalies, and it yields a number of new testable

empirical predictions.

JEL classification: D82, D83, G31, G32.


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I Introduction

Financial economists have documented at least three puzzling empirical regularities in the market

for initial public offerings of common stock: IPO underpricing, i.e. positive excess returns in the

short run; strong concentration of IPO activity in certain periods; and underperformance of IPO

shares in the long run.

The first empirical evidence on IPO underpricing dates back to a study by the U.S. Securi-

ties and Exchange Commission (SEC) in 1963. A large body of subsequent empirical research

has confirmed the finding that IPOs tend to be substantially underpriced in the U.S., as well as

internationally.1 Ibbotson and Jaffe (1975) present evidence of the existence of hot issue mar-

kets, which they define as periods during which the initial performance of IPOs is exceptionally

high. Moreover, they find evidence of a strong concentration of IPO activity in certain periods.

Loughran, Ritter, and Rydqvist (1994) report that similar patterns can be observed internation-

ally. Evidence of long-run underperformance of IPO shares was first presented by Ritter (1991),

who also finds that underperformance was most severe for firms that went public during certain

years of high IPO activity. Further evidence on IPO underperformance in the long run is presented

by Loughran and Ritter (1995).

Much of the theoretical research on IPOs has focused on explaining IPO underpricing. Possible

reasons for underpricing include self-interested investment bankers (Baron and Holmstrom (1980)

and Baron (1982)), the winners curse (Rock (1986)), lawsuit avoidance (Tinic (1988) and

Hughes and Thakor (1992)), signaling (Allen and Faulhaber (1989), Grinblatt and Hwang (1989),

and Welch (1989)), market incompleteness (Mauer and Senbet (1992)), bookbuilding (Benveniste

and Spindt (1989)), and informational cascades (Welch (1992)). Evidence suggests also that in

some countries IPO underpricing may be due to the regulatory environment (see Loughran, Ritter,

and Rydqvist (1994)), or because the allocation of IPO shares can be used as a bribe.

One possible explanation for the strong fluctuations in IPO volume is that the cost of issuing

1See Ibbotson and Ritter (1995) for a survey of the research on initial public o fferings. Loughran, Ritter, and

Rydqvist (1994) focus on the international evidence.

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equity varies across the business cycle. Choe, Masulis, and Nanda (1993) present a model in

which the business cycle causes adverse selection costs to vary over time. Adverse selection costs

are lower in periods of economic expansion which leads firms to issue equity primarily during

such times. Choe, Masulis, and Nanda also provide some evidence in support of their model.

Although their paper is about seasoned equity offerings, it is conceivable that varying degrees of

asymmetric information across the business cycle also affect the market for initial public offerings.

Chemmanur and Fulghieri (1999) point out another possible explanation for the strong clustering

of IPOs in certain periods, namely the existence of productivity shocks that are correlated across

firms. In Maug (2001) clustering of IPOs occurs because the first IPO(s) in a new industry

lead investors to developing a better understanding of this industry. This reduces their costs of

evaluating subsequent IPOs. Stoughton, Wong, and Zechner (2001) present a model in which the

market clearing price of an IPO conveys information with respect to the firms future prospects,

as well as with respect to the prospects of the entire industry. Clustering of IPOs can occur when

an IPO conveys very favorable information with respect to an industrys growth opportunities.

Lowry (2003) investigates empirically whether the fluctuations in IPO volume can be explained

by the aggregate capital demands of private firms, by the adverse selection costs of issuing equity,

or by the level of investor optimism. She finds that all three factors are statistically significant, but

that only the firms aggregate capital demand and investor sentiment appear to be economically

significant determinants of IPO volume. Lowry and Schwert (2002) report a significant positive

lead-lag relation between initial returns and future IPO volume. They provide evidence suggesting

that this lead-lag relationship between initial returns an IPO volume may be driven by information

that is learned by the underwriter during the registration period, but that is not fully incorporated

into the off

er price.If one is willing to accept the view that IPO shares tend to underperform the market in the long

run then rational explanations of this phenomenon appear hard to come by.2 Most of the literature

on this topic therefore empirical and points in the direction of limited investor rationality and

2Whether IPO shares really underperform the market in the long run has recently been subject to some debate.

See Brav and Gompers (1997), Eckbo and Norly (2000), and Loughran and Ritter (2000).

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psychological finance.3 However, one possible rational explanation is that IPO underperformance

is due to investor learning. Morris (1996) presents a model in which investors share a common

prior belief with respect to the value of IPO firms, but in which they have different prior beliefs

with respect to the distribution of the firm values. In this setting, speculative bubbles can

occur, in which IPOs are sold at a price that is above each individual investors valuation. In such

a bubble each investor anticipates to be able to sell his shares in the aftermarket to an investor

with a higher valuation. IPOs underperform as investors learn about the true distribution of the

firm values in the long run.

The to our knowledge only theory that links all three IPO market anomalies to a single cause is

due to Ljungqvist, Nanda, and Singh (2001). They attribute all three phenomena to the existence

of a class of investors who are, at times, irrationally exuberant about the prospects of IPOs. In

their model the selling policy that is in the issuers best interest usually involves staggered sales.

Underpricing happens to compensate the regular (institutional) investors for the risk of losing

money on their inventory in the event that the hot issue market ends prematurely.

We propose an alternative theory that is also capable of jointly explaining IPO underpricing,

fluctuations in IPO volume, and long-run underperformance of IPO shares. In our model firms

go public to raise capital for investment. IPO underpricing is a rational equilibrium outcome that

obtains as a result of the original shareholders personal wealth maximization under asymmetric

information. Periods of high IPO activity and long-run underperformance are attributable to the

presence of investors who learn about the IPO market based on publicly observable performance

of past IPOs.

The structure of our model is quite simple. We assume universal risk neutrality and model

IPOs as direct transactions between entrepreneurs and investors. Nature randomly creates in-vestment opportunities that are of one of two types (profitable or unprofitable). Each investment

opportunity is available to a single personal wealth maximizing entrepreneur who is privately in-

formed about its profitability. Undertaking the investment opportunity requires an IPO to raise

3Hirshleifer (2001) provides a comprehensive overview of the various psychological biases and their common root

causes.

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capital for investment. Depending on the terms at which equity can be sold the entrepreneurs

decide whether to go public, how much capital to raise and invest, and how to price an IPO.

Under plausible conditions some entrepreneurs find it optimal to augment the information that is

conveyed by their investment decisions to outsiders by underpricing their IPOs. In equilibrium,

the combination of IPO pricing and investment choice signals the quality of a firms investment

opportunities.

We then introduce a class of investors who learn about the IPO market only based on publicly

observable data, such as the performance of past IPOs. Intuitively, these investors are like sta-

tisticians who lack a structural model of the world, and who try to understand the IPO market

just from looking at the data. In the stationary equilibrium these investors can learn to infer the

value of an IPO from the issuing firms investment policy alone. The issue price of an IPO does

not convey any additional information.

Learning investors will eventually be able to infer which IPOs are underpriced, and also by

how much. However, while learning investors may eventually be able to forecast which IPOs are

underpriced, they cannot infer from the data why these IPOs are underpriced. It is consistent

with the beliefs of these investor to purchase any IPO that is issued at a price weakly below

its inferred expected value. This creates an opportunity for some issuing firms to sell equity at

a price closer to its fair value, but it also alters the incentives of those entrepreneurs that did

not previously find it profitable to go public. In our model even a slight increase in the price at

which firms can issue equity can alter the incentives of lower quality firms. This results in an

increase in IPO volume, and a sudden drop in the average IPO quality that leads to subsequent

IPO underperformance.4 After a period of high IPO activity the IPO market reverts back to

the rational equilibrium as long-run underperformance leads the learning investors to revise theirbeliefs downwards.

The main empirical implications of our model are as follows. (1) Consistent with the conven-

tional wisdom our model predicts that IPOs are, on average, underpriced, but that they under-

perform the market in the long run (if long run performance is measured based on the first market

4This suggests that underpricing may be a necessary condition for the absence of underperformance.

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interest rate is zero.

B Firms, Entrepreneurs, and IPOs

Nature creates investment opportunities in discrete time intervals, t > 0. At each point in

time, tj = t0 + jt, j {0, 1, 2, . . .}, nature creates a physical investment opportunity with

probability (0, 1). Any such investment opportunity is available only to a single entrepreneur.

An investment opportunity that opens up has to be undertaken immediately, or it ceases to exist.

We assume that firms are all-equity financed, and that additional investment can only be

financed by means of an initial public offering (IPO) of common stock. This could be, for example,

because entrepreneurs do not have any own funds to pay for the additional investment, asymmetric

information or agency problems rule out debt financing, and the wealth of each individual investor

is too small to allow a private equity placement.

The value of any firms assets in place prior to (or without) an IPO is equal to A > 0.5 An

investment opportunity is of one of two types that we refer to as type- G (good) and type-B

(bad) respectively.6 Firms differ only in terms of their investment opportunities so that we will

refer to a firm with access to a type-G (type-B) project as a type-G (type-B) firm and to its

original shareholder as a type-G (type-B) entrepreneur.

The (expected) marginal profit function of a type-T project is denoted by T(I). Marginal

profit functions are assumed to be continuously differentiable functions of investment with the

following properties:7

(A1) A type-G project yields higher marginal returns on investment than a type-B project,

5The assumption that firms do not differ in terms of the value of their assets in place is a simplifying assumption

that is not crucial for our results.6An extension of the model to more than two types is straightforward but tedious.

7None of these assumptions is crucial for the main results of the model. The assumptions (A1), (A2), and (A4),

are merely technical assumptions that allow us to keep the analysis simple. Even if we relaxed assumption (A3) by

assuming that both types have a positive NPV project we could obtain IPO underpricing, long-run underperfor-

mance, and fluctuations in the amount of equity that is issued. However, we could not obtain fluctuations in the

number of IPOs that take place.

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irrespective of the level of investment: B(I) < G(I) for all I 0.

(A2) Projects exhibit decreasing returns to scale: T(I) < 0 for all I > 0, T {G, B}.

(A3) Only type-G projects have a positive expected NPV: G(0) > 0, but B(0) 0.

(A4) Marginal gross returns on investment are non-negative. This means that marginal losses

from inefficient investment never exceed the additional investment outlay: T(I) > 1 for

all I > 0, T {G, B}.

The ex ante probability that an investment opportunity is type-G is equal to p (0, 1), and

the probability that it is type-B is equal to 1 p. The project type is initially private information

of the entrepreneur and it is not verifiable ex post.

We abstract from the existence of financial intermediaries and model IPOs as direct transac-

tions between entrepreneurs and investors. Our assumption of universal risk neutrality implies

that entrepreneurs have no incentive to sell equity for diversification purposes. The sole purpose

of the IPO in our model is to raise capital for investment, which means that the entire proceeds

from the issue will be invested in the firm.8 We assume that a firm without access to a physical

investment opportunity cannot go public.9 The value of a type-T firm that raises and invests I

dollars is equal to the expected value of its future cash flows:

VT(I) = A +

I0

1 + T(I) dI. (1)

Conditional on a physical investment opportunity being available, an entrepreneur can make

a take-it-or-leave-it offer I, [0, ) [0, 1] to outside investors, where I denotes the amount

of capital raised and invested, and denotes the fraction of equity retained by the entrepreneur.

The offer 0, 1 represents the special case that the entrepreneur does not raise any capital and

instead retains full ownership of her firm.

8We rule out the possibility that a firm invests the proceeds from the equity issue in marketable securities or

just holds cash. Moreover, the original shareholders informational advantage implies that outsiders would view

any attempt to sell more equity than necessary to finance the intended investment policy as a negative signal of

the firm quality.

9Hence, at most one IPO can take place at any given point in time.

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By assumption, an entrepreneur chooses the IPO that maximizes her expected terminal wealth.

As a tie breaker we assume that an entrepreneur who is indifferent between her first-best IPO and

some alternative IPO will stick to her first-best IPO. The total wealth of a type-T entrepreneur

who manages to sell an IPO I, is equal to the value of her retained equity,

WT(I,) = VT(I). (2)

An entrepreneurs payoff from unsuccessfully attempting an IPO I, is assumed to be equal to

zero. An entrepreneur whose IPO has failed cannot approach the market again.10

For any IPO, I,, we can calculate the firm valuation that is implicit in the terms of the

IPO as

Vi(I,) =I

1 . (3)

It is clear that Vi(I, ) need not be identical to the true firm value. If Vi(I, ) exceeds the true

firm value then the IPO is overpriced, and if Vi(I,) is smaller then the true firm value then the

IPO is underpriced. Outside investors just break even on purchasing (1 ) of a firms equity for

I dollars whenever the true firm value is identical to (3).

For each IPO I, we can now find an equivalent representation I, Vi(I, ), where Vi(I, )

denotes the firm value that is implicit in the terms of the IPO. For convenience we will refer to

Vi(I, ) in the following as the issue price.

Which of the two above IPO representations is going to be more convenient depends on

the questions that we will address. The representation I, captures the entrepreneurs trade-

off between raising more capital and retaining more equity. This representation will be very

intuitive when it comes to analyzing the decision problem of an entrepreneur. The representation

I, Vi(I,) emphasizes the link between investment policy and firm value. This representation

will be more intuitive when it comes to analyzing investor learning and the resulting IPO market

dynamics.

10Neither of these assumptions is crucial for our model. Our main results would still hold if an entrepreneur could

approach the market again after a failed first attempt to sell an IPO, or if we assumed a different (fixed) payoff

from a failed IPO.

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We assume that all uncertainty with respect to a firms cash flows is resolved exactly periods

after its IPO. At that time, the firm value is publicly revealed to be equal to VT(I) + T(I), where

T(I) is a random variable with zero expected mean.11

C Investors

Investors can either purchase stocks or hold cash, but they cannot short sell IPO shares.12 Whether

an investor decides to participate in a particular IPO depends on the terms of the IPO, and on

his belief with respect to the value of the offering firm, V.13 An investor will participate in anIPO whenever he expects to at least break even, that is, whenever

Vi(I, ) V . (4)A crucial assumption in our model is the existence of two types of investors that differ in terms

of their knowledge and their understanding of the IPO market.

C.1 R-Investors

The first group of investors, that we refer to as R-investors, are rational in the usual game

theoretic sense. These investors are assumed to have all the information about the firms with

the exception of the individual firm types. The R-investors can therefore infer each entrepreneur

types expected payoff from every feasible strategy, which means that they understand even the

not directly observable economic incentives of the entrepreneurs. They condition their beliefs with

respect to the firm value, VR, on the firms investment policy, as well as on the fraction of equityretained by the entrepreneur. Following (4), R-investors demand shares in an IPO whenever

Vi(I, ) VR(I, ). (5)11The only purpose of this stochastic component is to ensure that the firm type is not verifiable ex post. To

account for limited liability we have to assume that T(I) is bounded from below by VT(I).

12This assumption is necessary to obtain long-run underperformance of IPO shares in our model.

13It will be more convenient to specify the investors beliefs with respect to the firm value (V), rather than withrespect to the probabilities that they assign to the event that the firm is of the high type ().

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Note that R-investors may alternatively be viewed as conditioning their beliefs on I and

Vi(I, ), so that (5) may be stated equivalently as

Vi

(I, ) VR(I, Vi(I,)). (6)C.2 L-Investors

The second group of investors, that we refer to as L-investors, are not irrational, but they learn

about the IPO market only from the publicly observable performance of past IPOs. Intuitively,

an L-investor is like a statistician who lacks a structural model of the world, and who tries to

understand the world just from looking at the data.

We will see that L-investors may learn to predict the value of an IPO based on the issuing

firms investment policy alone. We therefore assume that L-investors condition their beliefs, VL,on the firms investment policy, and that they demand shares in an IPO as long as it is offered at

a price weakly below its inferred expected value,

Vi(I, ) VL(I). (7)L-investors are not ignoring how much of the firms equity is offered in exchange for the capital

that is raised in the IPO, but they differ from R-investors in that they view the choice of merely

as a pricing decision.

We will discuss the learning by L-investors in more detail in Subsection II.E, after the sequence

of events for an individual IPO has been specified.

C.3 IPO Market Regimes

We denote the number of R-investors and the number of L-investors in the economy by NR and

NL, respectively. It is crucial that we assume that NR and NL are sufficiently large so that each

group of investors by itself can purchase all IPOs in the economy. This assumption implies that

an entrepreneur may sell the IPO to the group of investors with the highest valuation for the

shares. Depending on whether IPOs are offered at a price (weakly) below or (strictly) above

the R-investors valuation we distinguish between two IPO market regimes that we refer to as

regime-R and regime-L, respectively.

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. . .tj + tj+1

. . .tj+

. . .

Vj,j Vj,j+1 Vj,j+

IPO takes place. The first market clearingprice is determined in pub-lic trading.

Nature reveals thetrue firm value.

Underpricing

Long run performance

Figure 1: Time line of events for an individual IPO

Definition 1 (Regime-R and Regime-L) At time-tj the IPO market is in regime-R if an

IPO is offered at a price weakly below the R-investors valuation. The market is in regime-L if

an IPO is offered at a price strictly above the R-investors valuation.

D Sequence of Events for an Individual IPO

Up to three events can take place in our model at any given point in time, tj: (1) there is a public

trading session in which the market clearing prices of all firms that previously issued equity are

determined, (2) if a firm issued equity at tj then the true value of this firms cash flows is

publicly revealed, and (3) a firm that is not yet publicly listed may undertake an IPO. To avoid

ambiguities we assume that these events take place in the aforementioned sequence. We may think

of the trading session, the revelation of the realized cash flows, and the IPO taking place at times

tj , tj , and tj + , respectively. This particular sequencing implies that no new information

becomes available between the time at which an IPO is sold to investors (tj + ), and the trading

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session in which its first market clearing price is determined (at tj+1 ). Moreover, the time-tj

market clearing prices of all publicly traded firms and the true fundamental value of any firm that

went public at tj are already common knowledge when the time-tj IPO is offered to the public.

The time line of events for an individual IPO firm is depicted in Figure 1.

Whenever we are primarily interested in the dynamicaspects of the IPO market we will denote

the time-tj+k price of the firm that went public at tj by Vj,j+k. For a successful IPO that takes

place at tj a time series offirm prices, Vj,j+k, k {0, 1, 2, 3, . . .}, is observed. The issue price, Vj,j ,

is set by the entrepreneur, and it is identical to (3). All subsequent prices are market clearing

prices that are determined by the investors with the highest valuation for the shares. At tj+ all

uncertainty with respect to the true firm value is resolved, and the true firm value is publicly

revealed. Hence, we have

Vj,j+k =

Vi(I,) for k = 0,

maxVL(I), VR(I,) for 1 k < , and

VT(I) + T(I) for k.

(8)

We measure the abnormal initial performance (i.e. underpricing) of an IPO by its stock price

performance from tj to tj+1. The long run performance of IPO shares is measured either based on

the offer price (i.e. from time tj to tj+), or based on the first market clearing price (i.e. from tj+1

to tj+), since our model yields separate predictions for the two time horizons.

E Learning by L-Investors

We denote the L-investors time-tj belief with respect to the value of a firm that raises and invests

I dollars by

VLj (I). No IPO took place before t0, so that

VL0 (I) denotes the L-investors prior

belief with respect to the value of a firm that raises and invests I dollars. We assume that this

prior is distributed according to some probability density function, g(), with compact support.

The probability density function is assumed to be common knowledge, but the prior itself is

private information of the L-investors. The (finite) weight that L-investors assign to their prior

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is denoted by 0(I).14 For convenience we assume that 0(I) is common knowledge.15

To formalize the updating of beliefs by L-investors we define the indicator functions

j(I) = 1 if an IPO of size I took place at t

j ,

0 otherwise,(9)

for each j {0, 1, 2, . . .}. In addition, we define

j(I) = j1(I) + j(I) (10)

= 0(I) +j10

j(I), (11)

which is the sum of the weight of the L-investors t0-prior, and of the number of IPOs of size I

that took place prior to tj.

L-investors update their beliefs whenever the first market clearing price for an IPO is observed,

and whenever new information on a publicly listed firm becomes available. By assumption, the

latter is the case at tj if and only if a firm went public at tj. The expected development of the

L-investors beliefs at a given point in time therefore depends not only on the current IPO market

regime, but also on the IPO market regime periods ago.

Assuming Bayesian updating, the L-investors updated time-tj belief may be expressed as

VLj (I) = j1(I)j(I) VLj1(I) +j1(I)

j(I) Vj1,j +

j(I)

j(I) (Vj,j Vj,j1) . (12)

The three terms in (12) represent the investors time-tj1 belief, the first market clearing price of

any time-tj1 IPO, and any change in the market value of a firm that issued equity at tj due

to the arrival of new public information. Note that (12) can alternatively be stated as

VLj (I) =0(I)

j(I) VL0 (I) +

j1

k=0k(I)

j(I) V

k,j, (13)

14A weight 0(I) = k means that the weight that type-L investors assign to their prior is equivalent to the weight

that they would assign to k independent observations.

15Assuming that only one of the parameters that determine the L-investors beliefs is private information implies

that all other agents in the economy fully understand the L-investors beliefs after observing only a single IPO in

regime-L. This limits the number of special cases that we have to consider (see Lemma 5) without affecting the

main insights that can be gained from our model.

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which reveals that the type-L investors time-tj belief with respect to the value of a firm that

raises and invests I dollars is just a weighted average of their time-t0 prior, and of the current

market values of all publicly listed firms that followed the same investment policy in the past.

III The Rational IPO Market Equilibrium in the Absence of

Learning: Regime-R

We begin our analysis by deriving the rational IPO market equilibrium that obtains in the absence

of learning. That is, we assume initially that the L-investors are not participating in the IPO

market. An in-depth analysis of this equilibrium can be found in a companion paper by the same

authors,16 so that we will skip most of the technical details and some of the proofs in this section

and instead focus mainly on the intuition.

A The First-Best Outcome

In the absence of informational asymmetries a type-T entrepreneur can raise I dollars by offering

1 F

T

(I) =I

VT(I)(14)

of her firms equity to outside investors. The IPO is priced correctly (i.e. Vi(I,FT(I)) = VT(I)),

so that outside investors just break even on purchasing the issue.

Since the equity is sold at the fair price, a type-T entrepreneur has no incentive to deviate

from her first-best investment policy, denoted by IFT . A type-B entrepreneur therefore does not

invest (IFB = 0), and a type-G entrepreneur undertakes the NPV-maximizing investment level

that is implicitly defined by G(IFG ) = 0.

The total payoff to a type-B entrepreneur is equal to WB(0, 1) = A, and the total payoff to a

type-G entrepreneur is equal to the sum of her project-NPV and the value of the assets in place:

WG(IFG ,

FG(I

FG)) = A +

IFG

0

T(I) dI. (15)

16Available from the authors, or from the Social Science Research Network at http://www.ssrn.com.

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Both the issue price and the first market clearing price are identical to the true (expected)

firm value. Hence, there is no underpricing, and no abnormal stock price p erformance is observed

in the long run.

B Equilibrium under Asymmetric Information

We assume that the first-best outcome is not feasible under asymmetric information. That is, we

assume that the bad types payoff from undertaking the good typess first-best offer exceeds the

bad types payoff from her own first-best strategy:

WB(IFG ,

FG(I

FG )) > WB(0, 1). (16)

The intuitive interpretation of (16) is that at the IPO IFG ,FG(I

FG) the bad types gain from

selling overvalued equity exceeds her imitation costs that arise from inefficient investment.

Whenever the first-best outcome is not feasible due to asymmetric information the good type

typically has an incentive to deviate from her first-best investment policy to signal her type.

Lemma 1 There exists an investment level I > IFG at which the good type could issue equity at

the fair price.

Lemma 1 states that, in principle, the good type could always distinguish herself from the bad

type by overinvesting to such an extend that it becomes too costly for the bad type to mimic.

However, doing so would not necessarily be optimal from the point of view of a personal wealth

maximizing entrepreneur: while the fact that the good type earns higher marginal returns on

investment than the bad type creates an incentive to signal with overinvestment, the necessity to

finance this investment with outside equity creates a counteracting incentive to underinvest.17

The direction in which the good type adjusts her investment policy can be determined by

looking at the two types respective marginal rates of substitution between the fraction of retained

17For all I > 0 the equity of a type-G firm is more valuable than that of a type-B firm since type-G firms earn

higher returns on investment than type-B firms. Under asymmetric information type-G firms are therefore faced

with higher marginal financing costs than type-B firms.

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equity, and the amount of capital that is raised and invested. From (2) we obtain

MRST(I,) =WT(I,)/I

WT(I, )/(17)

= V

T(I)VT(I)

. (18)

Based on (17) it can be shown that the good types second-best strategy entails undertaking

only those investments for which its marginal gross return, VG(I) = 1 + G(I), exceeds the bad

types marginal gross return, VB(I), by a factor greater than the ratio of the two types respective

marginal financing costs, VG(I)/VB(I).18 That is, the good type will undertake only investments

for which her marginal return on investment is sufficiently higher than the bad types to compen-

sate her for her relatively higher marginal cost of outside equity capital. However, the good typewill not increase investment beyond I since this investment level is already sufficiently informa-

tive to allow an equity issue at the fair price. The good-types second-best investment policy can

therefore be identified as the investment policy, I, that maximizes the ratio VG(I)/VB(I) on the

interval [0, I].19

There exists a (unique) pure-strategy equilibrium in regime-R. To be able to fully characterize

this equilibrium we need to calculate the fraction of equity that the entrepreneurs could retain

in a pooling equilibrium candidate IPO of size I, in which both firm types issue equity at the

average fair price,

P(I) = 1 I

(1 p)VB(I) +pVG(I), (19)

and the fraction of equity that a type-G entrepreneur could retain in an IPO of the same size that

is priced to prevent the bad type from mimicking,

SG(I) =A

VB(I). (20)

18The ratio of the firm values may be interpreted as the ratio of the two types respective marginal financing

costs since both types have to give up the same fraction of equity to raise another dollar of capital.

19Any equilibrium candidate in which the good type undertakes an investment policy I = I with strictly positive

probability fails the Intuitive Criterion of Cho and Kreps (1987): there exist an alternative IPO of size I that

makes the good type strictly better but the bad type strictly worse off than the IPO of size I. R-investors therefore

have to believe that a firm that offers the IPO of size I is of the good type, and the good type has no incentive to

undertake an IPO of size I in the first place.

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Proposition 1 (Equilibrium in Regime-R) Regime-R is characterized by the existence of a

unique equilibrium in pure strategies. Depending on the location of I on [0, I], and on the values

ofP(I) andSG(I

), this equilibrium is of the following type:20

a) If I = I then the equilibrium is a separating equilibrium in which type-G signals with

overinvestment and issues equity at the fair price, IeG,eG = I ,

FG(I), and in which the

bad type undertakes itsfirst-best IPO, IeB,eB = 0, 1.

b) If I < I and SG(I) < P(I

) then the equilibrium is a pooling equilibrium in which both

types raise and invest I, and in which equity is issued at the average value: IeG,eG =

IeB

,eB

= I,P(I).

c) IfI < I andSG(I) P(I

) then the equilibrium is a separating equilibrium in which the

bad type undertakes itsfirst-best IPO, IeB,eB = 0, 1, and in which the good type signals

with a combination of investment choice and IPO underpricing, IeG,eG = I

,SG(I).

The equilibrium can be supported by the R-investors beliefs that a firm that defects with an out-

of-equilibrium investment policy I = I is of the bad type with probability one, and that a firm

that defects with an out-of-equilibrium IPO I,, > eG

, is of the good type with probability

p, and of the bad type with probability 1 p. These beliefs are admissible under the Intuitive

Criterion of Cho and Kreps (1987).

The intuition underlying these different types of equilibria is as follows. Case a) corresponds

to a setting in which for any level of investment the good types marginal return on investment

is sufficiently higher than the bad types to ensure that good type always prefers a marginal

increase in equity financed investment over issuing equity below the fair price. The cases b) and

c) correspond to a setting in which this is only initially so. The latter cases are plausible since any

investment that the good type chooses to undertake increases the ratio of the firm values, and,

hence, increases the ratio of the two types respective marginal financing costs. Once I has been

reached the good types relative disadvantage on the financing side (vis-a-vis the bad type) begins

20The values of I, I, and SG(I), depend only on the two types respective marginal profit functions. The value

ofP(I) depends additionally on the probability distribution of the firm types.

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to outweigh its relative advantage on the investment side. At I the good type therefore prefers

issuing undervalued equity over changing the investment policy in either direction. Whether the

equilibrium in this case is a pooling equilibrium or a separating equilibrium in which the good

type issues deliberately underpriced equity depends only on the probability distribution of good

and bad firms in the economy. The pooling equilibrium exists only if the ex ante probability of

a firm being type-G is so high that P(I) > SG(I

). Otherwise the bad type prefers her own

first-best payoff over the payoff from the pooling equilibrium candidate IPO. In the latter case

the good type can retain SG(I) P(I

) of the equity without being mimicked by the bad type.

Only Proposition 1.c)s separating equilibrium with underpricing is of interest to us. We

therefore assume for the remainder of the paper that the model parameters are such that they

result in this type of equilibrium. Based on the equilibrium of Proposition 1.c) we obtain the

following properties of the IPO market in regime-R.

Proposition 2 (Issue Activity and IPO Performance in Regime-R) Suppose the equilib-

rium in regime-R is Proposition 1.c)s separating equilibrium with underpricing. Then the IPO

market is characterized by the following properties:

(1) The ex-ante probability that an IPO takes place at any given time, tj, j {0, 1, 2, . . .} is

equal to p.

(2) Any IPO that takes place is an issue I,SG(I) that is offered by a type-G firm. The

abnormal initial performance of IPOs from the issue price to the first market clearing price

(i.e. from tj to tj+1) is equal to

Vj,j+1

Vj,j 1 =

VG(I)

Vi

(I

,S

G(I

))

1 (21)

=FG(I

) SG(I)

1 FG(I)

> 0. (22)

(3) No abnormal stock price performance is observed after thefirst market clearing price. Hence,

there is no abnormal long run performance of IPO shares from tj+1 to tj+, but there is

positive abnormal long run performance from tj to tj+.

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The following subsection contains an example of Proposition 1.c)s IPO market equilibrium

with underpricing. We will extend this example in Subsection IV.C, where we present results

from a simulation of the IPO market dynamics.

C An Example

Suppose the value of any firms asset in place prior to (or without) the IPO is equal to VT(0) = 100,

T {G, B}. The ex-ante probability that a new project is of type-G is equal to p = 0.3, and the

two types marginal net profit functions are given by (23) and (24), respectively.21

G(I) = 2 I

50

(23)

B(I) = I

175(24)

In a world of symmetric information the good type undertakes her first-best IPO, IFG ,FG(I

FG) =

100, 23

, and the bad type does not go public IFB ,FB(I

FB ) = 0, 1. The corresponding first-best

payoffs to the entrepreneurs are equal to WG(100,2

3) = 200, and WB(0, 1) = 100.

The first-best outcome is not feasible under asymmetric information since the bad types

payoff from the good types first-best IPO exceeds her own first-best payoff: 23

VB(100) = 1142

7>

WB(0, 1) = 100. At the good types first-best investment level the good types marginal gross

return on investment exceeds the bad types by the factor (1 + G(100)) / (1 + B(100)) = 21

3,

which is greater than the ratio of the two types marginal financing costs, VG(100)/VB(100) =

1.75.22 Hence, the good type has an incentive to signal by increasing investment up to I = 124.5,

at which point

VG(I)

VB(I)=

1 + G(I)

1 + B(I)=

357

202 1.767.23 (25)

We obtain FG(I) = 77,599127,399 0.6091, SG(I) = 1,4002,523 0.4451, and P(I) = 129,599295,599 0.4384.

21Assuming linear marginal profit functions keeps the calculations tractable but also implies that we have to choose

marginal profit functions that violate the assumptions (A1) and (A4) for sufficiently high levels of investment. This

is not a problem, however, since neither assumption is necessary for the existence of the equilibrium of Proposition 1.

22VG(100) = 300, VB(100) = 1713

7, 1 + G(100) = 1, and 1 + B(100) =

3

7.

23VG(I) = 318 199

400 318.5, VB(I

) = 180 314 180.2, VP(I

) = 221 27974000

221.7, 1 + G(I) = 51

100= 0.51, and

1 + B(I) = 101

350 0.2886.

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This implies that the bad type would mimic an IPO in which the good type raises I dollars by

issuing equity at the fair price, but at the same time prefers her own fist-best strategy over the

pooling candidate IPO:

P(I) VB(I

) 79.01 < 100 < FG(I) VB(I

) 109.77. (26)

The pure-strategy equilibrium is consequently a separating equilibrium in which the bad type

does not go public, IFB ,FB(I

FB ) = 0, 1, and in which the good type undertakes the IPO

I,SG(I) = 124.5, 1,400

2,523, that is just sufficiently underpriced to prevent the bad type from

mimicking. Outside investors obtain 1 SG(I) of the good types equity in return for I dollars.

This translates into an issue price of Vi(I

,SG(I

)) 279.7, compared to a true firm value of

VG(I) 318.5. The good types IPO therefore experiences an abnormal initial return of +13.87%.

The equilibrium payoffto the good types original shareholders is equal to WG(I,SG(I

)) 176.7,

and the total amount of money left on the table is equal to

1 SG(I

)

VG(I) Vi(I,SG(I

))

17.3. (27)

Note that overinvesting up to I = 150 would enable the good type to issue equity at the

fair price: VB(I) = 1855

7, and VG(I) = 325, so that

FG(I) =

7

13, (1 FG(I))VG(I) = I, and

FG(I)VB(I) = 100. However, the good types payoff from doing so would only be equal to

WG(I ,SG(I)) = 175, which is less than its equilibrium payoff.

A graphic representation of the equilibrium is depicted in Figure 2. The three dotted curves

represent the (maximal) fraction of equity that the original shareholders can retain for a given

investment level for a good/bad/pooling IPO, so that outsiders still break even on purchasing

the offer. The solid black curve represents the bad types iso-payoff curve through her first-best

strategy, B = 0, 1. The solid grey curves are the good types iso-payoff curves through her

first-best IPO, G= IFG ,FG(I

FG), and through her equilibrium offer, U= I

,SG(I). At I the

good types iso-payoff curves are tangent to the bad types iso-payoff curves from above. Hence,

the good type prefers retaining less than the fair amount of equity over changing her investment

policy in either direction.

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I

B

G

F

U

P

1

0

FG(IFG)

FG(I)

SG(I)

P(I)

IFB = 0 IFG I

FB(I)

P(I)

FG(I)

Figure 2: The separating equilibrium of our example: type-G firms overinvest and issue under-

priced equity (U) and type-B firms do not go public (B).

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IV Learning by L-Investors, and Regime-L

We begin investigating what effects the presence of L-investors can have on the IPO market

equilibrium by studying how their beliefs evolve as they learn about the IPO market in regime-R.

A Learning by L-Investors in Regime-R

Suppose, without loss of generality, that the IPO market starts at t0 in regime-R. The L-investors

will then update their beliefs from t0 to t1 only based on the respective first market clearing

prices of any new IPOs. Beginning at t, L-investors will also update their beliefs whenever the

realized cash flows of a firm that went public in the past are observed.

The only IPOs that take place in regime-R are IPOs I,SG(I) that are undertaken by

good firms. Based on the first market clearing price, the market value of any such firm is equal

to VG(I). Thereafter, the market value does not change until the uncertainty with respect to

the firms cash flows is resolved. Firms that go public in regime-R are valued correctly by the

market, so that the new information that is contained in the realized cash flows is just noise

with an expected value of zero. If the IPO market was exogenously fixed in regime-R it would

therefore only be a matter of time until the L-investors learn to predict the value of IPOfi

rms

with arbitrary precision.

Lemma 2 (Learning by L-Investors in Regime-R) Suppose the IPO market is exogenously

fixed in regime-R. Then the L-investors belief with respect to the value of a firm that goes public

converges to the correct value:

VLj (I)VG(I). (28)Lemma 2 could be generalized in a straightforward manner to a setting in which two or more

firm types issue equity in regime-R. In the latter setting an equilibrium investment policy would

either be undertaken by only a single firm type, in which case Lemma 2 applies accordingly, or it

would be undertaken by two or more firm types that pool and issue equity at the identical terms,

in which case L-investors would learn to forecast the average value of these firms. In any case,

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L-investors would eventually be able to predict the value of a firm that goes public in regime-R

with the same accuracy as the R-investors.

Now recall that VG(I) > Vi(I,SG(I

)), so that Lemma 2 implies that the L-investors

valuation for IPO shares will sooner or later exceed the issue price. By assumption, the L-

investors demand shares whenever they expect to at least break even on purchasing an IPO. It is

therefore only a matter of time until the L-investors begin to participate in the IPO market.

Lemma 3 (Demand for IPO Shares) The L-investors demand IPO shares as soon as their

beliefs with respect to the value of a firm that raises and invests I exceeds Vi(I,SG(I)). At that

time the total demand for shares of IPOs I, Vi(I,SG(I)) increases from NR to NR + NL.

The L-investors participation in the IPO market is publicly observable due to the publicly

observable increase in the demand for IPO shares. While there was already excess demand for

underpriced IPOs when L-investors were absent, excess demand is even more severe now. The

increase in the demand for IPO shares reveals to the entrepreneurs that it may now be possible

to issue equity at a higher price.

Lemma 4 (Expected Payoff from Raising the Issue Price) Suppose that the IPO market is

in regime-R, and suppose that the L-investors are demanding shares of IPOs I, Vi(I,SG(I)).

Let V be such that Vi(I,SG(I)) < V < VG(I

). As time passes, the probability that an IPO

I, V would succeed converges to one. The good types expected payoff from offering I, V

converges to the l.h.s. of (29), and the bad types expected payoff from offering I, V converges

to the l.h.s. of (30).

WG

(I, 1 I/V) = VG

(I) VG(I

)

VI > W

G(I,S

G(I)) (29)

WB(I, 1 I/V) = VB(I

) VB(I

)

VI > WB(0, 1) (30)

It follows from Lemma 4 that it is only a matter of time until an entrepreneur will attempt

to sell an IPO of size I at a price V > Vi(I,SG(I)). The first such IPO(s) may fail since the

entrepreneurs do not know the L-investors t0-prior, so that they are uncertain with respect to

the L-investors exact willingness to pay. An IPO I, V that fails reveals that the L-investors

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valuation for an IPO of size I is less that V, but it does not lead to any updating of the L-

investors beliefs. The first successful IPO I, V with V > V i(I,SG(I)) marks the beginning

of regime-L.

B Regime-L

Lemma 5 (The First Successful IPO in Regime-L) Suppose the first IPO I, V that is

successfully offered at a price V > Vi(I,SG(I)) takes place at tk. Then this IPOsfirst market

clearing price, Vk,k+1, is equal to VLk (I) V, and it reveals the L-investors t0-prior, VL0 (I).The abnormal initial return of the IPO is non-negative, but strictly lower than the abnormal initial

return in regime-R:Vk,k+1

Vk,k 1 =

VLk (I) VV

0. (31)

The first market clearing price of any IPO that takes place in regime-L is equal to the L-

investors valuation for the shares. Once the market has entered regime-L the market clearing

prices of any new IPOs will therefore just confirm the L-investors beliefs. Moreover, since the

IPOs that previously took place in regime-R were valued correctly by the market no new in-

formation is expected to be incorporated in the L-investors beliefs for the first 1 periods.

Regime-L will consequently persist for a number of periods. The following proposition states the

entrepreneurs equilibrium strategies after the first successful IPO in regime-L.

Proposition 3 (Equilibrium in Regime-L) Suppose VLj (I) > V i(I,SG(I)) at tj > tk. Anentrepreneur with access to an investment opportunity will then successfully undertake the IPO

I, VLj (I), irrespective of her type. The IPO is undervalued if the issuingfirm is type-G, butovervalued if the issuingfirm is type-B.

Based on Proposition 3 it is straightforward to derive the following properties of the IPO

market in regime-L.

Proposition 4 (Issue Activity and IPO Performance in Regime-L) The IPO market

exhibits the following characteristics in regime-L:

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(1) The ex-ante probability that an IPO takes place at any given point in time, tj, is equal to .

(2) Any IPO that takes place is an issue I, VLj (I), where VLj (I) > Vi(I,SG(I)). Theissuingfirm is of type-G with probability p, and of type-B with probability 1 p. Thefirst

market clearing price of each IPO is equal to its issue price. Hence, there is no abnormal

initial performance (i.e. no underpricing) from tj to tj+1.

(3) The expected value of a firm that goes public is equal to VP(I) < Vi(I,SG(I

)). The

expected abnormal long run performance (from tj to tj+, or from tj+1 to tj+) is negative:

Vj,j+Vj,j

1 =Vj,j+Vj,j+1

1 =VP(I

)

VLj (I

) 1 < 0. (32)

A comparison of the IPO market characteristics in the two regimes reveals the following key

differences:

(1) The expected IPO volume in regime-L exceeds the expected IPO volume in regime-R by

the factor 1/p > 1.

(2) Issue prices are higher but the (first) market clearing prices are lower in regime-L than in

regime-R. Consequently there is less IPO underpricing in regime-L than in regime-R.

(3) On average, IPOs that take place in regime-L underperform in the long run. Based on

the first market clearing price there is no abnormal performance of IPOs that take place in

regime-R.

A remarkable feature of our model is that IPOs have to be underpriced to avoid under-

performance long run: just a small increase in the issue price changes the incentives of some

entrepreneurs and results in a (potentially dramatic) decline in the average IPO quality. It is thispoor performance of IPOs in the long run that eventually puts an end to regime- L.

Lemma 6 (Learning by L-Investors in Regime-L) Suppose the IPO market is exogenously

fixed in regime-L. Then the L-investors beliefs with respect to the value of an IPO firm converge

to the correct value:

VLj (I)VP(I). (33)

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If regime-L persisted for long enough then the L-investors would again learn to predict the

value of IPO firms with the same precision as the R-investors. However, the IPO market will

revert to regime-R as soon as the L-investors willingness to pay for IPOs of size I drops below

Vi(I,SG(I)). At that time entrepreneurs with access to bad projects find that it is no longer

profitable to issue equity, and there is a sudden decline in IPO volume that is accompanied by an

increase in IPO quality.

Proposition 5 (Lack of Convergence) The IPO market does not converge to the stationary

equilibrium that would obtain in the absence of L-investors. The IPO market is therefore charac-

terized by cyclical fluctuations in the IPO volume, in the degree of underpricing, and in the long

run performance of IPO shares.

That the IPO market in our model does not converge to the stationary (regime-R) equilibrium

may surprise somewhat. The reason for this lack of convergence lies in our assumptions that the

L-investors do not condition their beliefs with respect to the value of an IPO firm on the issue

price, and that they are willing to pay up to their full valuation for the shares of a new IPO.

However, for a number of reasons that are outside our model, we may not observe cyclical

variations in IPO volume within a single industry in the real world. One possibility is that investors

who do not understand the economic incentives of the entrepreneurs refrain from participating

in the IPO market after having lost a sufficient amount of money. Another possibility is that

investors may eventually learn that they have to condition their beliefs with respect to the value

of an IPO firm also on the issue price. In either case, no new period of high IPO volume will

be observed before a new generation of investors has entered the market. Finally, we may not

observe more than one period of high IPO activity in any single industry since the nature of the

physical investment opportunities in the economy changes over time.

C An Example (contd)

We now present the results from a simulation of the IPO market dynamics that was based on our

example from Subsection III.C. With respect to the parameters that have not been specified earlier

we make the following assumptions: (1) The probability that a new project becomes available at

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any given point in time is equal to = 0.5. (2) The L-investors t0-prior with respect to the value

of a firm that raises and invests I is equal to VL0 (I) = I. The weight that the investors assignto this prior is equal to 0(I

) = 30. (3) The probability distribution of the L-investors prior is

degenerate, with a single mass point at I. This implies that the L-investors beliefs are common

knowledge already at t0. The IPO market therefore enters regime-L for the first time as soon

as VLj (I) exceeds Vi(I,SG(I)). (4) The true cash flow of a firm is revealed exactly = 240periods after its IPO. (5) Finally, we assume that the probability distribution of the stochastic

component of a type-T firms cash flows (T(I)) is degenerate with a single mass point at zero.

This implies that a firms realized cash flows are always equal to its ex ante expected cash flows.

We simulate the evolution of the IPO market over a total of T = 5, 000 points in time. It

may be intuitive to view each point in time as a business day, a 5-day period as a week, a

4-week period as a month, and a 12-month period as a year. With this interpretation, the

true cash flow of each firm is revealed exactly one year after its IPO, and the simulation period

corresponds to a time span of 20 years and 8 months. The probability that a good investment

opportunity opens up on any given day is equal to p = 0.15, and the probability that a bad

investment opportunity opens up is equal to (1 p) = 0.35.

Our simulation resulted in the creation of 2, 541 investment opportunities, 758 of which were

of the good type, and 1, 783 of which were of the bad type. A total of 1, 017 IPOs took place,

which translates into an overall average of approximately one IPO per week.

We report the evolution ofL-investors beliefs (Figure 3), the monthly IPO volume (Figure 4),

the IPO prices and values (Figure 5), and the underpricing and the long run performance of IPOs

(Figure 6). While we have depicted the evolution of the L-investors beliefs and the monthly

IPO volume for the entire simulation period, we show the issue prices and values and the IPOperformance only for a 500-day window (from t751 to t1,250) to improve the clarity of the figures.

The IPO market starts at t0 by assumption in regime-R. The first market clearing price of

any firm that raises I = 124.5 dollars in regime-R is equal to VG(I) 318.5, so that initially

each IPO that takes place leads the L-investors to revise their beliefs upwards (Figure 3). As

soon as the L-investors beliefs exceed Vi(I,SG(I)) 279.7, firms with access to an investment

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VLj (I)

tj

Vi(I,SG(I))

279.7

240t1,000 t2,000 t3,000 t4,000 t5,000

Figure 3: The evolution of the L-investors beliefs over T = 5, 000 periods.

Number

of IPOs

tj

20= 10

20p= 3

0

Figure 4: The number of IPOs per month (i.e. per 20-day period).

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IPO Price,

IPO Value

tj

VB(I)

180.2

VP(I)

221.7

Vi(I, SG(I))

279.7

VG(I)

318.5

0

t751 t1,250

Regime-R Regime-L Regime-R

IPO Value

IPO Price

Expected IPO Value

Figure 5: Issue prices, values, and expected values for the 500-day period from t751 to t1,250.

IPO Underpricing

Long Run Performance

tj

tj

t751 t1,250

+13.9%

0

+13.9%

0

20.7%

35.6%

Regime-R Regime-L Regime-R

Figure 6: IPO underpricing and long run performance for the 500-day period from t751 to t1,250.

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opportunity are able to issue equity at a higher price. In our simulation the IPO market enters

regime-L for the first time at t800. The regime shift is accompanied by a slight increase in the

issue price from 279.71 to 279.96. Even though this price increase is too small to be noticeable in

Figure 5, it implies that IPOs are now no longer significantly underpriced, but rather significantly

overpriced (Figure 5). The reason for this is that firms with access to a bad project now also

find it profitable to go public. The fundamental value of a bad firm that raises and invests I

is only VB(I) 180.2, so that average IPO value drops from VG(I

) 318.5 in regime-R to

VP(I) 221.7 in regime-L. However, the true fundamental firm value is not revealed until some

time after the IPO, so that the drop in the average IPO quality is not immediately noticed by the

L-investors. The market clearing prices of all IPOs that take place in regime-L are identical to

the L-investors beliefs. The L-investors beliefs therefore do not change until the first cash flows

offirms that went public in regime-L are observed (Figure 3). The cash flow realizations of the

good firms that went public in regime-L will be unexpectedly high, and the cash flow realizations

of the bad firms will be unexpectedly low. Based on the first market clearing price, the expected

long run performance of good and bad firms that go public in regime-L is given by (34) and (35),

respectively.

VG(I)VL800(I) 1 =

318.5

279.96 1 = + 13.76% (34)

VB(I)VL800(I) 1 =

180.2

279.96 1 = 35.63%. (35)

The overall expected long run performance of IPOs that take place in regime-L is equal to 20.8%

(Figure 6), which leads L-investors to revise their beliefs downwards (Figure 3). The IPO mar-

ket reverts back to regime-R as soon as the L-investors valuation for IPO shares drops below

Vi(I,SG(I)).

The plot of the L-investors beliefs over time (Figure 3) reveals that the IPO market enters

regime-L at three distinct occasions, tj , j {800; 2, 294; 3, 831}, and reverts back to regime-R at

times tj, j {1, 043; 2, 536; 4, 074}. As we can see, a total of 1, 017 IPOs over a period of 5, 000

point in time does not result in convergence to a stationary equilibrium, even though there are

only two types offirms in our model.

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V Implications

Our model applies only to firms that go public to raise capital for investment. Moreover, it

is crucial in our theory that the different firm types are outwardly identical so that they are

indistinguishable to outside investors. Our model should therefore not be viewed as a model of

the IPO market as a whole, but rather as a model of the IPO market in a particular industry.

The following empirical predictions follow readily from our results:

Implication 1 An increase in the IPO volume in an industry is observed after a period during

whichfirms in this industry issued underpriced equity to raise capital for investment.

Implication 2 The prices at which IPOs are issued are higher during periods of high IPO volume

than during periods of low IPO volume. Thefirst market clearing prices of IPOs are lower during

periods of high IPO volume than during periods of low IPO volume. An increase in IPO volume

is accompanied by a reduction in underpricing.

Implication 3 Based on the first market clearing price, shares of IPOs that take place during

periods of low IPO volume show no systematic abnormal long run performance. Shares of IPOs

that take place during periods of high IPO volume underperform in the long run. Based on the

first market clearing price, the overall average long run performance of IPO shares is negative.

Note that our model does not yield any clear predictions with respect to the overall average

long run performance of IPOs if the latter is measured based on the issue price, rather than on

the first market clearing price. However, we obtain unambiguous predictions with respect to the

long run performance of IPOs in each of the two regimes.

Implication 4 The abnormal long run performance of IPOs that take place during periods of

high IPO volume is negative, even if long run performance is measured based on the issue price.

Based on the issue price the abnormal long run performance of IPOs that take place during periods

of low IPO volume is positive.

Implication 5 The variance of long run performance is higher amongfirms that go public during

periods of high IPO volume than forfirms that go public during periods with low IPO volume.

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Implication 6 After a period of high IPO volume issue activity decreases as a result of poor long

run performance.

Finally, our model also suggest that IPOs in periods of high and low IPO volume are bought

by different groups of investors. Our model predicts that IPOs that take place during periods of

high IPO volume are primarily purchased by investors who lack a deeper understanding of the

IPO market, and who invest in the IPOs only because they previously observed IPO underpricing.

However, this prediction may be difficult to test.

VI Conclusion

We have presented a stylized model of an IPO market in which firms go public to raise capital

for investment. The model is capable of jointly explaining three ma jor IPO market anomalies

that have been empirically documented in the literature: IPO underpricing, underperformance of

IPO shares in the long run, and strong concentration of issue activity in certain periods. In our

model, underpricing is observed even in the rational IPO market equilibrium. Some firms will

optimally use underpricing to augment the information that is conveyed by the firms investment

decisions to outside investors. This enables the firm to signal the profitability of its investment

opportunities to outsiders, and thereby maximizes the original shareholders personal wealth.

Long-run underperformance and periods of increased IPO volume arise from the presence of

investors who learn about the IPO market from publicly observable IPO market data. The latter

investors upset the rational equilibrium not because of ex ante unreasonable beliefs, but rather

because they learn to forecast the value of IPO firms based on the firms investment policy alone,

and because they are willing to purchase any IPO that is off

ered at a price below its inferredvalue.

While long-run underperformance may otherwise appear hard to reconcile with the abnor-

mal positive returns (underpricing) of IPOs in the short-run, the joint existence of these two

phenomena may be explained with relative ease if underpricing is indeed used to signal the prof-

itability of a firms investment opportunities. Even a minor increase in the price at which firms

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can issue equity is sufficient to alter the incentives offirms with lower quality investment oppor-

tunities. This results in a sudden increase in IPO activity that is accompanied by a significant

drop in the average IPO quality that leads to subsequent IPO underperformance.

A Mathematical Appendix

Proof of Lemma 1: The good type earns higher marginal returns on investment than the bad

type. Moreover, for I > IFG additional investment is unprofitable for either type. Hence, there

exists an investment level I > IFG that the good type is still able to finance, but at which the bad

type no longer prefers the payoff from the good types correctly priced IPO over her own first-best

payoff.

Proof of Proposition 1: It is straightforward to verify for each part of Proposition 1 that the

proposed pure strategy combination represents a perfect Bayesian equilibrium. We omit this part

of the proof an show only that the proposed equilibrium is the unique pure strategy equilibrium

that passes the Intuitive Criterion (IC).

By construction of the good types equilibrium offer there exists no out-of-equilibrium IPO

(I

,

), I

= I

, that would make the good type better off than its proposed equilibrium offer,

but that is not also strictly preferred by the bad type over its own equilibrium offer. The bad

type therefore cannot be ruled out a defector, and the belief that a firm that defects with (I,)

is of the bad type with probability one is admissible under the IC.

Now consider an equilibrium in which the good type undertakes some IPO (I,), with I = I.

No such equilibrium can pass the IC since we can construct an alternative IPO of size I that

investors would be willing to purchase if it was offered by the good type, and that makes the good

type strictly better off but the bad type strictly worse off than (I,). Hence, the bad type can

be ruled out as a defector for the IPO of size I, and the good type has no incentive to stick to

its proposed equilibrium strategy in the first place.

We have now established that the good type must raise and invest I in equilibrium. In each

case of Proposition 1 it is easy to see that the good type cannot raise the issue price, relative to

its proposed equilibrium IPO. Moreover, the bad type cannot make itself better off by defecting

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from its first-best strategy. This concludes the proof.

Proof of Proposition 2: Proposition 2 follows jointly from our assumptions and Proposition 1.

Proof of Lemma 2: For each point in time tk in regime-R we have either k(I) = 0, or

k(I) = 1 and

Vk,j =

VG(I) if j < k < j,

VG(I) + k if 0 k j .

(36)

Consequently, we can rewrite (13) as

VLj (I) = 0(I)j(I) VL0 (I) +j1k=0

k(I)

j(I) VG(I

) +jk=0

k(I)

j(I) k,j . (37)

As j the first and the third term in (37) converge to zero, and the second term converges

to VG(I). Hence, we obtain (28).

Proof of Lemma 3: Lemma 3 follows directly from our assumptions.

Proof of Lemma 4: The probability that an IPO I, V with Vi(I,SG(I)) < V < VG(I

)

succeeds is equal to the probability that the L-investors valuation for an IPOs of size I exceeds

V. The latter probability converges to one since VLj (I) converges to VG(I) > V for j byLemma 6. The results regarding the expected payoffs from undertaking the offer I, V follow

immediately.

Proof of Lemma 5: The L-investors valuation for the IPO I, V exceeds the R-investors

valuation for this IPO. No new information becomes available between the time of the IPO and

the time at which its first market clearing price is determined. The first market clearing price is

therefore equal to the L-investors valuation, VLk (I). The L-investors t0-prior can be inferredusing (13) based on this market clearing price and on the performance of past IPOs. Finally, that

the abnormal initial return of this IPO is smaller than the abnormal initial return in regime- R

follows from that facts that VLk (I) < VG(I), and V > Vi(I,SG(I)).Proof of Proposition 3: The proof is straightforward and therefore omitted.

Proof of Proposition 4: Proposition 4 follows directly from Proposition 3.

Proof of Lemma 6: The proof of Lemma 6 is analogous to the proof of Lemma 2.

Proof of Proposition 5: This proof is done by contradiction. Suppose the economy reaches

the stationary equilibrium of Proposition 1.c). Then it follows from Lemma 2, Lemma 3, and

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Lemma 4 that sooner or later a firm will successfully attempt an IPO at a higher price.

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