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Finance and Economics Discussion Series: 2006-12 Screen Reader version*

A Fully-Rational Liquidity-Based Theory of IPO Underpricing and Underperformance

Matthew Pritsker¹

First version: September 9, 2004

This version: March 24, 2006

Keywords: Liquidity, IPO, Asset Pricing, Market Microstructure

Abstract:

I present a fully-rational symmetric-information model of an IPO, and a dynamic imperfectly competitive model of trading in the IPO aftermarket. The model helps to explain IPO underpricing, underperformance, and why share allocations favor large institutional investors. In the model, underwriters need to sell a fixed number of shares at the IPO or in the aftermarket. To maximize revenue and avoid selling into the aftermarket where they can be exploited by large investors, underwriters distort share allocations towards investors with market power, and set the IPO offer price below the aftermarket trading price. Large investors who receive IPO share allocations sell them slowly afterwards to reduce their trade's price-impact. This curtails the shares that are available to small price-taking investors, causing them to bid up prices and bid down returns. In some simulations, the distorted share allocations and slow unwinding behavior generate post-IPO return underperformance that persists for several years.

1 Introduction

Two of the important functions of a financial system are to facilitate risk sharing among investors and capital formation by firms. The initial public offering (IPO) process performs both functions by allowing the initial owners of a firm to raise capital by transferring and sharing some of the firm's risk with the wider investing public. If the IPO process was fully efficient, an IPO should maximize the issuer's proceeds, the investors who most value the shares should receive them, and in the absence of news or private information, there should be little trade after the shares are allocated. Additionally, the fact that a stock is a new issue should not influence its risk-adjusted expected returns in aftermarket trading.

Relative to this benchmark, U.S. IPOs appear to be highly inefficient: post-IPO share trading is initially very heavy², and the allocation price of U.S. IPOs is on average nearly 19 percent below the closing price on the first day of trade [Ritter and Welch (2002)]. This underpricing is an apparent loss to issuers who would prefer to have sold at the higher price in the IPO aftermarket. The IPO process has other inefficiencies: allocations tends to favor institutional investors³ and, after the first trading day, the returns of new issues underperform on a market and characteristic adjusted basis for a period of time as long as three years [Loughran and Ritter (1991), Ritter and Welch (2002)].⁴

This paper presents a fully rational, symmetric information, theoretical model of IPO share allocation and price-setting, and of trading in the IPO aftermarket. The paper is built around the idea that trading conditions in the aftermarket may simultaneously explain underpricing, underperformance, and why allocations favor institutional investors. The model of the aftermarket is imperfectly competitive in the sense that there are some "large" investors who have market power, that is their trades move prices and they account for this when trading. The IPO is modeled as a bargaining game between the underwriter and the aftermarket investors: The underwriter must sell a fixed number of shares at the IPO or shortly afterwards in aftermarket trading. To do so, he sets a uniform IPO offer price and offers take it or leave it share allocations to the investors. Any shares that go unallocated are sold by the underwriter in aftermarket trading that follows the IPO. Large investors' market power in the aftermarket gives them bargaining power at the IPO because they can turn down their share allocation and force the underwriter to sell into the aftermarket, where large investors can influence (and lower) the price. To avoid this outcome, the underwriter distorts IPO asset allocations towards investors with market power, and gives them a favorable IPO offer price.⁵

From the initial asset allocations at the IPO, investors trade towards efficient asset allocations along an equilibrium transition path. Because large investors' trades move prices, the market is not perfectly liquid from their perspective, and this illiquidity influences returns and asset prices. In particular, when initial asset allocations favor large investors, along the equilibrium transition path large investors sell slowly through time to minimize the price impact of their trades. This restricts the supply of shares that is available to price-taking small investors. As a result small investors bid up the new issue's price, and because they don't expect to acquire many assets in the short-term, they bid down its expected returns, causing return underperformance. The magnitude and duration of the underperformance depends on the severity of illiquidity in the aftermarket, and on how the assets are allocated at the IPO.

There is a voluminous related literature on IPO underpricing, underperformance, and share allocations at the IPO.⁶ Most of the theoretical models in this literature explain only one or two of these phenomena. Only behavioral models can explain all three.⁷ Relative to this literature, this paper makes three contributions. First, to the best of my knowledge, it presents the only fully-rational theoretical model to date that can explain all three phenonomena. Second, the model shows that these phenomena are related to the competitiveness of the aftermarket, which in turn can be related to the distribution of the size of the investors that participate in the IPO or trade in the aftermarket (as measured by their wealth, or assets under management). Therefore, summary statistics of the distribution of investor size constitute a new set of state variables that can be used to test theories of IPOs, and to distinguish the predictions of this model from the related literature. Finally, this paper helps to fill a gap in the small literature on IPOs and liquidity. Within that literature, Ellul and Pagano (2003) present a theoretical model in which markets are illiquid, but competitive. Within their framework, they show that underpricing is required to compensate IPO participants for aftermarket illiquidity; they also find empirical support for the theory because aftermarket illiquidity has a large and positive correlation with more underpricing.⁸ Extending Ellul and Pagano's logic suggests IPOs should also earn a positive liquidity premium in aftermarket trading, but this is belied by the fact that IPO's underperform, and suggests that the Ellul and Pagano model cannot on its own explain both underpricing and underperformance. By contrast, this paper shows that a liquidity-based model can explain underpricing and underperformance provided one drops the perfect competition assumption in Ellul and Pagano and replaces it with the imperfect competition assumption that is used here.⁹

The rest of the paper proceeds in six parts. Section 2 provides a model overview; section 3 provides details on aftermarket trading; section 4 describes the process for share allocation and price setting at the IPO. Section 5 studies underpricing and underperformance using simulations; section 6 discusses the empirical implications of the model and provides a brief review of the most closely related empirical literature; a final section concludes.

2 Model Overview

The basic model is a stylized IPO in which a firm that wishes to raise capital by selling $X^{IPO}$ shares of stock enlists a single underwriting firm to market the issue. To abstract from agency issues, the underwriter is assumed to act on behalf of the issuer. The underwriter sells the issue to an investor base that consists of risk-averse investors who participate in the IPO and trade in the aftermarket. Investor 1 represents a continuum of small investors who each take prices as given. Investors through are large investors whose desired aftermarket trades are large enough to move asset prices. Because of differences in their size, the small investors can be viewed as representing the demands of retail investors, while the large investors represent the demands of institutional investors. The process for setting the IPO offer price and share allocations is modeled as a two-stage game. In the first stage, the underwriter assesses the demand for the new issue by learning about the characteristics of the investor base, and about aftermarket trading conditions. Based on his information, the underwriter sets a uniform IPO offer price and offers take-it or leave-it share allocations to the investors.¹⁰ In the second stage, investors decide whether to accept their allocations. Any shares that are turned down by investors at the IPO are sold by the underwriter in the aftermarket.

In much of the theoretical IPO literature, the primary explanation for underpricing is that it represents equilibrium compensation for various types of information asymmetries in the IPO process.¹¹ To establish that the channels for underpricing and underperformance in the present model do not rely on informational differences, asymmetric information is ruled out by assumption. More specifically, I assume that information on investors' risk preferences, asset holdings, all knowledge of asset values, and the entire model of aftermarket trading is publicly available at all points of time and is common knowledge. The next section formally models the IPO aftermarket; and the following section models the share allocation and price-setting process at the IPO.

3 The IPO aftermarket

The framework for aftermarket trading is a partial equilibrium extension of Pritsker's (2004) model of imperfect competition in asset markets.¹² Investors in the economy hold diversified portfolios, but also specialize in trading the assets that belong to a particular market-segment or industry-group. Although there are many market segments, most of the analysis focuses on the assets within a particular segment. For informational or other reasons, the investors in the model are the only investors in the economy that trade and hold the assets within this segment.¹³ Hereafter, these assets are referred to as segment-assets. All investors in the economy trade a riskless asset with gross return per period that is in perfectly elastic supply, and a broadly-held index that is in zero net supply.¹⁴ The index proxies for systematic risk and can be thought of as a futures market. Because the index is broadly held, the investors collective trades do not affect the price of the index.

Risky Assets

The prices of the segment and index assets are $P^{1}(t)$ and $P^{2}(t)$ ; and have stacked price vector . The segment-assets are in fixed supply $X^{1}$ . The assets pay i.i.d. dividends $D^{1}(t)$ , $D^{2}(t)$ , represented by stacked vector , that has distribution:

$LaTex Encoded Math: \displaystyle D(t) \sim$ i.i.d. $LaTex Encoded Math: \displaystyle \quad {\cal N}(\bar{D}, \Omega),$ where $LaTex Encoded Math: \displaystyle \quad \Omega = \left ( \begin{array}{cc} \Omega_{11} & \Omega_{12} \\ \Omega_{21} & \Omega_{22} \end{array} \right ).$

(B1)

Because prices can be negative, the assets' excess returns are best expressed on a per share basis, and are denoted:

$LaTex Encoded Math: \displaystyle Z(t) = P(t) + D(t) - r P(t-1)$

(B2)

For simplicity $P^{2}(.)$ is fixed, which implies $Z^{2}(t) \sim$ i.i.d. ${\cal N} (\bar{Z}^{2},\Omega_{22})$ . To solve investors' portfolio problem, it is useful to decompose the segment assets return into a component that is perfectly correlated with the index and into a residual return

$LaTex Encoded Math: \displaystyle Z^{1}(t) = \beta_{12} Z^{2}(t) + e(t)$

(B3)

where $\beta_{12} = \frac{\operatorname{Cov}[Z^{1}(t),Z^{2}(t)]}{\operatorname{Var}[Z^{2}(t)]} = \Omega_{21} \Omega_{22}^{-1}.$ Investors' portfolio problem can be represented as choosing their exposure to the index risk, and to the residual risk of the segment assets. Because a limited number of investors hold the residual risk, it is not diversifiable. Therefore, the expected return for holding the residual risk will not necessarily be equal to 0. The variance of

is denoted $\Omega_{e}$ ; in equilibrium it turns out to be constant through time and is given by:

$LaTex Encoded Math: \displaystyle \Omega_{e} = \Omega_{11} - \Omega_{21} \Omega_{22}^{-1} \Omega_{12}.$

Investors

Each investor chooses risky asset holdings $Q_{m}(t)$ and consumption $C_{m}(t)$ to maximize his discounted expected Constant Absolute Risk Aversion (CARA) utility of consumption:

$LaTex Encoded Math: \displaystyle U_{m}(C_{m}(1),...C_{m}(\infty)) = \sum_{t=1}^{\infty} -\delta^{t} e^{-A_{m} C_{m}(t)},$

(B4)

with discount factor $\delta$ and absolute risk aversion $A_{m}$ , subject to the standard intertemporal budget constraints:

$LaTex Encoded Math: \displaystyle W_{m}(t) = Q_{m}(t)'Z(t) + r[W_{m}(t-1) - C_{m}(t-1)] \quad t=1,\hdots T.$

(B5)

An investor's liquid wealth, $W_{ml}(t)$ , is that part of her wealth that can be liquidated at current prices. Because large investors' trades have price impact, their wealth and liquid wealth differ; and it is their liquid wealth that appears as an argument in their value functions. Therefore it is useful to express their budget constraints in terms of their beginning of period liquid wealth:

$\begin{displaymath}\begin{split}W_{ml}(t) = Q_{m}^{1} & (t)'D^{1}(t) + Q_{m}^{2}(t)'Z^{2}(t) \\ & + r \left [ W_{ml}(t-1) - \Delta Q^{1}_{m}(t-1)'P^{1}(t-1) - C_{m}(t-1) \right ] \quad t=1,\hdots T. \end{split}\end{displaymath}$

(B6)

Below, I show that investors who differ in their risk-aversion have different trading styles; this affects other investors ability to acquire illiquid assets. As a result, which investors hold the segment assets is a critical argument of investors value functions. It is summarized by $Q^{1}(t)$ ( vech $[Q^{1}_{1}(t)',\hdots, Q^{1}_{M}(t)']'$ ), the vector of all investors segment-asset holdings.

Trading Dynamics

Segment assets are traded over a total of time periods. The sequence of actions is as follows: At the beginning of each period $t \le T$ , investors enter the period with risky-asset holdings $Q_{m}(t), m=1,\hdots M$ ; they then receive dividends and choose their risky asset trades $\Delta Q_{m}(t)$ . These trades determine risky asset prices . Investors then choose consumption $C_{m}(t)$ ; the period ends; and the same sequence is repeated through time . After period , investors continue to consume and trade other assets, but trading of segment assets ends. The final period of trade in the segment-assets facilitates solution of the model via backwards induction.

The process of trade for the segment-assets is modeled as a dynamic Cournot-Stackelberg game of full information. In each period $t \le T$ , each small investor optimally computes his demand for the segment assets conditional on the state-variable $(Q^{1}(t),t)$ . Inverting the aggregated demands of the small investors defines a linear schedule of prices at which they are willing to absorb all possible quantities of the large investors trades for the segment assets:

$LaTex Encoded Math: \displaystyle P^{1}(.,t) = \frac{1}{r} \left ( \beta_{0}(t) - \beta_{Q^{1}}(t) Q^{1}(t) - \sum_{m=2}^{M}\beta_{m}(t) \Delta Q^{1}_{m}(t) \right ).$

(B7)

The matrix $\beta_{m}(t)$ measures the impact of large investor 's segment asset trades on the price of the segment assets.¹⁵ Large investors account for this price impact when choosing their trades. Given the price schedule in equation (7), large investors choose their time risky asset trades and consumption choices to solve the maximization problem:

$LaTex Encoded Math: \displaystyle \max_{\begin{array}{c}C_{m}(t),\\ \Delta Q_{m}(t) \end{array}} -e^{-A_{m} C_{m}(t)} + \delta \operatorname{E}_{t} V_{m}(W_{ml}(t+1),Q^{1}(t)+ \Delta Q^{1}(t),t+1)$

(B8)

subject to the budget constraint:

$LaTex Encoded Math: \displaystyle W_{ml}(t+1) = Q_{m}^{1}(t+1)'D^{1}(t+1) + Q_{m}^{2}(t+1)'Z^{2}(t+1) + r \left [ W_{ml}(t) - \Delta Q^{1}_{m}(t)'P^{1}(.,t) - C_{m}(t) \right ].$

(B9)

Large investors' equilibrium trades are found by solving for trades that are best-responses to each other under the Cournot-Stackelberg assumption that each large investor chooses his own trades while taking the price schedule and the trades of the other large investors as given. The resulting trades within a period are a Cournot-Nash equilibrium. The model of trade is solved by backwards induction from period , and the resulting equilibrium is subgame perfect. Additional details on model solution and investors' value functions are contained in the appendix.

Intuition for the main results on asset prices and trades comes from examining the first order condition for large investor 's optimal trade vector for the segment assets:

$LaTex Encoded Math: \displaystyle \frac{\partial \operatorname{E}V_{m}}{\partial \Delta Q^{1}_{m}} = \left [ \operatorname{E}V_{m,W}(.) ( D_{1}(t) - r P^{1}) + \operatorname{E}V_{m,Q^{1}_{m}}(.) \right ] - \left [\operatorname{E}V_{m,W}(.)r \Delta Q^{1}_{m}' P_{m}^1(.) \right ] = 0,$

(B10)

where $P_{m}^{1}(.) = \partial P^{1}(.)/\partial \Delta Q_{m}$ .

The first term in braces on the right hand side measures the expected net benefit from borrowing money to buy a bit more of a segment asset when trades have no price impact. This is the only term of the first order condition that should be present in a competitive setting. Therefore, in a competitive setting all investors would immediately trade to a point where the first term is equal to 0. When there is imperfect competition, it will never be optimal for a large investor to trade immediately to such a point because the second term in his first order condition would be non-zero, implying his trades had too much price impact. Instead, to reduce their price impact, large investors trade in a way that allows their positions to converge towards a point at which further trade is no longer optimal. Therefore, if a large investor believes an asset is overvalued, or provides too low a return, he will not liquidate its holdings immediately, but will instead sell the asset slowly over time. Because he sells slowly, prices will also adjust slowly, making it possible for low returns to persist in equilibrium, as will be discussed further below.

3.1 Asset Pricing

The intuition in the previous section and the results below both show that when markets are illiquid, asset returns will satisfy one equilibrium relationship when asset holdings are at their long-run equilibrium, and satisfy a different relationship during the transition to the long run. To illustrate the exact pricing relationship, additional notation is required. More specifically, let $Q_{m}^{1W}$ represent investor 's long-run holdings of the segment-assets; these holdings are also the same as if all investors were price-takers, and are also associated with the efficient sharing of risks. When risk sharing is efficient, each investor holds assets in proportion to his risk-tolerance as a fraction of the sum total of all investors' risk tolerances:

$LaTex Encoded Math: \displaystyle Q_{m}^{1W} = \frac{(1/A_{m})X^{1}}{\sum_{m=1}^{M} (1/A_{m})}\quad m=1,\hdots M$

(B11)

The following proposition shows that when markets are illiquid, the deviation from investors' efficient asset holdings behaves like a priced factor when computing one-period returns:

Proposition 1 The segment-assets equilibrium excess expected returns satisfy a linear factor model that provides a reward for assets covariances with the liquid index, for their covariances with segment-specific risk, and for covariances with deviations of large investors asset holdings from those associated with efficient sharing of the residual risk:

$LaTex Encoded Math: \displaystyle \bar{Z}^{1}(t) = \beta_{12} \bar{Z}^{2}(t) + \lambda_{[X^{1}]} \Omega_{e} X^{1} + \sum_{m=2}^{M} \lambda(m,t) \Omega_{e} (Q^{1}_{m}(t) - Q^{1W}_{m})$

(B12)

where

$LaTex Encoded Math: \displaystyle \lambda_{[X^{1}]} = \frac{1-(1/r)}{\sum_{m=1}^{M} 1/A_{m}}$

(B13)

Proof: See section D of the appendix.

The proposition is intuitive. Because investors can perfectly hedge the segment risk that is correlated with the index, the reward for bearing that risk is exactly the same as is provided by trading the index. Investors are left to share the segment assets portfolio of residual risk, $X^{1}'e_{t}.$ Segment-assets covariance with that residual risk is $\Omega_{e} X^{1}$ . I refer to this covariance as segment risk; segment risk is rewarded because the investors in the segment cannot diversify it away.¹⁶ In section D.1 of the appendix I show that in a competitive setting, index-risk and segment-risk are the only priced risks in the model.

When some investors trades move prices, imperfect risk sharing among investors introduces additional transient priced factors that vanish only when investors asset holdings converge to those associated with perfect risk sharing. Because of illiquidity, the convergence process takes time; therefore, imperfect risk sharing at period affects one period risk-premia at period $t+\tau$ , as shown in the following corollary:

Corollary 1 The segment assets $\tau$ period ahead 1-period excess returns follow a factor model in which deviations from perfect risksharing at period affect excess returns at period $t+\tau$ :

$LaTex Encoded Math: \displaystyle \operatorname{E}_{t} [Z^{1}(t+\tau + 1)] = \beta_{12} \bar{Z}^{2} + \lambda_{[X^{1}]} \Omega_{e} X^{1} + \sum_{m=2}^{M} \lambda_{m}(t,\tau) \Omega_{e} (Q^{1}_{m}(t) - Q_{m}^{1W})$

(B14)

Proof: See section D of the appendix.

In both the corollary and the proposition, the prices of risk of the transient factors (the $\lambda_{m}$ 's) are negative because when large investors hold more than their efficient share of risky assets, the marginal investors, in this case the small investors, hold less and hence require a smaller premium for holding the residual risk.

Potential Explanations for Underperformance

There are two notions of return underperformance after an IPO. The first is that an asset's expected returns in the short-run are lower than its expected returns in the long-run. The second are that its expected returns underperform after adjusting for some benchmark measure of risk. This section deals with the first type of underperformance; the second type of underperformance is discussed in section 5.3.

Corollary 1 illustrates how underperformance of the first type can occur. More specifically, because investors trade to efficient asset holdings, the third term in equation (14) is transient, while the first two terms are not. Therefore, short-run underperformance will occur whenever the third-term is negative. Because the market prices of risk $\lambda_{m}(t,\tau)$ are negative, the third term generates return underperformance when each large investor's IPO share allocation is greater than is consistent with optimal risk sharing; this is consistent with allocations being tilted towards institutional investors and away from retail investors at the IPO.

The magnitude of underperformance depends on whether allocations are distorted towards large investors, and it depends on the risk preferences of the large investors that receive the allocations. In a model with illiquidity, differences in investors' risk tolerances correspond to differences in investors' willingness to sell assets quickly and pay a high liquidity cost in order to share risk. The more risk tolerant large investors are less willing to pay a high a liquidity cost to share risk; therefore, risk sharing progresses more slowly when asset allocations are initially tilted towards them. This reduces the amount of risk that must be borne by the marginal (small) investors and reduces the assets required rate of return. Consistent with this reasoning, in simulations the $\lambda_{m}(t,\tau)$ functions (which measure the impact of allocation distortions towards investor at time on excess returns at time $t+\tau$ ) show that when allocations are distorted towards large investors with more risk tolerance, the effect on excess returns is longer-lived and larger in magnitude.

The amount and persistence of underperformance also depends on the competitiveness of the aftermarket. When the aftermarket is highly competitive, the $\lambda_{m}(t,\tau)$ functions rapidly asymptote toward 0 as $\tau$ increases, generating little underperformance. When the aftermarket is less competitive, the functions asymptote slowly, generating longer-lived underperformance.

Because IPO allocations affect post-IPO excess returns, they should also affect the initial trading price of the segment assets in the aftermarket. This is established in the appendix¹⁷, where I show that segment-asset prices at the beginning of each time period have the form:

$LaTex Encoded Math: \displaystyle P^{1}(t) = \frac{1}{r}(\alpha - \sum_{m=1}^{M} \gamma_{m}(t) \Omega_{e} Q_{m}^{1}),$

(B15)

where the scalars $\gamma_{m}(t)$ are positive numbers that are smaller for investors who are more risk tolerant. It follows that aftermarket prices will initially be higher than their long-run competitive values if the asset allocations are distorted away from efficient asset holdings towards relatively risk tolerant large investors. The initial "overpricing" in the aftermarket due to allocation distortions may help to explain part of the IPO underpricing puzzle. A full explanation depends on how shares are allocated at the IPO, and on how the IPO offer price is set. I turn to that topic below.

4 IPO Share Allocation and Price Setting

The motivation for the analysis in this section is based on Pritsker (2004). Pritsker studies a situation in which a distressed investor must rapidly sell a given number of shares into an imperfectly competitive aftermarket. Because the distressed investor is essentially selling to an oligopoly, the price he receives is discounted relative to the competitive price. The size of the discount is determined by the intensity of competition, which in turn depends on differences in large investors' risk tolerances. If one large investor is far more risk tolerant than the others, he has significant market power because by purchasing fewer shares, he can force the distressed sales to be absorbed by relatively risk averse investors who require a larger risk premium for taking the shares. Conversely, if investors risk tolerances are more similar, each investors market power is reduced, and the aftermarket becomes more competitive.¹⁸

The distressed investor analysis is applicable to the IPO setting. The issuing firm is like a distressed investor that needs to sell $X^{IPO}$ shares in a segment that is imperfectly competitive. The IPO is a mechanism that helps avoid selling shares into the imperfectly competitive aftermarket by instead allocating them beforehand at a fixed IPO price. I assume the underwriter acts on the issuer's behalf by lining up investors to buy the issue in order to maximize IPO proceeds. The IPO process resembles bookbuilding as practiced in the United States. The underwriter gathers demand information on the issue. In the model this information consists of knowledge about the other risky assets in the new issue's market segment, as well as knowledge about investors holdings of the segment-assets, the investors risk preferences, and whether there are investors who have market power in aftermarket trading. Based on this information, the underwriter sets an IPO offer price $P^{IPO}$ and makes take it or leave offers of share allocations to the large and small investors. The large investors allocations' are denoted by $X_{m}^{IPO}, {m=2,\hdots M}$ . The total number of shares offered to small investors is denoted $X_{1}^{IPO}$ . I assume that the small investors that are offered share allocations are offered identical amounts of shares. It will turn out that if shares are offered to some small investors, it will be optimal to offer them to all small investors. Therefore, I assume that when small investors are offered shares, they are all offered the same amount.

If a large or small investor turns down the share allocation that he is offered, then the underwriter is stuck with the shares. I assume that the underwriter sells any unallocated shares immediately in the aftermarket following the IPO. The possibility that an investor can force distressed sales in the aftermarket serves as a threat that constrains how the issuer allocates shares and chooses the IPO offer price.¹⁹ In particular, if an investor receives shares, the allocations and offer price must be set so that it cannot be in the interest of the investor to refuse their allocation and instead force the shares to be sold by the underwriter in the aftermarket. Of course, it is possible in theory that the underwriter might find it optimal to sell some shares in the aftermarket; denote these shares as $X_{U}^{IPO}$ and the aftermarket price on the first day of trading as $P_{A}^{IPO}$ .²⁰ This suggests that the underwriter chooses the vector of share allocations and aftermarket sales and IPO offer price $P^{IPO}$ to maximize

$LaTex Encoded Math: \displaystyle \Pi(P^{IPO},X) = P^{IPO} \times (\sum_{m=1}^{M} X_{m}^{IPO}) + P_{A}^{IPO} X_{U}^{IPO},$

(B16)

where $\Pi(.)$ represents the underwriters profit, the first term on the right hand side measures revenues raised at the IPO, and the second represents revenues raised by distressed sales in the IPO aftermarket.

The maximization takes place subject to the constraints that the total issue is allocated:

$LaTex Encoded Math: \displaystyle \sum_{m=1}^{M} X_{m}^{IPO} + X_{U}^{IPO} = X^{IPO},$

(B17)

that there are no short-sales by investors

through

$LaTex Encoded Math: \displaystyle X_{m}^{IPO} \ge 0, m = 1, \hdots M,$

(B18)

and subject to incentive compatibility (IC) constraints that those who receive allocations in the IPO will accept the allocations. The IPO allocations are evaluated based on investors ex-ante expected value of entering the first period of aftermarket trading when the asset allocation is

.²¹ Each small investor's

constraint is denoted $\psi_{s}$ , and takes the form:

$LaTex Encoded Math: \displaystyle \psi_{s} = V_{s}[Q_{s}^{IPO};Q^{IPO},t^{IPO}+1] - V_{s}[Q_{s}; Q^{IPO},t^{IPO} + 1] \ge 0$

(B19)

The segment-asset holdings of each small investor that chooses to and chooses not to participate in the IPO are denoted $Q_{s}^{IPO}$ and $Q_{s}$ respectively; and the vector of all investors post-IPO segment asset holdings is denoted $Q^{IPO}$ . Because each small investor is infinitesimal, his individual participation decision has no influence on the state vector $Q^{IPO}$ .

Large investors IC constraints are denoted $\psi_{m}$ and take the form:

$LaTex Encoded Math: \displaystyle \psi_{m} = V_{m}[Q^{IPO},t^{IPO}+1] - V_{m}[Q_{-m}^{IPO},t^{IPO}+1] \ge 0$

(B20)

where large investor

's share allocation is $Q_{m}^{IPO}$ if he participates in the IPO, and $Q_{-m}^{IPO}$ if he does not.

While other investors are not allowed to take a short position at the IPO, the underwriter can do so. If he does, then he is obligated to buy shares in the first period of aftermarket trade in order to cover his short position.²²

The assumption that any shares not sold at the IPO are sold immediately afterward by the underwriter in the form of distressed sales is very strong. A more realistic assumption is that any shares that the underwriter fails to sell at the IPO will instead be sold over a longer amount of time, $\tau_{S}$ trading periods, following the IPO. This modeling assumption is consistent with empirical evidence, reported in Ellis et. al. (2000), that IPO underwriters engage in price support activities in the IPO aftermarket, and with evidence reported by Ellis et. al. (2002) which shows that underwriters are active participants in the IPO aftermarket for long periods of time.²³.

I assume that when the underwriter sells shares over $\tau_{s}$ time periods he will sell them optimally. By optimality I mean that the underwriter buys shares at the IPO offer price, and then participates in the IPO aftermarket, trading as a large investor over the following $\tau_{S}$ time periods in order to maximize his own utility subject to the constraint that by time $\tau_{S}$ the underwriter holds no shares of the new issue. It is assumed that the certainty equivalent value of the underwriters utility from buying and trading the shares is turned over to the issuing firm at the time of the IPO. For tractability I assume that the underwriter has CARA utility like the other large investors. Let $CE_{U}(Q^{IPO},\tau_{s})$ denote the underwriters certainty equivalent. Then, under the less restrictive assumption, the underwriter maximizes:

$LaTex Encoded Math: \displaystyle P^{IPO} \times (X_{1}^{IPO} + \sum_{m=2}^{M} X_{m}^{IPO}) + CE_{U}(Q^{IPO},\tau_{s}),$

(B21)

subject to the constraints that the total issue is allocated [equation (17)], that there are no short sales [equation (18)], and subject to a new set of participation constraints that account for the underwriter's trading activity in the aftermarket.²⁴ My conjecture is that the underwriter's ability to spread out sales of unsold shares through time will reduce large investors bargaining power at the IPO and result in a higher IPO offer price, and higher profits for the issuer. That said, most of my recent simulations using the objective function in equation (21) show spreading out the underwriters sales has little effect on profit. Therefore, I do not report any results for this case in the current version; but hope to study it more in future work.

In closing this section I should emphasize that the optimal IPO allocation problem is difficult because it involves maximizing a nonlinear objective function subject to nonlinear inequality constraints. Details on how I solved the problem are provided in appendix E.

5 Simulation Analysis

The model's properties are studied for a single case in which an underwriter sells 40 shares to a continuum of small investors (investor 1) and five large investors (investors 2-6) under a variety of liquidity conditions. For simplicity, the only segment asset is the new issue, and investors risk tolerances are normalized so that they sum to 1.

Aftermarket liquidity depends on the concentration of risk bearing capacity among investors, and on the number of periods of aftermarket trade. An investors share of a segment's risk bearing capacity is his risk tolerance as a percentage of the sum total of all investors risk tolerances. Concentration of risk bearing capacity provides investors with market power in the aftermarket; and this makes the market more illiquid. Results on concentration of risk bearing capacity are reported using the Herfindahl index, which ranges from 10,000 when risk bearing capacity is concentrated with one investor, to 0 when all investors are small and there is no concentration.²⁵

The number of post-IPO trading periods consider ranges from 200 trading days (a bit less than a year) up to 2000 trading days (8 years).²⁶ To explain the relationship between the number of post-IPO trading periods and liquidity, note that the more shares an investor trades within a period, the more he moves the price. Therefore, he would prefer to break up his trades through time to minimize their price impact. When fewer periods of trade remain, opportunities to break up trades are limited; therefore risk sharing becomes more costly. Consequently, shares acquire a greater liquidity premium when fewer periods of trade remain. Conversely, as the number of tradings periods grow, the liquidity premium vanishes and the market becomes perfectly competitive.²⁷ Because my focus is on imperfect competition, I assume the number of trading periods is finite.

To illustrate the role of imperfect competition, the main results focus on two benchmark cases that differ only in whether the investors behave competitively. In both cases, investor 2 has more than a fifty percent share of the segment's risk bearing capacity (Table 1, Panel A); and the other investors differ in their risk tolerances.

When investors behave competitively, and the underwriter must sell all 40 shares, the IPO is no different than if the underwriter sold the shares directly in the aftermarket. Therefore, he raises $420 at the competitive price of $10.50 per share; additionally investors share allocations are proportional to their share of risk bearing capacity, which implies investor 2 should receive 21.82 shares.

When the aftermarket is imperfectly competitive, investor 2's large share of risk bearing capacity gives him enormous bargaining power at the IPO, because if he turns down his share allocation then the underwriter will have to sell shares into an aftermarket where investor 2 has substantial influence over price. Alternatively, if the underwriter instead chooses to allocate shares to other more risk averse investors, then they will be exploited by investor 2 in the aftermarket--and this will depress the price that other investors are willing to pay for shares. To illustrate these points, I studied a sequence of IPOs in which investor 2 can trade in the aftermarket but is restricted in the amounts that he can acquire at the IPO. More specifically investor 2's aquistion was progressively restricted to be no larger than $0,2,4,\dots X^{IPO}$ shares.²⁸If the aftermarket is perfectly competitive, the allocation restrictions are of little consequence because other investors can aquire the shares at the IPO and then quickly sell them to investor 2 in the aftermarket at the competitive price; therefore the IPO offer price is barely discounted from the competitive price. By contrast, if the aftermarket is imperfectly competitive, then the IPO offer price is severely discounted relative to the competitive price; moreover, the feasible offer price (Figure 1) and proceeds (not shown) are monotone increasing in the amount that can be offered to investor 2 up to the total outstanding supply of the issue. Investor 2's enormous bargaining power has implications for allocations and trading volume, as well as for underpricing, and return underperformance.

5.1 Allocation Distortions and Trading Volume

Because prices and proceeds are increasing in the amount that investor 2 can purchase, all of the shares are allocated to him at the IPO. This shows that allocations are distorted towards investors who have market power and away from retail investors (who each have none) (Table 1,Panel C). Additionally, among investors with market power, in the example the allocations are distorted towards the large investor with the most power. This illustrates that differences in investors market power provide a noninformation based explanation for why some large investors receive share allocations while others do not.

Because allocations at the IPO are not pareto optimal, the distortions create a basis for trade in the aftermarket. The resulting trading volume is heaviest on the first day of trading, constituting 1.2 percent of shares issued; daily volume drops off to about 0.1 percent of shares issued after the first week of trading (Figure 2). The example's pattern of heavy trading volume that rapidly drops off is qualitatively consistent with the empirical literature on post-IPO trading (Ellis et al. 2000), but the model fails to match the empirical magnitude of the first day's trading, which averaged 33.3 percent of shares issued in the 1980's, and 148.7 percent of shares issued in 1999-2000 (Ritter, 2005).

5.2 IPO Underpricing

The example shows the model is capable of generating substantial IPO underpricing. Underpricing ranges from a low of 40 percent when many trading periods remain to a high of 159 percent when relatively few periods remain (Table 1, Panel B). The underpricing has three pronounced features. First, the offer price is low relative to the asset's competitive price. Second, the aftermarket price is inflated relative to the asset's competitive price. If the competitive price is interpreted as the asset's "fundamental value" then it appears as if the IPO is associated with irrational price-overshooting even though the model is fully rational. The third feature is that the first two features are most pronounced when there are fewer periods of trade following the IPO.

Intuition for the first feature is related to investor 2's bargaining power; additional intuition for the underpricing comes from interpreting illiquidity as a tax on risk sharing. Because the tax makes risk sharing more costly, it pushes down the IPO offer price; in addition because of the tax less risk sharing takes place, which means investor 2 sells less in the aftermarket--and this constraint on supply pushes up the aftermarket price. When fewer trading periods remain, illiquidity is more severe, making the first two features more pronounced.

5.3 Underperformance

Recall that there are two types of return underperformance--underperformance relative to an assets long-run return--and underperformance relative to a risk adjusted benchmark. Section 3.1 shows that the former is associated with IPOs; to address the latter, post-IPO expected excess returns were computed for up to 2000 business days and then adjusted for market-risk by subtracting the asset's beta times the expected excess return on the market portfolio. To compute the excess return on the market portfolio, I aggregated up the excess returns across all market segments. Recall from equation (12) (reproduced below) that in each segment , assets excess expected returns can be decomposed into a component that is correlated with the index, a second that is associated with market segmentation (due to imperfect sharing of idiosyncratic risk), and a third that is associated with allocation distortions and illiquidity:

$LaTex Encoded Math: \displaystyle \bar{Z}^{j}(t) = \beta_{j2} \bar{Z}^{2}(t) + \lambda_{[X^{j}]} \Omega_{e_{j}} X^{j} + \sum_{m=2}^{M_{j}} \lambda(m,t) \Omega_{e_{j}} (Q^{j}_{m}(t) - Q^{jW}_{m}).$

Because investors trade towards efficient asset holdings, the third component is transitory. and is assumed to be small or zero for segments that have not had recent IPOs; but it is negative for those that have had them.

Under these circumstances, if investors trade the liquid index and specialize in trading the assets within different market-segments, then in the context of a one-period example (appendix F), the assets in each segment and the market portfolio earn a segmentation premium. Therefore, after adjustment for market-risk, the assets average segmentation premia are 0, and do not depend on whether the assets are new issues. Using similar reasoning, because relatively few firms are new issues, these firms will only have a minor effect on the market return. Therefore, after market adjustment, IPO firms will retain a large negative liquidity premium; that is they will underperform on a risk-adjusted basis.

Under simplifying circumstances (see appendix F), if $\phi$ is the fraction of firms that are new issues, then the average market adjusted underperformance at time for segments with IPO's should be approximately $(1-\phi)$ times the allocation distortion / liquidity component of excess returns:

$LaTex Encoded Math: \displaystyle (1-\phi ) \sum_{m=2}^{M} \lambda(m,t) \Omega_{e} (Q^{1}_{m}(t) - Q^{1W}_{m}).$

In the example, if $\phi$ is 0, and the market adjusted segmentation premium is approximately 0 for simplicity, then the CAR grows to -32 percent over 2000 periods. In the simulations which corresponds to about 8 years of underperformance (Figure 3, Panel A). More realistically, if $\phi$ is about 10 percent, then the expected market-adjusted CAR should be about -29 percent over 8 years. This confirms that the model can in theory generate both IPO underpricing and long-lived market adjusted return underperformance.

5.3.1 Adjusting for Characteristics

A stylized fact of measured underformance is that it is strongest when returns are market-adjusted, and weaken after adjustment for additional characteristics such as market-to-book (Ritter and Welch, 2002). Under some interpretations, these stylized facts are consistent with the liquidity-based explanation in the model; in particular, if assets' long-run competitive prices are interpreted as book-value, then the model predicts that because of illiquidity and allocation distortions at the IPO, then just after the IPO, market to book is high and consistent with the market-to-book effect, IPO returns underperform. It is important to bear in mind that in the empirical asset pricing literatature, market-to-book's ability to explain asset returns is an empirical regularity, not a theory of asset pricing. The contribution of the present model is that it provides a theoretical reason why market-to-book appears to "explain" post IPO return underperformance. This result has empirical implications; in particular, it shows that controlling for market-to-book can be problematic when testing the present model because a finding of no underperformance after adjustment for market-to-book is consistent with the model, and not evidence against it. This topic is discussed further in footnote 33 of section 6.1.

5.4 Money Left on the Table

IPO underpricing makes it appears as if the IPO should have raised more revenue. The loss in revenue is referred to as "money left on the table", hereafter MLOTT, and is usually measured as the differential between the IPO-offer price and closing price on the first day of trade times the number of shares issued. MLOTT often has the interpretation of a measure of the issuer's losses due to imperfections in the IPO process.²⁹ For the present model, the imperfection is illiquidity; and it should be clear that the usual MLOTT calculation overstates issuer losses because it incorrectly assumes that all of the shares could be sold at the artificially high price that prevails in an illiquid aftermarket. An alternative measure of MLOTT instead compares the revenues that were raised at the IPO against the revenues that could have been raised if the aftermarket was perfectly competitive, which in this model means perfectly liquid. Computing losses by this alternative metric shows that the usual calculation can very significantly overstate issuer losses; in the case of the example it overstates them by a factor of 5 to 10 (Table 1, Panel B).

5.5 Alternative market configurations

To study how differences in investors risk tolerances affect the results, I solved the model under two alternative market configurations. In the first, some of investor 2's risk tolerance is spread evenly among the other large investors (Table 2). This change makes the aftermarket more competitive, and has three further effects on the aftermarket. First, when the number of aftermarket trading periods is , or , the market is sufficiently competitive that differences in who receives shares at the IPO have a very small effect on prices (no more than 6 cents per share). In this circumstance, the optimal allocation problem is ill-posed. I simply assume that the competitive allocation results in these cases. Second, the magnitude and persistence of underperformance diminish somewhat, with the liquidity component of the CAR reduced to -16 percent over 5 years. Third, the prices charged by the underwriter at the IPO are actually above the competitive price. This occurs because when the aftermarket is a little bit more competitive, investor 2 has less market power in the aftermarket, and hence has less bargaining power at the IPO. In this circumstance, the underwriter can extract some of investor 2's surplus from acquiring the new issue. Because the aftermarket is still not perfectly liquid, the aftermarket price is inflated above the IPO offer price, but now the underwriter is actually generating more revenue for the issuer than if the aftermarket were competitive. Compared with the earlier results, this shows that the underwriter and issuer actually benefit if the aftermarket is a bit less than perfectly competitive, but is hurt if the aftermarket becomes too imperfectly competitive. Because a little bit of imperfect competition can sometimes help the underwriter, this result may help to explain underwriter practices that restrain trade in the aftermarket, such as restrictions on investors ability to flip shares.

The second alternative market configuration contained two dominant large investors that each have 30% of the risk bearing capacity.³⁰ Because this configuration was highly competitive, unless a small number of trading periods followed the IPO, results are only discussed for when there are 200 post-IPO trading periods. Unlike the previous examples in which the optimum involved allocating all of the shares to one investor, in this example the optimal share allocations involved splitting the shares evenly between investors 2 and 3 while distorting the share allocations away from all other investors. As in the previous example, the resulting IPO offer price was above the price that would prevail if the aftermarket was perfectly competitive. Nevertheless, the aftermarket price following the IPO was even higher, resulting in IPO underpricing of about 12%. Additionally, cumulative abnormal returns indicate short-lived underperformance; the liquidity component of the underperformance was 4.5% over 200 trading days.

6 Empirical Implications

The purpose of this section is to outline a rough strategy for testing the theory in this paper. The main implication of the theory is that IPO illiquidity and imperfect competition in the aftermarket following an IPO lead to allocation distortions towards large investors and IPO underpricing and underperformance. These implications lead to the following testable predictions:

Allocation distortions are associated with underperformance in the aftermarket.
Allocation distortions are associated with IPO underpricing.
The above two effects are associated with illiquidity and imperfect competition in aftermarket trading.

Prediction 1 imposes restrictions on the coefficients of empirical versions of equation (14):

$LaTex Encoded Math: \displaystyle Z_{i}(t+\tau + 1) = \beta_{i,2} \pi_{IND} + [\lambda_{[X^{1}]} \Omega_{e} X^{1}] \pi_{SEG} + [\sum_{m=2}^{M} \lambda_{m}(t,\tau) \Omega_{e} (Q^{1}_{m}(t) - Q_{m}^{1W})] \pi_{LIQ} + \epsilon_{i}(t+\tau)$

(B22)

The empirical version provided above differs from equation (14) because it is parameterized with additional $\pi$ coefficients, and because expected excess returns have been replaced by their realization on the left hand-side, which introduces an expectational error $\epsilon_{i}(t+\tau)$ on the right-hand side. The main coefficient of interest is $\pi_{LIQ}$ . Under the null hypothesis that allocation distortions at the IPO are unrelated to underperformance, $\pi_{LIQ}$ should be equal to 0; while under the alternative $\pi_{LIQ}$ should be positive and the coefficients $\lambda_{m,t}$ should be negative. Estimation of $\pi_{LIQ}$ in cross-section requires information on $\Omega_{e}$ and investors holdings of all assets within each segment. However, if $\Omega_{e}$ is nearly diagonal, or if allocation distortions are only significantly different from zero for new issues, then it is sufficient to create an allocation distortion measure for each new issue that only depends on its own idiosyncratic risk and own asset holdings. This simplified approach is outlined below.

The allocation distortion measure that I propose for firm has the form:

distort $LaTex Encoded Math: \displaystyle _{i} = \Omega_{ii} \sum_{m=2}^{M} g(RBC_{m}) \left [ \frac{x_{m}}{\sum_{m=1}^{M} x_{m}} - RBC_{m} \right ],$

(B23)

where $x_{m}$ is the share allocation of large investor

at the IPO; $\Omega_{ii}$ is the variance of the idiosyncratic component of firm

's return; $RBC_{m}$ is large investor

risk bearing capacity, which is his risk tolerance as a fraction of the sum total of all

investors risk tolerances; and

is a negative and monotone decreasing function, which serves as a proxy for $\lambda$ in equation (22). The distortion measure is 0 if allocations are efficient; positive if allocations are distorted towards small investors, and negative if distorted towards large investors. In addition, consistent with the results that show underperformance and the aftermarket price are maximized when allocations are distorted towards the most risk tolerant large investor, the distortion measure is maximized when all assets are allocated to the most risk tolerant large investor.

Proxies for $RBC_{m}$ are needed to operationalize the distortion measure. Because the model assumes that investors have CARA utility primarily for tractability, a sensible way to proceed is to use a more realistic assumption about investors utility. For example, if all investors have power utility with coefficient of relative risk aversion $\gamma$ , then investors absolute risk tolerance is $W_{m}/\gamma$ , and each investors risk bearing capacity is equal to their own wealth as a fraction of the sum total of the wealth of other investors in the segment ( $RBC_{m} = \frac{W_{m}}{\sum_{m=1}^{M} W_{m}}$ ).³¹ Since wealth is a measure of investors's size, the allocation distortion measure them becomes a measure of how assets are allocated relative to the size of the investors involved. Armed with this measure of allocation distortions it should be possible to test prediction 1 by estimating variants of equation (22) in cross-section and then test whether $\pi_{LIQ}$ is positive.³². A full analysis of how to estimate the equation is well beyond the scope of the present paper. To test prediction 2, the same measures of allocation distortions that are used to estimate equation (22) can be used to attempt to explain the cross-section of IPO underpricing.

To test the third prediction, measures of aftermarket illiquidity should be interacted with the allocation distortion variable. The theory predicts that the allocation distortions should only have an effect when there is illiquidity in the aftermarket. Hence, the interaction with the illiquidity variables should provide a sharper test of theory. The coefficients on the interaction terms are expected to have the same sign as the coefficients on the allocation distortions, and including these terms should cause the coefficients on the allocation distortions to weaken.

Allocation distortions are predicted by other theories, such as for example bookbuilding and adverse selection, in which the allocations are based on investors information. However, neither of those theories predicts a relationship between allocation distortions and IPO return underperformance. Therefore, if allocation distortions explain both underpricing and underperformance, it should be interpreted as evidence that an explanation based on illiquidity and imperfect competition in the aftermarket helps contribute to our understanding of IPOs.

To close this section, I briefly review the most closely related empirical literature on underpricing and underperformance.

6.1 Related literature

The empirical literature that is most closely related to this paper studies the relationship between after-market liquidity and underpricing or underperformance. The relationship between IPO underpricing and illiquidity has been empirically studied by Booth and Chua (1996), Hahn and Ligon (2004), and Ellul and Pagano (2003). In closely related work, Butler et. al. (2005) study the relationship between a stock's liquidity, and the underwriting fees that are paid during a seasoned equity offering.

Although the Booth and Chua model makes predictions about the relationship between underpricing and aftermarket liquidity, they don't test this implication of their model; instead their tests focus on underpricing as compensation for costs of information gathering. Because such costs could generate underpricing irrespective of illiquidity, the implications of their tests for the relationship between underpricing and aftermarket liquidity are unclear. Hahn and Ligon attempt to directly test the Booth and Chua hypothesis that underpricing is used to increase liquidity by running OLS regressions of market microstructure measures of aftermarket liquidity on IPO underpricing. In regressions that account for other determinants of illiquidity, their results are mixed; with coefficients on underpricing sometimes statistically significant and positive, sometimes statistically significant and negative, and sometimes not statistically significant at all. A potential difficulty with the Hahn and Ligon methodology is that causality may run from underpricing to illiquidity (as in Booth and Chua) as well as from illiquidity to underpricing (as in Ellul and Pagano). The possibility that causality runs in both directions suggests that an instrumental variable approach is needed. In Ellul and Pagano, they regress underpricing on a set of determinants for underpricing, including measures of aftermarket liquidity. Additionally, they recognize the potential for simultaneity bias and instrument for it in some of their regressions.³³ In all of Ellul and Pagano's regressions they find that more aftermarket illiquidity increases the amount of IPO underpricing. Butler et. al. find qualitatively similar relationship between illiquidity and underwriter fees in SEO's, but quantitatively the effects of illiquidity are much smaller than in Ellul and Pagano. This suggests the Ellul and Pagano results, while favorable for liquidity based theories, should be interpreted with caution. An additional reason for caution is if underpricing is a risk premium for aftermarket illiquidity, then the logical extension of Ellul and Pagano's theory would suggest that in the aftermarket, IPO's should earn a positive and significant risk premium for aftermarket illiquidity. If we believe the empirical evidence that IPO returns underperform in the aftermarket, this suggests that the mechanism driving aftermarket returns is more complicated than the theory of illiquidity considered by Ellul and Pagano. Eckbo and Norli (2002) take this argument one step further; they claim that newly issued stocks are more liquid than other stocks with similar risk characteristics; and thus their returns should underperform. To establish this point empirically, Eckbo and Norli compare the returns of a rolling portfolio of newly issued stocks that are held for up to five years against the returns a portfolio of more seasoned issues that are matched on size and book to market. They find that after adjusting for these factors, and controlling for differences in liquidity, new issues do not underperform.

The Eckbo and Norli and Ellul and Pagano findings, taken together are puzzling because the latter suggests that IPOs are very illiquid, while the former suggests the opposite. Both papers can only be correct if liquidity conditions change rapidly after the IPO, and if participants in IPOs are very concerned about a short-term need to liquidate. This paper points towards a different resolution in which illiquidity, when combined with imperfect competition, generates both underpricing and underperformance. To make a strong case that this is the correct resolution requires careful future research on how to measure underperformance and which measures of liquidity are relevant for large investors.³⁴ $^{,}$ ³⁵

7 Conclusions

In this paper I have presented a fully-rational, symmetric-information model to simultaneously explain IPO underpricing, underperformance, and a tilt in IPO share allocations towards institutional investors, and away from retail investors. The key model features that generate these results are illiquidity and imperfect competition in aftermarket trading. The model also generates a new set of testable predictions that tie the market structure of aftermarket trading and allocation distortions at the IPO, to the cross-sectional pattern of return underperformance following an IPO. Hopefully the results in this paper will stimulate new empirical research that studies the relationship between the structure of the IPO aftermarket trading environment and IPO underpricing and underperformace.

Appendix

8 Notation

There are investors and $N = N_{1} + N_{2}$ risky assets. The first $N_{1}$ assets are illiquid. The next $N_{2}$ assets are perfectly liquid. The risky asset holdings of investor at time are denoted by

$LaTex Encoded Math: \displaystyle Q_{m}(t) = \left ( \begin{array}{c} Q^{1}_{m}(t) \\ Q^{2}_{m}(t) \end{array} \right )$

where $Q^{1}_{m}(t)$ and $Q^{2}_{m}(t)$ are investor

holdings of illiquid and liquid risky assets respectively. $Q^{1}(t)$ denote the $N_{1} M \times 1$ vector of all investors illiquid asset holdings at time

where

$LaTex Encoded Math: \displaystyle Q^{1}(t) = \left ( \begin{array}{c} Q^{1}_{1}(t) \\ \vdots \\ Q^{1}_{M}(t) \end{array} \right ).$

$Q^{1}_{1}(t)$ represents the net asset holdings of a continuum of infinitesimal small investors indexed by :

$LaTex Encoded Math: \displaystyle Q^{1}_{1}(t) = \int_{0}^{1} Q^{1}_{s}(t) \mu(s) ds.$

The small investors are often collectively referred to as the competitive fringe. $Q^{1}_{2}(t)$ through $Q^{1}_{M}(t)$ denotes the net illiquid risky asset holdings of large investors, and is denoted by the $N_{1} \times (M-1)$ vector $Q^{1}_{B}(t)$ . The change in investors illiquid risky asset holdings from the beginning of time period

to the beginning of time period

is denoted by the $N_{1} M \times 1$ vector $\Delta Q^{1}(t)$ . Similarly, $\Delta Q^{1}_{1}(t)$ and $\Delta Q^{1}_{B}(t)$ denote the change in the competitive fringe's illiquid asset holdings, and the change in the illiquid asset holdings of the large investors.

The algebra which follows requires many matrix summations and the use of selection matrices. Rather than write summations explicitly, I use the matrix $S = \iota_{M}' \otimes I_{N}$ to perform summations where $\iota_{M}$ is an by vector of ones, and $I_{N}$ is the $N \times N$ identity matrix.³⁶ In some cases, the matrix may have different dimensions to conform to the vector whose elements are being added. In all such cases, will always have , or $N_{1}$ rows. The matrix $S_{i}$ is used for selecting submatrices of a larger matrix. $S_{i}$ has form

$LaTex Encoded Math: \displaystyle S_{i} = \iota_{i,M}' \otimes I_{N},$

where $\iota_{i,M}$ is an

vector has a

in its

'th element, and has zeros elsewhere.³⁷ As above $S_{i}$ will sometimes have different dimensions to conform with the matrices being summed, but it will always have

or $N_{1}$ rows.

In the rest of the exposition, I will occasionally suppress time subscripts to save space.

9 Proof of Proposition 2

Proposition 2 : Small investors value functions for entering period with liquid wealth $W_{s}$ , when investors' state vector of illiquid asset holdings is given by $Q^{1}$ is given by:

$\begin{displaymath}\begin{array}{rcl} V_{s}(W_{s},Q^{1},t) & = & -K_{1}(t) \ F(Q^{1},t) \ e^{-A_{s}(t)W_{s}}, \\ [0.06in] \mbox{where} \ F(Q^{1},t) & = & e^{-Q^{1}(t)'\bar{v}_{s}(t) - Q^{1}(t)'\theta_{s}(t) Q^{1}(t).} \end{array}\end{displaymath}$

(B1)

Large investor 's value function for entering period when the state vector of illiquid asset holdings is and his holdings of liquid wealth is $W_{m}$ is given by:

$LaTex Encoded Math: \displaystyle V_{m}(W_{m},Q^{1},t) = -K_{m}(t) e^{-A_{m}(t) W_{m} - A_{m}(t) Q^{1}' \Lambda_{m}(t) + .5 A_{m}(t)^{2} Q^{1}' \Xi_{m}(t) Q^{1}} \quad m = 2,\hdots M,$

(B2)

and the price function for illiquid assets has the functional form:

$LaTex Encoded Math: \displaystyle P^{1}(t) = \frac{1}{r}(\alpha(t) - \Gamma(t) Q^{1})$

(B3)

Proof: The proof is by induction. Part I of the proof establishes that if the value function has this form at time , then it has the same form at time . Part II of the proof establishes the result for time , the first period in which trade cannot occur.

9.1 Part I:

Suppose the form of the value function is correct for time . Then, to establish the form of the value function at time , I first solve for the competitive fringe's demand curve for absorbing the net order flow of the large investors. I then solve the large investors and competitive fringe's equilibrium portfolio and consumption choices, and then solve for the value function at time .

9.1.1 The competitive fringe's demand curve

The competitive fringe represents a continuum of infinitesimal investors that are distributed uniformly on the unit interval with total measure 1, i.e. $\mu(s) = 1$ for $s \in [0,1]$ . At time , each participant of the competitive fringe solves:

$LaTex Encoded Math: \displaystyle \max_{\begin{array}{c}C_{s}(t-1),\\ Q_{s},\\ q_{s} \end{array}} -e^{-A_{s} C_{s}(t-1)} - \delta \operatorname{E}[ K_{s}(t) F(Q^{1},t) e^{-A_{s}(t) W_{s}(t)} ]$

(B4)

where, $Q_{s}$ is the stacked vector of small investor 's holdings of illiquid ( $Q^{1}_{s}$ ) and perfectly liquid ( $Q^{2}_{s}$ ) risky assets:

$LaTex Encoded Math: \displaystyle Q_{s} = \left ( \begin{array}{c} Q^{1}_{s} \\ Q^{2}_{s} \end{array} \right );$

is the stacked vector of excess returns for the illiquid and liquid assets:

$LaTex Encoded Math: \displaystyle Z(t) = \left (\begin{array}{c} Z^{1}(t) \\ Z^{2}(t) \end{array} \right ) = \left ( \begin{array}{c} P^{1}(t) + D^{1}(t) - r P^{1}(t) \\ P^{2}(t) + D^{2}(t) - r P^{2}(t) \end{array} \right );$

(B5)

and small investors liquid wealth is given by

$LaTex Encoded Math: \displaystyle W_{s}(t) = Q_{s}'Z(t) + r [W_{s}(t-1) - C_{s}(t-1)].$

Note: Although I refer to the first set of assets as illiquid, they are only illiquid for large investors whose trades have price impact. Because each small investor is infinitesimal, their trades do not have price impact and hence both assets are perfectly liquid from their perspective.

In equation (B5),

$LaTex Encoded Math: \displaystyle \operatorname{E}Z(t) \equiv \bar{Z}(t) \equiv \left ( \begin{array}{c} \bar{Z}_{1}(t) \\ \bar{Z}_{2}(t) \end{array} \right ),$

and

$LaTex Encoded Math: \displaystyle \operatorname{Var}Z(t) \equiv \Omega \equiv \left ( \begin{array}{cc} \Omega_{11} & \Omega_{12} \\ \Omega_{21} & \Omega_{22} \end{array} \right ).$

Substituting the expression for $W_{s}$ in (B4) and taking expectations shows that small investors maximization becomes:

$LaTex Encoded Math: \displaystyle \max_{\begin{array}{c} C_{s}(t-1),\\ Q_{s} \end{array}} -e^{-A_{s} C_{s}(t-1)} - \delta F(Q^{1},t) e^{-A_{s}(t) r [W_{s}(t-1) - C_{s}(t-1)] -A_{s}(t) Q_{s}'\bar{Z}(t) + .5 A_{s}(t)^{2} Q_{s}' \Omega Q_{s} }$

(B6)

In solving the model, it is useful to break small investors maximization into pieces by first solving for optimal $Q^{2}_{s}$ as a function of $Q^{1}_{s}$ , and then solving for optimal $Q^{1}_{s}$ . For given $Q^{1}_{s}$ , the first order condition for optimal $Q^{2}_{s}$ shows that optimal $Q^{2}_{s}$ is given by

$LaTex Encoded Math: \displaystyle Q^{2}_{s} = \frac{1}{A_{s}(t)} \Omega_{22}^{-1} \bar{Z}_{2}(t) - \beta_{12} ' Q^{1}_{s},$

(B7)

where $\beta_{12} = \Omega_{12} \Omega_{22}^{-1}.$

Plugging the solution for $Q^{2}_{s}$ into the small investors value function and simplifying then shows that the small investors maximization problem reduces to:

$\begin{displaymath}\begin{split}\max_{\begin{array}{c} C_{s}(t-1),\\ Q^{1}_{s} \end{array}} -e^{-A_{s} C_{s}(t-1)} - & \delta F(Q^{1},t) K_{s}(t) \operatorname{Exp}\left \{ -.5 \bar{Z}_{2}'\Omega_{22}^{-1} \bar{Z}_{2} - A_{s}(t) r [W_{s}(t-1) - C_{s}(t-1)] \right \} \\ & \times \operatorname{Exp}\left \{ -A_{s}(t) Q^{1}_{s}'[\bar{Z}_{1}(t) - \beta_{12} \bar{Z}_{2}(t)] + .5 A_{s}(t)^{2} Q^{1}_{s}' \Omega_{e} Q^{1}_{s} \right \} \end{split}\end{displaymath}$

(B8)

where $\Omega_{e}$ is given by

$LaTex Encoded Math: \displaystyle \Omega_{e} = \Omega_{11} - \Omega_{12} \Omega_{22}^{-1} \Omega_{21}.$

To gain intuition for the above expression, note that the excess return on each illiquid asset can be decomposed into a component that is correlated with the liquid assets and into a second idiosyncratic component.

$LaTex Encoded Math: \displaystyle Z_{1}(t) = \beta_{12} Z_{2}(t) + \epsilon_{1}(t)$