Finance and Economics Discussion Series: 2006-12
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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.

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 heavy^{2}, 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
investors^{3}
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 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.

The prices of the segment and index assets are and ; and have stacked price vector . The segment-assets are in fixed supply . The assets pay i.i.d. dividends , , represented by stacked vector , that has distribution:

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

For simplicity is fixed, which implies i.i.d. . 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 :

where 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 ; in equilibrium it turns out to be constant through time and is given by:

Each investor chooses risky asset holdings and consumption to maximize his discounted expected Constant Absolute Risk Aversion (CARA) utility of consumption:

(B4) |

with discount factor and absolute risk aversion , subject to the standard intertemporal budget constraints:

(B5) |

An investor's liquid wealth, , 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:

(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 ( vech), the vector of all investors segment-asset holdings.

Segment assets are traded over a total of time periods. The sequence of actions is as follows: At the beginning of each period , investors enter the period with risky-asset holdings ; they then receive dividends and choose their risky asset trades . These trades determine risky asset prices . Investors then choose consumption ; 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 , each small investor optimally computes his demand for the segment assets conditional on the state-variable . 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:

The matrix
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:

subject to the budget constraint:

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:

where .

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 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:

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:

**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,
Segment-assets covariance with that residual risk is
. 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 , as shown in the following corollary:

**Proof:** See section D of the
appendix.

In both the corollary and the proposition, the prices of risk of the transient factors (the '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 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 functions (which measure the impact of allocation distortions towards investor at time on excess returns at time ) 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 functions rapidly asymptote toward 0 as 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 appendix^{17}, where I show that segment-asset
prices at the beginning of each time period have the form:

where the scalars 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.

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
shares.^{28}If 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.

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).

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:

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 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 times the allocation distortion / liquidity component of excess returns:

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.

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).

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):

The empirical version provided above differs from equation (14) because it is parameterized with additional coefficients, and because expected excess returns have been replaced by their realization on the left hand-side, which introduces an expectational error on the right-hand side. The main coefficient of interest is . Under the null hypothesis that allocation distortions at the IPO are unrelated to underperformance, should be equal to 0; while under the alternative should be positive and the coefficients should be negative. Estimation of in cross-section requires information on and investors holdings of all assets within each segment. However, if 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 | (B23) |

where is the share allocation of large investor at the IPO; is the variance of the idiosyncratic component of firm 's return; 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 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
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
, then investors
absolute risk tolerance is
, 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 (
).^{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 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}

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.

There are investors and risky assets. The first assets are illiquid. The next assets are perfectly liquid. The risky asset holdings of investor at time are denoted by

represents the net asset holdings of a continuum of infinitesimal small investors indexed by :

The algebra which follows requires many matrix summations and
the use of selection matrices. Rather than write summations
explicitly, I use the matrix
to perform summations
where is an
by vector of ones, and is the
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 rows. The matrix is used for selecting
submatrices of a larger matrix. has form

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

9 Proof of Proposition 2

**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.

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 .

The competitive fringe represents a continuum of infinitesimal investors that are distributed uniformly on the unit interval with total measure 1, i.e. for . At time , each participant of the competitive fringe solves:

where, is the stacked vector of small investor 's holdings of illiquid () and perfectly liquid () risky assets:

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

and small investors liquid wealth is given by

In equation (B5),

and

Substituting the expression for in (B4) and taking expectations shows that small investors maximization becomes:

In solving the model, it is useful to break small investors maximization into pieces by first solving for optimal as a function of , and then solving for optimal . For given , the first order condition for optimal shows that optimal is given by

(B7) |

where

Plugging the solution for into the small investors value function and simplifying then shows that the small investors maximization problem reduces to:

where is given by

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.

is the vector of expected returns on the idiosyncratic components at time and is the variance covariance matrix of the idiosyncratic returns. The expression shows that small investors portfolio maximization problem can equivalently be written in terms of choosing an exposure to the returns of the liquid assets, and to the idiosyncratic component of returns of the illiquid assets.

Solving for optimal then shows

The aggregate demand for at time by all small investors can be found by integrating both sides of equation (B9) with respect to , the density of small investors, yielding:

The price schedule faced by large investors at time maps large investors desired orderflow of the illiquid assets into the time prices at which the competitive fringe is willing to absorb the net orderflow. To solve for the price schedule, I solve for prices in equation (B10) such that when the large investors choose trade at time t-1, then the competitive fringe chooses trade .

Rearranging, equation (B10) while
making the substitutions

and

then produces the price schedule faced by large investors at time :

where,

Given the price schedule in equation (B11), large investors at time solve the maximization problem:

where, substituting in the budget constraint, liquid wealth at the beginning of time is given by

(B16) |

Note: Because dividends are paid in cash, the dividend payments received for holdings of illiquid asset are counted as part of liquid wealth even though the illiquid assets themselves are not counted.

Note that in equation (B15), and are deterministic functions of time that are parameters of the value function. Keeping this in mind, large investors holdings of the liquid assets are solved in the same way as for small investors. Taking expectations in equation (B15), solving for optimal given , and substituting the optimal choice back into the large investor's value function, transforms the large investors maximization problem so that it has the following form:

where,

The large investors play a Cournot game in which each choose his time trade in the illiquid assets to solve the maximization problem in (B17) while taking the trades of the other large investors as given, but while taking into account the effect that his own trades have on the prices of the illiquid assets. Recall the price impact function for the illiquid assets at time is given by equation (B11).

The first order condition for large investors illiquid asset choices is given by:

After substituting for from equation (B11), writing as and simplifying, this produces the following reaction function for large investor :

where,

(B22) |

Stacking the (M-1) reaction functions produces a system of linear equations in unknowns:

Assume that is invertible. Then the solution for is unique, and given by

The solution for is . Therefore, the solution for can be written as:

where,

With the above notation, the equilibrium purchases by large participant in period are given by

Additionally, the equilibrium transition dynamics for beginning of period illiquid risky asset holdings are given by:

where and .

Recall that the equilibrium price function in each time period maps investors beginning of period holdings of risky assets to an equilibrium price after trade. The equilibrium price function for period is found by plugging the solution for large investors equilibrium trades from equation (B26) into the price schedule faced by large investors (equation (B11)). The resulting price function for illiquid asset in period has form:

where,

Large investors optimal time consumption depends on optimal time trades. After plugging the expressions for equilibrium prices, and equilibrium trades [equations (B30), (B31), and (B27)] into equation (B17), large investors consumption choice problem has form:

where

(B35) |

The first order condition for choice of consumption implies that optimal consumption is given by:

Define as the value function to large investor from entering period when the vector of illiquid risky asset holdings is , and his liquid asset holdings are . After substituting the optimal consumption choice in (B36) into equation (B34), this value function is given by:

where,

Tedious algebra then shows that large investor 's value function at time has form:

where the parameters of the value function at time are given by the following Riccati difference equations.

The solution for each small investors consumption depends on small investors optimal trades. To solve for optimal consumptions, I first use equation (B9) to substitute out for in equation (B8). I then substitute out for with the expression:

(B44) |

where,

Finally I substitute out with . With these substitutions, small investors choice of optimal consumptions simplifies to:

where,

(B48) |

The first order condition for choice of optimal consumption implies that optimal consumption is given by:

Define as the value function to small investor from entering period when the vector of illiquid risky asset holdings is , and his liquid wealth is . After substituting the optimal consumption choice in (B49) into equation (B47), this value function is given by:

where,

(B51) |

Simplification then shows that the value function has form:

The parameters in the small investors value functions at time are a function of time parameters as expressed in the following Riccati equations:

(B53) |

(B54) |

This completes part I of the proof because equations (B39) and (B52) verify that the value functions at time have the same form as at time .

To establish part II of the proof, I need to show that investors value functions for entering entering period , the last period of trade, has the same functional form as given in the proposition. To establish this result, I first need to solve for investors value function at time , the first period when investors cannot trade the illiquid assets (recall they can continue to trade the riskless asset and the liquid assets indefinitely). Then, given this value function, I use backwards induction to solve for investors value function at time .

9.2.1 Investors Value Functions at Time T+1

Recall that investors are infinitely lived but that from time onwards they cannot alter their holdings of illiquid assets, but they can continue to alter their consumption, and their holdings of liquid and riskless assets. Because investors cannot trade in period and after, the distinction between small and large investors after this period is irrelevant. Hence, the index used below could be for either a large or small investor. Using the Bellman principle, the value function of entering period () with illiquid asset holdings and liquid wealth satisfies the functional equation:

where,

and,

Inspection shows that the function

with

exp

satisfies the Bellman equation (B57) for
all time periods .
Given the value function at time , to solve for investors value functions at time , I follow the same steps as in equations (B4) through equation (B56). Therefore, substituting in from equation (B59), small investors maximization problem at time has form:

such that,

(B61) |

where,

exp | (B62) | ||

(B63) | |||

(B64) | |||

(B65) | |||

(B66) | |||

(B67) |

Substituting the expression for into the value function, taking expectations, and then solving for optimal given , and substituting that into the value function shows that small investors optimal choice of and problem has form:

where

Integrating the solution for optimal over the set of small investors then reveals that the net demand for the illiquid assets by the competitive fringe is:

(B71) |

Following the approach that was used earlier to solve for the price schedule faced by large investors in equation (B11), inverting the small investors demand schedule for the illiquid assets reveals that the price schedule faced by large investors has the form:

(B72) |

Given the price schedule at time , and the value function in equation (B59), large investors maximization problem at time can be written in the form:

(B76) |

where,

Substituting in the budget constraint, liquid wealth at the beginning of time is given by

(B81) |

Large investors maximization problem at time has exactly the same form as given in equation (B15). Therefore, the optimal trades and consumption of large investors follow precisely the same equations as given in Part I of the proof. Large investors value function at time also has the same functional form as in part I. The equilibrium price function at time also has the same functional form as in part I. Therefore, to complete the proof, it suffices to solve for small investors consumption and then value function and verify that the value function has the appropriate functional form.

To do so, note that from equation (B68), it is straightforward to show that the optimal choice of is

(B82) |

where,

(B83) |

Using the same approach that was used to solve for large investors optimal consumption and then value function in part I of the proof, tedious algebra shows that small investors value function at time has form

(B84) | |||

(B85) |

(B86) |

(B88) |

This completes the proof by establishing that large and small investors value functions take the hypothesized form in all periods that involve trade.

**Proof:**

**For
and
:**

The proof is by induction. First, suppose that the results for and are true at time . Then, from equation (B12), . This implies that from equation (B23) that . As a result , which implies from equation (B32) that and from equations (B28) and (B30) that Substituting for and in equation (B42) and simplifying then shows:

(C6) |

Finally, substituting this result in equation (B18) proves the result for . To complete the induction, I use equations (B78) and (B18) to solve for ; I then substitute the resulting expression as well as the one for (equation (B73)) in equation (B23) and use it to show that , which implies . Substituting into equation (B32), then shows that , which confirms the result for . Finally, given the solutions for and , substitution in equations (B78) and (B18) confirms the result for and completes the induction.

**For and
:**

The proof is by backwards induction. We know from equation (B77). Using this expression, and iterating on equations (B40) and (B38) proves the result for all times

**For :**

The proof is by backwards induction. Equation (B80) establishes that it is true at time . Plugging the solution for into equation (B41) while using the solutions for and the result confirms the result for periods .

The next proposition provides information on the value functions of the small investors:

**Proof:**

**For and
**:
Plugging the solutions for and
from
proposition 3 into equation (B45) shows that
for all
times . Since
for all
times , it then follows from
equation (B55) that if
,
then so does
. To
complete the induction, note that substituting the solutions for
(equation (B69)) and (proposition 3) into equation (B87)
confirms the result.

**For ,
, and
**:

The form of the proof is identical to that given in proposition 3.

(C12) |

(C13) |

**Proof**: The proof is by induction. First, assume that the
theorem is true at time .
Then, from equations (B14) and (B13)
,
and
, where
is
and
is
. Applying
these substitutions in large investors reaction functions and then
stacking the results reveals that in equation (B25),
and
. The assumption
that the Nash Equilibrium trades in each period are unique implies
that
is
invertible. Solving for
and
then shows
that
and

(C14) | |||

(C15) | |||

(C16) |

where is a vector of ones, and is . Since , it follows that for . From here, substitution in equation (B33) shows that and substitution in equation (B43) and (B19) shows that . To complete the induction, I substitute the expression for (equation (B79)) into equation (B19) and show that the result is true for . Then, following steps similar to those in the first part of the induction, it is straightforward to show that the result holds for and , which completes the induction. .

**Proof**: Straightforward induction involving application of
the results from proposition 5.

11 Proofs of Asset Pricing Propositions

*Proof:* When investors risky asset holdings are
, then investors
asset holdings are identical to those associated with a competitive
equilibrium and complete markets in which trading is restricted to
the set of market participants that has been modeled. Hence, when
trade in the first set of assets is restricted to be among the
market participants, asset holdings are pareto optimal in all time
periods; and investors asset holdings will remain at because investors have no
basis to trade away from asset holdings that are associated with
perfect risk sharing. Because is the vector of asset holdings from a
competitive equilibrium, the resulting prices and expected returns
which support are the
same as in the competitive equilibrium.

**Proof**: Algebra shows that when asset holdings of asset 1
at time are
, then excess
returns of asset 1 are equal to:

**Proposition 1**: *When investors
asset holdings of the first asset are not , then equilibrium expected
asset returns satisfy a linear factor model in which one factor is
the returns on asset 2, another factor corresponds to perfect
risk-sharing, but imperfect diversification of the idiosyncratic
risk of asset 1, and the remaining factors correspond to the
deviation of large investors asset holdings from those associated
with the large investors perfectly sharing the idiosyncratic risk
of asset 1.*

**Proof**: Let
denote the vector of asset holdings of asset 1 that is associated
with perfect risk sharing among the investors that trade in asset
1. Manipulation of the equation for equilibrium prices given in
proposition 2, and substitution of
for shows:

Plugging in the solution for shows the first term in braces on the right hand side of the equation is equal to . The second term in braces is zero since proposition 3 shows that Adding and subtracting to , the above equation can be rewritten as:

Using the fact that , the vector can be expressed in terms of the deviations of large investors asset holdings from pareto optimal asset holdings:

Applying the substitution for , and the result of corollary 3 in equation (D1) shows

Finally, applying the algebra used in the derivation of proposition 5 shows

(D2) |

where is . Making this substitution then shows:

(D3) |

where .

**Corollary 1**: *When asset
holdings at time are
not efficient, then asset returns at time follow a factor model in which
the market portfolio, the portfolio of segment residual risk, and
the deviation of large investors time asset holdings from efficient asset
holdings are factors.*

**Proof**: Iterating equation (D1),
by periods shows:

Iterating the equation for equilibrium trades in each period shows

where,

11.1 Competitive Benchmark Model

It is useful to contrast the behavior in the multi-market model with large investors with the behavior of asset prices and trades in the same model when all investors are price takers and can trade forever.

In this infinite period set-up with competitive markets, the equilibrium risk-premium should be time invariant. Denote this risk premium by , where,

(D5) |

Note that is taken as exogenous. The goal is to solve for and that makes the prices of the first group of assets (the ones that are illiquid in the imperfect competition model) consistent with equilibrium in all time periods.

Solving the equation for forward while imposing the transversality condition shows that

Given the hypothesized behavior of prices, it remains to solve for and then to show that the hypothesized behavior of prices is consistent with equilibrium.

The function,

and the risk premium solution

where,

(D7) |

satisfies the Bellman equation,

In addition, in the competitive equilibrium, investors optimal choices of are constant through time, and are market clearing for the hypothesized . Investor competitive equilibrium holdings of is denoted by and is equal to

(D8) |

Substituting the hypothesized into the expression for equilibrium , it follows that in a competitive equilibrium, the equilibrium price is given by

(D9) |

12 Solving the IPO Allocation and Price-Setting Problem

The underwriter's problem in equations (16) - (18) problem requires that he maximize a nonlinear objective function subject to the equality constraint that the total issue is allocated at the IPO or sold in the aftermarket, and subject to a set of nonlinear participation constraints. To solve the maximation, the participation constraints were expressed in terms of investors certainty equivalent wealth. The transformed participation constraints are quadratic in the state variables. Additionally, the equality constraint was used to express in terms of the size of the issue, and the allocations to other investors:

To economize on notation, I will drop the "IPO" superscript below in what follows. The Lagrangian for the transformed maximization problem is:

(E1) |

The necessary conditions for an optimum are given by the Kuhn-Tucker conditions:

with c.s. | (E2) |

and for :

with c.s. | (E3) |

If an investor purchases any shares at the IPO, then his utility is decreasing in the IPO offer price. Therefore, if shares are issued at the IPO, then the incentive constraints must be binding for some investors, because if they were not the underwriter could profit by raising the offer price until the incentive constraints do bind.

Although it is clear the incentive constraints must bind for at
least one investor, it is not clear for how many other investors
the constraints will bind. I solved the model in the special case
when there are 6 investors () and the only asset in the segment is the new
issue. To solve the model, I assumed that the underwriters short
position is limited, i.e.
With
this added constraint, the feasible choices of lie on a simplex. I discretized
the simplex so that each investor's post IPO asset holdings could
take one of 21 values. Given a vector of asset holdings, I then
solved for the highest IPO offer price that satisfies all investors
incentive compatibility constraints,^{38}, and then evaluated the objective
function. The point on the discretized simplex for which the
objective function was highest was treated as being in a
neighborhood of the global optimum, and was used as a starting
point for a second stage optimization. Additionally, the investor
or investors for whom the incentive compatibility constraints were
binding, were deemed as the investors for whom the constraints will
be binding at the optimum.

The second stage optimization minimized the squared norm of the Kuhn-Tucker conditions while checking for changes in the set of investors for whom the constraints are binding. A quadratic hillclimbing algorithm with safeguards was used in the second stage optimization. When the investors all differ in their risk aversion, as they do in some of the examples in the paper, then the incentive compatability constraints tended to bind for a single investor. Under this circumstance, the Kuhn Tucker condition for price produce an analytical expression for , which simplifies the second-stage optimization. A complicating factor in the second stage is that the Hessians of the first and second stage objective functions are singular. I believe (but cannot show), that Gauss' nonlinear constrained optimization crashed because of this property. To compensate for this problem in the second stage, at each iteration I projected the direction vector onto the range and null spaces of the hessian matrix, and then solved for the projection coefficients that guarantee an increasing step in the objective function. Finally, I found the book Practical Optimization, by Gill, Murray, and Wright (1981) to be a very useful reference for solving nonlinear optimization problems.

13 Market Segmentation and Market Adjusted Abnormal Returns

The purpose of this section is to study the properties of market adjusted abnormal returns when there is market segmentation. The analysis and results are closely related to Merton (1987). The model contains two time-periods and blocks of risky assets, and a riskfree asset that is in perfectly elastic supply and has return . The supply of shares outstanding in block is denoted by the vector . Note that the number of assets can vary from block to block. Investors trade the assets in period 1 and consume in period 2. The period 2 payoff vector per share in block is equal to . The payoff of is further decomposed into its mean , a component that is sensitive to systematic risk factor , and an idiosyncratic component :

(F1) |

I assume that , and . The are uncorrelated across blocks and are uncorrelated with .

In addition to the blocks of assets, there is a single asset in zero net supply whose payoff vector is:

(F2) |

The N'th market can be viewed as a market for sharing systematic risk.

At time period 1, the price vector of the risky assets is denoted by , .

I assume that for some reason that is not specified here, asset markets are segmented. This means that each block of assets has its own set of investors that only take positions in the segment assets and in asset . For simplicity, the investors in each block are modelled as a single representative price-taking investor with CARA utility of period 2 wealth and risk tolerance .

The market portfolio has payoff , and price . Its excess return over the risk free rate is denoted by ; and the excess return of the assets in block over the risk free rate is denoted by .

Armed with this notation,

(F3) |

and,

(F4) |

If I use the primitive assets to construct assets with payoffs
, and a single asset with payoff
, then the new assets
will span the same space as the old, but they are easier to work
with. In particular, working from the new assets, it is
straightforward to show that for each block , under the assumption of market
segmentation,

(F5) | |||

(F6) |

where , and , and .

Using the above expression to solve for the market's excess expected return then shows:

(F7) |

The expression for contains two components. The second component is the standard risk premium for an assets covariance with market risk when investors fully diversify their asset holdings. The first component is an additional premium for imperfect diversification. Examination will show that each term of the imperfect diversification premium is positive, which implies that imperfect diversification increases the return on the market portfolio.

Below, I seek to examine how imperfect diversification affects market adjusted abnormal returns. Letting , algebra then shows:

The term on the right hand of equation (F8) is the vector of Jensen's alphas for block that is due to market segmentation. These alphas capture the risk premium for inefficiently sharing risk across market segments (or blocks). My goal here is to characterize the average behavior of the . Simple algebra shows . Because , it follows that some of the market corrected must be positive and some must be negative. Additionally, if the number of shares outstanding of each asset is the same, then the average value of is equal to 0. This suggests that to a first approximation when the market is perfectly competitive, and there is segmentation, then the average market adjusted is 0.

When there is imperfect competition, I showed that asset returns
have two components.^{39} The first component is compensation
for imperfect risk sharing across blocks (or segments), and is
identical to the component above for the competitive case. The
second component is for inefficient risk sharing among the
investors who trade within a segment. Because investors within each
segment eventually share risks optimally, this second component is
transitory. For simplicity here, I will write it as . should be close to 0 for
segments that have not experienced an IPO recently, but may be
nonzero if they did. Note: if the first asset in a segment is an
IPO, and the risks of the other assets are shared efficiently
within the segment, then the allocation distortions at the IPO will
cause to depart from
0 for all assets within the segment, but the size of the departure
for each asset is proportional to the first column of
. If
is
diagonal, then the other assets in the segment are not affected by
the IPO; but they will be otherwise. To examine the role that this
term plays in market adjusted excess returns, note that
should be
added to each segments excess returns, and
should be added to the excess return on the market. Arithmetic then
shows that:

The second term on the right hand side of the above equation
reflects how an IPO affects the returns within a particular market
segment. To get a feel for the magnitude of this term, with great
loss of generality suppose for a moment that an IPO occurs only in
segment 1, that there is only 1 risky asset per segment, and that
the characteristics of the assets in each segment are identical
(same number of shares, etc). Then is nonzero in segment , and 0 in the other segments. Algebra shows that
;
solving for and
substituting in the right hand side of (F10)
then shows that if segment 1 has an IPO then

where the last line follows because by assumption in this specialized case, , and for all . Generalizing a bit, since is 0 on average, this analysis suggests that averaging across IPOs through time and across segments, the average market adjusted returns on IPOs will be where is the fraction of segments that have an IPO. This implies that IPO's should underperform after adjusting for market risk.

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European IPO Bookbuilding,"
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Mutual Funds,"
*The Journal of Finance*, Forthcoming. - Ritter, J.R., 2005, "Some Factoids About the 2004 IPO Market," mimeo, the University of Florida, http://bear.cba.ufl.edu/ritter.
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Investor Number | Type | Risk Bearing Capacity |
---|---|---|

1 | Retail | 10.00 |

2 | Institutional | 54.56 |

3 | Institutional | 21.82 |

4 | Institutional | 8.73 |

5 | Institutional | 3.49 |

6 | Institutional | 1.40 |

Risk Bearing Concentration (Herfindahl Index) | 3543.26 |

**Table 1**: IPO Under-Pricing and Money Left on the Table: I.

B. IPO Under-Pricing and
Money Left on the Table (MLOTT)

Periods Liq | Prices P_Offer | Prices P_Open | Prices P_Comp | Prices % Und_Price | MLOTT Raw | MLOTT Liq. Adj. |
---|---|---|---|---|---|---|

2000 | 10.13 | 14.25 | 10.50 | 40.66 | 164.75 | 14.90 |

1800 | 10.06 | 14.83 | 10.50 | 47.38 | 190.73 | 17.38 |

1600 | 10.00 | 15.43 | 10.50 | 54.26 | 217.12 | 19.79 |

1400 | 9.94 | 16.04 | 10.50 | 61.33 | 243.93 | 22.23 |

1200 | 9.88 | 16.66 | 10.50 | 68.61 | 271.18 | 24.72 |

1000 | 9.82 | 17.29 | 10.50 | 76.10 | 298.87 | 27.24 |

800 | 9.75 | 17.93 | 10.50 | 83.82 | 327.01 | 29.81 |

600 | 9.69 | 18.58 | 10.50 | 91.76 | 355.60 | 32.41 |

400 | 9.62 | 19.24 | 10.50 | 99.94 | 384.65 | 35.06 |

200 | 7.95 | 20.62 | 10.50 | 159.36 | 506.89 | 101.74 |

**Table 1**: IPO Under-Pricing and Money Left on the Table: I.

C. IPO Allocation
Distortions (Percent)

Investor Number | ||||||

Post-IPO Trading Periods | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

2000 | -100.00 | 83.29 | -100.00 | -100.00 | -100.00 | -100.00 |

1800 | -100.00 | 83.29 | -100.00 | -100.00 | -100.00 | -100.00 |

1600 | -100.00 | 83.29 | -100.00 | -100.00 | -100.00 | -100.00 |

1400 | -100.00 | 83.29 | -100.00 | -100.00 | -100.00 | -100.00 |

1200 | -100.00 | 83.29 | -100.00 | -100.00 | -100.00 | -100.00 |

1000 | -100.00 | 83.29 | -100.00 | -100.00 | -100.00 | -100.00 |

800 | -100.00 | 83.29 | -100.00 | -100.00 | -100.00 | -100.00 |

600 | -100.00 | 83.29 | -100.00 | -100.00 | -100.00 | -100.00 |

400 | -100.00 | 83.29 | -100.00 | -100.00 | -100.00 | -100.00 |

200 | -100.00 | 83.29 | -100.00 | -100.00 | -100.00 | -100.00 |

Table 1 Continued

Notes: When there is a continuum of small retail investors (investor 1), and 5 large investors (investors 2-6), the table reports expected IPO underpricing and money left on the table for 10 IPOs that vary by the number of days of trading that take place in the IPO aftermarket (Periods Liq). Fewer trading days correspond to less aftermarket liquidity. Each investors risk bearing capacity (panel A) is his own risk tolerance as a percent of the sum total of all investors risk tolerances. Percentage underpricing (panel B) is equal to the net expected one-day return associated with purchasing at the offer price (P_Offer) and then selling at the first days closing price (P_Open). Raw money left on the table is the difference in P_Offer and P_Open times the number of shares issued. Liquidity adjusted money left on the table is equal to the shares issued times the one day return associated with purchasing at P_Offer and then selling into a competitive (and perfectly liquid) aftermarket at price P_Comp. In a competive market, each investors allocation at the IPO should be equal to risk capacity times the number of shares issued. Allocation distortions (panel C) measure are the percentage difference in investors allocations at the IPO relative to the allocation he would have received in an efficient market.

Investor Number | Type | Risk Bearing Capacity |
---|---|---|

1 | Retail | 10.00 |

2 | Institutional | 40.00 |

3 | Institutional | 12.50 |

4 | Institutional | 12.50 |

5 | Institutional | 12.50 |

6 | Institutional | 12.50 |

Risk Bearing Concentration (Herfindahl Index) | 2225 |

B. IPO Under-Pricing and Money Left on the Table (MLOTT)

Periods Liq | Prices P_Offer | Prices P_Open | Prices P_Comp | Prices % Und_Price | MLOTT Raw | MLOTT Liq. Adj. |
---|---|---|---|---|---|---|

10.50 | 10.50 | 10.50 | 0.00 | 0.00 | 0.00 | |

10.50 | 10.50 | 10.50 | 0.00 | 0.00 | 0.00 | |

1600 | 10.53 | 10.71 | 10.50 | 1.70 | 7.14 | -1.38 |

1400 | 10.69 | 11.65 | 10.50 | 8.97 | 38.33 | -7.62 |

1200 | 10.85 | 12.61 | 10.50 | 16.27 | 70.59 | -14.07 |

1000 | 11.01 | 13.60 | 10.50 | 23.49 | 103.48 | -20.65 |

800 | 11.18 | 14.61 | 10.50 | 30.62 | 136.94 | -27.34 |

600 | 11.35 | 15.63 | 10.50 | 37.65 | 170.96 | -34.14 |

400 | 11.52 | 16.66 | 10.50 | 44.59 | 205.55 | -41.06 |

200 | 11.70 | 17.72 | 10.50 | 51.43 | 240.70 | -48.09 |

C. IPO Allocation Distortions (Percent)

Investor Number | ||||||

Post-IPO Trading Periods | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

1600 | -100.00 | 147.50 | -100.00 | -100.00 | -100.00 | -100.00 |

1400 | -100.00 | 149.57 | -100.00 | -100.00 | -100.00 | -100.00 |

1200 | -100.00 | 149.77 | -100.00 | -100.00 | -100.00 | -100.00 |

1000 | -100.00 | 149.84 | -100.00 | -100.00 | -100.00 | -100.00 |

800 | -100.00 | 149.88 | -100.00 | -100.00 | -100.00 | -100.00 |

600 | -100.00 | 149.90 | -100.00 | -100.00 | -100.00 | -100.00 |

400 | -100.00 | 149.92 | -100.00 | -100.00 | -100.00 | -100.00 |

200 | -100.00 | 149.93 | -100.00 | -100.00 | -100.00 | -100.00 |

Table 2 Continued

Notes: This table is similar to Table 1 except that investors risk bearing capacity is less concentrated. When there are 2000, and 1800 post-IPO trading perids (marked with an asterisk), the aftermarket is sufficiently competitive that investors and the underwriter are nearly indifferent over how shares are allocated. I have assigned efficient share holdings in this case at competitive prices. When the aftermarket is slightly more competitive than in Table 1, but not perfectly competitive, the underwriter can allocate shares at a price that is higher than the competitive price (which is also the price with perfect liquidity). In such circumstances, liquidity adjusted money left on the table is negative.

*Notes*: For the market structure in panel A of Table
1, the figure presents optimal IPO offer prices
as a function of constraints on the amount of shares that investor
2 can acquire at the IPO. Results are presented for when the
aftermarket is perfectly competitive (nearly flat solid line), and
when it is imperfectly competitive and there are 2000 (short
dashes) or 400 (long dashes) periods of aftermarket trading
following the IPO. The figure shows that when there is imperfect
competition, the underwriter should distort share holdings towards
investor 2 in order to increase the revenue raised at the
IPO.

*Notes*: For the market structure in panel A of Table
1, when there are 2000 periods of trade
remaining following the IPO, the figure presents share turnover
[(buy volume + sell volume)/2] as a percent of shares
outstanding.

*Notes*: For the market structure in panel A of Table
1, the figure presents expected cumulative
abnormal returns (CARs) relative to the market portfolio for
differing numbers of periods of liquid trading in the aftermarket
following the IPO. The figure is constructed under the assumption
that IPO's are an infinitesimal fraction of the market portfolio.
If instead, new issues represent percent of the market, then the CAR should be
adjusted downward in magnitude by approximately percent. See appendix F for details.

1. Board of Governors of the
Federal Reserve System. The views expressed in this paper are those
of the author but not necessarily those of the Board of Governors
of the Federal Reserve System, or other members of its staff.
Address correspondence to Matt Pritsker, The Federal Reserve Board,
Mail Stop 91, Washington DC 20551. Matt may be reached by telephone
at (202) 452-3534, or Fax: (202) 452-3819, or by email at
[email protected]. Return to
Text

2. Ellis, Michaely, and O'Hara's (2000) study of NASDAQ
IPO's, found that turnover on the first trading day is equal to
1/3rd of the turnover that a NASDAQ stock experiences over a year.
Return to Text

3. Institutional investors
receive more favorable allocations than retail investors in the
most underpriced issues. Return to
Text

4. There is an ongoing
debate within the empirical literature concerning whether IPO
underperformance exists and whether it is statistically
significant. For a discussion of these issues see Viswanathan and
Wei (2004), de Jong and Dahlquist (2003), Schultz (2003), Loughran
and Ritter (2000), and Brav, Geczy, and Gompers (2000). Return to Text

5. In related work, Hoberg
(2004) presents a model in which underpricing results from
imperfect competition in securities underwriting. Return to Text

6. In the dominant strand of the underpricing literature,
underpricing occurs because information-asymmetries cause adverse
selection in allocating shares [Rock (1986)], or, because
underpricing is needed to entice informed investors to reveal
information to the underwriters, as in the bookbuilding models that
starts with Benveniste and Spindt (1989). Other theories of
underpricing are based on agency problems in which the underwriters
objective function departs from that of the issuer (Boehmer and
Fishe, 2000), (Bias et. al. 2002), (Loughran and Ritter, 2004). For
extensive reviews of this literature, see Ritter and Welch (2002),
and Ljungvist (2004). Return to
Text

7. Ritter and Welch (2002)
claim there are no rational theoretical models that explain
underperformance. Ljunqvist, Nanda, and Singh (2003) present a
behavioral model that generates all three phenomena. Return to Text

8. In the theoretical model
of Booth and Chua (1996), underpricing is used to increase the base
of investors that are interested in the new issue and increase
aftermarket liquidity. In Westerfield (2003) underpricing is also
used to influence the composition of the investor base. If Booth
and Chua are correct, then underpricing should be negatively
correlated with illiquidity, but this runs counter to the empirical
results in Ellul and Pagano, and hence casts doubt on the empirical
relevance of Booth and Chua's model. Return to Text

9. Other differences with
Ellul and Pagano are that their model has information asymmetry
while the present model does not, and in their model, the
aftermarket only lasts for one period whereas in the present the
model the aftermarket trading is fully dynamic and modelled over
thousands of periods. Return to
Text

10. The first stage share
allocation process resembles IPO bookbuilding: in both processes
the underwriter collects information about market demand, and then
allocates shares and sets an offer price based on the information
that he collects. Return to
Text

11. See footnote 5. Return to
Text

12. Closely related models
of imperfect competition in asset markets include Urosevic (2002a
& b), DeMarzo and Urosevic (2000), and Vayanos (2001). Return to Text

13. The results would be
very similar if some investors hold but do not trade the segment
assets. Return to Text

14. These assumptions are
loosely based on Merton (1987). Return to Text

15. It is a matrix because
his trades in one asset may affect the prices of other
segment-assets. Return to
Text

16. The presence of this
term suggests there is a CAPM-like relationship in the model for
sharing segment-risk. This is similar to Stapleton and Subrahmanyam
(1978), who derive circumstances in which the CAPM holds
dynamically through time when investors have CARA utility and trade
risky assets whose dividend payments are normally distributed.
Return to Text

18. In Pritsker (2004),
large investors can be interpreted as institutions that trade on
behalf of a base of small investors. In this interpretation, large
investors absolute risk tolerances are just the integrated risk
tolerances of their small investor base. If all small investors
have identical risk tolerances, then large investors risk
tolerances will differ substantially depending on the size of their
investor base. Return to
Text

19. There are many other
possible ways to model the threats that available to the large
investors and the threats that are available to the underwriter.
Return to Text

20. The aftermarket price
on the first day of trading was solved for in section 3; it is given by equation (B3). Because a distressed sale in period 1 is
tantamount to sales by investors that don't account for the price
impact of their trades, for the purposes of price determination if
is not equal to
0, then is added
to the initial allocation of investor 1 in order to determine the
aftermarket price. Return to
Text

21. The value functions
are derived from proposition 2, but
they contain an additional argument to account for the underwriters
net sales in the aftermarket . Return to
Text

22. The "Green Shoe" option
that is often given to underwriters allows the underwriter to cover
his short position following the offering by acquiring up to a
certain amount of additional shares from the issuer at the IPO
offer price. This option is valuable when there is uncertainty
about demand for the new issue. In the present model, there is no
demand uncertainty. Therefore, underwriters are not allowed to
short-cover by using the "Green Shoe" option. Return to Text

23. In Ellis et al.'s
(2002) sample of 313 NASDAQ IPOs, the lead underwriter participated
in an average of more than 90 percent of post IPO NASDAQ trades
during the first day of the IPO; this amount tapers down over the
next 140 days, but remained above 40 percent on average on the
140th day Return to Text

24. The solutions for
investors value functions and optimal trades when a distressed
investor optimally sells a position over time periods is similar to
Pritsker (2004), and is not presented here in order to save space.
Return to Text

25. The Herfindahl index is
equal to sum of each investors squared percentage share of risk
bearing capacity. Because each small investor is infinitesimal,
their contribution to the index is 0. Return to Text

26. The time scalefor the
trading periods is determined by the annualized per-period interest
rate. There are 250 trading periods per year, and the annualized
per-period interest rate is fixed at 2 percent. Return to Text

27. Alternative intuition
for the relationship between illiquidity and the number of post-IPO
trading periods is based on Coasian analysis of a durable goods
monopolist. Coase shows that a durable goods monopolist has more
market power if he can commit to selling over a single time period
instead of allowing for several periods of retrade. Kihlstrom
(2001) argues that stocks are durable goods and shows that the
Coasian analysis applies to a monopolist in stocks. The results
here are similar to Kihlstrom, but in an oligopoly setting.
Return to Text

28. The analysis used
approximate optimization over a discrete grid of allocations.
Return to Text

29. For example, in
informational theories of bookbuilding, the money left on the table
is equal to informational rents that investors receive for sharing
their information with the underwriter. Return to Text

30. The other 4 investors
each had 10% of the risk bearing capacity. Return to Text

31. In the case of
institutional investors, represents wealth under management, and the
measure of risk bearing capacity implies that more wealthy
investors have a greater capacity to bear risk. Return to Text

32. Getting data on IPO
allocations is difficult, but it is sometimes available on a
proprietary basis as in Cornelli and Goldreich (2003), or can be
constructed as in Reuter (2005). Return to Text

33. They do not report any
results on tests for the strength of the instruments, nor do they
report any results of tests for instrument validity. Return to Text

34. A difficult issue in assessing return
underperformance is whether to adjust returns for market-to-book
effects. The standard basis for adjustment is to determine whether
an empirical pricing anomaly is new, or is a manifestation of a
known anomaly such as market-to-book. Because market-to-book is an
empirical regularity, but not a theory, risk adjustment for it may
lead to spurious inference about IPO underperformance when testing
the present theory. To see what can go wrong, suppose that the
forces that drive market-to-book effects for other stocks don't
apply to IPO's, but that market-to-book is negatively correlated
with IPO returns for other theoretical reasons, such as those given
in this paper. In this circumstance, adjustment for market-to-book
effects will spuriously bias the test against detecting
underperformance when it is present. A better method to test the
present theory is to forgo a problematic market-to-book adjustment,
and instead test the predicted relationships between allocation
distortions and the returns on new issues. Return to Text

35. A difficult issue in
testing liquidity-based theories of IPO underpricing and
underperformance is determining an appropriate measure of
liquidity. Eckbo and Norli (2002) measure it based on trading
volume; but it is not clear that high volume after an IPO has the
same liquidity implications as high volume for other stocks. For
example, if IPO's extraordinarily high turnover (see footnote
1) implied they were very liquid for large
investors, then those investors should be selling out en masse on
the first day following an IPO because of the high first day
return. In fact, Aggarwal's (2003) study of sales by initial share
holders, which is also known as share flipping, shows that on
median only 7.34% of initial shares are sold during the first few
days following an IPO. This suggests the extraordinarily high
trading volume associated with IPOs does not necessarily translate
into a highly liquid environment in which large investors can sell
their shares. Post IPO volume may mismeasure liquidity because, as
noted by Aggarwal (2003) it is associated with the same small
number of shares being traded very frequently within the day, and
does not measure the price impact of a large number of shares being
sold into the market. This is consistent with Corwin et. al.
(2004), who find that NYSE-listed IPOs typically have low depth to
trading volume during the first several days of trading.
Additonally, for investors who risk punishment from underwriters
for flipping too many shares, the high volume does not measure
their ability to sell off their holdings. Finally, although volume
may be problematic as a measure of liquidity, other measures based
on bid-ask spreads and price impacts may be problematic because it
may not measure liquidity on the scale that large shareholders wish
to transact. Return to
Text

36. For example,
Return to Text

37. To illustrate the use
of the selection matrix,
.
Return to Text

38. This step is simple
because the price that makes each investors incentive constraint
bind has a closed form expression. Return to Text

39. When I speak of
imperfect competition here, it is only over a single period of
trade, which is a special case of the dynamic model in the paper. I
have full confidence that all of the reasoning in this subsection
can be written in terms of the general dynamic model in the text,
but for now it is easier not to do so formally. Return to Text

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