How does the Distribution of OwnershipAffect theProduction of Information ?

Joel Peress∗

INSEAD

August 26, 2004

JEL classification codes : D82, D83, G11, G12, G14

Abstract: A general equilibrium model is developed in which a stock’s ownership structure

determines how much information about its payoff is collected. Large shareholders have an

incentive to research stocks because they can afford a large number of shares. An investor

base expansion affects the production of information through three channels: (i) it enhances

risk sharing which reduces the value of information, (ii) it adds potentially informed investors

to the base, and (iii) it modifies the variance of liquidity trades. Expected returns fall but

their volatility may increase. Evidence from ADR listings supports the model.

∗INSEAD, Department of Finance, Boulevard de Constance, 77305 Fontainebleau Cedex, France. Telephone: (33)-1 6072

4035. Fax: (33)-1 6074 5500. E-mail: joel.peress@insead.edu. I am grateful for comments from Bernard Dumas, Lily Fang,

Harald Hau, Lucie Tepla, Steve B. Wyatt and seminar participants at INSEAD, the 2003 Midwest Finance Association

Meeting, the 2003 CEPR Summer Symposium in Financial Markets and the 2003 European Finance Association Meeting.

1 Introduction

In his presidential address to the American Finance Association, Merton (1987) argues that informational

frictions can explain observed departures from the CAPM. To illustrate his point, he develops an asset

pricing model under the assumption that investors are only aware of the existence of a subset of public

companies. Merton’s hypothesis, known as the “Investor Recognition Hypothesis” (IRH) implies that

the price of a stock depends (positively) on the size of its investor base. The IRH has received support

from several studies which identify events leading to a base expansion and document a price appreciation

around these events. Such events include US over-the-counter companies listing on the NYSE (Kadlec

and McConnell (1994)), non-US stocks listing on US exchanges as American Depositary Receipts (ADRs)

(Foerster and Karolyi (1999), Miller (1999)) and Japanese companies reducing their stock’s minimum

trading unit, i.e. the number of shares in a round lot (Amihud, Mendelson and Uno (1999)).

An aspect of the IRH that has received little attention concerns its impact on the quality of information.

Is a stock held by more investors more closely followed? Or do free-riding problems lead to less scrutiny?

These questions are not only interesting in their own right, answering them can also sharpen the IRH’s

predictions1. Indeed, Merton assumes that all investors aware of a stock share the same assessment of

its prospects. The entry of new stockholders does not modify this assessment but improves risk sharing,

thereby reducing return volatility. This prediction however is at odds with some of the evidence. ADR

listings, for example, are associated with a permanent increase in return volatility (Jayaraman, Shastri

and Tandon (1993), Domowitz, Glen and Madhavan (1998)). Stock splits, another example of a corporate

1Companies can take steps to expand their investor base or attract specific groups of investors. They may advertise theirbrand, increase media coverage through public relations activities, list on additional markets, split their stock or underwriteit through a reputable bank with wide distribution channels. Regulators, managers and shareholders want to account for theconsequences of such measures on the effectiveness of corporate control (through direct monitoring or incentive contracts) oron the stock’s risk-return trade-off.

1

event leading to a base expansion (Lamoureux and Poon (1987), Mukherji, Kim and Walker (1997)) and

a price appreciation (e.g. Grinblatt, Masulis and Titman (1984)) also generate a rise in return volatility

(Ohlson and Penman (1985), Lamoureux and Poon (1987), Sheikh (1989), Dubofsky (1991), Lynch Koski

(1998)). These observations suggest that a base expansion does more than improve risk sharing. We

argue here that it modifies an asset’s information structure2,3.

To investigate these issues, we extend the IRH by allowing stockholders to collect information about

their stocks. In equilibrium, how much information is collected about a stock, what we call the depth

of knowledge in this paper, depends on how the stock is distributed among investors. In particular, it

depends on how many investors recognize the stock, which we call the breadth of knowledge. The main

insight of the paper is that large shareholders have an incentive to collect costly information because

they can appropriate a large fraction of the benefits from their research. Large shareholders are those

with a high tolerance for risk and those investing in stocks in abundant supply relative to the size of

their investor base. Small shareholders free-ride large shareholders’ research which gets partially revealed

through prices. This insight echoes the familiar concept from the corporate control literature which argues

that a firm’s ownership structure influences its corporate governance (e.g. Shleifer and Vishny (1986)).

A crucial difference with this literature is that private information is acquired ex ante, i.e. before cash

flows are observed, and consequently is revealed by prices.

To formalize this insight, we build a model in which investors are endowed with a set of stocks they

recognize and choose whether to invest actively or passively in a competitive market. Active stockholders

2 In this paper, we dot not attempt to explain why companies listing ADRs or splitting their stock attract more investors.We take as given the base expansion and examine its impact on the stock’s information structure.

3Kadlec and McConnell (1994)) and Amihud, Mendelson and Uno (1999) do not explore the possible changes in returnvolatility. Ito, Lyons and Melvin (1998) provide another example of a rise in return variance generated by an increase in thenumber of traders through a change in information. They examine the change in exchange rate variance in 1994 when theTokyo foreign exchange market allowed trading over the lunch break. They find that lunch return variance doubles, whichthey attribute to a change in private information since the flow of public information did not change with the trading rule.

2

spend resources on predicting firms’ cash flows in order to improve their portfolio allocations. Passive

stockholders are not involved in research activities but they extract from prices part of the information

uncovered by active stockholders. The extraction is only partial because the presence of liquidity traders

makes the supply of stocks noisy as in Grossman and Stiglitz (1980). The key property of information is

that its cost, unlike its benefit, does not depend on the number of shares traded. Hence, more information

is produced by larger investors, that is by investors who can spread the cost over a larger number of shares.

Putting it differently, the production of information displays increasing returns to scale4. It follows that

the depth of information depends on a stock’s ownership structure. We assume in the model that investors

differ in their risk tolerance, for example because their differ in their wealth. An investor holds a large

position in a stock she recognizes if she accounts for a large fraction of the aggregate tolerance of those

who recognize the stock. To illustrate the importance of the distribution of risk tolerance, we show how

depth changes when risk tolerance is more equally distributed among investors aware of the stock, keeping

fixed aggregate risk tolerance and the number of shares5.

We also emphasize the role played by aggregate risk tolerance, and through aggregate tolerance, by

the breadth of knowledge. Their importance is shown in the context of an investor base expansion. We

identify three channels through which the depth of information can be affected in an expansion. The

first and most novel follows directly from the insight described above. When no shares are issued, an

expansion leads to a redistribution of ownership from incumbent to new shareholders. The redistribution

takes place through an average price appreciation, which forces incumbent investors to scale down their

4This increasing returns to scale property does not rely on assumptions regarding the cost of information. It holds inspite of a cost convex in the precision of the signal received.

5We show that a mean-preserving spread of the risk-tolerance distribution of investors aware of a stock increases the depthof information when the inverse of the marginal information cost is concave (lemma 4). This condition ensures that the costof information does not increase so steeply with its precision that it deters very risk tolerant investors from acquiring moreinformation.

3

holdings (aggregate risk tolerance increases). This reduces incumbents’ incentives to collect information.

Thus, information collection is limited by the extent of risk sharing : new investors unload some of the

risk from incumbent investors, and, in so doing, make information less valuable. This effect, which we

refer to as the ”negative information externality”, is the focus of the empirical tests we conduct in the

last section of the paper.

Though incumbent investors collect less information following a base expansion, the full effect on

depth is not straightforward because new stockholders may be active. Thus, they may contribute to the

depth of information. This constitutes the second channel. Finally, the variance of liquidity trading may

also change following the expansion, creating a third channel. If it increases, then private information

becomes easier to conceal and therefore more valuable6. The final effect on depth depends on the relative

importance of the three channels, which we quantify in the paper in terms of elasticities. We consider, as

an illustration, two scenarios. Both assume that there is no change in the variance of liquidity trading.

In the first, we add active stockholders to a passive base. Since incumbent shareholders are passive, the

negative information externality no longer operates. The depth of knowledge rises with its breadth, as

active investors join the base. In the second scenario, we add passive stockholders to an active base. The

only channel at work is the negative information externality so depth is reduced. In this case, the depth

of knowledge falls with its breadth.

We also study the implications for the distribution of returns. Returns are affected directly by a change

in breadth and indirectly by the induced change in depth. A larger breadth improves risk sharing while

a larger depth reduces risk (information on the stock’s payoff is more accurate). Both decrease the mean

6An alternative explanation for the return volatility increase following ADR listings or stock splits is that these firmsattract investors who are more prone to liquidity shocks, i.e. small investors. However, the evidence on volume of trade doesnot seem to fit this explanation (see the discussion at the end of section 6). The model presented here allows for but doesnot rely on increases in liquidity trading.

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and variance of returns. The effect of a base expansion is again complex. In the first scenario, breadth

and depth both increase, leading to a fall in the mean and variance of returns. In the second scenario,

breadth increases while depth falls. We show that expected returns fall but the variance of return rises

if the negative information externality is strong enough. In contrast, there is no information effect in the

standard IRH, so the variance of returns always falls.

The last section of the paper presents evidence showing that a negative information externality is at

work in equilibrium, i.e. that incumbent shareholders acquire less information following a base expansion.

We focus on this prediction of the model because it does not depend on how much information new

stockholders produce, if any. We draw on a sample of British companies issuing ADRs as ADR listings

generate a significant base expansion, with limited change in liquidity trading, and allow to disentangle

incumbent from new investors, i.e. those investing in the underlying stock from those investing in the

ADR. We use analyst coverage of the underlying stock to proxy for the amount of information produced.

Consistent with the model, we find that analyst coverage of the underlying stock falls in the year following

the listing. This finding is robust across specifications.

The remaining of the paper is organized as follows. Section 2 briefly reviews the related literature.

Section 3 describes the economy. Section 4 defines the equilibrium concept. Section 5 solves the model.

Section 6 analyzes the effects of an investor base expansion on the depth of information and the distribution

of returns. Section 7 examines ADR listings to provide evidence consistent with the model. Section 8

concludes. Proofs are relegated to the appendix.

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2 Related literature

Several papers establish a link between a asset’s investor base and its information structure. Early models

of economies in which investors have diverse information argue that a greater number of traders makes

prices more informative because the error from aggregating idiosyncratic signals shrinks (e.g. Grossman

(1976), Hellwig (1980), Verrecchia (1982)). Later papers incorporate a negative information externality

as we do. Stein (1987) shows that opening a futures market can create noise if the new traders are

badly informed. Hau (1998) relates the number of traders in currency markets to the mistakes they make

about expected return. In both models, the effect of information critically depends on its quality which

is exogenous. Moreover, idiosyncratic, i.e. trader specific, noise prevails in the aggregate: Stein assumes

all futures traders have the same signal and Hau that expectational errors are correlated across traders.

Consequently, as new traders enter, they bring some noise into the market, which, far from washing out,

adds to the existing noise. Thus, a broader base means greater aggregate risk. Our model differs in that

information may deteriorate even though new investors do not add to the equilibrium noise7. Instead,

the information effect is a rational response to a broader investor base: new investors force incumbents

to reduce their holdings and thus depress incentives to acquire information.

In a different context, Holmstrom and Tirole (1993) assume that liquidity trades increase with a

company’s float, leading to more information collection. As mentioned above, we allow for changes

in liquidity trading but our results do not depend on them. Brennan and Hughes (1991) develop a

model of stock splits. In their setup, the stock price determines the supply of information which in turn

determines the size of the investor base. In line with the IRH, they assume that investors only hold the

7Our results hold in particular when liquidity trading is not affected by the base expansion (e.g. lemmas 6 and 7).Furthermore, in our setup, private signals have independent disturbances which cancel out when aggregated.

6

stocks advertised by their broker through research reports. Brokers earn larger commissions on low-price

stocks so they produce more research about firms which split their stock. By construction, the breadth

of information is proportional to its depth. We complement their analysis with a model in which the

exogenously-given breadth of information determines its depth. Finally, another line of research initiated

by Miller (1977) and taken up by Chen, Hong and Stein (2002), takes as given investors’ information

and investigates how short-sales constraints interfere with the aggregation process. We do not consider

short-sales constraints but focus on information acquisition. In the next section, we describe the economy.

3 The economy

The model combines Merton (1987) with Verrecchia (1982). There are three periods, a planning period

(t = 0), a trading period (t = 1) and a consumption period (t = 2). Investors trade competitively stocks

and a riskless bond. As in Merton (1987), they can only hold positions in stocks of which they are aware.

As in Verrecchia (1982), they may purchase private information about these stocks. Some noise prevents

the equilibrium price from fully revealing private information. Table 1 lists the variables in the model

with their interpretation.

3.1 Assets

K+1 assets are traded competitively in the market, a riskless asset and K risky assets (the stocks). The

riskless asset has a gross rate of return of R and is in perfectly elastic supply. Stock k has a price P k and

a random payoff Πk. The Πk’s are independent from one another and normally distributed with identical

mean Π and variance σ2Π. Allowing for differences in means and variances or for correlations across stocks

would complicate the analysis without yielding new insights. P and Π denote the vectors of stacked P k

and Πk.

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3.2 Investors

There are two groups of stockholders, rational investors (investors or stockholders for short) and liquidity

traders8. Investors maximize a mean-variance objective function, U(W 0j , τ j) ≡ E(W 0

j) − 12τjV ar(W 0

j),

where τ j is investor j’s coefficient of absolute risk tolerance and W 0j is her final wealth

9. There is a

continuum of investors (in number J very large) ranked by risk tolerance. They can borrow freely at the

riskfree rate so their initial wealth is not directly relevant (it may be indirectly in determining τ j) and is

normalized to W . Let Xkj and Yj denote investor j’s holdings of stock k and the bond and let Xj denote

the vector of stacked Xkj ’s.

The liquidity traders are not modelled explicitly. They are subject to shocks that make their de-

mand for assets random and inelastic. Hence, the residual supply of stocks, i.e. the supply offered to

investors once liquidity traders have placed their inelastic orders, is random and unrelated to prices. This

assumption, standard in models where prices aggregate and transmit information, prevents equilibrium

prices from fully revealing stocks’ payoffs and preserves the incentives to purchase private information.

The (residual) supply of stock k is denoted θk and the vector of stacked θk is denoted θ. The θk’s are

identically and normally distributed with mean θ and variance σ2θ. They are independent from each other

and from the Πk’s. In the paper, we focus on the behavior of investors and abstract from the origin of the

randomness in the supply.

8The literature sometimes refers to rational investors as speculators and to liquidity traders as hedgers or noise traders.9Merton (1987) uses such preferences and assumes furthermore that absolute risk tolerance is proportional to initial

wealth. We do not take a stand on the determinants of risk tolerance which we take as given. Grossman and Stiglitz (1980)and Verrecchia (1982) use exponential (CARA or constant absolute risk aversion) preferences. Quadratic mean-variancepreferences are better suited than CARA to address the issues raised in this paper because the latter do not capture thescale effect of information. In particular under CARA preferences, the demand for information is independent of the supplyof shares and the number of investors (see footnote 12). In addition, Peress (2004) shows that locally quadratic preferencesare a good approximation to a wide class of preferences when risk is small.

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3.3 Information structure

Information is heterogenous in two ways. First, investors differ in the scope or breadth of their knowledge.

As in Merton (1987), investors are not aware of the existence of all stocks. Instead, they are endowed

with a set of stocks which they recognize. Let Kj be a set of integers such that k belongs to Kj if investor

j recognizes stock k. Obviously, Xkj ≡ 0 if k is not part of Kj . Symmetrically, define Gk(τ) (gk(τ)dτ)

as the number of investors with risk tolerance below τ (between τ and τ + dτ) who recognize stock k.

The number of investors aware of stock k, or the breadth of knowledge about stock k, is denoted Nk

(0 < Nk ≤ J). Formally, Nk ≡ limτ∞Gk(τ). N refers to the vector of stacked Nk10.

Second, investors differ in the quality or depth of their knowledge. In contrast to the former, this

source of heterogeneity is endogenous. Investors may spend time and resources to collect information

about the stocks of which they are aware. If investor j knows about stock k, she may purchase a signal

Skj about its payoff Πk:

Skj = Πk + υkj (1)

where V ar(υkj ) = 1/xkj , υ

kj is independent of Π

k, θk and across agents and stocks. xkj is the precision of

agent j’s signal about stock k. The signal costs C(xkj ). We assume C is continuous, increasing, strictly

convex and C(0) = 0. For example, we illustrate the model with the specification C(x) = cxb where c > 0

and b > 1. Let xj and Sj denote the vectors of stacked xkj and Skj (set S

kj = ∅ if no signal is acquired).

In addition to their private signals, investors use equilibrium prices to forecast payoffs. Let Fj denote

agent j’s information set: Fj = ∪k∈Kj{Skj , P k}. E(. | Fj) and Ej(.) refer to conditional (period 1) and

unconditional (period 0) expectations by investor j, i.e. where her private signals Sj have the precisions

10There are no trading costs so all investors aware of a stock trade it. Also note that other types of frictions such as legalconstraints banning some investors from holding some stocks can be assumed instead of the IRH. In that case, ”breadth ofknowledge” should be read as ”breadth of ownership”.

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listed in xj .

3.4 Timing

The timing is depicted in figure 1. There are 3 periods. Period 0 is the planning period. Investors decide

how much information, if any, to collect about the stocks they recognize. These decisions are made before

the signals Sj and the prices P are observed. The timing ensures that the information decisions neither

depend on Sj nor on P and hence that the equilibrium prices are linear in Π and θ.

The second period (t = 1) is the trading period. Investors observe their private signals Sj with the

precisions xj chosen in the previous period. At the same time, markets open and investors observe the

equilibrium prices of the stocks they recognize. Investors use public and private signals to estimate the

risk and return characteristics of stocks and then choose a position, Xj . The third period (t = 2) is a pure

consumption period: investors consume the proceeds from their investments, W 0j .

4 Equilibrium concept

4.1 Individual maximization

To solve for the equilibrium, we proceed in two stages, working from the end to the beginning of investors’

horizon. In the second period, investor j observes the signals available to her and forms her portfolio,

taking prices as given:

maxXj ,Yj

E£U(W 0

j , τ j) | Fj¤subject to

Π.Xj +RYj =W0

P.Xj + Yj =W −Pk∈Kj

C(xkj )

Xkj ≡ 0 if k /∈ Kj

(2)

Again, Xj and Yj are investor j’s stock and bond holdings and xj , the precision of private signals possibly

equal to 0, is inherited from the previous period. Call u(τ j ,Kj , Sj , P, xj) the value function for this

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problem.

In the planning period, investors choose the precision of their signals in order to maximize their

expected utility averaging over all the possible realizations of signals and prices and taking Kj and C(.)

as given:

maxxjEj [u(τ j ,Kj , Sj , P, xj)] subject to

xkj ≥ 0 for all k

xkj ≡ 0 if k /∈ Kj(3)

If xkj > 0 (xkj = 0) then investor j is actively (passively) investing in stock k.

4.2 Market aggregation

The optimal behavior of an investor (portfolio and precision choices) depends on the behavior of other

investors in the economy since the gains from trade and from private signals depend on how much risk

investors aware of a stock are willing to bear and how much information is revealed through prices.

Accordingly, define the aggregate risk tolerance for stock k as

nk ≡∞Z0

τdGk(τ) for k = 1, ...,K

Aggregate risk tolerance is directly related to the breadth of knowledge. For example, nk is proportional

to Nk if risk tolerance is identical across shareholders. Similarly, let

Dk ≡ 1

σθ

∞Z0

τxkdGk(τ) for k = 1, ...,K (4)

measure the depth of knowledge about stock k, implied by aggregating the precision choices of investors

who recognize it and normalized by the standard deviation of liquidity trades. Call µk its inverse scaled

by σθ, µk ≡ 1/(σθDk). µk measures the noisiness of the price of stock k, a variable that will prove useful in

the next sections. In the text, we refer indifferently to noisiness or the depth of information, one being the

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scaled inverse of the other. In equilibrium, depth and noisiness both depend on and determine individual

decisions. Let n, d and µ denote the vector of stacked nk, Dk and µk. We are now ready for the formal

definition of an equilibrium.

4.3 Definition of an equilibrium

A rational expectation equilibrium is given by investors’ demand for stocks Xj and information xj , prices

P and noisiness µ such that:

1. xj and Xj solve the maximization problem of an investor taking P , µ and n as given (equations 2

and 3).

2. For all stocks, the noisiness of the price µ implied by aggregating individual precision choices equals

the level assumed in the investor’s maximization problem (equation 4):

µk∞Z0

τxkdGk(τ) = 1 for k = 1, ...,K

3. For all stocks, prices clear the market:

∞Z0

XkdGk(τ) = θk for k = 1, ...,K

We solve for the equilibrium in the next section.

5 The breadth and depth of information

The first theorem solves for equilibrium prices for any given vectors of noisiness µ and precisions xj . The

next theorem will solve for the equilibrium values of µ and xj and show how the production of information

about stocks relates to their ownership structure, i.e. how the depth of knowledge depends on its breadth.

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Theorem 1 (Equilibrium prices and stockholdings)

Assume the information decisions have been made (i.e. µ and xj are given). There exists a uniquelinear rational expectations equilibrium.

• The equilibrium price of stock k is given by

RP k = P0(µk, nk) + PΠ(µ

k, nk)(Πk − µkθk) (5)

where h0(µk) ≡ hk0 ≡

1

σ2Π+

1

µk2σ2θhk ≡ hk0 +

1

µknk(6)

P0(µk, nk) ≡ P k0 ≡

1

hk

µΠ

σ2Π+

θ

µkσ2θ

¶and PΠ(µ

k, nk) ≡ P kΠ ≡1

hk

µ1

µk2σ2θ+

1

µknk

¶(7)

• Investor j’s holdings of stock k areXkj ≡ 0 if k /∈ Kj

Xkj = τ j

hXk0 + x

kj (S

kj −RP k)

iif k ∈ Kj

(8)

where Xk0 ≡

1

µknk

µP k0 −RP k

P kΠ+RP k

¶(9)

The characterization of the price is well known for this type of economy (e.g. He and Wang (1995) or

Peress (2004)). The proof of the theorem is by construction and is presented in the appendix. We make a

few comments. The equilibrium price depends on the payoff Πk and the random supply θk. θk enters the

price equation although it is independent of Πk because it determines the number of stocks to be held and

hence the total risk investors have to bear in equilibrium. Πk appears directly in the price function though

it is not known by any agent, because private signals Skj are aggregated and collapse to Πk. Observing P k

is equivalent to observing Πk − µkθk which acts as noisy signals for Πk with noise µkθk. For a given σ2Π,

the depth of information, Dk ≡ 1/(σθµk), captures the informativeness of the price signal. hk0 measures

the precision of the price signal and hk0 + xk measures the total precision for an investor using both the

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price and a private signal of precision xk. 1/(µknk) ≡ (∞R0

τxkdGk(τ))/(∞R0

τdGk(τ)) is a weighted average

of private precisions about stock k so hk ≡ hk0 + 1/(µknk) is the average total precision about stock k.

Two polar cases are instructive. If µk = 0 then there is no noise and the price reveals the true Πk.

There is no risk in this economy, the price function reduces to Πk/R and the asset has the same gross

return as the bond, R. On the other hand, if µk = ∞ then the price contains no information about the

payoff: Πk drops out of the price equation which becomes (Π− σ2Πnkθk)/R. The first term equals the present

value of the expected payoff discounted at the risk free rate and the second term is the discount on the

price to compensate investors for the risk incurred. This equilibrium corresponds to the one described in

Merton (1987).

Holders of stock k can be passive (xkj = 0) or active ( xkj > 0). Passive stockholders hold X

k0 shares.

Active stockholders choose the same number of shares except that they scale it by their risk tolerance

and tilt it according to their private signal. Indeed, Skj −RP k is stockholder j’s expectation of the excess

return on stock k based on her private signal Skj . The more accurate the signal (the larger xkj ), the greater

the weight on this expectation.

Theorem 1 takes as given the precision of private signals xj and noisiness µ. The next theorem shows

how investors make their precision choices and how noisiness and the depth of information are determined

in equilibrium.

Theorem 2 (The depth of information)

• Every stock k admits a threshold τ∗k ≡ τ∗(µk, nk) such that only investors aware of the stock whoserisk tolerance is above τ∗k are active. The threshold is given by

τ∗k ≡ 2RC 0(0)A(µk, nk)

where Ak ≡ A(µk, nk) ≡ θ2+ σ2θ + h

k0nk2 + 2nk/µk

(nkhk)2

. (10)

14

• An active investor with risk tolerance τ j who recognizes stock k (k ∈ Kj and τ j > τ∗k) chooses asignal of precision xkj ≡ x(τ j , µk, nk) such that

2RC 0(xkj ) = τ jA(µk, nk). (11)

• The noisiness, µk, of stock k is the unique solution to

µk∞Z

τ∗(µk,nk)

τx(τ , µk, nk)dGk(τ) = 1 (12)

where τ∗(µk, nk) and x(τ , µk, nk) are defined above.

• The depth of information about stock k, Dk, equals 1

µkσθ.

The theorem shows how the production of information about a stock depends on its ownership struc-

ture, that is how the depth of knowledge depends on its breadth. Its proof is presented in the appendix.

It first characterizes the composition of the investor base. Among investors aware of the stock, only those

whose risk tolerance is large enough (τ j > τ∗k) are active. The others are passive. In general, active

and passive shareholders coexist in equilibrium. Nevertheless, all shareholders are active if the least risk

tolerant of all investors aware of the stock has a tolerance larger than τ∗k. This is the case when C 0(0)

is small (for example equal to zero) or when the investor base is narrow (nk small)11. Alternatively, all

shareholders are passive if the most risk tolerant of all investors aware of the stock has a risk tolerance

smaller than τ∗(∞, nk) = 2RC 0(0)/((θ2+σ2θ)/nk2+σ2Π). This occurs when the investor base is broad (n

k

big). In that case, no information is collected about the stock and the equilibrium collapses to the one

described in Merton (1987).

The first order condition for active investors’ demand for information is given by equation 11. As

usual, it states that, at the optimum, the marginal cost of private information equals its marginal benefit.

11As we point out below, A is a decreasing function of nk.

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The cost equals the cost of the signal, C 0(xk), adjusted for the time value of money, R. The benefit is the

product of a scalar, Ak, by the investor’s risk tolerance τ j . Ak measures the marginal benefit of private

information to a holder of stock k. To see this, consider the excess return in dollars of one share of stock

k, Πk−RP k. The ratio of the mean excess return to its standard deviation is the stock’s Sharpe ratio (it

is investor-specific since it depends on the specific realization and precision of the private signal). Under

quadratic preferences, an investor’s expected utility is an increasing function of the expected squared

Sharpe ratio on her portfolio. As shown in the appendix, the squared Sharpe ratio an investor expects

to achieve on stock k in the planning period, if she acquires a signal of precision xkj , is (hk0 + x

kj )A

k − 1.

When an investor holds several stocks, the expected squared Sharpe ratios simply add up (the stocks are

independent from one another) and her expected utility is linear inPk∈Kj

[(hk0 +xkj )A

k − 1]. Thus Ak is the

marginal benefit of private information on stock k to investors aware of its existence.

As equation 11 shows, the benefit of information is scaled by risk tolerance. More risk tolerant investors

trade, on average, a larger number of shares (see equation 8) over which they can spread their informational

advantage. Because the benefit of information increases with the scale of investment, captured by τ j ,

while its cost does not, more risk tolerant investors collect more information. In short, the production of

information displays increasing returns to scale. This property also explains why only investors who are

sufficiently risk tolerant (τ j > τ∗k) are active. Those who are too risk averse (τ j ≤ τ∗k) invest on too

small a scale to make information profitable.

Ak plays an important role in the model. It is an increasing function of noisiness, µk. Indeed, bigger

noisiness implies that prices are less revealing, making private information easier to conceal. For the same

reason, Ak is an increasing function of the variance of supply shocks, σ2θ. Ak is also a decreasing function

of the aggregate risk tolerance for stock k, nk. This property is at the hart of our results. An increase

16

in nk leads to an increase in price (on average) and to a reduction in individual stockholdings. With less

scope to gain from information, stockholders reduce their purchase of information. Putting it differently,

an increase in risk sharing leads to a reduction in the risk borne by stockholders in equilibrium, which

makes information less valuable. We call this effect the ”negative information externality”. For the same

reason, Ak increases with the squared mean supply θ2. These properties imply that a shareholder acquires

a more precise signal about stock k when τ , µk, θ2or σ2θ increase or when n

k decreases12. Similarly, τ∗

is decreasing in µk, θ2and σ2θ and increasing in n

k.

Having characterized through τ∗(µk, nk) and x(τ , µk, nk) the precision choices of shareholders aware of

stock k for any level of noisiness µk, we can determine the equilibrium value of µk thanks to its definition.

µk is determined implicitly and uniquely by equation 12. The depth of information, Dk ≡ 1/(µkσθ),

follows. We summarize its main properties in the next two lemmas. They will prove useful in the

following section when we discuss the implications of an investor base expansion.

We begin with the distribution of liquidity trades, θ and σ2θ. The effect of θ is straightforward but

that of σ2θ is ambiguous. σ2θ generates conflicting effects, which we quantify through several elasticities.

For any function f(z1, z2, ...), let εf/z1 denote the elasticity of f with respect to z1 holding z2, ... fixed.

Formally, εf/z1(z1, ...) ≡ z1f(z1,...)

∂f∂z1. Whenever f is a function of one variable only, we write εf for short.

In particular, εkA/µ ≥ 0 and εkA/σ2θ

≥ 0 denote the elasticities of Ak with respect to µk and σ2θ and εkC0 > 0

denotes the elasticity of C 0 with respect to xk.

Lemma 3 (Depth and the distribution of liquidity trades)

• The depth of information is increasing in the squared mean supply of shares, θ2.

12Under CARA preferences, the optimal precision about stock k is independent of nk and θ. Indeed, the first order conditioncan be written as 2RC0(xkj ) = τ j/(h0(µ

k) + xkj ) from which nk and θ are absent. There is no trade-off between risk sharingand information collection.

17

• The depth of information is increasing in the variance of liquidity trades, σ2θ, if and only ifZ ∞

τ∗k

µ−12εkC0 −

1

2εkA/µ + εkA/σ2θ

¶xk

εkC0τdGk(τ) > 0.

The effect of θ on the depth of information follows directly from its effect on τ∗ and x.More information

is collected when more shares have to be borne in equilibrium. This property is illustrated by figure 4

(top panel). The effect of the variance of liquidity trades, σ2θ is more complicated because it has both a

direct and an indirect effect on the depth of information, 1/(µkσθ). On one hand, a greater σ2θ makes

prices directly more noisy (σθ increases in the denominator of 1/(µkσθ)). On the other, it makes private

information easier to conceal and therefore more valuable (µk decreases in the denominator of 1/(µkσθ)).

The direct effect is captured by the first two terms in the integral, −12εkC0 − 12εkA/µ ≤ 0, while the indirect

effect is reflected in the last term εkA/σ2θ

≥ 0. Depth rises if the indirect effect prevails. This happens

when the integral in the lemma is positive. For example, if εkA/σ2θ

is large while εkC0 and εkA/µ are small,

then an increase in σ2θ enhances the value of information greatly (εkA/σ2θ

large), leading to a large increase

in stockholders’ precisions (εkC0 small), only dampened by the implied increase in depth (εkA/µ small).

Otherwise, the direct effect dominates and depth falls. Note that we can drop the integral when εkC0

does not depend on xk (for example if C(x) = cxb for b > 1). In that case, the condition simplifies to

−12εkA/µ − 12εkC0 + εk

A/σ2θ> 0. Figure 4 (bottom panel) displays the effect of σ2θ when b = 1.1 (dashed) and

b = 2 (solid). In the first case, depth increases, while in the second, it decreases.

Next, we consider the effect of the distribution of risk tolerance. The aggregate level of risk tolerance,

nk, measures the extent of risk sharing and therefore plays an essential role in determining the depth of

information, as we emphasized above. Its effects are complex and depend on how the change in nk comes

about (does it stem from a change in the number of shareholders or from a change in their risk tolerance).

18

We analyze them in detail in the next section. Before we do so, we highlight that the aggregate level

of risk tolerance is not all that matters in determining the depth of information. Its whole distribution

matters.

Lemma 4 (Depth and the distribution of risk tolerance)

Suppose all stockholders are active and consider a mean preserving spread of the risk tolerance distri-

bution for stock k, i.e. nk ≡∞R0

τdGk(τ) and σθ are kept constant.

• If 1/C 0 is convex, then the depth of information about stock k increases.• If 1/C 0 is concave, then depth decreases.

Lemma 4 shows that the full distribution of risk tolerance is important to the depth of information.

We consider a mean-preserving spread of the distribution of risk tolerance of investors aware of stock

k. That is, we make the distribution more unequal while keeping aggregate risk tolerance, nk, constant.

This allows us to isolate a pure distributional effect, devoid of changes in the extent of risk sharing. The

implication of a mean-preserving spread depends on the curvature of τx as function of τ , which in turn

depends on the curvature of 1/C 0 as a function of x. If 1/C 0 is convex, then τx is convex. Shifting risk

tolerance from a poorly tolerant to a highly tolerant stockholder increases the precision of the latter more

than it decreases the precision of the former, enhancing the depth of information. If 1/C 0 is concave,

then τx is concave and the opposite happens13. Figure 5 illustrates the theorem thanks to two examples.

1/C 0(x) and τx(τ) are convex for C(x) = x2 + 0.5x (solid curves) while they are concave for C(x) =

0.5atanh(x) (dashed curves). In the next section, we study the consequences of a change in the number

of investors aware of stock k.

13C0 convex is a necessary but not sufficient condition for 1/C0 concave. Note also that the presence of passive stockholderscan alter the effect of a mean preserving spread if 1/C0 is concave. When risk tolerance is redistributed away from passivetowards active investors, the former do not reduce their collection of information (they are already at the minimum). Thisdampens and can overturn the effect of the spread. When 1/C0 is convex, it enhances the effect of the spread.

19

6 Implications of an investor base expansion

An investor base expansion can affect the depth of information through several channels. The first and

most novel channel is the negative information externality which we alluded to in the previous section.

The value of information A is a decreasing function of nk, reflecting the trade-off between risk sharing and

information collection. Therefore incumbents have less of an incentive to collect costly information when

new shareholders come on board. What complicates the analysis is that these new investors may be active

and therefore may contribute to the depth of information. This constitutes the second channel. Finally,

the variance of liquidity trades may also change following the expansion, creating a third channel (lemma

3). The final effect on the depth of information depends on the relative importance of each channel, which

we will express in terms of elasticities.

We analyze the addition to the base of α (≥ 0) shareholders of risk tolerance τα. The new equilibrium

values are function of the parameter α, the initial equilibrium corresponding to α = 0. The results

which follow apply to a generic stock. We drop the superscript k to simplify the notation. Following

the expansion, the total number of investors with risk tolerance below τ who recognize the stock is

G(τ ,α) = G(τ , 0)+α1{τ≥τα} where 1{τ≥τα} is the indicator function of the inequality τ ≥ τα (1{τ≥τα} = 1

if τ ≥ τα and 0 otherwise). Clearly, risk sharing is enhanced and n(α) = n(0) + ατα. The variance of

liquidity trades may change following the expansion. We denote σ2θ(α) the new variance and εσ2θthe

elasticity of σ2θ with respect to α. εσ2θis positive (negative) if new stockholders enhance (reduce) the

variance of supply shocks.

Theorem 5 (Depth following a base expansion)

Suppose a stock’s investor base expands by α shareholders of risk tolerance τα. The change in the depth

20

of information, D(α), is given by

dD

dα=

ταxα(1− εσ2θ/2) +

R∞τ∗(α)

hεA/nτα/n(α) +

³−εA/µ/2− εC0/2 + εA/σ2θ

´εσ2θ/αi x

εC0τdG(τ , 0)

σθ(α) + εA/µ/D(α)R∞τ∗(α)

x

εC0τdG(τ , 0)

(13)

where τ∗(α) is the threshold for information collection following the expansion (equation 10) and xα isthe precision of new stockholders’ signals (equation 11). If τα < τ∗(α), then new stockholders are passive(xα = 0). Otherwise, they are active (xα > 0).

The theorem highlights the three channels through which an investor base expansion can affect the

depth of information. (i) The negative information externality deters incumbent investors from collecting

private information. This effect is represented by the first term of the integral in the numerator of dD/dα.

This term is more negative when εA/n ≤ 0 is more negative (the benefit of information is more sensitive

to the entry of new investors), when τα/n(α) is larger (new investors account for a larger fraction of the

stock’s risk tolerance) and when εC0 ≥ 0 is smaller (the demand for information is more sensitive to change

in the benefit of information). It vanishes when all incumbent investors are passive (they cannot acquire

less information). (ii) New investors may contribute to the depth of information. This happens if they are

active, i. e. if τα > τ∗(α) where τ∗(α) is the threshold for information collection defined by equation 10.

In this case, the precision of their signal, denoted xα, increases with their risk tolerance τα (xα follows

from the first order condition 11). Otherwise, if τα < τ∗(α), they are passive and xα = 0. (iii) Finally, the

expansion may generate a change in the variance of liquidity trades. As discussed in lemma 3, an increase

in σ2θ affects the depth of information directly (−ταxαεσ2θ/2) ≤ 0 term and −12εA/µ − 12εC0 ≤ 0 term in

the integral) and indirectly through the incentives to collect information (εA/σ2θ ≥ 0 term in the integral).

If σ2θ does not change with the expansion, then both terms vanish (εσ2θ = 0). Finally, the denominator of

dD/dα reflects the fact that any change in depth is dampened by the response of investors (lower depth

increases the incentives to collect information, εA/µ ≥ 0).

21

The theorem implies that the depth of information can rise or fall depending on the importance of

the different channels. We illustrate this result and its implications thanks to two polar cases: first, the

addition of active stockholders to a passive base and second, the addition of passive stockholders to an

active base. In both scenarios, we assume that C 0(0) > 0 so that active and passive investors coexist

in equilibrium. We assume also that the variance of liquidity trades does not change. We discuss the

relevance of this assumption for corporate events such as ADR listings and stock splits at the end of the

section. We examine also how the stock’s return distribution is affected. We consider the unconditional

mean and variance of the excess return in dollars of one share of stock, Π−RP. The following two lemmas

summarize the results.

Lemma 6 (Addition of active stockholders to a passive base)

Suppose active investors join a stock’s investor base composed exclusively of passive investors. Supposethe variance of liquidity trades does not change. Then,

• The depth of information rises,• The expected return falls,• The variance of returns falls.

Lemma 7 (Addition of passive stockholders to an active base)

Suppose passive investors join a stock’s base which includes some active investors. Suppose the varianceof liquidity trades does not change. Then,

• The depth of information falls,• The expected return falls,• The variance of returns may rise or fall.

The expansion’s implications for the depth of information follow directly from theorem 5. In the first

scenario (lemma 6), incumbent shareholders are passive, so the negative information externality is no

longer at work. Therefore, depth rises when active investors join the base (set εA/n = εσ2θ= 0 and xα > 0

22

in equation 13). In this case, the depth of knowledge rises with its breadth. In the second scenario (lemma

7), new investors are passive, so the negative information externality is the only channel in operation and

depth falls (set xα = εσ2θin equation 13). In this case, the depth of knowledge falls with its breadth.

We now turn to the distribution of returns. A stock’s expected excess return is determined by two

factors, the extent of risk sharing and the amount of risk to bear. For a given level of risk, the greater

the breadth of knowledge, the smaller each stockholder’s exposure and the lower the risk premium. This

is the risk sharing effect considered in the IRH. The level of risk is also endogenous in our extended IRH.

It derives from the depth of knowledge, which in turn, adjusts to the breadth of knowledge. We refer to

this effect as the ”information effect”. The two effects can be identified by noting that, in equilibrium,

a stock’s expected excess return equals θ/(nh) which falls with both the aggregate risk tolerance for the

stock n (the risk sharing effect) and the average precision per investor h (the information effect).

The variance of returns is subject to the same risk sharing and information effects. It falls when

aggregate risk tolerance rises (holding fixed the depth of information). It also falls when risk falls (holding

fixed risk tolerance), i.e. when depth rises. Formally, V ar(Π − RP ) = (1 − PΠ)2σ2Π + (µPΠ)2σ2θ =

[¡σ2θ/n+ 1/µ

¢/(nh) + 1]/h. To appreciate the effect of depth on the variance of returns, start from

perfect information (D =∞ and µ = 0). The price tracks the future payoff perfectly. The return equals

the riskless rate so the variance equals zero. As information worsens (D decreases and µ increases), the

price is less sensitive to the payoff (PΠ decreases in equation 5) whereas the return is more sensitive

(1− PΠ increases). The effect on the noise term, µPΠ, is ambiguous but the payoff term leads.

In both scenarios, risk sharing is enhanced by the entry of new investors. In the first one, depth is

also enhanced. The information and risk sharing effects work in the same direction, reducing expected

returns and volatility, as illustrated in figure 6 depicting the entry of α active investors. In contrast, depth

23

decreases in the second scenario. This time, the information and risk sharing effects work in opposite

directions. lemma 7 establishes that the risk sharing effect dominates for expected returns but that this

need not be the case for the variance of returns. If the information effect is strong, then it can dominate

the risk sharing effect and lead to an increase in volatility. This configuration is illustrated in figure 7,

depicting the entry of α passive investors. The expected return falls (bottom left panel) but the variance

of returns first increases and then decreases (bottom right panel). The increase happens when the number

of new investors α is small because they reduce depth by a large amount (top right panel). Note that the

two scenarios coincide when both incumbent and new investors are passive. There is then no information

effect so both expected returns and volatility fall. This is the situation considered in Merton (1987).

We conjecture that the implications for volatility obtain in a dynamic version of the model. A difficulty

arises in a multiperiod setup because a stock’s return variance not only involves next period’s dividend

payment but also the stock’s resale price. When more information is acquired, the current price tracks

future dividends and prices more closely, thereby reducing the return variance (as in the static model).

But the variance of future prices also increases since future prices track more closely dividends even

further into the future. However, because future prices are discounted at the risk-free rate, the former

effect dominates the later as Pagano (1986) and West (1988) show14.

As noted in the introduction, the information effect helps reconcile the IRH with the observed increase

in return variance following breadth-enhancing events such as ADR listings and stock splits. An alternative

14The following example illustrates this point for risk neutral investors. Suppose investors know that a stock will pay threedividends, Π1, Π2 and Π3 at dates τ1, τ2 and τ3 where Π3 corresponds to the stock’s dividend. Nothing is known about thedividends except that they are distributed with identical mean Π and variance σ2Π. Suppose further that they are revealed∆ periods ahead, i.e. Π1 is revealed at τ1 −∆... The (cum dividend) stock price can easily be computed. For example att < τ1−∆, the price equals Π/Rτ1−t+Π/Rτ2−t+Π/Rτ3−t, at τ1−∆ ≤ t ≤ τ1, it equals Π1/R

τ1−t+Π/Rτ2−t+Π/Rτ3−t,at τ1 < t ≤ τ2 −∆, it equals Π/Rτ2−t +Π/Rτ3−t.... Importantly, early revelation of dividends does not simply shift pricesforward, it also rescales them through the discount factor. Returns, Pt + Πt − RPt−1, equal 0 on every date except on therevelation dates when they equal (Π1 − Π)/R∆, (Π2 − Π)/R∆ and (Π3 − Π)/R∆. The variance of returns therefore equalsσ2Π/R

2∆, a decreasing function of ∆: the earlier the dividends are revealed, the lower the return variance.

24

explanation is that these events attract small investors. If these small shareholders are more prone to

liquidity shocks, they will enhance return variance. This explanation, however, does not appear to match

the facts. ADRs are relatively sophisticated instruments that, most likely, do not attract small investors

(Karolyi (1996)). As for splits, Mukherji, Kim and Walker (1997) find that they do not increase the

proportion of individual shareholders. Moreover, as noted by Black (1986), small investors will also

increase the volume of trade. There is however no clear evidence supporting this implication. Domowitz,

Glen and Madhavan (1998) estimate the volatility change following ADR listings controlling for any

possible volume effect and again find a very strong volatility change15. Copeland (1979), Lamoureux and

Poon (1987), Lakonishok and Lev (1987), Conroy, Harris and Benet (1990) and Lynch Koski (1998) find

a decline in daily trading volume after adjusting for the split factor, while Schultz (2000) documents a

large number of small buy orders. Our model allows for but does not require changes in the variance of

liquidity trades (theorem 5 accounts for changes in σ2θ but lemma 7 does not) to rationalize the increase

in return volatility. In the next section, we confront the model to the data.

7 Empirical evidence

Assessing the empirical validity of the model is not straightforward given the multiplicity of effects.

Nevertheless, we try to provide some empirical tests drawing on data from the ADR market. As noted

in the introduction, Foerster and Karolyi (1999) and Miller (1999) document that non-US firms listing

in the US through ADRs earn positive abnormal returns. This observation is consistent with both an

investor base expansion and an improvement in liquidity. Foerster and Karolyi (1999) show further that

these abnormal returns are related to the number of shareholders rather than to liquidity proxies, lending

15Jayaraman, Shastri and Tandon (1993), Foerster and Karolyi (1999) and Miller (1999) do not examine volume.

25

support for the IRH. Building on these studies, we examine how the production of information changes

around ADR listings.

The predictions of the model regarding the depth of information are ambiguous. Depending on

how much information new shareholders collect, depth may rise (lemma 6) or fall (lemma 7). What is

unambiguous however, is that, absent changes in the variance of liquidity trades, incumbent shareholders

acquire less information16. This is the negative information externality we referred to throughout the

paper. We focus in this empirical section on this externality because it constitutes both a robust prediction

and the main contribution of the model. ADR listings provide the opportunity to test this prediction

for several reasons. First, they generate a significant base expansion (28.8% on average in Foerster and

Karolyi (1999)’s sample). Second, they allow to disentangle incumbent from new investors, i.e. those

investing in the underlying stock from those investing in the ADR. Third, because they are relatively

sophisticated instruments, primarily held by institutions, the base expansion is most likely not associated

with a change in liquidity trading (see the discussion at then end of the previous section).

Assessing how much information investors collect about a stock is a difficult task. A standard proxy

is analyst coverage (e.g. Bhushan (1989a, 1989b), Brennan and Hughes (1991), Brennan, Jegadeesh,

and Swaminathan (1993), Hong, Lim and Stein (2000)). While other factors may influence analyst

coverage, cross-sectional differences in coverage are certainly related to differences in investors’ demand

for information. We estimate the amount of information produced by incumbent shareholders by taking

16Formally, the change in incumbents’ production of information, Dinc, following the entry of α shareholders of risktolerance τα equals (set εσ2

θ= xα = 0 in equation 13)

dDinc

dα=

R∞τ∗(α) εA/nτα/n(α)

x

εC0τdG(τ , 0)

σθ(α) +R∞τ∗(α) εA/µ/D(α)

x

εC0τdG(τ , 0)

≤ 0.

26

the number of analysts following the underlying stock. The measurement procedure is discussed in the

next subsection. The testable hypothesis is therefore: does the number of analysts following the underlying

stock fall subsequent to the ADR listing?

Other effects associated with ADR listings may blur the analysis. For example, higher liquidity in the

ADR market may induce some investors to switch from trading the underlying to trading the ADR. Also,

the stricter disclosure and accounting standards imposed by the US Securities and Exchange Commission

(SEC) may have a direct effect on analyst following, independently from a change in stock ownership

(Lang and Lundholm (1996)). To mitigate these effects, we focus on a sample of companies from a highly

liquid market with comparable disclosure and accounting practices as the US, the British market. In

addition, we exploit institutional differences across ADRs to control for these possibilities.

Several types of ADRs exist. In line with the model, we restrict the empirical analysis to Level I and

Level II ADRs, the programs in which no new capital is raised17. The most basic ADR program is the

Level I program. Under such a program, the underlying company does not have to register with the SEC,

nor does it have to comply with US accounting standards. But the ADR cannot list on US exchanges and

trades in the over-the-counter (OTC) market. If the company wants to list its ADR on an exchange, it

must choose a Level II program. In that case, it must register, report fully to the SEC and reconcile its

financial statements to US GAAP. We focus in the empirical analysis on Level II ADRs but we use the

sample of Level I ADRs as a robustness check. Again, in both programs, no new shares are issued.

17 In the model, we keep the average residual supply of shares, θ, fixed. In addition to level I and II programs, there exist twotypes of capital-raising ADR programs: Level III ADRs listed on exchanges and “Rule 144A” ADRs traded over-the-counter.

27

7.1 Data and methodology

We start from a sample of ADR listings provided by J.P. Morgan (www.adr.com)18. We retain only

non-capital raising programs (Level I and II) initiated by British companies. We then retrieve from the

I/B/E/S International database the Summary History files on analyst forecasts for these companies. We

restrict the sample further to companies (i) that listed their ADRs between 1986 and 2002, and (ii) that

were followed by at least one analyst in their pre-listing year. The first requirement is imposed to obtain

analyst data in the years before and after the listing. The second avoids a truncation bias since the

number of analysts cannot drop below 019.

The I/B/E/S data distinguishes analysts following the underlying stock from those following the ADR.

Thanks to the I/B/E/S Detailed Reports on the underlying stock and the ADR, which identify analysts

and their employers, we check that, in our sample, the same analyst is never reported as following in

the same year both the underlying and the ADR, nor that she switches from following the underlying

to following the ADR around the listing. Therefore, changes in the number of analysts following the

underlying reflect terminations of coverage by incumbent analysts and initiations by new analysts20. We

can proxy for the production of information by incumbent shareholders by taking the number of analyst

following the underlying stock. For every year and stock in our sample, we measure analyst following as

18This list only features currently listed ADRs and therefore suffers from a survivorship bias. However, this bias is notlikely to affect our results as we are not interested in the effect of an ADR listing per se, but rather in the effect of an investorbase expansion, for which an ADR listing proxies. The delisted ADRs are those that failed to attract investors or that wereso successful that they were upgraded to higher level programs (e.g. a Level II instead of a Level I). In the first case, weare simply eliminating companies with no significant base expansion. In the second, we are wrongly attributing a large baseexpansion to the subsequent listing when it fact it began earlier with the original listing. This biases our analysis againstfinding any ADR listing effect.19 International I/B/E/S data starts in 1985. The results are a qualitatively unchanged if we do not impose the second

condition.20Changes in analyst coverage of the underlying do not reflect analysts switching from the underlying to the ADR in their

report to I/B/E/S. Nevertheless, some analysts initiating coverage may report to I/B/E/S that they follow the underlying.This will bias the analysis against finding a fall in coverage of the underlying subsequent to an ADR listing.

28

the average number of analysts making an earnings forecasts on the underlying stock for the fiscal year

end21. If a company is absent from the I/B/E/S file for a given year, then the number of analysts is

taken as 0 in that year. In addition, several variables reported in the literature as determinants of analyst

coverage are collected. Chief among them is company size: larger firms are followed by more analysts

(Bhushan (1989a, 1989b), Brennan and Hughes (1991), Hong Lim and Stein (2000)). We measure size

as the log of the year end market capitalization in British Pounds. We also consider earnings growth as

analysts tend to cover companies which they can recommend (McNichols and O’Brien (1997))22.

Table 2 displays an overview of the data. Stocks in our sample are fairly large: the average company

with a Level II program has a market capitalization of £ 1.4 billion in its listing year23. Firms with

a Level II ADR are more than three times the size of those with a Level I. Firms are evenly spread

across sectors, with an emphasis on consumer services for Level I programs. Consistent with the growing

globalization of capital markets, the number of ADR programs has increased over time, with more than

half of the listings occurring in the 1998-2002 period. About 80% of the Level II ADRs in our sample

trade on the NYSE and the remainder on the NASDAQ and AMEX. The average number of analysts

following the underlying stocks is 10.3 for a Level II program and 7.9 for a Level I. These figures are in

line with results found in other studies24. As expected, companies with a Level II program are followed

by more analysts than those with a Level I. The average number of analysts covering an underlying stock

falls from 14.3 to 11.8 in the years surrounding a Level II listing. However, as the tables show, there is

21Alternative measures such as the number of analysts providing a forecast in the seventh or eleventh month of the fiscalyear do not alter the results.22Moreover, growing firms may initiate an ADR program in order to use the ADR as currency for future mergers and

acquisitions.23 In comparison, the average market capitalization of the largest 100 companies on the NASDAQ at the end of 1996 was

roughly $ 6 billion or £ 3.7 billion.24For example, Frost & Kinney (1996) find that on average 29 analysts follow British stocks listed on US exchanges in

1990. Botosan and Frost (1998) report that on average 25 analysts follow non-US firms listed on US exchanges versus 18 forLevel I ADRs. These figures are not directly comparable to ours as we only count analysts following the underlying stock.

29

substantial variation across companies. To isolate the effect of the listing from that of other factors, we

now carry out a multivariate analysis.

7.2 Results

We begin with a cross-sectional analysis around ADR listing dates. We regress changes in the number of

analysts following the underlying stock from the year before to the year after the listing on a constant,

on changes in firm size and on changes in earnings growth. The results are displayed in table 3. In all

cases, the constant is negative and statistically significant at the 5% or 10% levels, indicating that analyst

coverage of the underlying falls around an ADR listing.

Next, we exploit the panel structure of the data to control for unaccounted differences in companies.

We run panel regressions with stock fixed-effects. The fixed-effects capture time-invariant firm charac-

teristics which are relevant to analyst following such as industry or year of listing on the London Stock

Exchange (LSE). Again, we include firm size and earnings growth as regressors. In some specifications,

we add lagged analyst following as following is very persistent (O’Brien and Bhushan (1990)), and a

time-trend variable defined as the annual average number of analysts per stock for the sample of Level II

programs. The key regressor is ADRdum, a dummy variable which equals 1 if the underlying stock has

an ADR listed and 0 otherwise.

Table 4 displays the results of these regressions. The coefficient on ADRdum is negative and statis-

tically significant at the 5% level in all cases and at the 1% level in four cases out of five: controlling

for other factors, a Level II ADR listing generates a reduction in the number of analysts following the

underlying stock ranging from 1.3 to 3.7 depending on the specifications in the year subsequent to the

listing.

30

These results could be partly driven by changes in disclosure and accounting rules. Indeed, Level

II programs impose strict requirements which may reduce the need for analysts’ research and lead some

analysts to terminate coverage. This seems unlikely for our sample of UK companies as UK rules resemble

those in the US in many aspects. Nevertheless, we exploit our sample of Level I ADRs to control for this

possibility as companies initiating a Level I program do not have to follow US disclosure and accounting

rules. We run for the sample of Level I listings the same panel regressions as for Level II listings. The

results, displayed in table 5, are similar: the coefficient on ADRdum is negative and statistically significant.

It is smaller than for a Level II listing (roughly a third smaller), consistent with the smaller abnormal

returns reported by Miller (1999) and a smaller base expansion.

To conclude, the significant decline in analyst coverage of the underlying stock subsequent to an ADR

listing lends support for the model: incumbent investors acquire less information when the investor base

expands25.

8 Conclusion

We show that a stock’s ownership distribution determines how much information is produced about its

payoff, i.e. determines the depth of knowledge about the stock. This dependence is the consequence of

a simple insight: large shareholders have an incentive to collect costly information about stocks because

they can appropriate a large fraction of the benefits from their research. Small shareholders prefer to free-

ride large shareholders’ research which gets partially revealed through prices. The same reason explains

why firms held by small shareholders are less closely monitored in the literature on corporate control. A

25This does not imply that the depth of information is reduced as ADR holders may also collect information (e.g. lemma 7).Baker, Nofsinger & Weaver (2002) actually document a doubling in the total number of analysts subsequent to a cross-listing.As for return volatility, it may rise or fall following the listing depending on the relative magnitudes of the information andrisk sharing effects (see lemma 7 and 8).

31

crucial difference with this literature is that information is acquired ex ante and therefore gets partially

revealed through prices.

We highlight the important role played by the number of investors who recognize the stock, i.e. by

the breadth of knowledge. In particular, we analyze the implications of an investor base expansion. We

identify three channels through which the depth of information can be affected. First and most novel, an

expansion leads to a redistribution of ownership from incumbent to new shareholders (assuming no shares

are issued). The redistribution reduces incumbents’ incentives to collect information. Thus, information

collection is limited by the extent of risk sharing: new investors unload some of the risk from incumbent

investors, and, in so doing, make information less valuable. We call this effect the negative information

externality. Second, new stockholders may collect information and thus contribute to the depth. Third, if

the variance of liquidity trading increases, then private information becomes easier to conceal and therefore

more valuable. The final effect on depth depends on the relative importance of the three channels, which

we quantify in the paper in terms of elasticities.

We consider, as an illustration, two scenarios, assuming no change in the variance of liquidity trading:

first, the addition of active stockholders to a passive base and second, the addition of passive stockholders

to an active base. Depth rises in the first scenario but falls in the second. Furthermore, we show how the

distribution of returns is affected. A wider breadth improves risk sharing while a larger depth reduces

risk. Both decrease the mean and variance of returns. In the first scenario, breadth and depth both

increase, leading to a fall in the mean and variance of returns. In the second scenario, breadth increases

while depth falls. We show that expected returns fall but the variance of return rises if the negative

information externality is strong enough. This result may explain why firms which expand their investor

base through stock splits or ADR listings see their return variance increase, an observation at odds with

32

the standard IRH.

We provide evidence consistent with the negative information externality by drawing on a sample of

British companies issuing ADRs. ADR listings generate a significant base expansion, with limited change

in liquidity trading, and allow to disentangle incumbent from new investors, i.e. those investing in the

underlying stock from those investing in the ADR. We use analyst coverage of the underlying to proxy

for the amount of information produced by incumbents. As predicted by the model, we find that analyst

coverage of the underlying stock falls in the year following the listing.

The model can be extended to address several issues. It can shed light on the determinants of firms’

ownership structure. In the current model, the structure of the investor base is taken as given to analyze

the depth of information. In fact, both are jointly determined in equilibrium. A model in which both

the ownership and information decisions are endogenous could help assess the ownership implications

of accounting standards and disclosure requirements. Such reforms are currently debated in several

countries. Another interesting avenue is to incorporate agency problems into the model. This would

provide a unified view of the production of information about companies, whether it is used for stock

selection or management monitoring. On the empirical front, it would be interesting to carry out a cross-

sectional analysis of stock returns, relating the return distribution to the characteristics of the shareholder

base.

A Proof of theorem 1 (Equilibrium prices and stockholdings)

To prove theorem 1, guess that the equilibrium price is given by equations 5 to 7 and solve for the optimal

portfolio of a investor (recall that precisions are taken as given at this stage). Let’s focus on stock k and

on investor j, aware of k. The first step is to estimate the mean and variance of the stock’s payoff using

the equilibrium price (or equivalently ξk ≡ Πk − µkθk) and the private signal Skj . The results of thisgaussian signal extraction problem are summarized below:

• Signal extraction

33

For the price function given in equation 5 (P k is linear in Πk and θk), the conditional mean and

variance of Πk are:

V ar(Πk | Fj) = 1

hk0 + xkj

≡ 1

hkjand E(Πk | Fj) = ak0l + a

k

ξlξk + a

kSjS

kj

where ak0lhkj≡Π

σ2Π+

θ

µkσ2θ= P k0h

k, akξlh

kj≡

1

µk2σ2θand akSjh

kj≡ xkj

Intuitively, V arj(Πk | Fj) falls as the precision of the private signal, xkj or the depth of information,1/(σθµ

k), increase. Similarly, Ej(Πk | Fj) is a weighted average of priors, public and private signalswhere the weight on the private signal (resp. the price) is increasing in xkj (resp. in 1/(σθµ

k)). If the

investor is passive, xkj = 0 and Skj vanishes from the equations.

• Individual portfolio choice

We can now compute investor j0s optimal portfolio. Under mean-variance preferences, the optimal

holding of stock k equals τ jEj(Π

k | Fj)−RP kV arj(Πk | Fj) . Plugging in the above expression for Ej(Π | Fj) and

V arj(Π | Fj) leads to equation 8 and 9. Obviously, an investor unaware of stock k has a 0 holding.

• Market clearing

The equilibrium price clears the market for stocks. Aggregating equations 8 and 9 over all stockholders

yields the aggregate demand for stock k:

∞Z0

XkdGk(τ) =

∞Z0

τhXk0 + x

k(Sk −RP k)ikdGk(τ) = nkXk

0 +Πk

µk− RP

k

µk

where µk∞R0

τxkdGk(τ) ≡ 1 and the term Πk/µk comes from applying the law of large numbers for

independent random variables to the sequence {τ jxkjυkj}. Indeed for a given k, {τ jxkjυkj } is a sequence of

independent random variables with the same mean 0 (conditional on Πk) and possibly different different

(finite) variances τ2jxkj so

∞R0

τxkυkdGk(τ) = 0. Therefore,∞R0

τxkSkdGk(τ) =∞R0

τxk(Πk + υk)dGk(τ) =

Πk∞R0

τxkdGk(τ)+∞R0

τxkυkdGk(τ) = Πk/µk (see He and Wang (1995) for more details). Finally, equating

aggregate demand to aggregate supply θk yields the equilibrium prices given by equations 5-7.

B Proof of theorem 2 (The depth of information)

• Investors’ demand for information

To prove theorem 2, we need to solve the information acquisition problem of a stockholder. By plugging

equation 8 and 9 into the formula for mean-variance utility and integrating over all possible values of Πk

34

and θk, we get the expected utility of an investor with risk tolerance τ j and signals of precision xj :

Ej [u(Sj , xj ;P )]=RW +Xk∈Kj

hτ j2Ej(z

2j )−RC(xkj )

i(14)

where zkj ≡ Ej(Πk|Fj)−RPk√

V arj(Πk|Fj)is investor j’s Sharpe ratio on stock k, a function of Skj and P

k (and xkj ).

Ej(zk2j ) is the squared Sharpe ratio stockholder j expects to achieve in the planning period on stock k.

It no longer depends on Skj and Pk but it is still a function of xkj . Integrating over the distributions of

Skj and Pk yields

Ej(zk2j ) = (h

k0 + x

kj )A

k − 1 (15)

where Ak ≡ A(µk, nk) is defined in equation 10. Ak measures of the marginal benefit of private in-

formation about stock k to investors aware of k. Differentiating A and rearranging yields ∂A∂nk

=

− 2

(nkhk)3

hσ2θσ2Π+ hk0θ

2i< 0, ∂A

∂µk= 2

(nkhk)3µk2

hθ2+ σ2θ + (

2θ2

σ2θ+ 3)n

k

µk+

hk0nk3

σ2θµk +

3nk2

σ2θµk2

i> 0 and ∂A

∂σ2θ=

1

(nkhk)2+ 1

(nkhk)3

nk

(σ2θµk)2(2θ

2+ 2σ2θ + 3

nk

µk+ hk0n

k2) > 0. A is a decreasing function of nk (smaller nk

implies bigger gains from trade) and an increasing function of µk and σ2θ (bigger µk or σ2θ imply that

prices are less revealing).

To solve for the optimal precision level, maximize the expected utility with respect to xkj taking µk

and nk (hence Ak) as given. If there is an interior solution, xkj , it must satisfy the first order condition

2RC 0(xkj ) = τ jAk. By assumption, C is convex in xkj so the right hand side is increasing while the left

hand side is constant. This implies that a unique interior solution exists if and only if 2RC 0(0) < τ jAk.

Otherwise the solution will be the corner 0, i.e. the investor is passive. It follows that investors aware of

stock k are active if and only if their risk tolerance is above the threshold τ∗k defined in equation 10.

• The depth of information

Noisiness µk is determined implicitly by equation 12, which is obtained by plugging the demand

for information, τ∗(µk, nk) and x(τ , µk, nk), into its definition. To prove that the equilibrium is unique

(within the class of linear equilibria), it suffices to show that µk is uniquely defined. Let Σ(µk) ≡µk

∞Rτ∗(µk,nk)

τx(τ , µk, nk)dGk(τ). µk is defined as the solution to Σ(µk) = 1. Differentiating Σ yields

Σ0(µk) = Σ(µk)/µk − µkτ∗kx∗kg∗k ∂τ∗

∂µk+ µk

∞Zτ∗(µk,nk)

τ∂x

∂µkdGk(τ)

where the second term comes from differentiating the lower bound in the integral and the third from differ-

entiating the integrand. The second term drops out because x∗k ≡ x(τ∗(µk, nk), µk, nk) = 0 (this followsfrom the assumption that C is continuous at 0). Differentiating equation 11 yields ∂x

∂µk= τ ∂A

∂µk/(2RC 00(x)).

C is strictly convex and, as shown above, ∂A∂µk

> 0 so ∂x∂µk

is positive. Therefore Σ0(µk) is positive andΣ increases over the real positive line. Furthermore, Σ(0) = 0 and Σ(∞) = +∞. Hence, there exists aunique µk satisfying equation 12.

35

C Proof of lemma 3 (Depth and the distribution of liquidity trades)

We begin with the effect of θ2on depth. Differentiating equation 12 with respect to θ

2yields 1

µkdµk

dθ2 +

µk∞R

τ∗k( ∂x

k

∂µkdµk

dθ2 +

∂x

∂θ2 )τdG

k(τ)− µkτ∗kx∗kg∗k(∂τ∗k∂θ

2 +∂τ∗k∂µk

dµk

dθ2 ) = 0 where x∗k and g∗k are evaluated at τ∗k.

The last term drops out because x∗k = 0. Therefore,

dµk

dθ2 = −

∞Zτ∗k

∂x

∂θ2 τdG

k(τ)

/ 1

µk2+

∞Zτ∗k

∂x

∂µkτdGk(τ)

.The numerator is positive because ∂x

∂µk> 0 (see appendix B). ∂x

∂θ2 is also positive, as can be seen by taking

the log of the first order condition 11 and differentiating it with respect to θ2( ∂x∂θ

2 =xk

θ2

εkA/θ

2

εkC0), and by

noting that ∂Ak

∂θ2 =

1

(nkhk)2> 0. Therefore dµk

dθ2 < 0 and depth rises with θ

2.

Similarly, to find the effect of σ2θ on depth, differentiate equation 12 with respect to σ2θ, drop the term

in x∗k and rearrange to obtain

dµk

dσ2θ= −µk

∞Zτ∗k

∂x

∂σ2θτdGk(τ)

/ 1

µk+ µk

∞Zτ∗k

∂x

∂µkτdGk(τ)

. (16)

Taking the log of the first order condition 11 and differentiating it with respect to µk and σ2θ yields

∂x∂µk

= xk

µkεkA/µ

εkC0

and ∂x∂σ2θ

= xk

σ2θ

εkA/σ2

θ

εkC0

where εkA/µ ≡ µk

Ak∂Ak

∂µk> 0 and εk

A/σ2θ≡ σ2θ

Ak∂Ak

∂σ2θ> 0. Substituting into

equation 16 yields

dµk

dσ2θ= −µk2

µZ ∞

τ∗kεkA/σ2θ

/σ2θx

εkC0τdGk(τ)

¶/

1 + ∞Zτ∗k

µkεkA/µx

εC0τdGk(τ)

(17)

Noisiness falls when σ2θ increases. This time, the effect on depth is not straightforward as it depends also

on σθ directly. Dk ≡ 1/(σθµk) which implies 1Dk

dDk

dσ2θ= − 1

µkdµk

dσ2θ− 1

2σ2θ. Replacing dµk

dσ2θwith expression 17

and rearranging yields

dDk

dσ2θ=1

σ3θ

µZ ∞

τ∗k(−εkC0/2− εkA/µ/2 + εkA/σ2θ

)x

εkC0τdGk(τ)

¶/

1 + ∞Zτ∗k

µkεkA/µx

εC0τdGk(τ)

.The denominator is positive so depth increases with σ2θ if and only if the numerator is also positive. This

is the condition stated in the lemma.

36

D Proof of lemma 4 (Depth and the distribution of risk tolerance)

Let β be a parameter indexing a mean-preserving spread of Gk(.), Gk(.,β). That is, the larger β, the more

unequal the distribution of risk tolerance for stock k, while its aggregate level,∞R0

τdGk(τ ,β), remains equal

to nk. Formally,∞R0

ϕ(τ)∂gk(τ ,β)∂β dτ > 0 (< 0) for any convex (concave) function ϕ, while

∞R0

τ ∂gk(τ ,β)∂β dτ = 0.

The final values of µk and τ∗k depend on β but we do not write them explicitly as functions of β to

simplify the notation. Differentiating equation 12 with respect to β yields 1µkdµk

dβ − µkτ∗kx∗kg∗k(∂τ∗k

∂β +

∂τ∗k∂µk

dµk

dβ ) + µk∞R

τ∗kτ ∂x∂µk

dµk

dβ gk(τ ,β)dτ + µk

∞Rτ∗k

τx∂gk(τ ,β)∂β dτ = 0 where x∗k and g∗k are evaluated at τ∗k.

The second term drops out because x∗k = 0. Therefore,

dµk

dβ= −

∞Zτ∗k

τx∂gk(τ ,β)

∂βdτ

/ 1

µk2+

∞Zτ∗k

τ∂x

∂µkgk(τ ,β)dτ

.The numerator is positive because ∂x

∂µk> 0 (see appendix B). If τx is a convex function of τ , then

dµk

dβ < 0. This happens if (τx(τ))00 = 2x0(τ) + τx00(τ) > 0 for all τ ≥ τ∗k. Differentiating the firstorder condition 11 twice and substituting yields x00(τ)/x0(τ) = −C 0(x(τ))C 000(x(τ))/[τC 00(x(τ))2]. Thusτx is convex if 2C 00(x)2 − C0(x)C 000(x) > 0 for all positive x. This condition in turn is equivalent to

(1/C 0(x))00 = [2C 00(x)2 − C 0(x)C 000(x)]/C 0(x)3 > 0. Intuitively, if 1/C 0 is convex, shifting risk tolerancefrom a low to a high risk tolerant investor increases the depth for information because the high tolerance

investor increases her demand for information by more than the low tolerance investor reduces hers.

Alternatively, if τx is a concave function of τ , then dµk

dβ > 0. This happens if 1/C 0 is concave. In thatcase, the cost of information increases so steeply with the precision (C 0 is convex) that highly risk tolerantinvestors hardly increase their precision as they become more risk tolerant. The depth of information

falls.

E Proof of theorem 5 (Depth following a base expansion)

The final values ofD, µ, n, σ2θ and τ∗ all depend on the number of new investors α. Except when ambiguous,

we do not write them explicitly as functions of α to simplify the notation. From the definition of µ,

1/µ =R∞τ∗ x(τ , µ, n,σ

2θ)τdG(τ ,α) where τ

∗ ≡ τ∗(µ, n,σ2θ) is the information collection threshold followingthe expansion. Substituting G(τ ,α) = G(τ , 0)+α1{τ≥τα} yields 1/µ = αxατα+

R∞τ∗ x(τ , µ, n,σ

2θ)τdG(τ , 0).

Differentiating with respect to α yieldsd(1/µ)

dα= xατα+

R∞τ∗ (

∂x∂n

dndα+

∂x∂µ

dµdα+

∂x∂σ2θ

dσ2θdα )τdG(τ , 0)−τ∗x∗g∗ dτ

∗dα

where x∗ ≡ x(τ∗, µ, n,σ2θ) and g∗ ≡ g(τ∗, 0). The last term drops out because x∗ = 0.Taking the log of the first order condition 11 and differentiating it with respect to n, µ and σ2θ yields

∂x∂n =

xn

εA/nεC0

, ∂x∂µ =

xµ

εA/µεC0

and ∂x∂σ2θ

= xσ2θ

εA/σ2

θεC0

. The definitions of n(α) = n(0) + ατα and εσ2θimply that

37

dndα ≡ τα and

dσ2θdα ≡

σ2θα εσ2θ

. Substituting back into the equation for d(1/µ)/dα and rearranging yields

d(1/µ)

dα=

ταxα +R∞τ∗(α)

hεA/nτα/n(α) + εA/σ2θ

εσ2θ/αi x

εC0τdG(τ , 0)

1 +R∞τ∗(α) µ(α)εA/µ

x

εC0τdG(τ , 0)

(18)

By definition, D ≡ 1/(σθµ) which implies dDdα = 1σθ(d(1/µ)dα −

εσ2θ

2µα). Substituting equation 18 and rearranging

yields equation 13.

F Proof of lemma 6 (Addition of active stockholders to a passive base)

Both µ and n change following the expansion: dndα = τα > 0 anddµdα = −µ2 d(1/µ)dα = −µ2xατα = −µ/α < 0.

The last equation obtains because only new investors are active so 1/µ = αxατα . Thus aggregates risk

tolerance and depth rise.

We turn to the distribution of returns. Substituting equations 5 to 7 into the definition for the

unconditional expected excess return Q ≡ E(Π − RP ), yields Q = θnh. Differentiating this expression

yields ∂Q∂n = −Qh0nh

< 0 and ∂Q∂µ =

Q

nhµ2

³1 + 2n

µσ2θ

´> 0. Q falls when the breadth or the depth of knowledge

rise so Q falls following the expansion. Formally, dQdα =∂Q∂n

dndα +

∂Q∂µ

dµdα < 0. Similarly, the unconditional

variance of excess returns and its derivatives equal V ≡ V ar(Π−RP ) = A−Q2 = 1h

h1nh

³σ2θn +

1µ

´+ 1i,

∂V∂n = − 2

(nh)3σ2θσ2Π< 0 and ∂V

∂µ =2

(nh)3µ3(3n+ h0n3

σ2θ+ 3n2

µσ2θ+µσ2θ) > 0. V falls when the breadth or the depth

of knowledge rise so V falls following the expansion. Formally, dVdα =∂V∂n

dndα +

∂V∂µ

dµdα < 0.

G Proof of lemma 7 (Addition of passive stockholders to an activebase)

We proceed in a similar fashion to lemma 6. Again, dndα = τα > 0 but this time dµdα = −µ2 d(1/µ)dα =

−µ2R∞τ∗(α) εA/nτα/n(α)

x

εC0τdG(τ , 0)

1 +R∞τ∗(α) µ(α)εA/µ

x

εC0τdG(τ , 0)

> 0 (recall that εA/n < 0 and εA/µ > 0). The last equation obtains

from setting xα = 0 (new investors are passive) and εσ2θ = 0 (the variance of liquidity trading is unchanged)

in equation 18. This time, aggregate risk tolerance rises but depth falls.

n and µ exert conflicting effects on returns. dQdα =

∂Q∂n

dndα +

∂Q∂µ

dµdα so Q falls if and only if −∂Q

∂n /∂Q∂µ >

− µ2

τα

d(1/µ)dα which can be written as

R∞τ∗(α)[−εA/n−(−∂Q

∂n /∂Q∂µ )(εA/µ+εC0)n/µ]

x

εC0τdG(τ , 0) < 0. A sufficient

condition for the integral to be negative is the integrand to be negative for all τ , for which it suffices, since

εC0 > 0, that −εA/n/εA/µ < (−∂Q∂n /

∂Q∂µ )n/µ. Substituting

∂Q∂n ,

∂Q∂µ , εA/n and εA/µwith their expressions

given in appendices B and F shows that this inequality in indeed satisfied. Therefore Q falls.

The procedure for V is identical. V falls if and only if −∂V∂n /

∂V∂µ > − µ2

τα

d(1/µ)dα which can be written asR∞

τ∗(α)[−εA/n − (−∂V∂n /

∂V∂µ )(εA/µ + εC0)n/µ]

x

εC0τdG(τ , 0) < 0. This time, −εA/n/εA/µ > (−∂V

∂n /∂V∂µ )n/µ so

38

V may rise. In particular, the previous integral is positive if εC0 = 0 for all x ≥ 0. The effect of a baseexpansion on the variance of returns is therefore ambiguous as figure 7 (bottom right panel) illustrates.

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42

Investors observe private signals and prices andform their porfolio.

Investors consume

t = 0 t = 2planning period trading period

t = 1consumption period

Investors choose how much information, if any, to collect about the stocks they recognize.

Investors observe private signals and prices andform their porfolio.

Investors consume

t = 0 t = 2t = 2planning period trading period

t = 1t = 1consumption period

Investors choose how much information, if any, to collect about the stocks they recognize.

Figure 1: Timing.

Figure 2: The optimal precision choice. 2RC 0(x) (solid line) and τ jA (dashed lines, top line for a highlyrisk tolerant investor and bottom line for highly risk averse investor). Risk tolerant investors acquireinformation of precision at the intersection of 2RC 0(x) and τ jA. Highly risk averse investors do notacquire information. The picture is drawn for C(x) = x2 + 0.5x, τ j equals 2 for the highly risk tolerantinvestor and 0.4 for the highly risk averse investor, E(π) = 1, σ2π = 1, E(θ) = 2, σ

2θ = 0.5, R = 1.1 and

A = 1.

43

Figure 3: The optimal precision for different levels of risk tolerance. The solid curve corresponds toC(x) = x2 + 0.5x, the dashed curve to C(x) = x3 + 0.5x and the dotted curve to C(x) = x1.5 + 0.5x.Investors with risk tolerance below τ∗ ≡ 1.1 do not acquire information. The picture is drawn for E(π) = 1,σ2π = 1, E(θ) = 2, σ

2θ = 0.5, R = 1.1 and A = 1.

Figure 4: The distribution of liquidity trades and the depth of information. The top panel shows the effectof the mean residual stock supply θ and the bottom panel the effect of the variance of liquidity trades σ2θ.The solid curves are drawn for C(x) = x2 and the dashed curve for C(x) = x1.1. The other parametersare E(π) = 1, σ2π = 1, E(θ) = 2 (bottom panel), σ2θ = 0.5 (top panel), n

k = 1, and R = 1.1.

44

Figure 5: The curvatures of the marginal information cost and the optimal precision choice. The marginalcost of information C 0(x) (top left panel) and its inverse 1/C 0(x) (top right panel) for different precisionchoices and the optimal precision choice x(τ) (bottom left panel) and its contribution to the depth ofinformation τx(τ) (bottom right panel) as a function of risk tolerance for C(x) = x2+0.5x (solid curves)and C(x) = 0.5atanh(x) (dashed curves). The other parameters are E(π) = 1, σ2π = 1, E(θ) = 2, σ

2θ = 0.5,

A = 1, and R = 1.1.

45

Figure 6: Addition of active stockholders to a passive base. The level of aggregate risk tolerance n (top leftpanel), the depth of information i (top right panel), the marginal benefit of private information A (middleleft panel), the information collection threshold τ∗ (middle right panel), the expected return (bottom leftpanel) and the variance of returns (bottom right panel) are represented when α active shareholders ofrisk tolerance 2 are added to a base composed of 5 passive shareholder of risk tolerance 0.1 (dashed linein middle right panel). The pictures are drawn for C(x) = x2+2x, E(π) = 1, σ2π = 1, E(θ) = 2, σ

2θ = 0.5,

and R = 1.1.46

Figure 7: Addition of passive stockholders to an active base. The level of aggregate risk tolerance n (topleft panel), the depth of information i (top right panel), the marginal benefit of private information A(middle left panel), the information collection threshold τ∗ (middle right panel), the expected return(bottom left panel) and the variance of returns (bottom right panel) are represented when α passiveshareholders of risk tolerance 0.3 (dashed line in middle right panel) are added to a base composed of oneactive shareholder of risk tolerance 1. The pictures are drawn for C(x) = x2 + 0.5x, E(π) = 1, σ2π = 1,E(θ) = 2, σ2θ = 0.5, and R = 1.1.

47

Table 1List of Variables and their Interpretation

K number of stocks, k running index of a stockR gross rate of return on the riskless assetP k price of stock k (stacked in P )¦k payo¤ of stock k (stacked in ¦)¦ mean of ¦k

¾2¦ variance of ¦k

J total number of investors, j running index of an investorW initial wealth, W 0

j investor j’s …nal wealth¿ j investor j’s absolute risk toleranceU utility function of an investor, u value function in trading periodKj set of stocks investor j recognizesGk(¿ ) number of investors with risk tolerance below ¿ aware of kgk (¿)d¿ number of investors with tolerance between ¿ and ¿ + d¿ aware of kNk number of investors aware of stock k (stacked in N )µk (residual) supply of stock k (stacked in µ)µ mean of µk

¾2µ variance of µk

Skj agent j’s private signal about stock k 0s payo¤ (stacked in Sj)

ºkj error in agent j’s private signal about stock k0s payo¤

xkj precision of Sk

jC (x) cost of signal of precision xFj = [k2Kj fSk

j ; P kg agent j’s information setEj (: j Fj ) agent j’s conditional (trading period) expectationsEj (:) agent j’s unconditional (planning period) expectationsXk

j investor j’s holdings of stock k (stacked in Xj )Yj investor j’s holdings of the bond¹k noisiness about stock k (stacked in ¹)Dk depth of information about stock k (stacked in D)hk

0 precision obtained from the price of stock k (stacked in h0)h

kaverage total precision about stock k (stacked in h)

P k¦ and P k

0 coe¢cients of the price function for k (stacked in P¦ and P0)Ak marginal bene…t of private information about stock k (stacked in A)¿ ¤k risk tolerance threshold for collection of information about stock k® number of investors joining the base¿ ® risk tolerance of investors joining the basex® precision of private signal of investors joining the base"A=n elasticity of Ak with respect to nk

"A=¹ elasticity of Ak with respect to ¹k

"A=¾2µ

elasticity of Ak with respect to ¾2µ

"C 0 elasticity of C0 with respect to x"¾2

µelasticity of ¾2

µ with respect to ®

Table 2 Summary Statistics

Firms listing American Depositary Receipts programs with their listing dates and levels are obtained from J.P. Morgan (www.adr.com). We retain only non-capital raising programs (Level I and II) initiated by British companies between 1986 and 2002 which were followed by at least one analyst in their pre- listing year. Capitalization is listing-year end market capitalization in millions of British Pounds and is obtained from Datastream. The number of analysts is measured as the average number of analysts making an earnings forecasts on the underlying stock for the fiscal year end and is obtained from I/B/E/S.

Capitalization Sector Level I Level II Level I Level II Mean (£ million) 384.1 1396.1 Basic industries 5 4 Standard deviation 712.0 2735.0 Capital goods 6 2 Maximum 1.0 19.2 Consumer goods 4 4 Minimum 4305.3 12128.8 Consumer services 15 4 Number 44 37 Energy 1 1 Finance 3 6 Health care 2 8 Public utilities 2 5 Technology 5 3 Transportation 1 0

Listing exchange Listing period Level I Level II Level I Level II NYSE 44 30 1988-1992 9 6 NASDAQ or AMEX 0 7 1993-1997 13 8 1998-2002 22 23

Number of analysts following the underlying stock Level I Level II Average over the complete period 7.9 10.3 In prelisting year 10.8 14.3 In postlisting year 10.3 11.8 Percentage change -4.7% -17.3%

Table 3 Analyst Coverage of the Underlying Stock for Level II ADRs

Cross-Section Regressions around Listing Date We estimate cross-section regressions of the change in the number of analysts following the underlying stock around ADR listing date (∆Analyst) on a constant, the change in company size (∆Size) and the change in earnings growth (∆Earnings_Growth). Changes are computed as differences between values in the years following and preceding the listing. Analysts is measured as the average number of analysts making an earnings forecasts on the underlying stock for the fiscal year end. Size is measured as the log of the year-end market capitalization in British Pounds. Earnings_Growth is the ratio of current earnings to previous year earnings minus one. Absolute value of t statistics is in parentheses.

(1) (2) (3) (4) ∆Analysts ∆Analysts ∆Analysts ∆Analysts

Constant -2.477 -2.482 -2.038 -2.134 (2.56)** (2.55)** (1.92)* (2.13)** ∆Size 0.922 -1.026 (1.14) (0.82) ∆Earnings_Growth 2.289 1.881 (2.10)** (2.41)** Observations 37 37 35 35 R-squared 0.00 0.01 0.15 0.13 * and ** indicate significance at 10 and 5 percent levels, respectively.

Table 4 Analyst Coverage of the Underlying Stock for Level II ADRs

Panel Regressions We estimate panel regressions with company fixed-effects of the number of analysts following the underlying stock (Analyst) on a dummy which equals 1 if the underlying stock has an ADR listed and 0 otherwise (ADRdum), company size (Size), earnings growth (Earnings_Growth), lagged number of analysts following the underlying (L_Analyst), a time trend (Time_Trend) and a constant. Analysts is measured as the average number of analysts making an earnings forecasts on the underlying stock for the fiscal year end. Size is measured as the log of the year-end market capitalization in British Pounds. Earnings_Growth is the ratio of current earnings to previous year earnings minus one. L_Analyst is the number of analysts following the underlying stock in the previous year. Time_Trend is constructed as the annual average number of analysts per stock for the sample of Level II programs. Absolute value of t statistics is in parentheses. (1) (2) (3) (4) (5) (6) Analysts Analysts Analysts Analysts Analysts Analysts ADRdum -1.268 -3.234 -3.725 -3.401 -1.995 -2.406 (2.10)** (6.86)** (7.51)*** (8.77)*** (4.76)*** (6.52)*** Size 2.437 0.780 0.323 -0.377 -0.275 (7.58)*** (2.75)*** (1.45) (1.55) (1.29) Earnings_Growth -0.065 -0.072 0.034 -0.047 0.020 (0.75) (0.83) (0.50) (0.67) (0.32) L_Analysts 0.555 0.399 (16.82)*** (11.60)*** Time_Trend 0.891 0.560 (15.03)*** (9.45)*** Constant -7.220 15.228 9.101 5.077 7.262 5.055 (2.91)*** (54.04)** (4.03)*** (2.85)*** (3.94)*** (3.11)*** Observations 617 487 481 481 481 481 Companies 37 37 37 37 37 37 R-squared 0.09 0.10 0.12 0.46 0.42 0.55 *, **, *** indicate significance at 10, 5 and 1 percent levels, respectively.

Table 5 Analyst Coverage of the Underlying Stock for Level I ADRs

Panel Regressions We estimate panel regressions with company fixed-effects of the number of analysts following the underlying stock (Analyst) on a dummy which equals 1 if the underlying stock has an ADR listed and 0 otherwise (ADRdum), company size (Size), earnings growth (Earnings_Growth), lagged number of analysts following the underlying (L_Analyst), a time trend (Time_Trend) and a constant. Analysts is measured as the average number of analysts making an earnings forecasts on the underlying stock for the fiscal year end. Size is measured as the log of the year-end market capitalization in British Pounds. Earnings_Growth is the ratio of current earnings to previous year earnings minus one. L_Analyst is the number of analysts following the underlying stock in the previous year. Time_Trend is constructed as the annual average number of analysts per stock for the sample of Level II programs. Absolute value of t statistics is in parentheses. (1) (2) (3) (4) (5) (6) Analysts Analysts Analysts Analysts Analysts Analysts ADRdum 0.102 -1.618 -2.070 -2.394 -1.331 -1.761 (0.25) (4.27)** (5.54)*** (8.47)*** (4.86)*** (7.26)*** Size 1.632 1.234 0.597 0.506 0.333 (8.72)*** (6.76)*** (4.21)*** (3.69)*** (2.75)*** Earnings_Growth 0.000 0.000 0.003 -0.004 -0.001 (0.07) (0.04) (0.90) (1.56) (0.58) L_Analysts 0.601 0.376 (19.15)*** (12.13)*** Time_Trend 1.126 0.778 (20.86)*** (14.07)*** Constant -1.746 12.293 3.991 1.672 -1.608 -1.327 (1.47) (50.26)** (3.18)*** (1.75)* (1.69)* (1.59) Observations 741 542 533 533 533 533 Companies 44 44 44 44 44 44 R-squared 0.11 0.04 0.12 0.50 0.53 0.64 *, **, *** indicate significance at 10, 5 and 1 percent levels, respectively.