Institutional Ownership, Share Price Levels, and the Value of the Firm

 
Institutional Ownership, Share Price Levels, and the Value of the Firm *

                                            Chitru S. Fernando
                                    Michael F. Price College of Business
                                         University of Oklahoma
                                             307 West Brooks
                                            Norman, OK 73019
                                            cfernando@ou.edu
                                              (405) 325-2906

                                           Vladimir A. Gatchev
                                          Department of Finance
                                    College of Business Administration
                                       University of Central Florida
                                         Orlando, FL 32816-1400
                                          vgatchev@bus.ucf.edu
                                              (407) 823-3694

                                               Paul A. Spindt
                                     A. B. Freeman School of Business
                                             Tulane University
                                             7 McAlister Drive
                                         New Orleans, LA 70118
                                             spindt@tulane.edu
                                              (504) 865-5413

                                              September 2007

Keywords: Share price level, firm value, institutional ownership, monitoring

JEL classification: C24, G12, G30

* We are grateful to Ed Dyl, Srini Krishnamurthy, Tom Noe, Russ Robins, Arturo Rodriguez, Wayne
Thomas, Martin Young, and seminar participants at Tulane University and the FMA European meetings for
useful discussions and comments. We remain responsible for any errors.
Institutional Ownership, Share Price Levels, and the Value of the Firm

                                          ABSTRACT

We theoretically and empirically examine the variation in institutional ownership and share price
levels across firms. By explicitly incorporating the role of institutional investors in monitoring
and information gathering, we show how they can substitute for costly information generation by
analysts. Ownership structure and share price levels emerge endogenously as an outcome of this
trade-off, in contrast to the causal relationship between price and institutional ownership that is
widely assumed in the literature. Holding size constant, our model predicts and empirical findings
confirm that higher valued firms have higher institutional ownership and higher share price
levels. We also show that analyst following and institutional ownership are inversely related and
that the positive association between share prices and institutional ownership exists independently
of liquidity and size considerations.
The potential for institutional shareholders to affect the stock market performance of a firm by

monitoring its managers and generating information has been widely recognized in the finance

literature.1 A standard assumption is that a firm’s share price level serves as a proxy for the stock market

liquidity of its shares,2 which has also been used to explain the documented monotonic relationship

between institutional ownership and share price levels.3 The relegation of share price level to being a

liquidity proxy belies the considerable attention paid by firms and investors to share price levels and the

documented links between share price levels and several other important firm characteristics.4 In this

paper, we develop and test a unified theoretical framework to show how a firm’s institutional ownership

and share price level are both endogenously determined based on a firm’s fundamental characteristics,

which permits us to explain several observed empirical regularities and provide new insights.

     Brennan and Hughes (1991) argue that the negative relation between brokerage commissions and

share prices provides an incentive for brokers to produce more information about low-priced stocks,

while Angel (1997) relies on the negative relation between share price and the relative tick size to

develop a model in which lower prices and higher relative tick sizes provide higher incentives for

analysts to promote the stock and help broaden the investor base of a firm. Merton (1987) predicts that

broadening the base of investors who are familiar with a stock will lower its required rate of return. Dyl

and Elliott (2006) empirically examine whether the cross-sectional variation of stock prices can be

explained by Merton’s (1987) model. They find that the share price level in their sample of firms

increases with both firm size and total equity per shareholder (a proxy for average investor size) and

1
  See, for example, Shleifer and Vishny (1986), Brickley, Lease and Smith (1988), Agrawal and Mandelker
(1990), McConnell and Servaes (1990), Smith (1996), Carleton, Nelson and Weisbach (1998), Gillan and Starks
(2000), and Allen, Bernardo and Welch (2000).
2
  See, for example, Brennan and Subrahmanyam (1996) and Gompers and Metrick (2001).
3
  See, for example, Falkenstein (1996) and Gompers and Metrick (2001).
4
  See, for example, Maloney and Mulherin (1992), Muscarella and Vetsuypens (1996), Falkenstein (1996), Angel
(1997), Seguin and Smoller (1997), Schultz (2000), Gompers and Metrick (2001), Fernando, Krishnamurthy and
Spindt (2004) and Dyl and Elliott (2006).

                                                      1
argue that their findings support the notion that firms select the trading ranges for their shares to enlarge

their investor base.

       Nevertheless, several questions remain. First, given the overwhelming evidence that lowering share

prices increases transactions costs,5 what is the trade-off between increased transactions costs and higher

information benefits associated with lower share prices? Second, what role do investors, specifically

institutional investors, play in substituting for the information generation role of stock market analysts?

While the aforementioned literature does not consider differences in investor characteristics, studies by

Falkenstein (1996) and Gompers and Metrick (2001) show that share price is significantly and

positively related to institutional ownership. Since institutions are considered to be more sophisticated

and better informed than retail investors, this finding appears inconsistent with the notion that firms who

have favorable private information maintain lower share prices to attract more analyst coverage and

promote their shares. It is possible that institutional ownership provides an alternative channel for firms

to disseminate information and increase their value through monitoring.6 Finally, the notion from

Merton (1987) that firms can increase their value by lowering share prices and broadening their

shareholder base seemingly conflicts with empirical evidence that suggests a positive association

between share price levels and the long term performance of the firm.7

       We develop a model to examine these questions. In our model, firms benefit by both institutional

ownership and analyst coverage, with institutional ownership having the potential to substitute for costly

information generation by analysts. Firms that anticipate smaller benefits from institutional ownership

5
  See, for example, Conroy, Harris and Benet (1990), McInish and Wood (1992), Stoll (2000), and Schultz (2000)
for evidence on the negative relation between price level and transactions costs. Benartzi et al. (2006) pose the
issue in stark terms by noting that if GE had not split its stock from 1935 to 2005, it would have saved investors
over $100 million in brokerage commissions in 2005 alone!
6
 Institutional investors employ their own buy-side analysts or are able to generate information through other
means without relying on sell-side analysts. Additionally, institutional investors can also add value to firms they
own by monitoring. See, for example, Shleifer and Vishny (1986), McConnell and Servaes (1990), Smith (1996),
Carleton, Nelson and Weisbach (1998), Gillan and Starks (2000), and Allen, Bernardo and Welch (2000) for a
discussion of the benefits to a firm brought about by institutional ownership.
7
    See, for example, Seguin and Smoller (1997) and Fernando, Krishnamurthy and Spindt (2004).

                                                         2
set lower share prices to increase the brokerage revenue associated with trading their shares and thereby

induce more information generation by analysts. In contrast, firms that anticipate large benefits from

institutional ownership place a lower value on analyst following and set higher share prices to decrease

the cost to investors of owning their shares. In equilibrium, higher priced firms have a higher

institutional ownership than lower priced firms. Additionally, our model predicts that higher priced

firms will have a higher value than lower priced firms.

     The association between institutional ownership and price documented in previous studies has been

justified as evidence that institutional investors favor high priced stocks to avoid transaction costs. This

explanation raises the question of why retail investors are more willing than institutional investors to

accept higher transactions costs. In contrast, while establishing a theoretical basis for the empirically

observed positive relationship between share prices and institutional ownership, we show that this

relationship exists independently of liquidity considerations.

     As in Brennan and Hughes (1991) and Angel (1997), in our model too lower share prices increase

the incentive for brokers to generate information and promote stocks. However, unlike Brennan and

Hughes (1991) and Angel (1997), who assume homogeneous populations of investors, we allow for

investor heterogeneity by separating investors into two clienteles, institutional and retail, and explicitly

account for the well-documented possibility that institutional investors can add value by generating

information about and monitoring the activities of firms they own. Our model is consistent with Merton

(1987) and the ensuing literature in the sense that, holding all else constant, lowering share prices will

increase a firm’s value provided the increased benefits of lower prices exceed the higher transactions

costs. Nonetheless, when the possibility that institutional ownership might also increase the value of the

firm is factored in, we show that in equilibrium the value of the firm will actually increase with share

price, holding all else constant. This theory of investor clienteles and share price determination is new to

the literature.

     In addition to providing new insights into the relationships among and endogeneity associated with

several key firm specific variables such as institutional ownership, analyst coverage, share price levels,

                                                     3
stock market liquidity and the value of the firm, our model also yields several other new findings and

empirical implications. First, by extending the previous literature to incorporate the role of institutional

investors, our model predicts that analyst coverage and institutional ownership will be inversely related.

Second, while establishing a theoretical basis for the empirically observed positive relationship between

share prices and institutional ownership, we show that this relationship exists independently of liquidity

considerations, which contradicts the widely-held belief that higher liquidity is what drives institutions

to hold higher-priced stocks. Additionally, our model also implies that the share price level will be an

indicator of a firm’s value. Therefore, we would expect to find a greater propensity for institutions to

invest in higher priced stocks even in the absence of “prudent-man” rules that constrain them to do so.8

Additionally, firms with higher levels of institutional ownership and higher values will choose higher

split-to prices when they split their shares. Surprisingly though, our model predicts that analyst coverage

of such firms will be lower than their coverage of other similar-sized firms.

     Our empirical findings strongly support our theoretical predictions. In particular, consistent with

Brennan and Hughes (1991), Angel (1997) and our theory, we find that while analyst coverage increases

with a firm’s market capitalization, it actually declines with the share price level after controlling for

size. Extending the work by Brennan and Hughes (1991) and Angel (1997), we show that the number of

analysts following a firm will decrease with the firm’s information quality (as measured by its S&P

quality rank) and with institutional ownership. We find that a firm’s share price level rises with

institutional ownership even after controlling for differences in stock market liquidity. We also find that

a firm’s value as measured by Tobin’s Q is positively related to institutional ownership or its share price

level, and that higher-valued firms will target higher price levels when they split their shares. Tobin’s Q

8
  See, for example, Del Guercio (1996). The argument here is similar to the “ownership clientele” effects
discussed by Allen, Bernardo, and Welch (2000), where higher quality firms attract relatively less-taxed
institutional investors. As noted by Allen, Bernardo and Welch (2000), such investors have a relative advantage in
ensuring that the firms they invest in are well managed.

                                                        4
is also positively related to the precision of analyst forecasts. Furthermore, share prices of firms with

higher information precision are less sensitive to broker information precision and vice versa.

     The rest of our paper is organized as follows. We develop our model in Section I and also discuss

its empirical implications. Section II describes our sample and methodology. Section III presents the

empirical results. Section IV concludes.

                              I.       The Model and Empirical Implications

     In this section, we develop a model that explains how firms optimally select their share price levels

to maximize firm value, taking into account the benefits of institutional ownership and the costs and

benefits of information generation by market intermediaries. We conclude the section by developing

several empirical implications of our model.

A. The Economy

     We consider a multi-stage economy with J risky assets (firms), a riskless asset, and a continuum

of risk-averse institutional and retail investors. Risky asset payoffs are independently distributed, with

each risky asset j having a random terminal payoff of Vj that is normally distributed with mean μ j and

precision h j . The riskless asset’s return is normalized at zero. All investors in the economy have

identical constant absolute risk aversion (CARA) preferences with CARA parameter γ .9 We examine

the investment decision of a representative investor of each type, assuming for convenience that in

aggregate both investor types are endowed with equal amounts of the riskless asset prior to trading in

the risky asset.

     Investment in the risky assets is facilitated by brokers. Brokers can produce signals of the final

payoff of each firm with precision hB . Conditional on the number of brokers producing information, N j ,

9
 Our model can be easily extended to the case where institutional and retail investors have different risk
preferences without altering its main predictions.

                                                         5
the precision of final payoff for each firm j becomes h j + N j hB . This specification assumes that broker

reports are independent of each other. As in Brennan and Hughes (1991), we assume that all brokers are

risk neutral, have identical abilities, have the same number of clients, and are perfectly competitive,

which implies that they will charge identical commissions. As suggested by the discussion and findings

in Benartzi, et al. (2006) and Goldstein, et al. (2006), we assume a commission structure where brokers

charge a fixed commission, c , per share.10

     In our model, institutional investors are differentiated from retail investors by their ability to affect

the ex ante payoff distribution of the firms they invest in by producing information about these firms

and/or monitoring their performance. For each firm j we aggregate these effects in a single institutional

influence parameter m j > 0 . Institutional investment modifies the precision of firm pay offs by the

multiplicative factor, (1 + m j θ I ) , where θI is the fraction of the firm held by institutions, resulting in a

precision of the distribution of the firm’s value of ( h j + N j hB )(1 + m j θ I ) . The precision increases with

both m j and θI , reflecting the increase in the benefits of institutional ownership with both the

institutional influence parameter and institutional ownership.

B. Stages

     The model consists of four stages. In the first stage firms determine the number of shares

outstanding to maximize the market value of the firm, which also determines the price per share. As

observed by Brennan and Hughes (1991) and Angel (1997), lower share prices (and a higher number of

shares outstanding) provide greater incentive for market makers to promote the firm’s shares since the

returns to market makers increase with lower share prices under prevailing transactions cost structures.

Therefore, a higher number of shares (and lower share price) will increase the information generated by

brokers about a firm but also the cost to shareholders of investing in the firm due to the higher

10
  Our results remain qualitatively unchanged when we assume that institutional and retail investors have different
commission structures.

                                                        6
brokerage commissions they will be required to pay. In the second stage, brokers produce information

about each firm’s terminal pay off. In the third stage, investors make their investment decisions

regarding each firm, which in turn determines the commissions earned by brokers and each firm’s

market-clearing valuation. In the final stage, firms’ payoffs are realized and firms liquidate. The four

stages are summarized below. We solve the model recursively.

 s=1                              s=2                                        s=3                    s=4

 • Firm j chooses                 • N j brokers produce                      • Ownership by         • Payoff is
   number of shares                                                            investor type          realized.
                                    information about
   outstanding S j .                firm payoff.                               i ∈ { I , R} is      • Firm
                                                                                                      liquidates.
                                  • Each broker                                determined.
 • Payoff Vj normally              produces                                 • Firm market-
    distributed with                independent                                clearing value Q j
    mean μ j and                    information with
                                                                               is determined.
                                    precision hB .
    precision   hj .                                                         • Broker
                                  • Payoff precision                           commissions are
                                    becomes                                    determined.
                                       h j + N j hB

C. Optimal Investor Holdings and Equilibrium Valuations

     An investor of type i ∈ { I ,R} has an initial endowment of the riskless asset denoted by W0 ,i . The

investor of type i ∈ { I ,R} solves:

                                                      max E ⎡⎣ −e− γW ⎤⎦ ,
                                                                     
                                                                         i

                                                       θij
                                                                                                                    (1)

     where Wi = W0 ,i + ∑ j =1 θijVj − ∑ j=1 θij ( Q j + cS j ) . Q j is the market valuation of firm j at the trading
                           J               J

stage and c is the per-share trading commission for all investors.

     In a perfectly competitive market for broker services, brokers will use all their commission

revenues to produce information. The total commission revenues earned by brokers from trading the

                                                              7
shares of firm j are equal to ( θ I + θ R ) cS j , and each broker incurs a cost of f per firm to produce

information with precision hB about that firm. Noting that ( θ I + θ R ) = 1, the expression for the optimal

number of brokers can be stated as follows:

     Lemma 1 (Brennan and Hughes, 1991): Under perfect competition, the number of brokers

producing information about firm j , N j , is given by:

                                                               cS j
                                                    Nj =               .                                  (2)
                                                                   f

     Therefore, firms with more shares outstanding will have a larger number of brokers producing

information about them.

     Given the previous assumptions regarding the distribution of final firm payoffs, the portfolio

optimization problem of each investor i can be stated as follows:

                                                      γ2
                                           θ ij
                                                   ( )
                                          max γE Wi − Var Wi ,
                                                      2
                                                                           ( )                            (3)

    where

                                       ( )
                                   Var Wi = ∑ j=1 θij2
                                               J

                                                          (h
                                                                           1
                                                                   + N j hB )(1 + m j θI )
                                                                                             .            (4)
                                                               j

     The expression in (4) indicates that all else equal, firms with more precise initial information ( h j ),

higher broker coverage ( N j ), more precise information produced by brokers ( hB ), and a higher benefit

from institutional investor participation ( m j ) have a lower variance of future values.

     Solving for the optimal ownership weights and the equilibrium valuation of the firm, subject to

market clearing, we can state our results for the equilibrium ownership structure of the firm as follows:

                                                          8
PROPOSITION 1: The equilibrium ownership structure for each firm j is given by

                                           1 2 3 + (1 + m j ) − ( 4 + m j )
                                                                    2

                                     θ Ij = +                                                                                (5)
                                           2            6m j

                                                    2 3 + (1 + m j ) − ( 4 + m j )
                                                                    2
                                              1
                                     θ Rj =     −                                                                            (6)
                                              2                 6m j

     (See Proof in Appendix A)

                                    2 3 + (1 + m j ) − ( 4 + m j )
                                                       2

     Since it is easily seen that
                                                    6m j
                                                                                            (
                                                                        lies in the interval 0 , 1
                                                                                                     6   ) for m   j   > 0 , we

can conclude that both institutional and retail ownership will be positive and institutional ownership will

be in the interval   ( 1 2 , 2 3 ) . Nonetheless, the institutional ownership share will increase with m                 j   and

the retail ownership share will be correspondingly reduced. It is interesting to note that trading

commissions, which are the outcome of the number of shares/share price decision of the firm, do not

directly affect the manner in which ownership of any given firm is shared between institutional and

retail investors. Therefore, in our model, there is no causal relation between trading commissions and

ownership structure. As noted previously, this result remains unchanged when we assume that

institutional and retail investors have different commission structures, provided the differential is

applied uniformly across all firms. Nonetheless, as we will show later, since firms optimize their share

price decision to maximize any potential benefit of institutional ownership, firms that have high (low)

values of m j will also have high (low) share prices in equilibrium, thereby leading to a positive

association between institutional ownership and share prices. These results are in stark contrast to the

existing empirical literature which assumes that the higher costs of trading low priced shares causes a

lower institutional ownership of these shares relative to high priced shares.

                                                            9
We can state our result for the market valuation of each firm at the trading stage, conditional on the

number of shares outstanding, as follows:

     Lemma 2: The market valuation of each firm j at the trading stage is given by

                                                                       (1 + m )   3 + (1 + m j ) − (1 + m j ) − 1
                                                                                               2            2

                                                         γf                  j
                         Q j = μ j − cS j −                                                                         (7)
                                              ( fh   j
                                                         + cS j hB )                      mj

     (See Proof in Appendix A)

     Conditional on the choice of number of shares outstanding, firm value increases with the benefit of

institutional ownership ( m j ), with the ex ante precision of the firm’s payoff distribution ( h j ), and the

precision of broker information ( hB ), while it decreases with investor risk aversion ( γ ).

D. The Choice of Shares Outstanding 11

     In the first stage, firms choose the number of shares outstanding so as to maximize their value at

the trading stage. In doing so, firms have to balance offsetting effects. On the one hand, setting a low

number of shares outstanding (leading to high share prices) reduces the cost of brokerage commissions

and increases the net returns to shareholders. On the other hand, a lower number of shares outstanding

induces less information production by brokers, which results in higher uncertainty and lower valuations

for the firm. Proposition 2 states the equilibrium result for the optimal number of shares that arises from

these tradeoffs.

11
  Since the mechanism by which firms select a share price is setting the number of shares outstanding, we pose
the firm’s optimization problem as one of selecting the number of shares rather than the share price, noting that
share price equals the firm’s value divided by the number of shares.

                                                                        10
PROPOSITION 2: The optimal number of shares outstanding, S *j , for each firm j , is given by

                                              ⎛                                           ⎞
                                     γfhB m j ⎜ (1 + m j ) 3 + (1 + m j ) − (1 + m j ) − 1⎟ − fh j m j
                                                                         2            2

                                              ⎝                                           ⎠
                            S *j =                                                                                                      (8)
                                                                 chB m j

        (See Proof in Appendix A)

        As firm specific quality ( h j ) and institutional benefits ( m j ) increase, the optimal number of shares

outstanding decreases. Additionally, for a high precision of broker information, the number of shares

outstanding decreases as the precision of broker information ( hB ) increases. For low precision of broker

information, however, as hB increases the optimal number of shares outstanding increases.12 We analyze

the relation between equilibrium share prices and these variables in Proposition 4 below.

        Solving the expression in (5) for m j we obtain that

                                                                         (
                                              m j = 2 ( 2θ Ij − 1) θ Ij ( 2 − 3θ Ij ) .     )                                           (9)

        Replacing for m j in the expression for Proposition 2 and for θ Ij in the interval 1 , 2                  (    2     3   ) , we find
that

                                            S *j =
                                                        ⎛
                                                     1 ⎜ γf
                                                             ×
                                                                (
                                                               2 − ( 5 − 3θ Ij ) θ Ij )
                                                                                      −
                                                                                             ⎞
                                                                                        fh j ⎟
                                                                                               .                                       (10)
                                                     c ⎜⎜ hB           θ Ij             hB ⎟⎟
                                                        ⎝                                    ⎠

        As we have shown previously, equilibrium values of institutional ownership ( θ Ij ) lie in the interval

( 1 2 , 2 3 ) and it is easily observed that for values of θ                 Ij   in this interval, the optimal number of shares

outstanding decreases with institutional ownership. As we prove later, this result gives rise to a positive

                                                       ⎛                  ⎛ γf ⎛                                           ⎞⎞⎞
     The point at which the relation changes is hB = 4 ⎜ ( fh j )              ⎜ (1 + m j ) 3 + (1 + m j ) − (1 + m j ) − 1⎟ ⎟⎟ ⎟⎟ .
                                                                  2                                       2            2
                                                                          ⎜⎜
12
                                                       ⎜
                                                       ⎝                  ⎝ mj ⎝                                           ⎠⎠⎠

                                                                    11
association between share prices and institutional ownership. We should note, however, that this

association is not causal.

     We can now extend the predictions of Brennan and Hughes (1991) presented in Lemma 1. We use

expression (8) to substitute for the shares outstanding in Lemma 1, which gives the equilibrium number

of brokers producing information.

     PROPOSITION 3: The number of brokers producing information is:

                                       ⎛                                           ⎞
                                         (1 + m j ) 3 + (1 + m j ) − (1 + m j ) − 1⎟
                                                                  2            2

                                  γ ⎝  ⎜
                        N *j =       ×                                             ⎠ − hj
                                                                                                     (11)
                                 fhB                        mj                         hB

     Replacing for m j in the expression for Proposition 3, we find that:

                                     N *j =
                                               γ
                                                  ×
                                                    (                   )
                                                    2 − ( 5 − 3θ Ij ) θ Ij
                                                                           −
                                                                             hj
                                                                                                     (12)
                                              fhB           θ Ij             hB

    The number of brokers producing information is a decreasing function of firm specific information

precision ( h j ) and of institutional ownership ( θ Ij ). Institutional ownership, however, should not be

viewed as causing brokerage coverage. Expression (12) simply shows that firms with higher monitoring

benefits will have higher institutional ownership and fewer brokers producing information as the

equilibrium outcome.

    The model also relates institutional ownership and firm characteristics to equilibrium firm value. By

substituting the optimal number of shares in (8) for the value of the firm expressed in Lemma 2, we

obtain the expression for the equilibrium value of the firm, which we state in Proposition 4:

     PROPOSITION 4: The equilibrium value of the firm is given by

                                              ⎛                                           ⎞
                                                (1 + m j ) 3 + (1 + m j ) − (1 + m j ) − 1⎟
                                                                         2            2

                                         γf ⎝ ⎜
                      Qj = μ j +
                       *
                                 fh j
                                      −2    ×                                             ⎠
                                                                                                     (13)
                                 hB      hB                        mj

                                                        12
Replacing for m j in the expression for Proposition 3, we find that:

                                      Q*j = μ j +
                                                     fh j
                                                              −2
                                                                   γf
                                                                      ×
                                                                               (
                                                                        2 − ( 5 − 3θ Ij ) θ Ij             )
                                                     hB            hB           θ Ij                               (14)

        It is evident from (14) that the maximized firm value is an increasing function of firm specific

information precision ( h j ) and of institutional ownership ( θ Ij ). Note that (14) does not imply that firms

can increase their values by increasing institutional ownership. What it shows is that firms with higher

monitoring benefits will have higher institutional ownership and higher values as the equilibrium

outcome.

        Our model also allows us to examine the relation between a firm’s share price and its expected cash

flows, institutional benefits, and parameters related to information precision of the firm. Proposition 5

provides the expression for a firm’s equilibrium share price (expression (14) divided by expression

(10)).

        PROPOSITION 5: The equilibrium share price of the firm is given by

                                                 ⎛                                                     ⎞
                                                 ⎜                                                     ⎟
                                                 ⎜                 μ j − fh j                          ⎟
                                         Pj* = c ⎜                                                  − 2⎟           (15)
                                                     γf ( 2 − ( 5 − 3θ ) θ )
                                                 ⎜                                                     ⎟
                                                 ⎜                                 Ij   Ij             ⎟
                                                 ⎜     ×                                     − fh j    ⎟
                                                 ⎝   hB                 θ Ij                           ⎠

        The equilibrium share price increases with the expected cash flows of the firm ( μ j ), institutional

ownership ( θ Ij , which measures institutional benefits), and the precision of the initial information

available about the firm ( h j ).13 Furthermore, a firm’s equilibrium value will increase with the precision

of broker information ( hB ) but the relation between the precision of broker information ( hB ) and share

13                                                        *                                                    *
     These relations hold for positive firm value ( Q j ) and positive number of shares outstanding ( S j ).

                                                                   13
price is not necessarily monotonic. For high precision of broker information, as precision of broker

information ( hB ) increases shares outstanding will decrease and share prices will increase. For low

precision of broker information, however, as the precision of broker information ( hB ) increases the

value of the firm increases but the number of shares outstanding also increases and no clear relation

exists unless one makes further assumptions about the remaining parameters. Further examination,

however, reveals that for sufficiently high μ j the cross-derivative of share price with respect to the

precision of broker information ( hB ) and the firm’s initial precision of information ( h j ) is negative.

Putting together the comparative statics results for the value of the firm and its share price, we can

conclude that if the parameters in the model are not perfectly measured in practice (as we expect) there

will be a positive association between the value of the firm and its share price level. This is not because

share price levels directly affect firm value but because the factors that lead to a higher firm value also

lead to a higher price per share.

E. Empirical Implications

     Our theoretical framework gives rise to several empirical hypotheses. The first hypothesis pertains

to the analyst coverage of the firm.

Hypothesis 1: The number of analysts following a firm will be positively related to the number of shares

outstanding. Alternatively, the number of analysts following a firm will be inversely related to share

price and positively related to firm market capitalization. Furthermore, the number of analysts

following a firm will be inversely related to the firm’s quality of information ( h j ) and institutional

ownership ( θ Ij ).Finally, as the precision of broker information ( hB ) increases, the effect of the firm’s

quality of information ( h j ) on the number of analysts following the firm will be less negative.

                                                     14
Hypothesis 1 follows from Lemma 1 (based on Brennan and Hughes, 1991) and the results

presented in Proposition 3. Empirically, we expect the number of analysts (the dependent variable) to

decrease as share price (the independent variable) increases. In addition, keeping share price fixed, we

expect the number of analysts to increase as the market capitalization of the firm increases. This is

because, for a given share price, increasing firm value means increasing the number of shares

outstanding. The model also predicts that the number of analysts will decrease with institutional

ownership and a firm’s quality of information -- predictions that extend Brennan and Hughes (1991). At

the same time, the model predicts that the interaction between a firm’s quality of information and broker

information precision will be positively related to the number of analysts following the firm.

     Empirically, we can also examine how the value of the firm is affected by the precision of

information and institutional monitoring benefit, which is our next empirical hypothesis.

Hypothesis 2: The value of the firm will increase with the firm’s quality of information ( h j ), the

precision of broker information ( hB ), and institutional ownership ( θ Ij ).

     Hypothesis 2 follows directly from Proposition 3 and expression (12).

     Our model predicts a positive relation between value and institutional ownership, and between

share price and institutional ownership (see also Hypothesis 4 below). This suggests a positive

association between value and share price that gives rise to our next empirical hypothesis:

Hypothesis 3: The value of the firm and its share price will be positively associated.

     It should be noted that the relation in Hypothesis 3 is not causal. It stems from the fact that share

price levels as well as firm value are both positively affected by monitoring benefits and the precision of

initial information about the firm.

                                                      15
Proposition 4 allows us to make empirical predictions about the cross-sectional differences in share

price levels of firms and gives rise to Hypothesis 4.

Hypothesis 4: A firm’s share price will increase with institutional ownership ( θ Ij ) and the precision of

the information available about the firm ( h j ). This relation should hold even after controlling for

measures of market liquidity.

     Examining expression (13), we do not find a clear relation between the share price of the firm and

the precision of broker information ( hB ). While it is true that the value of the firm monotonically

increases with the precision of broker information, the optimal number of shares outstanding first

increases and then decreases as the precision of broker information increases. However, we can propose

the following hypothesis about the interaction of the precision of broker information ( hB ) and the firm’s

information quality ( h j ) as far as firm share price is concerned.

Hypothesis 5: For sufficiently high firm cash flows ( μ j ), the precision of broker information ( hB ) and

the firm’s information quality ( h j ) act as substitutes in determining the share price of the firm.

     Empirically, we expect a negative interaction effect between the precision of broker information

( hB ) and the firm’s information quality ( h j ) in explaining the split-to price of the firm. This is because

as the precision of broker information increases the precision of information about the firm becomes

less important. As a direct consequence, split-to prices will be less (more) sensitive to firm information

precision for higher (lower) levels of broker information precision.

     We empirically examine these hypotheses in Section III after describing our sample and variable

definitions in the next section.

                                                      16
II.      Sample and Variables

        Our sample covers the years from 1985 to 2005 and includes all firms with common stock (CRSP

share codes 10 or 11) listed on NYSE/AMEX/Nasdaq (CRSP exchange codes 1, 2, or 3). We also

require that firms have data available in the Center for Research in Security Prices (CRSP) daily and

monthly files and the Compustat database.

        Share price levels and shares outstanding for year t are based on values at the end of June in year t

and come from the CRSP monthly files.14 Also from the CRSP monthly files we identify firms that split

their shares in year t and we use their split-to prices for that year as another measure of the firm’s

preferred share price. As measures of firm size we use firm market capitalization (June of year t closing

price times shares outstanding from CRSP) and total assets (Compustat item 6 for fiscal year t). In order

to control for trends in firms’ market capitalizations (asset sizes) over time, we divide firm market

capitalization (total assets) by median NYSE market capitalization (total assets).

        We use the S&P common stock ranking of the firm as a measure of firm specific information

precision ( h j ), which we obtain for each fiscal year from the Compustat annual files (Compustat item

282, available from 1985 onward). Each month, in its Security Owner's Stock Guide, S&P publishes

common stock rankings of firms that are listed on the NYSE/AMEX or are among the most active

Nasdaq firms. Firm rankings are based on historic (past ten years) stability and growth of earnings and

dividends, which makes the S&P common stock rankings a straightforward measure of firm-specific

uncertainty. The range of scores is aligned with the following rankings: A+ (Highest); A (High); A–

(Above Average); B+ (Average); B (Below Average); B– (Lower); C (Lowest); D (in Reorganization).

For the purposes of our quantitative analyses we translate the S&P rankings into the following scores:

(A+: 9), (A: 8), (A–: 7), (B+: 6), (B: 5), (B–: 4), (C: 3), (D: 2). Appendix B describes the S&P common

stock ranking methodology.

14
     Using CRSP average share prices for year t and Compustat share prices leads to similar results.

                                                          17
We measure the number of analysts producing information about the firm ( N j ) by the number of

analysts providing one year earnings forecasts from I/B/E/S. As a measure of the precision of analyst

information ( hB ), on the other hand, we use the inverse of the mean absolute error of one year earnings

forecasts from I/B/E/S. This variable is not available for a significant part of our sample. In order to

maximize sample size when estimating the coefficients for the remaining variables, in the subsequent

regressions we make this measure of analyst precision equal to zero while at the same time we also

include a dummy variable indicating whether analyst precision is missing. This approach effectively

allows us to estimate the coefficient for analyst precision only for firms with an available measure of

analyst precision. We use the standard deviation of daily returns for year t from the CRSP daily files as

a measure of overall firm risk.

     The model in the previous section shows that institutional ownership increases monotonically with

and is solely determined by institutional benefits ( m j ), which permits us to use institutional ownership

as a direct measure of institutional benefits ( m j ). We collect end-of-June percentage institutional

ownership data from the CDA Spectrum database of Thomson Financial, which consists of institutional

13F filings.

     We use two measures of Tobin’s Q, both from the existing literature, to measure firm value ( Q j ).

The first measure of Tobin’s Q is the ratio of the market value of assets to the book value of assets,

where asset market value is the sum of the book value of assets (Compustat item 6) and the market

value of common stock (Compustat item 199 times item 25) less the book value of common stock

(equity (Compustat item 216, or item 60 plus item 130, or item 6 minus item 181) minus preferred stock

(Compustat item 10, or item 56, or item 130)) and deferred taxes when available (Compustat item 35)

net of post-retirement benefits when available (Compustat item 330). Because of its ease of calculation

and because it is available for a large set of firms, this measure of Q is widely used in existing literature

(see, for example, Kaplan and Zingales, 1997; Gompers, Ishii, and Metrick, 2003; and the references

therein).

                                                     18
Lewellen and Badrinath (1997) show that using book values of assets in the denominator of the Q

ratio has shortcomings (e.g., downward biased Q ratios and incorrect ordering of Q ratios across firms).

They propose a different measure of Q that calculates and uses replacement costs (rather than book

values) of fixed assets and inventory. For our second measure of Tobin’s Q we calculate replacement

values using the approach of Lewellen and Badrinath (1997) as modified by Lee and Tompkins

(1999).15 This second measure of Tobin’s Q is equal to the market value of the firm’s common stock

plus the book values of preferred stock, short-term debt, and long-term debt divided by the replacement

value of firm assets. Asset replacement values are calculated as book value of total assets minus book

values of fixed assets and inventory plus replacement values of fixed assets and inventory minus all

liabilities other than long-term and short-term debt. The replacement value of fixed assets in year t

requires the estimation of historic investments in fixed assets year by year. Then the method uses

specific depreciation and inflation estimates to determine the replacement value in year t of each year’s

investment “vintage.” The replacement value of fixed assets is the sum of the replacement values of

historic investments. The calculation of inventory replacement values is simpler. If firms use FIFO to

account for inventory, then inventory replacement value is equal to its book value. If firms use a

different method to account for inventory (such as LIFO) they usually report what adjustment to make

to obtain the current (FIFO) value of inventory (Compustat item 240).16 Lewellen and Badrinath (1997)

and Lee and Tompkins (1999) provide further details on the calculation of asset replacement values.

Both measures of Q are adjusted for the median Q of the firm’s industry, where industries are defined as

in Fama and French (1997).

     The existing literature frequently uses share price level as a proxy for stock market liquidity. To

control for liquidity differences in our sample while also studying the relation of share price levels to

15
   To increase sample size, Lee and Tompkins (1999) propose an approach to calculate replacement values of fixed
assets for firms for which the Lewellen and Badrinath (1997) procedure leads to missing observations.
16
   Our results remain unchanged if we measure firms’ Q ratios based on Lindenberg and Ross (1981). Papers that
calculate Q ratios based on Lindenberg and Ross (1981) include McConnell and Servaes (1990) and Lang and
Stulz (1994).

                                                      19
other firm-specific variables, we employ the share turnover of the firm to control for market liquidity.

Our measure of liquidity is the natural logarithm of 0.01 plus the annualized share turnover for June of

year t. To address the overstatement of trading volume on Nasdaq compared to trading volume on

NYSE/AMEX and to also control for other differences between NYSE/AMEX and Nasdaq, in our

regressions we also use a dummy variable equal to one if an issue is traded on NYSE/AMEX and equal

to zero if it is traded on Nasdaq.

     Existing research has shown that firm value, as measured by the industry-adjusted Q ratio, is

positively related to the growth opportunities of the firm and S&P 500 membership. We control for

growth opportunities using the change in assets (Compustat item 6) from year t-1 to year t relative to

year t assets and the R&D expenses relative to total assets (Compustat item 46 divided by item 6).

Similar to our measure of the precision of analyst earnings forecasts, R&D expenses are not available

for a significant part of our sample. We again make R&D expenses equal to zero when not available and

at the same time we include a dummy variable indicating whether or not R&D expenses are missing.

This allows us to estimate the coefficient for R&D expenses relative to assets only for firms with

available R&D expenses and to estimate the other coefficients using all data for the remaining variables.

As an additional control for Q and to also control for a possible endogeneity problem due to indexing

affecting both firm value and institutional ownership, we also include a variable indicating whether the

stock is in the S&P 500 index (Compustat item 276 equals 0.1).

     We include two additional control variables when we analyze stock prices: the gross stock returns

for year t-1 (from June of year t-1 to June of year t) and for year t-2 (from June of year t-2 to June of

year t-1). We take the natural logarithm of both returns, which gives us two measures of past returns.

The rationale for controlling for recent stock returns is that a stock may have a high (low) price due to a

recent run-up (decline) rather than a systematic preference for high (low) price levels. Stock returns data

comes from the CRSP monthly files.

     Table I reports the mean, median, standard deviation, and number of observations for the variables

discussed above. The average (median) firm in our sample has approximately 7 (4) one-year analyst

                                                    20
earnings forecasts. The median firm has a share price of around $12 per share and a split-to price of

around $22 per share. The median firm in our sample is small relative to NYSE firms: its market

capitalization is approximately 13% of median NYSE market capitalization and its asset size is

approximately 12% of median NYSE asset size. Both average and median S&P common stock rankings

are around 5 (or B) while institutional ownership is 30% for the average firm and 24% for the median

firm.

                                              [Insert Table I about here]

                                       III.     Empirical Findings

     In this section we present the results of our empirical tests of the hypotheses developed in Section

I.E. In Section II.A we examine how analyst coverage is related to share prices and other variables. We

examine the relation between the value of the firm and several key variables in Section II.B. Finally, in

Section III.C we study the determination of share price levels.

A. Analyst Coverage

     Table II reports our estimates from regressions to explain the number of analysts providing

earnings forecasts. When the measure of share price is the actual share price of the firm then the

regression uses 66,568 firm-years. The adjusted R-square of the model is 63%, which is relatively high.

     We find strong support for Hypothesis 1. The number of analysts is negatively related to the share

price and positively related to the firm’s market capitalization, with both relations significant at the 0.01

level. The coefficient on share price shows that a firm that increases its share price by one percent, with

its market capitalization fixed, will incur a reduction in the number of its analysts by around 0.017

analysts or approximately 0.25% (0.43%) reduction in analyst coverage for the average (median) firm.

Alternatively, if a firm doubles its share price, the number of analysts declines by around 25% for the

average firm and 43% for the median firm. The results remain similar when we use the split-to price as

our measure of the firm’s target price level despite a significant reduction in sample size. In this case we

                                                     21
find that a one percent increase in the split-to price of the firm leads to a decline in the number of

analysts by around 0.3% for the average firm and by around 0.5% for the median firm. The adjusted R-

square is again around 64%. These findings are robust to controlling for the liquidity of a firm’s stock

market trading activity, measured by share turnover.

     Extending the work of Brennan and Hughes (1991), our model also makes predictions about the

relation between analyst coverage, and firm information quality and ownership structure. As predicted

in Hypothesis 1, we expect that the number of analysts following the firm will decrease with firm

information quality (as measured by S&P quality rank) and with institutional ownership. We further

expect that the effect of firm information quality on analyst coverage will be less negative as the broker

information precision increases. In other words, we expect a positive coefficient for the interaction

between S&P firm quality rank and the measure of broker information precision. The regression also

controls for the market capitalization of the firm, trading activity, standard deviation of returns, and

historic returns. The relevant coefficients are even larger in magnitude and have the same signs and

statistical significance if we do not control for standard deviation of returns and historic returns.

Consistent with Hypothesis 1, institutional ownership is negatively and significantly (at the 0.01 level)

related to the number of analysts following the firm. An increase in institutional ownership by one

percentage point is associated with a reduction in the number of analysts following the firm by around

0.09% for the average firm and by around 0.16% for the median firm. We also find that firms with

higher quality of information (as measured by their S&P quality rank) have fewer analysts following

their stock. The coefficient estimate for S&P quality rank in model (5) is significant at the 0.10 level and

implies that an increase in S&P rank by 1.00 leads to a 0.53% (0.92%) reduction in analyst coverage for

the average (median) firm in our sample. As predicted, the interaction term between S&P quality rank

and analyst information precision has a positive coefficient significant at the 0.01 level. The coefficient

for analyst information precision (model (5)) is significant at the 0.01 level and positive. Our theory

does predict a positive relation between analyst information precision and analyst coverage, but only for

sufficiently low analyst information precision.

                                                    22
[Insert Table II about here]

B. The Value of the Firm

        Table III reports our findings on the links between firm value (as measured by Tobin’s Q) and

other key variables of interest, such as the firm’s quality of information (measured by S&P quality

rank), the precision of analyst forecasts, institutional ownership, size, and share price levels. In Panel A

we compute Q as the ratio of the market value of assets to the book value of assets whereas in Panel B

we compute Q as the ratio of the market value of assets to the replacement value of assets.

        We obtain broadly consistent results across the two panels. Our findings provide strong support for

our theory in general and Hypothesis 2 in particular. We find that Tobin’s Q is positively related to

information quality as measured by the S&P rank. In Panel A this relation is significant at the 0.01 level

in models (1) to (3) while in Panel B it is significant at the 0.01 level in models (1), (2), and (4) and at

the 0.05 level in model (3). For example, model (2) of Panel A shows that a one unit increase in the

S&P ranking (e.g., from C to B) results in a 7.09 percent increase in firm value. When the share price is

included as an explanatory variable in the regressions, the relation between Tobin’s Q and S&P rank

remains significant although the economic effect of S&P rank on firm value is lower, with a unit

increase in S&P rank resulting in a 1.53 percent increase in firm value. For model (4), this relationship

remains significant at the 0.01 level in Panel B but is insignificant in Panel A despite having the correct

sign.

        Tobin’s Q is also positively related to the precision of analyst forecasts, a finding also supporting

Hypothesis 2. In both panels this relation is significant at the 0.01 level for all regression specifications.

For an economic interpretation of the coefficients we note that Table I shows that the standard deviation

of the measure of broker information precision is 1.53. Combined with the coefficient estimates in

model (2) of Panel A, for example, we find that a one standard deviation increase in the precision of

broker information leads to a 16.52 percent increase in firm value.

                                                      23
Further support for Hypothesis 2 is provided by the relation between Q and institutional ownership.

We find a positive and significant (at the 0.01 level) relation for all regression specifications in both

panels. Using the standard deviation of institutional ownership from Table I and the estimated

coefficient from model (2) of Panel A, we find that a one standard deviation (26.07) increase in

institutional ownership is associated with a 9.65% increase in firm value.

     Hypothesis 3 predicts a positive relation between a firm’s share price level and its value. Providing

strong support for this prediction and our theory, we find that the value of the firm is significantly (at the

0.01 level) and positively related to the share price using all specifications (share price or split-to price

and in both panels).

     When we examine the control variables we find that, consistent with existing literature, growth

opportunities (as measured by R&D expenses and asset growth) and S&P 500 membership are

positively related to the value of the firm, while size is negatively related to firm value (see, for

example, Lang and Stulz, 1994).

                                            [Insert Table III about here]

C. Share Price Levels

     Our last set of predictions concerns the determinants of cross-sectional differences in the share

price levels of firms as summarized in Hypotheses 4 and 5. These predictions are tested in Tables IV

and V. Table IV reports our empirical findings using share price levels and Table V repeats the analysis

using split-to prices. Our findings provide significant support for both these hypotheses. Regardless of

which specification we use, share price levels and split-to prices are positively and significantly (at the

0.01 level) related to institutional ownership and firm information quality as measured by the S&P rank.

It is especially important to note that this relation is robust to our control for stock market liquidity as

measured by share turnover. These results are also robust to controls for size, analyst forecast precision,

listing exchange, growth rates and returns volatility.

                                                     24
Examining the coefficient estimate of institutional ownership in model (3) of Table V, for example,

we find that a one standard deviation (26.07) increase in institutional ownership leads to an increase in

the split-to price of the firm by around 14.86 percent. At the median firm this implies an increase in the

split-to price from around $22 per share to around $25 per share. The economic effect of S&P rank on

share price levels is also notable. For example, the estimated model (3) of Table V implies that a one

unit increase in S&P rank (e.g., from C to B) leads to around a 7.01 percent increase in the split-to price

of the firm.

                                        [Insert Tables IV and V about here]

     Consistent with our prediction in Hypothesis 5, we also find a significant (at the 0.01 level)

negative interaction effect between the precision of broker information and the firm’s information

quality as measured by the S&P quality rank. This result is consistent with the hypothesis that the

precision of broker information and precision of information about the firm act as substitutes in

determining the share price level of the firm.

     Overall, the findings in this section provide strong support for our theory. Firm value is positively

related to firm specific information precision, broker information precision, and institutional ownership

while share price levels are positively related to institutional ownership and firm specific information

precision. Furthermore, share prices of firms with higher information precision are less sensitive to

broker information precision and vice versa.

                                           IV.      Conclusions

     We develop a model to examine cross sectional differences in institutional ownership and share

prices across firms. In contrast to the prior literature, we explicitly incorporate the role of institutional

investors in monitoring and information gathering and show how institutional investors can substitute

for costly information generation by analysts. We show that firms select share prices by trading off the

benefits of institutional ownership against the cost of information generation by market intermediaries.

Firms that anticipate small net benefits from institutional ownership set lower share prices to increase

                                                     25
the brokerage revenue associated with trading their shares and thereby induce more information

generation by market intermediaries. In contrast, firms that anticipate large net benefits from

institutional ownership set higher share prices to decrease the cost to investors of owning their shares. In

equilibrium, higher priced firms have a higher institutional ownership and higher valuations than lower

priced firms. Additionally, our findings also imply that the share price level will signal a firm’s quality.

In addition to establishing a theoretical basis for the empirically observed positive relation between

share prices and institutional ownership, we show that this relation exists independently of differences

in size and market liquidity, which are widely believed to be the drivers for institutions to hold higher

priced stocks. Firms with higher institutional ownership will choose higher split-to prices when they

split their shares. We show that analyst following and institutional ownership are inversely related and

therefore, that high-priced firms with high institutional ownership and high value have a lower analyst

following when size differences are controlled for. Overall, our findings provide several new insights

and point to a fruitful new line of research from both a theoretical and an empirical standpoint.

                                                    26
APPENDIX A

     Proof of Proposition 1: The first order condition for institutional investors is:

                                                ⎡       γθ Ij ( 2 + m j θ Ij )     ⎤
                             μ j − Q j − cS j − ⎢                                  ⎥=0
                                                ⎢ 2 (1 + m θ )2 ( h + N h ) ⎥                   (A1)
                                                ⎣          j Ij       j        j B ⎦

     and the first order condition for retail investors is:

                                                 ⎡              γθ Rj               ⎤
                              μ j − Q j − cS j − ⎢                                  ⎥=0
                                                 ⎢⎣ (1 + m j θIj )( h j + N j hB ) ⎥⎦
                                                                                                (A2)

     where

                                        Nj =
                                                 (θ    Ij
                                                            + θ Rj ) cS j
                                                                             =
                                                                                 cS j
                                                                                                (A3)
                                                               f                  f

     The market clearing condition for any firm j is:

                                                       θIj + θRj = 1
                                                                                                (A4)

     Substituting (A3) and (A4) into (A1) and (A2), and solving (A1) and (A2) for Q j we get:

                                                ⎡                                          ⎤
                                                ⎢
                                                ⎢
                             Q j = μ j − cS j − ⎢
                                                                     (
                                                      γ (1 − θ Rj ) 2 + m j (1 − θ Rj ) )  ⎥
                                                                                           ⎥
                                                                                           ⎥    (A5)
                                                ⎢ 2 1 + m (1 − θ ) ⎛⎜ h + ⎛ cS j ⎞ h ⎞⎟ ⎥
                                                   (                     )
                                                                       2

                                                ⎢         j        Rj
                                                                         ⎜ j ⎜ f ⎟ B ⎟⎥
                                                ⎣                        ⎝     ⎝        ⎠ ⎠⎦

    and

                                                 ⎡                                     ⎤
                                                 ⎢                                     ⎥
                                                 ⎢                γθ Rj                ⎥
                              Q j = μ j − cS j − ⎢                                     ⎥        (A6)
                                                 ⎢ 1 + m (1 − θ ) ⎜⎛ h + ⎛ cS j ⎞ h ⎞⎟ ⎥
                                                  (                      )
                                                 ⎢      j      Rj
                                                                    ⎜ j ⎜ f ⎟ B ⎟⎥
                                                 ⎣                  ⎝    ⎝      ⎠ ⎠⎦

     Equating (A5) and (A6) and then simplifying and solving the resulting expression yields the

following two solutions for θ Rj :

                                                              27
1 2 4 + mj (2 + mj ) − 4 − mj 1 2 4 + mj (2 + mj ) + 4 + mj
                          ⎛                                                                                     ⎞
                   θ Rj = ⎜ −                             , +                                                   ⎟                    (A7)
                          ⎜⎜ 2           6m j              2            6m j                                    ⎟⎟
                           ⎝                                                                                     ⎠

    The corresponding two solutions for θ Ij become:

                             1 2 4 + mj (2 + mj ) − 4 − mj 1 2 4 + mj (2 + mj ) + 4 + mj
                          ⎛                                                                                     ⎞
                   θ Ij = ⎜ +                             , −                                                   ⎟                    (A8)
                          ⎜⎜ 2           6m j              2            6m j                                    ⎟⎟
                           ⎝                                                                                     ⎠

    However, only the first solution in each case satisfies the second order conditions for institutional

and retail investors, which permits us to eliminate the second solution. Q.E.D.

    Proof of Lemma 2: From (A6) we have that:

                            ⎡                                     ⎤
                            ⎢                                     ⎥                ⎡                                            ⎤
                            ⎢                γθ Rj                ⎥                         γf                θRj
         Q j = μ j − cS j − ⎢                                       = μ j − cS j − ⎢                                            ⎥    (A9)
                                                                  ⎥
                            ⎢ 1 + m (1 − θ ) ⎛⎜ h + ⎛ cS j ⎞ h ⎞⎟ ⎥
                             (                      )
                                                                                   ⎢
                                                                                   ⎣                   (
                                                                                     ( fhj + cS j hB ) 1 + m j (1 − θRj )   )   ⎥
                                                                                                                                ⎦
                            ⎢      j      Rj
                                               ⎜ j ⎜ f ⎟ B ⎟⎥
                            ⎣                  ⎝    ⎝      ⎠ ⎠⎦

    Substituting for θRj and simplifying, we get:

                                                                  (1 + m )   3 + (1 + m j ) − (1 + m j ) − 1
                                                                                           2                2

                                                        γf              j
                  Q j = μ j − cS j −
                                         ( fh   j
                                                    + cS j hB )                      mj                                             (A10)

Q.E.D.

    Proof of Proposition 2: The first order condition for Q j with respect to S j is:

                    ⎛                                  ⎛                                          ⎞⎞
                  c ⎜ −m j ( fh j + cS j hB ) + f γhB ⎜ (1 + m j ) 3 + (1 + m j ) − (1 + m j ) − 1⎟ ⎟
                                             2                                   2            2

                    ⎝                                  ⎝                                          ⎠⎠
                                                                                                      =0                            (A11)
                                                   m j ( fh j + cS j hB )
                                                                          2

    Solving for S j yields two solutions,

                                                                   28
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