The Share Price Puzzle

Edward A. Dyl
William B. Elliott
University of Arizona




The Share Price Puzzle*


In frictionless markets, share prices per se do not affect         We document substantial
the value of the firm. Consider two identical firms,               variation in the prices of
each with a value of $1 billion. One firm has 20 million           common stocks in U.S.
                                                                   markets due to firms se-
shares outstanding and sells for $50 per share, and the            lecting particular price
other firm has 50 million shares outstanding that trade            ranges for their shares.
for $20 per share. If the firms’ values diverge, arbi-             Cross-sectional evidence
trageurs will quickly eliminate the discrepancy by si-             indicates that variables
                                                                   consistent with Merton’s
multaneously selling the overvalued shares and buying              model of capital market
the shares of the other firm.                                      equilibrium explain
   But market frictions exist, and corporate financial             roughly two-thirds of this
managers seem to believe that there is an optimal                  variation in share prices.
                                                                   In addition, measures of
trading range for a firm’s stock (Baker and Gallagher              trading range and share
1980). Even the New York Stock Exchange (NYSE)                     price appreciation predict
suggests that firms use stock splits to keep their shares          stock splits, and the “in-
in a preferred price range.1 The so-called optimal trad-           vestor base” of firms that
                                                                   split their stock increases
ing range for a firm’s shares seems to vary from firm              compared to other firms.
to firm. A glance at the stock prices published each               We conclude that firms
weekday in the Wall Street Journal reveals a wide                  manage share price levels
range of common stock prices in U.S. markets at any                to increase the value of
                                                                   the firm.
particular time. For example, on a given day at the
NYSE, Anheuser-Busch (BUD) sold for about $52 per
share, whereas Boston Beer (SAM) sold for only $15
per share. Similarly, on Nasdaq, Intel (INTC) sold for
$25 per share, but a share of Applied Micro Circuits
(APCC) cost about $5.2 If the total value of a firm is
primarily determined by the firm’s future earning

  * Contact the corresponding author, Edward A. Dyl, at edyl@
bpa.arizona.edu.
  1. New York Stock Exchange Listed Company Manual, 7–11.
  2. These prices refer to closing prices on August 5, 2003.

(Journal of Business, 2006, vol. 79, no. 4)
䉷 2006 by The University of Chicago. All rights reserved.
0021-9398/2006/7904-0013$10.00


                                                            2045
2046                                                                        Journal of Business


power, then the price per share of the firm’s stock simply depends on the
number of shares outstanding, a number that the firm determines.
   Is there an optimal trading range for a firm’s shares? If so, why do share
prices vary substantially among firms being traded in the same and/or similar
markets?3 That is, why do firms select different trading ranges for their firms’
shares? These questions constitute the “share price puzzle.” This article in-
vestigates whether Merton’s (1987) model of capital market equilibrium in
imperfect markets explains why firms target a particular trading range for
their shares, and why this target trading range varies among firms.
   Table 1 illustrates the dispersion in share prices of common stocks trading
in U.S. markets during 2001. Panel A contains aggregate results for the three
major U.S. stock markets. The mean and median stock prices of these 6,638
firms are $18.23 and $12.95 per share, respectively, but 40% of the firms
have average prices lower than $10 per share and 16% have average prices
higher than $30 per share, so more than 50% of all traded firms have share
prices that differ considerably from the average.4 Higher stock prices and the
firms’ market values are directly associated, but this relation is not merely
due to higher prices causing higher market values. The mean share price of
firms in the “greater-than-$70” group is only 29 times the mean price of firms
in the “less-than-$10” group, whereas the mean market value of firms in the
former group is 112 times the mean market value of the firms in the latter
group. Large firms have both higher stock prices and more shares outstanding.
   Panels B, C, and D show the same information separately for the three
major U.S. stock exchanges. The mean share price of firms trading on the
NYSE is roughly twice as high as that of firms trading on Nasdaq, and NYSE
firms are larger. Both the Amex and Nasdaq have a larger proportion of their
firms in the “less-than-$10” category.
   Evidently stock prices vary all over the place. This phenomenon may simply
be inexplicable if stock prices “just happen.” That is, firms’ shares begin
trading following an initial public offering (IPO), where the stock generally
has an average price of about $10 per share,5 and subsequently some firms
are more successful than others. These firms’ share prices increase, whereas
less successful firms’ prices may either stay the same or decrease. Differences
in the performance of firms over a long period could explain the cross-sectional
variation in stock prices at any particular time. The problem with this con-
jecture is that firms take actions to select a preferred trading range for their
shares (i.e., by manipulating the number of shares outstanding).
   Consider the evidence in figure 1, which shows the average annual price
of a share of common stock for 1,019 firms having continuous annual data

   3. Many items in our economy are, in fact, packaged in ways designed to reduce price dispersion
across companies and brands. Bonds typically have a par value of $1,000 per bond, most pasta
comes in 16-ounce packages, wine bottles generally hold 750 milliliters, and so forth.
   4. Berkshire Hathaway, an extreme outlier, is omitted from the analysis throughout the article.
   5. Affleck-Graves et al. (1996) study 2,096 IPOs from 1975–91 and find an average initial
offering price of $10.36 per share.
The Share Price Puzzle                                                                   2047


available from the Center for Research in Security Prices (CRSP) for the
period from 1976 through 2001.6 The average annual price for each firm is
an average of the monthly closing prices for the year. The solid line depicts
the average price over time for all the firms. The average annual share price
for these stocks over the 26-year period is $27.31, ranging from lows of
$19.84 and $20.21 in 1977 and 1976 to highs of $35.58 and $34.61 in 1997
and 1998, respectively. It is remarkable that the average price of these firms’
shares changes so little over a period of time when the S&P 500 Composite
index appreciated by 1,063% and the NYSE Composite Index appreciated by
1,238%. It is also prima facie evidence that these firms manage their share
prices.7 The broken line in the figure shows what the average share price
would have been in the absence of stock splits. The average price in 2001
would have been $405.39, not $28.48. Clearly, stock prices don’t “just
happen.”
   The article is organized as follows. In the next section we consider the
relation between Merton’s model (1987), trading range considerations, trans-
action costs, and share prices, and propose some hypotheses about why some
firms have higher share prices and vice versa. Section II presents cross-sec-
tional evidence about share ownership and share price levels from 1976 to
1996 that is consistent with these hypotheses. Section III shows that the
decisions of the firms in our sample to split their stock or not to split their
stock are consistent with these firms managing their share prices to increase
firm value. Conclusions are summarized in Section IV.


I.    Trading Ranges, Stock Splits, and Transaction Costs
The idea that the price of a firm’s stock is relevant is of long standing. In A
Study of Corporation Securities, Arthur Stone Dewing8 (1934) writes:

     When the stock of a corporation is quoted below $10 per share, there is
     an implied suggestion that the credit of the corporation is low; when the
     price is over $200 a share, there is a reluctance on the part of investors to
     buy the stock. . . . The author of this book is of the opinion that a market
     price somewhere between $30 and $60 a share meets best the various
     conditions. Below $30 there is the inarticulate implication of distressed
     credit; above $60 there is the handicap of mere price to the maximum
     width of market and diversification of stock ownership. (97)

Apparently, notions about the “best” stock price levels also change over time.
In 1953 Dewing opines:

   6. The rationale for examining this group of firms is discussed below.
   7. Our sample contains only survivors, so these 1,019 firms’ appreciation over the 26 years
from 1976 to 2001 was slightly less than that of the indices (957%).
   8. Arthur Stone Dewing (1880–1971) was arguably one of the first financial economists. He
received his PhD in philosophy from Harvard in 1905 and joined the faculty of the recently
founded Harvard Business School in 1919.
2048
TABLE 1          Stock Prices on the NYSE, Amex, and Nasdaq in 2001
                                 All Prices       Less than $10          $10 to $30          $30 to $50   $50 to $70   Greater than $70
                                              Panel A: All firms listed on the NYSE, Amex, and Nasdaq
Share price:
  Mean ($)                          18.23                4.92                17.90               37.99        58.10          140.34
  Median ($)                        12.95                4.44                16.68               37.29        57.26           87.85
  No. of firms                   6,638               2,682                2,869                 761          221             105
  Proportion of firms (%)          100                  40.4                 43.2                11.5          3.3             1.6
Market value (000,000):
  Mean ($)                       2,078                133                  1,221              6,777       14,551          14,880
  Median ($)                       167                 47                    304              1,570        3,280           1,898
                                                                Panel B: NYSE firms
Share price:




                                                                                                                                          Journal of Business
  Mean ($)                          25.29               6.33                 18.62               38.22        57.74          157.34
  Median ($)                        18.85               6.80                 17.68               37.90        57.16           81.92
  No. of firms                   2,497                537                 1,275                 477          150              58
  Proportion of firms (%)          100                 21.5                  51.1                19.1          6.0             2.3
Market value (000,000):
  Mean ($)                       4,380                300                 1,693               9,501       17,122          26,149
  Median ($)                       577                110                   508               2,356        5,760           5,872
The Share Price Puzzle
                                                                                Panel C: Amex firms
Share price:
  Mean ($)                                   17.98                     4.27                      16.36                   38.45                     59.33                     109.23
  Median ($)                                  8.93                     3.58                      15.02                   38.11                     58.61                      92.95
  No. of firms                              648                      346                        207                      31                        28                         36
  Proportion of firms (%)                   100                       53.4                       31.9                     4.8                       4.3                        5.6
Market value (000,000):
  Mean ($)                                  166                        56                   169                         581                      296                         742
  Median ($)                                 43                        26                    58                          88                      133                         133
                                                                               Panel D: Nasdaq firms
Share price:
  Mean ($)                                   13.24                     4.62                      17.47                   37.51                     58.55                     152.44
  Median ($)                                  9.52                     4.05                      16.26                   36.60                     57.98                      95.85
  No. of firms                            3,492                    1,798                      1,387                     253                        43                         11
  Proportion of firms (%)                   100                       51.5                       39.7                     7.2                       1.2                         .3
Market value (000,000):
  Mean ($)                                  789                        99                       945                   2,401                   14,862                      $1,729
  Median ($)                                 94                        40                       235                     955                   $3,102                        $510
   Note.—Includes all firms reported in the CRSP database for 2001 having an average price of at least $1.25 per share. Individual firms’ stock prices are the average of monthly closing
prices in 2001. Market value is the year-end market capitalization reported in the CRSP database.




                                                                                                                                                                                            2049
2050
                                                                                                                                                 Journal of Business
   Fig. 1.—Average common stock prices from 1976 to 2001. Average annual stock prices for 1,019 firms with continuous data available on CRSP
for 1976 through 2001. The annual price for each firm is the average of monthly closing prices. The solid line shows actual prices. The broken
line shows what the prices would have been in the absence of stock splits that occurred during this period.
The Share Price Puzzle                                                                    2051

   From a study of a wide variety of conditions affecting market price, one
   would gather that there prevails a general opinion that a market price
   somewhere between $15 and $40 a share meets best the various conditions.
   Below $15 there may arise an inarticulate implication of distressed credit—
   although this is not as true in 1953 as it was twenty years before; above
   $40 there is the handicap of mere price to the maximum width of the
   market and diversification of stock ownership. (1188–89)

   The notion that there is an optimal (or at least preferable) trading range
for a firm’s common stock is also widely accepted among finance practitioners.
In a survey of 100 chief financial officers of NYSE-listed firms, Baker and
Gallagher (1980) find almost universal agreement with the statement that
“stock splits keep a firm’s stock price in an optimal price range.” In a similar
vein, the New York Stock Exchange Listed Company Manual contains the
following statements:

   Exchange statistics indicate a preferential price range within which a sig-
   nificant percentage of Exchange round lot volume is generated. . . . Today,
   liquidity is probably the most important element in the investment decision,
   other than the financial condition or suitability of the security under con-
   sideration. . . . Consideration of a stock split is therefore justified when
   a company’s shares are selling at a relatively high price, and when such
   action is accompanied by healthy operating results and a strong financial
   condition. (7–11)

Apparently firms heed the NYSE’s advice. Numerous empirical studies report
that firms tend to announce a stock split when their share price is higher than
usual, and that stock splits convey positive information about firms’ future
prospects.9
   Despite the NYSE’s assertion regarding the importance of liquidity, how-
ever, stock splits and lower trading ranges affect the market for a firm’s shares
in various ways. For example, stock splits broaden the market for the firm’s
stock, but they increase certain transaction costs such as percentage bid-ask
spreads (Copeland 1979; Conroy et al. 1990; Schultz 2000; Easley et al. 2001).
Lower share prices are associated with higher trading costs such as bid-ask
spreads and commissions, and not vice versa (Benston and Hagerman 1974;
Stoll and Whaley 1983; Brennan and Copeland 1988; Brennan and Hughes
1991; Harris 1994). In a market that specifies a minimum price variation
between quotes (i.e., a tick size), an inverse relation between share prices and
bid-ask spreads is almost inevitable.10 When the minimum price variation in

   9. Lakonishok and Lev (1987), Lamoureux and Poon (1987), McNichols and Dravid (1990),
and Ikenberry et al. (1996). Whether or not the conveyance of information is intentional (i.e.,
so-called signaling) or simply inadvertent because managers behave according to the guidelines
in the NYSE Listed Company Manual is, fortunately, outside the scope of this article.
   10. Harris (1994), Angel (1997), and Anshuman and Kalay (1998, 2002). Angel (1997) doc-
uments an intriguing relation between share prices and tick sizes across markets and develops
a model of optimal tick size, but he does not examine price variation within these markets.
2052                                                            Journal of Business


a market is one-eighth of a dollar ($.125), as it was in U.S. stock markets for
most of the last century, then the smallest percentage bid-ask spread that can
occur for a stock priced at $5 is 2.5% and the smallest percentage spreads
possible for stocks priced at $12.50 and at $50 per share, respectively, are
1.0% and 0.25%. The dollar volume of trading in a stock also frequently
declines following a stock split (Copeland 1979; Lakonishok and Lev 1987;
Lamoureux and Poon 1987; Conroy et al. 1990). An inverse relation between
the cost of transactions and the demand for transaction services (i.e., trading)
is to be expected, and an extensive amount of literature addresses this issue
(Demsetz 1968; Tinic 1972; Benston and Hagerman 1974; Stoll 1978; Amihud
and Mendelson 1986; Constantanides 1986; Atkins and Dyl 1997). Amihud
and Mendelson (1986) demonstrate that high transaction costs may, ceteris
paribus, reduce the value of the firm.
   Why do firms use stock splits to move their share prices to trading ranges
that have higher transaction costs? Part of the answer to this question may
lie with Merton’s (1987) model of capital market equilibrium with incomplete
information. Merton observes that investors are generally aware of only a
subset of all available securities, that these subsets differ among investors,
and that investors can only invest in securities that they know about. Mounting
empirical evidence suggests that investors behave in a manner consistent with
Merton’s investor-recognition hypothesis (French and Poterba 1991; Falken-
stein 1996; Coval and Moskowitz 1999; Huberman 2001). Merton shows
analytically that an increase in the relative size of a firm’s investor base (i.e.,
the group of investors and potential investors who are aware of the firm)
increases the market value of the firm. Kadlec and McConnell (1994) and
Foerster and Karolyi (1999) study U.S. firms listing on the NYSE and non-
U.S. firms that list on U.S. exchanges, respectively, and present empirical
evidence that greater investor recognition for firms increases their share prices.
   A firm may therefore attempt to select a trading range for its shares that
enlarges the firm’s investor base. It is possible that attracting new investors
in this manner results in an increase in the market value of the firm that more
than offsets the costs of the higher bid-ask spreads following stock splits
reported by the studies cited above. In addition, other dimensions of liquidity,
such as the depth of the market and the volume of uninformed versus informed
trading, are enhanced when the firm has a larger shareholder base.
   Many studies support the notion that adjusting the price level of the firm’s
stock is a motive for stock splits. Lakonishok and Lev (1987) conclude that
stock splits are primarily aimed at moving stock prices to a “normal range.”
Muscarella and Vetsuypens (1996) provide evidence that splits of American
Depository Receipts (ADRs) move the price of the ADRs to a particular
trading range. Conroy and Harris (1999) show that managers design stock
splits to return their firms’ stock prices to a desired trading range (i.e., fre-
quently the price level after the firm’s last stock split). It is also well docu-
The Share Price Puzzle                                                                      2053


mented that the number of shareholders in a firm generally increases following
a stock split, so apparently a stock split does increase a firm’s investor base.11
   The specific mechanism through which a particular trading range for a
firm’s shares increases its investor base is not evident. The simplest expla-
nation derives from a combination of investors’ revealed preferences for trad-
ing in round lots, the benefits of holding a diversified portfolio (Markowitz
(1952), and the limited wealth of most individual investors.12 In particular, a
diversified portfolio of common stocks requires as many as 30 different
stocks.13 The NYSE’s Shareownership 2000 publication indicates that the
median and mean values of the common stock portfolios of all adult share-
holders are $28,000 and $148,500, respectively.14 Thus, the median (mean)
investor can only invest in about four (21) round lots of stocks in the $70–$75
per share trading range, whereas he/she could invest in 19 (99) round lots of
stocks priced at around $15 per share. It is likely that stock splits increase
the number of shareholders simply because the lower price facilitates the
formation of diversified portfolios by less wealthy individual investors. Schultz
(2000) documents a large number of small buy orders following stock splits.
Easley et al. (2001) report an increase in the number of uninformed trades
following stock splits and note that a new clientele for a firm’s shares may
tolerate higher spreads in order to gain increased diversification of their
holdings.
   Other explanations linking share prices and the firm’s investors have also
been suggested. Brennan and Hughes (1991) observe that brokerage com-
missions are inversely related to share price and suggest that, in a world with
incomplete information, brokers will produce and disseminate more infor-
mation (e.g., research reports and recommendations) about lower-priced firms.
Angel (1997) suggests that a single tick size in a market means that lower
share prices increase the minimum percentage bid-ask spread for a stock,
consequently encouraging more market makers to both make a market in the
firm’s stock and to promote that stock to investors. Firms are therefore able
to use low stock prices as a means to increase their investor base, which,
according to Merton (1987), will increase the firm’s value.
   Amihud et al. (1999) provide support for an even simpler explanation of
the trading-range phenomenon. They investigate reductions in the minimum
trading unit (MTU) in Japan, which is the minimum number of a firm’s shares

   11. Lamoureux and Poon (1987), Maloney and Mulherin (1992), Mukherji et al. (1997), and
Schultz (2000) report this finding for common stock splits. Fernando, Krishnamurthy, and Spindt
(1999) report a similar result for mutual fund splits.
   12. Although an additional fee is no longer charged for trading in odd lots, investors most
frequently trade in lots of 100 shares. One explanation for this behavior expounded in the social
science literature is that the practice has become ritualized (Knotterus 1997).
   13. Statman (1987). Earlier work by Evans and Archer (1968) suggests a smaller number.
   14. These data are based on the 1998 Survey of Consumer Finances, an ongoing survey
conducted by the Survey Research Center at the University of Michigan for the Federal Reserve
Board.
2054                                                                      Journal of Business


required for a transaction in that firm’s stock on an exchange in Japan. They
find an increase in the number of investors in a firm following a reduction
in a firm’s MTU, and a concomitant increase in stock price that is related to
the increase in the number of shareholders. In Japan, brokerage commissions
are a function of the value of the transaction, not the share price, and, in
addition, the reduction in the MTU does not change the stock price, so the
Brennan and Hughes (1991) and Angel (1997) conjectures do not apply. The
driving force behind the increase in shareholders and the accompanying in-
crease in market value found in Japan following a reduction in a firm’s MTU
is apparently new investment by less wealthy investors, whose limited wealth
did not enable them to purchase the minimum number of shares previously
required for these stocks.
   The explanation of the dispersion in share prices in U.S. markets that
emerges from Merton’s model, and from these other studies, is that firms
whose owners are less wealthy are most concerned with the trading range,
and so these firms opt for lower share prices to expand their investor bases.
The link between a firm’s stock price and its investor base does not depend
only on the information-based explanations of Merton (1987), Brennan and
Hughes (1991), and Angel (1997). Less wealthy investors also seek out lower-
priced stocks to diversify their portfolios.
   In addition, Merton’s model implies that changes in the investor base of
well-known, widely held firms, which already have very large investor bases,
have essentially no effect on the firm’s market value.15 For these firms, the
qualitative predictions of Merton’s model and Black’s (1972) “zero-beta”
Capital Asset Pricing Model coincide. These firms, therefore, care little about
the trading range of their stock, but instead emphasize attributes such as
liquidity and low transaction costs that are valued by large investors, primarily
institutions. This behavior explains why large firms tend to have higher share
prices.


II.    Ownership Characteristics and Share Prices
The studies described above lead to two testable hypotheses about the dis-
persion of share prices in U.S. markets: (1) lower share prices are characteristic
of firms owned by so-called small investors, and (2) higher share prices are
characteristic of large firms. This section presents cross-sectional evidence
regarding these hypotheses.

A.     Summary Data
Our initial sample includes all firms listed in the Center for Research in
Security Prices (CRSP) database that have both continuous data available over

  15. Badrinath, Gay, and Kale (1989) document the penchant of institutional investors for this
type of stock.
The Share Price Puzzle                                                                               2055

TABLE 2             Descriptive Statistics for Sample Firms, 1976–2001
                     1976           1981           1986           1991            1996            2001
                                        Panel A: Share price
Mean ($)              20.30         24.03        31.25        27.64         32.05                   28.48
SD                    19.52         16.55        26.99        22.09         27.31                   34.26
Median ($)            15.67         20.68        26.63        23.39         26.00                   23.72
No. of frms        1,019         1,019        1,019        1,019         1,019                   1,019
                                  Panel B: Market value (000,000)
Mean ($)             679           766        1,293        2,187         4,061                   7,175
SD                 3,809         3,714        3,855        6,414        12,081                  26,806
Median ($)            59           128          251          373           697                     642
No. of firms         818           984          947          947           947                     946
                                Panel C: No. of shareholders (000)
Mean                  25.72         27.99        23.02        23.92         25.19                   31.44
SD                   128.15        124.73        65.10        67.39         67.59                   93.68
Median                 4.38          5.00         4.66         4.94          4.96                    5.00
No. of firms         707           884          859          866           879                     815
                       Panel   D: Average investment per shareholder (000)
Mean ($)              24.53         39.56        88.23       137.54        250.50                  388.89
SD                    30.86         50.09       122.17       196.50        429.64                  709.09
Median ($)            12.03         23.68        53.67        72.82        125.94                  142.48
No. of firms         707           884          859          866           879                     815
  Note.—The share price sample includes all firms with a share price greater than or equal to $1.25 in 1976
and with continuous information available on CRSP from 1976 to 2001. Annual share price is the average of
monthly closing prices from CRSP. Market value is the year-end capitalization reported on CRSP. Number of
shareholders is from Compustat.


the 26-year period from 1976 to 2001 and a share price of at least $1.25 at
the end of January 1976.16 We examine only firms with continuous data over
the whole period to eliminate firms whose share price is largely a consequence
of either circumstance (e.g., newly public firms) or vicissitude (e.g., firms
experiencing financial distress). The initial sample consists of 1,019 firms. In
1976, 74.8% of the firms are listed on the NYSE and 26.2% are listed on the
Amex.17 In 1996, 84.6% of the firms are listed on the NYSE, 13.7% on the
Amex, and 1.7% on Nasdaq. Even survivors may not all prosper.
   Table 2 shows summary statistics for the sample firms at 5-year intervals
from 1976 to 2001. Panel A contains data on average annual share prices.
The annual share price is computed as the average of the monthly closing
prices reported on CRSP. The mean and median share prices are slightly
higher in the later years than in the earlier years, although, as we have already
noted in our discussion of figure 1, the increase in share prices pales in
comparison to the increase in the value of these firms during the period. Slight
differences between the mean figures reported in table 2 and the values for
the same years in figure 1 occur because the data in table 2 exclude common
stocks with a price of less than $1.25 per share in 1976, whereas these firms
are included in figure 1. Panel B in the table shows the mean and median

   16. The minimum price of $1.25 was chosen arbitrarily to eliminate so-called penny stocks.
It is simply 10 times the tick size that was in effect during this period.
   17. Data for Nasdaq firms are not available on CRSP until after 1976.
2056                                                                      Journal of Business


market values of the firms (i.e., the year-end market capitalization reported
by CRSP). The mean and median market values increase by 957% and 988%,
respectively, from 1976 to 2001.
   Panels C and D provide statistics about the number of shareholders per
firm and the average investment per shareholder, respectively. Although the
value of these firms increases greatly over the period, the average number of
shareholders remains remarkably constant at roughly 26,000 shareholders per
firm. The median number of shareholders per firm increases from 4,380 in
1976 to 5,000 in 2001, a 14% increase. The mean number of shareholders is
more than five times the median; obviously the distribution of shareholders
across the firms in our sample is skewed. Panel D shows the average in-
vestment per shareholder, which is calculated as the market value of the firm
divided by the number of shareholders. Overall, the combination of only a
small increase in the number of shareholders per firm and a large increase in
the market value of the firms results in a very large increase in the average
investment per shareholder from 1976 to 2001. The mean (median) investment
per shareholder increases 16-fold (12-fold) over the period. Skewness is again
evident in the distribution of this variable across firms.

B.     Regression Analysis
We test our hypotheses regarding share price levels by estimating the param-
eters of the following regression equation:

                  SharePricej,t p d 0 ⫹ d1 BVEquityj,t

                                     ⫹d 2 AvgHldgj,t ⫹ d 3 EPSj,t ⫹ ␧j,t .                 (1)

The dependent variable, SharePricej,t, is the logarithm of the average price
per share of firm j’s common stock during year t, where the average annual
share price is the average of the monthly closing prices, the d’s are the
parameters to be estimated, and ␧j,t is an error term.
   The first independent variable, BVEquityj,t, is the logarithm of the book
value of firm j’s equity at the end of year t. We use the book value of the
firm’s equity, rather than the firm’s market capitalization, as the measure of
firm size in equation (1) to avoid any spurious relation to share price that
might exist with the latter measure (i.e., because the dependent variable is
share price and market capitalization is share price times the number of shares
outstanding). The book value of the firm’s equity is obtained from
Compustat.18
   The second independent variable, AvgHldgj,t, is total equity per shareholder,
which we employ as a proxy for the relative size of the average investment
per shareholder. The average holding per shareholder is a proxy for average

  18. When we use the book value of total assets in place of BkValEquityj,t in equation (1) the
empirical results are qualitatively the same.
The Share Price Puzzle                                                                 2057


wealth of the firm’s ownership clientele. An alternative, and arguably more
precise, measure of the average owner’s wealth is market value per shareholder
as reported in table 2. However, since share price is one element of this ratio,19
including this variable on the right side of equation (1) would result in a
spurious correlation. We therefore use average equity per shareholder as an
indicator of cross-sectional differences in the size of the average shareholder’s
investment. The number of shareholders per firm is obtained from Compustat.20
   The number of shareholders provided by Compustat is the number from
the firm’s 10-K filing, reported under Item 201(b) of Securities and Exchange
Commission Regulation S-K, which is the only source of information re-
garding number of shareholders. Regulation S-K requires that firms “give the
approximate number of holders of record of each class of common equity.”
We use shareholders of record as a proxy for the actual number of shareholders,
which is greater than the number of shareholders of record due to accounts
in street names, shares held in joint accounts, etc. In addition, since the
regulation requires only that companies report the “approximate number hold-
ers,” the methodology for determining this number may vary from firm to
firm. To test for the sensitivity of our results to measurement error in this
variable, we replace measured values of AvgHldgj,t with an instrumental var-
iable estimated using the rank order of AvgHldgj,t sorted from lowest to highest.
The final results of our analysis are unchanged.
   The third independent variable in equation (1), EPSj,t, is the firm’s earnings
per share during year t. Although our regression equation is not intended to
be an equity valuation model, otherwise similar firms may have different share
prices due to differences in current earnings per share. That is, in a given
year the stock prices of firms whose earnings per share increase may appreciate
vis-à-vis the prices of firms whose earnings per share do not increase. Including
EPSj,t as an independent variable improves the specification of the model.
   The results of the six cross-sectional regressions estimating annual param-
eters of equation (1) are summarized in table 3. The numbers in parentheses
are White (1980) heteroskedasticity-consistent t-statistics, and the asterisks
denote statistical significance at the .01 level. The coefficient on firm size (d1)
is positive and significant in each regression, a finding consistent with the
hypothesis that larger, well-known firms have high share prices and vice versa.
The coefficient on average investment per shareholder (d2) is also positive
and significant at the .01 level in each regression, consistent with the hy-
pothesis that lower share prices are preferred by firms owned by “small”
investors and vice versa.21 The coefficient on earnings per share (d3) is also

   19. Average investment per shareholder p (Share Price # Shares Outstanding)/(Number of
Shareholders).
   20. Shareholder equity and/or number of shareholders are not available for every firm for
every year.
   21. When an instrumental variable—estimated using the rank order of AvgHldgj,t—is used in
place of AvgHldgj,t, d2 continues to be positive and significant in each regression.
2058
TABLE 3              Share Prices, Firm Size, and Ownership Characteristics
Year                      d0                            d1                            d2                           d3                    Adj. R2              F-test            No. of Firms
1976              .8548   (12.06**)             .2071   (15.36**)             .2185   (7.69**)             .1712   (11.60**)             .7200               601.02                   701
1981             1.4998   (16.61**)             .1044   (7.27**)              .2285   (7.85**)             .0994   (7.61**)              .5709               389.11                   876
1986             1.4294   (14.46**)             .1604   (11.08**)             .1999   (6.88**)             .1259   (11.16**)             .5837               397.33                   849
1991              .8705   (7.19**)              .2375   (17.28**)             .1703   (6.39**)             .1207   (6.91**)              .6060               436.82                   851
1996             1.1702   (10.41**)             .2233   (18.54**)             .1067   (5.00**)             .1233   (10.33**)             .6392               514.76                   871
2001              .9074   (6.40**)              .2526   (17.09**)             .1051   (4.20**)             .0728   (3.10**)              .4982               263.76                   795
  Note.—This table reports cross-sectional estimates of the parameters of the following regression variables:

                                                          SharePricej,t p d0 ⫹ d1 BVEquityj,t ⫹ d2 AvgHldgj,t ⫹ d3EPSj,t ⫹ ␧j,t,
where SharePricej,t is the logarithm of the average price per share of firm j’s common stock during year t, BVEquityj,t is the logarithm of the book value of firm j’s equity at the end of year
t, AvgHldgj,t is the logarithm of average book value of equity per shareholder, and EPSj,t is the firm’s earnings per share for the year of the cross section. The numbers in parentheses are
White (1980) heteroskedasticity consistent t-statistics.
   ** Significant at the .01 level or higher.




                                                                                                                                                                                                   Journal of Business
The Share Price Puzzle                                                        2059


positive and significant, indicating that including this variable improves the
specification of the regression model. The average R2 of the six regressions
is 0.60.
   The cross-sectional regression estimates in table 3 support both the hy-
pothesis that lower share prices generally characterize firms owned by
“smaller” investors and vice versa, and the hypothesis that higher share prices
are associated with large, well-known firms and vice versa. These findings
are consistent with corporations behaving according to Merton’s (1987) theory
of capital market equilibrium in a market where investors have incomplete
information. That is, firms appear to select trading ranges for their shares to
enlarge the firm’s investor base and thereby increase the value of the firm.
The findings are also consistent with the widespread belief among finance
practitioners that the choice of an appropriate trading range for a firm’s shares
matters because it affects the value of the firm. Our regression analysis is
missing any conclusive indication of the direction of causality. Do firms select
a trading range for their shares based on their size and on the wealth of their
owners, or do shareholders sort themselves into clienteles based on share
price? The association between share price and firm size is suggestive in this
regard, but not conclusive. We address this issue in the following section.


III.    Share Prices and Stock Splits
If firms manage the price of their shares to attempt to keep it within a so-
called optimal trading range, where “optimal” is defined in terms of the size
of the firm and the characteristics of its owners as in table 3, then firms whose
share prices rise above this range are presumably more likely to split their
stock than are firms whose share prices remain at or below the desired trading
range. We investigate this issue by testing whether a firm with a share price
that is “too high”—that is, higher than the price predicted by equation (1)—
is more likely to split its stock than a firm whose share price is not “too high.”
Merton’s model indicates that additional shareholders provide only a negligible
increase in firm value for firms that already have large shareholder bases,
which may also reduce the likelihood that these firms will split their stock.
Therefore, we incorporate a measure of the size of the firm’s shareholder base
in the tests described below.

A.     Methodology
We examine the propensity of firms to split their stock by estimating the
parameters of the following logit model:

         StockS plitj,T p F(b 0 ⫹ b1TradeRangej,t ⫹ b 2 StockAp prec j,t ),    (2)

where StockSplitj,T is the probability that firm j splits its stock during time
period T, F is the logistic cumulative density function, TradeRangej,t indicates
whether or not the price of the stock is “too high,” StockApprecj,t is the increase
2060                                                                               Journal of Business

TABLE 4             Sample Firms Splitting Their Stock
                               Stock Split                         No Stock Split
                                                                                                    Total
Years                   Number                 %              Number                %               Firms
                  Panel A: Firms splitting 2-for-1 or greater during the 4 years
1977–80                  209               21               810               79                    1,019
1982–85                  278               27               741               73                    1,019
1987–90                  214               21               805               79                    1,019
1992–95                  219               21               800               79                    1,019
1997–2000                201               20               818               80                    1,019
                 Panel B: Firms splitting 1.25-for-1 or greater during the 4 years
1977–80                  323               32               696               68                    1,019
1982–85                  387               38               632               62                    1,019
1987–90                  318               31               701               69                    1,019
1992–95                  313               31               706               69                    1,019
1997–2000                260               26               759               74                    1,019
  Note.—This table shows the proportion of the firms in our sample splitting their common stock during the
4-year periods following 1976, 1981, 1986, 1991, and 1996. Panel A shows the number of firms whose
cumulative stock splits were 2-for-l or greater during the 4 years. Panel B shows the number of firms whose
cumulative stock splits were 1.25-for-1 or greater.




in the firm’s stock price over the 5 years preceding year t, and the b’s are
the parameters of the logit model.22
   The dependent variable, StockSplitj,T, equals 1 if firm j has a cumulative
stock split of 2-to-1 or greater (or 1.25-to-1 or greater in a second model)
during time interval T, and 0 otherwise.23 These data are identified for 4-year
intervals following the years 1976, 1981, 1986, 1991, and 1996 by examining
the stock-split factors reported in the CRSP data. The proportion of firms in
our sample that have stock splits during each 4-year interval is shown in table
4. Panel A shows the incidence of 2-for-l or greater cumulative stock splits,
and Panel B shows 1.25-for-1 or greater splits. Clearly stock splits are not
exactly a rare event for the firms in our sample in any of the five subperiods.
Roughly 22% of the firms had at least a 2-for-1 split during each of the 4-
year periods, and roughly 32% had at least a 1.25-for-l split.
   The first independent variable, TradeRangej,t, is defined as follows:


                                             TradeRangej,t

          p SharePricej,t /E(SharePricej,tFBVEquityj,t , AvgHldgj,t , EPSj,t ),                        (3)


where E(SharePricej,tFBVEquityj,t, AvgHldgj,t, EPSj,t) is estimated for each of



   22. Using a normal cumulative density function (i.e., probit) yields qualitatively similar results.
   23. The results are not sensitive to the minimum level used to define whether or not a firm
has split its stock. We chose 1.25-to-1 as a minimum simply to exclude firms that use small
stock distributions in lieu of cash dividends.
The Share Price Puzzle                                                                           2061


the 5 years (1976, 1981, 1986, 1991, and 1996) using the regression relations
shown in table 3.24 That is,

                  E(SharePricej,tFetc) p d 0 ⫹ d1 BVEquityj,t

                                                ⫹ d 2 AvgHldgj,t , ⫹ d 3 EPSj,t ,                  (4)

so TradeRangej,t is the ratio of a firm’s actual share price in year t to its
predicted share price from equation (1), conditional on the firm’s size and
average holding per shareholder. A value greater than 1 for TradeRangej,t
suggests that firm j’s share price is “too high” given the firm’s size and average
investment per shareholder, and a value less than or equal to 1 indicates that
the firm’s share price is below or close to its appropriate trading range,
respectively.
   The second independent variable, StockApprecj,t, is the proportional increase
in the jth firm’s split-adjusted average stock price over the 4 years ending
with the estimation year, computed as follows:

                    StockAp prec j,t p SharePricej,t /SharePricej,t⫺4 .                            (5)

We include share price appreciation as an explanatory variable in our logit
model of stock splits because numerous studies report an increase in stock
prices (in excess of market returns) preceding stock splits. They also report
that this increase begins up to 5 years before the announcement of the stock
split (Fama et al. 1969; Lakonishok and Lev 1987; Lamoureux and Poon
1987; McNichols and Dravid 1990).

B.    Logit Results
The results of the logit analysis are shown in table 5, where the time interval
T in StockSplitj,T is the 4-year period following year t.25 Panels A and B show
the findings when the StockSplitj,T variable equals one if the firm had a cu-
mulative split of either 2-to-1 or greater or 1.25-to-1 or greater, respectively,
and zero otherwise. In both panels, the coefficient on TradeRange j,t (i.e., b1)
is generally positive and significant at the 1% level, indicating that a firm
whose share price in year t is above its predicted trading range is more likely
to split its stock during the next 4 years than is a firm whose price is near or
below its predicted trading range. The sign and significance of the b1 coef-
ficients are evidence that trading range considerations are important to firms
and that these firms use stock splits to manage share price levels.
   The coefficients on StockApprecj,t (i.e., the b2’s) are also positive, and
significant at the 5% level or higher, in seven of the 10 logit models, indicating
that during some periods firms were more likely to split their stock after a
price appreciation. The positive results are consistent with the findings of

  24. We require that E(SharePricej,t) be greater than $5 for the firm to be included in the logit
analysis. Inclusion of all observations does not qualitatively alter the results of the logit analysis.
  25. Replicating this analysis using a 2-year time interval yields similar results.
2062
TABLE 5             Results of a Logit Model of Stock Splits and Share Prices
Time Period                      b0                            b   1                        B2                  Likelihood Ratio             No. of Firms            Splitting Firms (%)
                                                                Panel A: Cumulative stock split of at least 2-to-1
1977–80                ⫺3.3697    (24.11**)            1.4606 (4.29*)         .5944 (9.34**)                   25.72                              426                         22.1
1982–85                ⫺2.2695    (32.37**)            1.2435 (10.05**)       .0366 (2.22)                     16.46                              876                         28.5
1987–90                ⫺3.1431    (42.30**)            1.5874 (11.78**)       .1015 (6.59**)                   23.06                              849                         21.7
1992–95                ⫺2.1072    (41.95**)             .8155 (7.02**)        .0808 (.19)                      11.40                              851                         23.7
1997–2000              ⫺2.3446    (41.78**)             .8617(6.30**)         .0779 (4.32*)                    13.93                              871                         21.2
                                                              Panel B: Cumulative stock split of at least 1.25-to-1
1977–80                ⫺2.7559    (20.30**)            1.2222 (3.67)          .7542 (14.10**)                  33.29                              426                         31.9
1982–85                ⫺1.4548    (16.59**)             .8594 (5.82*)         .0666 (5.82*)                    16.74                              876                         39.5
1987–90                ⫺2.4792    (34.31**)            1.4788 (12.83**)       .1001 (6.67**)                   26.06                              849                         32.0
1992–95                ⫺1.6597    (29.92**)             .8240 (7.96**)        .0845 (2.04)                     13.38                              851                         32.9
1997–2000              ⫺2.1441    (35.25**)             .9109 (6.98**)        .1008 (6.07**)                   18.35                              871                         26.5
  Note.—This table reports the results of the following logit model of stock splits and share prices:

                                                             StockSplitj,T p F(b0 ⫹ b1 TradeRangej,t ⫹ b2 StockApprecj,t ),
where StockSplitj,T equals 1 if firm j has a cumulative stock split of 2-to-1 or greater and 1.25-to-1 or greater in Panels A and B, respectively, during time period T (where T p 4 years).
Similar results are obtained when T p 2 years. F is the logistic cumulative density function, TradeRangej,t is the ratio of the actual stock price in year t to the predicted price from the
regression reported in Table 3, StockApprecj,t is a measure of the increase in the firm j’s stock price from year t ⫺ 4 to year t, and the b’s are parameters of the model. The numbers in




                                                                                                                                                                                               Journal of Business
parentheses are chi-square statistics.
  * Significant at the .05 level.
  ** Significant at the .01 level.
The Share Price Puzzle                                                       2063


earlier studies (see n. 24). The trading range variable, however, is more im-
portant than the price appreciation variable as a predictor of a firm’s propensity
to split its stock. Firms split their stock when their share prices become “too
high” relative to the appropriate trading range, and not due to price appre-
ciation per se. In summary, the results in both Panels A and B of table 5
suggest that when stock prices exceed some preferred trading range firms are
more likely to split their stock to move prices back toward a desired share
price habitat.

C.    Do Stock Splits Increase the Number of Shareholders?
A basic theme of this article is that firms manage share price levels to make
the firm’s common stock more attractive to investors. Why might one share
price be more attractive than another to a particular firm’s current and potential
shareholders? We have already summarized the explanations offered by Dew-
ing (1934, 1953), the NYSE Listed Company Manual (undated), Merton
(1987), Brennan and Hughes (1991), and Angel (1997), and have discussed
why the preferences of small investors might differ markedly from those of
large institutional investors. Indeed, it is likely that all of these explanations
partially account for the differences in share price preferences among firms.
   Although a unified theory of share price levels and stock splits is beyond
the scope of this article, like earlier studies, we also document that the stock
splits by the firms in our sample are associated with subsequent increases in
the number of shareholders (for whatever reason). These findings are sum-
marized in table 6. Firms that have a stock split of 2-for-1 or greater experience
an average of 59% greater increase in the number of shareholders during the
ensuing 4 years than nonsplitting firms. In addition, the median nonsplitting
firms average a 13% decrease in shareholders, while the median splitting firms
average a 10% increase in shareholders. Similarly, firms that have a 1.25-to-
1 or greater split exhibit an average increase in shareholders of 55% over that
of the nonsplitting firms. The results for our sample are thus consistent with
those of earlier studies that report an increase in the number of shareholders
following a stock split (see n. 11). We know that stock splits increase a firm’s
investor base, although the precise mechanism by which they increase it is
not entirely clear.


IV.    Conclusion

The wide variations among share prices in U.S. stock markets that we observe
do not “just happen.” Apparently firms manage their share prices. We present
cross-sectional evidence that variables consistent with Merton’s (1987) model
of capital market equilibrium with incomplete information explain approxi-
mately 66% of the variation among firms’ share prices. In particular, firms
owned primarily by so-called small investors have lower share prices and vice
versa, and large, well-known firms have higher share prices and vice versa.
2064                                                                                     Journal of Business

TABLE 6              Change in the Number of Shareholders of Splitting and Nonsplitting
                     Firms (Excluding Firms with an SEO or Merger during the Period)
                                       Panel A                                        Panel B
                                  (Splits x 2-for-1)                           (Splits x 1.25-for-1)
Time Interval
                       Stock Split       No Split         Diff.        Stock Split      No Split         Diff.
1977–80 (%)                ⫹40              ⫺2           ⫹42**            ⫹28              ⫺4           ⫹32**
                          (⫹18)            (⫺9)         (⫹27)**           (⫹9)           (⫺12)         (⫹21)**
No. of firms               131             489                            207             413
1982–85 (%)                ⫹45              ⫺6           ⫹51**            ⫹35              ⫺9           ⫹44**
                           (⫹4)           (⫺16)         (⫹20)**           (⫹2)           (⫺18)         (⫹20)**
No. of firms               178             471                            249             400
1987–90 (%)                ⫹26              ⫹0           ⫹26**            ⫹25              ⫺3           ⫹28
                           (⫹4)           (⫺14)         (⫹18)**           (⫹1)           (⫺15)         (⫹16)**
No. of firms               147             571                            215             503
1992–95 (%)                ⫹59              ⫹4           ⫹55**            ⫹54              ⫹0           ⫹54**
                           (⫺1)           (⫺14)         (⫹13)**           (⫹3)           (⫺16)         (⫹19)**
No. of firms               134             510                            184             460
1997–2000 (%)              ⫹59              ⫹3           ⫹56*             ⫹54              ⫹1%          ⫹53**
                           (⫹3)           (⫺21)         (⫹24)**           (⫹3)           (⫺21)         (⫹24)**
No. of firms               113             497                            138             472
   Note.—This table shows the mean (median) percentage change in the number of shareholders for firms
that split their stock and for firms that did not split their stock during four 4-year intervals from 1977 through
2000. The percentage change in firm j’s shareholders from 1977 to 1980 is computed as Nj,1980/Nj,1976 ⫺ 1, where
Nj,t is the number of shareholders at the end of each year. A stock split is defined as ≥ 2-or-1 in Panel A and
as x 1.25-for-1 in Panel B, respectively. The third item in each cell in the table is the number of firms in
that category. The number of firms varies across years; firms engaging in secondary equity offerings or mergers
and acquisitions during the 4-year interval are excluded.
   * Significant difference between the means or medians at the .05 level.
   ** Significant difference between the means or medians at the .01 level.



We find that share prices exceeding the expected trading based on equation
(1) predict stock splits, and those firms splitting their stock experience an
increase in shareholders compared to nonsplitting firms. We conclude that the
high dispersion in stock prices in U.S. markets is largely a manifestation of
firms tailoring their share prices to reflect the desires of the firm’s owners.


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