The Share Price Puzzle*

Transcription

1 Edward A. Dyl William B. Elliott University of Arizona The Share Price Puzzle* In frictionless markets, share prices per se do not affect the value of the firm. Consider two identical firms, each with a value of $1 billion. One firm has 20 million shares outstanding and sells for $50 per share, and the other firm has 50 million shares outstanding that trade for $20 per share. If the firms values diverge, arbitrageurs will quickly eliminate the discrepancy by simultaneously selling the overvalued shares and buying the shares of the other firm. But market frictions exist, and corporate financial managers seem to believe that there is an optimal trading range for a firm s stock (Baker and Gallagher 1980). Even the New York Stock Exchange (NYSE) suggests that firms use stock splits to keep their shares in a preferred price range. 1 The so-called optimal trading range for a firm s shares seems to vary from firm to firm. A glance at the stock prices published each weekday in the Wall Street Journal reveals a wide range of common stock prices in U.S. markets at any 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 We document substantial variation in the prices of common stocks in U.S. markets due to firms selecting particular price ranges for their shares. Cross-sectional evidence indicates that variables consistent with Merton s model of capital market equilibrium explain roughly two-thirds of this variation in share prices. In addition, measures of trading range and share price appreciation predict stock splits, and the investor base of firms that split their stock increases compared to other firms. We conclude that firms manage share price levels to increase the value of the firm. * Contact the corresponding author, Edward A. Dyl, at edyl@ bpa.arizona.edu. 1. New York Stock Exchange Listed Company Manual, These prices refer to closing prices on August 5, (Journal of Business, 2006, vol. 79, no. 4) 2006 by The University of Chicago. All rights reserved /2006/ $

2 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 investigates 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 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 conjecture 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 and find an average initial offering price of $10.36 per share.

3 The Share Price Puzzle 2047 available from the Center for Research in Security Prices (CRSP) for the period from 1976 through 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 $ 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, transaction costs, and share prices, and propose some hypotheses about why some firms have higher share prices and vice versa. Section II presents cross-sectional 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 Dewing 8 (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 ( ) 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.

4 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 $70 Panel A: All firms listed on the NYSE, Amex, and Nasdaq Share price: Mean ($) Median ($) No. of firms 6,638 2,682 2, Proportion of firms (%) Market value (000,000): Mean ($) 2, ,221 6,777 14,551 14,880 Median ($) ,570 3,280 1,898 Panel B: NYSE firms Share price: Mean ($) Median ($) No. of firms 2, , Proportion of firms (%) Market value (000,000): Mean ($) 4, ,693 9,501 17,122 26,149 Median ($) ,356 5,760 5, Journal of Business

5 Panel C: Amex firms Share price: Mean ($) Median ($) No. of firms Proportion of firms (%) Market value (000,000): Mean ($) Median ($) Panel D: Nasdaq firms Share price: Mean ($) Median ($) No. of firms 3,492 1,798 1, Proportion of firms (%) Market value (000,000): Mean ($) ,401 14,862 $1,729 Median ($) $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 Market value is the year-end market capitalization reported in the CRSP database. The Share Price Puzzle 2049

6 Fig. 1. Average common stock prices from 1976 to Average annual stock prices for 1,019 firms with continuous data available on CRSP for 1976 through 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 Journal of Business

7 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. ( ) 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 significant 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 consideration.... 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, however, 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) documents 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.

8 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; Falkenstein 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., frequently the price level after the firm s last stock split). It is also well docu-

9 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 explanation derives from a combination of investors revealed preferences for trading 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 shareholders 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 commissions are inversely related to share price and suggest that, in a world with incomplete information, brokers will produce and disseminate more information (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.

10 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 increase 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 lowerpriced 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 dispersion 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.

11 The Share Price Puzzle 2055 TABLE 2 Descriptive Statistics for Sample Firms, Panel A: Share price Mean ($) SD Median ($) No. of frms 1,019 1,019 1,019 1,019 1,019 1,019 Panel B: Market value (000,000) Mean ($) ,293 2,187 4,061 7,175 SD 3,809 3,714 3,855 6,414 12,081 26,806 Median ($) No. of firms Panel C: No. of shareholders (000) Mean SD Median No. of firms Panel D: Average investment per shareholder (000) Mean ($) SD Median ($) No. of firms 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 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 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 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.

12 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 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 investment 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 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 parameters of the following regression equation: SharePrice p d d BVEquity j,t 0 1 j,t d AvgHldg d EPS. (1) 2 j,t 3 j,t j,t The dependent variable, SharePrice j,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, BVEquity j,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, AvgHldg j,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 BkValEquity j,t in equation (1) the empirical results are qualitatively the same.

13 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 regarding 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 holders, 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 AvgHldg j,t with an instrumental variable estimated using the rank order of AvgHldg j,t sorted from lowest to highest. The final results of our analysis are unchanged. The third independent variable in equation (1), EPS j,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 EPS j,t as an independent variable improves the specification of the model. The results of the six cross-sectional regressions estimating annual parameters 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 (d 1 ) 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 (d 2 ) is also positive and significant at the.01 level in each regression, consistent with the hypothesis that lower share prices are preferred by firms owned by small investors and vice versa. 21 The coefficient on earnings per share (d 3 ) 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 AvgHldg j,t is used in place of AvgHldg j,t, d 2 continues to be positive and significant in each regression.

14 TABLE 3 Share Prices, Firm Size, and Ownership Characteristics Year d 0 d 1 d 2 d 3 Adj. R 2 F-test No. of Firms (12.06**).2071 (15.36**).2185 (7.69**).1712 (11.60**) (16.61**).1044 (7.27**).2285 (7.85**).0994 (7.61**) (14.46**).1604 (11.08**).1999 (6.88**).1259 (11.16**) (7.19**).2375 (17.28**).1703 (6.39**).1207 (6.91**) (10.41**).2233 (18.54**).1067 (5.00**).1233 (10.33**) (6.40**).2526 (17.09**).1051 (4.20**).0728 (3.10**) 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 SharePrice j,t is the logarithm of the average price per share of firm j s common stock during year t, BVEquity j,t is the logarithm of the book value of firm j s equity at the end of year t, AvgHldg j,t is the logarithm of average book value of equity per shareholder, and EPS j,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

15 The Share Price Puzzle 2059 positive and significant, indicating that including this variable improves the specification of the regression model. The average R 2 of the six regressions is The cross-sectional regression estimates in table 3 support both the hypothesis 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 socalled 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: StockSplit p F(b b TradeRange b StockApprec ), (2) j,t 0 1 j,t 2 j,t where StockSplit j,t is the probability that firm j splits its stock during time period T, F is the logistic cumulative density function, TradeRange j,t indicates whether or not the price of the stock is too high, StockApprec j,t is the increase

16 2060 Journal of Business TABLE 4 Years Sample Firms Splitting Their Stock Stock Split No Stock Split Number % Number % Total Firms Panel A: Firms splitting 2-for-1 or greater during the 4 years , , , , ,019 Panel B: Firms splitting 1.25-for-1 or greater during the 4 years , , , , ,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 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, StockSplit j,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, TradeRange j,t, is defined as follows: TradeRange j,t p SharePrice /E(SharePrice FBVEquity, AvgHldg, EPS ), (3) j,t j,t j,t j,t j,t where E(SharePrice j,t FBVEquity j,t, AvgHldg j,t, EPS j,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.

17 The Share Price Puzzle 2061 the 5 years (1976, 1981, 1986, 1991, and 1996) using the regression relations shown in table That is, E(SharePrice Fetc) p d d BVEquity j,t 0 1 j,t d AvgHldg, d EPS, (4) 2 j,t 3 j,t so TradeRange j,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 TradeRange j,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, StockApprec j,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 p SharePrice /SharePrice. (5) j,t j,t j,t 4 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 StockSplit j,t is the 4-year period following year t. 25 Panels A and B show the findings when the StockSplit j,t variable equals one if the firm had a cumulative 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., b 1 ) 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 b 1 coefficients 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 StockApprec j,t (i.e., the b 2 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(SharePrice j,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.

18 TABLE 5 Results of a Logit Model of Stock Splits and Share Prices Time Period b 0 b 1 B 2 Likelihood Ratio No. of Firms Splitting Firms (%) Panel A: Cumulative stock split of at least 2-to (24.11**) (4.29*).5944 (9.34**) (32.37**) (10.05**).0366 (2.22) (42.30**) (11.78**).1015 (6.59**) (41.95**).8155 (7.02**).0808 (.19) (41.78**).8617(6.30**).0779 (4.32*) Panel B: Cumulative stock split of at least 1.25-to (20.30**) (3.67).7542 (14.10**) (16.59**).8594 (5.82*).0666 (5.82*) (34.31**) (12.83**).1001 (6.67**) (29.92**).8240 (7.96**).0845 (2.04) (35.25**).9109 (6.98**).1008 (6.07**) 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 StockApprec j,t ), where StockSplit j,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, TradeRange j,t is the ratio of the actual stock price in year t to the predicted price from the regression reported in Table 3, StockApprec j,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 parentheses are chi-square statistics. * Significant at the.05 level. ** Significant at the.01 level Journal of Business

19 The Share Price Puzzle 2063 earlier studies (see n. 24). The trading range variable, however, is more important 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 appreciation 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 Dewing (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 summarized 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 approximately 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.

20 2064 Journal of Business TABLE 6 Time Interval Change in the Number of Shareholders of Splitting and Nonsplitting Firms (Excluding Firms with an SEO or Merger during the Period) Panel A (Splits x 2-for-1) Panel B (Splits x 1.25-for-1) Stock Split No Split Diff. Stock Split No Split Diff (%) ** ** ( 18) ( 9) ( 27)** ( 9) ( 12) ( 21)** No. of firms (%) ** ** ( 4) ( 16) ( 20)** ( 2) ( 18) ( 20)** No. of firms (%) ** ( 4) ( 14) ( 18)** ( 1) ( 15) ( 16)** No. of firms (%) ** ** ( 1) ( 14) ( 13)** ( 3) ( 16) ( 19)** No. of firms (%) * 54 1% 53** ( 3) ( 21) ( 24)** ( 3) ( 21) ( 24)** No. of firms 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 The percentage change in firm j s shareholders from 1977 to 1980 is computed as N j,1980/nj,1976 1, where N j,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. References Affleck-Graves, John, Shantaram Hegde, and Robert E. Miller Conditional price trends in the aftermarket for initial public offerings. Financial Management 25: Amihud, Yakov, and Haim Mendelson Asset pricing and the bid-ask spread. Journal of Financial Economics 17: Amihud, Yakov, Haim Mendelson, and Jun Uno Number of shareholders and stock prices: Evidence from Japan. Journal of Finance 54: Angel, James J Tick size, share prices, and stock splits. Journal of Finance 52: Anshuman, V. Ravi, and Avner Kalay Market making with discrete prices. Review of Financial Studies 11: Can splits create market liquidity? Theory and evidence. Journal of Financial Markets 5: Atkins, Allen B., and Edward A. Dyl Transactions costs and holding periods for common stocks. Journal of Finance 52: Badrinath, S. G., Gerald D. Gay, and Jayant R. Kale Patterns of institutional investment, prudence, and the managerial safety-net hypothesis. Journal of Risk and Insurance 56:

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