THE NUMBER OF TRADES AND STOCK RETURNS

Transcription

1 THE NUMBER OF TRADES AND STOCK RETURNS Yi Tang * and An Yan Current version: March 2013 Abstract In the paper, we study the predictive power of number of weekly trades on ex-post stock returns. A higher number of stock trades indicates a higher degree of slice and dice in stock trading and further a lack of stock liquidity in trading. Thus, we predict that a higher number of weekly trades is followed by a higher future stock return. Our findings are consistent with this prediction. Stocks whose number of trades in week t is in the highest quintile of the sample outperform those in the lowest quintile by 24.44% on the annualized basis in week t+1. The return differential is 19.24% based on the characteristic-adjusted returns after adjusting for size, book-to-market, and momentum; and it is 23.36% based on Carhart s four factor model. The positive predictive power of number of trades holds not only for all trades but also for the subsamples of the buyer-initiated and the seller-initiated trades. JEL classification code: G02, G10, G11, G12, G14. Keywords: The number of trades, liquidity risk, investor attention, expected stock returns * Schools of Business, Fordham University, 1790 Broadway, New York, NY ytang@fordham.edu. Phone: (646) Fax: (646) Schools of Business, Fordham University, 1790 Broadway, New York, NY ayan@fordham.edu. Phone: (212) Fax: (212)

2 Abstract In the paper, we study the predictive power of number of weekly trades on ex-post stock returns. A higher number of stock trades indicates a higher degree of slice and dice in stock trading and further a lack of stock liquidity in trading. Thus, we predict that a higher number of weekly trades is followed by a higher future stock return. Our findings are consistent with this prediction. Stocks whose number of trades in week t is in the highest quintile of the sample outperform those in the lowest quintile by 24.44% on the annualized basis in week t+1. The return differential is 19.24% based on the characteristic-adjusted returns after adjusting for size, book-to-market, and momentum; and it is 23.36% based on Carhart s four factor model. The positive predictive power of number of trades holds not only for all trades but also for the subsamples of the buyer-initiated and the seller-initiated trades. 2

3 1. Introduction It has been widely documented that stock liquidity can predict future stock returns. For example, Amihud and Mendelson (1986) predict in their theory that less liquid securities yield higher expected returns. Many empirical papers test this prediction using various measures of stock liquidity, such as trading volume, bid-ask spread, etc. They all find results consistent with this prediction. 1 In this paper, we use number of stock trades to measure stock liquidity. We study the predictive power of number of stock trades on future stock returns. We argue that number of stock trades can measure the degree of slice and dice in stock trading, which is not captured in the previous stock liquidity measures. Bertsimas and Lo (1998) suggest that the optimal execution strategy for certain traders involves breaking orders into pieces to minimize cost. Recently, such an execution strategy has become more prevalent with the technological progress in algorithmic trading and the rise of high frequency trading. Traders (or algorithms) frequently slice large orders into smaller chunks and spread the trades over time or across various equity trading venues. In particular, when stock liquidity is low (e.g., when bidask spread is high), traders tend to split large orders into smaller slices to minimize market impact and to reduce the opportunity cost. When stock liquidity is high (e.g., when bid-ask spread is low), traders could instead cluster their trades together and execute the clustered trades quickly. In other words, stock liquidity determines the degree to which traders slice or cluster their trades. 1 For example, Amihud and Mendelson (1986) use bid-ask spread to measure stock liquidity and find results consistent with their prediction (See also Amihud and Mendelson, 1989). Brennan, Chordia, and Subrahmanyam (1998) use trading volume to measure stock liquidity and they show a negative relation between trading volume and ex-post stock returns (See also Chordia, Subrahmanyam, and Anshuman, 2001). Amihud (2002) creates an illiquidity measure to capture the market impact of trading and they find a positive relation between their illiquidity measure and ex-post stock returns. For a comprehensive survey of this literature, see Amihud, Mendelson, and Pedersen (2006). 3

4 However, this new aspect of stock liquidity in slice and dice is not captured by the extant liquidity measures. A high trading volume could happen either to a liquid stock with several block trades or to a relatively illiquid stock with many sliced trades. Or if we consider bid-ask spread (or any market impact variable), it is possible that both a block trading of a liquid stock and multiple sliced trades of an illiquidity stock could cause a similar market impact and result in a similar level of bid-ask spread. Thus, it is important to measure the degree of slice and dice to complement the extant liquidity measures such as trading volume and bid-ask spread, and to better understand stock liquidity. We argue that number of trades can be such a measure: a higher number of stock trades indicates that traders slice their orders to a larger degree in their stock trading, thereby indicating a lack of stock liquidity. 2 Following the literature on stock liquidity and stock returns, we hypothesize that a higher number of stock trades is followed by a higher future stock return. We study the relation between number of trades and future stock returns, using portfolio sorts, Carhart s (1997) four-factor model, and the Fama-MacBeth s (1973) technique. We find that a decrease in number of stock trades in a contemporaneous week is followed by a lower stock return in the subsequent week. For example, in the results from the unconditional portfolio sorts of number of trades (based on raw stock returns and NYSE quintile breakpoints), we find that the stocks in the bottom quintile of number of trades in the current week t underperform the stocks in the top quintile by 0.47% in week t+1, which represent 24.44% on the annualized basis. The underperformance is 0.37% per week (or about 19.24% per year) based on the characteristic adjusted returns which adjust for size, book-to-market, and momentum. Similarly, the results from Carhart s four factor model also show that the bottom quintile underperforms the top 2 A higher number of trades could also arise from an increase in investor participation. We will discuss our control and test for this possibility later in the paper. 4

5 quintile by 0.43% on the weekly basis (or 22.36% on the annualized basis). This positive predictive power of number of trades holds not only for all trades but also for the subsamples of buyer-initiated trades and seller-initiated trades. It also remains statistically and economically significant even after we control for other common return predictors, such as beta, size, book-tomarket, momentum, stock reversals, idiosyncratic volatility, and other liquidity variables, such as Amihud s (2002) illiquidity ratio, trading turnover, trading volume, bid-ask spread, etc. Our controls for other liquidity variables are especially important. Number of trades is highly correlated to most of these liquidity variables. Thus, a natural concern is whether number of trades overlaps other liquidity variables and whether it provides a new aspect of stock liquidity. Our controls of other liquidity variables address this concern. They show that the predictive power of number of trades is above and beyond the predictive power of these extant liquidity variables. In this sense, number of trades captures a new dimension of stock liquidity that other liquidity variables do not capture, as we argued at the beginning of this introduction. We also run several robustness tests. We show that our results remain qualitatively the same even after eliminating the trades in the first and the last 30 minutes of the regular trading hours (9:30-16:00). We also show that the positive predictive power of number of trades on future stock returns exists even if we measure number of trades and future stock returns in the monthly measurement window rather than the weekly measurement window. Number of trades could increase either because traders break up their orders to small slices or because more investors participate in trading. The latter possibility of investor participation is more likely to happen when a stock attracts more investor attention. However, while investor attention or investor participation potentially could explain our results based on the buyerinitiated trades, it cannot explain our results on the sell-initiated trades. Investor attention would 5

6 affect investors trading only when investors are buying stocks from a large set of choices. It should have little impact on selling since sellers only have to choose from a small set of their limited portfolio holdings when they are selling (see Barber and Odean, 2008). To further test the investor attention argument and the slice and dice argument, we develop the following hypotheses. First, if the predictive power of number of trades is driven by investor attention, it should be greater when the stock attracts more investor attention. To test this conjecture, we use the market capitalization of the stock and the number of financial analysts following the stock to proxy for the degree of investor attention on the stock. 3 A large firm captures more investor attention. Financial analysts coverage also brings more visibility to the stock since investors follow closely analysts forecasts or recommendations (see, e.g., Womack (1996) and Barber, et. al. (2001)). We find that the predictive power of number of trades on future stock returns is stronger in the stocks with smaller market capitalizations or less analysts covering, presumably the stocks with less investor attention. The predictive power of number of trades disappears in the stocks with larger market capitalizations or more analysts covering. Thus these results are inconsistent with the investor attention argument. Second, number of trades is more likely to be an outcome of slice and dice rather than investor participation when the size of trades is smaller. Thus, we conjecture that the predictive power of number of trades on future stock returns is greater in the trades with smaller trade sizes. As expected, we find that the number of small trades demonstrates a stronger predictive power on future stock returns than the number of larger trades. In particular, for the number of the trades with trade size below $5,000, the stocks in the bottom quintile in week t underperform the 3 Merton (1987) argues that investors trade in the stock market only in the stocks that they recognize. Following this argument, some studies in the literature also use trading volume to proxy for investor attention. In the paper, we view trading volume as measuring a stock s liquidity. Under this view, we will study later in the paper how the predictive power of number of trades differs with stocks with different trading volume. Our results on trading volume are also consistent with the view that trading volume is a proxy for investor attention (details follow). 6

7 stocks in the top quintile by 0.62% in week t+1, which represent 32.24% on the annualized basis. In contrast, the predictive power disappears for the number of the trades with trade size above $50,000. These results support the slice and dice argument. Third, we also study how number of trades complements the other liquidity measures. We interact number of trades with the other liquidity measures. We conjecture that the information provided by a high number of trades is more valuable and relevant on stock liquidity when the other liquidity measures indicate a value of low liquidity for the stock. To test this conjecture, we double sort portfolios with number of trades and the other liquidity measures, such as bid-ask spread, Amihud s illiquidity ratio, and trading volume. As expected, we find that the predictive power of number of trades is stronger when bid-ask spread is higher, Amihud s illiquidity ratio is higher, or trading volume is lower. These results suggest that number of trades does complement the other liquidity variables in providing additional information on stock liquidity, especially when the other liquidity variables demonstrate a value of low liquidity. Finally, for the intellectual curiosity, we also study the cross-sectional difference in the predictive power of number of trades for stocks with different institutional investor holdings. The mix of institutional and retail investor holdings could affect the degree of slice and dice in different ways. When a trader trades a block of shares with retail investors, she may have to break the block into small chunks to facilitate retail investors. This possibility suggests that slice and dice is more pronounced and the predictive power of number of trades is stronger when there are more retail investors holding the stock. On the other hand, retail investors may not need to slice their orders given their small overall trading size. From this possibility, a stock with more retail investor holdings could be associated with less slice and dice. Thus, it is an empirical question how institutional and retail investor holdings could affect the predictive power of 7

8 number of trades. To answer this question, we double sort portfolios based on number of trades and the fraction of institutional investors holding the stock. We find that the predictive power of number of trades is stronger for the stocks with less institutional investor holdings and thus more retail investor holdings. These results are consistent with the first possibility that retails investors provide liquidity to institutional investors and institutional investors has to slice and dice to a larger degree when there are more retail investors holding the stock. As we discussed earlier, our paper is related to the literature on stock liquidity and stock returns. Some researchers in this literature view liquidity as a characteristic and suggest that investing in illiquid stocks is compensated by higher stock returns (see, e.g., Datar, Naik, and Radcliffe, 1998, Amihud and Mendelson, 1996). Under this view, many empirical studies use various measures of stock liquidity/illiquidity and find that high stock liquidity is indeed associated with lower future stock returns, see, e.g., Amihud and Mendelson (1986, 1989), Amihud (2002), Brennan and Subrahmanyam (1996), and Brennan, Chordia and Subrahmanyam (1998). 4 On the other hand, some researchers instead view liquidity as a priced risk factor. For example, Pastor and Stambaugh (2003) show that stocks with higher sensitivity to innovations in aggregate liquidity have higher expected returns (see also, Acharya and Pedersen, 2005, Sadka, 2006, Korajczyk and Sadka, 2008, and Charoenrook and Conrad, 2008). Our paper follows the first view. We argue that number of trades captures a new perspective of stock liquidity in slice and dice. We show that number of trades can predict future stock returns. Our paper is also related to a more general literature on the relation between trading behavior and stock returns. This literature starts to evolve rapidly, especially after the availability of the data on high-frequency trading orders and on stock holdings of institutional investors. For 4 There are also some empirical studies finding weak pricing effect of stock liquidity, see, e.g., Hasbrouck (2006), Spiegel and Wang (2005), etc. 8

9 example, Kaniel, Saar, and Titman (2008) study the trading by individual investors. Campell, Ramadorai, and Schwartz (2011) study the trading by institutional investors. Like some of these recent studies, we also use the data on high-frequency trading orders. However, unlike these studies, our focus in this paper is on number of stock trades and its predictive power on future stock returns. To the best of our knowledge, our paper is the first to document this predictive power. The remainder of the paper is organized as follows. Section 2 describes the data and the sample. Section 3 studies the relation between number of trades and future stock returns. Section 4 checks the robustness of our findings. Section 5 study the cross-section difference in the relation between number of trades and future stock returns. Section 6 concludes. 2. Data Our sample covers the period from January 1993 to December We follow the standard convention and limit our analysis to the common stocks traded on the NYSE, Amex, and Nasdaq and those that are identified by CRSP share type codes of 10 and 11. We exclude those stocks with market capitalization less than $10 million and with stock price less than $5 per share. Thus, our final sample consists of 10,546 stocks, with an average of 3,311 stocks per week. We obtain the return and volume data at the daily and monthly frequencies from the Center for Research in Securities Prices (CRSP) database, and the financial statement information from the Merged CRSP/Compustat database. The number of trades and the trade-order imbalance variables are computed using the tick-by-tick data from the Trade and Quotes (TAQ) database. Finally, we obtain the data on analysts earnings forecasts from the Institutional Brokers Estimate System (I/B/E/S) database. 9

10 Unless otherwise specified, we measure all control variables as of the end of portfolio formation week (i.e., week t), defined as the five days ending one day prior to the calendar week t+1. Hence, one day is skipped over which the cross-sectional predictors are measured and the weeks for which stock returns are predicted. We also require a minimum of 15 daily observations for the variables that are computed based on daily data over 21 trading days. Finally, we require a minimum of 200 daily observations for the variables that are computed based on daily data over 252 trading days The adjusted number of trades As we discussed in the introduction section, we use number of trades to measure stock liquidity. The larger (smaller) is the number of trades on a stock, the lower (higher) is the stock s liquidity. For most empirical studies in the paper, we calculate number of trades based on the weekly horizon. To check the robustness of our results based on the weekly horizon, we will also expand to the monthly horizon in the robustness section. We calculated the number of weekly trades using the tick-by-tick data obtained from the TAQ database. For each week t, we calculate the adjusted number of weekly trades, denoted by NUMTRDU, as the number of trades in the week, demeaned and scaled by the mean of the number of trades over the prior 26 weeks. 5 In the paper, we use the adjusted (demeaned) number of weekly trades NUMTRDU rather than the unadjusted number of weekly trades to purge stock fixed effect. The unadjusted number of trades could be effectively a permanent stock characteristic, with a weekly autocorrelation of 0.93 in our sample. The unadjusted number of trades is also highly correlated with many stock 5 Our finding remains robust to alternative reference window, e.g., past 12 or 52 weeks. They are also robust to the alternative measures such as the one scaled by the standard deviation the number of weekly trades over the past 26 weeks. 10

11 characteristics or the trading patterns of the stock. For example, the unadjusted number of trades is positively correlated with the volatility of trades: The average correlation coefficient between the unadjusted number of trades and its volatility over the past 26 weeks is 0.86 (unreported in the paper). These correlations could be attributed to the clientele effects that arise from the particular investor preferences and/or trading activities inherent to each individual stock. Thus, we calculate the adjusted number of weekly trades to control for these correlations as well as the unobserved stock characteristics. The adjusted number of trades has a low weekly autocorrelation of 0.05, and low correlation with its volatility in the past 26 weeks (the average correlation coefficient is 0.02) Expected returns and characteristic-adjusted returns In the paper, we will study the predictive power of the adjusted number of trades observed in week t (NUMTRDU) on stock return in week t+1 (RET). We study both raw stock returns (RET) and stock returns adjusted by size, book-to-market, and momentum (RETADJ). The study on adjusted stock returns can ensure that our results are not driven by the well-known cross-sectional return predictors, such as size, book-to-market (BM), and momentum. We calculate adjusted stock returns by following Daniel and Titman (1997) and forming characteristic matched portfolios at the end of prior June (defined as June of year k hereafter). In particular, at the end of June of year k, all stocks in our sample are sorted into size quintiles based on the NYSE size breakpoints; stocks within each size quintile are then sorted into BM quinitles using the NYSE BM breakpoints; stocks within each of the 25 size and BM groupings are further sorted into momentum quintiles based on the CRSP momentum breakpoints. The intersection of the size, BM, and momentum quintiles generates 125 benchmark portfolios at the 11

12 end of June of year k. The equal-weighted weekly benchmark returns for the 125 groupings are calculated over the following 52 weeks from July of year k to June of year k+1. A stock s weekly characteristic-adjusted return (RETADJ) at week t is defined as the difference between its raw weekly return and the weekly benchmark returns of one of the 125 benchmark portfolios to which the stock belongs as of the end of June of year k Control variables We construct the following control variables for our cross-sectional asset pricing tests. We estimate market beta ( ) by running a time-series regression using daily returns over the past 252 trading days if available. Following Fama and French (1992), we adjust our beta estimate for nonsynchronous trading (Dimson (1979)): (1) where is daily returns of stock i and is daily returns of the CRSP value-weighted index in excess of the daily return of the one-month Treasury bills. 6 A stock s market beta is the sum of the slope coefficients of the current and the lagged excess market returns, i.e.,. We calculate the stock s size ( ) as the natural logarithm of the product of the price per share and the number of shares outstanding (in million dollars). We calculate the book-tomarket equity ratio ( ) as the ratio of the book value to the market value of equity. The book value of equity is the book value of common equity plus the value of deferred tax and investment tax credit (if available) minus the value of preferred equity, where the value of preferred equity is calculated as the redemption, liquidating, or par value (in that order depending 6 The daily return for the one-month Treasury bills, as well as the daily and weekly size, book-to-market, and momentum factors as used in the later equations, are downloaded from Kenneth French s online data library. 12

13 on availability). The book value of equity is measured at the last fiscal year end prior to week t and the market value of equity is measured at end of December prior to week t. We follow Jegadeesh and Titman (1993) and calculate a stock s momentum ( ) as its cumulative return over a period of 252 trading days ending 21 trading days prior to the portfolio formation week t. The Amihud s (2002) liquidity measure ( ) is defined as the average daily ratio of the absolute stock return to the dollar trading volume over the prior 21 trading days. We also use stock return over the prior five and 21 trading days, denoted and, respectively, to control for the short-term reversals as in Jegadeesh (1990). Following Ang, Hodrick, Xing, and Zhang (2006), we calculate the idiosyncratic volatility of stock i ( ) as the standard deviation of the residuals from the regression using daily observations over the prior 21 trading days: (2) where and are defined as in the previous equation (1); and and are the daily size and book-to-market factors, respectively, as in Fama and French (1993). Following Harvey and Siddique (2000), we define the stock s co-skewness ( ) as the estimate of in the regression using the daily return observations over the prior 21 trading days: (3) Following Barber, Odean and Zhu (2006), we calculate the fraction of buyer-initiated trades ( ) as the number of trades that are classified as buyer-initiated to the sum of trades 13

14 that are classified as either buyer- or seller-initiated in week t. The identification of a trade as buyer- or seller-initiated is based on the procedure described in Lee and Ready (1991). 7 Following Diether, Malloy, and Scherbina (2002), we calculate forecast dispersion ( ) based on financial analysts forecasts on the stock s one-year-ahead earnings. In particular, DISP is the standard deviation of earnings forecasts scaled by the absolute value of the average forecast in the month of week t. Next, we control for several other liquidity variables. We calculate the average daily turnover ( ) in week t. We calculate the normalized weekly trading volume ( ) and normalized weekly dollar trading volume ( ) as: and (4) (5) and are trading volume and dollar trading volume, respectively. and are the averages of the weekly trading volume and dollar volume over the past 26 weeks, respectively. For the other liquidity variables, we also calculate Amihud's illiquidity ratio (ILLIQ) as the ratio of the absolute value of daily return to the value of daily trading volume, averaged over week t. We calculate effective bid-ask spread (SPRD) as the equal-weighted average of daily effective spread in week t. 7 Lee and Ready (1991) suggest two rules for the identification. The quote rule identifies a trade as buyer- (seller-) initiated if the trade price is above (below) the midpoint of the most recent bid-ask quote. The tick rule classifies a trade as buyer- (sell-) initiated if the trade price is above (below) the last executed trade price. Following the recommendation of Ellis, Michaely, and O Hara (2000), for Nasdaq-listed stocks, the tick rule is used for trades that are executed within the bid-ask bounds the quote rule is used for all other trades. For NYSE/Amex stocks, the tick rule is used for trades that execute at the midpoint of posted bid-ask quotes and the quote rule is used for all other trades. Moreover, the opening auctions for the NYSE/Amex exchanges are removed from the analysis. All trades are equal-weighted. 14

15 Finally, we also calculate the standardized unexpected quarterly earnings as (SUE) as (E q E q-4 c q )/s q, where E q and E q-4 are earnings in the current quarter of week t and in the same quarter a year ago, respectively; and c q and s q are the mean and standard deviation, respectively, of (E q E q 4 ) over the preceding eight quarters. INST is the fraction of stock holdings by institutional investors. CVRG is the number of financial analysts following the stock Summary statistics Panel A of Table 1 reports the time-series averages of the cross-sectional descriptive statistics for the aforementioned variables. The average adjusted number of trades (NUMTRDU) is around NUMTRDU exhibits substantial cross-sectional variation with an average crosssectional standard deviation of 1.41, about nine times of its mean. Panel B of Table 1 presents the time-series average of the cross-sectional correlation coefficients for the variables. The correlation coefficients between the adjusted number of trades (NUMTRDU) and one-week-ahead raw stock return (RET) and characteristic-adjusted return (RETADJ) are 1.60% and 1.28%, respectively. Both correlations are significant at the 1% level. As we discussed earlier, number of trades can be viewed as a proxy for stock liquidity. Thus, the positive correlations between NUMTRDU and our future return variables are consistent with the results in the literature, suggesting that a low level of stock liquidity (i.e., a high number of trades) is associated with high future stock returns. NUMTRDU is also highly correlated with many of the other liquidity variables. For example, the correlation coefficient between NUMTRDU and share turnover (TURN) is 19.05%. The correlation coefficient is 52.06% between NUMTRDU and normalized trading volume (VOLU) and 79.26% between NUMTRDU and normalized dollar trading volume (VOLDU). 15

16 Finally, NUMTRDU is also positively correlated with many contemporaneous cross-sectional predictors, including size (LNME), momentum (MOM), idiosyncratic volatility (IVOL), shortterm reversal (REVM), and fraction of buyer-initiated trades (BUY). All the correlations are significant. 3. Cross-sectional Relation between Number of Trades and Stocks Returns In this section, we study the predictive power of number of trades (NUMTRDU) on future stock returns. We use number of trades to measure stock liquidity. Following the literature, we predict that a lower number of trades is followed by a lower future stock return Univariate portfolio-level analysis We begin our empirical analysis with univariate portfolio sorts. For each week t over the period of July 1993 to December 2008, we sort the stocks in our sample into quintile portfolios based on their adjusted number of weekly trades (NUMTRDU) using the CRSP NUMTRDU breakpoints. 8 We then calculate the equal-weighted portfolio returns over the subsequent five weeks, starting from week t+1 to t+5. Figure 1 depicts the Carhart s (1997) four-factor alphas of the return differentials between the highest and the lowest NUMTRDU quintile portfolios in week t+1 to t+5, and their 95% confidence bounds, calculated using the Newey-West (1987) robust standard errors. 9 It shows that the positive pricing effect of NUMTRDU on future stock returns diminishes through time. The predictive power of NUMTRDU on future stock returns is the strongest in the first 8 Note that we lose the observations in the first 6 months or 26 weeks in computing the adjusted number of trades. This is why our study on portfolio sorts starts from July In other words, the four-factor alpha is the intercept of the time-series regression of the weekly return differentials between the highest and the lowest NUMTRDU quintiles against the contemporaneous market, size, book-to-market, and momentum factors. 16

17 week after portfolio formation, i.e., week t+1. The corresponding four-factor alpha is close to 0.43% per week in the first week t+1. It drops significantly from 0.31% per week in the week t+2 and to 0.12% per week in week t+5, though all alphas are significant at the 5% or better level. In Panel A of Table 2, we report the average stock returns in week t+1 for NUMTRDU (adjusted number of trades) quintiles, the return differential between the highest and the lowest NUMTRDU quintiles, and the corresponding four-factor alphas. We report the results on both raw stock returns in Column RET and characteristic-adjusted portfolio returns in Column RETADJ. Consistent with the correlation coefficients reported in Panel B of Table 1, our results from the univariate portfolio sorts also show that NUMTRDU is positively correlated with future stock returns. Moving from the lowest to the highest NUMTRDU quintile, the average weekly raw return (RET) increases monotonically from -0.10% per week to 0.45% per week, or from -5.20% to 23.40% on an annualized basis. The average raw return differential between the highest- and the lowest-quintile portfolios is 0.56% per week, or about 29.12% per year, with a Newey-West adjusted t-statistic of The Carhart s (1997) four-factor alpha is 0.50% per week, or 26.00% per year, with a Newey-West adjusted t-statistic of The characteristicadjusted returns after controlling for risk premiums associated with the size, book-to-market, and momentum factors also increase monotonically from the lowest to the highest NUMTRDU quintile, yielding an average characteristic-adjusted return differential of 0.43% per week or about 22.36% per year. This characteristic-adjusted return differential is significant at the 1% level as well with a Newey-West adjusted t-statistic of These results of the univariate portfolio-level analysis suggest that the positive pricing effect of number of trades is statistically and economically significant. A portfolio strategy that buys stocks in the top NUMTRDU quintile 17

18 and sells short stocks in the bottom NUMTRDU quintile, on average, generates a positive return around 29% on an annualized basis. We also report in the remaining columns of Panel A of Table 2 the average NUMTRDU and the average market share of the NUMTRDU quintile portfolios. By construction, the average NUMTRDU increases from (the lowest NUMTRDU quintile) for Quintile 1 to 1.13 for Quintile 5 (the highest NUMTRDU quintile). On the other hand, the market share is heavily skewed to the median NUMTRDU quintiles, with the third and the fourth quintile portfolios collectively accounting for more than 60% market share, while the average market share of stocks in the lowest quintile is about 5%. To alleviate the concern that the CRSP NUMTRDU quintile breakpoints are distorted by the large number of small-cap Nasdaq- and Amex-traded stocks, we reconstruct the NUMTRDU quintile portfolios using the NYSE breakpoints. We first define the NUMTRDU breakpoints for each week based on the subsample that is made up of all NYSE stocks and meet our data requirements. We then sort all NYSE, Amex, and Nasdaq stocks in our sample into quintiles based on the NYSE NUMTRDU breakpoints. We report the results based on the NYSE NUMTRDU breakpoints in Panel B of Table 2. As can be seen, the positive relation between NUMTRDU and future stock returns remains intact. Similar to the results based on the CRSP NUMTRDU breakpoints, the results based on the NYSE NUMTRDU breakpoints show that the average raw returns and characteristic-adjusted returns increase monotonically from the lowest to the highest NUMTRDU quintiles. The average differentials of raw and characteristic-adjusted returns between the highest and the lowest quintile portfolios are 0.47% (22.36% per year) and 0.37% per week (19.24% per year), respectively, both of which are significant at the 1% level. The four-factor alpha of the weekly raw return differential is 0.43% per week (22.36% per year), 18

19 with a Newey-West adjusted t-statistic of Furthermore, the market share is fairly evenly distributed among the five portfolios, with a slight decrease from the lowest to the highest quintile. This finding, together with the strictly monotonic cross-sectional return patterns associated with NUMTRDU, suggests that the positive pricing effect is not driven by extremely small stocks that are economically insignificant. For the remainder of the paper, we form NUMTRDU portfolios based on the NYSE breakpoints to alleviate the concern that the CRSP decile breakpoints are distorted by the large number of small NASDAQ and Amex stocks. Finally, we investigate whether the positive and significant return difference between high and low NUMTRDU deciles is due to underperformance by stocks in the low NUMTRDU decile, or outperformance by stocks in the NUMTRDU decile, or both. We compare the performance of the low NUMTRDU decile to the performance of the rest of deciles as well as the performance of rest of deciles to the performance of the high NUMTRDU decile, both in terms of raw returns, Carhart s alphas, and characteristic-adjusted returns. Panel B of Table 2 shows that, on average, high NUMTRDU stocks produce 0.28% more raw returns, 0.26% more risk-adjusted returns, and 0.23% more characteristic-adjusted returns per week than the stocks in the other deciles (with t- statistics of 9.26, 11.06, and 10.46, respectively), and low NUMTRDU stocks generate 0.31% less raw returns, 0.27 less risk-adjusted returns, and 0.24 characteristic-adjusted returns per week than the stocks in the other deciles (with t-statistics of 11.60, 10.86, and 13.45, respectively). These results suggest that the positive and significant return difference between high NUMTRDU and low NUMTRDU stocks is due to both outperformance by high NUMTRDU stocks and underperformance by low NUMTRDU stocks. Overall, these results show that number of trades has a positive predictive power on future stock returns at the weekly frequency. 19

20 3.2. Bivariate portfolio-level analysis In Table 3, we report the averages of our control variables in all five NUMTRDU quintiles. The trends of the averages from the lowest to the highest NUMTRDU quintile are consistent with the pairwise correlations reported in panel B of Table 1. They show that NUMTRDU is significantly positively correlated with many well-known characteristics that predict future stock returns in the cross section. For example, stocks with a higher adjusted number of trades, on average, have a larger size (LNME), a higher momentum (MOM), higher short-term turn reversals (REVW and REVM), a larger fraction of buyer-initiated trades (BUY), a higher share turnover (TURN), and higher trading volumes (VOLU and VOLDU). Given these trends, there could be a concern that the portfolios sorted on NUMTRDU alone could still be affected by the other well-known return predictors. In the previous subsection, we use the four-factor model to calculate alphas and we use the matched portfolio method to compute characteristic-adjusted returns so that we can control for the size, market-to-book, and momentum factors. However, even with these two methods, our results could still be affected by the other factors such as short-term reversals, share turnover, etc. To further control for the other well-known return predictors, we form bivariate portfolio sorts on NUMTRDU in combination with market beta (BETA), size (LNME), book-to-market equity ratio (LNBM), momentum (MOM), illiquidity (ILLIQ), idiosyncratic volatility (IVOL), coskewness (COSKEW), five-day reversal (REVW), 21-day reversal (REVM), share turnover (TURN), fraction of buyer-initiated trades (BUY), normalized trading volume (VOLU), normalized dollar trading volume (VOLDU), effective bid-ask spread (SPRD), standardized unexpected quarterly earnings (SUE), institutional investor holdings (DISP), analyst earnings dispersion (DISP), and number of analysts covering the stock (CVRG). Our method of bivariate 20

21 portfolio sorts is as follows. For each week t, we first sort stocks into quintile portfolios based on one of the aforementioned control variables, and then further into NUMTRDU quintiles within each control variable quintile. The intersection of the control variable quintiles and the NUMTRDU quintiles yields 25 portfolios. We then group together the stocks in the same NUMTRDU quintiles and report the average quintile stock returns in week t+1, the return differential between the highest and the lowest NUMTRDU quintiles, and the corresponding four-factor alpha. We report the results based on the conditional bivariate sorts in Table 4. Panel A of Table 4 reports the results based on raw stock returns (RET). As can be seen, the predictive ability of NUMTRDU remains intact. With no exception, the average returns increase monotonically from the lowest to the highest NUMTRDU quintile. The average raw return differentials range from 0.19% per week (9.88% per year) to 0.53% per week (27.56% per year). The Newey-West adjusted t-statistics on the raw return differentials are all above 6, indicating a high statistical significance. The corresponding four-factor alphas are also significant at the 1% level based on the Newey-West adjusted t-statistics, ranging from 0.19% per week (9.88% per year) to 0.49% per week (25.48% per year). Panel B of Table 4 presents the same set of results based on characteristic-adjusted returns (RETADJ). The results are similar to those based on raw stock returns. Moving from the lowest to the highest NUMTRDU quintile, the average portfolio returns again increase in a monotonic fashion. The average return differentials between the highest and the lowest NUMTRDU quintile are all significant at the 1% level, ranging from 0.16% per week (8.32% per year) to 0.44% per week (22.88% per year). 21

22 Overall, our results based on conditional bivariate portfolio sorts show that each control variable alone fails to subsume the pricing effect of the adjusted number of trades. The positive predictive power of the adjusted number of trades on future stock returns remains significant both statistically and economically after each individual control Stock-level cross-sectional regressions While the portfolio-level analysis has an advantage of being nonparametric, it does not allow us to control for the simultaneous effects of the control variables on future stock returns. To examine whether the predictive power of NUMTRDU remains robust after simultaneous controls, we run the following weekly cross-sectional predictive regressions: (6) where is the realized weekly return on stock i in excess of the weekly return on the onemonth Treasury bills in week t+1; is the adjusted number of trades of stock i, is a vector of control variables for stock i in week t. We use the Fama-MacBeth (1977) technique for the above regression. In particular, we run a separate cross-sectional regression for each week. We then average the regression coefficients across weeks as in Fama and MacBeth (1973) and estimate the statistical inference based on the Newey-West standard errors. We report in Table 5 the results from the above weekly Fama-MacBeth regressions. We begin with the regression without any controls and report the results in column (1). We then estimate the three-factor model in column (2) that controls for the market beta (BETA), the natural logarithm of market capitalization (LNME), and the natural logarithm of book-to-market equity ratio (LNBM), followed by the four-factor model in column (3) which further controls for the price momentum as an additional factor. Finally in columns (4) and (5), we add all other 22

23 control variables simultaneously to control for the large set of cross-sectional predictors. To avoid multicollinearity, we add normalized trading volume (VOLU) and normalized dollar trading volume (VOLDU) one at a time in these two columns. Table 5 shows that the coefficients of NUMTRDU are positive in all specifications, ranging from to The coefficients are all significant at the 1% level based on the Newey- West adjusted t-statistics. They are economically significant as well. For example, consider a long-short equity portfolio that buys stocks in the top NUMTRDU quintile and sells short stocks in the bottom NUMTRDU quintile. As reported in Table 2, the average NUMTRDU is for the bottom NUMTRDU quintile and 1.13 for the top NUMTRDU quintile, a difference of 1.54 between the top and the bottom NUMTRDU quintiles. Hence, the coefficients of to reported in Table 5 imply that ceteris paribus, the long-short portfolio will on average generate an abnormal one-week ex-post stock return in the range of 0.18% per week (9.36% per year) to 0.23% per week (11.96% per year). This return magnitude is in line with the average return differentials and the corresponding four-factor alphas obtained from the univariate and bivariate portfolio studies. They again show a positive pricing effect of number of trades on ex-post stock returns. In general, the coefficients of the control variables are also consistent with the earlier studies. For example, the coefficients of LNME, IVOL, REVW, REVM, and DISP are negative in all specifications. The coefficients of LNBM, MOM, ILLIQ, VOLU, VOLDU, and SPRD are positive and significant Buyer-initiated and seller-initiated trades 23

24 In the following, we study separately the buyer-initiated trades and the seller-initiated trades. In this study, we first identify a trade as buyer- or seller-initiated based on the procedure described in Lee and Ready (1991). We then calculate the adjusted number of trades NUMTRDU separately for the buyer- and seller-initiated trades every week. We present in Panel A of Table 6 the results from univariate portfolio sorts based on the adjusted number of the buyer-initiated trades (in the left panel) and the adjusted number of the seller-initiated trades (in the right panel). Our results show that the positive pricing effect remains significant statistically and economically for both the buyer- and seller-initiated trades. For the buyer-initiated trades, the raw and characteristic-adjusted return differentials between the highest and the lowest NUMTRDU quintile portfolios are 0.43% per week (22.36% per year) and 0.33% per week (17.16% per year), respectively. Both are significant at the 1% level. The corresponding four-factor alpha is 0.39% per week (20.28% per year) with a Newey-West adjusted t-statistic of The results from the seller-initiated trades are similar to those from the buyer-initiated trades. The raw and characteristic-adjusted return differentials for the sellerinitiated trades are 0.43% per week (22.36% per year) and 0.34% per week (17.68% per year), respectively. Both are also significant at the 1% level. The corresponding four-factor alpha is 0.38% per week (19.76% per year) with a Newey-West adjusted t-statistic of We report in Panel B of Table 6 the coefficients of the adjusted number of the buyer- (in the left panel) and the seller- (in the right panel) initiated trades. The specification of the regression is the same as that in the weekly Fama-MacBeth regression (6). The coefficients of NUMTRDU based on the buyer-initiated trades are in the range of to 0.13 and they are all significant at the 1% level. These coefficients imply that ceteris paribus, a long-short portfolio strategy by buying stocks in the top buyer-initiated NUMTRDU quintile and selling short stocks in the 24

25 bottom buyer-initiated NUMTRDU quintile will produce an abnormal return ranging from 0.16% per week (8.42% per year) to 0.22% per week (11.41% per year). Similarly, the coefficients of seller-initiated NUMTRDU are also significant at the 1%, ranging from 0.10 to These coefficients imply that ceteris paribus, the long-short portfolio strategy described above will produce an abnormal return ranging from 0.15% per week (7.91% per year) to 0.22% per week (11.60% per year). Overall, our results using buyer- and seller-initiated NUMTRDU are consistent with those based on the original measure defined on all trades. The consistency holds in both univariate portfolio sorts and weekly Fama-MacBeth regressions. 4. Robustness In this section, we check the robustness of our finding. We check whether or not our results are robust to (1) eliminating the first and last 30 minutes of regular trading hours and (2) extending from the weekly to the monthly investment horizon Eliminating the first and last 30 minutes of regular trading hours First, we recalculate the adjusted number of trades NUMTRDU by eliminating all trades executed before 10:00am and after 3:30pm. We do so to alleviate the potential concern that NUMTRDU can be distorted by abnormal trading activity during the first and last 30 minutes of regular trading hours (9:30am 4:00pm). We present in Panel A of Table 7 the average returns for the NUMTRDU quintile portfolios, excluding all trades executed in the first and the last 30 minutes of regular trading hours. The average raw and characteristic-adjusted return differentials are 0.48% per week (24.96% per year) 25

26 and 0.38% per week (19.76% per year), respectively, and they are all significant at the 1% level. The four-factor alpha of the raw return differential is 0.43% per week (22.36% per year). In Panel B of Table 7, we present the coefficients of NUMTRDU estimated from the weekly Fama-MacBeth regressions. The coefficients of NUMTRDU range from to 0.154, implying that the long-short premiums are in the range of 0.195% per week (10.16% per year) to 0.24% per week (12.41% per year). Thus, our results remain qualitatively intact after eliminating trades executed in the first and the last 30 minutes of the regular trading hours. They are consistent with those based on the original measure of the adjusted number of trades Extending to the monthly investment horizon In the following, we replicate our study based on the monthly investment horizon. For each month, we count the number of trades for each individual firm. We calculate the adjusted number of monthly trades in the same way as we calculate the adjusted number of weekly trades. Panel A of Table 8 reports the average returns in the next month for the quintile portfolios formed on the adjusted number of monthly trades, the return differential between the highest and the lowest quintiles, the four-factor alpha, the characteristic-adjusted returns, and the differential in the characteristic-adjusted returns. The raw monthly return differential between the highest and the lowest NUMTRDU quintiles and the corresponding four-factor alpha are, 0.92% per month (or 11.04% per year) and 0.57% per month (or 6.84% per year), respectively. The characteristic-adjusted return differential is 0.76% per month or 9.12% per year. All the above statistics are significant at the 1% level. 26

27 In Panel B of Table 8, we present the coefficients of monthly NUMTRDU, estimated from the monthly Fama-MacBeth regressions. The coefficients of NUMTRDU range from 0.19 to 0.31, implying that the long-short premiums are in the range of 0.30% per month (3.6% per year) to 0.49% per month (5.84% per year). It is worth noting that our monthly evidence is weaker than our weekly evidence. It is consistent with the diminishing predictive power of the adjusted number of weekly trades in week t+1 through t+5, as shown in Figure Cross-sectional difference in the predictive power of number of trades There are two possibilities that number of trades increases: (1) traders could slice their trades into small chunks, or (2) more investors could participate in trading. To test whether the predictive power of number of trades is driven by slice and dice or investor participation, we study the cross-sectional differences in this predictive power. We intend to find whether the slice and dice argument or the investor participation argument better explain these cross-sectional differences. In Section 5.4, we also study a cross-sectional difference just for the intellectual curiosity Investor attention Investor participation is directly related to investor attention. The investor attention literature suggests that investors purchase only stocks that have caught their attention (see, e.g., Barber, 27