LIQUIDITY AND ASSET PRICING UNDER THE THREE-MOMENT CAPM PARADIGM. Abstract

Transcription

1 The Journal of Financial Research Vol. 30, No. 3 Pages Fall 2007 LIQUIDITY AND ASSET PRICING UNDER THE THREE-MOMENT CAPM PARADIGM Duong Nguyen University of Massachusetts, Dartmouth Suchismita Mishra and Arun Prakash Florida International University Dilip K. Ghosh Institute of Policy Analysis Abstract We examine whether the use of the three-moment capital asset pricing model can account for liquidity risk. We also make a comparative analysis of a four-factor model based on Fama French and Pástor Stambaugh factors versus a model based solely on stock characteristics. Our findings suggest that neither of the models captures the liquidity premium nor do stock characteristics serve as proxies for liquidity. We also find that sensitivities of stock return to fluctuations in market liquidity do not subsume the effect of characteristic liquidity. Furthermore, our empirical findings are robust to differences in market microstructure or trading protocols between NYSE/AMEX and NASDAQ. JEL Classification: G12 I. Introduction Studies show that liquidity is an important attribute of an asset that investors consider when making investment decisions. Less liquid stocks demand a higher rate of return than more liquid stocks. Amihud and Mendelson (1986) formalize this intuition and propose a positive return illiquidity relation. Since then, numerous studies investigate this relation. The evidence, however, is not unanimous. 1 We are extremely indebted to Gerald Gay, the editor of this journal; Gady Jacoby, Faculty of Management University of Manitoba Winnipeg; and Gordon V. Karels, University of Nebraska, Lincoln; and the participants of Finance Workshop of the College of Business Administration at Florida International University for their many helpful suggestions and comments while the article was in preparation. All remaining errors and omissions, however, are ours. 1 Some studies find evidence supporting the liquidity premium theory (e.g., Amihud and Mendelson 1986; Datar, Naik, and Radcliffe 1998; Amihud 2002; Chan and Faff 2005). Other studies, however, find inconsistent results. For example, Fama and French (1992) argue that liquidity is important but the combination 379

2 380 The Journal of Financial Research Brennan and Subrahmanyam (1996) suggest that one explanation for the mixed results is that the asset pricing models used in those studies do not adjust for risk adequately. For example, the three-factor model of Fama and French (1993) is used widely as a model for risk adjustment, but its validity is questioned. 2 Furthermore, many studies show that the three-moment capital asset pricing model (three-moment CAPM) fits the return distribution better than does the traditional two-moment CAPM (e.g., Samuelson 1970; Kraus and Litzenberger 1976; Harvey and Siddique 2000; Chung, Johnson, and Schill 2006). Based on these findings, one of the major goals of our study is to investigate whether liquidity still needs to be incorporated in asset pricing if the three-moment CAPM is used as a model for risk adjustment. Another goal is to examine whether well-known determinants of stock returns such as size, book-to-market ratio, and the Fama French factor loadings can explain the liquidity premium. These variables are commonly known to capture variations in stock return. Limited evidence, however, is available about the relation of these variables and liquidity. 3 Using turnover ratio as a measure for liquidity, we find evidence supporting the findings of Amihud and Mendelson (1986), Amihud (2002), and others in favor of the liquidity premium theory. More specifically, we find that an increase of 1% in the turnover rate is associated with a reduction in the rate of return of about 3 to 4 basis points per month, on average. We also show that the Fama French factors, and the well-known stock characteristics of size and book to market ratio do not capture the liquidity premium. Based on the findings of Pástor and Stambaugh (2003) and Acharya and Pedersen (2005), it is accepted that marketwide liquidity is a state variable and is important for asset pricing. That is, stocks with higher sensitivities to market liquidity demand a higher required rate of return than those with low sensitivities to market liquidity. However, few studies investigate whether the liquidity beta (i.e., sensitivity of returns to market liquidity) can subsume the effect of characteristic liquidity in the spirit of Amihud and Mendelson (1986). We address this issue and find that the sensitivities of stock returns to Pástor Stambaugh market liquidity factor do not proxy for the effect of characteristic liquidity. Our results are consistent in both time-series and cross-sectional framework as well as robust to differences in of size and book-to-market ratio can subsume the effect of liquidity. Brennan and Subrahmanyam (1996) do not document that the liquidity premium is significant when using the fixed-cost component of transaction cost to proxy for liquidity. According to Eleswarapu and Reinganum (1993), the liquidity premium is confined only to January. 2 Fama and French (1993, 1995, 1998) present evidence that largely supports the three-factor model. Several studies do not reach the same agreement, however (e.g., Kothari, Shanken, and Sloan 1995; Kim 1995; Chung, Johnson, and Schill 2006). 3 Fama and French (1993) argue that factor loadings explain stock return, whereas Daniel and Titman (1997) argue that stock characteristics such as size and book-to-market, not factor loadings, explain stock returns.

3 Liquidity and Asset Pricing 381 market microstructure or trading protocols between NYSE/AMEX and NASDAQ exchanges. II. Prior Research Liquidity Literature Amihud and Mendelson (1986) are one of the first to examine the role of liquidity in asset pricing. They use the bid ask spread as a proxy for illiquidity and find a positive relation between expected stock return and illiquidity. Brennan and Subrahmanyam (1996) examine the liquidity premium by separating transaction cost into variable- and fixed-cost components. They find a concave relation between return and variable transaction costs. However, the relation between asset returns and estimated fixed costs is convex, which is not consistent with Amihud and Mendelson s findings. They attribute this conflict to either a poor proxy for liquidity or an incomplete adjustment for risk in the Fama French model. The concerns about the bid ask spread being a poor proxy for liquidity in the studies of Peterson and Fialkowski (1994) and Brennan and Subrahmanyam (1996) lead to many other investigations of liquidity and asset returns using alternative measures of liquidity, such as trading volume (Brennan, Chordia, and Subrahmanyam 1998), turnover ratio (Datar, Naik, and Radcliffe 1998; Chan and Faff 2005), and the illiquidity ratio (Amihud 2002). In general, these studies support the liquidity premium notion as in Amihud and Mendelson (1986). Recently, the literature has shifted its focus from liquidity as a characteristic of a stock to liquidity as an aggregate risk factor. For example, Jacoby, Fowler, and Gottesman (2000) develop a static one-period CAPM-based model to demonstrate that the true measure of systematic risk, when considering liquidity costs, is based on net (after bid ask spread) returns. Pástor and Stambaugh (2003) document that stocks whose returns are more sensitive to the market liquidity factor demand higher required returns than stocks that are less sensitive to the market liquidity factor. Acharya and Pedersen (2005) decompose Jacoby, Fowler, and Gottesman s liquidity-adjusted beta into four components: the standard CAPM beta and three additional betas that capture commonality in liquidity with market liquidity, return sensitivity to market liquidity, and liquidity sensitivity to market return. Using an illiquidity measure as in Amihud (2002), they find that their model significantly improves the performance of a standard CAPM. Keene and Peterson (2007) find liquidity to be an important factor affecting portfolio returns, even after the effects of the market, size, book-to-market equity, and momentum are considered. Three-Moment CAPM and Liquidity The three-moment CAPM was first derived and tested empirically by Kraus and Litzenberger (1976). The intuition behind the model is that risk-averse investors

4 382 The Journal of Financial Research who want to maximize their expected utility would choose, ceteris paribus, higher expected return to lower return, lower variance to higher variance, and higher (and positive) than lower (and negative) coskewness. Therefore, investors are willing to accept a lower expected return for higher positive coskewness if the market is also positively skewed. Kraus and Litzenberger assume that security returns do not follow symmetric distributions and derive the three-moment CAPM as follows: R i R F = b 0 + b 1 β i + b 2 γ i, (1) where b 1 and b 2 are, respectively, the prices of covariance (β i ) and coskewness (γ i ) risk, and b 0 is the intercept term. Recently, Harvey and Siddique (2000) show that conditional skewness helps explain the cross-sectional variation of expected returns. Chung, Johnson, and Schill (2006) suggest that Fama French factors appear to proxy for higher order comoments. Chen, Hong, and Stein (2001) document a relation between skewness and the relative change in the turnover ratio. These findings lead to a pertinent question from the perspective of asset pricing. Does coskewness risk capture liquidity? If it does, liquidity should not be significant in explaining the cross-section of asset returns when the three-moment CAPM is used to adjust for risk. If liquidity preference is still significant despite the use of the three-moment CAPM, Amihud and Mendelson s (1986) argument holds. Therefore, examination of the relation between a higher order comoment (i.e., coskewness), in addition to covariance risk and liquidity, will resolve the adequacy or inadequacy of asset pricing models in explaining liquidity risk. Time-Series Test III. Method The motivation for time-series testing is to investigate whether the asset pricing model can explain stock returns after controlling for liquidity. If the time-series intercepts are jointly equal to zero (not zero), the model captures (does not capture) the liquidity effect. We control for liquidity by sorting stocks into liquidity portfolios. To test whether the intercepts are jointly equal to zero, we use the test developed by Gibbons, Ross, and Shanken (1989). Portfolio-Formation Procedure. Using NYSE and AMEX stock data, each year we construct 25 portfolios, each based on size and turnover ratio, book-tomarket and turnover ratios, and the turnover ratio only. Specifically, at the end of each calendar year from 1963 to 2004, we rank all the U.S. common stocks listed on the NYSE and AMEX by market capitalization and divide the sample into five portfolios of equal size. We define the annual turnover ratio for each stock in

5 Liquidity and Asset Pricing 383 the sample as the average number of shares traded during the year divided by the average number of shares outstanding during the same period. Using the definition of the annual turnover ratio as described previously, we can measure the liquidity of each security for each calendar year. After computing the turnover ratios, we assign each stock to one of the five portfolios with an equal number of securities, within each size quintile, based on its annual turnover ratios. After forming portfolios using the size and turnover ratio criteria, the portfolio-formation procedure is repeated using book-to-market value and turnover ratio. For this sorting procedure, all common stocks in NYSE and AMEX from 1963 to 2004 are ranked by the beginning-of-period book-to-market ratio and then divided into five portfolios of equal size. Within each book-to-market quintile, each stock is assigned to one of five portfolios of an equal number of securities based on annual turnover rates. Finally, all the NYSE and AMEX stocks are sorted into 25 portfolios based on their turnover ratios alone. Because the pre-sorting on both size and bookto-market ratio may be interpreted as liquidity portfolios controlled by size and book-to-market value, as a check for robustness, the analysis is further conducted with 25 portfolios sorted by turnover ratios only. Using the portfolios constructed previously, the equally weighted monthly returns are computed for each of the 25 portfolios. The 30-day Treasury bill yield is subtracted to obtain the excess portfolio return. The portfolios are rebalanced every year from 1963 to Time-Series Regressions. The ordinary least squares (OLS) time-series regressions are estimated for each of the 25 portfolios (sorted by size and turnover ratio, by book-to-market ratio and turnover ratio, and by turnover ratio alone) on the four- factor model as follows: r(i,t) = α i + β i (R mt R ft ) + δ i SMB t + γ i HML t + ψ i LIQ t + e it, (2) where r(i,t) is the excess return on portfolio i in month t; (R mt R ft ), SMB t, and HML t are the Fama and French (1993) three factors related to market premium, firm size, and book-to-market ratio; and LIQ t is the Pástor and Stambaugh (2003) liquidity factor in month t. 4 For the three-moment model, the OLS time-series regressions are estimated for each portfolio using the quadratic characteristics line model: r(i,t) = C 0i + C 1i (R mt R ft ) + C 2i (R mt R m ) 2 + e it, (3) 4 We are grateful to the Wharton Research Database Services for providing the data on the Pástor and Stambaugh (2003) market liquidity factor.

6 384 The Journal of Financial Research where r(i,t) is the excess return on portfolio i in month t; R mt and R ft are the market return and risk-free rate in month t, respectively; and R m is the average stock market return. Finally, to take into account all of the risk factors, we perform the timeseries regressions for each portfolio using the five-factor model (including the four factors from equation (2) and the coskewness factor in equation (3)) as follows: r(i,t) = α i + β i (R mt R ft ) + δ i SMB t + γ i HML t + ψ i LIQ t + φ i (R mt R m ) 2 + e it. (4) We test the null hypothesis that liquidity, if proxied by the turnover ratio, has no effect on expected stock returns and that the intercepts in these time-series regressions are jointly equal to zero using the Gibbon, Ross, and Shanken (1989) F-test. 5 Cross-Sectional Test We use cross-sectional regressions to directly investigate the relation between liquidity and stock returns after controlling for other variables. In particular, for each month t in the sample period, we perform cross-sectional regressions as follows: Liquidity and stock characteristics: R it = γ 0t + γ 1t Beta + γ 2t Size + γ 3t BM + γ 4t Turnover + ε it (5) Liquidity and market factors: covariance (beta) and coskewness (gamma): R it = γ 0t + γ 1t Beta + γ 2t Gamma + γ 3t Turnover + ε it (6) Liquidity and factor loadings: R it = γ 0t + γ 1t F Rm Rf + γ 2t F SMB + γ 3t F HML + γ 4t F LIQ + γ 5t turnover + ε it, (7) where Beta, Gamma, Size, BM, and Turnover are, respectively, the market beta, market coskewness, market value of equity, book-to-market ratio of firm i, and turnover ratio as a proxy for the liquidity of the firm. F Rm Rf,F SMB, and F HML are the factor loadings of firm i on the Fama French common factors, and F LIQ is the factor loading of firm i on the Pástor Stambaugh market liquidity factor. The coefficients from the cross-sectional regressions are averaged over time using the Litzenberger and Ramaswamy (1979) method. This method weights 5 See Gibbons, Ross, and Shanken (1989) for details on the test procedure.

7 Liquidity and Asset Pricing 385 the coefficients by their precision when summing across the cross-sectional regressions and thus corrects for the inefficiency under time-varying volatility with the standard Fama MacBeth (1973) procedure. 6 Our data set consists of all stocks on the NYSE and AMEX from January 1963 to December Monthly data on returns are collected from the Center for Research in Security Prices (CRSP) and the book values are extracted from the Compustat tapes. We measure the turnover ratio of every stock as suggested by Datar, Naik, and Radcliffe (1998). In particular, for each month t, we calculate the average monthly trading volume (the average number of shares traded during the previous three months (i.e., months t 3, t 2, t 1) and divide it by the number of shares outstanding at month t. 7 If the number of shares outstanding in a stock changes because of stock splits, we exclude that stock for three months. We construct the book-to-market variable (natural logarithm of book value to market value for individual firms) as suggested by Fama and French (1992). We define the log of firm size as the natural logarithm of total market capitalization of firm i, at the end of the prior month (month t 1). In our sample, the book-to-market variable has a minimum value of 7.81 and a maximum value of 4.40 with a mean of The size variable ranges from 5.46 to with a mean of We estimate betas and gamma for each security similar to Amihud and Mendelson (1986). First, at the end of each year, we sort all stocks into 25 portfolios based on their individual beta rankings. Once the portfolios are formed, we estimate betas and gammas for each portfolio and assign the betas and gammas of the portfolio to all the stocks in that portfolio. Thus, we eliminate potential measurement errors that may occur if we estimate betas and gammas at the individual firm level. IV. Results We first estimate the intercepts from the time-series regressions using the four-factor and three-moment models for our 25 portfolios sorted by size and the turnover ratio, by the book-to-market ratio and the turnover ratio, and by turnover only. We argue that if liquidity has no effect on expected stock returns, the 25 intercepts from the regressions should not be jointly different from zero. Tables 1, 2, and 3 report the results. We document a consistent pattern in the intercepts in Tables 1 and 2. The intercepts are consistently decreasing from the lowest liquidity groups (group 1) 6 See Litzenberger and Ramaswamy (1979) for details on the procedure. Briefly, instead of putting equal weight on all monthly slope coefficients as in the traditional Fama MacBeth procedure, the Litzenberger Ramaswamy procedure places more (less) weight on parameters that are estimated more (less) precisely. The weights are inversely proportional to the variances of the parameter estimates. 7 We also compute turnover ratio on the basis of 6- and 12-month trading volume. The results are qualitatively similar.

8 386 The Journal of Financial Research TABLE 1. Intercepts from Time-Series Regressions of the 25 Portfolios Sorted by Size and Turnover Ratio for the Four-Factor and Three-Moment Models. Liquidity Group SizeGroup Panel A. Four-Factor Model (2.57) (2.07) (1.23) ( 0.31) ( 1.58) ( 4.87) (0.91) ( 0.26) ( 0.89) ( 2.51) ( 7.32) ( 7.50) (1.98) ( 0.38) ( 1.50) ( 3.84) ( 5.95) ( 6.34) (0.68) (0.29) ( 0.69) ( 2.12) ( 3.47) ( 3.77) ( 0.05) ( 0.26) (0.38) ( 0.75) ( 2.31) ( 1.79) F-value for Gibbons, Ross, and Shanken test that the intercepts are jointly equal to zero is 4.51% Panel B. Three-Moment Model (4.47) (4.42) (3.79) (2.47) (1.12) ( 4.34) (4.14) (3.49) (3.09) (1.72) ( 1.31) ( 5.76) (5.03) (3.00) (2.46) (0.89) ( 0.91) ( 5.02) (3.74) (3.03) (2.17) (1.50) ( 0.55) ( 3.18) (0.73) (0.34) (1.34) (0.71) ( 1.55) ( 1.53) F-value for Gibbons, Ross, and Shanken test that the intercepts are jointly equal to zero is 4.52% Note: This table reports the value of the intercepts obtained in the four-factor and three-moment models for 25 portfolios of NYSE and AMEX stocks sorted according to size and average turnover ratio. The average turnover ratio is defined as the average number of shares traded divided by average number of shares outstanding during the year. Portfolios are formed yearly for Within each calendar year, all stocks in the sample are allocated into five size portfolios based on their market equity ranking. Each size quintile is then subdivided into five liquidity portfolios using the average turnover ratio. Panel A presents intercepts from the time-series regression of four-factor model as in the following equation r(i,t) = α i + β i (R mt R ft ) + δ i SMB t + γ i HML t + ψ i LIQ t + e it, where r(i,t) is the excess return on portfolio i in month t, and (R mt R ft ), SMB t, and HML t, are the Fama and French (1993) factors related to market premium, firm size, and the book-to-market ratio in month t. LIQ t is the Pástor and Stambaugh (2003) liquidity factor. Panel B presents intercepts from the time-series regression of the three-moment model as in the following equation: r(i,t) = C 0i + C 1i (R mt R ft ) + C 2i (R mt R m ) 2 + e it, where r(i,t) is the excess return on portfolio i in month t, and R mt,r ft, are market return and the risk free rate at month t. R m is the average stock market return. The last column represents the difference between liquidity group 5 (highest liquidity group) and liquidity group1 (lowest liquidity group). The bottom of each panel presents the Gibbons, Ross, and Shanken (1989) test of the hypothesis that the intercepts jointly equal zero for the four-factor model and three-moment model. Intercepts are reported in percentage terms (t-statistics are in parentheses). Significant at the 1% level. Significant at the 10% level.

9 Liquidity and Asset Pricing 387 TABLE 2. Intercepts from Time-Series Regressions of the 25 Portfolios Sorted by Book-to-Market and Turnover Ratios for the Four-Factor and Three-Moment Models. Liquidity Group Book-to-Market Group Panel A. Four-Factor Model (0.13) (0.18) (0.02) ( 1.65) ( 3.29) ( 2.82) (0.83) ( 1.06) ( 1.40) ( 1.21) ( 2.02) ( 2.16) (0.62) (0.49) ( 0.44) ( 1.31) ( 3.50) ( 3.09) (2.99) (0.57) (2.04) ( 0.39) ( 1.55) ( 3.10) (3.61) (2.58) (1.95) (0.75) ( 1.09) ( 4.34) F-value for Gibbons, Ross, and Shanken test that the intercepts are jointly equal to zero is 2.07 Panel B. Three-Moment Model (2.14) (1.63) (0.46) ( 0.42) ( 2.42) ( 3.62) (4.09) (1.79) (1.49) (1.73) (0.65) ( 2.13) (3.88) (3.52) (2.72) (2.32) (0.86) ( 2.18) (5.17) (3.03) (4.33) (2.78) (2.21) ( 2.36) (5.68) (4.82) (4.54) (2.71) (2.20) ( 4.62) F-value for Gibbons, Ross, and Shanken test that the intercepts are jointly equal to zero is 3.18 Note: This table reports the value of the intercepts obtained in the four-factor and three-moment models for 25 portfolios of NYSE/AMEX stocks sorted according to book-to-market and average turnover ratio. The average turnover ratio is defined as the average number of shares traded divided by average number of shares outstanding during the year. Portfolios are formed yearly for Within each calendar year, all stocks in the sample are allocated into five portfolios based on their book-to-market ratios ranking. Each book-to-market quintile is then subdivided into five liquidity portfolios using the average turnover ratio. Panel A presents intercepts from the time-series regression of four-factor model as in the following equation r(i,t) = α i + β i (R mt R ft ) + δ i SMB t + γ i HML t + ψ i LIQ t + e it, where r(i,t) is the excess return on portfolio i in month t, and (R mt R ft ), SMB t, HML t, are the Fama and French (1993) factors related to market premium, firm size, and book-to-market ratio in month t. LIQ t is the Pástor and Stambaugh (2003) liquidity factor. Panel B presents intercepts from the time-series regression of the three-moment model as in the following equation r(i,t) = C 0i + C 1i (R mt R ft ) + C 2i (R mt R m ) 2 + e it where r(i,t) is the excess return on portfolio i in month t, and R mt,r ft, are market return and the risk-free rate at month t. R m is the average stock market return. The last column represents the difference between liquidity group 5 (highest liquidity group) and liquidity group1 (lowest liquidity group). The bottom of each panel presents the Gibbons, Ross, and Shanken (1989) test of the hypothesis that the intercepts jointly equal zero for the four-factor model and three-moment model. Intercepts are reported in percentage terms (t-statistics are in parentheses). Significant at the 1% level. Significant at the 5% level.

10 388 The Journal of Financial Research TABLE 3. Intercepts from Time-Series Regressions of the 25 Portfolios Sorted by Turnover Ratio for the Three-Moment and Four-Factor Models. Turnover-Sorted Group Panel A. Three-Moment Model (5.47) (4.22) (4.62) (4.92) (3.54) (3.15) (2.98) (1.93) (1.86) (1.69) (0.04) ( 0.57) ( 1.98) F-value for Gibbons, Ross, and Shanken test that the intercepts are jointly equal to zero is 3.40 Panel B. Four-Factor Model (2.78) (0.77) (0.94) (1.93) (0.95) (0.13) ( 0.04) ( 1.60) ( 1.95) ( 2.24) ( 4.49) ( 4.64) ( 5.04) F-value for Gibbons, Ross, and Shanken test that the intercepts are jointly equal to zero is 3.15 Note: This table reports the value of the intercepts obtained for the three-moment and the four-factor models for 25 portfolios of NYSE/AMEX stocks sorted according to average turnover ratio. To save space, only the odd-numbered portfolios are reported. The average turnover ratio is defined as the average number of shared trading divided by average number of shares outstanding during the year. Portfolios are formed yearly for Within each calendar year, all stocks in the sample are allocated into 25 portfolios based on their average turnover ratio ranking. Panel A presents intercepts from the time-series regression of three-moment model as in the following equation r(i,t) = C0i + C1i (Rmt Rft) + C2i (Rmt Rm) 2 + eit, where r(i,t) is the excess return on portfolio i in month t, and Rmt and Rft are market return and the risk-free rate at month t. Rm is the average stock market return. Panel B presents intercepts from the time-series regression of the four-factor model as in the following equation r(i,t) = αi + βi (Rmt R ft) + δi SMBt + γi HMLt + ψi LIQ t + e, where r(i,t) is the excess return on portfolio i in month t, and (Rmt R ft), SMBt, and HMLt, are Fama and French (1993) factors related to market premium, firm size, and the book-to-market ratio in month t. LIQt is the Pástor and Stambaugh (2003) liquidity factor. The bottom of the each panel presents the Gibbons, Ross, and Shanken (1989) test of the hypothesis that the intercepts jointly equal zero for the three-moment and four-factor models. Intercepts are reported in percentage terms (t-statistics are in parentheses). Significant at the 1% level.

11 Liquidity and Asset Pricing 389 to the highest liquidity groups (group 5) except in one case. 8 These results imply that more liquid stocks demand higher expected returns than less liquid stocks after controlling for risk in the four-factor and three-moment models. As can also be seen in Tables 1 and 2, within each size or book-to-market group, we document a consistent decrease in the intercepts from low-liquidity to high-liquidity portfolios. This finding suggests that size and book-to-market ratio do not relate to liquidity. As a robustness check, we perform the same analysis for the 25 portfolios sorted by turnover only. The results are strikingly similar, as can be seen in Table 3. The results here also indicate that the systematic liquidity measure of Pástor and Stambaugh (2003) does not explain characteristic liquidity in the spirit of Amihud and Mendelson (1986). If the liquidity beta subsumes the liquidity level per se, we should not see systematic differences in the intercepts from the timeseries regressions for liquidity portfolios. However, the evidence in Tables 1, 2, and 3 shows a monotonic decrease in the intercepts from low-liquidity to high-liquidity portfolios. The differences in the intercepts between the highest liquidity and lowest liquidity groups are statistically significant and negative. The Gibbon, Ross, and Shanken (1989) statistics are also reported in Tables 1, 2, and 3. In all cases, the F-tests strongly reject the null hypothesis that intercepts are jointly equal to zero for both the four-factor and three-moment models at the 1% level. To investigate whether the liquidity premium remains after taking into account all risk factors namely, the Fama French factors, the Pástor Stambaugh market liquidity factor, and the coskewness factor we perform time-series regressions for the 25 portfolios using equation (4). 9 The results are similar (see Table 4). The intercepts consistently decrease from the low-liquidity to the high-liquidity groups in all cases. The Gibbon, Ross, and Shanken statistics again reject the null hypothesis that all the intercepts are jointly equal to zero for portfolios sorted by size and turnover, book-to-market and turnover, and turnover alone. 10 Overall, our result, on the one hand, confirms Brennan and Subrahmanyam s (1996) finding that the Fama French factors do not proxy for liquidity effects. On the other hand, our result contributes to the literature because we show that the second and third systematic comoments do not proxy for liquidity either. Furthermore, we document that the sensitivities of stock returns to the market liquidity factor in Pástor and Stambaugh (2003) do not subsume the characteristic liquidity effect according to Amihud and Mendelson (1986). 8 See Table 1, Panel B, row 5. However, this is the only case where the intercept is not decreasing although the difference between the intercepts of portfolio 5 (most liquid) and 1 (least liquid) is negative but not statistically significant. 9 We are grateful to the referee for bringing this point to our attention. 10 The result for 25 portfolios sorted by turnover alone is not reported for space consideration but is available on request.

12 390 The Journal of Financial Research TABLE 4. Intercepts from Time-Series Regressions of the 25 Portfolios Sorted by Size and Turnover Ratio, Book-to-Market and Turnover Ratio for the Five-Factor Model. Panel A. Portfolios Sorted by Size and Turnover Liquidity Group Size Group (1.60) (1.51) (0.72) ( 0.90) ( 1.94) ( 4.28) ( 0.13) ( 1.18) ( 1.58) ( 3.44) ( 7.39) ( 6.88) (1.49) ( 2.19) ( 2.49) ( 4.57) ( 5.89) ( 6.00) ( 0.13) ( 1.31) ( 2.41) ( 2.63) ( 3.45) ( 3.27) ( 1.05) ( 2.17) ( 1.28) ( 1.88) ( 2.88) ( 1.71) F-value for Gibbons, Ross, and Shanken test that the intercepts are jointly equal to zero is 3.91 Panel B. Portfolios Sorted by Book-to-Market and Turnover Liquidity Group Book-to-Market Group (0.39) (0.39) ( 0.74) ( 1.87) ( 4.19) ( 3.72) (1.17) ( 1.52) ( 1.81) ( 1.71) ( 1.97) ( 2.32) (0.39) ( 1.05) ( 1.91) ( 1.86) ( 3.19) ( 2.70) (1.70) ( 1.87) ( 0.00) ( 1.97) ( 1.83) ( 2.52) (2.67) (1.53) (0.91) ( 1.85) ( 1.89) ( 4.18) F-value for Gibbons, Ross, and Shanken test that the intercepts are jointly equal to zero is 2.60 Note: This table reports the value of the intercepts obtained by the five-factor model for 25 portfolios of NYSE/AMEX stocks sorted according to size and turnover ratio, book-to-market and turnover ratio. The turnover ratio is defined as the average number of shares trading divided by average number of shares outstanding during the year. Portfolios are formed yearly for Each panel presents intercepts from the time series regression of five-factor model as in the following equation r(i,t) = α i + β i (R mt R ft ) + δ i SMB t + γ i HML t + ψ i LIQ t + φ i (R mt R m ) 2 + e it, where r(i,t) is the excess return on portfolio i in month t, and (R mt R ft ), SMB t, HML t, are the Fama and French (1993) factors related to market premium, firm size, and the book-to-market ratio in month t. LIQ t is the Pástor and Stambaugh (2003) liquidity factor. R mt and R ft are market return and the risk-free rate at month t. R m is the average stock market return. The last column represents the difference between liquidity group 5 (highest liquidity group) and liquidity group1 (lowest liquidity group). Panel A reports the results for 25 portfolios sorted by size and turnover. Panel B reports the results for 25 portfolios sorted by book-to-market and turnover. The bottom of each panel presents the Gibbons, Ross, and Shanken (1989) test of the hypothesis that the intercepts from times-series regressions are jointly equal zero. Intercepts are reported in percentage terms (t-statistics are in parentheses). Significant at the 1% level. Significant at the 5% level. Significant at the 10% level.

13 Liquidity and Asset Pricing 391 TABLE 5. Summary Statistics. Standard 25th 50th 75th Variable Mean Deviation Minimum Percentile Percentile Percentile Maximum Book-to-market Turnover Size Beta Gamma Return Note: This table reports basic statistics on variables of concern for NYSE and AMEX stocks from 1963 to Return is the percentage change over one month in the value of $1 of investment in a common stock. Book-to-market is the natural logarithm of the book-to-market ratio. Size is the natural logarithm of the firm s total market capitalization in the prior month. Turnover is computed as the average monthly trading volume during the previous three months divided by number of shares outstanding. Beta and Gamma are the firm s assigned portfolio beta and gamma, respectively. In Tables 5 and 6 we report the basic statistics of the variables of interest. Table 5 reports the basic summary statistics such as mean, standard deviations, minimum, and maximum. In Table 6, we present the correlations among turnover, stock characteristics, beta, and gamma. Panel A reveals that return is positively correlated with the book-to-market ratio but negatively correlated with size, beta, and turnover ratio. Size is strongly and negatively correlated with the book-tomarket ratio whereas beta is strongly and positively correlated with the turnover ratio. Their correlations are 0.35 and 0.26, respectively. Panel B reveals that return is negatively correlated with all other variables, although the level of correlation with gamma is smaller than that of the other variables. Beta and gamma have a correlation of 0.64, which is to be expected as they are both comoments of security return with market return. The turnover ratio has correlation of 0.26 and 0.16 with beta and gamma, respectively. Table 7 summarizes the results of our cross-sectional regressions of stock returns on turnover after controlling for various stock characteristic variables, the Fama French and Pástor Stambaugh factor loadings, and market risk factors (i.e., beta and gamma). We find that the turnover ratio is significantly and negatively related to stock returns in all models. The negative sign implies that stocks with low turnover ratios will demand higher returns than stocks with high turnover ratios. This result confirms the liquidity premium notion as previously found in the literature. We next perform several regressions of stock returns on well-known determinants of stock returns with and without the turnover ratio. We find that the magnitude and direction of the coefficient, when estimated without the turnover ratio, on some variables are consistent with previous findings (specifically, Datar, Naik, and Radcliffe 1998; Brennan and Subrahmanyam 1996 findings for Fama French factor models). In particular, the coefficient on size is negative ( 0.009) and

14 392 The Journal of Financial Research TABLE 6. Simple Correlations. Panel A. Correlations Among Turnover and Stock Characteristics Return Beta Book-to-Market Size Turnover Return Beta Book-to-market Size Turnover Panel B. Correlations Among Return, Beta, Gamma, and Turnover Return Beta Gamma Turnover Return Beta Gamma Turnover Panel C. Correlation Between Gamma Residual and Return, Beta, and Turnover Return Beta Turnover Gamma residual Note: This table reports time-series averages of monthly cross-sectional correlations of variables in asset pricing tests for all NYSE and AMEX stocks from 1963 to Return is the percentage change over one month in the value of $1 of investment in a common stock. Book-to-market is the natural logarithm of the book-to-market ratio. Size is the natural logarithm of the firm s total market capitalization at the prior month. Turnover is computed as the average monthly trading volume during the previous three months divided by number of shares outstanding. Beta and Gamma are the firm s assigned portfolio beta and gamma, respectively. Gamma residual is the residual from the cross-sectional regression of gamma on beta. significant (t-statistic = 13.21), whereas the coefficient of the book-to-market variable is positive (0.0043) and significant (t-statistic = 23.27). In the case of the three-moment CAPM, gamma is significantly and negatively correlated with stock return (t-statistic = 3.46). This confirms the skewness preference as found in Kraus and Litzenberger (1976). We find the slope on beta is negative and significant (t-statistic = 16.87), which is similar to Datar, Naik, and Radcliffe (1998). Those authors claim that the negative sign may come from measurement errors while estimating the betas for securities. We then perform several multivariate regressions of stock return on its determinants to examine the effect of liquidity while controlling for other variables. In all cases, the turnover ratio remains negative and significant in the presence of other variables. In particular, in the presence of size, the t-statistic of the coefficient on turnover ratio is 12.50, and the t-statistic in the presence of the book-tomarket variable is also negative and significant ( 13.07). When size, beta, and book-to-market ratio are all present, the turnover ratio remains highly significant (t-statistic = 11.14). The results are similar when controlling for beta and gamma. The turnover ratio is remains negatively and significantly correlated with stock

15 Liquidity and Asset Pricing 393 returns in the presence of beta or gamma or both. Furthermore, the strength of this relation (magnitude and significance of the coefficient) is not affected by the presence of stock characteristics (size and book-to-market ratio) or market factors (covariance and coskewness). This implies that liquidity is a distinct effect. TABLE 7. Average Slopes of Monthly Cross-Sectional Regressions. Panel A. Turnover and Characteristic Variables: Book-to-Market, Size, and Beta Constant Turnover Book-to-Market Size Beta (19.16) ( 15.80) (20.72) (23.27) (15.30) ( 13.21) (22.86) ( 16.87) (22.10) ( 13.29) ( 12.40) (21.64) ( 13.07) (21.29) (17.01) ( 12.50) ( 12.50) (9.46) ( 11.14) (17.29) ( 4.73) (15.22) ( 9.15) (15.21) ( 7.38) ( 10.51) Panel B. Turnover and Factor Loadings on Four Factors Constant Turnover SMB HML R m R f LIQ (15.44) ( 9.28) ( 12.03) (14.40) ( 12.54) ( 0.23) (14.54) ( 14.60) (16.23) ( 16.00) ( 1.08) ( 4.53) ( 14.59) (4.97) ( 11.76) (11.96) (4.74) (2.92) ( 8.99) (4.70) (7.00) ( 10.01) (2.47) (15.62) ( 9.64) ( 11.43) (14.62) ( 12.73) (2.45) ( 13.80) (10.42) (4.28) (10.26) ( 11.62) (12.55) ( 9.96) (1.08) (15.59) ( 11.92) ( 8.78) ( 8.42) (1.48) (Continued)

16 394 The Journal of Financial Research TABLE 7. Continued. Panel C. Turnover, Beta, and Gamma Constant Turnover Beta Gamma (22.10) ( 13.29) ( 12.40) (12.01) ( 3.46) (21.48) ( 7.51) (1.40) (14.92) ( 14.03) ( 2.20) (20.83) ( 13.29) ( 4.98) (1.29) Panel D. Turnover, Beta, and Gamma Residual Constant Turnover Beta Gamma Residual (22.00) ( 13.29) ( 12.41) (1.29) (22.76) ( 16.88) (1.40) (18.79) ( 15.81) (1.40) Note: This table reports average slopes of monthly cross-sectional regressions of returns on turnover ratio using monthly individual security data of NYSE and AMEX stocks from 1963 to 2004 after controlling for stock characteristics as well as for market factors. In each month, a cross-sectional regression is estimated wherein the individual stock return is the dependent variable and the explanatory variable set comprises various combinations of the turnover ratio with other variables corresponding to each asset pricing model. Book-to-market is the natural logarithm of the book-to-market ratio. Size is the natural logarithm of the firm s total market capitalization at the prior month. Turnover is computed as the average monthly trading volume during the previous three months divided by number of shares outstanding. Beta and Gamma are the firm s assigned portfolio beta and gamma, respectively. (R mt R ft ), SMB t, and HML t are the Fama and French (1993) three factors related to market premium, firm size, and book-to-market ratio, and LIQ t is the Pástor and Stambaugh (2003) liquidity factor in month t. The generalized least squares estimates of average slopes and associated t-statistics (in parentheses) are calculated using the Litzenberger and Ramaswamy (1979) procedure. Panel A presents the results for turnover and size, book-to-market ratio, beta. Panel B presents the results for turnover and factor loadings on three common factors of Fama-French (1993) and the Pástor and Stambaugh (2003) market liquidity factor. Panel C presents the results for turnover and market factors (beta, gamma) in the three-moment model. Panel D presents the results for turnover, beta, and the gamma residual (calculated from cross-sectional regression). Note that we consider only firm characteristics such as size and the book-tomarket ratio, not the Fama French and Pástor Stambaugh market liquidity factor loadings. Fama and French (1993) argue that it is the factor loadings on the common factors that explain variation in stock returns, not the firm s characteristic variables. In fact, the characteristics are proxies for the nondiversifiable factor risk. However, according to Daniel and Titman (1997), it is the characteristics rather than the comovement between stock returns and common factors (i.e., factor loadings) that

17 Liquidity and Asset Pricing 395 explain the cross-sectional variation in stock returns. To test this conjecture, we compute the factor loadings for each stock by regressing the stock returns over the past five years on the factors (R m R f ), SMB, HML, and LIQ according to model (7) and then perform a cross-sectional regression of stock returns on these factor loadings and the turnover ratio. The results are reported in Table 7, Panel B. The findings are similar to those in Panel A a liquidity premium remains even after controlling for Fama French and Pástor Stambaugh factors. More interesting, we find the significant level of the turnover ratio remains almost the same in all cases. The result implies the sensitivities of stock returns to Fama French common factors and market liquidity do not proxy for the characteristic liquidity effect as in Amihud and Mendelson (1986). Another finding is also noteworthy. The correlation between beta and gamma is high (0.64) as reported in Table 6. This may be the reason gamma is not priced when beta is present in Panel C. To remove the effect of multicollinearity, we orthogonalize gamma on beta in the cross-sectional analysis. For each month, gamma is regressed on beta cross-sectionally and the residuals from the regression are used as a measure for gamma in the cross-sectional regression of stock returns on beta, turnover ratio, and gamma. The results, reported in Panel D, are similar to those in Panel C. The implication is that our findings are not driven by the high correlation between beta and gamma. Our analysis up to this point has been restricted to NYSE and AMEX stocks. We separate NASDAQ stocks from NYSE/AMEX stocks because we are interested in determining whether our results are driven by the design of the trading process. NASDAQ volume can be considered overstated relative to NYSE/AMEX volume because of interdealer trading on the NASDAQ. Trading volume for NASDAQ stocks is not available on CRSP until the beginning of We perform both time-series and cross-sectional regressions for NASDAQ stocks (the results are not reported for space consideration but are available on request). The results are similar to those for NYSE/AMEX stocks. In particular, in all 25 portfolios sorted by size and turnover, by book-to-market and turnover, and by turnover alone, the intercepts consistently decrease from the low-liquidity to the high-liquidity group. The Gibbon, Ross, and Shanken (1989) statistics reject the null hypothesis that the 25 intercepts are jointly equal to zero for both the four-factor and three-moment models. The cross-sectional analyses also indicate that turnover is negatively related to stock returns after controlling for stock characteristics, Fama French and Pástor Stambaugh factor loadings, and market risk factors (i.e., covariance and coskewness). 11 Therefore, our original results, obtained using NYSE and AMEX 11 The coefficients on the Pástor Stambaugh (2003) liquidity factor are negative, which is not consistent with the results for NYSE AMEX stocks. The reason may be the way Pástor and Stambaugh construct the liquidity factor. They use NYSE AMEX stocks and exclude NASDAQ stocks to construct the factor. However, in their analysis of whether stocks with higher liquidity beta demand higher returns than those

18 396 The Journal of Financial Research stocks, are not driven by market microstructure or trading protocols. Also, and more important, our results are robust to differences in the measurement of trading volume on the NYSE/AMEX versus the NASDAQ. V. Conclusion We address the question of whether adjusting for risk using the three-moment CAPM captures liquidity risk. We also look at the issue from a comparative perspective using the four-factor model, which includes the Fama French and Pástor Stambaugh market liquidity factors, as this model represents an alternative to the established models based solely on market factors such as the three-moment CAPM. We find the Gibbons, Ross, and Shanken (1989) statistics reject the null hypothesis that the intercept terms from time-series regressions of the 25 liquidity portfolios for both four-factor and three-moment models are jointly equal to zero. In addition, we document a consistent increase in the magnitude of the intercepts from high-liquidity portfolios to low-liquidity portfolios within all size or book-tomarket groups. The time-series findings imply the existence of a liquidity premium and suggest that the Fama French and Pástor Stambaugh market liquidity factors do not explain the traditional liquidity premium as in Amihud and Mendelson (1986). We also find that the second and third systematic comoments, as well stock characteristics (size and book to market), do not proxy for liquidity. Our investigation of the direct relation between stock returns and liquidity using the two-step generalized least squares method developed by Litzenberger and Ramaswamy (1979) shows that stocks with a low turnover ratio (less liquid) will demand a higher return than the stocks with a high turnover ratio (more liquid). Furthermore, we find that the strength of the relation between turnover and stock return is not affected by the presence of other variables. The results confirm the previous findings that liquidity is a distinct effect and is not influenced by other determinants of stock returns. Our cross-sectional results also show that neither the characteristic variables nor the Fama French factor loadings relate to liquidity. The sensitivities of stock returns to Pástor Stambaugh market liquidity do not proxy for the effect of characteristic liquidity. This is also consistent with our time-series results. Our findings obtained using NASDAQ stocks are similar to those obtained using NYSE/AMEX stocks. This implies our results are not being driven by market microstructure or trading protocols. More important, our results are robust to the with lower liquidity beta, they use all stocks from NYSE AMEX and NASDAQ, even though the factor is constructed using NYSE and AMEX stocks only. For the sake of comparison, we also perform crosssectional regressions for all stocks in the three exchanges from 1968 to 2004 and get similar results: the coefficients of the liquidity factor are positive and significant.