Price Impact or Trading Volume: Why is the Amihud (2002) Illiquidity Measure Priced? XIAOXIA LOU TAO SHU * September 2014 * Lou is at the Alfred Lerner College of Business, University of Delaware. Email: lous@udel. Shu is at the Terry College of Business, University of Georgia, and is visiting at the HKUST Business School. Email: taoshu@terry.
We appreciate the helpful comments from Yakov Amihud, Michael Brennan, Kalok Chan, Darwin Choi, Sudipto Dasgupta, Amit Goyal, Ronnie Sadka, Johan Sulaeman, Liyan Yang, Tong Yao, Chu Zhang, and the participants at the 2014 China International Conference of Finance, HKUST, University of Hong Kong, Chinese University of Hong Kong, City University of Hong Kong, University of South Carolina, and Zhejiang University. Price Impact or Trading Volume: Why is the Amihud (2002) Illiquidity Measure Priced? September 2014 The return premium associated with the widely used Amihud (2002) illiquidity measure is generally considered liquidity premium that compensates for price impact or transaction cost. We find that the pricing of the Amihud measure is not attributable to the construction of return-to-volume ratio that is intended to capture price impact, but entirely due to the trading volume component, the pricing of which has been explained in various ways unrelated to liquidity. Additionally, using the high- frequency measure of price impact, we find little evidence that stocks with greater price impact earn higher expected return as predicted by conventional theory.
The illiquidity measure developed by Amihud (2002) is one of the most widely used liquidity proxies in the finance literature. During 2009-2013, over one hundred papers published in the Journal of Finance, the Journal of Financial Economics, and the Review of Financial Studies use the Amihud (2002) measure for their empirical analyses. 1 The Amihud measure has two advantages over many other liquidity measures. First, the Amihud measure has a simple construction that uses the absolute value of the daily return-to-volume ratio to capture price impact.
Second, the measure has a strong positive relation to expected stock return (see, e. The positive return premium of the Amihud measure is generally considered to be a liquidity premium that compensates for price impact or transaction cost. Despite the strong empirical evidence, it is not clear ex ante that the Amihud measure would be priced because of the compensation for price impact. While the Amihud measure intends to capture price impact through the ratio of absolute return to trading volume, this construct is not precisely mapped to theory.
As discussed in Chordia, Huh, and Subrahmanyam (2009), “Although many microstructure theories have been developed, extant economic models are unable to map precisely onto the Amihud (2002) construct of the ratio of absolute return to volume. Why is the Amihud (2002) illiquidity measure priced despite its lack of full theoretical support? Is it because the construct of the daily return-to-volume ratio captures price impact? This is an important research question for two reasons. First, because the Amihud illiquidity measure is widely used by researchers to examine liquidity premium, construct liquidity factor, or control for liquidity, it is necessary for us to know whether the pricing of the Amihud measure is indeed due to price impact (stock liquidity) or other reason(s). 2 Second, the answer to this question also has 1 We count only published papers and exclude any forthcoming papers.
2 Besides the Amihud (2002) measure, the finance literature has also proposed many other liquidity measures (see Holden, Jacobsen, and Subrahmanyam (2014) for a survey). 1 important general implications for how we measure liquidity and how liquidity affects security prices. For example, the examination of this question can provide evidence on whether investors, as predicted by theory, demand compensation for the price-impact component of the transaction cost. In this paper, we examine the pricing of the Amihud (2002) measure from a new perspective.
Our study is motivated by the close connection between the Amihud measure and trading volume, which is illustrated by the construction of the measure: 1 Diy | rit | Aiy = ∑ Diy t =1 Dvolit , (1) where Aiy is the Amihud measure of firm i estimated in year y; rit and Dvolit are daily return and daily dollar trading volume for stock i on day t; Diy is the number of days with available ratio in year y. 3 Everything else equal, higher trading volume will lead to a lower Amihud illiquidity measure. This linkage is particularly strong because the trading volume component has a much greater cross- sectional variation than the stock return component. For example, the 75th percentile cutoff of the trading volume component is over 100 times its 25th percentile cutoff, but the 75th percentile cutoff of the return component is just two times its 25th percentile cutoff.
4 Many studies have documented that stocks with higher trading volume earn lower returns subsequently, although they offer vastly different explanations (e., Brennan, Chordia, and Subrahmanyam (1998), Lee and Swaminathan (2000)). We therefore examine whether the pricing of the Amihud measure is due to its association with trading volume. Our sample includes NYSE/AMEX-listed companies from 1964 to 2012, and we first confirm the previously documented strong relation between the Amihud (2002) illiquidity measure and expected return. 3 Some studies further adjust the Amihud measure for inflation.
The approaches of our analyses are such that we need not to do so. For sorting analysis, we sort stocks into portfolios every month. For the Fama-MacBeth regression analysis that uses the Amihud measures as independent variables, we follow the literature (e., Brennan, Huh, and Subrahmanyam (2013)) and transform the measures into natural logs, which makes the scaling irrelevant. 4 The corresponding statistics are presented in Table I and discussed in Section I.
2 Stocks in the top quintile portfolio of the Amihud measure outperform those in the bottom quintile portfolio by 0.38) per month in raw return and 0.83) in four- factor alpha that controls for the three Fama-French factors and the momentum factor. To focus on the trading volume component of the Amihud measure, we construct a “constant” version of the Amihud measure, A_C, by replacing absolute return in the Amihud measure with one: 1 Diy 1 A _ C iy = ∑ Diy t =1 Dvolit , (2) where A_Ciy is the “constant” measure of firm i estimated in year y, and all the other variables are as defined in equation (1). We find that the A_C measure has a correlation of 0.94 with the original Amihud measure. This result suggests that the variation in the Amihud illiquidity measure is likely driven in large part by the variation in the trading volume component.
We further find that the A_C measure is priced similarly to the Amihud (2002) measure. Stocks in the top quintile of A_C outperform those in the bottom quintile by 0.95) per month in raw return and 0.50) in four-factor alpha. These return spreads are very similar to those using the Amihud illiquidity measure. Next, we test whether it is the trading volume component that drives the pricing of the Amihud (2002) measure.
For the first approach, we construct a residual Amihud measure as the residual from cross-sectional regressions of the A measure on the A_C measure. The residual measure is therefore the component of the Amihud measure that is orthogonal to the “constant” measure. 5 We find that the residual Amihud measure no longer leads to a positive return premium. In fact, stocks in the top quintile of the residual measure underperform those in the bottom quintile by 5 We do not use two-dimensional sorting because the very high correlation between the A and A_C measures leads to insufficient numbers of stocks in the two-dimensional portfolios.12) per month in raw return and 0.40) in four-factor alpha.
These results indicate that the pricing of the Amihud measure is explained by its trading volume component but not by its construct of return-to-volume ratio. For the second approach, we construct a monthly factor, IMLA_C (“illiquid minus liquid”), as the return of the top tercile portfolio of A_C minus that of the bottom tercile portfolio. 6 We then examine the return spread associated with the Amihud measure but report the IMLA_C alpha calculated by regressing return spread on the IMLA_C factor, and the five-factor alpha calculated by regressing return spread on the IMLA_C factor in addition to the four-factor model. The spread between the top and the bottom quintiles of the A measure is -0.46) per month in terms of IMLA_C alpha, and -0.74) in terms of five-factor alpha.
Therefore, the results of the factor approach are consistent with those of the residual approach in that the trading volume component drives the pricing of the Amihud (2002) measure. We also conduct a multivariate analysis by estimating the firm-level Fama-MacBeth (1973) regressions of monthly stock returns on the Amihud measure controlling for size, book-to-market ratio, momentum, and short-term return reversal. The results of the regression analyses are consistent with the sorting analyses, in that the coefficient on the “constant” measure is significantly positive but the coefficient on the residual Amihud measure is either insignificant or significantly negative. Our results are robust when we use the turnover-based Amihud measure proposed by Brennan, Huh, and Subrahmanyam (2013) that is constructed using the absolute return-to-turnover ratio instead of the absolute return-to-volume ratio, or construct the Amihud measures monthly instead of annually.
The results are also robust to using the sample of NASDAQ stocks, using the sub-periods, and controlling for idiosyncratic return volatility. 6 The results are similar when we construct the factor by sorting stocks into two or four groups instead of three groups. 4 Brennan, Huh, and Subrahmanyam (2013) decompose the Amihud (2002) measure into the turnover-based Amihud measure and firm size (market capitalization) and examine the relations of these two metrics with expected return separately. We extend their analysis and decompose the Amihud (2002) measure further into the absolute return component, the turnover (volume) component, and the firm size component.
We estimate regressions of stock returns on these components. The coefficient on the turnover component is significantly positive, indicating that the trading volume component contributes to the pricing of the Amihud illiquidity measure. By contrast, the coefficient on the absolute return component is either insignificant or significantly negative in the regressions. To further examine the role of price impact in the pricing of the Amihud measure, we follow the literature and construct a high-frequency price impact benchmark (Hasbrouck (2009), Goyenko, Holden, and Trzcinka (2009)).
The price impact benchmark is estimated as the slope coefficient of five-minute stock return regressed on signed square-rooted five-minute trading volume for a firm- year. We construct the measure for NYSE/AMEX stocks from 1983 to 2012 using the ISSM and TAQ transaction data. Previous studies document a strong positive relation between the Amihud (2002) measure and the high-frequency price impact benchmark (Hasbrouck (2009), Goyenko, Holden, and Trzcinka (2009)). We find that this relation is also mainly due to the trading volume component of the Amihud measure rather than the construct of return-to-volume ratio.
Additionally, the high- frequency price impact measure has no significant relation with expected return, nor does it explain the pricing of the trading volume component of the Amihud measure. We further examine the return premium of the “constant” Amihud (2002) measure (the trading volume component) in the earnings announcement period and non-earnings-announcement period separately. If the pricing of the trading volume component is liquidity premium, then we 5 expect such premium to be relatively evenly distributed across the trading days. If, as suggested by the existing literature, the pricing of trading volume is due to high volume stocks initially being overpriced and therefore earning lower returns subsequently, then the return premium may concentrate in the earnings announcement window, as the release of earnings information helps correct mispricing (e., La Porta, Lakonishok, Shleifer, and Vishny (1997)).