MAXIMUM BID-ASK SPREADS AND EXPECTED
STOCK RETURNS
By
Benjamin M. Blaua and Ryan J. Whitbyb
Abstract:
The return premium associated with the illiquidity of stocks is well documented. In this study, we
focus our attention on the uncertainty of liquidity. We test whether brief but significant liquidity
droughts, as measured by the maximum daily bid-ask spread during a particular month, are
associated with the illiquidity premium. Results show a robust return premium associated with
stocks with the largest maximum bid-ask spread. We find that stocks with the largest maximum
spreads generate alphas of approximately 1% per month. These results are distinct from premiums
associated with the bid-ask spread and hold in a multi-factor setting after controlling for the PastorStambaugh liquidity risk factor.
Keywords: Illiquidity Premium, Bid-Ask Spreads, Liquidity Risk
a
Blau is an Associate Professor in the Department of Economics and Finance at Utah State
University, Logan Utah, 84322. Email: [email protected]. Phone: 435-797-2340. Fax: 435-7972301.
b
Whitby is an Assistant Professor in the Department of Economics and Finance at Utah State
University, Logan Utah, 84322. Email: [email protected]. Phone: 435-797-9495. Fax: 435797-2301.
Electronic copy available at: http://ssrn.com/abstract=2712513
1. INTRODUCTION
Since the capital asset pricing model (CAPM) was first presented in Sharpe (1964), Litner
(1965), and Mossin (1966), many articles have attempted to provide meaningful extensions to this
theory. Among these extensions is the well-documented relationship between liquidity risk and
expected returns. Amihud and Mendelson (1986) show both theoretically and empirically that
expected returns are a positive, concave function of bid-ask spreads. Additional theory in Acharya
and Pedersen (2005) shows that expected illiquidity costs are also associated with higher expected
returns.1 Several empirical studies provide evidence of this relationship, which is often
characterized as the illiquidity premium (Brennan and Subrahmanyam (1996), Datar, Naik, and
Radcliffe (1998), Liu (2006), and Han and Lesmond (2011), among others).
While the return premium associated with illiquidity has been well documented, this study
focuses on a less commonly identified dimension of liquidity, namely, the potential for future
illiquidity droughts or, what we denote as liquidity uncertainty. In particular, we test whether
stocks with the largest, maximum, daily bid-ask spread will have higher expected returns. The
framework in Acharya and Pedersen (2005) motivates these tests. They allow for a liquidity
adjustment to the traditional CAPM and show that a security’s expected return depends on its
expected liquidity as well as how that security’s liquidity covaries with its own return, the return
of the market, and the liquidity of the market. Although Acharya and Pederson (2005) do not
focus much of their attention on liquidity uncertainty, they clearly state that “uncertainty about the
illiquidity cost is what generates the liquidity risk in this model”. Given their model, and that their
results stem from this assumption about liquidity uncertainty, it is important to better understand
1
In particular, Acharya and Pedersen (2005) show first that illiquidity costs are directly related to expected returns,
which is similar to Amihud and Mendelson (1986). Second, they show that the sensitivity of an individual stock’s
illiquidity relative to market illiquidity also affects expected returns.
1
Electronic copy available at: http://ssrn.com/abstract=2712513
how liquidity uncertainty relates to liquidity risk and thus affects asset prices. Motivated by this
notion, we argue that liquidity uncertainty can be properly captured by our measure of the
maximum bid-ask spread. Although they may be brief events that occur with low probability,
large illiquidity shocks are likely to create uncertainty about future liquidity. We therefore contend
that the size of the maximum daily bid-ask spread (max spread hereafter) will increase investors’
expected illiquidity costs and thus directly affect expected returns. Similar to the intuition in Bali,
Cakici, and Whitelaw (2011), which suggests that the maximum daily return signals to investors
the expected distribution of returns, we argue that the maximum daily spread will provide an
equally meaningful signal to investors about the expected distribution of liquidity. To the extent
that investors presume that stocks with unusually large bid-ask spreads, even if those spreads are
relatively short lived, face greater liquidity risk, investors will command a return premium. The
objective of this study is to provide tests of this contention.
After obtaining daily spreads for a broad cross-sectional sample during the time period of
1993 to 2012, we find that average daily spreads are approximately 2.35%. However, the standard
deviation of daily spreads is more than 3.5%, indicating that spreads have a large degree of
variation. In fact, the average stock in our sample has a max spread of 4.12% which is more than
75% larger than the mean spread. Further, we find that both mean spreads and max spreads are
markedly larger during the 1990s than during the 2000s. However, during the latter time period,
we find that the max spread for the average stock is 140% larger than the mean spread suggesting
that, although liquidity has dramatically improved over the last of couple decades, short-lived
liquidity droughts may have become more severe. Perhaps the more recent presence of these
droughts provides the context for the assertions made in Persuad (2003) that there is broad belief
among practitioners and regulators that the principal concern about liquidity in financial markets
2
is not the average liquidity level, which has improved over time, but the uncertainty of liquidity.
In our univariate tests, we find that max spreads are larger for stocks that have larger mean spreads,
which is to be expected. However, we also find that max spreads are also greater in stocks with
lower share prices, less trading activity, lower market capitalization, higher return volatility, and
lower levels of systematic risk.
When examining the return premium associated with max spreads, we find that next-month
returns increase monotonically across portfolios sorted by max spreads. These results are robust
to various measures of next-month returns, such as returns in excess of the risk-free rate and returns
less the value-weighted market return. We also estimate three-factor, four-factor, and five-factor
models, where the fifth factor is the Pastor and Stambaugh (2003) liquidity risk factor. We find
that alphas from each of these models increase monotonically across portfolios sorted by max
spreads. In economic terms, next-month alphas are approximately 1% for stocks with the largest
max spread. These results are robust to both the 1990s and the 2000s. However, we find that
alphas during the latter time period are markedly lower than during the former time period. In
particular, we find that five-factor alphas, while monotonically increasing across max spread
portfolios, are .74% for stocks with the largest max spreads. These findings indicate that the
magnitude of these next-month returns are still economically meaningful and support the idea that
stocks with the largest max spreads have the highest expected returns.
We recognize the possibility that the relation between next-month returns and max spreads
is being driven by other stock characteristics, such as firm size, volatility, and, perhaps most
importantly, other measures of liquidity. To address this potential concern, we control for a
number of stock characteristics, including mean spreads, firm size, and volatility, in a variety of
multivariate tests. The results from our Fama–MacBeth (1973) regressions show a strong, positive
3
relation between max spreads and next-month returns that is both statistically significant and
economically meaningful.
For instance, after controlling for a number of other stock
characteristics that might directly affect next-month returns, we find that a one standard deviation
increase in max spreads during month t results in an increase in next-month returns that ranges
from .32% to .55%.
We note that these multivariate tests hold for both the 1990s and 2000s. For instance, when
including the various controls, Fama–MacBeth regression coefficients on max spread are both
positive during the 1990s and during the 2000s. Perhaps more importantly, we find that when we
include max spreads in the Fama–MacBeth regressions, the coefficient on mean spreads becomes
statistically insignificant during the most recent time period suggesting that the traditionally
positive relation between mean spreads and future returns no longer exists when accounting for
max spreads.
We explore an additional possibility that the observed return premium in stocks with the
largest max spreads is driven by other measures of illiquidity. Following Amihud (2002), we
include in our analysis additional controls for price impact, which is the ratio of the absolute value
of daily returns scaled by daily volume. With this additional control for illiquidity, our Fama–
MacBeth regressions produce coefficients on max spread that are both positive and significant.
Further, the magnitude of the coefficients on max spread are similar whether or not we include
Amihud’s illiquidity. Interestingly, while the estimate for max spread is reliably positive, both the
coefficients on Amihud’s illiquidity and mean spreads are statistically close to zero, indicating that
relative to other more traditional measures of illiquidity, the return predictability of max spreads
seems to dominate in our multivariate analysis. The results from this set of tests seem to suggest
4
that the observed return premium associated with max spreads explains the return premium above
and beyond that which is documented using other measures of illiquidity.
In our final set of tests, we attempt to disentangle the potential for collinearity issues
between max spreads and mean spreads. First, instead of including max spreads as our variable of
interest, we include the skewness of bid-ask spreads. We note that while estimated variance
inflation factors for both max spreads and means spreads were not unusually high in the regressions
described above, the variance inflation factors for both spread skewness and mean spreads are
virtually negligible. Results, however, show that both mean spreads and spread skewness produce
positive estimates that are reliably different from zero suggesting that the non-normality of the
distribution of spreads contains predictability in monthly stock returns. In our second attempt to
account for the possibility of collinearity issues, we estimate a simply regression where the
dependent variable is max spreads and the independent variable is mean spreads. We obtain the
residuals from this first-stage regression, which is estimated for each stock. The residual from this
regression is the portion of the max spread that is orthogonal to the mean spread. We then replace
the max spread with this residual as the independent variable of interest in our Fama-MacBeth
(1973) regressions. As expected, variance inflation factors are negligible. However, we find that
the residuals from our first-stage regressions are still positive and significant. These findings
indicate that the max spread return premium is independent of the traditional return premium
associated with bid-ask spreads (Amihud and Mendelson (1986)).
Combined, our findings contribute to the literature by examining the role of liquidity
uncertainty in explaining expected returns. If larger max spreads are associated with greater
liquidity uncertainty, then our findings suggest that an expectation of future liquidity droughts are
indeed an important aspect of liquidity risk and should be considered when examining the
5
traditional illiquidity premium (Amihud and Mendelson (1986), Acharya and Pederson (2005),
Brennan and Subrahmanyam (1996), Datar, Naik, and Radcliffe (1998), Liu (2006), and Han and
Lesmond (2011)).
2. DATA DESCRIPTION
The data used in this analysis come from several sources. From the Center for Research
on Security Prices (CRSP), we obtain daily prices, returns, volume, and market capitalization.
From Compustat, we gather balance sheet information, such as assets, liabilities, and owners’
equity, in order to calculate the book-to-market ratios. From CRSP, we obtain the universe of
stocks from 1993 to 2012 and merge the data to Compustat.2 We restrict our sample to stocks that
have prices greater than $2. We note, however, that the conclusions that we are able to draw are
similar without this restriction.3 The variable of interest is the largest daily maximum bid-ask
spread for a particular stock in a particular month.4 The spreads are obtained from closing ask
prices and closing bid prices from CRSP.5 Our sample includes 17,826 unique securities and
1,348,513 stock-month observations.
Table 1 reports some statistics that describe our sample. Panel A presents the summary
statistics for stocks during the entire time period. Panels B and C report the summary statistics for
2
While we would prefer to use historical data that extends before 1990, the CRSP closing ask and bid prices, which
were recently added, are not available for every stock during the 1980s.
3
In unreported tests, we conduct additional tests of the return premium associated with max spreads while
restricting our sample to stocks with prices less than $5. We still find that results consistent with our findings
below.
4
In a somewhat related study, Bali, Cakici, and Whitelaw (2011) examine the preferences by investors for lottery-like
returns. To do so, they examine the maximum daily return during a particular for each stock and show that stocks
with the largest max return underperform those stocks with the lowest maximum daily return. These results are
consistent with the idea that investors’ preferences for lottery-like stocks leads to price premiums and subsequent
underperformance. In a similar manner, we test whether investors perceive that those stocks with the largest maximum
daily spread have greater liquidity risk given the possibility of future liquidity droughts. Our objective is to determine
whether there is a return premium associated with max spreads.
5
Roll and Subrahmanyam (2010) find that closing bid-ask spreads in CRSP properly approximate intraday bid-ask
spreads. Similarly, Chung and Zhang (2013) show that the use of CRSP closing bid-ask spreads is a very close
approximation to using high frequency data when calculating bid-ask spreads.
6
the 1990s and the 2000s, respectively. MaxSpread is the maximum daily percentage bid-ask
spread during a particular month. Spread is the average daily percentage bid-ask spread, which is
calculated as the ask-price minus the bid-price scaled by the spread midpoint. Price is the CRSP
closing price on the last day of the month. Turn is the share turnover or the average daily volume
scaled by shares outstanding (in percent). Size is the market capitalization on the last day of each
month. B/M is the book-to-market ratio, where the market value is gathered from CRSP and the
book value is obtained from Compustat. Volt is the standard deviation of daily returns over a
rolling six-month period. Beta is the beta estimate obtained from the daily capital asset pricing
model over a six-month rolling period.
Panel A shows that the average stock has a max spread of 3.94% and a mean spread of
2.19%. This indicates that there is large variation in bid-ask spreads as max spreads are nearly
80% higher than mean spreads for the average stock. We also find that the average stock has a
share price of $34.35, turnover of .82%, market capitalization of $2.2 billion, a book-to-market
ratio of .3822, volatility of 3.08%, and a beta of .8030.
In Panel B we find that the average stock during the 1990s (1993 to 1999) has a much
larger max spread than the average stock across the entire sample reported in Panel A (.0587 versus
.0394). The average stock also has a much larger mean spread (.0387 versus .0219). The average
stock during the 1990s also has a share price of $25.12, turnover of .48%, market cap of $1.25
billion, a book-to-market ratio of .3334, volatility of 3.29%, and a beta of .8662.
Panel C shows the summary statistics for the latter time period (2000 to 2012). We find a
marked decrease in max spreads and mean spreads as the average stock has a max spread of .0274
and a mean spread of .0114. While both max and mean spreads have declined substantially during
the latter portion of our time period, we do find that, for the average stock, max spreads as a
7
percentage of mean spreads increased dramatically. For example, column [1] shows that the max
spread is 2.4 times larger than the mean spread in Panel C. In Panel B column [1], the max spread
is only 1.5 times larger than the mean spread. This simple comparison suggest that while liquidity
has generally improved over time, there exists a greater degree of extreme variation in spreads
relative to the mean. We also note that the average stock in Panel C has a share price of $40.09,
turnover of 1.0295%, a market capitalization of $2.79 billion, a book-to-market ratio of .4174,
volatility of 2.96%, and a beta of .7636.
As an additional way to provide some description of the data used throughout the study,
we report mean spreads and max spreads in each year across our sample time period. The results
are reported in Figure 1. According to the figure, we find that both mean spreads and max spreads
have decreased across time. However, as mentioned previously, the decline in both mean and max
spreads has not been systematic. It appears that max spreads have declined at a slower pace than
mean spreads. This is likely why we observe larger max spreads – relative to mean spreads –
during the more recent time period.
3. EMPIRICAL RESULTS
In this section, we present the results from our empirical tests. We first look at the relation
between max spreads and other stock characteristics. Then we examine the relation between max
spreads and future returns in a variety of settings. In particular, we explore next-month returns in
a multifactor model framework and compare raw returns, excess returns, as well as 3-, 4-, and 5factor alphas for stocks with the largest max spreads and stocks with the smallest max spreads. We
also examine next-month returns while specifically controlling for other dimensions of liquidity,
such as the Pastor and Stambaugh (2003) liquidity factor and the average bid-ask spread. We then
turn our attention to Fama and MacBeth (1973) regressions so that we can control for firm
8
characteristics more directly. To demonstrate the robustness of our findings, we also examine the
relation between max spreads and future returns by time period and after specifically controlling
for Amihud’s measure of liquidity.
3.1 Univariate Analysis – Firm Characteristics and Max Spreads
We begin by examining the relation between the max spreads and other stock
characteristics. Table 2 reports the means of several stock characteristics sorted by the max spread.
Panel A of Table 2 reports means for the entire sample while Panel B examines the period from
1993 to 1999 and Panel C examines the period from 2000 to 2012. In the first column of Table 2,
we can see that MaxSpread ranges from 0.0083 in the lowest quintile to 0.1062 in the highest
quintile. Not surprisingly, MaxSpread is positively related to the average spread. The average
bid-ask spread increases monotonically across MaxSpread quintiles with a difference of 0.0519
between the fifth and first quintiles. The positive correlation between max spreads and mean
spreads will require us to carefully control for possible multicollinearity bias in our tests below.
Stocks with the largest max spread tend to have lower prices as evidenced in column [3] and
smaller market values of equity as seen in column [5]. Firms with larger max spreads have less
turnover, are more volatile, but have less systematic risk. The only firm characteristic that we
include that is not strongly associated with MaxSpread in the entire sample is the book-to-market
ratio. Panel B of Table 1 looks at the relation between MaxSpread and firm characteristics from
1993 to 1999. The correlations in Panel B are the same as in the full sample with one exception.
In the 1990s, MaxSpread is negatively related to the book to market ratio. Firms with higher max
spreads have lower book-to-market ratios. Panel C, which examines the sub period from 2000 to
2012 is also similar to the full sample except for the book to market ratio. However, in the 2000s,
the book-to-market ratio is positively related to the MaxSpread. Given the strong relation between
9
MaxSpread and other firm level characteristics, it will be important for us to adequately control
for these factors in a multivariate setting.
3.2 Multivariate Analysis – The Max Spread Return Premium
Although the relation between the max spread of a stock and other firm-level characteristics
is informative, the importance of liquidity uncertainty is best highlighted by examining whether
there is a return premium associated with max spreads. In this section, we closely examine the
relation between max spreads and future returns. We begin by estimating a simple multifactor
model for portfolios sorted by max spreads. Table 3 reports various measures of next-month
returns (or returns in month t+1) across quintiles sorted by max spreads in month t. Specifically,
we estimate the following equation using pooled stock-month data.
Excess Returni,t+1 = α + β1MRPt+1 + β2SMBt+1 + β3HMLt+1 + β4UMDt+1 +
(1)
β5LIQt+1 + εi,t+1
The dependent variable is the excess return – or the difference between CRSP monthly raw returns
and monthly estimates of the risk-free rate – for stock i in month t+1. We include independent
variables that have been shown to explain expected returns (see Fama and French (1996), Carhart
(1997), and Pastor and Stambaugh (2003)). The independent variables include the market risk
premium (MRP); the small-minus-big (SMB) risk factor, which captures the size premium; the
high-minus-low (HML) risk factor, which accounts for the value premium; the up-minus-down
(UMD) risk factor, which captures the momentum premium; and the Pastor and Stambaugh
liquidity risk factor (LIQ). Table 3 reports various measures of returns along with the estimated
alphas from various specifications of the above model. The five-factor alpha is obtained from the
full specification of equation (1), which we label as 5F Alpha. The four-factor alpha excludes LIQ
and the three-factor alpha excludes LIQ and UMD. We denote these estimated alphas as 4F Alpha
10
and 3F Alpha, respectively. Panel A of Table 3 reports results for the full sample while Panels B
and C report results for the 1990s and the 2000s, respectively. A clear pattern emerges across all
of the reported specifications. Each measure of return/alpha increase monotonically across max
spread quintiles. For instance, column [1] shows that raw returns increase across quintiles. The
difference between extreme quartiles is 0.73% per month. Differences of similar size are reported
for excess returns and adjusted returns. While these differences are statistically different than zero
(p-value = <.0001), the results are also economically meaningful. In annual terms, a long-short
portfolio where stocks with the highest max spreads are bought and stocks with the lowest max
spreads are shorted results in a return premium of nearly 9% per year.
We find even stronger results when focusing on our estimated alphas in columns [4]
through [6]. Again, alphas increase monotonically across increasing max spreads quintiles.
Further the differences between extreme quintiles are reliably different from zero and range from
.79% to 1.06%, depending on the model specification. Results in column [6] suggest that, after
controlling for some of the traditional risk factors including the Pastor-Stambaugh liquidity risk
factor, the max spread return premium is slightly more than 12.5% in annual terms.
When we partition our analysis into separate time periods, we find that the return premium
is driven by the 1990s decade although a significant return premium exists during the 2000s. For
example, Panel B, which examines the time period from 1993 to 1999, shows that the return
premium ranges from 1.11% per month to 1.97% per month. In the more recent time period, Panel
C again shows that our various measures of returns generally increase across max spread quintiles.
Further the differences between extreme quintiles are markedly smaller than the corresponding
differences in Panel B. However, the differences are still statistically significant (p-values =
<.0001), and economically meaningful.
In economic terms, the monthly return premium
11
associated with max spreads is .46% in column [6] suggesting that, after controlling for various
risk factors, a long-short portfolio based on max spreads is slightly larger than 5.5% per year.
As mentioned above, our analysis consists of stocks with prices greater than or equal to $2.
We note, however, that when we replicate our analysis without the $2 price restriction, the
unreported results are similar to those that we report in this study. In other unreported tests, we
increase the price restriction from $2 to $5. While these price cut offs are admittedly ambiguous,
we are simply attempting to provide some robustness to our results. When we eliminate stocks
with prices less than $5 and replicate Table 3, we still observe a significant return premium that is
robust to both subperiods. Our results are also robust to double-sorted portfolios where we sort by
max spreads and then mean spreads. In unreported tests, we construct double-sorted portfolios by
max spreads and mean spreads andshow that the return premium associated with max spreads,
while increasing in the size of mean spreads, is still robust to the group of stocks with the smallest
mean spreads. For example, in the quintile of stocks with the smallest mean spreads, the difference
in five-factor alphas between extreme max spread quintiles is 25 basis points per month.
3.3 Multivariate Analysis – A Fama-MacBeth Approach
Thus far, we have identified a return premium associated with max spreads in a multifactor
framework. In this section, we continue in our objective to determine whether a cross-sectional
relationship between max spreads and future returns exist after controlling for a variety of firmspecific variables in a number of Fama-MacBeth regressions. We estimate the following model
using pooled stock-month observations.
Returni,t+1 = β0 + β1MaxSpreadi,t + β2Spreadi,t + β3Turni,t + β4ln(Sizei,t) + β5B/Mi,t +
β6Betai,t + β7Volti,t + β8Momi,t + εi,t+1
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(2)
The dependent variable is the raw return for stock i in month t+1. The independent variables are
measured in month t and include: the maximum daily (closing) percentage bid-ask spread during
the month (MaxSpread), the percentage bid-ask spread (Spread), the average daily volume of
shares traded scaled by the total shares outstanding (Turn), the size of the firm measured as the
natural log of market capitalization measured on the last day of each month (Ln(Size)), the book
to market ratio (B/M), the beta from the capital asset pricing model estimated over rolling sixmonth windows (Beta), the standard deviation of daily returns calculated over a rolling six-month
window (Volt), and the cumulative return for stock i from month t-6 to t-1 (Mom).
The Fama-MacBeth approach estimates cross-sectional regressions each month and then
averages the coefficients and standard errors across months. We note that we provide p-values
obtained from Newey-West standard errors below each mean coefficient estimate. Column [1] of
Table 4 reports the simple regression with MaxSpread as the only explanatory variable. Column
[2] is similar, but includes the mean spread (Spread) instead of MaxSpread. In both instances, the
coefficients on MaxSpread and Spread are positive and statistically significant in columns [1] and
[2]. Columns [3] and [4] include the control variables described above that have been shown to
impact the cross section of expected returns.6 After controlling for these factors, column [3] shows
that the estimate for MaxSpread remains reliably different from zero. For instance, the coefficient
is 0.0818 with a p-value less than 0.0001. Similarly, column [4] shows that the coefficient on
Spread (coefficient = 0.1412) is statistically significant after controlling for these additional
variables. Column [5] provides the results when we estimate the full model specification.
Interestingly, we find that when we control for mean spreads, the coefficient on MaxSpread
6
Banz (1981) show that market cap affects future returns. Similar results are found in Fama and French (1992), who
also show that book-to-market ratios are important predictors of future returns. Ang et al. (2006, 2009) show that
volatility explains the cross-section of expected returns while Jegadeesh and Titman (1993) discuss the return
premium associated with return momentum.
13
decreases by approximately 42% (coefficient = 0.0476).
However, the coefficient remains
statistically significant at the 0.01 level (p-value = 0.004). Perhaps, what is more interesting is
that the coefficient on Spread decreases by more than 56%. Further, this estimate for mean spreads
is no longer reliably different from zero (p-value = 0.207). These results seem to indicate that when
controlling for a variety of firm-specific factors, max spreads contains predictability in future
returns while mean spreads do not.
We recognize the possibility that the conclusions we draw are adversely affected by the
presence of multicollinearity. Given the strong correlation between max spreads and mean
spreads, it is possible that our results are biased. In unreported tests, we estimate a pooled OLS
regression replicating the specification in Column [5] and find that variance inflation factors are
3.43 for the coefficient on MaxSpread and 3.44 for the coefficient on Spread. The small size of
these factors suggest that multicollinearity bias is not much of an issue. However, we attempt to
carefully account for this possibility in some of the tests that follow.
We continue our analysis by examining the time-varying component of the max spread
return premium by estimating equation (2) using Fama-MacBeth regressions for two sub-periods.
Table 5 reports the results from this analysis. As before, we report mean estimates from the crosssectional regressions with p-values obtained from mean standard errors that account for the
Newey-West adjustment. Columns [1] through [3] shows the results for the period from 1993 to
1999. Columns [4] through [6] present the results for the period from 2000 to 2012. As before,
columns [1] and [2] show that coefficients on MaxSpread and Spread are positive and statistically
significant. When controlling for other firm-specific factors, we find positive and statistically
significant estimates for mean spreads in column [3]. However, we do not find a significant
coefficient for max spreads in this earlier time period. A closer looks suggests that when
14
controlling for mean spreads, the coefficient on MaxSpread decreases by more than 73%. On the
other hand, when controlling for max spreads, the estimate for Spread only decreases by 21%.
Given our results in Table 3 that shows that the return premium associated with max spreads is
markedly larger during the 1990s, the findings in Table 5 seem to suggest that when controlling
for other factors, including mean spreads, the max spread premium is negligible.
Next, we focus our tests on the more recent time period. Similar to findings in columns
[1] and [2], results in columns [4] and [5] again report positive and significant estimates for max
spreads and mean spreads in the simple regressions. Further, column [6] shows the results for the
full specification. Here, we find that the estimate for MaxSpread is positive and significant
(estimate = 0.0555, p-value = 0.039) while the coefficient on Spread is not reliably different from
zero (estimate = 0.0300, p-value = 0.675). Further, the coefficient on MaxSpread only decreases
by 5.6% when controlling for mean spreads along with other firm characteristics in Table 6.
Combined with our earlier results, these findings suggest that the traditional return premium
associated with mean spreads no longer exists when accounting for max spreads during the most
recent time period. These findings have important implications about the common illiquidity
premium. Results from Tables 4 and 5 seem to suggest that, over time, investors have sought to
identify liquidity risk by examining the distribution of liquidity instead of simply examining the
levels of liquidity. Perhaps a more thorough examination of the preferences for distributional
liquidity across time might be a fruitful area for future research.
3.5 Robustness Tests
Thus far, we have controlled for illiquidity using mean spreads. However, previous
research has proxied for liquidity in other ways. Thus, it is only natural that questions would arise
concerning the robustness of our results with respect to other measures of liquidity. Although mean
15
spreads are one of the most common proxies for liquidity, we have also controlled for the PastorStambaugh liquidity factor in our analysis. Another common proxy for illiquidity is shown in
Amihud (2002), which measures the price impact of trading volume at the daily level. In this
section, instead of controlling for liquidity using mean spreads, we control for liquidity using
Amihud’s measure of illiquidity, which is given below:
 | returni ,t | 
Illiqi,t = 
 × 100,000 (3)
 volumei ,t 
We estimate equation (3) at the daily level and average daily illiquidity to the monthly level for
each stock. This measure of illiquidity captures the price impact for a given unit of trading volume
and suggests that stocks with higher ratios of absolute returns to trading volume represent those
stocks that are least liquid. Including this additional measure illiquidity can also ease our concerns
about multicollinearity bias. For instance, while the Pearson correlation between max spreads and
mean spreads is 0.8329, the correlation between max spreads and illiquidity is only 0.0492.
We continue our analysis by estimating the following equation using pooled stock-month
observations.
Returni,t+1 = β0 + β1MaxSpreadi,t + β2Illiqi,t + β3Spreadi,t + β4Turni,t + β5ln(Sizei,t) + β6B/Mi,t +
β7Betai,t + β8Volti,t +β9Momi,t + εi,t+1 (4)
As before, the dependent variable is the raw return for stock i in month t+1. The independent
variables are the same as those in equation (2) except we include Amihud’s illiquidity measure
(Illiq) in addition to mean spreads. Similar to our analysis in Tables 4 and 5, we estimate the above
equation using a Fama-MacBeth approach and report average coefficients with p-values obtained
from Newey-West (1987) standard errors.
16
Table 6 reports results from Fama-MacBeth regressions but includes both Spread and
Amihud’s Illiquidity measure (Illiq). Column [1] shows the simple regression results and suggest
that mean illiquidity contains predictability about next-month returns. We are able to draw similar
inferences in column [2] when we include both max spreads and illiquidity although the coefficient
on illiquidity is only marginally significant at the 0.10 level. We also note that we find that max
spreads produce a positive and significant estimate that is similar in sign and magnitude to the
corresponding estimate in Table 4 columns [1] and [3]. Similar to our findings in Table 4, once we
control for other firm-specific characteristics, including both mean spreads and illiquidity
(columns [3] and [4]), the other liquidity measures are dominated by MaxSpread. Column [3]
shows that when including both max spreads and illiquidity (along with several other independent
factors), we find that the coefficient on max spreads is positive and significant (coefficient =
0.0795, p-value = <.0001) while the coefficient on illiquidity is not reliably different from zero
(coefficient = 0.0021, p-value = 0.120). Column [4] shows the results for the full model
specification. Results show that the coefficient on max spread is 0.0446 (p-value = 0.026) while
the coefficients on Illiq and Spread are positive but not reliably different from zero. These results
support our findings in Table 4 and suggest that the return premium associated with max spreads
is robust to other controls for liquidity levels. In unreported tests, we use a double-sorted portfolio
approach – by max spreads and illiquidity – to determine whether the max spread return premium
is robust to stocks with the least illiquidity. Again, we find evidence that the max spread return
premium is increasing in Amihud’s measure of illiquidity. However, we still observe a significant
max spread return premium in stocks with the least illiquidity. For instance, in the quintile of
stocks with the least illiquidity, the difference in five-factor alphas between extreme max spread
quintiles is 55 basis points per month.
17
At the end of the previous section, we contend that investors might be more concerned with
the distribution of liquidity rather than mean levels of liquidity when identifying liquidity risk. In
Table 7, we re-estimate the equation (4) but instead of including max spreads, we include the
skewness of bid-ask spreads (SpreadSkew). Roll and Subrahmanyam (2010) examine spread
skewness and argue that over time, mean spreads have declined while spread skewness has
increased. The explanation for the increasing skewness in bid-ask spreads is consistent with the
idea that increased competition among market makers have reduced the ability of market makers
to cross-subsidize periods of high information asymmetry. Stated differently, more competition
has lowered the ability of market makers to increase spreads during periods of low information
asymmetry to offset the lower spreads they set during periods of high information asymmetry.
However, there are low probability instances when extreme large spreads are set by market makers
despite the increased competition. We test whether such low probability instances influence future
returns.
Table 7 reports the results from estimating equation (4) but replacing max spreads with
spread skewness. Column [1] reports the Fama-MacBeth (1973) results without controlling for
Amihud’s illiquidity while column [2] reports the full model specification. In both cases, we find
that spread skewness produces a positive estimate (0.0005) that is reliably different from zero at
the 0.05 level. We note that including spread skewness instead of max spreads also provides some
relief to the concern of multicollinearity bias in our previous tests. Pooled OLS estimates of
variance inflation factors for spread skewness and mean spreads are just 1.0813 and 1.6390,
respectively. The small size of these factors suggest that multicollinearity does not have much of
an effect on the conclusions we draw in this study.
18
We continue with our robustness tests by providing a unique test to account for the
possibility of multicollinearity bias. We begin by estimating a simple regression where the
dependent variable is max spreads and the independent variable is mean spreads. We estimate this
equation for each stock and obtain the residual from this regression, which we denote as
MaxSpread*. MaxSpread* can be thought of as the portion of max spreads that is orthogonal to
mean spreads. Therefore, we attempt to take out any collinearity that might exist between max
spreads and mean spreads. We then re-estimate equation (4) and replace MaxSpread with
MaxSpread*. Table 8 reports the results from the estimation using the traditional Fama-MacBeth
(1973) approach. As in Table 7, column [1] presents the estimates without including Amihud’s
illiquidity while column [2] provides the results for the full model specification. We see in Table
8 that the coefficients on MaxSpread* are 0.0710 and 0.0683 in columns [1] and [2]. Both
estimates are statistically significant (p-values = <.0001) suggesting that the portion of max
spreads that is independent of mean spreads reliably predicts future returns. We also note that
mean spreads produce reliable estimates in both columns. As expected, we find that variance
inflation factors for MaxSpread* and mean spreads are just 1.0044 and 1.5761, respectively.
Combined with previous tests, these results support the idea that low probability events such as
one day with extremely large bid-ask spreads produce illiquidity risk that is not explained simply
by mean levels of illiquidity.
In other unreported tests, we also conduct other robustness tests by including price in our
regressions. We replicate Tables 4 through 8 but include price as an additional independent
variable. The controls for price allow us to hold the share price constant while testing for the return
premium associated with max spreads. We find that the results we report in these tables only
19
change marginally as the variable of interest – max spreads – produces a positive and significant
coefficient in each of the specifications.
4. CONCLUSION
In this study, we develop and test the hypothesis that stocks with larger maximum, daily
bid-ask spreads will require a return premium to compensate for liquidity uncertainty. The idea
behind this hypothesis is that investors might perceive that stocks with unusually large bid-ask
spreads, even if those spreads are relatively short lived, will have a higher likelihood of liquidity
droughts in the future. To the extent that this is true, investors will expect greater illiquidity costs
for these stocks and, according to theory in Acharya and Pederson (2005), these stocks will have
higher expected returns.
Our empirical tests confirm our hypothesis as stocks with the largest, maximum bid-ask
spreads have the largest expected returns. For instance, after sorting stocks into portfolios based
on max spreads, we find that next-month returns increase monotonically across portfolios. After
estimating more common multifactor models, we show that stocks in the largest max spread
portfolio generate next-month alphas that are approximately 1% per month. These results are
robust to different time periods and additional controls for levels of liquidity, including the mean
spread.
In a variety of multivariate tests, we find that max spreads are directly associated with nextmonth returns. The results from these Fama-MacBeth regressions are not only statistically
significant, but the results are also economically meaningful. Further, these results hold when we
control for firm size, volatility, and other measures of illiquidity like mean bid-ask spreads and
Amihud’s (2002) illiquidity. In addition, the direct relation between max spreads and next-month
returns, which we observe in our multivariate analysis, are robust to different time periods. Perhaps
20
more importantly, we find some evidence that, when including max spreads, the traditionally
positive relationship between mean spreads and future returns is no longer statistically significant.
These results have important implications for the current literature examining the illiquidity
premium. It appears that while the recent changes in financial markets have greatly increased the
liquidity of the average asset over the last couple of decades, the uncertainty of liquidity has gained
more prominence and is an important piece in identifying the return premium associated with
illiquidity, which is described in Persuad (2003). If indeed investors perceive that larger max
spreads represent a greater potential for future liquidity droughts, then our findings suggest that
such droughts are an important aspect of liquidity risk and should be considered part of the
traditional illiquidity premium (Amihud and Mendelson (1986), Acharya and Pederson (2005),
Brennan and Subrahmanyam (1996), Datar, Naik, and Radcliffe (1998), Liu (2006), and Han and
Lesmond (2011)).
21
REFERENCES
Acharya, Viral V., Lasse H. Pedersen, 2005. Asset pricing with liquidity risk. Journal of Financial
Economics 77 (2), 375–410.
Amihud, Yakov, 2002. Illiquidity and stock returns: cross-section and time- series effects. Journal
of Financial Markets 5, 31–56.
Amihud, Yakov and Haim Mendelson, 1986, Asset pricing and the bid-ask spread, Journal of
Financial Economics 17, 2233249.
Ang, Andrew, Robert J. Hodrick, Yuhang Xing, Xiaoyan Zhang, 2006. The cross-section of
volatility and expected returns. Journal of Finance 61 (1), 259–299.
Ang, Andrew, Robert J. Hodrick, Yuhang Xing, Xiaoyan Zhang, 2009, High idiosyncratic
volatility and low returns: International and further U.S. evidence, Journal of Financial Economics
91, 1-23.
Bali, Turan, Nusret Cakici, and Robert Whitelaw, 2011, Maxing Out: Stocks as Lotteries and the
Cross-Section of Expected Returns, Journal of Financial Economics 99, 427-446.
Banz, Rolf W., 1981, The relationship between return and market value of commons stocks,
Journal of Financial Economics 9, 3-18.
Brennan, Michael J. and Avanidhar Subrahmanyam, 1996, Market microstructure and asset
pricing: On the compensation for illiquidity in stock returns, Journal of Financial Economics 41,
441-464.
Carhart, Mark M., 1997, On Persistence of mutual fund performance, Journal of Finance 52, 5782.
Chung, Kee, H., and Hao Zhang, 2013, A simple approximation of intraday spreads with daily
data, Journal of Financial Markets, Forthcoming.
Datar, Vinay T., Narayan Naik, and Robert Radcliffe, 1998, Liquidity and Stock Returns: An
Alternative Test, Journal of Financial Markets 1, 203-219.
Fang, Vivian W., Thomas H. Noe, and Sheri Tice, 2009, Stock market liquidity and firm value,
Journal of Financial Economics 94, 150-169.
Fama, Eugene and Kenneth French, 1992, The cross-section of expected stock returns, Journal of
Finance 47, 427-465.
Fama, Eugene and Kenneth French, 1996, Multifactor explanations of asset pricing anomalies,
Journal of Finance 51, 55-84.
22
Fama, Eugene and James Macbeth, 1973, Risk, return, and equilibrium: Empirical tests, Journal
of Political Economy 71; 607-636.
Han, Yufeng and David Lesmond, 2011, Liquidity biases and the pricing of cross-sectional
idiosyncratic volatility, Review of Financial Studies, 1590-1629.
Litner, J., 1965, The valuation of risk assetss and the selection of risky investments in stock
portfolios and capital budgets, Review of Economics and Statistics 47, 13-37.
Liu, Weimin, 2006, A liquidity-augmented capital asset pricing model, Journal of Financial
Economics 82, 631-671.
Mossin, J., 1966, Equilibrium in a capital asset market, Econometrica 34, 768-783.
Pastor, Lubos, and Robert Stambaugh, 2003, Liquidity risk and expected stock returns, Journal of
Political Economy 111, 642–685.
Persaud, Avinash. (2003) Liquidity Black Holes: Understanding, Quantifying and Managing
Financial Liquidity Risk. Risk Books, Page xvi (Introduction).
Roll, Richard, and Avandidhar Subrahmanyam, 2010, Liquidity skewness, Journal of Banking and
Finance 34, 2562-2571.
Sharpe, W.F., 1964, A theory of market equilibrium under conditions of risk, Journal of Finance
19, 425-442.
23
Table 1. Summary Statistics
The table reports statistics that describe the sample. Panel A reports the summary statistics for the entire sample
time period (1993 to 2012), while Panels B and C show the summary statistics for the period of 1993 to 1999 and
2000 to 2012, respectively. MaxSpread is the maximum daily (closing) percentage bid-ask spread during a
particular month. Spread is the percentage bid-ask spread, which is calculated as the ask-price minus the bid-price
scaled by the spread midpoint. Price is the closing month price from CRSP. Turn is the share turnover of the
average daily volume scaled by shares outstanding (in percent). Size is the market capitalization on the last day of
each month. B/M is the book-to-market ratio, where the market value is gathered from CRSP and the book value
of obtained from Compustat. Volt is the standard deviation of daily returns over a rolling six-month period. Beta
is the beta estimate obtained from the daily Capital Asset Pricing Model over a six-month rolling period.
Panel A. Entire Time Period
Mean
Median
Std. Deviation
Min
Max
[1]
[2]
[3]
[4]
[5]
MaxSpread
0.0394
0.0225
0.0564
0.0005
1.9998
Spread
0.0219
0.0108
0.0328
0.0003
1.9975
Price
34.35
15.10
1,057.59
2.00
141,600.00
Turn
0.0082
0.0034
0.0458
0.0001
0.0630
Size
2,199,829,898
218,889,456
11,625,689,547
13,700.00
626,550,334,496
B/M
0.3822
0.1627
1.05236
-0.4569
2.2286
Volt
0.0308
0.0259
0.0213
0.0001
0.1011
Beta
0.8030
0.7914
2.2673
-1.2167
3.0542
Panel B. 1993-1999
MaxSpread
0.0587
0.0438
0.0531
0.0070
1.3158
Spread
0.0387
0.0268
0.0399
0.0038
1.0000
Price
25.12
14.13
526.21
2.00
78305.00
Turn
0.0048
0.0025
0.0113
0.0001
0.0360
Size
1,249,918,378
129,630,375
7,305,814,500
47,562.50
602,432,918,750
B/M
0.3334
0.1680
0.5970
-0.3180
2.2286
Volt
0.0329
0.0282
0.0216
0.0017
0.1011
Beta
0.8662
0.8007
3.5445
-1.2167
3.0542
Panel C. 2000-2012
0.0551
0.0005
1.9998
MaxSpread
0.0274
0.0102
0.0003
1.9975
Spread
0.0114
0.0039
0.0217
141,600.00
40.09
15.74
1,281.55
0.0001
Price
0.0044
0.0575
2.0000
0.0630
Turn
0.0103
13,607,367,039
13,700.00
626,550,334,496
Size
2,791,219,955
304,019,734
-0.4569
2.2124
0.4174
0.1590
1.2837
B/M
0.0995
0.0247
0.0210
0.0001
Volt
0.0296
Beta
0.7636
0.7868
0.7178
-0.9413
2.5661
24
Table 2. The Relation Between Maximum Bid-Ask Spreads and Other Stock Characteristics
The table reports the means of several stock characteristics across quintiles sorted by maximum bid-ask spreads. Panel A reports the summary statistics for the
entire sample time period (1993 to 2012), while Panels B and C show the summary statistics for the period of 1993 to 1999 and 2000 to 2012, respectively.
MaxSpread is the maximum daily (closing) percentage bid-ask spread during a particular month. Spread is the percentage bid-ask spread. Price is the closing
month price from CRSP. Turn is the share turnover of the average daily volume scaled by shares outstanding (in percent). Size is the market capitalization on
the last day of each month. B/M is the book-to-market ratio. Volt is the standard deviation of daily returns over a rolling six-month period. Beta is the beta
estimate obtained from the daily Capital Asset Pricing Model over a six-month rolling period. We report at the bottom of each panel, the difference between
extreme quintiles with corresponding p-values. *,**, and *** denote statistical significance at the 0.10, 0.05, and 0.01 levels.
Panel A. Entire Sample Time Period
MaxSpread
Spread
Price
Turn
Size
B/M
Volt
Beta
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
0.9860
7,772,834,233
0.3562
0.0234
0.0139
0.0083
0.0049
71.40
QI
0.0259
0.9390
1,802,648,775
0.3909
38.71
0.0103
0.0162
0.0093
Q II
0.8508
0.0284
0.0077
864,430,596
0.3446
32.47
Q III
0.0257
0.0147
0.7321
453,766,182
0.4525
0.0322
0.0056
0.0407
0.0235
17.98
Q IV
0.3694
0.0444
0.5072
108,610,447
0.0568
11.19
0.0035
0.1062
QV
QV–QI
Panel B. 1993-1999
QI
Q II
Q III
Q IV
QV
QV–QI
Panel B. 2000-2012
QI
Q II
Q III
Q IV
QV
QV–QI
0.0979
(<.0001)
0.0519
(<.0001)
-60.21
(<.0001)
-0.0104
(<.0001)
-7,664,223,786
(<.0001)
0.0132
(0.585)
0.0210
(<.0001)
-0.4788
(<.0001)
0.0152
0.0286
0.0442
0.0661
0.1392
0.0098
0.0180
0.0281
0.0437
0.0936
68.00
22.45
16.13
11.91
7.12
0.0068
0.0055
0.0048
0.0041
0.0031
4,669,632,807
938,487,705
395,591,386
206,262,255
42,107,303
0.3529
0.3839
0.3543
0.3229
0.2602
0.0218
0.0248
0.0293
0.0354
0.0533
0.9849
0.9587
0.9014
0.8284
0.6578
0.1240
(<.0001)
0.0838
(<.0001)
6.44
(<.0001)
-0.0037
(<.0001)
-4,627,525,504
(<.0001)
-0.0909
(<.0001)
0.0315
(<.0001)
-0.3271
(<.0001)
0.0041
0.0084
0.0141
0.0248
0.0856
0.0019
0.0039
0.0064
0.0110
0.0339
73.51
48.83
42.64
21.76
13.73
0.0183
0.0134
0.0096
0.0066
0.0037
9,704,439,377
2,340,781,448
1,156,115,363
607,878,412
150,034,108
0.3585
0.3956
0.3373
0.5529
0.4461
0.0243
0.0266
0.0278
0.0303
0.0388
0.9866
0.9267
0.8193
0.6721
0.4134
0.0815
(<.0001)
0.0320
(<.0001)
-59.78
(<.0001)
-0.0146
(<.0001)
-9,554,405,269
(<.0001)
0.0876
(0.023)
0.0145
(<.0001)
-0.5732
(<.0001)
25
Table 3. Max Spread Return Premium
The table reports various measures of next-month returns across quintiles sorted by maximum bid-ask spreads in
month t. Panel A reports the summary statistics for the entire sample time period (1993 to 2012), while Panels B
and C show the summary statistics for the period of 1993 to 1999 and 2000 to 2012, respectively. Column [1]
reports CRSP raw returns. Column [2] shows the results for excess returns or the difference between raw returns
and monthly risk-free rates. Column [3] presents the results from adjusted returns, or the difference between raw
returns and value-weighted market returns. In columns [4] through [6], we report estimated intercept from the
different variants of the following equation.
Excess Returni,t+1 = α + β1MRPt+1 + β2SMBt+1 + β3HMLt+1 + β4UMDt+1 + β5LIQt+1 + εi,t+1
The dependent variable is excess return for stock i in month t+1. The independent variables include the market
risk premium (MRP), the small-minus-big risk factor (SMB), the high-minus-low risk factor (HML), the Carhart
(1997) up-minus-down risk factor (UMD), and Pastor-Stambaugh liquidity risk factor (LIQ). 3F Alpha is the
intercept from estimating the above equation but excluding UMD and LIQ. 4F Alpha is the estimated intercept
from the above equation without including LIQ. 5F Alpha is the intercept from estimating the full version of the
above equation. We report these measures of returns across quintiles sorted by max spreads along with differences
between extreme quintiles with corresponding p-values. *,**, and *** denote statistical significance at the 0.10,
0.05, and 0.01 levels.
Panel A. Entire Time Period
Raw Returns
Excess Rets
Adj. Rets
3F Alpha
4F Alpha
5F Alpha
[1]
[2]
[3]
[4]
[5]
[6]
0.0011
0.0014
0.0010
0.0012
QI
0.0102
0.0076
0.0015
0.0021
0.0019
Q II
0.0113
0.0087
0.0025
0.0030
0.0120
0.0093
0.0032
0.0019
0.0030
Q III
0.0099
0.0037
0.0028
0.0042
0.0043
Q IV
0.0125
0.0088
0.0089
0.0105
0.0117
QV
0.0175
0.0149
0.0073
(<.0001)
Panel B. 1993-1999
QI
0.0148
Q II
0.0137
Q III
0.0136
Q IV
0.0153
QV
0.0260
0.0073
(<.0001)
0.0074
(<.0001)
0.0079
(<.0001)
0.0093
(<.0001)
0.0106
(<.0001)
0.0110
0.0098
0.0098
0.0114
0.0221
-0.0018
-0.0030
-0.0030
-0.0014
0.0094
-0.0010
-0.0016
-0.0008
0.0018
0.0146
-0.0008
0.0003
0.0022
0.0050
0.0182
-0.0008
0.0003
0.0023
0.0050
0.0189
0.0112
(<.0001)
Panel C. 2000-2012
QI
0.0073
Q II
0.0098
Q III
0.0109
Q IV
0.0107
QV
0.0122
0.0111
(<.0001)
0.0112
(<.0001)
0.0156
(<.0001)
0.0190
(<.0001)
0.0197
(<.0001)
0.0055
0.0080
0.0091
0.0089
0.0104
0.0035
0.0060
0.0071
0.0069
0.0084
0.0027
0.0036
0.0041
0.0044
0.0068
0.0028
0.0036
0.0041
0.0044
0.0068
0.0028
0.0033
0.0037
0.0043
0.0074
0.0049***
(<.0001)
0.0049***
(<.0001)
0.0041***
(<.0001)
0.0040***
(<.0001)
0.0046***
(<.0001)
Q V-Q I
Q V-Q I
Q V-Q I
0.0049***
(<.0001)
26
Table 4. Fama-MacBeth Regressions
The table reports the results from estimating the following equation using pooled stock-month observations.
Returni,t+1 = β0 + β1MaxSpreadi,t + β2Spreadi,t + β3Turni,t + β4ln(Sizei,t) + β5B/Mi,t + β6Betai,t + β7Volti,t +
β8Momi,t + εi,t+1
The dependent variable is the raw return for stock i in month t+1. The independent variables, which are measured
in month t, including the following: MaxSpread is the maximum daily (closing) percentage bid-ask spread during a
particular month. Spread is the percentage bid-ask spread. Turn is the share turnover of the average daily volume
scaled by shares outstanding (in percent). Ln(Size) is the natural log of market capitalization on the last day of each
month. B/M is the book-to-market ratio. Beta is the beta estimate obtained from the daily Capital Asset Pricing
Model over a six-month rolling period. Volt is the standard deviation of daily returns over a rolling six-month period.
Mom is the cumulative return for stock i during month t-12 to t-2. We estimate the above equation using a FamaMacBeth approach and report average coefficients with p-values obtained from Newey-West (1987) standard errors.
*,**, and *** denote statistical significance at the 0.10, 0.05, and 0.01 levels.¥
[1]
[2]
[3]
[4]
[5]
Intercept
0.0094***
0.0094***
0.0160***
0.0160***
0.0149***
(0.006)
(0.006)
(0.002)
(0.003)
(0.005)
MaxSpreadi,t
0.0812***
0.0818***
0.0476**
(<.0001)
(<.0001)
(0.017)
Spreadi,t
0.1524***
0.1412***
0.0618
(<.0001)
(0.0003)
(0.207)
Turni,t
-0.0817
-0.0713
-0.0695
(0.187)
(0.246)
(0.256)
Ln(sizei,t)
-0.0006*
-0.0006*
-0.0006
(0.062)
(0.072)
(0.104)
B/Mi,t
-0.0001***
-0.0001***
-0.0001***
(0.007)
(0.006)
(0.006)
Betai,t
0.0004
0.0004
0.0005
(0.713)
(0.679)
(0.631)
Volti,t
-0.0360
-0.0260
-0.0325
(0.636)
(0.732)
(0.670)
Momi,t
0.0076***
0.0074***
0.0075***
(0.0001)
(0.0002)
(0.0002)
¥
VIFs for MaxSpread and Spread are 3.43 and 3.44, respectively
27
Table 5. Fama-MacBeth Regressions by Time Period
The table reports the results from estimating the following equation using pooled stock-month observations for the
time period 1993 to 1999 (columns [1] and [2]) and the time period 2000 to 2012 (columns [3] and [4]).
Returni,t+1 = β0 + β1MaxSpreadi,t + β2Spreadi,t + β3Turni,t + β4ln(Sizei,t) + β5B/Mi,t + β6Betai,t + β7Volti,t +
β8Momi,t + εi,t+1
The dependent variable is the raw return for stock i in month t+1. The independent variables, which are measured
in month t, including the following: MaxSpread is the maximum daily (closing) percentage bid-ask spread during
a particular month. Spread is the percentage bid-ask spread. Turn is the share turnover of the average daily volume
scaled by shares outstanding (in percent). Ln(Size) is the natural log of market capitalization on the last day of each
month. B/M is the book-to-market ratio. Beta is the beta estimate obtained from the daily Capital Asset Pricing
Model over a six-month rolling period. Volt is the standard deviation of daily returns over a rolling six-month
period. Mom is the cumulative return for stock i during month t-12 to t-2. We estimate the above equation using a
Fama-MacBeth approach and report average coefficients with p-values obtained from Newey-West (1987) standard
errors. *,**, and *** denote statistical significance at the 0.10, 0.05, and 0.01 levels.
1993-1999
2000-2012
[1]
[2]
[3]
[4]
[5]
[6]
Intercept
0.0098*
0.0108**
0.0085
0.0092**
0.0091***
0.0184***
(0.051)
(0.037)
(0.280)
(0.045)
(0.055)
(0.009)
MaxSpreadi,t
0.1223***
0.0329
0.0588***
0.0555**
(<.0001)
(0.240)
(0.001)
(0.039)
Spreadi,t
0.1518***
0.1204***
0.1528***
0.0300
(<.0001)
(0.008)
(0.001)
(0.675)
Turni,t
0.0497
-0.1342**
(0.697)
(0.037)
Ln(sizei,t)
-0.0003
-0.0007
(0.595)
(0.103)
B/Mi,t
-0.0002***
-0.0001
(0.0004)
(0.233)
Betai,t
0.0006
0.0004
(0.541)
(0.768)
Volti,t
0.0312
-0.0671
(0.770)
(0.514)
Momi,t
0.0125***
0.0048*
(<.0001)
(0.085)
28
Table 6. Fama-MacBeth Regressions while including Amihud’s (2002) Illiquidity
The table reports the results from estimating the following equation using pooled stock-month observations for the
time period 1993 to 1999 (columns [1] and [2]) and the time period 2000 to 2012 (columns [3] and [4]).
Returni,t+1 = β0 + β1MaxSpreadi,t + β2Illiqi,t + β3Spreadi,t + β4Turni,t + β5ln(Sizei,t) + β6B/Mi,t + β7Betai,t + β8Volti,t
+β9Momi,t + εi,t+1
The dependent variable is the raw return for stock i in month t+1. The independent variables, which are measured
in month t, including the following: MaxSpread is the maximum daily (closing) percentage bid-ask spread during a
particular month. Illiq is average of Amihud’s (2002) illiquidity, which is the ratio of the absolute value of daily
returns scaled by daily volume (multiplied by 100,000). Spread is the percentage bid-ask spread. Turn is the share
turnover of the average daily volume scaled by shares outstanding (in percent). Ln(Size) is the natural log of market
capitalization on the last day of each month. B/M is the book-to-market ratio. Beta is the beta estimate obtained
from the daily Capital Asset Pricing Model over a six-month rolling period. Volt is the standard deviation of daily
returns over a rolling six-month period. Mom is the cumulative return for stock i during month t-12 to t-2. We
estimate the above equation using a Fama-MacBeth approach and report average coefficients with p-values obtained
from Newey-West (1987) standard errors. *,**, and *** denote statistical significance at the 0.10, 0.05, and 0.01
levels.
[1]
[2]
[3]
[4]
Intercept
0.0128***
0.0095***
0.0162***
0.0153***
(0.0002)
(0.006)
(0.002)
(0.004)
MaxSpreadi,t
0.0784***
0.0795***
0.0446**
(<.0001)
(<.0001)
(0.026)
Illiqi,t
0.0031***
0.0013*
0.0021
0.0016
(<.0001)
(0.068)
(0.120)
(0.221)
Spreadi,t
0.0607
(0.239)
Turni,t
-0.0794
-0.0655
(0.199)
(0.283)
Ln(sizei,t)
-0.0007*
-0.0006*
(0.057)
(0.093)
B/Mi,t
-0.0001***
-0.0001***
(0.007)
(0.007)
Betai,t
0.0004
0.0005
(0.691)
(0.642)
Volti,t
-0.0398
-0.0366
(0.602)
(0.632)
Momi,t
0.0076***
0.0075***
(0.0001)
(0.0001)
29
Table 7. Fama-MacBeth Regressions while including the Skewness of Bid-Ask Spreads
The table reports the results from estimating the following equation using pooled stock-month observations for the
time period 1993 to 1999 (columns [1] and [2]) and the time period 2000 to 2012 (columns [3] and [4]).
Returni,t+1 = β0 + β1SpreadSkewi,t + β2Spreadi,t + β3Illiqi,t + β4Turni,t + β5ln(Sizei,t) + β6B/Mi,t + β7Betai,t + β8Volti,t
+β9Momi,t + εi,t+1
The dependent variable is the raw return for stock i in month t+1. The independent variables, which are measured
in month t, including the following: SpreadSkew is the skewness of percentage bid-ask spread during a particular
month. Spread is the percentage bid-ask spread. Illiq is average of Amihud’s (2002) illiquidity, which is the ratio of
the absolute value of daily returns scaled by daily volume (multiplied by 100,000). Turn is the share turnover of the
average daily volume scaled by shares outstanding (in percent). Ln(Size) is the natural log of market capitalization
on the last day of each month. B/M is the book-to-market ratio. Beta is the beta estimate obtained from the daily
Capital Asset Pricing Model over a six-month rolling period. Volt is the standard deviation of daily returns over a
rolling six-month period. Mom is the cumulative return for stock i during month t-12 to t-2. We estimate the above
equation using a Fama-MacBeth approach and report average coefficients with p-values obtained from Newey-West
(1987) standard errors. We include all observations in columns [1] and [2] and estimate the above equation for the
1990s in column [3] and the 2000s in column [4]. *,**, and *** denote statistical significance at the 0.10, 0.05, and
0.01 levels.¥
Intercept
SpreadSkewi,t
Spreadi,t
Illiqi,t
Turni,t
Ln(sizei,t)
B/Mi,t
Betai,t
Volti,t
Momi,t
¥
All Observation
[1]
[2]
0.0156***
0.0159***
(0.004)
(0.003)
0.0005**
0.0005**
(0.014)
(0.011)
0.1449***
0.1422***
(0.0002)
(0.001)
0.0015
(0.248)
-0.0704
-0.0663
(0.251)
(0.277)
-0.0006*
-0.0007*
(0.077)
(0.070)
-0.0001***
-0.0001***
(0.007)
(0.007)
0.0004
0.0004
(0.691)
(0.704)
-0.0280
-0.0339
(0.710)
(0.655)
0.0074***
0.0074***
(0.0002)
(0.0002)
VIFs for SpreadSkew and Spread are 1.0813 and 1.6390, respectively
30
Table 8. Fama-MacBeth Regressions while including Max Spread Residuals
The table reports the results from estimating the following equation using pooled stock-month observations for the
time period 1993 to 1999 (columns [1] and [2]) and the time period 2000 to 2012 (columns [3] and [4]).
Returni,t+1 = β0 + β1MaxSpread*i,t + β2Spreadi,t + β3Illiqi,t + β4Turni,t + β5ln(Sizei,t) + β6B/Mi,t + β7Betai,t + β8Volti,t
+β9Momi,t + εi,t+1
The dependent variable is the raw return for stock i in month t+1. The independent variables, which are measured
in month t, including the following: MaxSpread* is the residual from a simple regression of MaxSpread on Spread
by stock. Spread is the percentage bid-ask spread. Illiq is average of Amihud’s (2002) illiquidity, which is the ratio
of the absolute value of daily returns scaled by daily volume (multiplied by 100,000). Turn is the share turnover of
the average daily volume scaled by shares outstanding (in percent). Ln(Size) is the natural log of market
capitalization on the last day of each month. B/M is the book-to-market ratio. Beta is the beta estimate obtained
from the daily Capital Asset Pricing Model over a six-month rolling period. Volt is the standard deviation of daily
returns over a rolling six-month period. Mom is the cumulative return for stock i during month t-12 to t-2. We
estimate the above equation using a Fama-MacBeth approach and report average coefficients with p-values obtained
from Newey-West (1987) standard errors. We include all observations in columns [1] and [2] and estimate the above
equation for the 1990s in column [3] and the 2000s in column [4]. *,**, and *** denote statistical significance at the
0.10, 0.05, and 0.01 levels.¥
Intercept
MaxSpread*i,t
Spreadi,t
Illiqi,t
Turni,t
Ln(sizei,t)
B/Mi,t
Betai,t
Volti,t
Momi,t
¥
All Observation
[1]
[2]
0.0162***
0.01065***
(0.002)
(0.002)
0.0710***
0.0683***
(<.0001)
(0.0001)
0.1467***
0.1439***
(0.0002)
(0.001)
0.0015
(0.286)
-0.0737
-0.0694
(0.232)
(0.258)
-0.0007*
-0.0007*
(0.066)
(0.059)
-0.0001***
-0.0001**
(0.009)
(0.010)
0.0005
0.0005
(0.610)
(0.616)
-0.0282
-0.0337
(0.712)
(0.659)
0.0075***
0.0075***
(0.0001)
(0.0001)
VIFs for MaxSpread* and Spread are 1.0044 and 1.5761, respectively
31
0.10
Max Spreads and Average Spreads Across Sample Time Period
0.09
0.08
Max Spread
0.07
Spread
0.06
0.05
0.04
0.03
0.02
0.01
0.00
1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
Figure 1. The figure shows average maximum daily bid-ask spreads and average percentage bid-ask spreads during
the sample time period.
32