DBJ Discussion Paper Series, No. 1802

Liquidity Risk?

Jeffrey Pontiff(Boston College)
Rohit Singla(University of Michigan)

February 2019

Discussion Papers are a series of preliminary materials in their draft form. No quotations,reproductions or circulations should be made without the written consent of the authors in orderto protect the tentative characters of these papers. Any opinions, findings, conclusions or
recommendations expressed in these papers are those of the authors and do not reflect the viewsof the Institute.

Liquidity Risk?ψ
Jeffrey Pontiff
andRohit Singla
February 5, 2019
Abstract
We revisit the role of liquidity risk. We successfully replicate Pastor andStambaugh’s (2003) gamma liquidity risk index and, within their time period,
concur with their risk premium estimate. An out-of-their-time-period analysis finds
post-time-period returns that are higher and pre-time-period returns that are lower
than in-time-period returns. Modest variations to the index that are intended to
improve power—such as value weighting, including zero volume days, including all
stock price levels, and a modification intended to reduce estimation error—all cast
doubt on whether the gamma premium is compensation for liquidity risk. We create
five alternative liquidity risk indices from various popular liquidity proxies. Using
time-series that start in either 1932 or 1968, none of the ten specifications produce
statistically significant risk premia.
Keywords: Liquidity, Risk, Factor Model, Replication.

JEL Code: G00, G14, L3, C1
ψPontiff (pontiff@bc.edu) is at Boston College. Singla (singlaro@umich.edu) is at the

University of Michigan. Part of this work was completed by Pontiff while he was a Shimomura Fellow

at the Development Bank of Japan. We thank George Aragon, Joel Hasbrouck, Jennifer Koski, Lubos
Pastor, Ronnie Sadka, Robert Stambaugh, Kumar Venkataraman (referee), and Ivo Welch (editor) for

valuable conversations. Lubos Pastor and Robert Stambaugh deserve special thanks for providing uswith their original code. We thank Ming Lu for coding advice and Maximilan Papile for research
assistance. SAS code that was used to generate this paper’s results is available on Jeffrey Pontiff’s

Boston College website. Both co-authors are aware of concurrent work by Professor Robert NovyMarx. We want our analysis to be independent and uninfluenced by Professor Novy-Marx’s paper. As
such, we have not read his paper and we are unaware of his results.

“Liquidity” describes the extent to which an asset can be quickly tradedwithout too much of a price concession. Although the topic of liquidity is ubiquitousin the finance and economics literature, there is much disagreement on a precise
measurement of it (see, for example, Goyenko, Holden, and Trzcinka, 2009)

Pastor and Stambaugh (2003) spawned a new literature by introducing theconcept of “liquidity risk.” If liquidity contains a systematic component, security
covariance with systematic liquidity risk might be priced. Investors demand higher

expected returns for securities that deliver poor performance in in bad times. Thus,

securities with lower returns in times of low liquidity will have higher equilibrium
expected returns. Pastor and Stambaugh introduce a new measure of liquidity,

gamma, which measures securities’ recovery from volume-related return shocks.

They show that stocks with higher covariance to innovations of average, marketwide gamma have higher expected returns.

Using the Pastor-Stambaugh gamma index as a launching pad and focusing

on simple measures of priced risk, our broad goal is to better understand the role ofliquidity risk in asset pricing. Our investigation is comprised of two steps. First, werevisit the specifics of Pastor and Stambaugh’s liquidity index. Throughout ourinvestigation we focus on direct implications that are likely to broadly steer liquidity
risk-research. As the original, impactful liquidity risk paper, Pastor and Stambaugh(PS) provide a good starting place. The PS gamma index requires only daily CRSP

data and the methodology can produce an index that starts 1920’s. Liquidity risk

indices such as Sadka (2006), although appealing, require intraday data that is only
available starting in the 1980’s. We also do not consider Acharya and Pedersen1(2005), which is more grounded in theory, since they produce a more complicatedmulti-measure expected return that has not been adopted by empirical research to
the extent of Pastor and Stambaugh.

Our estimation precisely replicates the Pastor and Stambaugh gamma index(and we make this code available). Keeping with our focus, we evaluate their index

based on direct tests of whether unconditional beta risk with respect to the index
commands a return premium. We examine both the robustness of their index with
their time period as well as the robustness of the index to construction decisions
that are intended to increase statistical power. We find that gamma-risk continues
to be associated with positive returns after the end of the Pastor and Stambaugh’s

original time period. On the other hand, our findings pose challenges forinterpreting gamma-risk-related returns as compensation for liquidity risk. We

consider four simple modifications of the gamma index that are expected tostrengthen the detection of priced liquidity risk. Pastor and Stambaugh’s estimationof gamma omits data on zero-volume days and does not include stocks with prices

under $5 or over $1,000. Our first modification includes days with no trading

volume. Our second modification includes all firms, regardless, of share price level.
Pastor and Stambaugh’s gamma index is an equal-weighted average of firm-level

gammas. Our third modification constructs a value-weighted index. The original
gamma estimation consumes a degree of freedom estimating an intercept. A wide

range of theories imply that this estimate should be equal to zero. Our fourthmodification imposes this restriction on the estimation. Although we expect all four

modifications to strengthen the detection of priced liquidity risk, they all result in2

lower return premium estimates and none of the modifications command statisticalsignificance at the 5 percent level.

The second step of our investigation departs from the gamma index andconsiders liquidity risk more broadly. Guided by the literature, we construct four
liquidity risk indices based on popular liquidity measures and we construct a fifth
new liquidity risk measure that avoids theoretical pitfalls of common measures. The

liquidity indices and the returns associated with these indices display remarkably

low correlations. Ten out of ten specifications fail to generate a statisticallysignificant liquidly risk premium. Our findings suggest that it is premature toconclude liquidity risk is priced or that we even have a reliable liquidity risk
measure.

1. Replicating the Pastor-Stambaugh gamma index
Pastor and Stambaugh estimate the following daily return-generatingprocess,𝑒𝑒𝑒𝑒𝑟𝑟𝑖𝑖,𝑑𝑑+1,𝑡𝑡= 𝜃𝜃𝑖𝑖,𝑡𝑡 + ∅𝑖𝑖,𝑡𝑡 𝑟𝑟𝑖𝑖,𝑑𝑑,𝑡𝑡 + 𝛾𝛾𝑖𝑖,𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �𝑟𝑟𝑖𝑖,𝑑𝑑,𝑡𝑡
� ∙ 𝑣𝑣𝑖𝑖,𝑑𝑑,𝑡𝑡 + 𝜖𝜖𝑖𝑖,𝑑𝑑,𝑡𝑡 𝑑𝑑 = 1, … 𝐷𝐷,(1)This regression is estimated at the firm-month level. d denotes the day in month t.

𝑒𝑒
We utilize Pastor and Stambaugh’s notation. 𝑟𝑟𝑖𝑖,𝑑𝑑+1,𝑡𝑡= 𝑟𝑟𝑖𝑖,𝑑𝑑,𝑡𝑡 − 𝑟𝑟𝑚𝑚,𝑑𝑑,𝑡𝑡 , where 𝑟𝑟𝑖𝑖,𝑑𝑑,𝑡𝑡 and𝑟𝑟𝑚𝑚,𝑑𝑑,𝑡𝑡 are the respective returns on stock i and the CRSP value-weighted index. 𝑣𝑣𝑖𝑖,𝑑𝑑,𝑡𝑡
is the dollar volume and 𝜖𝜖𝑖𝑖,𝑑𝑑,𝑡𝑡 is the residual of stock i on day d in month t.

Our estimation follows Pastor and Stambaugh’s lead. Individual regressions

are estimated for each stock-month. Thus, the three parameters estimated fromequation (1) will tend to be noisy, since the typical month contains twenty trading3

days. If a stock’s first or last month’s data in CRSP is partial, it is excluded. All stocks

must have a CRSP share code of 10 or 11, and the must be listed on either the New

York Stock (NYSE) or American Stock (AMEX) exchanges. We (like Pastor andStambaugh) do not include NASDAQ stocks to estimate gammas during this period
since there was variation in the conventions that NASDAQ used to compute volume.

All stock-months must contain 16 or more observations. The gamma estimate foreach month does not have a look-ahead bias, in that data used is from the current
month and the last trading day of the previous month. Following the constructionused to post data to Professor Stambaugh’s and Professor Pastor’s websites (andWRDS), we make an exception for September of 2001. During this month only 11 ormore stock-month level observations are required. This accommodates exchange

closing due to the New York City terrorist attacks on September 11.

Our estimation in this section uses two filters that were used in the original

estimation, although Section 3 reconsiders these filters. First, we only use stocks

with closing stock prices in the previous month that are greater than or equal to $5
and less than or equal to $1,000. Second, we only use stocks with non-zero trading
volume. The non-zero trading volume restriction is not explained in the original
paper, but this restriction is not unheard of in the liquidity literature. The Amihud(2002) measure also only uses observations from non-zero volume days.

Gamma critique. The crucial parameter that equation (1) estimates is stock-level monthly liquidity, 𝛾𝛾𝑖𝑖,𝑡𝑡 . γ is usually negative. On days where a stock’s return inexcess of the market is negative (positive), higher volume is associated with a higher(lower) excess return in the following day. γ’s are intended to measure liquidity. The
4

more negative a stock’s γ, the more illiquid the stock. Although Pastor andStambaugh focus on Grossman and Miller (1988) and Campbell, Grossman, andWang (1993), their framework fits broadly into the dealer inventory models that goback as far as Ho and Stoll (1981, 1983). Liquidity provider risk aversion is the coreassumption behind these models. Market clearing implies a lower (higher) price forthe risky asset, accompanied with higher (lower) expected returns. γ reflects thereaction of expected returns to volume-related price movements. Put another way,high expected returns induce the risk-averse market maker to provide liquidity. The

decision to estimate γ using return bounce-back over the next day as the dependent

variable is a judgment call: the parameter could have been estimated with rolling
multi-day returns as the dependent variable.

Another class of liquidity models results from asymmetric information

between the liquidity provider and informed traders (such as Kyle, 1985). Bothinventory models and asymmetric information models predict contemporaneous

relation between absolute returns and volume. This relation is at the core of thedefinition in the first sentence of this paper. Negative gammas are artifacts ofinventory models but not asymmetric information models. For example, in Kyle

(1985) and Glosten-Milgrom (1985)—gamma is precisely equal to zero. In thecontext of asymmetric information models, a non-zero gamma implies a trading
strategy. In this case, negative gamma is a market inefficiency, and an investor can
expect to profit by buying (selling) high volume stocks with negative (positive)
returns5

Like PS, we compute a series of cross-sectional averages of γ, γ�𝑡𝑡 , by dividing

the sum of γ’s in a given month by the number of cross-sectional observations.
These cross-sectional averages are used to construct an innovation measure(following PS’s equation 6).
𝑚𝑚

1

𝑁𝑁

𝑡𝑡
∆𝛾𝛾�𝑡𝑡 = �𝑚𝑚 𝑡𝑡 � �𝑁𝑁 � ∑𝑖𝑖=1
�𝛾𝛾�𝑖𝑖,𝑡𝑡 − 𝛾𝛾�𝑖𝑖,𝑡𝑡−1 �
1

𝑡𝑡

(2)

mt is the market value of all stocks used in the index in the previous month, and m1 isthe value of all stocks used in the index in August of 1962. Gamma communicatesthe return-reversal cost of a $1 trade in stock i. Scaling the difference by the total

market capitalization makes it easier to compare gamma shocks in different periods.
For example, this adjustment rids the differences from variation that is mechanicallyattributable to market movements.

We use PS’s equation (7), to estimate the following time series process,𝑚𝑚∆𝛾𝛾�𝑡𝑡 = 𝑎𝑎 + 𝑏𝑏∆𝛾𝛾�𝑡𝑡−1 + 𝑐𝑐 � 𝑚𝑚𝑡𝑡−1 � 𝛾𝛾�𝑡𝑡−1 + 𝜇𝜇𝑡𝑡
𝑡𝑡(3)This specification produces the series, 𝜇𝜇𝑡𝑡 , which is an estimate of the innovation to
the liquidity series. The Pastor-Stambaugh gamma index, Lt, is the fitted value of 𝜇𝜇𝑡𝑡 ,scaled by 100. We investigate various liquidity measures and we also investigate theextent to which the PS index proxies for liquidity risk. In the interest of clarity, we
refer to the PS index as the “gamma index.”

Using the data from August, 1962 through December 2017, we compare ourestimates of the level of gamma (the cross-sectional average of 𝛾𝛾�𝑖𝑖,𝑡𝑡 ’s) and the gamma

index, Lt, to the data available on Professor Lubos Pastor’s website. Our estimatesare virtually identical, with a correlation of one (up to 5 significant digits).
6

Next, we estimate individual stocks’ sensitivities to gamma risk. Following

Pastor and Stambaugh, we apply the same exclusions that were used to create theliquidity index. Since estimating sensitivity to gamma risk does not require volume
data, we include NASDAQ stocks and we require that stocks have complete returndata for the last 60 months. The liquidity index is re-estimated, as per equation (3),each year such that no future information is used. The following factor model isestimated at each year-end.

𝑟𝑟𝑖𝑖,𝑡𝑡 = 𝛽𝛽𝑖𝑖0 + 𝛽𝛽𝑖𝑖𝐿𝐿 𝐿𝐿𝑡𝑡 + 𝛽𝛽𝑖𝑖𝑀𝑀 𝑀𝑀𝑀𝑀𝑀𝑀𝑡𝑡 + 𝛽𝛽𝑖𝑖𝑆𝑆 𝑆𝑆𝑆𝑆𝑆𝑆𝑡𝑡 + 𝛽𝛽𝑖𝑖𝐻𝐻 𝐻𝐻𝐻𝐻𝐻𝐻𝑡𝑡 + 𝜖𝜖𝑖𝑖,𝑡𝑡(4)Equation (4) is the well-known Fama-French three factor model with theaddition of gamma innovation risk. For each year-end, we use 𝛽𝛽𝑖𝑖𝐿𝐿 to sort stocks into

10 value-weighted portfolios. If the difference in these portfolio returns is non-zero,
this is evidence that liquidity risk is “priced.” This exercise follows pages 673 to 677
of Pastor and Stambaugh.

Table 1 compares our estimate of the return to this portfolio during the sametime period as Pastor and Stambaugh’s paper and with the data available on
Professor Lubos Pastor’s website. We estimate a liquidity premium of 3.91% per

year, with a t-statistic of 2.00. Both parameters are to identical to estimatescalculated from the website. These set of results are very similar to the finding
reported in Table 8 of Pastor and Stambaugh, which reports a liquidity premium of

4.15% with a t-statistic of 2.08. This minor difference is almost certainly

attributable to year-to-year corrections to the CRSP data.
(Table 1)

7

Pastor and Stambaugh devote considerable effort to construct predictedliquidity betas. They estimate time-series and cross-sectional variation in liquidity

betas as a function of historical liquidity betas and various other right hand side

variables such as stock momentum, volume, and return volatility. We have notpursued this avenue for two reasons. First, studies influenced by Pastor andStambaugh almost never follow this approach. We read the 40 most cited papers(according to Google Scholar) that, in turn, cite Pastor Stambaugh. None of these

papers use the predicted beta approach. Second, some of the conditioning variables
used by Pastor and Stambaugh have already been shown to predict cross-sectional
returns. As such, our analysis to focuses on the simple, unconditional liquidity betas
used by the literature.

2. Out-of-Time-Period Stability
We compare the impact of gamma during the time period of Pastor andStambaugh’s, before their data starts, and after their time period ends. Our before-time-period investigation is possible since CRSP daily is currently available before

1962, whereas when Pastor and Stambaugh was written, this data was unavailable.

To conduct this analysis, we re-estimate the gamma innovation measure

every year, only using historic data. Thus, in the 1960’s estimation of equation (3)reflects data going back to the 1920s. To the extent, that the equation (3)parameters are stable, this is expected to produce a better estimate of gamma risk
than the estimate in Section 1.

8

Figure 1 plots the aggregate gamma index for the entire series. Pastor andStambaugh considered many specification variations. They settled on the equation(1) specification, in part, based on the resulting index’s low level during the stock

market crash of 1987. Therefore, pre- and post-time-period evidence is particularly

valuable. Figure 1 shows that the gamma index is much less volatile, after their time

period, and in particular, during the pre-1962 period. Despite equation 3’s attempt
to whiten the gamma process, the series has a first-order autocorrelation of 0.18.

The financial drama of the great crash of 1929 and the great depression fail toregister liquidity levels that could be construed as being low during the 60’s through90’s. Post 1999, the financial crisis of 2007 to 2008 received incredible attention
from the press and academia as being a period of market failure. The good news forthe gamma index is that low index levels during this period are noticeable. The lowlevels are not confined to one month, rather they span several months.

The most negative level of the post-1999 gamma index is October 2002. This

period is challenging, in that nothing in the media during this time seems consistent
with a liquidity explanation. 1(Figure 1 goes here)

An analysis outside of Pastor-Stambaugh’s original time period is presented

in Table 2. Our estimate of the price of gamma-risk during the Pastor-Stambaugh

time period has a negligible drop of 9 basis points. This occurs since the liquidity
index varies slightly from the Table 1 index since the estimation now starts in 1926.
Before the start of the Pastor and Stambaugh time period, we estimate the price of
1 We thank Brian Weller for insightful feedback about the behavior of the gamma index during thisperiod.

9

gamma risk as -2.72 percent and statistically insignificant from zero. After the end of

their time period, we estimate a 5.56 percent gamma risk premium with a t-statisticof 1.84. Using the entire time-series, we estimate an annual gamma risk premium of

1.45 percent with a t-statistic of 1.19. Thus, using data for the longest possible timeseries, we are unable to reject the null that gamma risk is not priced. The difference

between the early data and the more-recent data is dramatic. For example, focusing

on the 1968 to 2017 time period, the liquidity premium yields a hefty 4.45 percent

annual returns and clearly rejects the null (t-statistic=2.67). Kamara et al. (2016)

also document a time trend of increasing liquidity premium.
(Table 2)

McLean and Pontiff (2016) examine 97 variables documented to predict

returns in academic papers. Using indicator variables, they estimate statistical bias

and predictability decay from publication-informed trading. Their sample did notinclude the risk premium on the Pastor and Stambaugh liquidity measure. McLean

and Pontiff estimate a decay of 26% after the original sample ends and a total decayof 58% post publication. Pastor and Stambaugh’s study is unusual, in that Table 2

shows that the estimate of the return premium on gamma innovations increase
post-sample by 46% [(5.56-3.82)/3.82)].

Readers will have different interpretations of Table 2. A reader who thinksthat compensation for gamma-risk should be relatively stable over time will focus

on the longest time-series and conclude that the price of gamma risk is insignificant

from zero and likely small. A reader who believes that market participants’ concerns
about liquidity risk should lead to financial innovation that, in turn, mitigates this10

risk (such as Alchian, 1950), is likely to be skeptical about whether gamma risk isliquidity risk, since the data before Pastor-Stambaugh’s time period displays anegative price of gamma-risk. A reader who believes that financial innovation (orperhaps government polices) has enabled liquidity risk to be an increasingly

systematic problem over time (for example, Taleb, 2012), will gravitate towards themore recent data and conclude that gamma risk is priced liquidity risk.

PS show that their gamma index is negatively correlated with market return

volatility. They report a correlation between their gamma index and the within-month daily standard deviation of the value-weighted market (presumably
restricted to AMEX and NYSE firms) of -0.57. During PS’s original time period, ourindex has a correlation -0.56 with within-month daily standard deviations of the

CRSP value-weighted index. Outside of PS’s original time period, we find a lesspronounced, albeit still negative, correlation of -0.38.

PS consider the extent to which correlations between stock market returns

and bond returns vary in normal periods versus periods that the gamma index

denotes as being illiquidity periods. They are interested in whether duringapparently illiquid periods there is a “flight to quality.” In low liquidity months they

expect correlations of stock returns with low-risk bond returns to be negative as

money flows between these investment classes.

Table 3 compares our in-time-period and out-of-time-period results withthose that PS report in their Table 1 2. Two caveats are in order. First, our analysis,
2 Table 1, panel A, in PS refers to their main time period as starting in January 1962. This is almost

certainly a typo, since availability of CRSP data would have prevented them from starting before

11like PS’s is asking a lot from the data. The number of observations in low gamma
subsamples is sparse—a little more than a dozen observations. Also, our results,

although in the spirit of Pastor and Stambaugh, should not be considered a

replication. Our index is constructed using an autocorrelation adjustment that is

estimated starting in 1926—whereas PS start in 1962. We use CRSP bond data,

whereas PS use Ibbotson Associates bond data. Our market return index is the CRSP

value-weighted index, whereas their value-weighted index is confined to NYSE and
AMEX listings.

(Table 3 goes here)

Like Pastor and Stambaugh, in Table 3 we call a month a “low liquidity”

month, if the gamma index is more than two standard deviations lower than theindex mean during the original PS time period Using CRSP data, we measure −∆𝑅𝑅𝑓𝑓,𝑡𝑡

as the next period’s return minus this period’s return on 30 day T-bills. Pastor andStambaugh’s version of this variable is last month’s treasury yield minus thismonth’s. Again, using CRSP we measure 𝑅𝑅𝐺𝐺𝐺𝐺,𝑡𝑡 as the return on 20 year Treasury
bonds, whereas Pastor and Stambaugh use the Ibbotson Associate return on long-term government bonds. We are unable to retrieve corporate bond returns, so they
are not included in our Table 3.

Panel A of Table 3 considers the same time period as PS. The estimates are

similar, but not exact. Like PS, we find that in low gamma months the correlation

between −∆𝑅𝑅𝑓𝑓,𝑡𝑡 and market returns is negative and in other months it is positive.

Using Fisher’s r-to-z transformation, our p-value on this difference is 0.19, whereas
August 1962. The number of observations that we report (starting in August 1962) corresponds towhat PS report.

12

PS report a bootstrap p-value of 0.09. We suspect that the difference in p-values

levels is attributable to the fact that our p-values assume a two-tailed test and PS

likely report values for a one-tail test. We find differences in correlations between

20 year bond returns and market returns in normal and low-gamma months thatare similar to PS, although not quite as pronounced. Our two-sided p-value on the
difference is 0.10, roughly twice the 0.045 reported by PS.

Column (3) considers the correlation between the changes in volume andmarket returns. PS report that their change is the average percentage equallyweighted change for NYSE-AMEX stocks. We report the median percentage changeof this variable on for all CRSP stocks with share codes of 10 or 11 3. Like PS, we findnegative correlations between change of volume and returns in low-gamma months,

and positive correlations in other months. This difference is statistically significant
at all standard levels.

Panel B repeats the Panel A statistics for data outside PS’ original time

period. We continue to use the mean and standard deviation during their time
period to sort gammas into normal and low months. The differences in correlation

between market returns and changes in Treasury bill prices are very similar to the

PS results during the same time period. This is consistent with PS’ appeal to “flightto quality,” although the 0.24 p-value is unable to reject the null. On the other hand,the difference in correlations switches signs for 20 year government bonds. Outsideof the PS time period, in low-gamma months, stock market returns are positively
correlated with long-term government bond returns, and in other months they are
3We focus on medians since percentage changes in volume are susceptible to extreme outliers, sincehigh volume can follow periods with no volume or very low volume.

13

negatively correlated. We are unable to use all the months outside of the PS time
period for this test, since long-term government bond data is not always available.

The volume evidence also varies. Both low-gamma and normal months exhibit

positive correlations with changes in volume.

What do these results tells us about PS’ liquidity risk measure? Regarding

capital flows from stocks to Treasury bills, although the inside/outside period point
estimates of are very close, both fail to reject the null. Regarding capital flows fromstocks to long-term government bonds, the outside period result flips. This result

challenges a “flight to quality” interpretation. The inside/outside volume differences
appear similar, suggesting consistency. This being noted, PS’ original investigationof volume is descriptive and not a test of a specific story.

3. Modifying the gamma index
In this section, we consider four simple modifications in the construction of

the Pastor-Stambaugh gamma index and the extent to which these modifications

influence our estimate of the gamma risk premium. We focus on modifications thatare expected to improve estimates of average returns based on the presumptionthat these returns are compensation for liquidity risk. These 4 modifications to the

gamma index (and the modification in Section 4) were selected before we knew how

the estimated price of risk would be affected, and all modifications that we
estimated are presented in this paper. 4

4Although we did not know how the modifications would affect the price of risk, for two

modifications we knew something about their correlation with the base-line PS gamma index. First,
PS report that the correlation between gamma index innovations and a value-weighted version is14

Including observations on zero-volume days. The estimation of equation (1)does not include data from zero volume days. While not discussed in Pastor andStambaugh, this decision is consistent with other liquidity measures such as that ofAmihud (2002). Since August 1962, zero volume days account for 7.8 percent of
CRSP daily data (with non-missing returns, non-missing prices, and a share code of

10 or 11). Lesmond, Holden, and Trczinka (1999) argue that zero volume days are a

critical consideration for measuring liquidity. On zero volume days, we follow CRSP

in calculating returns with bid-ask averages instead of trade prices. We expect that
inclusion of zero volume days will improve equation (1)’s estimation of liquidity
since doing so allows the estimation to use more data over a wider liquidity range.

Inclusion of all price levels. As noted by Demsetz (1968) Stock price levels

proxy for transaction costs and liquidity. Pastor and Stambaugh exclude stocks withprices that are either less than 5 dollars or greater than 1,000 dollars. Although PS

do not provide an explanation for this restriction, this restriction is common instudies that use stock price data. Authors such as Amihud (2002) attribute therestriction to concerns about relative bid-ask spreads being too big, especiallyduring periods of minimum tick sizes, and some authors (such as Boni and Womack,
2006) attribute the restriction to difficulties that traders have shorting stocks withprices under $5. The under $5 restriction eliminates 24.5% of the CRSP daily data(with non-missing returns, non-missing prices, and a share code of 10 or 11).

Including this data should improve the index for two reasons. First, additional data

0.77. Our own specification produces a correlation of 0.95. Second, when we first attempted toreplicate the gamma index we did not exclude zero volume days, and we realized that this produced
an index with very little correlation to the original index. We learned about this exclusion fromconversations with professors Pastor and Stambaugh.

15

should decrease estimation error. Second, stocks from these price levels are likely to
be very sensitive to changes in market-wide liquidity.

Value-weighted Index. Pastor and Stambaugh’s liquidity index is equal

weighted. They consider a value-weighted index, but they do not estimate the priceof liquidity risk relative to a value-weighted index. Their decision to focus on an
equal-weighted index is based on the equal-weighted index’s low levels during

times of purported low liquidity—October 1987 and September 1998. Almost all

theories of market equilibrium produce value-weighted pricing implications. For
example, representative agent models imply that the agent holds the value-weighted

market. Similarly, market clearing with multiple agents implies a value-weighted

equality. As such, we consider a value-weighted version of the Pastor andStambaugh index.

𝜃𝜃𝑖𝑖,𝑡𝑡 restricted to zero. Estimation of equation (1) produces parameter

estimates that are very noisy. Since August 1962, the average month has 21.0
trading days. Thus, in an average month (for stocks with a full array of data), 21

observations are used to estimate three parameters: 𝜃𝜃𝑖𝑖,𝑡𝑡 , ∅𝑖𝑖,𝑡𝑡 , and 𝛾𝛾𝑖𝑖,𝑡𝑡 . For somestock-months (even if we disregard September 2011), 16 observations are used toestimate these parameters. One way to estimate more efficient liquidity parameters,

𝛾𝛾𝑖𝑖,𝑡𝑡 , is to impose theoretically sensible restrictions on one of the other two
coefficients. We consider specifications where the intercept, 𝜃𝜃𝑖𝑖,𝑡𝑡 , is restricted to be

zero. This restriction implies that a stock with a zero return in given day, is expected
to have a return in the following day that equates to the return of the valueweighted market. This restriction is in line with previous research. Using monthly
16data, Simin (2008) shows that forecasts of individual stock returns that are equatedto the market return have lower mean-square-errors than estimates from asset

pricing models.

Using data from August 1962 to December 2017, Table 4 reports correlations

between the Pastor-Stambaugh index and the four modified indices (panel A),
correlations between long-short decile portfolio returns for this set of indices, andreturn premiums for long-short portfolio based on these indices. Panel A shows the

Pastor-Stambaugh index, the value-weighted modification, and the intercept

restricted version, all have correlations with one another in excess of 0.90. The
correlation between the PS index and value-weighted modified index is 0.95, which

is substantially higher than the 0.77 correlation that Pastor and Stambaugh
document between these indices using data that ends in 1999.
(Table 4 goes here)

Including all stocks regardless of price and including zero volume days

creates indices that are largely orthogonal to the remaining indices. For example,

the highest correlation, 0.27, is between the PS index and the modification that
includes zero volume days. The lowest correlation, 0.15, is between the modificationthat includes all stocks regardless of price with the intercept restricted modification,

Panel B looks at correlations between long-short extreme decile portfolio

returns, that are formed based on slope coefficients with these indices. The
population of stocks used to form these populations is the same for all indices,

except the indices that includes all stocks regardless of price. For this specification,

the long-short portfolio, like the index, includes stocks regardless of price level. The
17

correlations in panel B, with a few exceptions, tend to shrink compared with panel
A.

Panel C compares estimates of annual return premia from the long-shortdeciles portfolios. All modifications result in lower estimates of the liquidity risk
premium than the premium estimated from the PS index. All modifications produce

positive estimates of the risk premium, although the only modification withstatistically significant estimate is the intercept-restricted modification (t-statistic of1.89). The news from this panel is mixed. On one hand, a modification intended to
improve the power of the gamma estimation, continues to command a statisticallysignificant risk premium, albeit with a lower point estimate. On the other hand,modifications that are expected to do a better job measuring liquidity risk (such as

including all prices and zero volume days), and a specification expected to do a

better job capturing priced risk (value-weighted index), are unable to produce
statistically significant estimates of risk premia. An interpretation of these results isthat the Pastor-Stambaugh gammas convey information about expected returns, but

this information is either unrelated to liquidity or unrelated to priced risk. The
specifications that include all prices and zero volume days, challenge the notion thatthe gamma index is liquidity risk, while the value-weighted results cast doubt onwhether gamma risk is priced.

4. Liquidity risk beyond Pastor-Stambaugh
The literature has developed many proxies for liquidity. In this section, we

explore three liquidity proxies from the literature, a version of the gamma index
18

with two modifications, and a fifth proxy that we develop ourselves that is intended
to avoid some of the pitfalls of other measures.

In selecting liquidity measures, we focus on measures that only require CRSP

data. Measures that use TAQ data, such as Sadka (2006) are appealing, but data
availability prevents the creation of a long-time series of liquidity shocks. We seek
an even-playing field to compare measures. As such, the indices are calculated bytaking the market-capitalization-weighted average of each stock’s monthly change

in the three liquidity measures from the month before. Market capitalization is
calculated with last month’s price and shares outstanding. This decision to weightby market capitalization recognizes that illiquidity among large market

capitalization stocks is more likely to be a cause of economy-wide risk than small

stocks. Stocks are only excluded from each index if their CRSP share code is notequal to 10 or 11, if we are unable to calculate their liquidity measure in the current
or past month, if we are unable to calculate their market capitalization in theprevious month, or if they are not traded on the NYSE or AMEX exchanges.

Similar to the methodology used in the previous section to estimate the priceor risk of the PS gamma index, betas on the indices’ innovations are calculated inannual factor models that include the Fama-French three factors. Stocks areincluded from all exchanges as long as they have a full 60 months of data. Betas areused to allocate stocks in decile portfolios in the following year. The price of risk isthe annual return of the value-weighted portfolio that holds the top decile andshorts the value-weighted portfolio that holds the bottom decile. Our first four

liquidity measures are: the proportion of zero returns (Lesmond, Ogden, and19

Trcizinka, 1999), the Amihud liquidity measure (Amihud, 2002), relative bid-ask
spreads, and a hybrid index that avoids common pitfalls of other measures. For
comparison, we also include a variation of the PS gamma index that is constructedusing the methodology in Section 1, except that two modifications are included

simultaneously--value-weighting and inclusion all stocks regardless of price. For the
non-gamma indices, high values are associated with illiquidity, whereas for themodified gamma index high values are associated with liquidity. As such, we

multiply index innovation for non-gamma indices by negative one, such that thebetas of all indices can be interpreted as liquidity betas.

Proportion of zero returns. Lesmond, Ogden, and Trczinka (1999) develop atransaction cost proxy that is the proportion zero return days. Their insight is that

trade occurs when the value of information exceeds transaction costs. Variation intransaction costs causes illiquid securities to have a higher a proportion of zero
return days. Following Lesmond et al., the zero return index is based on the monthly
percentage of zero return days at the stock level.

Amihud Liquidity Measure. Amihud (2002) proposes a measure of priceimpact that is based on the ratio of absolute return and trading volume. Thismeasure is calculated as,
𝐴𝐴𝑖𝑖,𝑚𝑚 = 𝐷𝐷

1

𝑖𝑖,𝑚𝑚

𝐷𝐷

𝑖𝑖,𝑚𝑚
∑𝑡𝑡=1

�𝑟𝑟𝑖𝑖,𝑡𝑡 �

𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑖𝑖,𝑡𝑡

.

(xx)

𝐴𝐴𝑖𝑖,𝑚𝑚 is stock i’s Amihud measure for month m, Di,m is the number of days in month mwith both return and dollar volume data, ri,t is stock i’s return on day t, and Volumei,tis stock i’s dollar trading volume on day t. Like the Pastor and Stambaugh measure,

the Amihud measure does not use information from zero volume days. Thus, a20

disadvantage of the Amihud measure is that it assigns the same liquidity measure to

stocks with the same average ratio of absolute return to volume, despite the factsome stocks might trade much more frequently than the other. An advantage of the
Amihud measure is that it is constructed with contemporaneous returns andvolume. As mentioned earlier, a necessary condition for all theories of illiquidity is acontemporaneous relation between absolute returns and volume.

Bid-ask spread index. Amihud and Mendelsohn (1996) develop a transaction

cost asset pricing model. Clientele groups with longer holding periods buy securities
with higher relative bid-ask spreads. In equilibrium, securities with higher levels of

relative bid-ask spreads have higher expected returns. Using NYSE bid-ask spreaddata, Amihud and Mendelsohn find empirical support for their model.

Bid-ask spread data is not generally available. Corwin and Schultz (2012)

propose a procedure that estimates levels of bid-ask spreads based on high and low
daily prices. They show that their estimates have a correlation of 0.9 with actual bid-

ask spreads. Using Corwin and Schultz’s method, we estimate bid-ask spreads for all

CRSP stocks, and create a relative spread by dividing the estimated spread of a stock
in a given month by the average price of that stock in that month.

Hybrid index. A hybrid index is constructed based on a liquidity measure thatavoids the pitfalls of other measures. The hybrid measure is defined as,
𝐷𝐷𝑖𝑖,𝑚𝑚�𝑟𝑟𝑖𝑖,𝑡𝑡 − 𝑟𝑟𝑚𝑚,𝑡𝑡 �
𝑀𝑀𝑀𝑀𝑀𝑀𝑚𝑚
1
𝐻𝐻𝑖𝑖,𝑚𝑚 = �����𝐷𝐷𝑖𝑖,𝑚𝑚
𝐷𝐷𝑖𝑖,𝑚𝑚
𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑖𝑖,𝑡𝑡
𝑡𝑡=1

where 𝐷𝐷𝑖𝑖,𝑚𝑚 is the number of days in month m for which stock i has turnover andreturn data. Maxm is the total number of trading days in month m. 𝑟𝑟𝑖𝑖,𝑡𝑡 and 𝑟𝑟𝑚𝑚,𝑡𝑡 are the21respective returns of the stock i and the market for day t. For estimation, we use the

CRSP equal-weighted index return with dividends as our market proxy. Turnoveri,t isthe number of shares stock i that are traded in day t as a fraction of shares

outstanding. The second and third expressions are similar to the Amihud measure,in that they attempt to measure the contemporaneous volume-related price impact.

The numerator of the last expression recognizes that return movements that
correspond to broad market movements are unlikely to provide information about
illiquidity. The first expression magnifies the illiquidity measure for stocks that do
not trade every day. Thus, a stock that only trades on half of the days, will beassigned an H measure that is twice that of stock which trades every day. Thisadjustment measures liquidity under the assumption that a trader is unable to tradeon a zero volume day, and proportionately increases trading on non-zero volume
days.

The decision to use turnover instead of dollar volume in the denominator of

the Hybrid index avoids a mechanical relation between the liquidity measure andmarket capitalization. This characterization dovetails with the decision to valueweight the measures to create the index.

Results. Table 5 and 6 present results for the investigation of alternativeliquidity risk indices. Table 5 focuses on the most recent time period and Table 6

reports on the full time period. Correlations between innovations and the indices

are low. These correlations are even lower than those calculated on a different set of
liquidity measures (detrended using a second-order auto-regressive process) by

Koraczyck and Sadka (2008). Post 1962 (Table 5, panel A) the highest correlation,22

0.39, is between the Amihud index and the Hybrid index. The fact that this is thehighest correlation among the group is not too surprising since the construction ofboth involves the division of an absolute return measure by a trading intensity
measure. The second highest correlation is 0.05. The lowest correlation, -0.12, isbetween the zero return index and the bid-ask spread index. The full time period,

Table 6 results demonstrate higher correlations—the strongest being 0.95, again

between the Amihud and Hybrid index. The second highest correlation is 0.37. The
message is clear—these indices share little commonality. Correlations between
long-short portfolio returns based on these indices tell the same story.

The low correlations in tables 5 and 6 reinforce a re-occurring challenge inthe liquidity literature. Different liquidity measures often seem at odds with one

another (for example, Goyenko et al., 2009). Although there is no unifying
explanation for these differences, a contributing factor is the non-observability of

uncompleted trades. Market participants make decisions to not trade whenilliquidity is high. In days of high illiquidity, all trades may be prohibitively
expensive. In turn, this results in no volume and unobservable returns based ontransaction prices—which are inputs to most liquidity measures. 5

Panel C of Table 5 and Panel F of Table 6, present average annualized risk

premiums. None of the ten estimates is significant at the 5% level. Using recent data,
the best performer is the zero return index, with an annualized risk premium of 3.44

percent (t-statistic of 1.74) and using the full time series, the best performer is the
hybrid index with a risk premium of 2.25 percent (t-statistic of 1.52).
5 We thank the referee, Kumar Venkataraman, for pointing this out.

23

5. Conclusions
Pastor and Stambaugh correctly estimate a return premium on their gamma

index that that is positive and statistically significant. Although the result does nothold before their time period, it is stronger after their time period. This is unusual
given that the typical finding of return predictability decays by nearly 60% once a
paper is published (McLean and Pontiff, 2016).

Is gamma risk priced illiquidity risk? On one hand, the aforementioned

predictability continues after the end of Pastor and Stambaugh’s original sample.

Despite this, we fail to find other supporting evidence that gamma risk return
predictability is compensation for liquidity risk. Four modifications to the gamma

index that are designed to strengthen the measurement of priced liquidity risk all

yield lower returns that fail to reject the null of no premium at the 5% level. Outsideof Pastor and Stambaugh’s time period, we fail to find that low gamma-periods
exhibit a “flight-to-quality.” Departing from the gamma index and consideringliquidity risk more broadly, we fail to find any evidence that indices created from
common liquidly variables capture priced risk.

24

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Pricing, Journal of Financial and Quantitative Analysis 51. 1769-1793.
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Korajczyk, Robert and Ronnie Sadka, 2008, Pricing the commonality acrossalternative measures of liquidity, Journal of Financial Economics 87, 42-72.
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Finance 32, 1151-1168.

Sadka, Ronnie, 2006, Momentum and post-earnings-announcement drift anomalies:
The role of liquidity risk, Journal of Financial Economics 80, 309-349.
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Taleb, Nassim Nicholas, 2012, Antifragile: things that gain from disorder, Random
House (New York, New York).

26

Figure 1:

27

Table 1: Replication of PS Decile Return Spread
Description: This table compares annualized alphas for a long-short portfolio thatis long a value-weighted portfolio of stocks in the highest decile of gamma betas andis short a value-weighted portfolio of stocks in the lowest decile of gamma betas. tstatistics are in parentheses. The time period is 1968:1 to 1999:12.

Interpretation: Our code perfectly replicates the results produced on Lubos Pastor
and Robert Stambaugh’s website. Our code produces estimates that are very close to
those produced in their paper.
Annual alpha of long-short liquidity beta portfolio
(1)
(2)
(3)
Replication results
Estimate from
Website estimate
Table 8 of
Pastor Stambaugh
3.91
4.15
3.91
(2.00)(2.08)(2.00)

28

Table 2: Outside of PS time period comparison
Description: This table compares annualized alphas for a long-short portfolio thatis long a value-weighted portfolio of stocks in the highest decile of gamma betas andis short a value-weighted portfolio of stocks in the lowest decile of gamma betas. tstatistics are in parentheses.
Interpretation: Before the start of Pastor and Stambaugh’s original time period,
gamma-risk is associated with a negative statistically insignificant risk premium. In
contrast to the typical return-predictability findings (McLean and Pontiff, 2016),
after the end of Pastor and Stambaugh’s original time period, gamma risk
commands a higher return premium.
Note: The column (2) risk premium estimate differs from the Table 1, column (2)risk premium estimate because the gamma index this table is whitened using
information that starts in the 1932. Table 1, like the Pastor and Stambaugh paper,
whitens the series using data that starts in 1968.
(1)
Pre-PS-time
periodStartEnd

1932:1

1967:12-2.72(-1.36)(2)
PS timeperiod1968 :1

1999:12
3.82(1.93)(3)
Post-PS time
period(4)
Entire CRSP

2000:1

1932: 1

2017:12
5.56(1.84)

29

2017:121.45(1.13)(5)
PS plus postPS time
period1968:1
2017:1
4.45(2.67)

Table 3: Comparison of correlations of PS Index innovations during andoutside the original PS period
Description: The table reports correlations between monthly returns on the CRSP
aggregate value-weighted index. 𝑅𝑅𝑆𝑆,𝑡𝑡 , and (i) the current month’s return on 30 day
Treasury bills minus the next month’s return on 30 day Treasury bills, −∆𝑅𝑅𝑓𝑓,𝑡𝑡 ,; (ii)the return on 20 year Treasury bonds; and (iii) the median percentage change inmonthly dollar volume.

Interpretation: This estimation produces two-tailed p-values that are roughly
twice that reported by PS. PS does not report whether their test is two- or onetailed, so it is likely that their test is one-tailed. There are not many low gamma
months and as such, our results and PS results suffer from low statistical power.
Although the correlation differences between low gamma month and other month
are similar during and outside of the PS time period for both the change in the risk
free return and the median change of monthly dollar value, the correlationdifferences with the return on 20 year Treasury bonds switch sign.
Panel A: Within PS time period correlation of monthly CRSP value-weighted index
return with other variables. 1962:8 to 1999:12
Correlation of 𝑅𝑅𝑆𝑆,𝑡𝑡 withAll months
Low-gamma months
Other months
p-value (difference)(1)−∆𝑅𝑅𝑓𝑓,𝑡𝑡
-0.01
-0.35
0.03
0.19

(2)
𝑅𝑅𝐺𝐺𝐺𝐺,𝑡𝑡
0.30
-0.13
0.35
0.10

(3)
𝑉𝑉𝑉𝑉𝑉𝑉𝑡𝑡
0.47
-0.21
0.52
0.01

Number of

Observations

449
14
435

449
14
435

449
14
435

Panel B: Outside PS time period correlation of monthly stock returns with othervariables in months with large liquidity drops. 1927:1–1962:7 and 2000:1–2017:12
Correlation of 𝑅𝑅𝑆𝑆,𝑡𝑡 withAll months
Low-gamma months
Other months
p-value (difference)(1)−∆𝑅𝑅𝑓𝑓,𝑡𝑡
-0.02
-0.30
0.02
0.24

(2)
𝑅𝑅𝐺𝐺𝐺𝐺,𝑡𝑡
-0.22
0.19
-0.24
0.14

30

(3)
𝑉𝑉𝑉𝑉𝑉𝑉𝑡𝑡
0.45
0.26
0.46
0.41

Number of

Observations

654
16
638

463
15
448

654
16
638

Table 4: Performance of modified PS gamma indices
Description: This table examines the PS gamma index and four modifications of the gamma
index. The four modifications are including zero volume days in the gamma estimation,
including all prices regardless of level, value-weighting gammas to create the index instead ofequal-weighting, and estimating the gammas in an equation that restricts the intercept to beequal to zero. Correlations between the innovations, correlations between correlations between
long-short extreme deciles portfolios, and the mean returns on these portfolios are reported. tstatistics are in parentheses.
Interpretation: Panel A and B show that the zero volume and all price modifications create
results with low correlation with PS. Panel C shows that these produce liquidity portfolio
returns that are lower than the PS estimate and that fail to reject the null.

Panel A. Correlations between PS variant liquidity innovations: 1962:8 to 2017:12
Value weighted
All prices
Zero volume
Baseline PS
Index
includedincluded
index
Zero volume included
0.27
All prices included

Value weighted index
𝜃𝜃𝑖𝑖,𝑡𝑡 restricted to zero

0.17

0.92

0.15

0.11

0.17

0.24

0.97

0.26

Panel B. Correlations between PS variant liquidity returns: 1968:1 to 2017:12
Value weighted
All prices
Zero volume
Index
includedincluded
Zero volume included
All prices included

Value weighted index
𝜃𝜃𝑖𝑖,𝑡𝑡 restricted to zero

0.15

0.77

0.18

0.02
0.25
0.25

0.95

Baseline PS
index
0.26
0.10
0.82
0.87

Panel C. Average annualized returns of PS variant traded liquidity factors: 1968:1 to
2017:12𝜃𝜃𝑖𝑖,𝑡𝑡 restricted to zero
Value weighted
All prices
Zero volume
Baseline PS
Index
includedincludedindex
3.13(1.89)

1.02(0.59)

0.78(0.15)

31

0.82(0.50)

4.45(2.67)

Table 5: Performance of alternative liquidity risk indices—recent data
Description: This table examines five indices constructed from liquidity measures.
Correlations between the innovations, correlations between correlations between longshort extreme deciles portfolios, and the mean returns on these portfolios are reported. tstatistics are in parentheses

Interpretation: Panel A and B show that there tends to be low correlation between thealternative indices and the returns of alternative indices. Panel C reports that none of theindices produce a liquidity portfolio return that is statistically significant from zero.

Panel A. Correlations between alternative liquidity innovations: 1962:8 to 2017:12
Zero return index

Bid-ask spread
index

Amihud index

Amihud index

Bid-ask spread index

PS dual modified index

0.05

0.04

0.03

Zero returnindex

Hybrid index

0.04

0.39

0.05

-0.12

0.04

-0.01

-0.01

Zero returnindex

Hybrid index

-0.09

0.30

Panel B. Correlations between alternative liquidity returns: 1968:1 to 2017:12
Zero return index

Bid-ask spread
index

Amihud index

Amihud index

Bid-ask spread index

PS dual modified index

0.12

-0.16

-0.06

-0.20

0.27

-0.08
0.13
0.01

Panel C. Average annualized returns of alternative traded liquidity factors: 1968:1 to
2017:12PS dual modified index-1.26(-0.71)

Bid-ask spread
index

Amihud index-1.76(-0.89)0.47(0.24)

32

Zero return
index
3.44
(1.74)

Hybrid index
1.57(0.90)

Table 6: Performance of alternative liquidity risk indices—entire time period
Description: This table examines five indices constructed from liquidity measures.
Correlations between the innovations, correlations between correlations between longshort extreme deciles portfolios, and the mean returns on these portfolios are reported. tstatistics are in parentheses

Interpretation: Panel A and B show that there tends to be low correlation between thealternative indices and the returns of alternative indices. Panel C reports that none of theindices produce a liquidity portfolio return that is statistically significant from zero.

Panel A. Correlations between alternative liquidity innovations: 1927:1 to 2017:12
Zero return index

Bid-ask spread
index

Amihud index

Amihud index

Bid-ask spread index

PS dual modified index

0.27

0.02

Non-zero
return index

Hybrid index

0.04

0.95

0.08

0.05

0.37

0.01

-0.00

Amihud index

Zero returnindex

Hybrid index

0.12

0.51

Panel B. Correlations between alternative liquidity returns: 1932:1 to 2017:12
Zero return index

Bid-ask spread
index

Amihud index

Bid-ask spread index

PS dual modified index

0.25

0.07

0.03

0.12
0.32

0.01

0.18
0.35

0.10

Panel C. Average annualized returns of alternative traded liquidity factors: 1932:1 to
2017:12PS dual modified index-0.65(-0.45)

Bid-ask spread
index

Amihud index-1.60(-1.01)

1.86(1.21)

33

Zero return
index
1.63
(1.08)

Hybrid index
2.25(1.52)