WORKING PAPER · NO. 2019-57

Liquidity Risk After 20 Years
Lubos Pástor and Robert F. Stambaugh
APRIL 20195757 S. University Ave.
Chicago, IL 60637
Main: 773.702.5599
bfi.uchicago.edu

Liquidity Risk After 20 Years

Ľuboš Pástor
Robert F. Stambaugh*

April 14, 2019

Abstract

The Critical Finance Review commissioned Li, Novy-Marx, and Velikov (2017)and Pontiff and Singla (2019) to replicate the results in Pástor and Stambaugh(2003). Both studies successfully replicate our market-wide liquidity measureand find similar estimates of the liquidity risk premium. In the sample period
after our study, the liquidity risk premium estimates are even larger, and the liquidity measure displays sharp drops during the 2008 financial crisis. We respondto both replication studies and offer some related thoughts, such as when to use
our traded versus non-traded liquidity factors and how to improve the precisionof liquidity beta estimates.

*Pástor is at the University of Chicago Booth School of Business, 5807 S Woodlawn Ave, Chicago,
IL 60637. Stambaugh is at the Wharton School of the University of Pennsylvania, 3620 Locust Walk,
Philadelphia, PA, 19104. Both authors are also at the NBER. Pástor is also at the CEPR and the National
Bank of Slovakia. Email: lubos.pastor@chicagobooth.edu, stambaugh@wharton.upenn.edu. The views inthis paper are the responsibility of the authors, not the institutions with which they are affiliated. For
helpful comments, we thank Doug Diamond, Ralph Koijen, Stefan Nagel, Robert Novy-Marx, Jeff Pontiff,
Jeff Russell, Rob Vishny, and the seminar audience at the University of Chicago.

1. Introduction
We began our work on liquidity risk 20 years ago, shortly after the 1998 collapse of theLong-Term Capital Management (LTCM) hedge fund. LTCM held multiple large trading
positions that were exposed to the risk that market liquidity might deteriorate. One famous
example was their long-short position in off-the-run versus on-the-run Treasury bonds. These
positions performed well in normal times, but they suffered when liquidity dried up in thesummer of 1998. The LTCM episode highlighted the facts that market liquidity fluctuates
over time and that asset prices respond to these fluctuations.
Twenty years ago, the role of liquidity in asset pricing was viewed largely in static terms,
based on differences in liquidity across assets. As in Amihud and Mendelson (1986), lessliquid assets were understood to offer higher expected returns to compensate for their lower
liquidity. Inspired by the LTCM episode, we took a dynamic view, focusing on time variationin liquidity. Our idea was that the price of an asset can depend not only on the asset’s
liquidity but also on the sensitivity of the asset’s return to pervasive liquidity shocks.
Our published study, Pástor and Stambaugh (2003), makes three primary contributions.
First, it advances the hypothesis that aggregate liquidity is a state variable relevant for
asset pricing. Motivated by the evidence of commonality in liquidity (Chordia, Roll, andSubrahmanyam, 2000), we hypothesize that liquidity risk is systematic, or non-diversifiable,
and thus potentially priced. Second, to test the above hypothesis, we construct a measureof aggregate stock market liquidity. From that measure, we derive a liquidity factor that
captures innovations in aggregate liquidity. Finally, we find that the liquidity factor is priced,in that stocks with higher sensitivities to this factor—i.e., higher liquidity betas—have higher
average returns. Liquidity betas are defined as slope coefficients βiL from the regression
ri,t = βi0 + βiL Lt + βiM MKTt + βiS SMBt + βiH HMLt + i,t ,(1)

where ri,t denotes asset i’s excess return, Lt is the liquidity factor, and MKT, SMB, and
HML are the three factors of Fama and French (1993).
While our study focuses on equities, subsequent studies find liquidity risk to be priced
in other asset classes, such as hedge funds (e.g., Sadka, 2010), private equity (e.g., Franzoni,
Nowak, and Phalippou, 2012), emerging markets (e.g., Bekaert, Harvey, and Lundblad,2007), and corporate bonds (e.g., Bongaerts, de Jong, and Driessen, 2017). Important theoretical insights into liquidity risk are provided by Acharya and Pedersen (2005), Brunnermeierand Pedersen (2009), and others. The literature on liquidity risk has grown tremendously,
and we do not attempt to review it here.
1

A decade after LTCM, academic interest in liquidity risk was further stimulated by the2008 financial crisis, during which markets experienced dramatic reductions in liquidity. The
crisis highlighted the effects of liquidity shocks on not only asset prices but also real activity.
When liquidity dries up in the market, other bad things tend to happen. Liquidity shocks
can destroy value, for example, by creating runs on financial institutions or by impeding
the financing of viable investment projects. The crisis clearly showed why investors may bewilling to pay more for assets whose values better withstand market-wide liquidity shocks.
Our liquidity factor picks up the key events of the crisis, as we show in Section 2.2. Because it
happened five years after the publication of our paper, the crisis provides some out-of-sample
validation for the methodology behind our liquidity factor.
We are grateful to the editor of the Critical Finance Review, Ivo Welch, for commissioning
two studies to assess the extent to which our empirical results can be replicated. We areespecially grateful to the authors of those studies, namely Hongtao Li, Robert Novy-Marx,
Mihail Velikov, Jeffrey Pontiff, and Rohit Singla, for undertaking this task and executing itwith the utmost competence and professionalism. We have learned a lot from their work.
In the rest of this paper, we first review key results of the replicators and respond tosome of their claims. We then present a few thoughts on our liquidity factor, such as when
to use its traded versus non-traded version, why we pay more attention to historical rather
than predicted liquidity betas, and why our liquidity measure is more useful at the aggregate
rather than the individual-stock level. Finally, we discuss the implications of the asymmetricnature of liquidity shocks. For example, we explain why liquidity betas are relatively hardto estimate and what can be done to improve their precision.

2. Replication Results
Both Li, Novy-Marx, and Velikov (2017) and Pontiff and Singla (2019) successfully replicate
the construction of our liquidity factor over the time period used in our study (1962 through1999). Li, Novy-Marx, and Velikov report a correlation of 98.9% between their liquidity
series and ours. Pontiff and Singla report a correlation of one, up to five significant digits.
The above correlations pertain to our main liquidity factor, Lt from equation (1). Bothstudies also replicate the traded version of our factor, which we discuss in more detail in
Section 3.1. Li, Novy-Marx, and Velikov (2017) report a 95% correlation between theirtraded liquidity factor and ours in 1968 through 1999. Both their factor and ours produce
very similar alphas (see their Table 2). For example, both factors produce in-sample four2

factor Carhart alphas that round to 0.37% per month and five-factor Fama-French alphas
that round to 0.20% per month in 1968 through 1999. Pontiff and Singla (2019) also reportsimilar results for the traded factor. Their in-sample estimate of the three-factor FamaFrench alpha of the traded factor is 3.91% per year (t = 2.00; see their Table 1), which is
close to the estimate of 4.15% (t = 2.08) reported in Table 8 of our 2003 study. Overall,both studies have managed to closely replicate our main in-sample results.
In general, a successful outcome of in-sample replication is not guaranteed even if themethodology used by the replication study is identical to that used in the original study.
Even if the code is the same, the data need not be. Our 2003 study relies heavily on datafrom the Center for Research in Security Prices (CRSP). In an effort to keep its database asclean as possible, CRSP performs corrections to its historical data anytime it identifies, or
becomes aware of, a data error. As a result, the CRSP dataset analyzed in a given year is not
identical to the CRSP dataset covering the same period but downloaded years later. The twodatasets are very similar, but the cumulative effect of CRSP updates over the interim period
need not be negligible, especially if a long time has elapsed. A good example is provided inTable 1 of Pontiff and Singla. Their estimated in-sample alpha of 3.91% per year (t = 2.00),
mentioned earlier, is identical to the alpha they obtain when they replace their replicatedtraded factor series with the updated series downloaded from our own websites. In otherwords, the difference between 3.91% and 4.15%, discussed in the previous paragraph, is nota result of a discrepancy between our methodology and that of Pontiff and Singla. Instead,
it is driven by the corrections in the 1962–1999 CRSP dataset since we first used the dataalmost twenty years ago. Pontiff and Singla themselves write that “This minor difference isalmost certainly attributable to year-to-year corrections to the CRSP data.”

2.1. Out-of-Sample PerformanceOur 2003 study estimates the liquidity risk premium based on the sample period of 1968
through 1999. If our estimate were a fluke, it would likely disappear out of sample. In
contrast, both replication teams report that the magnitude of the liquidity risk premium has
increased since 1999.
Li, Novy-Marx, and Velikov (2017) compare the in-sample (1968–1999) and the outof-sample (2000–2015) performance of the traded liquidity factor, for both their replicatedversion and ours. They find that the out-of-sample performance is substantially stronger
than the in-sample performance, for all three factor models they use. For example, the fourfactor Carhart alpha of our factor increases from 0.37% per month (t = 2.14) in 1968–1999 to3

0.68% per month (t = 2.31) in 2000–2015. Pontiff and Singla (2019) also report an improved
post-sample performance of the traded factor, going through 2017 (see their Table 2). Thisevidence shows that had we written our 2003 study today, our main results would have beenquite a bit stronger!
A possible reason why the evidence is stronger post-sample than in-sample is the 2008
financial crisis. This post-sample crisis produced wide fluctuations in liquidity, allowing more
precise estimation of liquidity betas. In periods without liquidity crises, liquidity betas are
estimated with less precision. We elaborate on this point in Section 4.
Given the strong post-sample performance of our traded liquidity factor, it is not surprising that the factor’s full-sample performance in 1968 through 2015 is highly significant. In
their Table 3, Li, Novy-Marx, and Velikov report that our factor’s alphas with respect to thethree-, four-, and five-factor models range from 0.44% to 0.46% per month, with t-statistics
ranging from 2.85 to 3.02. The premium for liquidity risk seems alive and well.
Pontiff and Singla (2019) estimate not only the post-sample but also the pre-sample
magnitude of the liquidity risk premium. Such analysis was not feasible when we wrote
our 2003 study because CRSP added the pre-1962 daily data only in 2005. In the sample
period of 1932 through 1967, Pontiff and Singla find no significant premium associated withliquidity risk: their traded factor has a premium of -2.72% per year, with the t-statistic of-1.36. As a result of this negative pre-sample performance, their estimate of the full-sample
premium in 1932 through 2017 is positive but insignificant (see their Table 2).
Why is the pre-1962 evidence so different from the post-1962 evidence? Figure 1 ofPontiff and Singla (2019) shows that the pre-1962 fluctuations in liquidity are much smaller
in magnitude than the more recent fluctuations. The largest pre-1962 fluctuations in liquidity
are associated with the Great Depression, which seems plausible, but the scale of thesefluctuations is substantially smaller than the scale of the post-1962 fluctuations.
One possible reason why the pre-1962 liquidity series is so much less volatile is a different stock universe. Whereas the post-1962 CRSP daily data include stocks from boththe New York Stock Exchange (NYSE) and American Stock Exchange, the pre-1962 daily
data includes only NYSE stocks. The larger average size of NYSE stocks could very well be
responsible for the significantly lower pre-1962 volatility of the liquidity series. The source ofthe pre-1962 daily data is also different. The primary source is The New York Times, withthe secondary source being The Wall Street Journal. The main source of post-1962 datais the Interactive Data Corporation. Given the different stock universes and data sources,
the pre-1962 results may not be directly comparable to the post-1962 results. Additional
4

analysis of the volatility differences in the pre- and post-1962 samples seems warranted.

2.2. The 2008 Financial Crisis
Our 2003 study came out five years before a major financial crisis, during which liquidity
evaporated from many financial markets. The 2008 crisis provides a valuable out-of-sample
perspective on the behavior of our liquidity factor. If the factor truly captures liquidity, it
should have large negative realizations during the crisis. Luckily for the factor, it does.
The 2008 financial crisis is commonly considered to have begun in the summer of 2007
and ended in the spring of 2009. Between July 2007 and April 2009, our liquidity factor
experienced negative innovations in 17 out of 22 months. Three of the five largest negative
innovations during this period are associated with the defining moments of the crisis, namely,
the demise of Lehman Brothers (September 2008), the fall of Bear Stearns (March 2008), andthe quant crisis (August 2007). The two remaining largest drops in our liquidity measureoccurred in December 2008 and January 2009, two of the worst months of the crisis. Allfive of these drops rank among the 13 biggest drops in our liquidity measure since 1962, andfour of them are among the top 9. This evidence suggests that our liquidity factor, whose
methodology was designed well before the 2008 crisis, does a good job capturing the mostsalient events of the crisis.
The large drops in liquidity are clearly visible in Figure 1, which plots the time seriesof our aggregate liquidity measure in August 1962 through December 2017. The drops
in liquidity experienced during the 2008 crisis are not among the three largest drops in
the full sample, which occurred in October 1987, September 1998, and November 1973.
However, all three of those liquidity drops were short-lived. In contrast, during the 2008
crisis, liquidity diminished month after month, with devastating cumulative effects. For
example, our liquidity measure experienced four large drops during the five-month period
between September 2008 and January 2009; as a result, the cumulative drop in liquidity
during this period was very large.
Our traded liquidity factor, which offers a long-short portfolio exposure to liquidity risk,
also performed poorly during the financial crisis. A simple way to see this effect is in Figure 2of Li, Novy-Marx, and Velikov (2017), which plots the cumulative performance of the tradedfactor in 1968 through 2015. The factor’s performance has been strong overall, as indicatedby the upward trend. Across the whole sample, the sharpest drop in the factor’s performanceoccurs in the second half of 2008. Consider a strategy that invests $1 in a Treasury bill and5takes a long-short position of $1 each in our traded factor, rebalanced monthly. The strategy’s
cumulative excess return over the six-month period (July through December 2008) is -34.3%,
with negative returns in five of the six months (all except August). This is what one would
expect from a strategy designed to capture the liquidity risk premium—it performs well on
average, but it loses money in periods in which the risk materializes. These results provide
additional out-of-sample support for our methodology.

2.3. Excluding Zero-Volume Days
To successfully replicate our liquidity factor, one must be aware of the fact that when we
estimate equation (1) in our 2003 study, we exclude days with zero trading volume. That
equation (1) takes the form of the regressioneeri,d+1,t= θi,t + φi,t ri,d,t + γi,t sign(ri,d,t) · vi,d,t + i,d+1,t ,d = 1, . . . , D,(2)where γi,t is the liquidity measure for stock i in month t and the remaining quantities aredefined as follows:
ri,d,t :the return on stock i on day d in month t,eri,d,t: ri,d,t − rm,d,t , where rm,d,t is the return on the CRSP value-weighted market
return on day d in month t, andvi,d,t :the dollar volume for stock i on day d in month t.

When we run this regression, we impose the screen vi,d,t > 0, thereby excluding days with
vi,d,t = 0 from the analysis. Unfortunately, we inadvertently neglected to mention thisimportant detail in our paper, as pointed out to us by Li, Novy-Marx, and Velikov (2017).
If one replicates our analysis without imposing the positive-volume screen, the resulting
series has only a modest correlation with our factor: 39.2% when computed over the periodof August 1962 through December 1999, according to Li, Novy-Marx, and Velikov (2017),
and 27% over the period of August 1962 through December 2017, according to Pontiff andSingla (2019). Our failure to mention the positive-volume screen may have caused some past
replication attempts by other researchers to fail. We sincerely apologize for this omission.
We are grateful to both replication teams for recognizing this oversight in a gracious manner.
Without excluding zero-volume days, the estimates from the regression in equation (2)would be difficult to interpret. As we explain in our 2003 study, the coefficient of interest fromthis regression, γi,t , captures volume-related return reversals. To capture return reversals thatare not volume-related, such as the bid-ask bounce, we include a second independent variable:
lagged stock return, ri,d,t . However, controlling for a bid-ask bounce is inappropriate on days
6

with zero trading volume because on such days, the stock price reported by CRSP is simply
the average of the bid and the ask. The inclusion of an inappropriate control would resultin a biased estimate of γi,t because this control, ri,d,t , is positively correlated with signedevolume sign(ri,d,t) · vi,d,t . In other words, if we did not impose vi,d,t > 0, the regression inequation (2) would be misspecified. Our imposition of this screen was appropriate, but our
failure to mention it in the paper obviously wasn’t.

2.4. Modifications of Our Liquidity Measure
Li, Novy-Marx, and Velikov (2017) analyze numerous modifications of our factor construction methodology, such as different approaches to portfolio sorts, different rebalancing frequencies, and different rebalancing months. While these modifications are interesting, ourmethodological choices seem reasonable a priori. For example, while one could certainly
consider other numbers of portfolios, our choice to sort stocks into 10 decile portfolios seems
natural. Our choice to allocate an equal number of stocks to each decile portfolio seems
equally plausible. Li, Novy-Marx, and Velikov show that allocating stocks differently, by
equalizing total market capitalization in each portfolio, produces somewhat weaker results.
Yet such “market cap breaks” are not typically used in cross-sectional asset pricing tests,
and it is not clear why they should possess more power than the “name breaks” employedin our study. Applying market cap breaks could result in a large imbalance in the numberof stocks across portfolios, to the point where some of the decile portfolios could containa very small number of large stocks.1 The diversification of idiosyncratic risk within suchsmall portfolios is quite imperfect, making the results noisier.
How often should one rebalance the portfolios of stocks sorted on liquidity betas? We
rebalance annually, at the end of the calendar year, which seems perfectly plausible to us.
Li, Novy-Marx, and Velikov argue that rebalancing at the monthly frequency would be morenatural. We agree that monthly rebalancing would also be plausible, but compared to annual
rebalancing, it would significantly increase the portfolio’s trading costs. Li, Novy-Marx, andVelikov state in their abstract that their liquidity factor constructed with monthly rebalancing “exhibits significantly weaker performance” than our factor. Yet even under monthly
rebalancing, the performance of the liquidity factor remains statistically and economicallysignificant. The full-sample (1968–2015) alphas of the resulting traded factor are all significant, ranging from 0.36% to 0.45% per year, with t-statistics ranging from 2.24 to 2.84 (see1

For example, as of this writing, three stocks—Microsoft, Apple, and Amazon—together account for morethan 10% of the S&P 500 index. If we were to apply market cap breaks to stocks in this index, we could in
principle end up with a decile portfolio containing only two stocks. The CRSP value-weighted market index
contains more stocks than the S&P 500, so the problem is somewhat less severe in our context.

7

Table 3 of Li, Novy-Marx, and Velikov (2017)).
Pontiff and Singla (2019) consider four modifications of our liquidity measure: (1) inclusion of zero-volume days, (2) inclusion of stocks of all price levels, (3) value-weighted index,
and (4) zero-intercept restriction. They do not find any of these measures to be priced.
Like our 2003 study, Pontiff and Singla test the joint hypothesis that liquidity risk is
priced and that they have the right measure of liquidity. We believe that their rejection ofthis joint hypothesis is driven by the fact that their four liquidity measures are inadequate.
We plot the four series in Figure 2. The first three series, plotted in Panels A through C,
do not appear to be good representations of market-wide liquidity. We base this judgment
on a visual examination of these series, invoking the “you know it when you see it” doctrine
mentioned in Li, Novy-Marx, and Velikov’s introduction. Despite Pontiff and Singla’s goodintention to improve the estimates of the liquidity series, it seems hard to believe that anyof the three series are indeed improvements. On the contrary, they look less appealing as
measures of market-wide liquidity. Therefore, the evidence based on these modified series
does not imply that liquidity risk is without a premium.
The three series look unappealing not only ex post but also ex ante. The misspecificationinduced by using zero-volume days is discussed in Section 2.3. The inclusion of stocks
with prices below $5 and above $1,000 is relatively uncommon. Value-weighting seems lessappealing than equal-weighting from the perspective of diversifying away the estimation
errors in the γi,t estimates across firms. Additional arguments pertaining to value-weighting
are discussed in Section II.C of our 2003 study.
The only series in Figure 2 that exhibits liquidity-like features is the series plotted inPanel D, which corresponds to the zero-intercept modification. Perhaps not surprisingly, theresults for that series are more supportive of priced liquidity risk than the results for theother three series. In Panel C of their Table 4, Pontiff and Singla (2019) report that theresulting traded liquidity factor has an average return of 3.13% per year in 1968 through2017, with a t-statistic of 1.89. While these numbers are lower than the 4.45% return witha t-statistic of 2.67 that Pontiff and Singla report for our baseline index, they still indicate
marginal significance of liquidity risk, with a p-value of 0.06.
More important, we are not convinced by the argument that imposing the zero-intercept
restriction improves our procedure. It is true that imposing this restriction saves a degree of
freedom, but it also introduces misspecification. Pontiff and Singla argue that the restriction
is reasonable because it generates one-day-ahead forecasts of individual stock returns that8are equal to market returns when the independent variables in regression (2) are zero. While
market returns are indeed plausible unconditional forecasts of individual stock returns, it is
not clear that they are plausible conditional on the very unlikely event that the independent
variables—lagged return and signed volume—are both equal to zero. Zeroing out the intercept could result in biased regression estimates. We think of the zero-intercept-modification
series as a slightly misspecified version of our baseline series. Given that, we do not find it
surprising that this series produces slightly weaker results.
Besides the four modifications plotted in Figure 2, Pontiff and Singla consider five additional alternative liquidity measures: (1) the proportion of zero returns, (2) the Amihudindex, (3) estimated relative bid-ask spread, (4) a hybrid measure, and (5) a double variationon our original index. They find that none of these measures are priced. Again, the joint
hypothesis problem mentioned earlier rears its head. Pontiff and Singla’s findings do not
imply that liquidity risk is not priced; they could simply mean that the five series are poor
time-series proxies for aggregate liquidity. All five series are plotted in Figure 3. Their visual
examination indeed casts doubt on whether any of them capture liquidity. Most of theseseries fail to pick up well-known liquidity-crisis months. For example, with the exception of
the bid-ask spread, all other alternative series fail to pick up the undisputed evaporation ofliquidity during the 2008 financial crisis. At the same time, the alternative series do pickup months that are not known for being low-liquidity based on other sources. These series
exhibit both Type I and Type II errors, so to speak.
The only series whose plot reveals some resemblance to liquidity is the bid-ask spread.
While this spread could be a good measure of liquidity from the perspective of small retailinvestors, who might trade at the bid and ask quotes, it is less useful to large investors, whose
trades have price impact. We measure liquidity by the temporary price impact resulting froma given amount of order flow. The bid-ask spread cannot capture price impact because it
does not take the market’s depth into account. The correlation between the levels of ourseries and the bid-ask spread in August 1962 through December 2017 is 29.6%. The twoseries clearly capture different dimensions of liquidity. Not all dimensions of liquidity need to
be priced. If asset prices are determined primarily by large investors, our liquidity measure,which aims to capture price impact, is more likely to be priced than bid-ask spread.

2.5. Momentum
Pástor and Stambaugh (2003) find that the liquidity risk factor accounts for half of the
profits of the momentum strategy in stocks. Subsequent studies provide further evidence
9

of a link between liquidity risk and momentum. Sadka (2006) also finds that a substantial
portion of the profit in stock momentum is explained by liquidity risk. To estimate the latter,he constructs an alternative measure of market-wide liquidity using intraday data. Asness,
Moskowitz, and Pedersen (2013) conclude that a link between momentum and liquidity risk
exists quite broadly. Using multi-country data, they find that, within stocks as well as otherasset classes, momentum returns exhibit positive sensitivities to measures of market-wide
liquidity.
Li, Novy-Marx, and Velikov (2017) argue that our finding of a link between momentum
and liquidity risk is due to the inclusion of momentum-related variables, namely past returnand share price, among the predictors of liquidity betas. They note that if the predictedbeta liquidity factor is constructed by excluding cumulative return and stock price, it canno longer explain much of momentum profit. That is useful to know. However, we did notinclude these two variables in order to explain momentum. We included them because theyshow up as significant determinants of liquidity betas (see Table 2 in Pástor and Stambaugh(2003)). Excluding a significant determinant would cause an omitted-variable problem in
the analysis. For that reason, we believe it makes sense to include both cumulative returnand stock price in the construction of the predicted-beta liquidity factor.
Li, Novy-Marx, and Velikov (2017) also argue that explaining half of momentum is oneof our most important results. We do not share that view. Our main results instead relate
to the measurement of market-wide liquidity and the pricing of liquidity risk more generally.
Our objective was not to investigate momentum, which is not central to our study. Evidenceof a momentum link emerged simply when investigating the performance of traded liquidity
factors in the context of other popular traded factors.

3. How to Use the Liquidity Factors
3.1. Traded and Non-Traded Factors
Our liquidity factor comes in two versions: traded and non-traded. The non-traded factor,
Lt , is in equation (8) of Pástor and Stambaugh (2003). The traded factor, LIQ, is the payoffon the 10–1 portfolio that is long stocks with the highest historical liquidity betas and shortstocks with the lowest historical liquidity betas. We are sometimes asked by colleagues whichof the two factors they should use. The answer depends on the objective of the analysis.
For most objectives, the appropriate liquidity factor is the non-traded one, Lt . This is the10primary liquidity factor, designed to capture innovations in market liquidity. If one wishes
to estimate an asset’s liquidity risk, one should use Lt .
The traded factor, LIQ, is useful in estimating alpha with respect to a multifactor modelthat includes a role for liquidity risk. In order to interpret the intercept from a factor
model regression as alpha, all factors must be payoffs on tradable positions. Because Lt is
not traded, it would be inappropriate to interpret the intercept βi0 from the regression inequation (1) as alpha. However, if we replace Lt in equation (1) by LIQ, the intercept canbe interpreted as an alpha, because all four factors are then traded.
Compared to other traded factors in popular multifactor models, LIQ is rather unique inhaving an identified non-traded factor underlying it. Moreover, our traded factor is formedby sorting on risk estimates, unlike other long-short factors such as SMB and HML of Fama
and French (1993, 2015). The latter factors are often described as returns on mimicking
portfolios, but the non-traded factors (state variables) that those portfolios presumably
mimic are not specified. That slack cannot be afforded to LIQ.
One definition of a mimicking portfolio is advanced by Huberman, Kandel, and Stambaugh (1987), hereafter HKS. In their setting, portfolios “mimic” non-traded factors if betas
with respect to the former can be used instead of betas with respect the latter to explain the
cross-section of expected returns. HKS provide the general characterization for the weights
in mimicking portfolios. One admissible specification constructs portfolios maximally correlated with the non-traded factors. As HKS explain, however, many other constructions arealso admissible mimicking portfolios. Our traded factor, LIQ, is not designed to maximize
its correlation with innovations in liquidity. Indeed, the simple correlation between LIQ
and Lt is only 5.3%, based on Pontiff and Singla’s data in the longest possible time period
January 1932 through December 2017.
Using LIQ in a model with traded factors has a simple investment-based motivation, as
we explain in Section IV of Pástor and Stambaugh (2003). Adding LIQ to an investment
opportunity set consisting of other popular factors substantially increases the maximum
obtainable Sharpe ratio. Therefore, other assets are less likely to produce alphas with respect
to factors that include LIQ. If LIQ reduces alphas, then betas with respect to LIQ help
explain expected returns. Thus LIQ behaves as a mimicking portfolio in the sense of HKS,
though it is not constructed to obey the formal conditions derived by that study, which does
not address empirical implementation. Instead, LIQ has a simple long-short construction
typical of other traded factors. Therefore, except when a traded factor is needed, one should
use the underlying non-traded factor, Lt , rather than its imperfect stand-in, LIQ.

11

3.2. Aggregate versus Firm-Level Liquidity
To construct our measure of aggregate market liquidity in month t, we average the slope
estimates γ̂i,t across stocks i (see equations (1) and (5) of Pástor and Stambaugh (2003)).
One might be tempted to use the individual-firm estimates γ̂i,t on their own as firm-levelmeasures of liquidity. However, that is not an appropriate use of our liquidity measure.
As we explain in Section III.D of Pástor and Stambaugh (2003), the γ̂i,t estimates are too
noisy to be useful at the individual firm level. Despite our warnings, several studies, such as
Goyenko, Holden, and Trzcinka (2009), examine the cross-section of the γ̂i,t estimates. Not
surprisingly, they find that these estimates are noisy. We do not use the firm-level estimates
γ̂i,t for any purpose other than to construct a market-wide measure of liquidity. When weaverage these estimates, much of the noise diversifies away, resulting in what we believe isan appealing measure of market-wide liquidity.

3.3. Historical versus Predicted Liquidity Betas
In our 2003 study, we produce two versions of the traded liquidity factor. Both versions golong high-liquidity-beta stocks and short low-liquidity beta stocks, but they differ in how
the betas are estimated. One version uses “historical” liquidity betas, which are estimatedvalues of βiL from regression (1) estimated over the previous 60 months of data. The otherversion uses “predicted” liquidity betas, which are estimated as a linear function of sevenfirm-level characteristics (see equation (10) in Pástor and Stambaugh, 2003). The results
based on both versions of the traded factor suggest that liquidity risk is priced.
In our 2003 study, we put a larger emphasis on predicted betas because the associatedprocedure produced a larger spread in post-ranking liquidity betas. Specifically, we sortstocks into portfolios in two different ways: based on pre-ranking predicted betas and basedon pre-ranking historical betas. At the time when we wrote our 2003 study, the sort on
predicted betas produced a larger spread in post-ranking liquidity betas. Comparing Tables
3 and 7 in Pástor and Stambaugh (2003), the 10-1 portfolio’s post-ranking liquidity beta is
8.23 (t = 2.37) when the portfolios are formed based on predicted betas, but it is only 5.99
(t = 1.88) when the portfolios are formed based on historical betas.
Years later, we noticed that the beta pattern flipped in the expanded sample: historicalbetas now do a better job than predicted betas in creating dispersion in the post-ranking
betas. For example, when we extend Tables 3 and 7 through December 2018, the 10-1
portfolio’s post-ranking liquidity beta is only 3.24 (t = 0.91) when the portfolios are formed
12

based on predicted betas, but it is 8.04 (t = 3.16) when the portfolios are formed basedon historical betas. It appears that our model for predicting liquidity betas is somewhat
unstable over time. To maintain the quality of our traded liquidity factor, we decided about
ten years ago to focus on historical betas, and we have been doing that ever since. A
side benefit of this re-focus is that historical betas are substantially easier to estimate than
predicted betas. Estimating historical betas requires only CRSP data, whereas estimating
predicted betas requires also Compustat data.
Li, Novy-Marx, and Velikov (2017) report some difficulty in replicating our traded factor
based on predicted betas. We are not sure what the source of this difficulty is. Given ourpresent focus on historical betas, we have decided not to allocate time to this issue. It is
comforting that despite its nontrivial differences from our factor, the predicted-beta-based
liquidity factor constructed by Li, Novy-Marx, and Velikov earns positive alphas both inand out of sample. Their Table 5 shows that the factor performs most strongly against the
five-factor Fama-French model, with alphas that are positive and significant both in and outof sample (0.59% per month (t = 4.04) in 1966–1999 and 0.73% per month (t = 2.56) in
2000–2015). That is, even though their predicted-beta-based traded liquidity factor differs
somewhat from ours, both factors appear to be priced.

4. Asymmetric Liquidity Shocks: Implications
The liquidity factor sleeps most of the time but wakes up during financial crises and othersignificant disruptive events. This asymmetric nature of liquidity shocks, which is apparent
from Figure 1, has two important implications.
First, liquidity risk can explain, at least in principle, why asset return correlations tend torise sharply during crises. Most risky assets have positive exposures to liquidity fluctuations:
their prices tend to fall when liquidity dries up. In normal times, these positive liquidity
exposures contribute little to asset returns because they are multiplied by small liquidity
shocks. However, during crises, large negative realizations of the liquidity factor are multiplied by positive liquidity betas to produce large negative contributions to asset returns. As
a result of this common exposure to liquidity risk, the prices of many risky assets fall andtheir correlation rises sharply. Formally, in equation (1), βiL Lt ≈ 0 when Lt ≈ 0, but βiL Lt islarge and negative for most assets when Lt << 0, resulting in highly correlated asset returns
during liquidity crises.
Second, liquidity betas are relatively hard to estimate, especially in normal times. Run13

ning a regression of asset returns on liquidity shocks, whose values in normal times are close
to zero, produces noisy estimates of liquidity betas. To achieve some precision, it helps toestimate liquidity betas over periods that include substantial shocks to liquidity, such as
financial crises. For example, liquidity betas estimated over a period that includes the 2008
financial crisis are likely to be more precise than those estimated in normal times. That the2008 crisis occured after our initial sample period could be the reason why the post-sample
evidence on liquidity risk is even stronger than the in-sample evidence. Another way of
improving the precision of liquidity beta estimates is to form portfolios. To illustrate these
implications of skewed liquidity shocks, we conduct a simulation analysis.

4.1. Simulation Analysis
We assume that excess stock returns are determined by the two-factor modelri,t = βiLIQ FtLIQ + βiM KT FtM KT + i,t ,(3)where FtLIQ and FtM KT are the realizations of two hypothetical factors that we refer to as
liquidity and market factors, respectively. The liquidity factor’s realizations exhibit negativeskewness, whereas the market factor’s realizations are symmetrically distributed.
We simulate monthly time series of both factors, which we make uncorrelated with eachother. Each factor realization is the sum of a constant premium and a zero-mean innovation.
We assign a premium of 0.33% per month, or 4% per year, to each factor. We assume
both factors contribute equally to the volatility of stock returns and that the volatility of
FtLIQ + FtM KT is 20% per year, implying a monthly volatility of 4.08% for each factor. We
draw each monthly realization of FtM KT from a normal distribution with a mean of 0.33%
and standard deviation of 4.08%. We construct monthly realizations of the liquidity factor
as FtLIQ = St + Jt , where St represents “small” shocks and Jt captures occasional jumps.
Specifically, St is normally distributed with a mean of 0.33% and a standard deviation of
1.22%, whereas Jt = −30% when a jump occurs and Jt = 0 otherwise. We set the probabilityof a jump in any given month to 1/60.
Figure 4 plots simulated innovations of both factors over a 50-year period. Panel A showsthat the liquidity shocks are strongly negatively skewed, with eight jumps (“liquidity crises”)
occurring in this typical simulated history. These asymmetric dynamics of simulated liquidity
shocks are reminiscent of actual liquidity realizations, plotted in Figure 1. In contrast, Panel
B shows a symmetric pattern of simulated market shocks resembling that of actual stock
market returns.
14

We assume that stocks’ liquidity betas and market betas, βiLIQ and βiM KT , are both
uniformly and independently distributed within the range of 0.5 to 1.5. For each individual
stock, we assume that the two factors together explain 30% of the variance of the stock’s
return, ri,t . Given the stock’s betas, this assumption implies the value for the varianceof the stock’s i,t , which is normally distributed, independently across stocks and months.
The draws of i,t are combined with the stock’s betas and the draws of FtLIQ and FtM KT
to construct ri,t as in equation (3). Fixing the sample length at 60 months, we estimate
equation (3) as an OLS regression with an intercept. For each simulated sample, we collectthe sample estimates of βiLIQ and βiM KT , denote them by β̂iLIQ and β̂iM KT , and construct
the estimation errors
eLIQ= β̂iLIQ − βiLIQi
KTeM= β̂iM KT − βiM KT .
i

(4)
(5)

We simulate many 60-month samples of factors and returns on individual stocks. PanelKT
A of Figure 5 plots the distributions of eLIQ
and eM
across the simulated samples. Thereiiis a stark difference between the two distributions, especially in the tails. The distribution
KT
tails off around −3
tails off around the values of −1 and 1, whereas that of eLIQ
of eM
i
i

and 3. In other words, it is extremely rare for the estimates of market betas to be off by
more than 1, whereas the estimation errors in liquidity beta can be as large as 3. In thatsense, liquidity betas are clearly harder to estimate than market betas.
The difficulty in estimating liquidity betas stems from the rarity of liquidity crises. The
betas are especially hard to estimate in “calm” periods in which no crises occur. To establish
this result, we divide our simulated samples into two subsets: those containing at least one
large liquidity shock (i.e., at least one non-zero value of Jt ) and those with no large shocks.
The first subset includes about 64% of all simulated samples. Panel A of Figure 6 shows thatthe distribution of eLIQ
is substantially wider in samples without large liquidity shocks. Inisuch samples, we sometimes observe estimation errors as large as 3 in magnitude, whereas
errors larger than 1 are extremely rare in samples with at least one large liquidity shock.
Therefore, a simple way to improve the precision of liquidity beta estimates is to compute
them over periods that include at least one liquidity crisis.
Another effective way to reduce the estimation error in liquidity betas is to form portfolios.
To illustrate this point, we expand our simulation exercise by adding portfolio formation.
Following the same procedure as before, we simulate many 60-month samples of factor andreturn realizations for 2,000 individual stocks. For each sample, we sort stocks into 10
portfolios by their true liquidity betas, and also separately by their true market betas. We
15compute the liquidity (market) beta of each portfolio as the average of the liquidity (market)
betas across all stocks in that portfolio, for both true and estimated betas.
Panels B of Figures 5 and 6 plot the distributions of the estimation errors in portfolio
betas. Compared to Panels A of the same figures, the distributions are narrower by an orderof magnitude, indicating that beta estimates are much more precise when computed at theportfolio level rather than the individual stock level. At both levels, the estimation errors inliquidity betas are larger than those in market betas because the underlying problem—thescarcity of liquidity crises—cannot be completely eliminated by forming portfolios. Nonetheless, these plots show that portfolio formation is especially valuable when estimating liquidity
betas, given the large amount of noise in their individual-stock estimates.

5. Conclusion
Twenty years after LTCM, our profession has a deeper understanding of liquidity risk andits effects on asset prices. Hundreds of studies have examined the pricing of liquidity risk
in various markets and the exposures of professional money managers to this risk. This
literature has also been among the few beneficiaries of the 2008 financial crisis.
We commend Li, Novy-Marx, and Velikov (2017) and Pontiff and Singla (2019) for theirthorough and successful replication of our 2003 study. In addition to handling their task
professionally and gracefully, they offer numerous interesting insights on this topic. Theyfind the liquidity risk factor to be priced not only in sample but also post sample, and thepost-sample performance they find is even stronger than the in-sample performance. They
also report that various alternative liquidity measures are not priced, which leads them tovoice skepticism about the cross-sectional relation between liquidity risk and expected stock
returns. However, many of these alternative measures do not resemble liquidity when plotted,
while others may not be capturing the price-relevant dimension of liquidity. Upon reviewing
their evidence, we continue to believe it is reasonable to infer that liquidity risk is priced.
Besides addressing replication, we also offer recommendations regarding the usage of ourliquidity factor, such as when to use its traded versus non-traded version and whether to use
historical or predicted liquidity betas. We explain why liquidity betas are inherently difficult
to estimate and offer tips on how to improve their precision.
Despite the immense growth in research on liquidity risk, important questions remainunanswered. For example, what predicts the periodic seizures in market liquidity? Our

16

work shows how one can measure market-wide liquidity and provides clues about how tohedge it, but not how to predict it. The origins of liquidity shocks are of obvious interest
not only to investors but also to policymakers.
The replication initiative of the Critical Finance Review raises a number of interestingquestions about replication in general. For example, should the replications that are published be a subset of those conducted, and, if so, how should the former be selected? Galiani,
Gertler, and Romero (2017) report evidence that editors are more likely to publish replication
studies that overturn previous results rather than support them. Is such a preference desirable? If journals designate ex ante the studies to be replicated, should editors select studies
that they suspect are more likely to be overturned, or should they instead select randomly
from studies deemed influential by an objective standard? The latter selection could be more
informative, for example, about the profession’s error rate. If a study goes beyond replicating
the original study and introduces additional results from alternative specifications or sample
periods, who replicates the replicators? If replicators try enough variations, they will almost
surely find some by chance that fail to reproduce the original result. What are the incentivesof a replicator? It is increasingly common to make the computer code underlying the paper’s
analysis publicly available. Could such public posting jeopardize the independence of code
written for subsequent replications? These questions, and many others, are open for debate
in our academic community.2

2

All of these questions apply to replications in general. None of them should reflect in any way on thetwo replications of our study, which we believe were conducted with the utmost care and integrity.

17

0.2

0.1

0

-0.1

-0.2

-0.3

-0.4

-0.5
1962

1969

1976

1983

1990

1997

2004

2011

2017

Year

Figure 1. The Original PS Liquidity Series. This figure plots the aggregate liquidity series of Pástor
and Stambaugh (2003) extended through 2017 by Pontiff and Singla (2019).

18

2

PS: Use Zero Volume Days

10

PS: No Stock Price Screen

1
0

5

-1
-2
0

-3
-4
-5
1962

1973

1984

1995

2006

-5
1962

2017

1973

Year
5

1984

1995

2006

2017

Year

10-6 PS: Value-Weighted

PS: Zero Intercept
0.1

0

0
-5

-0.1

-10

-0.2
-0.3

-15
-20
1962

-0.4
1973

1984

1995

2006

-0.5
1962

2017

Year

1973

1984

1995

2006

2017

Year

Figure 2. Modifications of the PS Liquidity Series. This figure plots the four modifications of thePástor and Stambaugh (2003) aggregate liquidity series constructed by Pontiff and Singla (2019).

19

0.05

Number of Zero Return Days

10

Amihud

0

-0.05
1962

5

10-5

1973

10

-4

1984

1995

2006

-1
1962

2017

Year
Bid-Ask Spread

1984

1995

2006

2017

2006

2017

2006

2017

Year
Hybrid

0.5

0

0

-5
1962

1973

1973

1984

1995

2006

-0.5
1962

2017

Year
Value-Weighted, All Price

1973

1995

Year
PS: Original

0.2

10

1984

0
0

-0.2
-0.4

-10
1962

1973

1984

1995

2006

2017

1962

Year

1973

1984

1995

Year

Figure 3. Alternative Liquidity Series. Panel A plots the original Pástor-Stambaugh aggregate liquidity
series. The remaining panels plot the five alternative liquidity series constructed by Pontiff and Singla (2019).

20

Panel A. Simulated innovations: Liquidity factor
0.05

0

-0.05

Innovation

-0.1

-0.15

-0.2

-0.25

-0.3

-0.35
0

100

200

300

400

500

600

500

600

Month

Panel B. Simulated innovations: Market factor
0.15

0.1

Innovation

0.05

0

-0.05

-0.1

-0.15
0

100

200

300

400

Month

Figure 4. Simulated innovations for a hypothetical sample. For one simulated 50-year sample, Panel
A displays the liquidity innovations and Panel B displays the market innovations.

21Panel A. Individual stocks
1.5
Liquidity betas
Market betas

Sampling density

1

0.5

0
-3

-2

-1

0

1

2

3

1

2

3

Estimation error

Panel B. Portfolios
20
18

Liquidity betas
Market betas

16

Sampling density

14
12
10
8
6
4
2
0
-3

-2

-1

0

Estimation error

Figure 5. Distributions of beta estimation errors. The plots display simulation-based sampling
densities of estimation errors in liquidity betas and market betas estimated with five years of monthly data.
Panel A displays the densities of errors in betas estimated for individual stocks; Panel B displays the densities
of errors in estimated portfolio betas.

22

Panel A. Individual stocks
1.8
1.6

Large shocks present
Large shocks absent

1.4

Sampling density

1.2
1
0.8
0.6
0.4
0.2
0
-3

-2

-1

0

1

2

3

1

2

3

Estimation error

Panel B. Portfolios
25
Large shocks present
Large shocks absent

Sampling density

20

15

10

5

0
-3

-2

-1

0

Estimation error

Figure 6. Effect of liquidity shocks on liquidity beta estimates. The plots display simulation-based
sampling densities of errors in liquidity betas estimated during five-year periods with and without large
liquidity shocks. Panel A displays the densities of errors in betas estimated for individual stocks; Panel B
displays the densities of errors in estimated portfolio betas.

23

REFERENCES
Acharya, Viral V., and Lasse Heje Pedersen, 2005, Asset pricing with liquidity risk, Journalof Financial Economics 77, 375–410.
Amihud, Yakov, and Haim Mendelson, 1986, Asset pricing and the bid-ask spread, Journalof Financial Economics 17, 223–249.
Asness, Clifford S., Tobias J. Moskowitz, and Lasse Heje Pedersen, 2013, Value and momentum everywhere, Journal of Finance 68, 929–985.
Bekaert, Geert, Campbell R. Harvey, and Christian Lundblad, 2007, Liquidity and expected
returns: Lessons from emerging markets, Review of Financial Studies 20, 1783–1831.
Bongaerts, Dion, Frank De Jong, and Joost Driessen, 2017, An asset pricing approach toliquidity effects in corporate bond markets, Review of Financial Studies 30, 1229–1269.
Brunnermeier, Markus K., and Lasse Heje Pedersen, 2009, Market liquidity and funding
liquidity, Review of Financial Studies 22, 2201–2238.
Chordia, Tarun, Richard Roll, and Avanidhar Subrahmanyam, 2000, Commonality in liquidity, Journal of Financial Economics 56, 3–28.
Fama, Eugene F., and Kenneth R. French, 1993, Common risk factors in the returns on
stocks and bonds, Journal of Financial Economics 33, 3–56.
Fama, Eugene F., and Kenneth R. French, 2015, A five-factor asset pricing model, Journalof Financial Economics 116, 1–22.
Franzoni, Francesco, Eric Nowak, and Ludovic Phalippou, 2012, Private equity performanceand liquidity risk, Journal of Finance 67, 2341–2373.
Galiani, Sebastian, Paul Gertler, and Mauricio Romero, 2017, Incentives for replication ineconomics, NBER Working Paper No. 23576, Cambridge, MA.
Goyenko, Ruslan Y., Craig W. Holden, and Charles A. Trzcinka, 2009, Do liquidity measures
measure liquidity? Journal of Financial Economics 92, 153–181.
Huberman, Gur, Shmuel Kandel, and Robert F. Stambaugh, 1987, Mimicking portfolios andexact arbitrage pricing, Journal of Finance 42, 1–9.
Li, Hongtao, Robert Novy-Marx, and Mihail Velikov, 2017, Liquidity risk and asset pricing,
Critical Finance Review, forthcoming.
Pástor, Ľuboš, and Robert F. Stambaugh, 2003, Liquidity risk and expected stock returns,
Journal of Political Economy 111, 642–685.
Pontiff, Jeffrey, and Rohit Singla, 2019, Liquidity risk?, Critical Finance Review, forthcoming.
Sadka, Ronnie, 2006, Momentum and post-earnings-announcement drift anomalies: The roleof liquidity risk, Journal of Financial Economics 80, 309–349.
Sadka, Ronnie, 2010, Liquidity risk and the cross-section of hedge-fund returns, Journal of
Financial Economics 98, 54–71.

24