Electronic copy available at: http://ssrn.com/abstract=439122

Measuring Market Liquidity

R. Burt Porter* Warrington College of Business

University of Florida

October 2003

*PO Box 117168, 321 Stuzin Hall, Gainesville, FL 32611-7168. email: burt.porter@cba.ufl.edu. Phone: (352)392-8928. I would like to thank Rob Stambaugh for providing additional information about the construction of his liquidity measure and Amy Edwards, Mark Flannery, M Nimalendran, and Jay Ritter for their many helpful comments. Any remaining errors are my own.

Electronic copy available at: http://ssrn.com/abstract=439122

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Abstract

Recent research has suggested that aggregate market liquidity varies over time and that the covariance of returns with innovations in market liquidity is priced. However, liquidity has multiple dimensions which incorporate key elements of volume, time and transaction costs. An ideal measure of market-wide liquidity should therefore incorporate elements of depth, breadth and resiliency. This paper estimates measures of market-wide liquidity along each of these dimensions and finds that each measure's innovations are correlated, that covariance of stock returns and innovations in each measure is priced, and combining the information in each measure improves the precision of estimates of liquidity risk premia. I estimate the liquidity risk premiu to be approximately 2-5% per year and show that this premium is distinct from firm size, a security’s individual liquidity, and the covariance between changes in a security's individual liquidity and market-wide liquidity. As a byproduct, I also document that the liquidity risk premium has a strong January seasonal, which is unrelated to firm size.

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I. Introduction

Asset liquidity occupies an important, but elusive, position in the study of asset pricing.

Market microstructure research has made it clear that liquidity providers offer a real service.

Buyers and sellers may not arrive in the market simultaneously, creating a role for liquidity

providers to transact and hold securities on a temporary basis1. Liquidity providers are

compensated for their expense and risk exposure via the bid/ask spread. This cost of liquidity

may be viewed as an added transaction cost and investors might require a higher expected gross

return to compensate for this added cost.

At the level of individual securities, Amihud and Mendelson (1986), Brennan and

Subrahmanyam (1996), Brennan, Chordia, and Subrahmanyam (1998), and Datar, Nail, and

Radcliffe (1998) have all found a negative relationship between a security's characteristic

liquidity and its average gross return2. Other researchers have established that the characteristic

liquidity of individual stocks covary with one another (Chordia, Roll, and Subrahmanyam

(2000), Hasbrouck and Seppi (2001) and Huberman and Halka (2001)). Commonality in

characteristic liquidity raises the question of whether shocks to aggregate or market-wide

liquidity comprise a source of nondiversifiable risk that is compensated with expected return.

When market-wide liquidity is low the probability of a seller completing a large transaction

in a timely manner without making a significant price concession is low relative to times of high

market liquidity. However, the definition of terms such as "large", "timely", and "significant"

tend to be subjective. Fernandez (1999) points out that "liquidity, as Keynes noted, is not

defined or measured as an absolute standard but on a scale, which incorporates key elements of

volume, time and transaction costs. Liquidity then may be defined by three dimensions which

incorporate these elements: depth, breadth (or tightness) and resiliency."

Standard asset pricing theory says that covariance between stock returns and any state

variable that investors care about in aggregate should be priced. If the market-wide liquidity is 1 NYSE specialists and NASDAQ market makers perform this function, however individual investors may also provide liquidity via limit orders. 2 Amihud and Mendelson use the bid-ask spread as a proxy for liquidity, Brennan and Subrahmanyman use fixed and variable components of transactions costs estimated from microstructure data, Brennan, Chordia and Subrahmanyam use trading volume, and Datar, Naik, and Radcliffe use share turnover.

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such a state variable and securities differ in their return covariances with market liquidity, then

liquidity betas should be priced. One natural approach to investigating this question is to follow

the majority of the characteristic stock liquidity literature and estimate measures of systemic

liquidity by aggregating microstructure data, but this approach suffers from at least two practical

problems. First, the large volume of data per unit of time makes it difficult to compute even the

most basic aggregate liquidity measure. Second, even the longest time series of transaction data

is short compared to the availability of lower frequency data.

Pastor and Stambaugh (2002) devise a measure of the price reversal (resiliency) dimension

of market-wide liquidity utilizing daily returns over a long time period (1962-1999). Controlling

for the usual risk factors, they find a positive relationship between stock returns and the

covariance of return with their measure of market-wide liquidity. Using other dimensions of

liquidity such as depth and breadth to construct market-wide liquidity measures appears to

remain an unexplored area of research.

This study asks three questions. First, are measures of aggregate liquidity using depth and

breadth priced, as Pastor and Stambaugh (2002) find for their resiliency measure? In addition, is

it possible to aggregate exposure to measures derived using the three dimensions of liquidity to

derive an estimate of the price of liquidity risk? Second, is it possible for investors who do not

care about return sensitivity to liquidity shocks to invest in a portfolio that is sensitive to liquidity

shocks but hedged against other common sources of systematic risk, using only prior

information, and earn a liquidity risk premium? Finally, does the premium associated with high

liquidity beta stocks survive after controlling for market capitalization, the covariance of a

stock's characteristic liquidity with changes in aggregate liquidity, and the level of the stock's

characteristic liquidity? To preview, I find that estimates of the liquidity risk premium of

approximately 2-5% per year are not sensitive to the approach used for measuring market-wide

liquidity, that a feasible investment strategy earns approximately this return before transaction

costs, and that the result survives after controlling for the three alternatives listed above.

I investigate the pricing of alternative liquidity measures by first calculating two variations

of each of three types of aggregate liquidity measures based on market resiliency, depth, and

breadth. The resiliency measure relies on the principle that order flow induces greater return

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reversals when market-wide liquidity is low, as in Pastor and Stambaugh (2002). The second

type of liquidity measure attempts to capture the depth of the market and reflects the average

price impact per unit of trading volume. This measure is closely related to that used by Amihud

(2002). The third type reflects the breadth of the market and is derived from microstructure data

on individual stock bid/ask spreads. Although this measure is available for only part of the

sample period (1983-2001) and is computationally intensive, it is important to understand how

breadth measures derived from transaction level data compare to the two alternative approaches

that are estimated using daily frequency data.

The innovations in each time series of market-wide liquidity measures are highly correlated

with each other and reflect periods of especially low measured liquidity corresponding to

commonly accepted low liquidity periods in recent U.S. history. I find that return covariance

with shocks to aggregate liquidity is priced for all three types of liquidity measures. The

estimated risk premium is positive, however there is a significant negative January seasonal in

the liquidity premium that is not related to firm size. A feasible investment strategy constructed

to have positive exposure to systemic liquidity shocks but hedged against other common risk

factors earns positive returns on average and negative returns when there is a shock to liquidity.

The relationship between liquidity beta and return remains after controlling for market

capitalization, the covariance between stock liquidity and market-wide liquidity, and the liquidity

level of the individual stocks. In other words, the higher return earned by stocks with large

liquidity betas is not due to these stocks being small, being themselves illiquid, or becoming

particularly illiquid when there is a market-wide liquidity shock.

The rest of this paper is organized as follows. Section II describes each of the measures of

aggregate liquidity examined in this paper. Section III tests whether liquidity risk as measured

by return covariance with shocks to the aggregate liquidity measures is priced. Section IV

investigates the relationship between liquidity betas and the covariance of individual stock

liquidity with aggregate liquidity, stock's characteristic liquidity, and firm size. Section V

concludes.

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II. Measures of Aggregate Liquidity

This section defines two versions for each of three types of market liquidity measure. I

show that each of the measures is correlated with the others, with market returns, and with the

size and book-to-market factor returns of Fama and French.

A. The Price Reversal Measure

Pastor and Stambaugh estimate a liquidity measure based on the idea that price changes

accompanying large volume tend to be reversed when market-wide liquidity is low. This view of

volume related return reversals arising from liquidity effects is motivated by Campbell,

Grossman, and Wang (1993), where risk-averse market makers (in the sense of Grossman and

Miller (1988)) accommodate order flow from liquidity motivated traders and are compensated

with higher expected return. For this type of measure, low market-wide liquidity refers to those

states where market makers require a higher expected return to accommodate a given order flow.

Using daily data for NYSE and AMEX listed stocks from 7/1962 through 12/2001, the

following regression is estimated for each stock for each month:

, 1, , , , ,, , ,, , 1,, ( ) , 1,...,PRVe ei d t i d t i d ti t i d ti t i d ti t sign d Dvr r rγ εφθ+ += + + ⋅ + = (2.1)

where

ri,d,t: the return on stock i on day d in month t,

rei,d,t: ri,d,t – rm,d,t, where rm,d,t

is the return on the NYSE/AMEX CRSP value-weighted weighted market return on day d in month t, and

vi,d,t: the dollar volume for stock i on day d in month t.

The ordinary least squares estimate of ,PRVi tγ is a proxy for stock i's liquidity in month t.

Superscripts on liquidity measures are used to differentiate between the various measures used.

An upper case X denotes a generic liquidity measure.

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Following Pastor and Stambaugh (2002), a stock's liquidity is computed in a given month

only if there are more than 15 observations from which to estimate the regression (2.1), it is not

the first or last month that the stock appears on CRSP, and the share price at the end of the

previous month is between $5 and $1000. The market-wide liquidity measure is then

constructed from the individual stock measures by averaging all of the individual measures

during the month and inflating by the ratio of total market capitalization at the end of month t-1

to total market capitalization at month 0. See Pastor and Stambaugh (2002) for a detailed

discussion of their measure.

The rational for inflating the average liquidity measure by the ratio of market

capitalizations may not be clear. Pastor and Stambaugh (2002) argue that ,

PRV

i tγ can be viewed as

"the liquidity cost, in terms of return reversal, of trading $1 million of stock i, averaged across all

stocks." Since $1 million was a relatively larger trade in the 1960s than in the 1990s, the simple

average coefficient will fall through time. Inflating the coefficient adjusts for this condition.

One drawback of the PRV liquidity measure is the use of dollar volume since equal size

trades may have different impact due to differences in, for example, the number of shares

outstanding, differences in float, and differences in the number and types of shareholders. One

possible alternative is to substitute turnover (dollar volume divided by end of previous month

market capitalization) for dollar volume although, as Pastor and Stambaugh point out, this is

similar to simply value weighting their measure. However even this measure would miss

variation in return impact of order flow due to, for example, differences in float.

A second alternative, unexplored in previous studies, is to substitute turnover scaled by

average daily turnover during the previous month for dollar volume in equation (2.1). This

measure of transaction volume will be high when daily volume is high relative to a recent time

series average. I calculate this modified version of the PRV measure as:

, 1, , , , ,, , ,, , 1,, ( ) , 1,...,mPRVe ei d t i d t i d ti t i d ti t i d ti t sign d DSTOr r rγ εφθ+ += + + ⋅ + = (2.2)

where:

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, ,, ,

, 1

i d ti d t

i t

TOSTO

TO −= ,

, , :i d tTO turnover defined as dollar volume for stock i on day d in month t divided by the market capitalization of firm i at the end of month t-1.

, 1 :i tTO − the average daily turnover for stock i in month t-1. The estimate of the aggregate liquidity measure is the average of ,

mPRV

i tγ across all stocks i in

month t. This measure of order flow will capture any unusual volume at the expense of ease of

interpretation but without the need to inflate the average coefficient by total market

capitalization.

Figure 1a plots the time series of scaled ,

PRV

i tγ and Figure 1b plots ,

mPRV

i tγ . The series are

very similar with large negative liquidity levels in months where liquidity is generally considered

to be low including October of 1987 (the crash, which is the largest negative value in both

series), November of 1973 (the Arab oil embargo, 2nd and 12th largest negative levels

respectively), September of 1998 (the Russian debt and LTCM crisis, 4th and 2nd), and October

of 1997 (the height of the Asian financial crisis, 13th and 9th). The overall correlation between

the two series is 0.713. Table 1 reports that both series display significant autocorrelation (0.21

and 0.16 for PRV and mPRV respectively).

B. The Price Impact Measure

Amihud (2002) estimates a liquidity measure based on price impact. Kyle (1985) argues that

spreads are an increasing function of the probability of facing an informed trader, and since the

market-maker cannot distinguish between order flow from informed traders and order flow from

noise traders, she sets prices that are an increasing function of the order imbalance that may

indicate informed trading. This implies an inverse relationship between price impact and

liquidity. Alternatively, price impact measures for a particular stock may be large for reasons

unrelated to asymmetric information issues or liquidity. For example, when there is a news

3 Omitting October of 1987 from both series reduces the estimated correlation to 0.67. Although influential, omitting this observation does not have a large impact on the reported correlations of the levels or innovations of the liquidity measures.

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release which impacts firm value but about which there is little disagreement, price change can

be large and volume small resulting in a large estimated price impact.

My use of the price impact measure follows the spirit of Amihud but is different from the

individual stock or characteristic liquidity approach. When market-wide liquidity is low, price

concessions required from Grossman-Miller market makers are larger per unit of volume than

when market-wide liquidity is high. By averaging price impact measures across all stocks the

idiosyncratic effects should diversify leaving only systematic liquidity. Whether this is

measurable in practice is an empirical issue.

For every NYSE/AMEX stock meeting the requirements outlined above, I calculate:

, ,,

1 , ,

1 nD i d tPIi t

dn i d t

rD v

γ=

= − ∑ (2.3)

where:

ri,d,t: the return of stock i on day d in month t.

vi,d,t: the dollar volume for stock i on day d in month t.

Dn: the number of trading days in month t.

The measure is defined as the negative of the daily average so that large negative values signify

'low liquidity' consistent with the interpretation of the PRV and mPRV measures. The market-

wide measure is the simple average of the individual stock measures. The resulting time series is

then inflated by the ratio of total market capitalization at the end of month t-1 to total market

capitalization at the end of month 0.

The same criticisms that apply to the use of dollar volume in the PRV measure also apply

here; therefore I also examine a modified version of the price impact measure:

( ), ,

,1 , ,

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nD i d tmPIi t

dn i d t

rD STO

γ=

= − ∑+

(2.4)

The aggregate measure is the simple average of the individual stock measures and is not rescaled

by market capitalization.

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Figure 1c plots the time series of scaled ,PIi tγ and Figure 1d plots ,

mPIi tγ . Scaled ,

PIi tγ is highly

serially correlated (first order serial correlation, ρ1=0.89) with periods of especially low liquidity

in the early 1970s and again in 1999-2000. ,mPIi tγ also shows similar periods of illiquidity

although not as severe as ,PIi tγ . The correlation between the two measures is 0.54. The

correlation matrix for all of the aggregate liquidity measures is shown in Table 1. All of the

measures are positively correlated with each other and we can reject the null that each correlation

is zero although the correlation between scaled ,PRVi tγ and scaled ,

PIi tγ is only 0.09.

C. Measures Based on Bid/Ask Spread

Amihud and Mendelson (1986), Chordia, Roll, and Subrahmanyam (2000), Hasbrouck and

Seppi (2001), Huberman and Halka (2001), Jones (2001), Baker and Stein (2002) and many

others examine the bid-ask spread as a measure of the characteristic liquidity of individual

stocks. An investor wishing to trade immediately may always sell (buy) at the quoted bid (ask)

price that includes a concession (premium) for immediate execution. Therefore the spread

between the bid and the ask prices, which is the sum of the concession and premium, divided by

the midpoint of the spread, is a natural measure of liquidity.

Using ISSM data from 1983-1992 and TAC data from 1993-2001, I calculate aggregate

liquidity measures using all NYSE/AMEX stocks as follows.4 First, define RQSi,d,t as the daily

average relative quoted spread for stock i on day d in month t. RQSi,d,t is the average of every

best bid and offer (BBO) eligible quote from the open until just prior to the market close divided

by the quote midpoint. RESi,d,t is defined as the daily average relative effective spread and is the

average of the absolute value of the difference between each transaction price and the midpoint

of the most recent quote, which is at least five seconds prior to the trade, divided by the quote

midpoint. The aggregate liquidity level during month t is:

tD

i,d,t1 d=1t

1 RQS D

γ=

= − ∑ ∑tNRQS

t itN (2.5)

4 I am grateful to M. Nimalendran for providing the quote and effective spread data.

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tD

i,d,t1 d=1t

1 RES D

γ=

= − ∑ ∑tNRES

t itN (2.6)

where Nt is the number of firms in month t and Dt is the number of days in month t. Increasing

spreads are associated with decreasing liquidity, therefore the leading negative sign is added so

that smaller values of γ are associated with lower liquidity, consistent with the other measures.

Figure 1e plots the time series of RQS

tγ and Figure 1f plots RES

tγ for the period 1983-2001.

Both plots show an upward trend reflecting the falling quoted and effective spreads during the

period. There are also large negative changes in the liquidity measure in October of 1987 and

September of 1998. Consistent with the positive time trend, both series are strongly positively

serially correlated.

D. Innovations in Aggregate Liquidity

For asset pricing purposes it is the covariance of asset returns with innovations in the

aggregate liquidity measure that is important. This is in contrast to characteristic liquidity where

the difference in liquidity levels implies differences in transaction costs that must be

compensated with expected return. To estimate innovations from levels, I calculate the first

difference of each liquidity measure as:

( ), , 11

1ˆ ˆ ˆ .γ γ γ −=

∆ = −∑tN

X X X Xt t i t i t

it

MN

(2.7)

where XtM = (mt-1 /m0), the ratio of total market capitalization at time t-1 and time zero, for X =

PRV and X=PI and MtX =1 for all others. I then regress ˆtγ∆ on its lag as well as the lagged value

of the scaled series:

1 1 , 1ˆ ˆ ˆ .X X X X Xt t j i t ta b cM uγ γ γ− − −∆ = + ∆ + + (2.8)

Thus, the predicted change depends on the lag level and the lag change. The innovation in

aggregate liquidity is Xtu . To ease comparison of results between liquidity measures in later

sections, I rescale Xtu so that the standard deviation of the innovations is of the same order of

magnitude for each measure.

11

100 0.10

10 100 100

PRV PRV mPRV mPRV PI PIt t t t t t

mPI mPI RQS RQS RES RESt t t t t t

L u L u L u

L u L u L u

= = ⋅ = ⋅

= ⋅ = ⋅ = ⋅ (2.9)

Table 2 shows that (2.8) yields innovations that are serially uncorrelated for all measures in the

full sample and in both subperiods.

If the three dimensions of market-wide liquidity are related, then we might expect the

innovations to be correlated. Panel A of Table 2 shows the correlation matrix of the innovations

over the full sample period. All of the innovations are significantly positively correlated, both in

the full sample and in each subperiod. The only exception is the correlation between the PRV

and PI measures in the second subperiod, which is a statistically insignificant 0.06. The

significant correlations for the later subperiod in Panel B between the breadth measures

estimated from microstructure data and the resiliency and depth measures estimated from daily

data are particularly encouraging. If market-wide liquidity measures can be estimated using low

frequency data then the cost of estimation is greatly reduced and the measures can be estimated

over much longer time periods and for markets for which transaction level data is not available.

Figure 2 plots the time series of the innovations in aggregate liquidity. All show large

negative values on similar dates, October of 1987 in particular, although the magnitude of these

shocks varies. Panel B of Table 2 shows us that the LmPI measure is highly correlated with each

of the microstructure based measures with an estimated correlation of 0.74 with each. The LPRV,

LmPRV, and LPI measures are also significantly correlated with LRQS and LRES. The correlation

matrices in Table 2 suggest a similarity among proxies but do not by themselves imply that

market liquidity is a priced state variable.

E. Empirical Features of the Liquidity Measures

Pastor and Stambaugh (2002) describe a "flight to quality" effect when their measure of

market liquidity is low. Months in which liquidity is exceptionally low tend to be months in

which stock returns and bond returns move in opposite directions. Table 3 reports the correlation

between the value-weighted CRSP index of NYSE-AMEX stocks and three fixed income

variables: minus the change in the rate on one-month Treasury bills, the return on the thirty year

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government bond, and the return on a portfolio of long term corporate bonds5. Over the full

sample period 1962-2001, the correlation between minus the change in the rate on one-month

Treasury bills and the market return is near zero and between the market return and the bond

returns is positive. In months of low liquidity, defined as a liquidity shock more than two

standard deviations below the mean, the correlation between the market return and both minus

the treasury bill return and the government bond return is negative, regardless of the liquidity

measure used to identify months of low liquidity. The correlation between the corporate bond

return and the market return is near zero when low liquidity is defined using LPRV or LmPRV and

negative when using LPI or LmPI. Panel B reports similar figures for the 1983-2001 period and

include the spread based (breadth) measures of liquidity. The results are very similar to those in

Panel A.

Also shown in Table 3 is the correlation between the market return and the equally weighted

average percentage change in monthly dollar volume for NYSE-AMEX stocks. The

unconditional correlation between volume changes and market returns is positive; however,

regardless of the measure used to identify months of low liquidity, when liquidity is low, market

returns and changes in volume are negatively correlated.

Table 4 reports correlations between innovations in each liquidity measure and the value-

weighted CRSP index, the equal-weighted CRSP index, and the Fama French factors SMB and

HML. Each measure is positively correlated with both CRSP indices; however the correlation is

driven by months in which the market falls. For example, the LPRV measure has a correlation of

0.29 with the value-weighted index, but the correlation is –0.02 in months in which the index

return is positive and 0.44 when negative. Each of the measures is positively correlated with

SMB and negatively correlated with HML. When liquidity is low, large stocks outperform small

stocks and value outperforms growth. The correlations are larger in magnitude and significance

for the price impact and spread based measures than for the reversal-based measures.

It is remarkable is that the six liquidity measures that address the three separate dimensions

of liquidity appear so similar. Months of low liquidity are months in which stock market returns

fall, large stocks outperform small stocks, and value outperforms growth. The next step is to

5 The corporate bond data is from Ibbotson Associates.

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examine the pricing implications of a stock's return covariance with each of these measures,

controlling for other commonly used sources of risk.

III. The Liquidity Risk Premium

This section investigates whether a stock's expected return is related to the covariance of its

return with innovations in each of the liquidity measures after controlling for other variables that

have been found to be important in asset pricing. To accomplish this I use a portfolio-based

approach where the portfolios are formed on the basis of predicted sensitivity to liquidity shocks.

Each month, the universe of available stocks is sorted into ten portfolios by predicted liquidity

beta and held for one month. The portfolio returns are linked through time to form a single

return series for each decile portfolio. These post formation returns are then regressed on return

based factors that are commonly used in empirical asset pricing studies. To the extent that the

intercepts are different from zero, liquidity sensitivity explains a component of returns not

captured by exposure to other factors.

Specifically, for each month t, I regress the excess stock return on the liquidity innovation,

LX, in a regression that also includes the Fama and French (1993) factors:

0,

M S H X Xi t i i t i t i t i t tr RMRF SMB HML Lβ β β β β ε= + + + + + (3.1)

where XtL is the innovation calculated using one of the six methods described above. For every

month t between December 1965 and December 2001, the regression is run for every stock

whose end of month price at month t-1 is between $5 and $1000 and which has valid return data

in at least 36 months between t-1 and t-60.

Although it seems natural to use the estimated regression coefficient βX

i to sort stocks into

portfolios, it is well known that sorting on regression coefficients in this manner is problematic,

especially when the standard errors of the regression coefficients are large. This is of particular

concern for the regression (3.1) since the standard errors of βX

i for individual stocks are very

large and therefore sorting on βX

i , in effect, leads to sorting on estimation errors.

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To mitigate this problem I use a Bayesian approach to form the estimates of Xiβ then sort

into portfolios based on these estimates. The Bayesian estimates of Xiβ are only used to sort

stocks into portfolios, all point estimates reported in the tables are the result of classical

econometric techniques. Specifically, I estimate (3.1) for every stock at month t, then treat each

estimate of the vector β = [β0, βM, βS, βH, βX] as a draw from a multivariate normal distribution

with estimated covariance Σ . The Bayesian estimate of β, ib , is then estimated as:

( )( ) ( )( )β β−− −

− −= Σ + Σ +11 12 2' 'i ii ib X X X Xs s (3.2)

where ( ) 12 'i X Xs− is the estimated covariance matrix of iβ . The Bayesian estimate of β is a

weighted average of the OLS estimate of βi and the average of β across all stocks at time t where

the weight on βi is the inverse of the covariance matrix of βi. This estimator "shrinks" the

estimate of the coefficient vector for each stock towards the population average with the amount

of shrinkage an inverse function of the precision of the estimate for the individual stock.

Pastor and Stambaugh (PS) use a similar approach to infer that their liquidity measure is

priced, although they use a different method of sorting stocks into portfolios. They model the

time variation in β Xi explicitly using the full sample up to time t to estimate the parameters. I

prefer my method for sorting into portfolios for three reasons. First, although PS model time

variation in the liquidity beta, they assume the other factor loadings and the parameters of the

model for time variation in liquidity beta do not change over a sample period of up to 35 years.

Second, the model of time variation proposed by PS captures very little of the variation in

liquidity betas as measured by R2, and the coefficients are unstable through time. Third, the in-

sample loadings on innovations in liquidity are not as the model predicts. Appendix A discusses

the PS methodology, elaborates on the above points, and compares it to the method used in this

paper.

A. Asset Pricing Tests

I test for the existence of a liquidity risk premium in two ways. First I estimate the abnormal

return to each predicted liquidity beta sorted decile portfolio using the three-factor model of

Fama and French and examine the intercepts. The difference in abnormal return between the

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extreme deciles provides information about a component of expected returns not captured by the

three-factor model. The second test uses the information in all ten decile return series to estimate

the liquidity risk premium directly.

1. Fama French Alphas

The time series returns for each liquidity beta decile portfolio are regressed on the three

Fama-French factors that are commonly used in empirical asset pricing studies6. To the extent

that the regression intercepts, or alphas, differ from zero, βX explains a component of expected

returns not captured by exposure to the other factors.

Table 5, panel A shows the alphas from Fama-French regressions of the excess return on

each equal-weighted decile portfolio for each liquidity measure. The intercepts are generally

negative for the portfolio of those stocks with the least sensitivity to liquidity shocks and

increasing as we move to portfolios with a greater sensitivity. The spread in intercepts between

the most and least sensitive portfolios is positive, ranging from 0.32 to 3.11 percent per year.

None of the spreads differs significantly from zero. Panel B reports results from value-weighting

the decile portfolios. The pattern in intercepts in Panel A repeats although the pattern is less

apparent. Five of the six spreads in intercept are positive and none of the spreads are statistically

significant.

Since there is a size component to liquidity, smaller firms are concentrated in the extreme

deciles, it seems appropriate to check for a January seasonal in order to verify that the "liquidity

premium" is not a rediscovery of the size/January effect7. To isolate the seasonal effect in the

intercepts I estimate (3.1) and (3.2) separately for Januaries and for all other months. The

portfolio returns are the same as those above. Table 6 reports the results.

When the portfolios are equal-weighted, the spread in annualized intercepts for non-January

months ranges from 0.93 to 2.67 percent. Few of the individual portfolio intercepts differ

significantly from zero, but both price reversal measures and one of the two price impact

6 For each liquidity measure, the ten equations are stacked and estimated using GMM. The point estimates will be identical to those from equation by equation OLS but the standard errors are corrected for autocorrelation and conditional heteroskedasticity. This method makes tests of cross-equation restrictions simple. 7 Unreported tests fail to identify a January seasonal in any of the six market-wide liquidity measures.

16

measures' portfolio intercept spreads are statistically significant. Although the spread in

intercepts for the microstructure based measures are of similar magnitude (annualized spreads of

2.65 and 1.87), the shorter time series results in an inability to reject that the spreads are zero.

Interestingly, the spreads in January are negative for five of the six measures. Only the RQS

measure, which is based on quoted spreads and does not include transaction prices, has a positive

spread in January. Panel B of Table 6 reports value-weighted results similar to the equal-

weighted results. Five of six measures have positive intercept spreads in non-January months

although none are statistically significant. Only the RQS measure is associated with a positive

intercept spread in January.

The difference in intercept spreads in Januaries vs. Non-Januaries presents an interesting

puzzle. Even after including the SMB factor, the model of Fama and French does not fully

explain the January effect. Since small firms are concentrated in the lowest and highest liquidity

beta sorted deciles, we might be tempted to argue that the inability of the three factor model to

explain the January effect in small stock returns is confounding any liquidity effects. However,

as I will show later, the negative January seasonal in the liquidity premium is not confined to

small stocks but exists across all size quintiles.

2. Direct Estimation of the Liquidity Risk Premium

The previous section infers the existence of a liquidity risk premium from the spread in

abnormal returns between the highest and lowest decile of predicted liquidity sensitivity. It is

also possible to estimate the liquidity risk premium directly using information from all ten

portfolios. For each measure of aggregate liquidity X, define the time series regression:

0X X

t t t tr BF L eβ β= + + + (3.3)

where rt is a 10x1 vector of excess returns on the decile portfolios, Ft is a 3x1 vector containing

the realizations of the Fama-French factors RMRF, SMB, and HML, B is a 10x3 matrix of factor

loadings, βX is a 10x1 vector of liquidity betas, and XtL is the innovation in aggregate liquidity

measure X. Assume the portfolios are priced by:

( ) ,F X XtE r Bλ β λ= + (3.4)

17

where ( )E i denotes the unconditional expectation, and λi is the risk premium for factor i. Since

F are returns on portfolios, let ( )FTE Fλ = . Taking expectations of both sides of (3.3),

substituting (3.4), and solving for β0 gives:

( )( )0X X X

tE Lβ β λ= − (3.5)

I estimate the vector of parameters b = [β0 B βx λX] using the General Method of Moments of

Hansen (1982). The GMM estimator of b minimizes g(b)'W-1g(b) where g(b) is the sample

average of ft(b),

( ) ( )( )' '

0

1

,

t t

t X Xt t

Xt t t

X Xt t t t

h ef b

L E L

h F L

e r BF Lβ β

⊗ = −

=

= − − −

(3.6)

and W is a consistent estimator of the spectral density of ft8.

The estimates of the liquidity risk premium λX for each of the liquidity measures as well as

the associated t-statistics for both equal-weighted and value-weighted portfolios are reported in

Table 7. The magnitude of λX depends on the arbitrary scaling of LX, but the scaling does not

affect the t-statistics or the product βλ, therefore Table 7 also reports (β10-β1)λ for each of the

liquidity measures. The first column uses the full time series of predicted liquidity beta portfolio

returns and ignores the January seasonal, and is comparable to Table 5. The second column uses

the same time series but drops all January observations from the sample and is comparable to

Table 6.

When portfolios are equal-weighted and the full sample is used, the estimated liquidity risk

premium λ is positive for all six measures and is statistically significant for four of six. When

Januaries are dropped from the sample, five of six are significant, and the point estimates are

generally larger. (β10-β1)λ is always positive, ranging from 0.40 to 4.29 percent per year using

the full sample and 1.17 to 4.30 percent per year using only non-January months. All of the

8 I use an iterated GMM estimator where the moment conditions are equally weighted in the first step and the value of b that minimizes the objective function used with the QS kernel to estimate the spectral density of ft.

18

values of (β10-β1)λ are statistically significant with the exception of the modified price impact

measure, mPI. Splitting the sample shows the price of liquidity to be approximately constant

through time.

Panel B of Table 7 reports results using value-weighted portfolios. The results are similar

to the equal-weighted results. Estimates of the return for bearing systematic liquidity risk as

measured by (β10-β1)λ in non-Januaries ranges from 1.72 to 5.02 percent per year.

B. Hedged Portfolio Returns

If a portfolio constructed to have a positive sensitivity to liquidity risk earns a risk premium

then we would expect the portfolio to do well on average and to do poorly when there is a market

liquidity shock. To test whether this is in fact the case, I form a feasible portfolio for each

liquidity measure that is long decile 10 (high predicted liquidity beta) and short decile 1 (low

liquidity beta). The returns to this portfolio are then hedged for the usual Fama-French risk

factor exposure using factor loadings estimated using data from months t-1 through t-60.

Table 8 reports the results. When the Fama-French factor-neutral portfolios are equal-

weighted, the liquidity trading strategy earns from 14 to 42 basis points (1.68% and 5.04%

annualized) per month. The profits from portfolios formed on the four non-microstructure data

based measures that are available for the full sample period are all statistically significant. Five

of the six portfolios have negative returns in months when liquidity is low, and all six have

smaller returns in low liquidity months than in other months. The last three columns drop

Januaries from the sample with little effect. When the feasible Fama-French factor-neutral

portfolios are value weighted, all six earn positive returns, four of six are negative in low

liquidity months and five of six earn lower returns on average when liquidity is low than in other

months. Statistical significance is generally lower than when portfolios are equal-weighted.

C. Combining Liquidity Measures

If the three dimensions of liquidity are related, then it should be possible to combine the

information contained in each variable to improve our estimates of liquidity risk factor exposures

and liquidity premia. To this end I form a new set of decile portfolios based on the sum of the

19

portfolio assignments from each of the six individual liquidity measures. For each stock that has

a portfolio assignment for each of the six measures (four prior to 1987), I sum the portfolio

numbers to which each is assigned then sort this summary statistic into deciles. I then repeat the

experiment outlined in Section A using the summary deciles as test assets. Table 9 presents the

results.

The spread in annualized intercepts is a statistically significant 2.64% when portfolios are

equal-weighted and 2.11% when portfolios are value weighted. We are unable to reject that the

spread in intercepts is zero when portfolios are value-weighted. Both point estimates are within

the range of those estimated in Table 5 using the individual liquidity measures. An inspection of

the t-statistics associated with the individual intercepts shows that the estimation error associated

with the intercepts is much smaller with the aggregate measure than with individual measures.

The annualized spread in intercepts is a statistically significant 3.13% when portfolios are equal-

weighted and Januaries are omitted and 2.65% when value-weighted.

Direct estimation of the liquidity risk premia using the method of Table 8 is difficult since

the portfolios have been formed based on information contained in all six liquidity measures.

However, it is possible to estimate the returns to an investment strategy long decile 10 and short

decile 1 formed using the aggregate measure and hedged against any exposure to the Fama-

French risk factors. The results to such a strategy are reported in Panel B of Table 9. The

strategy using equal-weighted portfolios earns a statistically significant return of 43 basis points

per month (5.16% annualized), larger than the return earned by any of the six individual liquidity

measure based strategies in Table 8. When the portfolio is value-weighted, the investment

strategy earns a statistically significant 50 basis points per month, again larger than that earned

by the feasible investment strategy based on any of the six individual liquidity measures.

If a "low liquidity" month is defined as a month in which all available liquidity measures are

greater than two standard deviations below their mean, the equal weighted strategy earns an

average –242 basis points in low liquidity months and the value weighted strategy an average of

+33 basis points per month. Although the average monthly return for the value weighted

strategy is positive, it is the average of only three observations. The median return of these three

observations is -324 basis points and the average return is below that of the other months. If

20

"low liquidity" is defined as any liquidity measure being more than two standard deviations

below is average, then both equal and value-weighted strategies earn negative returns in low

liquidity months and significantly positive returns in other months.

IV. Individual Stock Liquidity

The previous sections ask whether stocks whose return covaries with each of several market-

wide liquidity measures earn higher returns. This section asks whether the covariance between

stock return and market-wide liquidity shocks (liquidity return beta) is a proxy for the covariance

between changes in a stock's characteristic liquidity and market-wide liquidity shocks (liquidity

spread betas), or a proxy for the stock's characteristic liquidity level, or simply a proxy for

market capitalization.

A. Liquidity Spread Beta

Amihud and Mendelson (1986) develop a model in which expected returns are an increasing

function of the bid/ask spread. Because illiquid stocks are more expensive to trade, investors

must be compensated with higher expected returns. The question examined here is similar to

that of Amihud and Mendelson, stocks that become relatively more illiquid when aggregate

liquidity falls are particularly unattractive members of a portfolio. To see why this might be the

case recall that aggregate liquidity, regardless of which measure is used, and market returns

covary strongly when returns are negative, therefore a mutual fund manager selling to meet

redemption requests or an investor selling to meet a margin call would incur a particularly large

transaction cost associated with liquidity for holding stocks which become particularly illiquid

when market liquidity falls.

This was of particular importance during the Long Term Capital Management (LTCM)

experience of 1998. (See Lowenstein (2000) for a description of events surrounding the takeover

of LTCM by a consortium orchestrated by the New York Federal Reserve.) The hedge fund was

very highly levered in often very illiquid securities. When the Russian debt crisis precipitated a

fall in market liquidity, the value of the fund's portfolio value dropped triggering a need to

liquidate positions to meet margin calls. The anticipation of LTCM's need to liquidate further

eroded the value of the fund's positions. Prior to 1998, did LTCM earn a liquidity premium for

21

holding illiquid securities9, holding securities whose returns were sensitive to liquidity shocks, or

both?

To investigate whether the covariance of individual stock liquidity with aggregate liquidity is

priced, each month I regress changes in individual stock liquidity measures on lag changes in

individual stock liquidity measures, the lag level of the individual stock's liquidity, and shocks to

aggregate liquidity:

, , 1 , 1 ,ˆ ˆ ˆ .X X X X Xi t i t i t t i ta b c dL uγ γ γ− −∆ = + ∆ + + + (4.1)

where

,ˆ Xi tγ∆ : The change in characteristic liquidity of stock i from month t-1 to t using liquidity

measure X.

, 1ˆ Xi tγ − : The characteristic liquidity of stock i using liquidity measure X.

XtL : The shock to market-wide liquidity at time t using liquidity measure X.

and X corresponds to one of the six liquidity measures: PRV, mPRV, PI, mPI, RQS, or RES.

The coefficient vector β=[a b c d] is adjusted using the Bayesian technique described in section

III. I then sort the stocks into deciles by the Bayesian estimate of the coefficient on aggregate

liquidity shocks, LX. I refer to this coefficient as a "liquidity spread beta" and refer to the

liquidity beta discussed in the first three sections as a "liquidity return beta". The resulting decile

portfolios are then regressed on the Fama-French risk factors and the difference in annualized

intercepts between deciles 10 and 1 is examined for evidence of variation in alpha across the

decile portfolios in a manner similar to Tables 5 and 6.

Table 10 reports the results. For brevity the individual decile intercepts have been omitted

and only the annualized difference in the extreme deciles is reported. The first three columns

represent the difference in alphas for equal-weighted portfolios for the full sample and for the

sample split by January vs. Non-January. Examining the Non-January column there is some

evidence, particularly for the mPI (modified price impact) measure and the RQS (relative quoted

9 This argument applies to long positions. Many of LTCM's trading strategies involved the simultaneous purchase of long and short positions in similar securities whose prices were expected to converge.

22

spread) measure that stocks which become relatively more illiquid when the market becomes

more illiquid, conditional on the previous months change in liquidity and liquidity level, earn

negative abnormal returns, precisely the opposite of what we might expect.

These results must be interpreted with caution. Sorting on the sensitivity of changes in

individual stock liquidity to changes in aggregate liquidity is also a sort on market capitalization.

For example, the ratio of the average market capitalization of decile 1 to decile 10 for the mPI

measure reported in the first six columns of Table 10 is 4.20 and market capitalization decreases

nearly monotonically between decile 1 and decile 10. The size relative for the RQS measure is

much larger (38.70) and market capitalization is also monotonically decreasing across deciles10.

B. Characteristic Liquidity

To investigate the role of the level of characteristic liquidity, each month I sort all stocks

with available data into ten portfolios by the average of their characteristic liquidity over months

t-2 to t-4. I skip month t-1 to avoid issues associated with bid/ask bounce. The deciles are then

linked through time, regressed on the Fama-French risk factors, and the difference in intercepts

between the extreme portfolios examined. The results are reported in the right six columns of

Table 10. There is some evidence that the most liquid stocks in decile 10 earn higher risk

adjusted returns than the less liquid stocks in decile 1, especially when the mPI (modified price

impact ) or the RES (relative effective spread) measures are used as a measure of liquidity.

Again, we must interpret the results with caution since sorting on characteristic liquidity is

similar to a sort on market capitalization with the most illiquid stocks in decile 1 also being the

smallest stocks.

Table 10 provides weak evidence that more liquid stocks have higher risk adjusted returns

than illiquid stocks. Although this result contradicts Amihud and Mendelson (1986), it is

consistent with Eleswarapu and Reinganum (1993). In particular, Eleswarapu and Reinganum

find stocks that are particularly illiquid as measured by relative quoted bid/ask spread earn higher

size-adjusted average returns in January and lower size-adjusted returns in non-Januaries,

consistent with Table 10. 10 This is in contrast to the liquidity return beta deciles from Section III that have smaller firms concentrated in the lower and higher deciles with larger firms in the middle deciles.

23

C. Two-Way Sorts

To disentangle liquidity effects from pure size effects, I first sort all stocks into size quintiles

using NYSE derived breakpoints, then within each size quintile I sort into quintiles by liquidity

measure, either liquidity return beta, liquidity spread beta, or characteristic liquidity, to form 25

portfolios. These portfolios are linked through time and regressed on the Fama-French risk

factors and, as before, the difference in alphas is examined for evidence of abnormal return

associated with liquidity while controlling for market capitalization. I also examine the

relationship between liquidity return betas and both liquidity spread betas and characteristic

liquidity by first sorting into quintiles by either spread beta or characteristic liquidity and then

sorting on liquidity return beta within each quintile. In the interest of brevity, the results reported

in Table 11 are only for equal-weighted portfolios and only report the difference in alpha by

control variable quintile.

Examining the spread in alphas from liquidity return betas while controlling for market

capitalization, there is no obvious relationship between size quintile and spread in abnormal

return. The quintile with the largest spread in intercepts varies by measure with the quintile of

the largest stocks having the biggest spread in intercepts (in non-Januaries) for three of six

measures. The most surprising result comes from the January months. The negative spread in

abnormal returns is not confined to the smallest stocks, indeed the biggest negative spread in

intercepts is always in one of the biggest three of the five quintiles. While the classic January

effect is closely related to market capitalization, the relative underperformance of high liquidity

return beta stocks in January is not limited to smaller firms.

Consistent with Table 10, when using the reversal-based liquidity measures PRV and mPRV,

there is no evidence that liquidity spread betas or characteristic liquidity is priced after

controlling for market capitalization. The price impact measure mPI's significant negative

spread in alphas when sorted by liquidity spread beta continues when controlling for size

although the effect is larger for the smaller quintiles. The similar results using the spread based

measure RQS also persists across size deciles. The significant positive spread in alphas between

a portfolio of stocks with low characteristic liquidity as measured by mPI and high characteristic

liquidity is largest among the smallest stocks and virtually disappears by the largest quintile

24

because there is little variability in the measure for larger stocks. The significant positive spread

using the spread-based RQS as a measure of characteristic liquidity almost disappears after

controlling for size.

To verify that liquidity return betas are not proxies for characteristic liquidity or liquidity

spread betas, I use the latter variables as control variables and sort stocks into quintiles by the

control variable before sorting by liquidity return beta. For each liquidity measure, sorting first

by liquidity spread beta or characteristic liquidity then by liquidity return beta has little effect.

The point estimates by quintile are similar to those of the univariate sort in Table 5 and Table 6.

In summary, there is weak evidence that liquidity spread beta and characteristic liquidity are

priced, but the risk premia do not have the expected sign. Stocks with a high covariance between

changes in individual liquidity and shocks to market-wide liquidity (after controlling for the

lagged level and lagged change in characteristic liquidity) earn lower risk adjusted returns than

those with a low covariance. There is also weak evidence that stocks which are relatively liquid

earn higher average risk adjusted returns than stocks which are relatively illiquid in non-January

months. There is strong evidence that illiquid stocks (as measured by characteristic liquidity) do

earn larger abnormal returns in January, but the effect is concentrated in the smallest two size

quintiles. Most important, there does not appear to be any relationship between the higher

abnormal returns earned by high liquidity return beta stocks and either liquidity spread betas or

the stock's characteristic liquidity.

V. Conclusion

Numerous authors have found that illiquid stocks earn higher average returns, presumably to

compensate for the higher costs of transacting. This paper addresses the related but separate

issue of market-wide liquidity. If the ability to transact a given volume with minimal price

concession varies through time and there exist cross sectional differences in a stock’s return

covariance with measures of market-wide liquidity, then this covariance should carry with it a

higher expected return.

Measures of market-wide liquidity designed to capture three related, but separate

dimensions of liquidity yield similar results, namely that covariance with market-wide measures

25

of liquidity carries a risk premium. This risk premium varies from approximately 2 to 5% per

year depending on the measure. This estimate of the liquidity risk premium associated with

covariance with market-wide liquidity shocks is much lower than that estimated by Pastor and

Stambaugh (2002) but is still statistically significant. Four of the six measures used do not

require transaction level data and thus can be constructed over much long time spans and in

markets for which transaction level data is unavailable.

Covariance of return with market-wide liquidity does not appear related to the covariance

between changes in a stock’s characteristic liquidity and market-wide liquidity. In other words,

stocks whose price is sensitive to market-wide liquidity shocks do not necessarily become

themselves more illiquid when market liquidity is low. This is consistent with fund managers

selling more liquid stocks to meet margin calls or redemption requests when market-wide

liquidity is low.

Several surprising results provide avenues for future research. First, there is a strong

January seasonal in the liquidity risk premia but not in the liquidity measures themselves.

Although high liquidity beta stocks earn higher risk-adjusted returns on average, they earn lower

returns in January. This January effect in liquidity is not related to firm size, rather it exists

across all size quintiles. Second, liquidity spread betas, or the covariance between changes in a

stock's own characteristic liquidity and shocks to market-wide liquidity, is associated with lower

average returns. In other words, stocks that become particularly illiquid when markets become

illiquid earn below average returns, a counterintuitive result.

The negative average return associated with liquidity spread betas is particularly surprising

because one might have expected that an individual stock's liquidity is particularly important

when the market is less liquid overall. If aggregate liquidity falls when market returns are large

and negative, then investors who must sell will sell those investments that have the best

individual characteristic liquidity so as to minimize the transaction costs associated with

liquidity. This reasoning is consistent with the financing of margin investors by uninformed

outside lenders who react to losses by cutting lending (see Shleifer and Vishny (1997) and Xiong

(1999)). This would result in a high covariance between the return of stocks whose individual

liquidity is high and the aggregate liquidity state variable. Why then should investors demand a

26

premium for holding these stocks? Why do investors not require a premium for holding stocks

whose characteristic liquidity worsens when aggregate liquidity falls? These questions provide

an interesting avenue for further research.

27

References Amihud, Yakov, 2002, "Illiquidity and Stock Returns: Cross-Section and Time-Series Effects," Journal of Financial Markets, 5, 31-56. Amihud, Yakov and Haim Mendelson, 1986, "Asset Pricing and the Bid-Ask Spread," Journal of Financial Economics, 17 223-249. Andrews, D. W. K., 1991, “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation,” Econometrica, 59, 817-858. Baker, Malcolm and Jeremy C. Stein, 2002, "Market Liquidity as a Sentiment Indicator," Harvard Institute of Economic Research Discussion Paper Number 1977. Brennan, Michael J. and Avanidhar Subrahmanyam, 1996, "Market Microstructure and Asset Pricing: On the Compensation for Illiquidity in Stock Returns," Journal of Financial Economics, 41, 441-464. Campbell, John Y., Sanford J. Grossman and Jiang Wang, 1993, "Trading Volume and Serial Correlation in Stock Returns," The Quarterly Journal of Economics, 108, 905-939. Chordia, Tarun, Richard Roll, and Avanidhar Subrahmanyam, 2000, "Commonality in Liquidity," Journal of Financial Economics, 56, 3-28. Cochrane, John H., 2001, Asset Pricing, Princeton University Press, Princeton N.J. Datar, Vinay T., Narayan Y. Naik, and Robert Radcliffe, 1998, "Liquidity and Stock Returns: An Alternative Test" Journal of Financial Markets, 1, 203-219. Eleswarapu, Venkat R., and Marc R. Reinganum, 1993, "The Seasonal Behavior of the Liquidity Risk Premium in Asset Pricing", Journal of Financial Economics, 34, 373-386. 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 James D. MacBeth, 1973, "Risk, Return, and Equilibrium: Empirical Tests," Journal of Political Economy, 81, 607-636. Fernandez, Frank A., 1999, "Liquidity Risk: New Approaches to Measurement and Monitoring," Securities Industry Association Working Paper. Grossman, Sanford J. and Merton H. Miller, 1988, "Liquidity and Market Structure," The Journal of Finance, 43, 617-633. Hansen, L. P., 1982, “Large Sample Properties of Generalized Method of Moments Estimators,” Econometrica, 50, 1029-1054.

28

Hasbrouck, Joel and Duane J. Seppi, 2001m, "Common Factors in Prices, Order Flows, and Liquidity," Journal of Financial Economics, 59, 383-411. Huberman, Gur and Dominika Halka, 2001, "Systematic Liquidity," The Journal of Financial Research, 24, 161-178. Jones, Charles M., 2001, "A Century of Stock Market Liquidity and Trading Costs," Columbia University Working Paper. Kyle, Albert S., 1985, "Continuous Auctions and Insider Trading," Econometrica, 53 1315-1335. Lowenstein, Roger, 2000, When Genius Failed, New York, Random House. Pastor, Lubos and Robert F. Stambaugh, 2002, "Liquidity Risk and Expected Stock Returns," Journal of Political Economy, forthcoming. Shleifer, Andrei and Robert Vishny, 1997, "Limits of Arbitrage," The Journal of Finance, 52, 35-55. Shanken, Jay, 1990, "Intertemporal Asset Pricing: An Empirical Investigation," Journal of Econometrics, 45, 99-120. Shanken, Jay, 1992, "On the Estimation of Beta Pricing Models", Review of Financial Studies 5, 1-24. Xiong, Wei, 2001, "Convergence Trading with Wealth Effects: An Amplification Mechanism in Financial Markets," Journal of Financial Economics, 62 247-292.

29

Appendix A

This appendix compares two methods for predicting liquidity betas. The first is that used by

Pastor and Stambaugh (2002) and the second is the Bayesian regression approach used in this

paper.

I. A Two-Stage Approach For Predicting Liquidity Betas

The liquidity beta is defined as the coefficient on the liquidity innovation, LX, in a regression that

also includes the Fama and French (1993) factors:

0,

m S H X Xi t i i t i t i t i t tr RMRF SMB HML Lβ β β β β ε= + + + + + (A.1)

where XtL is the innovation calculated using one of the six methods described in Section II. One

method of predicting future values of βX is to run the regression (A.1) using past data and

estimate the future value of the liquidity beta as the estimate of X

iβ from (A.1).

An alternative used by Pastor and Stambaugh (PS) is to model explicitly the time variation in

liquidity betas using the method of Shanken (1990). Specifically, the liquidity beta is modeled as

a linear function of a set of instruments Z:

', 1 1, 2, , 1X

i t i i i tZβ ψ ψ− −= + . (A.2)

The vector Zi,t-1 used by PS contains the following characteristics: (i) the historical liquidity beta

estimated using all data available for stock i from months t-60 through t-1 (if at least 36 months

are available), (ii) the average value of the liquidity level for the individual stock from month t-6

through t-1, (iii) the natural log of the stock's average dollar volume from months t-6 through t-1,

(iv) the cumulative return on the stock from month t-6 through t-1, (v) the standard deviation of

the stocks' monthly price per share from month t-1, (vi) the natural log of the price per share

from month t-1, and (vii) the natural log of the number of shares outstanding from month t-1.

Each characteristic is "demeaned" by subtracting the time series average through month t-1of the

characteristic's cross-sectional average in each previous month.

30

Substituting (A.2) into (A.1):

( )0 ', 1, 2, , 1

m S H Xi t i i t i t i t i i i t t tr RMRF SMB HML Z Lβ β β β ψ ψ ε−= + + + + + + (A.3)

where XtL , the innovation in the liquidity series, is estimated from the residuals in (2.8) using

only information through t-1. This is in contrast to the series reported in Figure 2 that used the

full time series to estimate the shocks. ψ1,i and ψ2,i are restricted to be the same across all stocks

for each measure of liquidity. Specifically, at the end of each year from 1965 through 2000, PS

construct for each stock which has at least 36 observations the historical series of

, ,m S H

i t i t i t i t i tr RMRF SMB HMLε β β β= − − − (A.4)

where the βs are estimated from the regression of the stock's excess return on the Fama-French

factors and the liquidity innovation using all data through t-1. Then they run a pooled time-

series cross-sectional regression of εi,t on the characteristics,

( )', 0 1 2 , 1 ,

X Xi t t i t t i tL Z L vε ψ ψ ψ −= + + ⋅ + (A.5)

A stock is excluded for any month in which it has any missing characteristics.

At the end of each year using the coefficient estimates from (A.5) and the end of year values

of Z, the predicted liquidity beta is calculated from (A.2). Stocks are sorted by their predicted

liquidity betas and assigned to ten portfolios. Portfolio returns are calculated for each decile for

each month and linked through time generating a single series of returns for each predicted

liquidity beta decile.

II. A Bayesian Approach For Predicting Liquidity Betas

An alternative to the two-pass approach is simply to use the liquidity betas from (A.1), calculated

using 60 prior months of data as the point estimate of the future liquidity beta. Using five years

of prior data, I calculate (A.1) for each stock each month and sort stocks into ten decile

portfolios. I link the equal-weighted decile portfolio returns through time and regress the return

series on the three Fama French risk factors. The results are reported in Panel A of Table A1.

31

The intercepts are increasing as we move from decile 1 to 10 although the abnormal return

is negative for the portfolio with the highest predicted liquidity betas. Running a second

regression of the ten portfolios and including liquidity shocks along with the Fama-French

factors we see that the in-sample coefficient on decile 10 is negative. Recall that the portfolios

have been formed explicitly to have increasing sensitivity to liquidity shocks as we move from

decile 1 to 10, therefore the fact that the in-sample coefficient on liquidity shocks for decile 10 is

the second lowest of all ten portfolios shows that sorting on βX does not adequately predict the

sensitivity of individual stocks to market-wide liquidity shocks. The reason why the in-sample

performance is so poor is due to the manner in which we formed the portfolios. Sorting on a

regression coefficient also sorts on the estimation error. If the estimated coefficient has a large

standard error, which is almost always the case when running regressions of this type with

individual stocks, then the portfolio sort does a poor job of sorting by sensitivity to LX.

The alternative to sorting on the raw coefficient is to shrink the coefficients to the

population average in inverse proportion to the estimation error. This is the Bayesian estimator

used in this paper. Sorting on the Bayesian estimates, forming decile portfolios, and regressing

portfolio returns on the Fama-French factors, we see in Panel B of Table A1 that the intercepts

increase in a nearly monotonic fashion from decile 1 to 10. When we regress the portfolio

returns on the Fama-French factors and the liquidity shock we see that the in-sample coefficients

on liquidity increase as we move from decile 1 to 10.

III. The Second Pass Regression

PS run regressions (A.4) and (A.5) using "all data available up to current year end". This implies

that the Fama-French coefficients are constant through time, over periods of up to 35 years.

Although this may not be an onerous assumption for portfolios formed on predicted liquidity, the

assumption that the coefficients in the second pass regression (A.5) are constant are not reflected

in the data.

Column 1 of Table A2 reports the coefficient estimates from (A.5) using data up to

10/1969. As in PS, each value reported is equal to the coefficient estimate multiplied by the

time-series average of the annual cross-sectional standard deviation of the characteristic. Note

that I have mimicked the PS methodology with the exception of running the sorting procedure

32

each month rather than once a year. Columns 2 and 3 report the results from the same regression

in 10/1985 and 10/2000. Similar to Table 2 in Pastor and Stambaugh (2002), the coefficients

vary through time with some switching signs. I also include the sample size and R2 for each

regression. Although many of the individual coefficients are statistically significant, the very

large sample size (as large as 410,518 observations) implies that the coefficients may be

statistically significant but economically uninteresting. Columns 4 through 6 in Panel B are the

results from running the regression in (A.5) using a maximum of sixty months of data. As

expected the coefficients are even more unstable. The R2 shows that the regression explains

almost none of the variance in stock return adjusted for the Fama-French factors.

Panel A of Table A3 shows the results from sorting on the predicted liquidity betas using the

coefficients from the second pass regression (A.5) and the current values of Zi,t-1. There is an

impressive spread in intercepts however the in sample coefficients on liquidity shocks are not

increasing across deciles. In fact, decile 10 has a large negative coefficient on liquidity but a

large positive intercept. Panel B reports the results from only using five years of data and is

broadly consistent with the results in Panel A.

Any method for predicting liquidity sensitivity must produce portfolios whose in-sample

sensitivity to liquidity shocks increases as predicted liquidity sensitivity increases. Also, given

the changing nature of financial markets over time, it is preferable not to use to long a time series

in estimating liquidity sensitivities. Of the two methods discussed, sorting by the coefficient

from the Bayesian regression does a better job of predicting future market-wide liquidity shock

sensitivity.

Tabl

e A

1

Abn

orm

al R

etur

ns fr

om P

ortfo

lios

Sor

ted

by th

e Fi

rst P

ass

Reg

ress

ion

Coe

ffici

ent

Pan

el A

: Por

tfolio

s S

orte

d by

Raw

Coe

ffici

ent

1 2

3 4

5 6

7 8

9 10

In

terc

ept

-1.5

9 -0

.35

-0.6

4 0.

32

0.66

0.

76

0.74

0.

70

0.58

-0

.43

(-

1.47

) (-

0.41

) (-

0.82

) (0

.44)

(0

.93)

(1

.13)

(1

.07)

(1

.04)

(0

.86)

(-

0.48

) M

kt-R

f 1.

13

1.06

1.

00

0.99

0.

98

0.97

0.

97

0.99

1.

03

1.06

(52.

16)

(63.

54)

(64.

18)

(68.

28)

(69.

22)

(72.

07)

(70.

57)

(74.

02)

(76.

14)

(60.

16)

SM

B

1.06

0.

73

0.58

0.

52

0.48

0.

49

0.57

0.

65

0.71

1.

02

(3

6.00

) (3

1.72

) (2

7.13

) (2

6.49

) (2

5.22

) (2

6.32

) (3

0.55

) (3

5.80

) (3

8.72

) (4

2.34

) H

ML

0.07

0.

25

0.31

0.

30

0.30

0.

33

0.32

0.

27

0.21

-0

.01

(2

.28)

(1

0.98

) (1

4.46

) (1

5.21

) (1

5.80

) (1

8.12

) (1

6.84

) (1

5.56

) (1

1.48

) (-

0.25

) ---

------

------

Liqu

idity

-2

.57

-0.0

7 1.

16

-0.7

4 0.

37

0.52

0.

96

1.44

2.

17

-1.1

4 S

hock

(-

1.56

) (-

0.06

) (0

.97)

(-

67.0

0)

(0.3

5)

(0.5

0)

(0.9

2)

(1.4

1)

(2.1

2)

(-0.

85)

Pan

el B

: Por

tfolio

s S

orte

d by

Bay

esia

n R

egre

ssio

n C

oeffi

cien

t

1

2 3

4 5

6 7

8 9

10

Inte

rcep

t -0

.58

-0.5

5 -0

.12

-0.5

4 -0

.19

-0.2

6 0.

69

0.13

0.

57

1.49

(-0.

57)

(-0.

64)

(-0.

16)

(-0.

79)

(-0.

28)

(-0.

40)

(1.1

3)

(0.1

9)

(0.8

4)

(1.9

2)

Mkt

-Rf

1.06

1.

04

1.03

1.

02

1.02

1.

02

0.99

0.

98

1.01

1.

02

(5

2.44

) (6

0.61

) (6

9.26

) (7

4.78

) (7

7.12

) (7

9.20

) (8

1.90

) (7

3.36

) (7

5.01

) (6

6.08

) S

MB

0.

76

0.65

0.

63

0.67

0.

64

0.64

0.

70

0.72

0.

70

0.69

(27.

53)

(27.

90)

(31.

40)

(35.

89)

(35.

33)

(36.

25)

(42.

34)

(39.

62)

(37.

93)

(32.

79)

HM

L 0.

21

0.27

0.

27

0.26

0.

24

0.28

0.

21

0.19

0.

20

0.21

(7.5

0)

(11.

61)

(13.

46)

(13.

95)

(13.

41)

(15.

85)

(12.

89)

(15.

59)

(11.

01)

(10.

16)

----

----

----

- Li

quid

ity

-0.5

7 -0

.05

0.65

-1

.19

-0.7

5 -0

.36

-0.0

4 0.

51

1.52

1.

76

Sho

ck

(-0.

37)

(-0.

63)

(0.5

7)

(-1.

14)

(-0.

73)

(-0.

37)

(-0.

04)

(0.5

0)

(1.4

8)

(1.4

9)

Table A2

Determinants of Predicted Liquidity Betas in Two-Stage Methodology

Panel A: All data through t-1 Panel B: 60 month window 1 2 3 4 5 6 10/1969 10/1985 10/2000 10/1969 10/1985 10/2000

Intercept -2.22 -2.08 -0.32 -2.22 0.50 0.36 (-2.41) (-5.31) (-0.86) (-2.41) (0.45) (0.40) Historical Beta 6.59 3.44 2.48 6.59 3.39 2.60 (7.41) (8.59) (9.10) (7.41) (4.20) (4.98) Average Volume -1.43 -2.35 0.05 -1.43 7.90 0.87 (-0.90) (-2.81) (0.08) (-0.90) (3.49) (0.62) Average Liquidity -2.60 0.69 -0.46 -2.60 9.10 0.68 (-3.68) (2.32) (-2.33) (-3.68) (5.24) (1.74) Cumulative Return -0.25 1.20 0.51 -0.25 6.78 1.29 (-0.27) (3.36) (2.08) (-0.27) (7.86) (3.37) Return Volatility 1.25 -2.08 0.38 1.25 -8.32 -3.27 (1.02) (-5.01) (1.26) (1.02) (-7.02) (-6.79) Price 7.66 3.11 -1.51 7.66 -6.49 -4.41 (5.81) (5.39) (-3.27) (5.81) (-4.07) (-4.90) Shares Outstanding -1.92 0.82 -1.13 -1.92 -6.43 -0.76 (-1.44) (1.16) (-1.73) (-1.44) (-3.11) (-0.61) N 62,206 267,988 410,518 62,206 103,883 144,503 Adjusted R2 0.0021 0.0010 0.0004 0.0021 0.0014 0.0006

Tabl

e A

3

Abn

orm

al R

etur

ns fr

om P

ortfo

lios

Sor

ted

by P

redi

cted

Liq

uidi

ty B

eta

Usi

ng th

e Tw

o-S

tage

Met

hodo

logy

Pan

el A

: Sec

ond

Sta

ge U

ses

All

Ava

ilabl

e D

ata

1

2 3

4 5

6 7

8 9

10

Inte

rcep

t -8

.02

-7.7

9 -4

.49

-1.8

3 -0

.76

1.19

0.

98

2.59

3.

91

5.75

(-2.

83)

(-4.

43)

(-3.

58)

(-2.

01)

(-1.

03)

(1.5

9)

(1.1

5)

(2.4

2)

(2.7

6)

(2.8

2)

Mkt

-Rf

1.32

1.

30

1.22

1.

12

1.05

0.

98

0.94

0.

88

0.83

0.

78

(2

3.71

) (3

7.89

) (4

9.48

) (6

2.86

) (7

1.95

) (6

7.13

) (5

6.53

) (4

2.06

) (2

9.98

) (1

9.45

) S

MB

1.

70

1.27

1.

04

0.86

0.

72

0.65

0.

61

0.62

0.

66

0.84

(22.

09)

(26.

56)

(30.

59)

(34.

68)

(35.

38)

(32.

36)

(26.

49)

(21.

34)

(17.

31)

(15.

22)

HM

L 0.

95

0.79

0.

57

0.48

0.

37

0.24

0.

18

0.07

-0

.11

-0.3

9

(12.

56)

(16.

79)

(17.

15)

(19.

58)

(18.

71)

(12.

22)

(7.9

4)

(2.3

9)

(-2.

84)

(-7.

09)

------

------

---

Liqu

idity

-1

0.95

-3

.63

-1.2

5 -0

.26

0.64

-1

.12

-1.1

2 -0

.68

-2.1

5 -6

.45

Sho

ck

(-2.

55)

(-1.

36)

(-0.

66)

(-0.

19)

(0.5

6)

(-0.

98)

(-0.

87)

(-0.

42)

(-1.

00)

(-2.

09)

P

anel

B: S

econ

d S

tage

Use

s M

axim

um o

f 5 Y

ears

of D

ata

1

2 3

4 5

6 7

8 9

10

Inte

rcep

t -6

.93

-5.9

8 -3

.95

-2.7

2 -0

.23

0.68

1.

97

2.41

3.

53

5.48

(-3.

03)

(-4.

08)

(-3.

46)

(-3.

12)

(-0.

29)

(0.8

3)

(2.2

4)

(2.2

5)

(2.4

7)

(2.2

8)

Mkt

-Rf

1.18

1.

20

1.18

1.

12

1.06

1.

02

0.98

0.

95

0.89

0.

88

(2

6.05

) (4

1.36

) (5

2.10

) (6

5.30

) (6

7.08

) (6

2.63

) (5

6.34

) (4

4.73

) (3

1.45

) (1

8.54

) S

MB

1.

37

1.04

0.

85

0.76

0.

72

0.68

0.

69

0.74

0.

91

1.30

(21.

91)

(26.

05)

(27.

27)

(32.

06)

(32.

81)

(30.

16)

(28.

67)

(25.

14)

(23.

25)

(19.

79)

HM

L 0.

41

0.43

0.

47

0.42

0.

37

0.35

0.

29

0.17

0.

09

0.02

(6.6

5)

(10.

82)

(15.

22)

(17.

88)

(17.

04)

(15.

53)

(12.

37)

(5.8

9)

(2.4

5)

(0.2

8)

------

------

---

Liqu

idity

-5

.56

-2.2

5 -1

.00

-1.3

6 -0

.20

-0.8

3 -2

.33

-2.9

9 -4

.42

-9.6

3 S

hock

(-

1.61

) (-

1.01

) (-

0.58

) (-

1.03

) (-

0.16

) (-

0.66

) (-

1.75

) (-

1.85

) (-

2.05

) (-

2.67

)

36

Figures 1a & 1b The PRV (Price Reversal) market-wide liquidity measure estimates the tendency of price changes accompanied by large volume to reverse as defined by equation (2.1). mPRV is the modified version of PRV defined by (2.2). PI is the price impact liquidity measure defined by (2.3) and mPI is the modified version of the price impact measure defined by (2.4). RQS is the market wide average quoted spread described in section II.c. and equation (2.5). RES is the market wide average effective spread described in II.c and equation (2.6).

PRV Liquidity Levels

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

08/1

962

12/1

963

04/1

965

08/1

966

12/1

967

04/1

969

08/1

970

12/1

971

04/1

973

08/1

974

12/1

975

04/1

977

08/1

978

12/1

979

04/1

981

08/1

982

12/1

983

04/1

985

08/1

986

12/1

987

04/1

989

08/1

990

12/1

991

04/1

993

08/1

994

12/1

995

04/1

997

08/1

998

12/1

999

04/2

001

mPRV Liquidity Levels

-0.0050

-0.0040

-0.0030

-0.0020

-0.0010

0.0000

0.0010

0.0020

08/1

962

12/1

963

04/1

965

08/1

966

12/1

967

04/1

969

08/1

970

12/1

971

04/1

973

08/1

974

12/1

975

04/1

977

08/1

978

12/1

979

04/1

981

08/1

982

12/1

983

04/1

985

08/1

986

12/1

987

04/1

989

08/1

990

12/1

991

04/1

993

08/1

994

12/1

995

04/1

997

08/1

998

12/1

999

04/2

001

37

Figures 1c & 1d

PI Liquidity Levels

-7.00

-6.00

-5.00

-4.00

-3.00

-2.00

-1.00

0.00

08/1

962

12/1

963

04/1

965

08/1

966

12/1

967

04/1

969

08/1

970

12/1

971

04/1

973

08/1

974

12/1

975

04/1

977

08/1

978

12/1

979

04/1

981

08/1

982

12/1

983

04/1

985

08/1

986

12/1

987

04/1

989

08/1

990

12/1

991

04/1

993

08/1

994

12/1

995

04/1

997

08/1

998

12/1

999

04/2

001

mPI Liquidity Levels

-0.025

-0.020

-0.015

-0.010

-0.005

0.000

08/1

962

12/1

963

04/1

965

08/1

966

12/1

967

04/1

969

08/1

970

12/1

971

04/1

973

08/1

974

12/1

975

04/1

977

08/1

978

12/1

979

04/1

981

08/1

982

12/1

983

04/1

985

08/1

986

12/1

987

04/1

989

08/1

990

12/1

991

04/1

993

08/1

994

12/1

995

04/1

997

08/1

998

12/1

999

04/2

001

38

Figures 1e & 1f

RQS Liquidity Levels

-0.0250

-0.0200

-0.0150

-0.0100

-0.0050

0.000008

/196

2

08/1

963

08/1

964

08/1

965

08/1

966

08/1

967

08/1

968

08/1

969

08/1

970

08/1

971

08/1

972

08/1

973

08/1

974

08/1

975

08/1

976

08/1

977

08/1

978

08/1

979

08/1

980

08/1

981

08/1

982

08/1

983

08/1

984

08/1

985

08/1

986

08/1

987

08/1

988

08/1

989

08/1

990

08/1

991

08/1

992

08/1

993

08/1

994

08/1

995

08/1

996

08/1

997

08/1

998

08/1

999

08/2

000

08/2

001

RES Liquidity Levels

-0.0090

-0.0080

-0.0070

-0.0060

-0.0050

-0.0040

-0.0030

-0.0020

-0.0010

0.0000

08/1

962

08/1

963

08/1

964

08/1

965

08/1

966

08/1

967

08/1

968

08/1

969

08/1

970

08/1

971

08/1

972

08/1

973

08/1

974

08/1

975

08/1

976

08/1

977

08/1

978

08/1

979

08/1

980

08/1

981

08/1

982

08/1

983

08/1

984

08/1

985

08/1

986

08/1

987

08/1

988

08/1

989

08/1

990

08/1

991

08/1

992

08/1

993

08/1

994

08/1

995

08/1

996

08/1

997

08/1

998

08/1

999

08/2

000

08/2

001

39

Figures 2a & 2b

The PRV (Price Reversal) market-wide liquidity measure estimates the tendency of price changes accompanied by large volume to reverse as defined by equation (2.1). mPRV is the modified version of PRV defined by (2.2). PI is the price impact liquidity measure defined by (2.3) and mPI is the modified version of the price impact measure defined by (2.4). RQS is the market wide average quoted spread described in section II.c. and equation (2.5). RES is the market wide average effective spread described in II.c and equation (2.6). Innovations are calculated from levels using (2.7)

PRV Liquidity Shocks

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

09/1

962

01/1

964

05/1

965

09/1

966

01/1

968

05/1

969

09/1

970

01/1

972

05/1

973

09/1

974

01/1

976

05/1

977

09/1

978

01/1

980

05/1

981

09/1

982

01/1

984

05/1

985

09/1

986

01/1

988

05/1

989

09/1

990

01/1

992

05/1

993

09/1

994

01/1

996

05/1

997

09/1

998

01/2

000

05/2

001

mPRV Liquidity Shocks

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

09/1

962

01/1

964

05/1

965

09/1

966

01/1

968

05/1

969

09/1

970

01/1

972

05/1

973

09/1

974

01/1

976

05/1

977

09/1

978

01/1

980

05/1

981

09/1

982

01/1

984

05/1

985

09/1

986

01/1

988

05/1

989

09/1

990

01/1

992

05/1

993

09/1

994

01/1

996

05/1

997

09/1

998

01/2

000

05/2

001

40

Figures 2c & 2d

PI Liquidity Shocks

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

09/1

962

01/1

964

05/1

965

09/1

966

01/1

968

05/1

969

09/1

970

01/1

972

05/1

973

09/1

974

01/1

976

05/1

977

09/1

978

01/1

980

05/1

981

09/1

982

01/1

984

05/1

985

09/1

986

01/1

988

05/1

989

09/1

990

01/1

992

05/1

993

09/1

994

01/1

996

05/1

997

09/1

998

01/2

000

05/2

001

mPI Liquidity Shocks

-0.15

-0.10

-0.05

0.00

0.05

0.10

09/1

962

09/1

963

09/1

964

09/1

965

09/1

966

09/1

967

09/1

968

09/1

969

09/1

970

09/1

971

09/1

972

09/1

973

09/1

974

09/1

975

09/1

976

09/1

977

09/1

978

09/1

979

09/1

980

09/1

981

09/1

982

09/1

983

09/1

984

09/1

985

09/1

986

09/1

987

09/1

988

09/1

989

09/1

990

09/1

991

09/1

992

09/1

993

09/1

994

09/1

995

09/1

996

09/1

997

09/1

998

09/1

999

09/2

000

09/2

001

41

Figures 2e & 2f RQS Liquidity Shocks

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

09/1

962

08/1

963

07/1

964

06/1

965

05/1

966

04/1

967

03/1

968

02/1

969

01/1

970

12/1

970

11/1

971

10/1

972

09/1

973

08/1

974

07/1

975

06/1

976

05/1

977

04/1

978

03/1

979

02/1

980

01/1

981

12/1

981

11/1

982

10/1

983

09/1

984

08/1

985

07/1

986

06/1

987

05/1

988

04/1

989

03/1

990

02/1

991

01/1

992

12/1

992

11/1

993

10/1

994

09/1

995

08/1

996

07/1

997

06/1

998

05/1

999

04/2

000

03/2

001

RES Liquidity Shocs

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

09/1

962

08/1

963

07/1

964

06/1

965

05/1

966

04/1

967

03/1

968

02/1

969

01/1

970

12/1

970

11/1

971

10/1

972

09/1

973

08/1

974

07/1

975

06/1

976

05/1

977

04/1

978

03/1

979

02/1

980

01/1

981

12/1

981

11/1

982

10/1

983

09/1

984

08/1

985

07/1

986

06/1

987

05/1

988

04/1

989

03/1

990

02/1

991

01/1

992

12/1

992

11/1

993

10/1

994

09/1

995

08/1

996

07/1

997

06/1

998

05/1

999

04/2

000

03/2

001

RES Liquidity Shocks

42

Table 1 Correlation of Liquidity Measures

This table reports the correlation in levels and the autocorrelation of the six liquidity measures. The PRV (Price Reversal) market-wide liquidity measure estimates the tendency of price changes accompanied by large volume to reverse as defined by equation (2.1). mPRV is the modified version of PRV defined by (2.2). PI is the price impact liquidity measure defined by (2.3) and mPI is the modified version of the price impact measure defined by (2.4). RQS is the market wide average quoted spread described in section II.c. and equation (2.5). RES is the market wide average effective spread described in II.c and equation (2.6). All correlations except those in bold are significant at the 5% level.

Panel A: Full Sample

Liquidity Measure n PRV mPRV PI mPI RQS RES

ρ

PRV 473 1.00 0.71 0.09 0.40 0.21 0.26 0.21 mPRV 472 1.00 0.25 0.48 0.02 0.05 0.16 PI 473 1.00 0.54 -0.41 -0.46 0.89 mPI 472 1.00 -0.09 -0.04 0.69 RQS 228 1.00 0.95 0.96 RES 228 1.00 0.95

Panel B: Subperiod Results

Subperiod 1: 8/1962-12/1982 PRV 245 1.00 0.75 0.49 0.54 0.27 mPRV 244 1.00 0.35 0.53 0.25 PI 245 1.00 0.55 0.88 mPI 244 1.00 0.69

Subperiod 2: 1/1983-12/2001

PRV 228 1.00 0.71 -0.10 0.26 0.21 0.26 0.09 mPRV 228 1.00 0.17 0.44 0.02 0.05 0.07 PI 228 1.00 0.64 -0.41 -0.46 0.87 mPI 228 1.00 -0.09 -0.04 0.69 RQS 228 1.00 0.95 0.96 RES 228 1.00 0.95

43

Table 2 Correlation of Liquidity Shocks

This table reports the cross-correlation and autocorrelation in market-wide liquidity measure. The PRV (Price Reversal) market-wide liquidity measure estimates the tendency of price changes accompanied by large volume to reverse as defined by equation (2.1). mPRV is the modified version of PRV defined by (2.2). PI is the price impact liquidity measure defined by (2.3) and mPI is the modified version of the price impact measure defined by (2.4). RQS is the market wide average quoted spread described in section II.c. and equation (2.5). RES is the market wide average effective spread described in II.c and equation (2.6). Innovations are calculated from levels using (2.8). All correlations except those in bold are significant at the 5% level.

Panel A: Full Sample

Liquidity Measure n PRV mPRV PI mPI RQS RES

ρ

PRV 472 1.00 0.73 0.26 0.46 0.34 0.37 0.00 mPRV 470 1.00 0.34 0.49 0.34 0.36 0.00 PI 472 1.00 0.57 0.64 0.55 0.01 mPI 470 1.00 0.74 0.74 -0.01 RQS 226 1.00 0.95 0.03 RES 226 1.00 0.01

Panel B: Subperiod Results

Subperiod 1: 9/1962-12/1982 PRV 244 1.00 0.73 0.46 0.54 0.06 mPRV 242 1.00 0.51 0.53 0.10 PI 244 1.00 0.63 0.08 mPI 242 1.00 -0.01

Subperiod 5: 1/1983-12/2001 PRV 228 1.00 0.75 0.06 0.38 0.34 0.37 -0.06 mPRV 228 1.00 0.16 0.45 0.34 0.36 -0.08 PI 228 1.00 0.49 0.64 0.55 -0.08 mPI 228 1.00 0.74 0.74 -0.04 RQS 228 1.00 0.95 0.03 RES 228 1.00 0.01

44

Table 3 Correlations of Stock Market Returns with Other Variables in Months with Large Liquidity Shocks

The table reports the correlation between the monthly return on the CRSP value-weighted NYSE-AMEX index and (i) minus the change in the rate on one-month Treasury bills, -∆Rf, (ii) the return on long term government bonds, RGB, (iii) the return on long-term corporate bonds, RCB, and (iv) the equally weighted average percentage change in monthly dollar volume for NYSE-AMEX stocks, Vol. "Low" liquidity months are those in which the innovation in the liquidity series is at least two standard deviation below zero. The bootstrapped p-values for the hypothesis that the correlation during these months is equal to those in other months are in brackets. The PRV (Price Reversal) market-wide liquidity measure estimates the tendency of price changes accompanied by large volume to reverse as defined by equation (2.1). mPRV is the modified version of PRV defined by (2.2). PI is the price impact liquidity measure defined by (2.3) and mPI is the modified version of the price impact measure defined by (2.4). RQS is the market wide average quoted spread described in section II.c. and equation (2.5). RES is the market wide average effective spread described in II.c and equation (2.6). Innovations are calculated from levels using (2.8)

Panel A: 1962-2001

Number of Correlation of RS,t with Liquidity Measure Observations -∆Rf,t RGB,t RCB,t Volt All months 471 -0.01 0.27 0.36 0.41 PRV Low Liq. 18 -0.37 -0.12 0.04 -0.36 Other 453 0.01 0.32 0.39 0.44 [0.11] [0.07] [0.09] [0.00] mPRV Low Liq. 15 -0.27 -0.19 0.04 -0.44 Other 454 0.03 0.35 0.40 0.45 [0.16] [0.06] [0.10] [0.00] PI Low Liq. 13 -0.35 -0.70 -0.61 -0.48 Other 458 0.02 0.34 0.40 0.43 [0.13] [0.00] [0.00] [0.00] mPI Low Liq. 16 -0.40 -0.59 -0.61 -0.27 Other 453 0.04 0.35 0.41 0.43 [0.09] [0.00] [0.00] [0.01]

45

Table 3 (continued)

Panel B: 1983-2001 number of Correlation of RS,t with Liquidity Measure observations -∆Rf,t RGB,t RCB,t Volt All months 225 -0.12 0.24 0.28 0.28 PRV Low Liq. 7 -0.82 -0.28 -0.13 -0.84 Other 218 -0.01 0.33 0.35 0.37

[0.00] [0.05] [0.06] [0.00] mPRV Low Liq. 7 -0.73 -0.42 0.00 -0.63 Other 218 0.00 0.39 0.37 0.37

[0.02] [0.02] [0.10] [0.00] PI Low Liq. 5 -0.69 -0.98 -0.84 -0.52 Other 220 0.00 0.39 0.39 0.34

[0.03] [0.00] [0.00] [0.01] mPI Low Liq. 7 -0.72 -0.69 -0.68 -0.36 Other 218 0.00 0.38 0.37 0.35

[0.02] [0.00] [0.00] [0.02] RQS Low Liq. 4 -0.82 -0.61 -0.65 -0.35 Other 221 -0.02 0.32 0.35 0.35

[0.05] [0.09] [0.07] [0.14] RES Low Liq. 5 -0.63 -0.66 -0.64 -0.18 Other 220 0.00 0.37 0.36 0.36

[0.04] [0.02] [0.01] [0.09]

46

Tabl

e 4

Cor

rela

tion

of L

iqui

dity

Sho

cks

with

Fam

a-Fr

ench

Fac

tors

The

tabl

e re

ports

the

corr

elat

ion

betw

een

the

valu

e-w

eigh

ted

CR

SP

inde

x, th

e eq

ual-w

eigh

t CR

SP

inde

x an

d th

e Fa

ma-

Fren

ch H

ML

and

SMB

fa

ctor

s w

ith s

hock

s to

the

six

mar

ket-w

ide

liqui

dity

mea

sure

s ex

amin

ed in

this

pap

er.

The

PR

V (P

rice

Rev

ersa

l) m

arke

t-wid

e liq

uidi

ty m

easu

re

estim

ates

the

tend

ency

of p

rice

chan

ges

acco

mpa

nied

by

larg

e vo

lum

e to

reve

rse

as d

efin

ed b

y eq

uatio

n (2

.1).

mP

RV

is th

e m

odifi

ed v

ersi

on

of P

RV

def

ined

by

(2.2

). P

I is

the

pric

e im

pact

liqu

idity

mea

sure

def

ined

by

(2.3

) an

d m

PI i

s th

e m

odifi

ed v

ersi

on o

f the

pric

e im

pact

mea

sure

de

fined

by

(2.4

). R

QS

is th

e m

arke

t wid

e av

erag

e qu

oted

spr

ead

desc

ribed

in s

ectio

n II.

c. a

nd e

quat

ion

(2.5

). R

ES

is th

e m

arke

t wid

e av

erag

e ef

fect

ive

spre

ad d

escr

ibed

in II

.c a

nd e

quat

ion

(2.6

).

Inno

vatio

ns a

re c

alcu

late

d fro

m le

vels

usi

ng (

2.8)

p-v

alue

s fo

r th

e hy

poth

esis

that

the

corr

elat

ion

is z

ero

are

repo

rted

in b

rack

ets.

Pan

el A

: 196

2-20

01

P

anel

B: 1

983-

2001

PR

V

mP

RV

P

I m

PI

P

RV

m

PR

V

PI

mP

I R

QS

R

ES

R

VW,t

0.29

0.

34

0.50

0.

54

0.

29

0.33

0.

44

0.51

0.

55

0.53

[0.0

0]

[0.0

0]

[0.0

0]

[0.0

0]

[0

.00]

[0

.00]

[0

.00]

[0

.00]

[0

.00]

[0

.00]

R

EW,t

0.31

0.

34

0.58

0.

55

0.

29

0.30

0.

56

0.56

0.

73

0.70

[0.0

0]

[0.0

0]

[0.0

0]

[0.0

0]

[0

.00]

[0

.00]

[0

.00]

[0

.00]

[0

.00]

[0

.00]

SM

B t

0.22

0.

18

0.46

0.

38

0.

10

0.05

0.

41

0.32

0.

46

0.41

[0.0

0]

[0.0

0]

[0.0

0]

[0.0

0]

[0

.15]

[0

.43]

[0

.00]

[0

.00]

[0

.00]

[0

.00]

H

ML t

-0

.09

-0.0

5 -0

.20

-0.1

6

-0.0

6 -0

.08

-0.2

6 -0

.20

-0.1

5 -0

.12

[0

.05]

[0

.29]

[0

.00]

[0

.00]

[0.4

0]

[0.2

5]

[0.0

0]

[0.0

0]

[0.0

2]

[0.0

8]

RVW

,t(≥

0)

-0.0

2 0.

02

0.24

0.

15

-0

.02

0.02

0.

22

0.05

0.

24

0.19

[0.7

6]

[0.7

8]

[0.0

0]

[0.0

1]

[0

.82]

[0

.81]

[0

.01]

[0

.55]

[0

.01]

[0

.02]

R

VW,t(

< 0)

0.

44

0.49

0.

45

0.67

0.48

0.

53

0.47

0.

73

0.65

0.

67

[0

.00]

[0

.00]

[0

.00]

[0

.00]

[0.0

0]

[0.0

0]

[0.0

0]

[0.0

0]

[0.0

0]

[0.0

0]

47

Tabl

e 5

Alp

has

of P

ortfo

lios

Sor

ted

on P

redi

cted

Liq

uidi

ty B

etas

Eve

ry m

onth

from

12/

1965

thro

ugh

12/2

001,

elig

ible

sto

cks

are

sorte

d in

to 1

0 po

rtfol

ios

acco

rdin

g to

pre

dict

ed li

quid

ity b

etas

. Pr

edic

ted

beta

s ar

e fro

m B

ayes

ian

regr

essi

ons

of e

xces

s re

turn

s on

the

thre

e Fa

ma-

Fren

ch fa

ctor

s an

d th

e liq

uidi

ty in

nova

tions

dur

ing

the

prev

ious

five

yea

rs.

The

estim

atio

n an

d so

rting

pro

cedu

re e

ach

mon

th u

ses

only

dat

a av

aila

ble

at th

at ti

me.

Th

e po

rtfol

io r

etur

ns fo

r th

e po

st-ra

nkin

g m

onth

s ar

e lin

ked

acro

ss y

ears

to fo

rm o

ne s

erie

s of

pos

t-ran

king

retu

rns

for e

ach

deci

le.

The

tabl

e re

ports

the

deci

le p

ortfo

lio p

ost r

anki

ng a

lpha

s fro

m re

gres

sion

s us

ing

the

thre

e Fa

ma-

Fren

ch fa

ctor

s.

Ann

ualiz

ed a

lpha

s an

d t-s

tatis

tics

are

repo

rted.

Th

e P

RV

(P

rice

Rev

ersa

l) m

arke

t-wid

e liq

uidi

ty m

easu

re

estim

ates

the

tend

ency

of p

rice

chan

ges

acco

mpa

nied

by

larg

e vo

lum

e to

reve

rse

as d

efin

ed b

y eq

uatio

n (2

.1).

mPR

V is

the

mod

ified

ver

sion

of

PR

V d

efin

ed b

y (2

.2).

PI

is t

he p

rice

impa

ct li

quid

ity m

easu

re d

efin

ed b

y (2

.3)

and

mP

I is

the

mod

ified

ver

sion

of

the

pric

e im

pact

mea

sure

de

fined

by

(2.4

). R

QS

is th

e m

arke

t wid

e av

erag

e qu

oted

spr

ead

desc

ribed

in s

ectio

n II.

c. a

nd e

quat

ion

(2.5

). R

ES

is th

e m

arke

t wid

e av

erag

e ef

fect

ive

spre

ad d

escr

ibed

in II

.c a

nd e

quat

ion

(2.6

). T

he p

ortfo

lios

form

ed u

sing

pre

dict

ed li

quid

ity b

etas

rela

tive

to th

e R

QS

and

RES

mea

sure

s be

gin

in 1

987.

t-s

tatis

tics

are

in p

aren

thes

is, p

-val

ues

in b

rack

ets.

P

anel

A:

Full

Sam

ple

– E

qual

Wei

ghte

d po

rtfol

ios

Liqu

idity

1 2

3 4

5 6

7 8

9 10

10-1

M

easu

re

(lo

w)

(hig

h)

PR

V

-0

.58

-0.5

5 -0

.12

-0.5

4 -0

.19

-0.2

6 0.

69

0.13

0.

57

1.48

2.

07

(-0.

57)

(-0.

63)

(-0.

16)

(-0.

76)

(-0.

27)

(-0.

39)

(1.1

1)

(0.1

8)

(0.8

2)

(1.8

0)

[0.1

1]

mP

RV

-0.8

5 -0

.79

-0.8

0 0.

80

-0.2

5 0.

40

0.33

0.

42

0.22

1.

15

2.00

(-

0.90

) (-

1.01

) (-

1.05

) (1

.06)

(-

0.39

) (0

.60)

(0

.47)

(0

.60)

(0

.30)

(1

.39)

[0

.08]

P

I

-1.1

5 -0

.28

0.02

0.

48

0.01

-0

.31

-0.2

4 1.

32

0.47

0.

31

1.46

(-

1.38

) (-

0.38

) (0

.03)

(0

.71)

(0

.01)

(-

0.48

) (-

0.33

) (1

.85)

(0

.59)

(0

.40)

[0

.17]

m

PI

-0

.47

0.01

-0

.08

0.29

0.

49

0.32

0.

01

-0.0

3 0.

25

-0.1

5 0.

32

(-0.

43)

(0.0

2)

(-0.

12)

(0.3

9)

(0.7

4)

(0.4

4)

(0.0

1)

(-0.

04)

(0.2

7)

(-0.

15)

[0.8

5]

RQ

S

-1

.05

0.40

0.

63

-0.1

7 1.

74

2.53

1.

57

2.35

3.

66

2.06

3.

11

(-0.

61)

(0.3

2)

(0.5

4)

(-0.

16)

(1.5

8)

(2.0

8)

(1.1

5)

(1.8

6)

(2.5

0)

(1.2

5)

[0.1

9]

RE

S

-0

.19

0.32

0.

08

1.16

1.

72

2.01

2.

62

1.97

2.

43

1.61

1.

80

(-0.

11)

(0.2

6)

(0.0

8)

(1.0

2)

(1.4

2)

(1.7

1)

(2.0

7)

(1.3

7)

(1.5

8)

(0.9

8)

[0.4

4]

48

Tabl

e 5

(con

tinue

d)

Pan

el B

: Fu

ll S

ampl

e –

Val

ue W

eigh

ted

Por

tfolio

s.

Liqu

idity

1 2

3 4

5 6

7 8

9 10

10-1

M

easu

re

(lo

w)

(hig

h)

PR

V

-0

.59

0.05

0.

39

1.26

0.

39

0.54

0.

79

0.44

0.

05

0.23

0.

81

(-0.

54)

(0.0

5)

(0.4

5)

(1.4

0)

(0.4

2)

(0.6

6)

(0.9

5)

(0.4

6)

(0.0

5)

(0.2

1)

[0.6

4]

mP

RV

-0.5

8 1.

74

0.01

1.

42

0.19

0.

70

-0.6

6 1.

34

-1.2

2 0.

46

1.04

(-

0.50

) (1

.70)

(0

.01)

(1

.57)

(0

.22)

(0

.74)

(-

0.74

) (1

.39)

(-

1.17

) (0

.38)

[0

.59]

P

I

0.20

1.

76

-0.6

2 1.

06

0.63

0.

18

0.73

1.

01

-0.1

4 -1

.38

-1.5

8

(0

.19)

(1

.83)

(-

0.69

) (1

.16)

(0

.72)

(0

.23)

(0

.96)

(1

.07)

(-

0.14

) (-

1.13

) [0

.39]

m

PI

-0

.97

-0.2

7 0.

99

1.93

0.

17

-0.1

5 -0

.27

1.66

0.

60

-0.6

8 0.

29

(-0.

76)

(-0.

24)

(0.9

2)

(2.4

2)

(0.2

3)

(-0.

19)

(-0.

30)

(1.5

3)

(0.5

3)

(-0.

50)

[0.8

9]

RQ

S

-1

.86

-1.2

9 -0

.32

-1.9

1 0.

37

3.95

2.

85

1.55

5.

82

0.70

2.

56

(-0.

85)

(-0.

79)

(-0.

26)

(-1.

26)

(0.2

5)

(2.8

5)

(1.5

4)

(0.8

6)

(2.9

2)

(0.3

7)

[0.4

2]

RE

S

-2

.07

-0.6

5 0.

63

-0.9

8 0.

79

-0.4

9 4.

78

1.90

3.

00

0.25

2.

32

(-1.

07)

(-0.

39)

(0.4

1)

(-0.

73)

(0.5

7)

(-0.

34)

(3.2

0)

(1.0

4)

(1.6

7)

(0.1

4)

[0.4

2]

49

Tabl

e 6

Alp

has

of P

ortfo

lios

Sor

ted

on P

redi

cted

Liq

uidi

ty B

etas

, Jan

vs.

Non

-Jan

.

Eve

ry m

onth

from

12/

1965

thro

ugh

12/2

001,

elig

ible

sto

cks

are

sorte

d in

to 1

0 po

rtfol

ios

acco

rdin

g to

pre

dict

ed li

quid

ity b

etas

. P

redi

cted

bet

as a

re

from

Bay

esia

n re

gres

sion

s of

exc

ess

retu

rns

on th

e th

ree

Fam

a-Fr

ench

fact

ors

and

the

liqui

dity

inno

vatio

ns d

urin

g th

e pr

evio

us fi

ve y

ears

. Th

e es

timat

ion

and

sorti

ng p

roce

dure

at e

ach

mon

th u

ses

only

dat

a av

aila

ble

at th

at ti

me.

The

por

tfolio

retu

rns

for t

he p

ost-r

anki

ng m

onth

s ar

e lin

ked

acro

ss y

ears

to fo

rm tw

o se

ries

of p

ost-r

anki

ng r

etur

ns fo

r ea

ch d

ecile

, one

of o

nly

Janu

arie

s an

d th

e se

cond

with

all

othe

r m

onth

s.

The

tabl

e re

ports

the

dec

ile p

ortfo

lio p

ost

rank

ing

alph

as f

rom

reg

ress

ions

usi

ng t

he t

hree

Fam

a-Fr

ench

fac

tors

. A

nnua

lized

alp

has

and

t-sta

tistic

s ar

e re

porte

d.

The

PR

V (

Pric

e R

ever

sal)

mar

ket-w

ide

liqui

dity

mea

sure

est

imat

es t

he t

ende

ncy

of p

rice

chan

ges

acco

mpa

nied

by

larg

e vo

lum

e to

re

vers

e as

def

ined

by

equa

tion

(2.1

). m

PR

V is

the

mod

ified

ver

sion

of P

RV

def

ined

by

(2.2

). P

I is

the

pric

e im

pact

liqu

idity

mea

sure

def

ined

by

(2.3

) and

mP

I is

the

mod

ified

ver

sion

of t

he p

rice

impa

ct m

easu

re d

efin

ed b

y (2

.4).

RQ

S is

the

mar

ket w

ide

aver

age

quot

ed s

prea

d de

scrib

ed in

se

ctio

n II.

c. a

nd e

quat

ion

(2.5

). R

ES

is th

e m

arke

t wid

e av

erag

e ef

fect

ive

spre

ad d

escr

ibed

in II

.c a

nd e

quat

ion

(2.6

). T

he p

ortfo

lios

form

ed u

sing

pr

edic

ted

liqui

dity

bet

as re

lativ

e to

the

RQ

S a

nd R

ES

mea

sure

s be

gin

in 1

987.

t-s

tatis

tics

are

in p

aren

thes

is, p

-val

ues

in b

rack

ets.

50

Tabl

e 6

(con

tinue

d)

Pan

el A

: Fu

ll S

ampl

e - E

qual

Wei

ghte

d P

ortfo

lios.

Li

quid

ity

1

2 3

4 5

6 7

8 9

10

10

-1

Mea

sure

(low

)

(h

igh)

P

RV

N

on-J

an

-0.9

1 -0

.30

0.04

-0

.57

-0.1

0 -0

.09

0.75

0.

20

0.84

1.

77

2.67

(-

0.87

) (-

0.32

) (0

.05)

(-

0.78

) (-

0.14

) (-

0.14

) (1

.25)

(0

.29)

(1

.25)

(2

.10)

[0

.04]

Jan

10.5

4 0.

01

2.11

1.

51

0.50

-2

.09

0.75

-1

.03

-2.7

7 -0

.72

-11.

27

(2.2

2)

(0.0

0)

(0.7

2)

(0.6

3)

(0.2

1)

(-1.

07)

(0.3

2)

(-0.

38)

(-1.

13)

(-0.

25)

[0.0

5]

mP

RV

N

on-J

an

-1.0

4 -0

.63

-0.7

8 0.

95

-0.3

0 0.

62

0.54

0.

70

0.39

1.

22

2.26

(-

1.10

) (-

0.76

) (-

0.99

) (1

.22)

(-

0.46

) (0

.94)

(0

.74)

(1

.03)

(0

.51)

(1

.35)

[0

.05]

Jan

5.23

2.

94

2.06

1.

37

0.93

-3

.75

0.70

-3

.07

0.07

2.

18

-3.0

4

(1

.36)

(0

.99)

(0

.86)

(0

.54)

(0

.35)

(-

2.34

) (0

.27)

(-

1.26

) (0

.03)

(0

.71)

[0

.52]

P

I N

on-J

an

-1.5

9 -0

.45

0.15

0.

45

0.04

-0

.12

-0.0

3 1.

53

0.86

0.

80

2.40

(-

1.80

) (-

0.58

) (0

.19)

(0

.62)

(0

.06)

(-

0.19

) (-

0.04

) (2

.17)

(1

.17)

(1

.01)

[0

.03]

Jan

7.41

2.

34

0.34

3.

75

-0.6

5 0.

83

1.38

0.

03

-3.1

9 -3

.40

-10.

81

(1.9

5)

(0.6

4)

(0.1

2)

(1.6

4)

(-0.

31)

(0.3

6)

(0.6

3)

(0.0

1)

(-1.

25)

(-1.

33)

[0.0

1]

mP

I N

on-J

an

-0.6

5 -0

.44

-0.0

4 0.

25

0.34

0.

50

0.32

0.

43

0.67

0.

28

0.93

(-

0.64

) (-

0.59

) (-

0.05

) (0

.35)

(0

.49)

(0

.66)

(0

.39)

(0

.48)

(0

.73)

(0

.27)

[0

.56]

Jan

7.81

11

.56

0.67

1.

22

4.32

0.

60

-3.8

2 -4

.81

-4.4

6 -4

.43

-12.

23

(2.0

6)

(3.2

1)

(0.2

2)

(0.4

6)

(1.3

5)

(0.2

3)

(-1.

54)

(-1.

67)

(-1.

79)

(-1.

55)

[0.0

2]

RQ

S

Non

-Jan

-0

.19

0.71

1.

08

0.09

2.

01

2.81

1.

97

2.93

4.

04

2.46

2.

65

(-0.

11)

(0.5

7)

(0.9

4)

(0.0

8)

(1.8

5)

(2.3

1)

(1.5

4)

(2.4

9)

(2.9

4)

(1.5

2)

[0.2

7]

Ja

n -4

.49

1.76

-2

.38

0.20

-0

.97

0.87

-1

.67

-2.1

0 1.

00

-0.6

0 3.

89

(-0.

53)

(0.2

5)

(-0.

42)

(0.0

5)

(-0.

24)

(0.2

6)

(-0.

46)

(-1.

13)

(0.3

7)

(-0.

13)

[0.6

9]

RE

S

Non

-Jan

0.

13

0.64

0.

67

1.21

2.

21

2.10

3.

35

2.51

3.

09

2.00

1.

87

(0.0

7)

(0.5

1)

(0.6

4)

(1.1

4)

(1.8

6)

(1.8

3)

(2.8

0)

(1.8

3)

(2.2

0)

(1.2

5)

[0.4

3]

Ja

n 3.

94

0.61

-3

.80

1.81

-3

.38

3.89

-4

.37

-2.4

2 -7

.26

2.59

-1

.34

(0.4

8)

(0.1

1)

(-0.

64)

(0.4

0)

(-0.

84)

(0.8

0)

(-1.

34)

(-0.

90)

(-4.

84)

(0.5

9)

[0.8

8]

51

Tabl

e 6

(con

tinue

d)

Pan

el B

: Fu

ll S

ampl

e - V

alue

Wei

ghte

d P

ortfo

lios.

Li

quid

ity

1

2 3

4 5

6 7

8 9

10

10

-1

Mea

sure

(low

)

(h

igh)

P

RV

N

on-J

an

-0.9

5 0.

36

0.15

1.

26

0.24

0.

53

1.30

0.

48

0.66

0.

82

1.77

(-

0.85

) (0

.37)

(0

.17)

(1

.45)

(0

.27)

(0

.68)

(1

.59)

(0

.51)

(0

.63)

(0

.76)

[0

.31]

Jan

6.56

-3

.25

6.56

4.

32

-3.4

2 1.

92

-6.5

0 1.

04

-5.1

0 -1

2.20

-1

8.76

(1

.19)

(-

0.98

) (1

.83)

(1

.41)

(-

0.75

) (0

.44)

(-

1.53

) (0

.35)

(-

2.05

) (-

3.21

) [0

.02]

m

PR

V

Non

-Jan

-0

.65

1.94

0.

12

1.78

0.

08

0.40

-0

.64

1.60

-1

.10

0.89

1.

54

(-0.

57)

(2.0

0)

(0.1

3)

(1.8

8)

(0.1

0)

(0.4

2)

(-0.

71)

(1.6

2)

(-1.

06)

(0.7

4)

[0.4

3]

Ja

n 3.

34

6.71

1.

73

-0.4

2 -1

.96

5.58

-1

.26

-5.4

8 -5

.06

-8.4

7 -1

1.82

(0

.74)

(2

.46)

(0

.49)

(-

0.13

) (-

0.47

) (1

.55)

(-

0.50

) (-

1.72

) (-

1.18

) (-

1.91

) [0

.09]

P

I N

on-J

an

0.32

0.

94

-0.4

1 0.

83

0.75

0.

64

0.82

1.

36

0.09

-1

.01

-1.3

3

(0

.29)

(1

.03)

(-

0.49

) (0

.95)

(0

.87)

(0

.82)

(1

.02)

(1

.49)

(0

.09)

(-0

.82)

[0

.47]

Jan

0.57

11

.47

-3.3

2 2.

71

-2.2

0 -9

.39

0.82

-3

.40

1.68

0.

25

-0.3

2

(0

.13)

(2

.71)

(-

0.86

) (0

.74)

(-

0.78

) (-

4.00

) (0

.35)

(-

0.92

) (0

.59)

(0

.07)

[0

.96]

m

PI

Non

-Jan

-1

.09

-0.9

7 1.

32

2.14

0.

16

0.18

0.

25

2.07

0.

59

-0.2

9 0.

79

(-0.

86)

(-0.

92)

(1.1

9)

(2.6

6)

(0.2

0)

(0.2

3)

(0.2

7)

(2.0

4)

(0.5

4)

(-0.

21)

[0.7

1]

Ja

n 3.

40

12.5

3 -4

.34

-0.5

4 4.

17

-3.7

1 -1

0.13

-4

.78

-2.5

2 -4

.46

-7.8

6

(0

.76)

(2

.56)

(-

1.41

) (-

0.18

) (1

.59)

(-

1.24

) (-

3.35

) (-

1.15

) (-

0.61

) (-

0.93

) [0

.32]

R

QS

N

on-J

an

-2.0

5 -0

.77

0.30

-1

.10

0.24

3.

90

2.41

2.

49

5.45

0.

56

2.61

(-

0.94

) (-

0.50

) (0

.23)

(-

0.76

) (0

.17)

(2

.72)

(1

.41)

(1

.38)

(2

.86)

(0

.29)

[0

.43]

Jan

2.41

-0

.90

-3.0

0 -8

.86

-8.1

2 -2

.72

8.37

-9

.14

15.7

2 12

.13

9.72

(0

.44)

(-

0.11

) (-

0.67

) (-

1.86

) (-

1.50

) (-

0.82

) (1

.83)

(-

2.74

) (3

.51)

(1

.98)

[0

.36]

R

ES

N

on-J

an

-2.7

5 0.

24

1.47

-0

.52

0.72

-0

.72

4.73

2.

74

2.21

0.

44

3.18

(-

1.42

) (0

.15)

(0

.95)

(-

0.37

) (0

.49)

(-

0.51

) (3

.29)

(1

.51)

(1

.23)

(0

.23)

[0

.28]

Jan

8.72

-8

.79

-5.5

8 -2

.34

-0.3

1 -2

.09

-3.0

5 -2

.53

11.9

4 6.

63

-2.0

8

(1

.13)

(-

1.49

) (-

0.90

) (-

0.69

) (-

0.06

) (-

0.64

) (-

0.49

) (-

0.79

) (1

.74)

(1

.47)

[0

.84]

52

Tabl

e 7

Li

quid

ity R

isk

Pre

mia

and

Con

tribu

tions

to E

xpec

ted

Ret

urn

Th

is t

able

rep

orts

the

est

imat

es o

f th

e ris

k pr

emiu

m a

ssoc

iate

d w

ith t

he s

ix li

quid

ity f

acto

rs a

s w

ell a

s th

e co

ntrib

utio

n of

liqu

idity

ris

k to

the

ex

pect

ed r

etur

n on

the

"10

-1"

spre

ad.

Sto

cks

are

sorte

d in

to 1

0 po

rtfol

ios

by t

heir

pred

icte

d liq

uidi

ty b

etas

eac

h m

onth

. Th

e pr

emiu

m λ

is

estim

ated

usi

ng p

ost-r

anki

ng re

turn

s on

all

10 p

ortfo

lios.

The

dec

iles

are

equa

l-wei

ghte

d in

pan

el A

and

val

ue-w

eigh

ted

in p

anel

B.

The

prem

ium

on

the

thre

e Fa

ma-

Fren

ch ri

sk fa

ctor

s is

con

stra

ined

to th

e re

turn

on

the

long

-sho

rt fa

ctor

por

tfolio

retu

rns.

The

PR

V (P

rice

Rev

ersa

l) m

arke

t-wid

e liq

uidi

ty m

easu

re e

stim

ates

the

tend

ency

of p

rice

chan

ges

acco

mpa

nied

by

larg

e vo

lum

e to

reve

rse

as d

escr

ibed

in e

quat

ion

(2.1

). m

PRV

is th

e m

odifi

ed v

ersi

on o

f PR

V d

escr

ibed

in (2

.2).

PI i

s th

e pr

ice

impa

ct li

quid

ity m

easu

re d

escr

ibed

by

(2.3

) and

mP

I is

the

mod

ified

ver

sion

of t

he p

rice

impa

ct m

easu

re d

escr

ibed

in (2

.4).

RQ

S is

the

mar

ket w

ide

aver

age

quot

ed s

prea

d de

scrib

ed b

y (2

.5).

RE

S is

the

mar

ket w

ide

aver

age

effe

ctiv

e sp

read

des

crib

ed b

y (2

.6).

The

por

tfolio

s fo

rmed

usi

ng p

redi

cted

liqu

idity

bet

as r

elat

ive

to t

he R

QS

and

RE

S m

easu

res

begi

n in

198

7.

Full

Sam

ple

refe

rs to

the

full

time

serie

s of

pre

dict

ed li

quid

ity b

eta

sorte

d po

rtfol

ios

as in

Tab

le 5

. N

on-J

an r

efer

s to

the

sam

e tim

e se

ries

omitt

ing

Janu

arie

s as

in T

able

6.

t-sta

tistic

s ar

e in

par

enth

esis

, p-v

alue

s in

bra

cket

s.

53 Ta

ble

7 (c

ontin

ued)

Pa

nel A

: Equ

al-W

eigh

ted

Portf

olio

s

Non

-Jan

Non

-Jan

Li

quid

ity

Mea

sure

Full

Sam

ple

Fu

ll S

ampl

e

1965

-19

86

1987

-20

02

19

65-

1986

19

87-

2002

P

RV

λ P

RV

3.

54

5.

35

2.

79

5.23

2.81

9.

72

(2.2

1)

(2

.22)

(2.5

4)

(2.2

6)

(2

.59)

(2

.02)

(β10

-β1)

λ PR

V

1.65

2.62

1.97

1.

27

2.

85

1.31

[0

.05]

[0.0

2]

[0

.06]

[0

.37]

[0.0

1]

[0.4

5]

m

PR

V

λ mP

RV

34

.20

4.

85

4.

88

2.93

2.97

39

.94

(0.6

7)

(2

.27)

(2.2

3)

(1.5

4)

(2

.45)

(1

.22)

(β10

-β1)

λ mP

RV

1.77

2.18

2.03

0.

89

2.

52

1.67

[0

.00]

[0.0

2]

[0

.10]

[0

.49]

[0.0

4]

[0.3

3]

PI

λ P

I 1.

43

1.

94

2.

65

2.00

3.21

2.

31

(2.4

7)

(2

.95)

(2.7

9)

(2.7

1)

(3

.41)

(3

.07)

(β10

-β1)

λ PI

2.64

3.32

4.20

3.

32

4.

34

3.75

[0

.02]

[0.0

0]

[0

.00]

[0

.29]

[0.0

0]

[0.2

9]

m

PI

λ mP

I 0.

08

0.

19

-0

.47

0.51

0.07

1.

21

(0.2

6)

(0

.73)

(-1.

15)

(1.7

3)

(0

.26)

(1

.98)

(β10

-β1)

λ mP

I 0.

40

1.

17

-1

.04

4.12

0.23

5.

45

[0.8

0]

[0

.46]

[0.3

6]

[0.0

5]

[0

.86]

[0

.03]

RQ

S

λ RQ

S

3.15

3.67

3.

15

3.67

(4

.15)

(4.0

5)

(4.1

5)

(4.0

5)

(β

10-β

1)λ R

QS

4.

29

4.

30

4.29

4.

30

[0.0

1]

[0

.02]

[0

.01]

[0

.02]

RE

S

λ RE

S

1.17

1.34

1.

17

1.34

(3

.09)

(3.2

8)

(3.0

9)

(3.2

8)

(β

10-β

1)λ R

ES

3.

65

4.

19

3.65

4.

19

[0.0

1]

[0

.01]

[0

.01]

[0

.01]

54

Tabl

e 7

(con

tinue

d)

Pane

l B: V

alue

-Wei

ghte

d Po

rtfol

ios

N

on-J

an

N

on-J

an

Liqu

idity

M

easu

re

Fu

ll S

ampl

e

Full

Sam

ple

19

65-

1986

19

87-

2002

1969

-19

86

1987

-20

02

PR

V

λ PR

V

8.04

6.49

-7.0

1 16

.12

-6

.60

18.5

6

(1

.24)

(1.9

8)

(-

2.61

) (1

.79)

(-

2.61

)(1

.94)

(β

10-β

1)λ P

RV

2.

14

3.

58

-1

.36

3.02

-0

.43

5.12

[0.2

0]

[0

.02]

[0.5

1]

[0.3

1]

[0.8

2][0

.11]

m

PR

V

λ mP

RV

41

7.95

1374

.74

-6

.45

16.5

5

-4.7

014

.37

(0.0

7)

(0

.02)

(-2.

14)

(2.1

7)

(-2.

79)

(3.0

6)

(β10

-β1)

λ mP

RV

1.13

1.72

-1.6

4 5.

17

-2.0

86.

43

[0

.53]

[0.3

7]

[0

.41]

[0

.05]

[0

.31]

[0.0

2]

PI

λ PI

-1.3

4

1.73

2.94

-3

.89

2.

832.

88

(-

1.57

)

(1.9

3)

(2

.14)

(-

2.99

) (1

.72)

(2.4

7)

(β10

-β1)

λ PI

-2.1

8

2.64

2.38

-4

.88

1.

514.

79

[0

.11]

[0.1

3]

[0

.13]

[0

.07]

[0

.44]

[0.1

9]

mPI

λ m

PI

0.83

1.00

1.23

1.

10

1.72

1.28

(2.0

4)

(2

.18)

(1.4

6)

(2.8

2)

(2.5

3)(2

.81)

(β

10-β

1)λ m

PI

2.76

3.59

1.46

5.

80

2.74

5.95

[0.0

9]

[0

.03]

[0.3

3]

[0.0

9]

[0.1

4][0

.14]

R

QS

λ R

QS

3.

22

6.

87

3.22

6.

87

(3.2

9)

(2

.89)

(3

.29)

(2

.89)

(β10

-β1)

λ RQ

S

4.57

4.74

4.

57

4.74

[0

.02]

[0.1

2]

[0.0

2]

[0.1

2]

R

ES

λ R

ES

0.

95

1.

69

0.95

1.

69

(3.1

8)

(3

.97)

(3

.18)

(3

.97)

(β10

-β1)

λ RE

S

4.70

5.02

4.

70

5.02

[0

.01]

[0.0

4]

[0.0

1]

[0.0

4]

55

Tabl

e 8

Ris

k Ad

just

ed R

etur

ns fr

om Z

ero

Inve

stm

ent P

ortfo

lios

Form

ed b

y P

redi

cted

Liq

uidi

ty B

eta

E

very

mon

th b

etw

een

1968

and

200

1, e

ligib

le s

tock

s ar

e so

rted

into

10

portf

olio

s ac

cord

ing

to p

redi

cted

liqu

idity

bet

as.

Pre

dict

ed b

etas

are

be

tas

from

Bay

esia

n re

gres

sion

s of

exc

ess

retu

rns

on th

e th

ree

Fam

a-Fr

ench

fact

ors

and

the

liqui

dity

inno

vatio

ns d

urin

g th

e pr

evio

us fi

ve

year

s. T

he e

stim

atio

n an

d so

rting

pro

cedu

re a

t eac

h m

onth

use

s on

ly d

ata

avai

labl

e at

that

tim

e. F

ama-

Fren

ch c

oeffi

cien

ts a

re e

stim

ated

fo

r eac

h st

ock

usin

g 60

mon

ths

of p

rior r

etur

n da

ta (m

inim

um o

f 36

mon

ths)

and

use

d to

cal

cula

te p

ortfo

lio c

oeffi

cien

ts fo

r the

10

portf

olio

s.

The

hold

ing

perio

d re

turn

for

the

follo

win

g m

onth

is r

isk-

adju

sted

usi

ng th

ese

prio

r pe

riod

fact

or lo

adin

gs a

nd th

e ac

tual

fact

or re

aliz

atio

ns.

Mon

thly

ris

k ad

just

ed r

etur

ns a

re r

epor

ted.

Th

e PR

V (

Pric

e R

ever

sal)

mar

ket-w

ide

liqui

dity

mea

sure

est

imat

es t

he t

ende

ncy

of p

rice

chan

ges

acco

mpa

nied

by

larg

e vo

lum

e to

reve

rse

as d

escr

ibed

in e

quat

ion

(2.1

). m

PR

V is

the

mod

ified

ver

sion

of P

RV

des

crib

ed in

(2.2

).

PI i

s th

e pr

ice

impa

ct li

quid

ity m

easu

re d

escr

ibed

by

(2.3

) an

d m

PI is

the

mod

ified

ver

sion

of t

he p

rice

impa

ct m

easu

re d

escr

ibed

in (

2.4)

.

RQ

S is

the

mar

ket w

ide

aver

age

quot

ed s

prea

d de

scrib

ed b

y (2

.5).

RE

S is

the

mar

ket w

ide

aver

age

effe

ctiv

e sp

read

des

crib

ed b

y (2

.6).

Th

e po

rtfol

ios

form

ed u

sing

pre

dict

ed li

quid

ity b

etas

rela

tive

to th

e R

QS

and

RE

S m

easu

res

begi

n in

198

7. "

No

Janu

arie

s" u

ses

the

sam

e sa

mpl

e as

the

left

colu

mns

with

Jan

uarie

s om

itted

. t-s

tatis

tics

are

in p

aren

thes

es.

P

anel

A: E

qual

ly W

eigh

ted

Por

tfolio

s

N

o Ja

nuar

ies

A

ll M

onth

s

By

mag

nitu

de o

f rea

lized

liqu

idity

sho

ck

A

ll M

onth

s

Sor

ted

by m

agni

tude

of

real

ized

liqu

idity

sh

ock

L<-2

σ L

L>-2

σ L

L<

-2σ L

L>

-2σ L

Liqu

idity

M

easu

re

Ave

rage

R

etur

n n

Av

erag

e

Ret

urn

n Av

erag

e

Ret

urn

n

Ave

rage

R

etur

n

Ave

rage

R

etur

n A

vera

ge

Ret

urn

PR

V

0.30

42

0 -0

.49

14

0.33

40

6 0.

30

-0.4

9 0.

33

(2

.68)

(-

0.96

) (2

.86)

(2

.63)

(-

0.96

) (2

.83)

m

PR

V

0.42

42

0 0.

25

16

0.43

40

4 0.

40

0.13

0.

41

(4

.25)

(0

.45)

(4

.25)

(4

.14)

(0

.23)

(4

.20)

P

I 0.

28

420

-2.6

1 10

0.

35

410

0.32

-2

.61

0.39

(2.6

7)

(-2.

56)

(3.4

3)

(2.9

7)

(-2.

21)

(3.8

2)

mP

I 0.

28

420

-0.4

9 15

0.

31

405

0.25

-0

.49

0.28

(2.3

7)

(-0.

47)

(2.6

4)

(2.0

7)

(-0.

47)

(2.3

6)

RQ

S

0.20

18

0 -0

.94

3 0.

22

177

0.19

-0

.94

0.21

(1.1

1)

(-0.

88)

(1.2

0)

(1.0

7)

(-0.

88)

(1.1

8)

RE

S

0.14

18

0 -1

.02

5 0.

17

175

0.16

-1

.02

0.19

(0.7

7)

(-0.

86)

(0.9

4)

(0.9

0)

(-0.

86)

(1.1

0)

56

Ta

ble

8 (c

ontin

ued)

Pan

el B

: Val

ue W

eigh

ted

Por

tfolio

s

No

Janu

arie

s

A

ll M

onth

s

By

mag

nitu

de o

f rea

lized

liqu

idity

sho

ck

A

ll M

onth

s

Sor

ted

by m

agni

tude

of

real

ized

liqu

idity

sh

ock

L<

-2σ L

L>

-2σ L

L<-2

σ L

L>-2

σ L

Liqu

idity

M

easu

re

Ave

rage

R

etur

n n

Av

erag

e

Ret

urn

n Av

erag

e

Ret

urn

n

Ave

rage

R

etur

n

Ave

rage

R

etur

n A

vera

ge

Ret

urn

PR

V

0.20

42

0 -0

.36

14

0.22

40

6 0.

24

-0.3

6 0.

26

(1

.25)

(-

0.35

) (1

.35)

(1

.57)

(-

0.35

) (1

.71)

m

PR

V

0.40

42

0 -0

.41

16

0.43

40

4 0.

38

-0.6

3 0.

42

(2

.57)

(-

0.41

) (2

.75)

(2

.45)

(-

0.60

) (2

.71)

P

I 0.

09

420

-1.8

7 10

0.

13

410

0.12

-1

.87

0.17

(0.5

6)

(-1.

53)

(0.8

7)

(0.7

6)

(-1.

53)

(1.0

9)

mP

I 0.

34

420

1.52

15

0.

30

405

0.33

1.

52

0.28

(2.0

8)

(1.5

7)

(1.7

9)

(1.9

9)

(1.5

7)

(1.6

8)

RQ

S

0.41

18

0 0.

38

3 0.

41

177

0.37

0.

38

0.37

(1.7

1)

(0.2

0)

(1.6

9)

(1.5

3)

(0.2

0)

(1.5

1)

RE

S

0.37

18

0 -0

.15

5 0.

38

175

0.45

-0

.15

0.46

(1.5

4)

(-0.

16)

(1.5

6)

(1.8

6)

(-0.

16)

(1.8

9)

57

Tabl

e 9

P

ortfo

lios

Sor

ted

on th

e S

um o

f Por

tfolio

Ass

ignm

ents

Usi

ng In

divi

dual

Liq

uidi

ty M

easu

res

Eve

ry m

onth

from

12/

1965

thro

ugh

12/2

001,

elig

ible

sto

cks

are

sorte

d in

to 1

0 po

rtfol

ios

acco

rdin

g to

pre

dict

ed li

quid

ity b

etas

rela

tive

to e

ach

of s

ix

mar

ket-w

ide

liqui

dity

mea

sure

s. P

redi

cted

bet

as a

re fr

om B

ayes

ian

regr

essi

ons

of e

xces

s re

turn

s on

the

thre

e Fa

ma-

Fren

ch fa

ctor

s an

d th

e liq

uidi

ty

inno

vatio

ns d

urin

g th

e pr

evio

us fi

ve y

ears

. Th

e es

timat

ion

and

sorti

ng p

roce

dure

at e

ach

mon

th u

ses

only

dat

a av

aila

ble

at th

at ti

me.

The

por

tfolio

as

sign

men

ts fo

r eac

h st

ock

for e

ach

mon

th a

re s

umm

ed to

cre

ate

an a

ggre

gate

sco

re.

Sto

cks

are

assi

gned

to d

ecile

por

tfolio

s ea

ch m

onth

by

thei

r ag

greg

ate

scor

e. T

he p

ortfo

lio re

turn

s fo

r the

pos

t-ran

king

mon

ths

are

linke

d ac

ross

yea

rs to

form

one

ser

ies

of p

ost-r

anki

ng re

turn

s fo

r eac

h de

cile

. P

anel

A r

epor

ts th

e de

cile

por

tfolio

pos

t ran

king

alp

has

from

reg

ress

ions

usi

ng th

e th

ree

Fam

a-Fr

ench

fact

ors.

A

nnua

lized

alp

has

and

t-sta

tistic

s ar

e re

porte

d.

Pan

el B

rep

orts

hol

ding

per

iod

retu

rns

to a

stra

tegy

that

is lo

ng p

ortfo

lio 1

0 an

d sh

ort p

ortfo

lio 1

with

any

res

idua

l exp

osur

e to

the

thre

e Fa

ma-

Fren

ch fa

ctor

s fe

asib

ly h

edge

d in

the

man

ner

of T

able

8.

The

six

indi

vidu

al m

easu

res

are:

PR

V (

Pric

e R

ever

sal)

whi

ch e

stim

ates

the

tend

ency

of p

rice

chan

ges

acco

mpa

nied

by

larg

e vo

lum

e to

rev

erse

as

desc

ribed

in e

quat

ion

(2.1

), m

PRV

whi

ch is

the

mod

ified

ver

sion

of P

RV

desc

ribed

in (2

.2),

PI w

hich

is th

e pr

ice

impa

ct li

quid

ity m

easu

re d

escr

ibed

by

(2.3

), m

PI w

hich

is th

e m

odifi

ed v

ersi

on o

f the

pric

e im

pact

mea

sure

de

scrib

ed in

(2.

4), R

QS

whi

ch is

the

mar

ket w

ide

aver

age

quot

ed s

prea

d de

scrib

ed b

y (2

.5),

and

RE

S w

hich

is th

e m

arke

t wid

e av

erag

e ef

fect

ive

spre

ad d

escr

ibed

by

(2.6

). T

he p

ortfo

lios

form

ed u

sing

pre

dict

ed li

quid

ity b

etas

rel

ativ

e to

the

RQ

S a

nd R

ES

mea

sure

s be

gin

in 1

987.

t-s

tatis

tics

are

in p

aren

thes

is, p

-val

ues

in b

rack

ets.

P

anel

A: F

ama-

Fren

ch A

lpha

s

1

2 3

4 5

6 7

8 9

10

10

-1

(low

)

(h

igh)

E

qual

- A

ll -1

.44

-0.3

3-0

.10

-0.4

40.

740.

30

0.67

-0.0

30.

541.

202.

64

Wei

ghts

M

onth

s (-

1.52

) (-

0.40

)(-

0.14

)(-

0.69

)(1

.09)

(0.4

0)

(0.8

8)(-

0.03

)(0

.64)

(1.3

8)[0

.04]

N

on-J

an

-1.5

9 -0

.55

-0.1

8-0

.42

0.85

0.31

0.

750.

221.

271.

543.

13

(-

1.65

) (-

0.66

)(-

0.25

)(-

0.61

)(1

.18)

(0.4

4)

(0.9

9)(0

.29)

(1.5

1)(1

.70)

[0.0

1]

Jan

7.79

7.

193.

46-0

.04

-2.1

62.

39

0.39

-1.3

0-7

.68

-2.5

3-1

0.32

(1.7

7)

(1.9

8)(1

.17)

(-0.

02)

(-1.

13)

(0.9

4)

(0.1

7)(-

0.47

)(-

3.25

)(-

1.00

)[0

.05]

Val

ue-

-1

.37

0.78

2.20

0.46

-0.4

01.

36

0.29

-0.1

30.

560.

742.

11 W

eigh

ts

(-

1.35

) (0

.89)

(2.7

5)(0

.55)

(-0.

47)

(1.6

9)

(0.3

0)(-

0.13

)(0

.51)

(0.6

2)[0

.21]

N

on-J

an

-1.6

1 0.

642.

280.

49-0

.26

1.35

0.

500.

410.

631.

042.

65

(-

1.59

) (0

.74)

(2.7

7)(0

.63)

(-0.

32)

(1.7

2)

(0.5

5)(0

.39)

(0.5

9)(0

.85)

[0.1

3]

Jan

6.19

7.

783.

13-1

.68

-7.1

7-2

.58

-2.0

2-9

.27

2.80

-3.3

8-9

.57

(1.6

3)

(2.3

5)(1

.06)

(-0.

47)

(-2.

19)

(-0.

48)

(-0.

88)

(-2.

52)

(0.6

4)(-

0.69

)[0

.20]

58

Tabl

e 9

(con

tinue

d)

Pan

el B

: Ris

k A

djus

ted

Ret

urns

Li

quid

ity S

hock

by

All

Mea

sure

s

Liqu

idity

Sho

ck b

y A

ny M

easu

re

A

ll M

onth

s

L <

-2σ

L >

-2σ

L

< -2

σ L

> -2

σ

A

vera

ge

Ret

urn

n

Ave

rage

Ret

urn

n A

vera

geR

etur

n n

A

vera

geR

etur

n n

Ave

rage

Ret

urn

n E

qual

- 0.

43

420

-2

.42

3 0.

45

417

-0

.85

34

0.54

38

6 W

eigh

ts

(3.9

3)

(-1.

66)

(4

.12)

(-

1.72

)

(4.9

7)

Val

ue-

0.50

42

0

0.33

3

0.50

41

7

-0.1

3 34

0.

56

386

Wei

ghts

(3

.28)

(0

.09)

(3.3

0)

(-0.

18)

(3

.65)

59

Tabl

e 10

Spr

ead

in A

lpha

of P

ortfo

lios

Sor

ted

on L

iqui

dity

Spr

ead

Bet

a an

d C

hara

cter

istic

Liq

uidi

ty

E

very

mon

th f

rom

12/

1965

thr

ough

12/

2001

, el

igib

le s

tock

s ar

e so

rted

into

10

portf

olio

s by

liq

uidi

ty s

prea

d be

ta o

r by

the

sto

ck's

ave

rage

ch

arac

teris

tic li

quid

ity d

urin

g m

onth

s t-2

thro

ugh

t-4.

Liqu

idity

spr

ead

beta

s ar

e th

e co

effic

ient

s on

mar

ket w

ide

liqui

dity

sho

cks

from

Bay

esia

n re

gres

sion

s of

cha

nges

in s

tock

s' c

hara

cter

istic

liqu

idity

on

the

prev

ious

mon

ths

chan

ge in

cha

ract

eris

tic li

quid

ity, t

he p

revi

ous

leve

l, an

d sh

ocks

to

mar

ket-w

ide

mea

sure

s of

liqu

idity

usi

ng d

ata

from

mon

ths

t-1 th

roug

h t-6

0. A

lpha

s ar

e th

e in

terc

epts

from

regr

essi

ons

of th

e po

rtfol

io re

turn

s on

th

e Fa

ma-

Fren

ch r

isk

fact

ors.

Th

e di

ffere

nce

in a

lpha

bet

wee

n de

cile

10

(larg

est)

and

deci

le 1

(sm

alle

st)

is r

epor

ted.

A

ll in

terc

epts

hav

e be

en

annu

aliz

ed a

nd t

-sta

tistic

s ar

e in

par

enth

eses

. T

he s

ix in

divi

dual

mea

sure

s ar

e: P

RV

(P

rice

Rev

ersa

l) w

hich

est

imat

es t

he t

ende

ncy

of p

rice

chan

ges

acco

mpa

nied

by

larg

e vo

lum

e to

reve

rse

as d

escr

ibed

in e

quat

ion

(2.1

), m

PR

V w

hich

is th

e m

odifi

ed v

ersi

on o

f PR

V d

escr

ibed

in (2

.2),

PI w

hich

is th

e pr

ice

impa

ct li

quid

ity m

easu

re d

escr

ibed

by

(2.3

), m

PI w

hich

is th

e m

odifi

ed v

ersi

on o

f the

pric

e im

pact

mea

sure

des

crib

ed in

(2.4

), R

QS

whi

ch is

the

mar

ket w

ide

aver

age

quot

ed s

prea

d de

scrib

ed b

y (2

.5),

and

RE

S w

hich

is th

e m

arke

t wid

e av

erag

e ef

fect

ive

spre

ad d

escr

ibed

by

(2.6

). T

he p

ortfo

lios

form

ed u

sing

pre

dict

ed li

quid

ity b

etas

rela

tive

to th

e R

QS

and

RE

S m

easu

res

begi

n in

198

7.

Liqu

idity

Spr

ead

Bet

a

Cha

ract

eris

tic L

iqui

dity

Li

quid

ity

E

qual

-Wei

ghte

d

Val

ue-W

eigh

ted

E

qual

-Wei

ghte

d

Val

ue-W

eigh

ted

Mea

sure

All

Non

Jan

Ja

n

All

Non

Jan

Jan

A

ll N

on J

anJa

n

All

Non

Jan

Jan

P

RV

-0.1

0 -0

.36

4.93

0.

36

-0.1

8 8.

40

-0.2

6 -0

.17

-0.2

5 0.

09

-0.0

1 1.

40

[0.9

1]

[0.6

8]

[0.0

3]

[0.7

6]

[0.8

8]

[0.0

1]

[0.7

6]

[0.8

4]

[0.9

4]

[0.9

3]

[0.9

9]

[0.7

7]

mP

RV

-0.6

2 -0

.90

1.43

-2

.87

-2.0

2 -1

8.89

-1

.56

-1.0

2 -6

.71

-3.3

1 -2

.60

-18.

90

[0.4

2]

[0.2

7]

[0.5

3]

[0.0

2]

[0.0

9]

[0.0

0]

[0.0

9]

[0.2

7]

[0.0

9]

[0.0

4]

[0.1

2]

[0.0

0]

PI

0.

87

-0.4

7 23

.49

-0.1

0 -0

.68

14.5

2 -0

.28

1.76

-3

1.07

0.

36

1.43

-1

8.04

[0

.57]

[0

.76]

[0

.00]

[0

.96]

[0

.71]

[0

.00]

[0

.86]

[0

.23]

[0

.00]

[0

.80]

[0

.27]

[0

.00]

m

PI

-3

.40

-4.5

1 8.

77

-3.9

3 -4

.49

-0.8

7 7.

86

9.81

-1

2.96

7.

97

8.72

10

.56

[0.0

6]

[0.0

1]

[0.2

4]

[0.1

0]

[0.0

5]

[0.9

1]

[0.0

0]

[0.0

0]

[0.1

0]

[0.0

1]

[0.0

1]

[0.2

4]

RQ

S

-1

.16

-4.0

0 23

.01

-2.5

3 -3

.12

2.03

-2

.89

-0.1

5 -3

4.18

-1

.07

0.63

-1

5.44

[0

.59]

[0

.05]

[0

.00]

[0

.39]

[0

.28]

[0

.77]

[0

.53]

[0

.97]

[0

.00]

[0

.81]

[0

.89]

[0

.30]

R

ES

0.39

-2

.75

29.7

0 0.

98

-0.9

3 14

.74

3.84

6.

59

-27.

41

5.57

9.

02

-26.

32

[0.8

6]

[0.1

8]

[0.0

0]

[0.7

6]

[0.7

5]

[0.0

9]

[0.2

3]

[0.0

4]

[0.0

0]

[0.1

3]

[0.0

1]

[0.1

3]

60

Table 11

Spread in Alphas from Two-Way Sorted Portfolios

Each month, all stocks with available data are sorted into quintiles by the "control variable", then sorted within quintile by the difference variable. The variables are: Liquidity Return Beta – the coefficient on market-wide liquidity shocks from a Bayesian regression of stock return on the three Fama-French risk factors and liquidity shocks, Liquidity Spread Beta – the coefficient on market-wide liquidity shocks from a Bayesian regression of change in the stock's own liquidity measure from t-1 to t on the lagged change in the stock's own liquidity measure, the lagged level, and the market-wide liquidity shock at t, Characteristic Liquidity – the average of the stock's liquidity measure at months t-2 through t-4, and Market Capitalization: the market capitalization at the end of month t-1. Each two-by-two sort yields 25 portfolios that are then equal-weighted and linked through time to form 25 return time series. These series are then regressed on the Fama-French Factors and the spread in alphas between the extreme values of the difference variable within the control variable quintile is reported. The six individual measures are: PRV (Price Reversal) which estimates the tendency of price changes accompanied by large volume to reverse as described in equation (2.1), mPRV which is the modified version of PRV described in (2.2), PI which is the price impact liquidity measure described by (2.3), mPI which is the modified version of the price impact measure described in (2.4), RQS which is the market wide average quoted spread described by (2.5), and RES which is the market wide average effective spread described by (2.6). The portfolios formed using predicted liquidity betas relative to the RQS and RES measures begin in 1987. t-statistics are in parentheses.

61

Table 11 (continued)

Control Variable Quintile

Liquidity Measure

Difference Variable

Control Variable Sample 1 2 3 4 5

PRV Liquidity Market All 0.45 2.42 2.79 -0.14 0.78 Return Capitalization (0.42) (1.74) (1.87) (-0.09) (0.56) Beta Non Jan 0.61 2.65 3.07 0.34 1.56 (0.56) (1.88) (2.02) (0.22) (1.11) Jan -1.22 -7.56 -6.45 -10.60 -18.70 (-0.23) (-1.15) (-0.84) (-1.31) (-3.08) Liquidity Market All -0.42 0.49 -0.72 0.83 -0.41 Spread Capitalization (-0.45) (0.45) (-0.65) (0.71) (-0.41) Beta Non Jan -0.34 0.95 -0.56 1.51 -0.31 (-0.35) (0.85) (-0.50) (1.31) (-0.31) Jan 0.72 -3.33 -0.24 -1.90 -0.90 (0.16) (-0.67) (-0.05) (-0.27) (-0.14) Characteristic Market All -0.34 -0.45 -0.72 0.44 -1.17 Liquidity Capitalization (-0.35) (-0.45) (-0.71) (0.45) (-1.13) Non Jan -0.16 -0.65 -1.11 0.43 -1.07 (-0.16) (-0.64) (-1.09) (0.42) (-1.00) Jan -2.89 0.71 7.24 -0.95 -3.93 (-0.48) (0.12) (1.42) (-0.24) (-0.91) Liquidity Liquidity All -0.64 1.42 1.12 2.38 2.29 Return Spread (-0.42) (1.07) (0.85) (1.72) (1.62) Beta Beta Non Jan -0.37 2.07 1.67 2.79 2.18 (-0.24) (1.56) (1.26) (2.03) (1.50) Jan -13.20 -14.80 -8.93 -9.44 2.95 (-1.59) (-2.05) (-1.23) (-1.22) (0.45) Liquidity Characteristic All 1.84 1.95 1.33 1.07 2.06 Return Liquidity (1.33) (1.41) (0.97) (0.74) (1.56) Beta Non Jan 1.90 2.32 1.77 1.21 2.37 (1.37) (1.64) (1.29) (0.84) (1.79) Jan -2.05 -6.48 -11.60 -10.00 -2.89 (-0.28) (-0.93) (-1.56) (-1.42) (-0.37)

62

Table 11 (continued)

Control Variable Quintile Liquidity Measure

Difference Variable

Control Variable Sample 1 2 3 4 5

mPRV Liquidity Market All 1.35 2.93 2.00 1.15 0.01 Return Capitalization (1.28) (2.17) (1.49) (0.83) (0.01) Beta Non Jan 1.51 3.18 2.17 1.48 0.46 (1.40) (2.30) (1.58) (1.07) (0.32) Jan 1.46 -3.19 -3.17 -8.64 -18.40 (0.29) (-0.49) (-0.48) (-1.08) (-2.99) Liquidity Market All -1.29 -1.02 -1.27 0.77 -1.43 Spread Capitalization (-1.45) (-0.93) (-1.27) (0.75) (-1.51) Beta Non Jan -1.47 -1.14 -1.18 1.31 -1.10 (-1.61) (-1.02) (-1.12) (1.28) (-1.16) Jan -0.80 -0.31 -0.32 -7.78 -11.20 (-0.19) (-0.06) (-0.08) (-1.34) (-2.08) Characteristic Market All -1.17 -0.80 -0.78 0.45 -1.73 Liquidity Capitalization (-1.10) (-0.69) (-0.63) (0.41) (-1.54) Non Jan -0.72 -0.70 -0.81 0.51 -1.72 (-0.67) (-0.59) (-0.63) (0.44) (-1.51) Jan -7.43 0.39 3.61 -2.94 -3.75 (-1.31) (0.06) (0.76) (-0.77) (-0.68) Liquidity Liquidity All 0.86 2.15 1.49 1.11 1.96 Return Spread (0.65) (1.68) (1.12) (0.89) (1.46) Beta Beta Non Jan 1.06 2.09 2.32 1.26 1.88 (0.80) (1.62) (1.71) (1.01) (1.35) Jan -6.70 -0.62 -14.20 -4.70 5.29 (-1.01) (-0.09) (-2.48) (-0.68) (0.95) Liquidity Characteristic All 1.59 1.87 0.63 1.10 1.15 Return Liquidity (1.15) (1.53) (0.54) (0.90) (0.85) Beta Non Jan 1.78 2.03 0.75 1.51 1.06 (1.26) (1.66) (0.64) (1.21) (0.76) Jan -3.85 -2.76 -7.12 -6.15 0.29 (-0.55) (-0.39) (-1.15) (-0.98) (0.05)

63

Table 11 (continued)

Control Variable Quintile Liquidity Measure

Difference Variable

Control Variable Sample 1 2 3 4 5

PI Liquidity Market All 2.07 1.99 -0.69 -0.60 0.90 Return Capitalization (1.96) (1.53) (-0.52) (-0.40) (0.66) Beta Non Jan 2.59 2.49 0.36 0.42 1.67 (2.43) (1.87) (0.26) (0.27) (1.20) Jan -2.57 -3.93 -16.60 -14.50 -7.14 (-0.46) (-0.66) (-3.43) (-2.34) (-1.11) Liquidity Market All 0.29 2.31 4.69 0.69 0.34 Spread Capitalization (0.22) (1.48) (2.73) (0.46) (0.25) Beta Non Jan 0.33 2.86 5.46 1.99 1.01 (0.24) (1.83) (3.15) (1.31) (0.74) Jan 7.46 -0.84 1.06 -14.40 -4.56 (1.27) (-0.11) (0.12) (-2.15) (-0.81) Characteristic Market All -1.43 -5.12 -0.63 -1.17 2.12 Liquidity Capitalization (-0.74) (-2.58) (-0.34) (-0.66) (1.45) Non Jan 0.77 -5.12 -0.91 -2.12 0.76 (0.42) (-2.66) (-0.48) (-1.22) (0.55) Jan -38.80 -9.09 -1.95 10.51 17.49 (-3.40) (-0.69) (-0.21) (0.96) (1.78) Liquidity Liquidity All 0.97 0.08 0.08 1.37 1.70 Return Spread (0.76) (0.07) (0.07) (1.01) (1.17) Beta Beta Non Jan 1.51 1.20 1.24 1.72 2.62 (1.17) (0.93) (1.04) (1.26) (1.82) Jan -8.78 -14.20 -19.80 1.74 -3.65 (-1.24) (-2.89) (-3.90) (0.22) (-0.48) Liquidity Characteristic All 1.17 2.22 1.77 -0.29 0.54 Return Liquidity (0.90) (1.70) (1.31) (-0.23) (0.38) Beta Non Jan 1.82 3.35 2.28 0.55 1.27 (1.37) (2.59) (1.68) (0.42) (0.86) Jan -3.37 -13.80 -7.79 -14.90 -6.03 (-0.59) (-1.85) (-1.00) (-3.36) (-0.87)

64

Table 11 (continued)

Control Variable Quintile Liquidity Measure

Difference Variable

Control Variable Sample 1 2 3 4 5

mPI Liquidity Market All -0.30 -0.98 -0.01 0.11 1.33 Return Capitalization (-0.21) (-0.63) (-0.01) (0.06) (0.79) Beta Non Jan 0.37 -0.26 1.20 1.11 2.02 (0.26) (-0.16) (0.70) (0.59) (1.18) Jan -9.81 -12.70 -18.90 -16.40 -11.00 (-1.35) (-1.95) (-2.65) (-2.09) (-1.33) Liquidity Market All -3.66 -4.43 -1.68 -2.14 -0.74 Spread Capitalization (-2.24) (-2.57) (-0.92) (-1.09) (-0.38) Beta Non Jan -4.49 -5.10 -2.20 -2.56 -1.36 (-2.80) (-2.94) (-1.20) (-1.34) (-0.72) Jan 1.34 1.09 4.48 -2.73 2.65 (0.14) (0.12) (0.43) (-0.21) (0.21) Characteristic Market All 9.65 7.00 5.35 0.57 -0.09 Liquidity Capitalization (5.02) (3.35) (2.62) (0.25) (-0.04) Non Jan 11.51 7.63 6.18 1.28 0.68 (6.01) (3.69) (2.93) (0.58) (0.31) Jan -11.10 12.77 -0.03 -0.83 2.82 (-1.13) (1.13) (0.00) (-0.06) (0.20) Liquidity Liquidity All 1.41 0.26 2.14 -1.57 -1.73 Return Spread (1.25) (0.20) (1.48) (-0.87) (-0.86) Beta Beta Non Jan 1.89 0.91 2.76 -1.01 -0.93 (1.66) (0.66) (1.95) (-0.56) (-0.44) Jan -10.20 -13.80 -15.10 -14.70 -14.60 (-1.80) (-2.32) (-1.75) (-1.49) (-1.98) Liquidity Characteristic All 0.15 -1.96 -0.94 0.90 1.06 Return Liquidity (0.07) (-1.19) (-0.71) (0.82) (1.08) Beta Non Jan 0.86 -1.37 0.15 0.92 1.34 (0.38) -0.82 0.11 0.85 1.35 Jan -12.40 -14.00 -23.20 -9.23 -6.79 (-1.32) (-1.70) (-2.88) (-1.56) (-1.38)

65

Table 11 (continued)

Control Variable Quintile Liquidity Measure

Difference Variable

Control Variable Sample 1 2 3 4 5

RQS Liquidity Market All 3.16 0.24 0.47 3.00 5.52 Return Capitalization (2.08) (0.10) (0.19) (0.97) (2.17) Beta Non Jan 3.69 -0.83 0.67 4.08 5.50 (2.33) (-0.34) (0.27) (1.27) (2.09) Jan -3.50 3.64 -2.15 -11.40 3.15 (-0.54) (0.24) (-0.18) (-0.76) (0.26) Liquidity Market All -0.28 -2.95 -1.28 -0.11 -1.36 Spread Capitalization (-0.14) (-1.11) (-0.41) (-0.03) (-0.41) Beta Non Jan -2.52 -5.37 -3.63 -2.70 -2.98 (-1.37) (-2.09) (-1.12) (-0.79) (-0.91) Jan 18.66 15.18 20.61 13.94 5.28 (1.73) (1.43) (2.56) (1.40) (0.37) Characteristic Market All 0.75 -1.71 -1.47 -1.67 1.14 Liquidity Capitalization (0.29) (-0.63) (-0.58) (-0.71) (0.48) Non Jan 2.81 -0.75 -0.55 -0.62 0.18 (1.09) (-0.26) (-0.21) (-0.26) (0.08) Jan -28.60 -13.30 -11.50 -3.51 13.08 (-2.32) (-1.27) (-1.57) (-0.40) (0.98) Liquidity Liquidity All 1.29 4.19 0.13 3.41 2.37 Return Spread (0.69) (1.94) (0.05) (1.46) (0.97) Beta Beta Non Jan 0.99 4.51 0.59 3.39 2.22 (0.52) (1.96) (0.22) (1.39) (0.88) Jan 0.39 -0.32 -9.22 -0.13 7.94 (0.04) (-0.05) (-0.64) (-0.01) (0.68) Liquidity Characteristic All 0.97 5.15 1.80 1.01 4.90 Return Liquidity (0.51) (2.47) (0.74) (0.42) (1.93) Beta Non Jan 1.24 5.75 0.89 1.29 4.94 (0.63) (2.67) (0.37) (0.53) (1.84) Jan 1.03 -0.76 5.25 -9.28 -0.42 (0.13) (-0.08) (0.40) (-0.70) (-0.05)

66

Table 11 (continued)

Control Variable Quintile Liquidity Measure

Difference Variable

Control Variable Sample 1 2 3 4 5

RES Liquidity Market All 2.99 -1.42 0.33 -1.50 4.08 Return Capitalization (2.02) (-0.62) (0.14) (-0.51) (1.66) Beta Non Jan 3.76 -1.75 1.50 -0.41 3.91 (2.45) (-0.77) (0.61) (-0.13) (1.55) Jan -4.80 -2.99 -16.90 -18.80 4.15 (-0.80) (-0.21) (-2.13) (-1.33) (0.35) Liquidity Market All 1.70 -2.20 -0.84 0.14 -2.06 Spread Capitalization (0.91) (-0.82) (-0.26) (0.04) (-0.60) Beta Non Jan -0.49 -4.70 -2.90 -2.13 -4.08 (-0.28) (-1.81) (-0.89) (-0.64) (-1.20) Jan 23.20 20.99 16.71 12.78 8.64 (2.70) (1.83) (1.54) (1.22) (0.50) Characteristic Market All 0.98 -1.91 0.24 -2.11 1.04 Liquidity Capitalization (0.39) (-0.75) (0.09) (-0.75) (0.48) Non Jan 3.25 -0.19 1.71 0.26 1.11 (1.31) (-0.07) (0.59) (0.10) (0.49) Jan -29.60 -15.70 -6.65 -11.70 9.38 (-2.70) (-1.68) (-0.72) (-0.93) (1.28) Liquidity Liquidity All 3.87 0.67 0.95 0.56 0.58 Return Spread (2.13) (0.31) (0.37) (0.24) (0.22) Beta Beta Non Jan 3.71 0.66 2.66 1.15 1.23 (2.02) (0.30) (1.04) (0.49) (0.45) Jan 3.17 -4.94 -23.00 -11.70 -6.08 (0.32) (-0.49) (-1.74) (-0.94) (-0.68) Liquidity Characteristic All 0.29 3.25 1.13 -1.19 3.49 Return Liquidity (0.15) (1.50) (0.47) (-0.47) (1.61) Beta Non Jan 1.40 4.09 0.87 -0.50 3.81 (0.69) (1.80) (0.37) (-0.19) (1.70) Jan -9.47 -4.90 -3.12 -12.10 -4.97 (-1.17) (-0.63) (-0.25) (-1.14) (-0.57)