Petkova Paper

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Idiosyncratic Volatility of Liquidity and

Expected Stock Returns

Ferhat Akbas, Will J. Armstrong, Ralitsa Petkova

December, 2011

ABSTRACT

We show that idiosyncratic liquidity risk is positively priced in the cross-section ofstock returns. Our measure of idiosyncratic liquidity volatility is based on a marketmodel for stock liquidity. Idiosyncratic volatility of liquidity is priced in the presenceof systematic liquidity risk: the covariance of stock returns with aggregate liquidity,the covariance of stock liquidity with aggregate liquidity, and the covariance of stockliquidity with the market return. Our results are puzzling in light of Acharya andPedersen (2005) who develop a model in which only systematic liquidity risk affectsreturns.

We thank Kerry Back and seminar participants at the Bank of Canada, Purdue University, Texas A&MUniversity, University of Georgia, and University of Oklahoma for helpful comments and suggestions.

KU School of Business, University of Kansas, Lawrence KS 66049, USA.; E-mail: [email protected] Business School, Texas A&M University, College Station TX 77845, USA.; E-mail:

[email protected] author. Krannert School of Management, Purdue University, West Lafayette IN 47906,

USA. Tel.: 765 494 4397; E-mail: [email protected].


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There is increasing evidence that liquidity affects asset returns. Numerous studies

examine the liquidity characteristics of stocks and show that illiquid assets earn higher

expected returns.1 Furthermore, the evidence also shows that the liquidity of individual

stocks varies over time. If a stocks liquidity is very volatile this will increase the uncertainty

attached to the stock position and limit the investors flexibility at the time he chooses to

trade.2 For example, an investor who needs to reduce his exposure in a stock may have

to sell at fire-sale prices or unbalance his portfolio by selling his most liquid securities. In

extreme cases, a stocks liquidity may suddenly dry up eliminating the opportunity for the

investor to enter or exit the position at all.

Part of the variation in a stocks liquidity is systematic since it is driven by factors that

affect all stocks, such as variation in market-wide liquidity. The sensitivity to aggregate

market liquidity creates commonality in the liquidity variation across stocks and most

stocks become more (less) liquid when aggregate market liquidity increases (decreases).3 It

seems reasonable that investors might require compensation for being exposed to systematic

liquidity variation. Consider, for example, an investor who has experienced a large drop

in wealth and must liquidate some assets to raise cash. Since a decline in wealth is likely

to occur at times when aggregate liquidity is low, all the assets held by the investor are

likely to become less liquid due to commonality. Therefore, the commonality in liquidity is

a systematic risk that cannot be diversified away. Since liquidation is costlier when liquidity

is low, the investor would require higher expected returns from assets whose liquidity has

1See, among others, Amihud and Mendelson (1986, 1989), Brennan and Subrahmanyam (1996),

Eleswarapu (1997), Brennan, Chordia and Subrahmanyam (1998), Chalmers and Kadlec (1998), Chordia,Roll and Subrahmanyam (2001), Amihud (2002), Hasbrouck (2009), Chordia, Huh, and Subrahmanyam(2009).

2Persaud (2003) observes that there is a broad belief among users of financial liquidity-traders, investorsand central bankersthat the principal challenge is not the average level of financial liquidity, but its variabilityand uncertainty.

3The commonality in liquidity has been documented by Huberman and Halka (2001), Chordia, Roll, andSubrahmanyam (2000), and Hasbrouck and Seppi (2001), among others.

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stronger covariance with market liquidity. Acharya and Pedersen (2005) confirm that this

type of liquidity risk due to commonality is priced in the cross-section of stock returns.

Another part of the variation in individual stock liquidity comes from variation in

idiosyncratic sources of liquidity. These include information uncertainty, idiosyncratic

information content of trades (adverse selection), volatility in depth of sources of supply and

demand, demand from different parties, dealer inventories, and others. Since it is not likely

that these idiosyncratic sources of variation always move together across stocks, idiosyncratic

variation in liquidity might make some stocks more liquid and others less liquid at a time

when the investor needs to raise cash. Therefore, idiosyncratic liquidity variation is a source

of risk that could potentially be diversified away. Standard intuition suggests that if investors

hold well-diversified portfolios, idiosyncratic liquidity variation should not affect expected

returns.

In this paper we show, however, that idiosyncratic volatility of liquidity is priced in the

cross-section of stock returns. We use a simple model that decomposes individual liquidity

into systematic and idiosyncratic components. The volatility of the idiosyncratic component

commands a significant and positive price of risk in the cross-section of stocks. This result

suggests that investors require a risk premium for holding stocks with high idiosyncratic

variation in liquidity.

In this study we consider a stock to be illiquid when trading induces negative price

impact. If investors want to sell large amounts in a short period of time, the price impact

is of special concern (e.g. Brennan et al. (2011)). This is the case since price impact

decreases the potential return from investing in a stock by reducing the price received when

the investor attempts to sell the stock. This price impact view of liquidity is theoretically

motivated in Kyles (1985) model which predicts that there is a linear relation between net

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order flow and price changes. In addition, Brunnermeier and Pedersen (2009) define liquidity

theoretically as the difference between transaction prices and fundamental values as a result

of buying or selling pressure. Therefore, in our empirical analysis we use the price impact of

trade based on Amihud (2002) as a measure of liquidity. According to this measure, stocks

are considered to be liquid if a large volume of shares can be traded without affecting the

price substantially. For each stock, we compute its daily Amihud measures across time. We

regress the daily Amihud measures within a month on contemporaneous market liquidity

and the market return. The standard deviation of the residuals from this regression proxies

for idiosyncratic volatility of liquidity.4 We find reliable evidence that stocks with high

idiosyncratic variability in liquidity command higher expected returns. This finding persists

across a wide range of robustness checks, which include standard control variables, exposure

to common risk factors, and different sub-periods.

Furthermore, we show that idiosyncratic volatility of liquidity is priced in the presence of

the level of liquidity and systematic liquidity risk. In our empirical analysis we consider three

types of systematic liquidity risk. The first one is related to commonality in liquidity and

is measured by the covariance of stock liquidity with aggregate liquidity. The second type

is measured by the covariance of stock returns with aggregate market liquidity. Pastor and

Stambaugh (2003) observe that market liquidity is an important feature of the investment

environment and they show that differences in expected returns are significantly related to

the sensitivities of returns to fluctuations in aggregate liquidity. The third type of systematic

liquidity risk is measured by the covariance of stock liquidity with the market return. This

risk reflects the difficulty of selling illiquid stocks during market downturns.

4More precisely, the volatility of liquidity is measured as the standard deviation of the residuals scaledby the average level of liquidity. We do this since the mean and standard deviation of liquidity are highlycorrelated due to the presence of dollar volume in the Amihud liquidity measure.

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Acharya and Pedersen (2005) develop a model that incorporates all three types of

systematic liquidity risk. Their model provides a unified theoretical framework that explains

previous empirical findings that return sensitivity to market liquidity is priced (Pastor and

Stambaugh (2003)), that average liquidity is priced (Amihud and Mendelson (1986)), and

that liquidity comoves with returns and predicts future returns (Amihud (2002), Chordia et

al. (2001), Jones, (2002), and Bekaert et al. (2007)). In their model, the expected return of a

security is increasing in its expected illiquidity, its market beta, and three betas representing

different forms of liquidity risk. These liquidity risks are associated with commonality in

liquidity with market liquidity, return sensitivity to market liquidity, and liquidity sensitivity

to market returns.

Other studies also examine the pricing of systematic liquidity risk. Bekaert et al. (2007)

study nineteen emerging markets using a model that extends Acharya and Pedersen (2005).

They use country-specific and global liquidity factors and show that the price of local liquidity

risk is positive and significant. Sadka (2006) examines the pricing of liquidity risk in a

factor model that includes the three Fama-French factors and a liquidity factor, calculated

as an average of the stocks permanent market impact coefficients. The results show that

the liquidity factor is priced, with a positive risk premium. Watanabe and Watanabe (2008)

propose that the effect of liquidity on stock returns varies over time across identifiable states.

They find that liquidity loadings are higher in states when investors may expect liquidity

needs, especially when turnover is abnormally high. In states of high liquidity betas, the

price of liquidity risk is higher. Liu (2004) uses a factor-mimicking portfolio for aggregate

liquidity. He shows that a model that contains the market return and the tracking portfolio

for liquidity performs well in explaining stock returns. Chordia et al. (2001) examine another

form of liquidity risk, measured by the total volatility of trading activity. They show that

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the level and volatility of trading activity have a negative effect on stock returns. Their

findings are not directly comparable to ours or the studies mentioned above since they focus

on the total volatility of trading activity.

All the models mentioned above predict that only systematic liquidity risk should affect

expected return. Therefore, our finding that the idiosyncratic volatility of liquidity is priced,

in addition to systematic liquidity variation, represents a puzzle within the context of these

models. If investors hold well-diversified portfolios, idiosyncratic liquidity volatility should

not affect expected returns. This is the case since high variation in liquidity implies that

some stocks may become more liquid and others less liquid at a time when the investor

needs to trade. This suggests that the investors will be able to raise cash by choosing to

liquidate the more liquid securities. For example, Brown, Carlin, and Lobo (2010) develop a

one-period model in which the investor faces a margin constraint and experiences an urgent

need for liquidity. They find that, for a given portfolio and price impact parameters, the

investor optimally sells assets that are more liquid to meet pending obligations. Therefore,

in sufficiently diversified portfolios, idiosyncratic liquidity variance is likely to be diversified

away.

In contrast, we find that idiosyncratic liquidity variation significantly affects expected

returns, and the effect is positive. We offer a possible explanation for the positive relation

between average returns and idiosyncratic liquidity risk. If an investor faces an immediate

liquidity need due to exogenous cash needs, margin calls, dealer inventory rebalancing, forced

liquidations, or standard portfolio rebalancing, he needs to unwind his positions in a short

period of time. In case of such a liquidity need the investor may not be able to wait for

periods of high liquidity to sell the stock, and thus the level of liquidity of the stock on the

day the investor closes his position is important. This effect will be reinforced if investors are

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subject to borrowing constraints and cannot borrow easily in case of an urgent consumption

need (e.g., see Huang (2003)). The higher a stocks idiosyncratic volatility of liquidity, the

more likely it is that the stock will be very illiquid at a time when it is traded. Therefore, the

investor might end up unwinding his position at a low level of liquidity for the stock, which

induces a significant loss of wealth due to a large price impact of trade. Thus, investors will

require compensation for being exposed to this risk. All else equal, a risk-averse investor

may be willing to pay a higher price for a stock that has a lower risk of becoming less liquid

at the time of trading, i.e., a stock with a low idiosyncratic liquidity variation.

In addition, when the investor faces an immediate consumption need, it may not be

optimal to sell the more liquid securities to meet his short term obligations. This is the case

if the investor expects further liquidity shocks in the future. For example, Scholes (2000)

notes, In an unfolding crisis, most market participants respond by liquidating their most

liquid investments first to reduce exposures and to reduce leverage ... However, after the

liquidation, the remaining portfolio is most likely unhedged and more illiquid. Without new

inflows of liquidity, the portfolio becomes even more costly to unwind and manage. Brown,

Carlin, and Lobo (2010) develop a model that captures this intuition. They point out that

selling the more liquid assets first will limit the immediate loss in portfolio value. However,

the remaining portfolio will be more exposed to adverse conditions in the future. Selling the

less liquid assets first will result in a portfolio that is less exposed to future liquidity shocks.

However, this could result in unnecessary loss in portfolio value if the subsequent liquidity

shock does not materialize. Brown, Carlin, and Lobo (2010) obtain a theoretical solution

for this tradeoff and show that if the expected future liquidity shock is sufficiently large, the

investor would prefer to retain more of the assets with low price impact in order to hedge

against future conditions. Therefore, when the investor faces an immediate liquidity need

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he might end up selling stocks that have become very illiquid at the time of trading due to

high volatility in their liquidity.

In summary, our paper contributes to the literature by documenting that the positive

effect on returns of idiosyncratic volatility of liquidity is different from previously documented

effects such as the mean level of liquidity and systematic liquidity risk. We conjecture that

the volatility of liquidity matters most for investors who may face an immediate liquidity need

over a relatively short horizon and are unable to adapt their trading to the state of liquidity

of their stocks. For example, a mutual fund manager faced with unexpected investors

redemptions will be forced to engage in liquidity-motivated trading. The manager may be

forced to liquidate securities that are highly illiquid at the time. Edelen (1999), among

others, documents that the common finding of negative return performance at open-end

mutual funds could be attributed to the costs of liquidity-motivated trading. Furthermore,

the volatility of liquidity is also important for investors who might not be professional traders.

For example, a household may have to liquidate its illiquid assets due to consumptions needs.

Similarly, a firm may have to liquidate certain assets to undertake a surprise investment

opportunity.

The rest of the paper is organized as follows. In section I we discuss the benchmark model

that defines systematic liquidity risk. Then we describe the construction of our idiosyncratic

volatility of liquidity measure and the data sample. Section II documents the main results.

Robustness tests are presented in section III, and section IV concludes.

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I. Empirical Methods

A. Benchmark Model

In this section we define a benchmark model for systematic liquidity risk. We will use

this model as a starting point to show that idiosyncratic volatility of liquidity is priced in

addition to systematic liquidity risk. We use the dynamic overlapping-generations model of

Acharya and Pedersen (2005) who study the effects of variations in liquidity on asset prices

under risk aversion. The illiquidity cost, ci in the model is defined as the cost of selling

security i. Uncertainty about the illiquidity cost is what generates the liquidity risk in the

model. When investors are risk averse and illiquidity and dividends are risky, Acharya and

Pedersen (2005) show that the conditional expected net return of security i in the unique

linear equilibrium is

Et(rit+1 cit+1) = rf + tCovt(rit+1 cit+1, RMt+1 CM t+1)

V ar(RMt+1 CMt+1), (1)

where rit+1 cit+1 is the return of security i net of liquidity cost ci, RM t+1 CMt+1 is the

return of the market portfolio net of the aggregate liquidity cost CM, and rf is the risk-free

rate. Equivalently, equation (1) can be written as

Et(rit+1 rf) = Et(cit+1) +tCovt(rit+1, RM t+1)

V ar(RMt+1 CMt+1)+ t

Covt(cit+1, CMt+1)

V ar(RM t+1 CMt+1)

tCovt(rit+1, CM t+1)

V ar(RM t+1 CMt+1) t

Covt(cit+1, RMt+1)

V ar(RMt+1 CM t+1). (2)

Equation (2) states that the required excess return is the expected relative illiquidity cost,

Et(ci), plus four betas (covariances) times the price of risk . For convenience, we denote

the four covariance terms above as Rr , Cc ,

Cr , and

Rc , respectively. As in the standard

CAPM, the model shows that the excess return on an asset increases with market beta Rr .

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The model of Acharya and Pedersen contains three additional betas which represent three

different types of liquidity risk.

The first liquidity beta Cc is positive for most assets due to commonality in liquidity.

Since investors want to be compensated for holding a security that becomes illiquid when

the market in general becomes illiquid, the expected excess return increases with Cc in the

model. The second liquidity beta Cr measures the sensitivity of asset returns to market-

wide illiquidity. It is usually negative since an increase in market illiquidity implies that

asset values will go down (e.g., Amihud (2002)). This liquidity beta has a negative effect on

excess returns since investors are willing to accept a lower return on an asset whose return

is higher in states of high market illiquidity. The third liquidity beta Rc is also negative for

most stocks (e.g., Acharya and Pedersen (2005) and Chordia et al. (2006)). It has a negative

effect on excess returns since investors are willing to accept a lower expected return on a

security that is liquid in a down market.

The model in equation (2) implies that only systematic liquidity risk commands a

risk premium in the cross-section of expected returns. Our objective is to test whether

idiosyncratic variation in liquidity is also priced in addition to systematic variation. We

are motivated by previous studies that find that types of idiosyncratic risk are priced in

the cross-section of returns. For example, numerous studies have documented that the

idiosyncratic volatility of returns is a significant determinant of average returns.5 Since

liquidity affects the level of prices, liquidity volatility can affect asset price volatility itself.

Therefore, idiosyncratic volatility of liquidity may affect expected returns through its effect

on return volatility. Before we proceed to test whether idiosyncratic volatility of liquidity is

priced in the cross-section of returns, we define the measure of liquidity that we use.

5See, among others, Ang et al. (2006, 2009), Fu (2009).

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B. The Main Measure of Liquidity

Liquidity is a stock characteristic that is difficult to define. Usually, a stock is thought

to be liquid if large quantities can be traded in a short period of time without moving the

price too much. If an investor faces an immediate need to sell a stock, he may not be able to

adapt his trading to the liquidity state of the stock. If he needs to unwind his position in the

stock in a short time he might sell at a very unfavorable price due to the high price impact of

trade. Therefore, the price impact of trade dimension of liquidity becomes the most relevant

part of liquidity. Thus, we use price impact of trade as our main measure of liquidity.

Studies that use price impact as a measure of liquidity include Brennan and Subrahmanyam

(1996), Bertsimas and Lo (1998), He and Mamayasky (2001), Amihud (2002), Pastor and

Stambaugh (2003), Acharya and Pedersen (2005), and Sadka (2006).6

We follow Amihud (2002) and use a measure of liquidity which captures the relation

between price impact and order flow. A key benefit of using Amihuds (2002) measure is

that it can be estimated over a long sample period at relatively high frequencies. Measures

of price impact that use intraday data also provide high frequency observations of liquidity.

These measures have high precision, but are not available prior to 1988. Since we require a

long sample period for our asset-pricing tests, we use Amihuds measure which is available

for a longer time period. Hasbrouck (2009) compares price impact measures estimated from

daily data and intraday data, and finds that the Amihud (2002) measure is most highly

correlated with trade-based measures. For example, he finds that the correlation between

Kyles lambda and Amihuds measure is 0.82.7 Similarly, comparing various measures of6The bid-ask spread has also been used as a measure of liquidity, starting with Amihud and Mendelson

(1986). However, it is a less useful measure of liquidity for large investors since large blocks of shares usuallytrade outside the bid-ask spread (see, e.g., Chan and Lakonishok (1995) and Keim and Madhavan (1996)).In addition, Eleswarapu (1997) finds that the bid-ask spread does not predict returns for NYSE/AMEXstocks, but only for NASDAQ stocks.

7Kyles (1985) lambda is first estimated by Brennan and Subrahmanyam (1996) using intraday trade and

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liquidity, Goyenko, Holden, and Trzcinka (2009) conclude that Amihuds measure yields

significant results in capturing the price impact of trade. They find that it is comparable

to intraday estimates of price impact such as Kyles lambda.8 Therefore, we use Amihuds

ratio as the main liquidity proxy in our study.

C. Constructing Idiosyncratic Volatility of Liquidity

We calculate the daily price impact of order flow, following Amihud (2002) 9:

cid =|rid|

dvolid, (3)

where rid is the return of stock i on day d and dvolid is the dollar trading volume for stock

i on day d.10 The higher the daily price impact of order flow is, the less liquid the stock is

on that day. Therefore, Amihuds ratio measures illiquidity. We denote it as c to keep the

notation comparable to the model described in section I.A.

The mean level of illiquidity for month t is calculated as follows:

illiqit =

1

Dit

Dit

d=1

|ridt|

dvolidt , (4)

where Dit is the number of trading days in month t.

We use a market model time-series regression for each stock to decompose daily

variation in individual stock illiquidity into systematic and idiosyncratic components. More

quote data. Brennan and Subrahmanyam (1996) estimate lambda by regressing trade-by-trade price changeon signed transaction size. Lambda measures the price impact of a unit of trade size and, therefore, it islarger for less liquid stocks. Hasbrouck (2009) uses a similar method to estimate Kyles lambda.

8

They also compare Pastor and Stambaughs (2003) gamma and the Amivest liquidity ratio, and concludethat these measures are ineffective in capturing price impact.9Acharya and Pedersen (2005) also use daily Amihud measures to construct total volatility of liquidity.

They use the volatility of liquidity as a sorting variables for portfolios. They do not examine its pricing inthe cross-section of stock returns

10We have also tried adjusting c for inflation as cid =|rid|

dvolidinfdt, where infdt is an inflation-adjustment

factor. We obtain similar results.

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specifically, using daily data within a month, we regress daily firm-level illiquidity on daily

aggregate market illiquidity and the excess market return:

cid = b0i + b1iCMd + b1i(RMd rf d) + eid, (5)

where RM d rf d is the excess market return on day d and CMd is a measure of aggregate

market illiquidity on day d. Aggregate illiquidity is computed as an equally-weighted average

of the illiquidities of all stocks. This model specifies two sources of commonality in illiquidity.

The first one, CM, is motivated by Chordia et al. (2000) who show that market-wide

liquidity drives variation in the liquidity of all stocks. The second one, RM, is motivated by

the observation that liquidity tends to change with the market return (e.g., Hameed et al.

(2010)).

The standard deviation of the residuals from equation (5), (eid), measures the variation

in individual illiquidity which is not related to movements in aggregate illiquidity or the

market return. We use it as a measure of idiosyncratic variation in liquidity.11 The mean

level of illiquidity, illiq from equation (4) and the standard deviation of the residuals from

equation (5) are highly correlated. In our empirical analysis we control for the mean level of

illiquidity and therefore, it is important to have a measure of the volatility of liquidity which

is not highly correlated with the mean. Therefore, every month we compute a coefficient of

variation by dividing the idiosyncratic volatility of liquidity by the mean level of illiquidity:

ivolliqit =

(eid)t

illiqit . (6)

We use the coefficient of variation ivolliqit as our measure of the idiosyncratic volatility of

11Even though we refer to it as volatility of liquidity, it is actually the volatility of illiquidity since Amihudsratio measures illiquidity. The higher the volatility of the Amihud ratio within a month, the riskier the stockwill be.

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liquidity for stock i in month t. We examine the relation between this variable and average

stock returns and show that they are significantly positively correlated.

D. Data and Descriptive Statistics

Our main data sample consists of NYSE-AMEX common stocks for the period from

January 1964 to December 2009.12 Following Avramov, Chordia and Goyal (2006), we

exclude stocks with a month-end price of less than one dollar to ensure that our results are

not driven by extremely illiquid stocks. We also require that each stock has at least 15 days

with trades each month in order to calculate its volatility of liquidity. Stocks with prices

higher than one thousand dollars are excluded. Stocks that are included have at least 12

months of past return data from CRSP and sufficient data from COMPUSTAT to compute

accounting ratios as of December of the previous year.

We compute several other stock characteristics in addition to illiquidity and the

idiosyncratic volatility of liquidity. The variable definitions are as follows:

SIZE is the market value of equity calculated as the number of shares outstanding times

the month-end share price;

BM is the ratio of book value to market value of equity. Book value is calculated as in

Fama and French (2002) and measured at the most recent fiscal year-end that precedes

the calculation date of market value by at least three months.13 We exclude firms with

negative book values.

RET12M is the cumulative return from month t-13 to t-2;

12We exclude NASDAQ stocks from the analysis for two reasons. First, Atkins and Dyl (1997) arguethat the volume of NASDAQ stocks is inflated as a result of inter-dealer activities. Second, volume data onNASDAQ stocks is not available prior to November 1982.

13Book value is defined as total assets minus total liabilities plus balance sheet deferred taxes andinvestment tax credit minus the book value of preferred stock. Depending on data availability, the book valueof preferred stock is based on liquidating value, redemption value, or carrying value, in order of preferences.

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RET1M is the return in the previous month;

IVOL is idiosyncratic return volatility calculated as the standard deviation of the residuals

from the Fama-French (1993) model, following Ang, Hodrick, Xing, and Zhang (2006).

We require at least 15 days of return data to compute IVOL.

SKEW is the monthly skewness of daily returns;

Cov(r, c) is the covariance between daily stock returns and daily stock illiquidity over month

t. This variable is motivated by previous studies that show that liquidity comoves with

returns (e.g., Amihud (2002)).

TURNis the turnover ratio measured as the number of shares traded divided by the number

of shares outstanding in month t.14

We also compute several measures of systematic liquidity risk motivated by the model of

Acharya and Pedersen (2005). First, using daily data within a month, we regress firm-level

illiquidity on changes in market illiquidity and the excess market return:

cid = ai + CciCMd +

Rci (RMd rf d) + uid. (7)

Then, using daily data within a month, we regress firm-level excess returns on changes

in market illiquidity and the excess market return:

rid rf d = i + CriCMd +

Rri(RM d rf d) + vid. (8)

The two slope coefficients from equation (7) and the first slope coefficient from (8) are

used as measures of systematic liquidity risk. The second slope coefficient from (8) is a

14Our results are robust to including dollar volume among the set of control variables. However, we excludedollar volume from the reported results since it is highly correlated with both illiq and SIZE.

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measure of market beta. Note that the four betas defined above are very similar to the ones

in model (2) discussed previously. The difference between the two sets of betas is that the

ones in model (2) are univariate, while the ones that we estimate are not. Acharya and

Pedersen (2005) observe that there is a strong multicollinearity among their estimates of the

three univariate liquidity betas from model (2). Using multivariate regressions as in (7) and

(8) alleviates the multicollinearity problem to an extent. In addition, the intuition behind

the interpretation of the liquidity betas is preserved in the multivariate regressions. We also

note that using changes in aggregate market illiquidity in (7) and (8) allows for capturing

the effects of lagged values of illiquidity.

In our analysis, we match stock returns in month t to idiosyncratic volatility of liquidity

and other stock characteristics in month t 1. However, in order to avoid potential

microstructure biases and account for return autocorrelations, we measure stock returns

as the cumulative return over a 22-day trading period that begins a week after the various

stock characteristics are measured. Skipping a week between measuring stock characteristics

and future returns also allows us to use the most recent information about the stocks.

This is important since we want to capture the dimension of liquidity related to short-term

variability in trading costs. In addition, skipping a week assures that there is no overlap

between the returns used as dependent variables and the returns used to derive our liquidity

measures. Since liquidity varies over time, skipping a longer time interval might result in

loss of information relevant for future returns. However, our results are robust to skipping a

month and matching stock returns in month t to stock characteristics in month t 2.

Panel A of Table I presents time-series averages of monthly cross-sectional statistics for

all stocks. There are on average 1,635 firms each month and the total number of observations

is 902,308. Our sample of firms exhibits significant variation in market capitalization and the

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after portfolio formation. Monthly portfolio returns are calculated as equally-weighted or

value-weighted averages of the returns of all stocks in the portfolio.

Table II present the average returns of portfolios sorted by ivolliq alone and by

characteristics and ivolliq. The first panel contains the results for the univariate sort on

ivolliq using both equal- and value-weighted returns. According to the results, as ivolliq

increases the average returns also increase. The difference between the highest and lowest

ivolliq quintiles (P5-P1) is 31 (25) basis points per month for equally-weighted (value-

weighted) returns. The difference is significant with a t-statistic of 2.82 (2.73). We also

calculate the abnormal returns of the high-minus-low volatility of liquidity strategy (P5-P1)

using the Fama-French (1993) model augmented with the momentum factor (FF4). The

FF4 alpha is 31 basis points and significant at the 1% level. Similar results hold for value-

weighted returns. In untabulated results we also use the Fama-French (1993) 3-factor model

(FF3) and the FF3 model augmented with momentum and aggregate liquidity. The results

are qualitatively identical.15

In the second panel of Table II, we first sort stocks into three groups, S1, S2, and

S3, based on SIZE, where S1 represents small stocks and S3 represent large stocks. We

then sort stocks independently into quintiles based on ivolliq. The intersection of the two

sorts creates 15 portfolios which are held for a month after skipping a week after portfolio

formation. The results show that the difference between the extreme ivolliq quintiles, P5

and P1, decreases as firm size increases. While the difference between P5 and P1 for small

stocks is 32 basis points per month and significant, it decreases to an insignificant 7 basis

15The aggregate liquidity factor is constructed using 9 equally-weighted portfolios sorted on size andilliquidity. Every month, we sort stocks into 3 groups (Small, Medium, and Big) according to their end-of-previous-month market capitalization. Then we further sort stocks into three groups (High, Medium,and Low) according to their average monthly Amihud illiquidity. Each portfolio is rebalanced monthly. Theliquidity factor is the average return on three high illiquidity portfolios minus the average return on three lowilliquidity portfolios: ILL = 1/3(HighSmall+HighMedium+HighBig)1/3(LowSmall+LowMedium+LowBig).

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points per month for large stocks. However, the positive relation between the volatility of

liquidity and returns is not confined to the smallest size group, it is also present among

medium cap stocks. The FF4 alpha of the P5-P1 strategy is 42 basis points per month for

small stocks. Overall, the results suggest that the volatility of liquidity effect is strongest

among small stocks.

In the remainder of Table II, we perform additional double-sorts using control variables

that have been shown to affect returns: illiquidity (illiq), momentum (RET12M), book-to-

market (BM), and idiosyncratic volatility of returns (IVOL). The result suggest that the

average return of the high-minus-low volatility of liquidity strategy (P5-P1) is higher for

less liquid stocks (ILL3), value stocks (BM3), and stocks with higher idiosyncratic return

volatility (IV3). While past performance over the previous 12 months does not seem to

be related to the volatility of liquidity when we use raw returns, the effect appears to be

more pronounced among winners when we use the Fama-French model augmented with

momentum.

Overall, the portfolio approach suggests that the positive relation between idiosyncratic

volatility of liquidity and average returns is a separate effect which is different than the

well-documented size, illiquidity, momentum, book-to-market, and idiosyncratic volatility of

return effects. In addition, the ivolliq effect does not seem to be concentrated only among a

small portion of the sample of stocks. In untabulated tables, we also use monthly turnover,

monthly dollar volume, and contemporaneous monthly returns, and the ivolliq effect is

robust to controlling for these additional variables.

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B. Regression Approach

In this section we extend the portfolio analysis from before by performing cross-sectional

regressions. These regressions allow us to control for various other stock characteristics that

may potentially affect the relation between idiosyncratic volatility of liquidity and returns.

More precisely, we use Fama-MacBeth (1973) regressions in which the dependent variables

are excess stocks returns. We adjust the Fama-MacBeth t-statistics for heteroskedasticity

and autocorrelation of up to 8 lags. Asparouhova et al. (2010) show that microstructure-

induced noise in prices can lead to biases in empirical asset pricing tests. Following their

recommendations, we correct for this bias by estimating all regressions using weighted

least squares where the weights are based on past month gross returns. When we use the

Asparouhova et al. (2010) adjustment, we exclude the variable RET1M from the analysis.

However, our results are similar if we use ordinary least squares and include RET1M in the

regressions.

We use two specifications for the independent variables. In the first one, we transform

the independent variables into percentile ranks and then standardize the ranks with values

between zero and one. This rank transformation has two advantages: it makes the coefficient

interpretation more intuitive and comparable across variables, and it minimizes the effect of

outlier observations. In the second specification, we use the real values of the independent

variables.

The benchmark model we examine is:

rit+1 rit+1 = 0 + 1illiqit + 2Rrit + 3

Ccit + 4

Crit + 5

Rcit + it+1. (9)

This model is interesting since it corresponds to the empirical implementation of Acharya

and Pedersen (2005). While they test their model on liquidity-sorted portfolios, we use the

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cross-section of individual stocks. In addition, Acharya and Pedersen use full-sample liquidity

betas, while we use liquidity betas estimated with short-window regressions of daily data

within each month.

The results from estimating regression (9) are presented in Panel A of Table III, Columns

1 and 4. When we use the ranks of the independent variables in Column 1, the level of

illiquidity illiq is positively related to average returns, but the relation is not significant.

This result is consistent with the findings of Acharya and Pedersen (2005). Among the

three liquidity betas, the one that measures the covariance between an assets illiquidity

and the market return is significantly negatively priced. This is also in line with the results

in Acharya and Pedersen. None of the other betas in the model are significantly priced.

When we use the real values of the independent variables in Column 4, the significance ofRc

disappears. A possible explanation for this is the fact that the cross-sectional distribution of

Rc exhibits considerable skewness and the presence of outliers might influence the results.

The only risk factors in Acharya and Pedersen are the market return and market liquidity.

In order to take into account the exposure of asset returns to other risk factors that have been

shown to affect returns, we also look at risk-adjusted excess returns as dependent variables.

The risk-adjustment is based on the Fama-French model augmented with a momentum factor

following Brennan, Chordia, and Subrahmanyam (1998). The factor loadings with respect

to the risk model are estimated using 60-month rolling windows.16

The results for model (9) using risk-adjusted returns are presented in Panel B of Table

III, Columns 1 and 4. The results are largely consistent with the ones reported in Panel

A. When ranked independent variables are used, the beta that measures commonality in

liquidity also becomes significant. Overall, the results from estimating model (9) suggest

16We achieve similar results if we use Dimson (1979) betas with one lag. We also use the Fama-Frenchthree-factor model to adjust for risk and obtain similar results.

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that at least one type of systematic liquidity risk is priced in the cross-section of stocks.

Next, we directly test whether idiosyncratic volatility of liquidity, ivolliq, is priced in the

presence of systematic liquidity risk. We examine the following specification:

rit+1 rit+1 = 0 +1illiqit + 2Rrit + 3

Ccit + 4

Crit + 5

Rcit

+6Covt(rit+1, cit+1) + 7IVOLit + 8ivolliqit + it+1. (10)

We include the variable Cov(r, illiq) since previous studies have shown that returns and

liquidity tend to move together (e.g., Amihud (2002)). In addition, we include the

idiosyncratic volatility of returns, IVOL to account for the fact that the measure of liquidity

depends on returns. If systematic liquidity risk is the only part of total liquidity variation

that affects returns, then the coefficient 8 should be equal to zero.

The results from estimating (10) are presented in Panel A of table III, Columns 2 and 5.

When the independent variables are ranked in Column 2, the null hypothesis that 8 = 0 is

rejected. The coefficient in front ofivolliq is 0.22 implying that an increase in idiosyncratic

volatility of liquidity from the 1st to the 99th percentile leads to an increase of 22 basis points

per month in expected returns. The level of illiquidity and idiosyncratic return volatility are

also significantly priced. Systematic liquidity risk, as measured by the covariance between

individual liquidity and the market return, is also significantly related to returns. Its price of

risk is -0.20. Therefore, the price of risk for idiosyncratic liquidity risk (0.22) is of a similar

magnitude as the price of systematic liquidity risk. The market beta of the stocks is also

significantly priced. Similar conclusions hold in Column 5 in which the real values of the

independent variables are used.

Under additional risk-adjustment, in Panel B of Table III, Columns 2 and 5, the

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idiosyncratic volatility of liquidity remains significantly positively correlated with expected

returns. Note that its price of risk in Column 2 of Panel B, 0.21, is very close to the one

reported in Panel A where no further risk-adjustment is used. Therefore, it is not likely

that our measure of idiosyncratic liquidity risk is simply a proxy for assets exposure to the

Fama-French and momentum factors.

To test the robustness of these results, we introduce further control variables in the cross-

sectional Fama-MacBeth regression (10). These variables are stock-specific characteristics

described in Section I.D. The results are presented in Panel A of Table III, Columns 3 and

6. When we use the ranks of the independent variables, the price of idiosyncratic volatility

risk is still positive and significant. Its magnitude increases to 27 basis points per month.

Idiosyncratic return volatility and market beta are also significant, and only one type of

systematic liquidity risk is priced, Rc . Among the other stock characteristics, book-to-

market, past returns, and skewness have significant coefficients.

The illiquidity level, illiq, is not significant in Column 3 of Panel A. This lack of

significance may be due to a multicollinearity problem generated by the high correlation

between illiq and SIZE. In untabulated results, we exclude SIZE from the model and

the coefficient on illiq becomes significantly positive. When we exclude illiq instead,

the coefficient on SIZE is negative and significant. The relation between returns and

idiosyncratic liquidity risk is not affected by these modifications.

Idiosyncratic liquidity risk is significantly positively priced when we use the real value of

the independent variables in Column 5 of Panel A, and risk-adjusted returns in Columns 3

and 5 of Panel B.

Overall, the results in Table III suggest that idiosyncratic volatility of liquidity, ivolliq

is significantly positively related to expected returns. This relation persists over and

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above the positive correlation between the level of illiquidity and returns, and the relation

between systematic liquidity risk and returns. The results imply that investors want to be

compensated for holding stocks with high idiosyncratic liquidity variation. This might be

the case since it is not always possible for investors to time their trades according to the

liquidity state of their stocks.

C. Regression Approach within Size and Illiquidity Groups

As mentioned earlier, the high correlation between size and illiquidity may cause potential

multicollinearity problems and bias our results. In this section we perform additional tests

to ensure that the main results are not driven by this correlation. Every month we sort

stocks into three groups based on size and run Fama-MacBeth regressions within each size

group. This way we control for size, allowing illiquidity to vary within each size group. For

the sake of brevity we report the results using the ranks of the independent variables. The

results using the real values of the independent variables are similar and they are available

upon request.

In Panel A of Table IV, we report Fama-MacBeth regressions within each size category,

with excess returns as the dependent variables. We examine the specification that includes all

control variables. The results show that the positive relation between ivolliq and returns is

stronger among small stocks. The price of ivolliq risk in the small category is 0.48, implying

that an increase in ivolliq from the 1st to the 99th percentile leads to an increase in expected

returns of 48 basis points per month. Idiosyncratic liquidity risk is significantly positively

related to average returns for medium stocks as well. The relation is not significant for large

stocks. A possible explanation for this finding might be that smaller stocks have low average

levels of liquidity and therefore, a high volatility of the liquidity distribution implies that

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investors in illiquid stocks may face even lower levels of liquidity at a point when they need

to trade. Larger stocks, on the other hand, may expose investors to this risk to a lesser

extent since their liquidity distributions have higher means.

Panel A of Table IV also shows that the level of illiquidity is significant and positive

in the small and medium size groups. The illiquidity effect is stronger for smaller stocks.

Systematic liquidity risk does not appear to be priced. The only two stock characteristics

that are systematically priced across all categories of stocks are book-to-market and past

returns.

In Panel B of Table IV, the Fama-MacBeth regressions within each size category are

performed using risk-adjusted returns. For small stocks, the coefficient on ivolliq is positive

and significant. Its magnitude decreases to 0.35. The significance level of ivolliq decreases as

size increases, but it remains positive among the largest stocks. Overall, the results suggest

that, after controlling for the size effect, both the mean and the second moment of illiquidity

are positively related to expected stock returns. Therefore, it is not likely that our previous

findings are driven by the high multicollinearity between size and illiquidity.

D. Interpretation of the Positive Price of Risk for ivolliq

In the context of existing models about liquidity risk, the pricing of idiosyncratic liquidity

variation presents a puzzle. If investors hold well-diversified portfolios, idiosyncratic liquidity

volatility should not affect expected returns. This is the case since high variation in liquidity

implies that some stocks may become more liquid and others less liquid at a time when

the investor needs to raise cash. This suggests that the investors will be able to raise cash

by choosing to liquidate the more liquid securities. Therefore, in sufficiently diversified

portfolios, idiosyncratic liquidity variance is likely to be diversified away.

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However, several studies document that investors might not be as well-diversified as

theory would suggest. For example, Barber and Odean (2000) report that a typical individual

investor holds a portfolio with only four stocks. Goetzmann and Kumar (2008) analyze the

diversification choices of more than 60,000 individual investors at a large U.S. discount

brokerage house during a six-year period (1991 to 1996). They show that more than 25%

of the investor portfolios contain only one stock, over half of the investor portfolios contain

no more than three stocks, and less than 10% of the investor portfolios contain more than

10 stocks. Additionally, using data from the Survey of Consumer Finances, Polkovnichenko

(2005) provides evidence of under-diversification among U.S. households.

If investors are not well-diversified, idiosyncratic volatility of liquidity might become

important. If an investor faces an immediate liquidity need due to exogenous cash

needs, margin calls, dealer inventory rebalancing, forced liquidations, or standard portfolio

rebalancing, he needs to unwind his positions in a short period of time. In case of such a

liquidity need the investor may not be able to wait for periods of high liquidity to sell his

stock, and thus the level of liquidity of the stock on the day the investor closes his position is

important. This effect will be reinforced if investors are subject to borrowing constraints and

cannot borrow easily in case of an urgent consumption need (e.g., see Huang (2003)). The

higher a stocks idiosyncratic volatility of liquidity, the more likely it is that the stock will

be very illiquid at a time when it is traded. Therefore, the investor might end up unwinding

his position at a low level of liquidity for the stock, which induces a significant loss of wealth

due to a large price impact of trade. Thus, investors will require a compensation for being

exposed to this risk. All else equal, a risk-averse investor may be willing to pay a higher

price for a stock that has a lower risk of becoming less liquid at the time of trading, i.e., a

stock with a low idiosyncratic liquidity variation.

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Even if investors hold well-diversified portfolios, idiosyncratic volatility of liquidity may

become important. When the investor faces an immediate consumption need, it may not be

optimal to sell the more liquid securities to meet his short term obligations. This is the case

if the investor expects further liquidity shocks in the future. Brown, Carlin, and Lobo (2010)

point out that selling the more liquid assets first will limit the immediate loss in portfolio

value. However, the remaining portfolio will be more exposed to adverse conditions in the

future. Selling the less liquid assets first will result in a portfolio that is less exposed to

future liquidity shocks. However, this could result in unnecessary loss in portfolio value if

the subsequent liquidity shock does not materialize. Brown, Carlin, and Lobo (2010) obtain

a theoretical solution for this tradeoff and show that if the expected future liquidity shock

is sufficiently large, the investor would prefer to retain more of the assets with low price

impact in order to hedge against future conditions. Therefore, the investor might end up

selling stocks that become very illiquid at the time of trading due to high volatility in their

liquidity.

III. Robustness Tests

A. Alternative Measurement Periods for ivolliq and illiq

So far our results are based on idiosyncratic volatility of liquidity measured using daily

data within a month. However, since the Amihud ratio includes returns, it is possible that our

measure of ivolliq captures short-term return autocorrelations that cannot be adjusted for

with our control variables. In addition, it might be possible to obtain more precise estimates

of idiosyncratic liquidity variation by using a larger sample of daily Amihud ratios. Therefore,

in this section we investigate whether our results are robust to alternative measurement

periods for our key variable, ivolliq. Instead of using only one month of daily data, we use

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three and six months of daily Amihud ratios to compute the ivolliq measure. The measure

in each case is denoted as ivolliq3M and ivolliq6M, respectively.

Table V presents the results using the regression specification that includes the full set of

control variables. In Panel A of Table V, using excess returns, the coefficient on ivolliq3M is

significantly positive. Its magnitude is higher than the magnitude of the coefficient in front

ofivolliq in Panel A of Table III, Column 3. When the ivolliq6M measure is used, the price

of risk for idiosyncratic volatility of liquidity increases even further to 0.37. These results

suggest that the attenuation bias in the estimation of the ivolliq price of risk is smaller when

the measure of idiosyncratic liquidity risk is based on more observations and is, therefore,

more precise.

The results in Panel A of Table V also show that the covariance between individual

liquidity and the market return is significantly priced. So is market beta. The other stock

characteristics consistently related to returns are idiosyncratic return volatility, book-to-

market, past returns, and skewness. Similar results hold for risk-adjusted returns.

Since the individual illiquidity measure for each stock is also estimated from daily data,

the volatility of liquidity might capture the imprecision in estimating the mean level of

illiquidity. Therefore, we use an alternative measure of illiquidity which is more precisely

estimated to test the robustness of our results. Namely, for each stock we compute its

illiquidity by using daily Amihud ratios within the last one year. This measure is denoted

by illiq1Y. Our objective is to test whether idiosyncratic volatility of liquidity, ivolliq, is

still significant in the presence of illiq1Y. Table VI presents the results using the full set

of control variables. When the ranks of the independent variables are used, the coefficient

in front of illiq1Y is significantly positive. This is in contrast to column 3 in Table III in

which illiq is not significantly priced. In both cases, the idiosyncratic volatility of liquidity is

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significantly positively priced and the magnitude of the ivolliq coefficient is similar. Similar

results hold for the real values of the independent variables and for risk-adjusted returns.

Therefore, using a more precise measure of illiquidity affects the pricing of the mean level

of illiquidity. However, the pricing of idiosyncratic volatility of liquidity is not affected.

Therefore, the effect of ivolliq on expected returns seems to be different than the effect of

the mean level of illiquidity.

B. Expected Idiosyncratic Volatility of Liquidity

We are interested in the relation between expected returns and ex-ante idiosyncratic

volatility of liquidity. However, it is not straightforward to test this relation empirically.

Our analysis so far uses lagged idiosyncratic volatility of liquidity as a proxy for the ex-

ante variable. If the volatility of liquidity is time-varying, lagged volatility of liquidity alone

may not adequately forecast expected volatility of liquidity. Therefore, we estimate a cross-

sectional model of expected idiosyncratic volatility of liquidity that uses additional predictive

variables. Specifically, we run a cross-sectional regression of ivolliq, measured over the same

holding period as returns, on firm characteristics measured at the end of the previous month.

In the cross-sectional regressions we use two lags of ivolliq, SIZE, BM, IVOL, RET1M,

RET12M, illiq, and TURN. Then we use the fitted values of ivolliq from the cross-sectional

regressions as independent variables in the subsequent Fama-MacBeth regressions.

The results are presented in Table VII. The predicted value of ivolliq, Fivolliq, is

significantly positively related to average returns in all specifications. Therefore, our main

results are robust to this alternative estimate of the volatility of liquidity.

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C. Alternative Specification of Residual Liquidity

So far we calculate idiosyncratic volatility of liquidity using the residuals of a model that

includes aggregate liquidity and the market return. However, if variation in a stocks liquidity

is related to factors missing from the model, such as HML and SMB, then our results might

capture liquidity covariance with these additional factors rather than idiosyncratic volatility

of liquidity. To address this issue, we include the Fama-French factors HML and SMB in

the model in equation (5) and estimate ivolliq accordingly. In untabulated results, we also

add the momentum factor and obtain similar results.

The results are presented in table VIII. The coefficient on ivolliq is significant and positive

in all specifications. In addition, its magnitude is almost identical to the ones reported in

Table III. Overall, the results suggest that using this alternative specification in calculating

ivolliq does not affect the conclusion of our tests.

D. Volatility of Liquidity Effect across Business Cycles

In this section, we split the sample into good and bad states of the business cycle. The

motivation for doing this comes from recent theoretical research that relates crisis periods

to declines in asset liquidity. Several models predict that sudden liquidity dry-ups may

occur due to demand effects such as market participants engaging in panic selling, supply

effects such as financial intermediaries not being able to provide liquidity, or both.17 These

models predict that the demand for liquidity increases in bad times as investors liquidate

their positions across many assets. At the same time, the supply of liquidity decreases in badtimes as liquidity providers hit their funding constraints. In addition, borrowing constraints

are tighter in bad times. Investors, who cannot borrow easily in case of an emergency

17See Gromb and Vayanos (2002), Morris and Shin (2004), Vayanos (2004), Garleanu and Pedersen (2007),and Brunnermeier and Pedersen (2009), among others.

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consumption need, would have to liquidate their positions. As a result, the uncertainty

associated with an assets liquidity is likely to increase around crisis periods and become a

stronger concern for investors. Therefore, we conjecture that the idiosyncratic volatility of

liquidity effect will be stronger during bad economic times.

We use the growth rate of industrial production as an indicator of good or bad economic

times. The advantage of this variable is that it is a contemporaneous indicator of the

business cycle. Data on the level of industrial production comes from the website of the

Federal Reserve Bank of St. Louis. Industrial production growth (IND) is defined as the

first difference in the log of industrial production. To capture crisis periods, we split the

sample into two parts: one corresponding to the 10% lowest observations of IND (bad

times), the other corresponding to the rest of the observations. We compute the average

return of the equally-weighted high-minus-low ivolliq strategy (P5-P1) within each sub-

sample. Untabulated results show that the average P5-P1 return is 1.38% per month in

bad times and 0.19% per month the rest of the time. The difference between the two is

statistically significant. If we define bad times as the 25% lowest observations ofIND, the

average P5-P1 return is 0.78% in bad times and 0.19% the rest of the time. The results are

similar when we use risk-adjusted returns. Therefore, the results suggest that the expected

return premium for stocks with high ivolliq is higher in bad times and increases with the

severity of the crisis period.18

E. Additional Robustness Checks

In this section we address some remaining concerns about the main results. First, since

our findings are stronger among small stocks, it might be the case that the results are

18We obtain similar results by using the default premium as an indicator of good/bad times. The defaultpremium is defined as the spread in yields between a BAA and a AAA bond. The default premium is directlyrelated to the tightness of borrowing constraints.

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driven by the January effect documented by Keim (1983) (see also Tinic and West (1986),

Eleswarapu and Reinganum (1993), and Amihud (2002)). In separate regressions and sorting

analysis we control for the January effect and find similar results.

Second, the idiosyncratic volatility of liquidity measure ivolliq might capture an

interaction effect between past returns and trading volume. For example, Cooper (1999) and

Lee and Swaminathan (2001) document that return continuations accentuate with volume,

while Avramov et al. (2006) show that the short-term return reversals accentuate with

volume. Accordingly, we include an interaction term between trading volume and past

returns and trading volume and contemporaneous returns in the Fama-MacBeth regressions.

We find that the coefficient on ivolliq remains positive and significant.

Third, since Amihuds measure of illiquidity includes the absolute value of the return in

the numerator, the volatility of this measure might be correlated with the kurtosis of stock

returns. When we include kurtosis in our analysis the coefficient on ivolliq is still significant

and positive.

Finally, to ensure that our results are not driven by a non-linear relation between

illiquidity and future returns, we include illiq-squared in the regressions and find similar

results.

IV. Conclusion

In this paper we find that the idiosyncratic volatility of liquidity is positively related

to future returns. The positive correlation between idiosyncratic liquidity risk and expected

returns suggests that risk averse investors require a risk premium for holding stocks that have

high idiosyncratic variation in liquidity. Our results are robust to various control variables,

systematic liquidity risk, and different sub-periods. Higher variation in liquidity implies that

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a stock may become illiquid with higher probability at a time when it is traded. This is

important for investors who may face an immediate liquidity need due to exogenous cash

needs, margin calls, dealer inventory rebalancing, or forced liquidations. In case of such

liquidations, the investor may not be able to time the market by waiting for periods of high

liquidity and thus, the level of liquidity on the day of the liquidity need is important.

Our results are puzzling in light of recent models that suggest that only systematic

liquidity risk is priced in the cross-section of stock returns. Idiosyncratic liquidity risk may

proxy for an omitted systematic source of liquidity risk. Additional work is necessary to

identify such a source. Alternatively, the pricing of idiosyncratic liquidity volatility may

indicate that not all investors are well-diversified. Finally, future theoretical work might

identify a mechanism that allows for the pricing of idiosyncratic liquidity risk.

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Table I: Summary Statistics

This table presents time-series averages of cross-sectional summary statistics (Panel A) and monthly cross-sectional correlations

(Panel B) for various stock characteristics. The sample consists of common stocks listed on AMEX and NYSE from January1964 to December 2010. illiq is the Amihud measure of illiquidity, ivolliq is the coefficient of variation of liquidity calculated

as standard deviation of residual liquidity, obtained by regressing daily stock illiquidity on daily market illiquidity and daily

market return over a month, normalized by mean level of liquidity, SIZE is end-of-month price times shares outstanding (in

billion dollars), BM is the book-to-market ratio, I V O L is the standard deviation of the residuals from the Fama-French model,

TURN is the turnover ratio measured by the number of shares traded divided by the number of shares outstanding, SKEW

is the monthly skewness of daily returns, RET12M is the cumulative return over the past twelve months, RET1M is the

return during the previous month. Rc is the coefficient estimate RMd rfd and Cc is the coefficient estimate on CMd

obtained by regressing daily firm-level illiquidity on daily excess market return and daily change in market illiquidity. Rr is the

coefficient estimate on RMd and Cr is the coefficient estimate on CMd obtained by regressing daily firm-level excess return on

daily excess market return and daily change in market illiquidity. COV (r, c) is the covariance between stock level daily return

and daily illiquidity over a month. Panel In Panel B, we apply log transformations to SIZE, BM, and TURN. Spearman

correlations are reported above the diagonal and Pearson correlations are reported below the diagonal.

A: Descriptive Statistics

MEAN MEDIAN STD P5 P95

ivolliq 1.009 0.926 0.36 0.61 1.70illiq 1.551 0.173 5.92 0.01 6.78SIZE 2.277 0.404 7.94 0.02 9.01BM 0.912 0.725 0.93 0.18 2.19TURN 0.710 0.512 0.80 0.09 1.94I V O L 0.020 0.017 0.01 0.01 0.04SKEW 0.264 0.228 0.87 -1.05 1.71RET1M 0.014 0.006 0.12 -0.14 0.19RET12M 0.166 0.096 0.49 -0.39 0.93Rc 1.858 -0.060 378 -102 102

Cc 0.470 0.004 9.01 -1.20 1.91Rr 0.859 0.779 1.07 -0.63 2.63

Cr 0.000 0.000 0.03 -0.04 0.04CO V(r, c) -0.003 0.000 0.28 -0.03 0.02

B: Cross-Sectional Pearson (lower diag.) and Spearman (upper diag.) Correlations

ivolliq illiq SIZE BM TU RN IV OL SKEW RET12M RET1M Rc Cc

Rr

Cr CO V(r, c)

ivolliq 1 0.48 -0.43 0.19 -0.25 0.10 0.03 -0.11 -0.02 0.00 0.03 -0.23 -0.01 -0.06illiq 0.27 1 -0.94 0.31 -0.37 0.48 0.07 -0.19 -0.06 -0.03 0.08 -0.16 0.00 -0.10SIZE -0.42 -0.38 1 -0.32 0.14 -0.53 -0.08 0.16 0.06 0.03 -0.07 0.13 0.00 0.11BM 0.16 0.14 -0.30 1 -0.15 0.06 -0.02 -0.34 -0.11 -0.01 0.02 -0.11 0.01 -0.04TURN -0.24 -0.19 0.13 -0.15 1 0.27 0.07 0.09 0.09 -0.01 -0.02 0.30 0.01 -0.01I V O L 0.15 0.38 -0.49 0.07 0.23 1 0.18 -0.15 0.04 -0.03 0.05 0.17 0.01 -0.07SKEW 0.04 0.02 -0.08 -0.02 0.06 0.19 1 -0.03 0.31 0.05 0.01 0.04 0.00 0.15RET12M -0.09 -0.11 0.09 -0.33 0.14 -0.08 -0.02 1 0.03 0.00 -0.02 0.04 -0.01 0.00RET1M 0.02 0 .00 0.02 -0.10 0 .12 0.17 0.34 0.01 1 0.10 0.00 0.04 -0.01 0.23Rc 0.01 0 .04 -0.01 0.00 0.00 0.02 0.00 0.00 0.01 1 0.05 -0.03 0.01 0.21Cc 0.12 0.52 -0.10 0.04 -0.05 0.14 0.01 -0.04 0.01 0.07 1 0.00 0.00 -0.01Rr -0.19 -0.08 0.11 -0.10 0.28 0.11 0.04 0.06 0.03 -0.01 -0.02 1 0.05 0.00Cr -0.01 -0.01 0.00 0 .01 0.01 0.00 0.00 -0.01 -0.01 0.00 -0.01 0.05 1 0.02COV (r, c) -0.03 -0.09 0.03 -0.01 0.00 -0.04 0.00 0.01 0.00 0.05 -0.06 0.00 0.06 1

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Table II: Average Portfolio Returns

This table presents average returns (in % form) for various portfolios. The first set of portfolios involves a

single sort on the volatility of liquidity, ivolliq. The other sets of portfolios involve a double sort on a stockcharacteristic (size, illiquidity, momentum, book-to-market and idiosyncratic volatility ) and the volatility

of liquidity. The stock characteristics and volatility of liquidity are computed as described in Table 1. The

sample consists of common stocks listed on AMEX and NYSE from January 1964 to December 2010. The

portfolios are rebalanced every month and we skip a week between portfolio formation and the holding

period. The table also presents the average returns of the high-minus-low volatility of liquidity strategy, P5-

P1, within each sort, together with the corresponding Fama-French four factor alphas (FF4). Newey-West

t-statistics are shown below the average returns.

Mean Portfolio Returns

All Stocks Size IlliquidityEW VW S1 S2 S3 IL1 IL2 IL3

P1 1.05 0.84 1.14 1.10 0.96 0.99 1.07 1.15P2 1.15 0.86 1.30 1.23 1.00 1.03 1.19 1.32P3 1.24 0.96 1.29 1.33 1.09 1.13 1.35 1.22P4 1.31 1.03 1.44 1.37 1.07 1.09 1.33 1.44P5 1.36 1.16 1.46 1.39 1.03 1.20 1.35 1.45

P5 P1 0.31 0.31 0.32 0.29 0.07 0.20 0.28 0.30t-statistic 2.82 2.73 2.05 2.81 0.74 2.26 2.58 2.27

FF4 alphas 0.31 0.25 0.42 0.41 0.10 0.27 0.39 0.39

t-statistic 3.02 2.86 2.22 4.39 1.09 3.13 4.06 2.54

Momentum Book-to-Market Idiosyncratic Vol.

M1 M2 M3 BM1 BM2 BM3 IV1 IV2 IV3

P1 0.71 1.04 1.31 0.90 1.05 1.32 1.08 1.20 0.88P2 0.87 1.08 1.44 0.97 1.14 1.43 1.11 1.31 1.07P3 0.88 1.25 1.55 1.10 1.19 1.41 1.24 1.37 1.09P4 1.09 1.25 1.61 1.10 1.21 1.56 1.19 1.42 1.31P5 1.07 1.47 1.74 1.00 1.32 1.59 1.24 1.50 1.31

P5 P1 0.36 0.43 0.43 0.11 0.28 0.26 0.16 0.31 0.44t-statistic 2.30 3.97 3.89 0.89 2.33 2.01 1.89 2.64 2.69

FF4 alphas 0.12 0.35 0.57 0.10 0.29 0.29 0.16 0.36 0.42t-statistic 0.78 3.49 5.58 0.93 2.47 2.03 2.09 3.42 2.39

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Table I II: Fama-MacBeth Regression Estimates using Individual Security Data

This table presents the results from Fama-MacBeth regressions in which the dependent variables are stock returns and the

independent variables are various stock characteristics. The sample consists of common stocks listed on AMEX and NYSE

from January 1964 to December 2010. The stock characteristics are defined in Table 1. In Panel A the dependent variables are

excess stock returns, while in Panel B the dependent variables are risk-adjusted stock returns. Risk-adjustment is based on the

Fama-French 3-factor model augmented with a momentum factor. In both panels the independent variables are various stock

characteristics in both percentile ranks (standardized between zero and one) and real values. When real values of independent

variables used we apply log transformations to SIZE, BM, and TURN. To minimize microstructure issues, one week is skipped

between measurement of the independent and dependent variables and all models are estimated using Weighted Least Squares

(weight equals 1 + gross lagged stock return). Coefficient estimates are multiplied by 100. Newey-West t-statistics are reported

below the coefficient estimates.

A: Excess Returns B: Risk-adjusted ReturnsRanks Ranks Ranks Real Real Real Ranks Ranks Ranks Real Real Real

ivolliq 0.22 0.27 0.32 0.17 0.21 0.26 0.29 0.153.48 4.49 3.99 2.71 3.12 3.88 4.06 2.32

illiq 0.41 0.58 0.12 0.04 0.06 0.04 0.21 0.54 -0.50 0.05 0.08 0.051.36 2.60 0.37 1.29 2.04 2.61 1.13 3.03 -1.11 2.08 3.16 2.83

Cov(r, c) -0.05 0.01 1.37 1.31 -0.10 -0.02 1.77 1.56-0.75 0.18 1.16 1.06 -1.23 -0.25 1.27 1.18

IVOL -0.59 -0.50 -19.54 -27.12 -0.79 -0.50 -25.66 -28.90-2.06 -2.92 -3.31 -7.17 -4.80 -3.83 -6.82 -7.84

Rc -0.20 -0.20 -0.10 0.00 0.00 0.00 -0.30 -0.20 -0.20 0.00 0.00 0.00-3.03 -2.65 -2.49 0.78 0.39 0.39 -3.11 -2.91 -2.69 -0.03 -0.08 0.04

Rr 0.24 0.44 0.25 0.09 0.13 0.09 0.02 0.29 0.17 0.03 0.09 0.071.00 2.50 2.37 1.08 1.95 2.18 0.14 1.94 1.45 0.39 1.39 1.38

Cr 0.12 0.09 0.08 2.76 1.70 1.64 0.16 0.15 0.12 2.11 2.18 1.681.55 1.28 1.19 1.21 0.95 1.02 1.45 1.38 1.16 1.05 1.19 0.95

Cc 0.09 0.07 0.09 -0.01 -0.01 0.00 0.13 0.12 0.12 0 .00 0.00 0.001.36 1.09 1.61 -1.29 -1.13 -0.56 2.06 1.82 1.92 -0.29 -0.47 -0.21

SIZE -0.69 -0.16 -1.09 -0.14-1.91 -3.61 -2.44 -6.64

BM 0.69 0.23 0.54 0.173.89 3.60 4.75 4.17

RET12M 1.34 0.68 0.98 0.495.18 3.36 3.92 3.07

SKEW -0.30 -0.07 -0.50 -0.12

-4.75 -2.76 -5.66 -3.53TURN 0.20 0.10 -0.10 0.04

1.00 1.92 -0.66 0.78Adj.R2 0.03 0.04 0.07 0.03 0.05 0.07 0.01 0.02 0.03 0.02 0.03 0.04

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Table IV: Fama-MacBeth Regression Estimates by Size Group

This table presents the results from Fama-MacBeth regressions within three separate size groups. The sample consists of

common stocks listed on AMEX and NYSE from January 1964 to December 2010. The stock characteristics are defined in

Table 1. In Panel A the dependent variables are excess stock returns, while in Panel B the dependent variables are risk-adjusted

stock returns. Risk-adjustment is based on the Fama-French 3-factor model augmented with a momentum factor. In both

panels the independent variables are various stock characteristics in percentile ranks (standardized between zero and one). To

minimize microstructure issues, one week is skipped between measurement of the independent and dependent variables and all

models are estimated using Weighted Least Squares (weight equals 1 + gross lagged stock return). Coefficient estimates are

multiplied by 100. Newey-West t-statistics are reported below the coefficient estimates.

A: Excess Returns B: Risk-adjusted ReturnsSmall Medium Large Small Medium Large

ivolliq 0.48 0.19 0.10 0.35 0.19 0.153.81 2.13 1.38 2.79 1.89 1.85

illiq 1.41 0.73 0.60 1.19 0.47 0.262.33 2.15 1.80 2.02 1.55 1.00

COV(r, c) 0.21 -0.20 -0.69 0.21 -0.30 -0.892.42 -1.45 -3.36 1.99 -1.71 -4.06

IVOL -0.79 -0.59 -0.40 -0.79 -0.69 -0.50-3.35 -3.86 -2.27 -3.74 -4.51 -3.03

Rc -0.10 -0.10 -0.08 -0.20 -0.20 -0.20-1.55 -1.19 -0.54 -1.64 -1.84 -0.86

Rr 0.13 0.35 0.25 -0.01 0.30 0.16

1.11 2.55 1.70 -0.05 1.73 0.99Cr 0.03 0.18 0.06 0.08 0.15 0.13

0.39 1.79 0.62 0.67 1.21 1.02Cc 0.08 0.06 0.05 0.09 0.13 0.13

0.96 0.76 0.46 0.98 1.42 0.93BM 1.01 0.60 0.42 0.93 0.46 0.23

4.35 3.07 2.26 4.47 3.17 1.52RET12M 1.99 1.13 0.87 1.68 0.78 0.44

7.47 3.87 3.12 6.52 2.75 1.53SKEW -0.59 -0.10 0.01 -0.79 -0.30 -0.02

-4.45 -1.77 0.12 -5.51 -3.35 -0.28TURN 0.59 0.44 0.42 0.35 0.20 0.15

1.77 2.04 1.99 0.97 1.03 0.93Adj.R2 0.04 0.06 0.10 0.02 0.03 0.05

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Table V: Fama MacBeth Regression Estimates: Using Different Measurement

Periods for the Volatility of Liquidity

This table presents the results from Fama-MacBeth regressions in which the dependent variables are stock returns and the

independent variables are various stock characteristics. The sample consists of common stocks listed on AMEX and NYSE

from January 1964 to December 2010. The stock characteristics are defined in Table 1. ivolliq3M is ivolliq measured over

3 months, while ivolliq6M is measured over 6 months. In Panel A the dependent variables are excess stock returns, while

in Panel B the dependent variables are risk-adjusted stock returns. Risk-adjustment is based on the Fama-French 3-factor

model augmented with a momentum factor. In both panels the independent variables are various stock characteristics in both

percentile ranks (standardized between zero and one) and real values. When real values of independent variables used we apply

log transformations to SIZE, BM, and TURN. To minimize microstructure issues, one week is skipped between measurement

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