Is Liquidity Risk Priced in the Corporate Bond Market?

Chris Downing, Shane Underwood, and Yuhang Xing∗

June 11, 2005Preliminary and Incomplete: Not for Citation

Abstract

This paper employs a new, comprehensive dataset on corporate bond transaction prices and

volumes to test whether liquidity is a priced factor in bond returns. We show in the context

of a linear APT-style factor model that liquidity is an important risk factor in bond returns;

this result appears to be robust to the liquidity proxy that is employed. Our results indicate

that liquidity likely accounts for some, but not all, of the discrepancy between the bond prices

produced by structural models and observed bond prices. More generally, the results lend

further support to the notion that market liquidity is a pervasive risk factor affecting all asset

returns.

Key words: Liquidity; Asset Pricing; Corporate Bonds

Classification: G12

Please address correspondence to the authors at Jones Graduate School of Management, Rice University, 6100

Main Street, Houston, TX 77005.

Is Liquidity Risk Priced in the Corporate Bond Market?

Abstract

This paper employs a new, comprehensive dataset on corporate bond transaction prices and

volumes to test whether liquidity is a priced factor in bond returns. We show in the context

of a linear APT-style factor model that liquidity is an important risk factor in bond returns;

this result appears to be robust to the liquidity proxy that is employed. Our results indicate

that liquidity likely accounts for some, but not all, of the discrepancy between the bond prices

produced by structural models and observed bond prices. More generally, the results lend

further support to the notion that market liquidity is a pervasive risk factor affecting all asset

returns.

1 Introduction

Liquidity, as the term is used in finance, generally refers to the ability to trade large volumes of

assets quickly, at low cost, with little price impact. This much seems relatively non-controversial.

What is far less certain is whether liquidity is priced—that is, are the cross-sectional differences in

asset returns related to the sensitivities of these returns to movements in market liquidity? In this

paper, we take up this question with respect to corporate bond prices.

Why should we care if corporate bond returns incorporate a compensation for liquidity? There

are good reasons to care, in fact. First, it is well known in the finance literature that structural

models of corporate bond prices based on default risk alone tend to seriously miprice long-term

corporate bonds. Recent empirical studies, including Lyden and Saraniti (2000) and Eom, Helwege

and Huang (2002), find that extensions of the basic Merton (1974) structural model, such as Leland

and Toft (1996) and Collin-Dufresne and Goldstein (2001), significantly over-price bonds issued

by large, well capitalized firms, and under-price bonds issued by risky firms.

A number of papers have offered explanations for why structural models based only on default

risk might fail to produce realistic bond prices. For example, Duffie and Lando (2001) argues

that asymmetric information about default risk between issuers and investors boosts observed risk

spreads, and Giesecke (2003) suggests that uncertainty about model parameters might also play a

role. Recently, an increasing amount of research effort has been devoted to understanding the role

that liquidity plays in determining bond prices. For example, Chacko (2005) finds that liquidity risk

is an important factor determining bond prices in the context of an APT-style linear factor model.

de Jong and Driessen (2004) employ a similar APT-based approach and find that fluctuations in

Treasury bond and equity market liquidity affect corporate bond returns. Chordia, Sarkar and

Subrahmanyam (2005) provide evidence of a link between money flows and transactions liquidity.

More broadly, our investigation of liquidity in the corporate bond market adds a new dimension

to the well-established literature on liquidity and equity returns (see, for example, Amihud and

Mendelson (1991), Brennan and Subrahmanyam (1996), Brennan, Chordia and Subrahmanyam

(1998), and Datar, Naik and Radcliffe (1998)). An examination of bond market liquidity potentially

1

has much to add to this line of inquiry because, in general, bonds are orders of magnitude less

frequently traded than equities, transaction costs are much higher, and all of the available evidence

suggests bonds are more narrowly held by investors. Hence for bonds all of the key dimensions of

liquidity—ease of trade, transaction costs, and price impact—are first-order concerns. Based on

these observations, it is logical to expect that any effect of market liquidity on asset returns ought

to be stronger in the bond market. Given this, it seems clear that the study of liquidity began in the

equity market at least in part due to data availability: until fairly recently, comprehensive data on

bond market transactions were not available.

In this paper, we exploit newly available data on corporate bond transactions—the “TRACE”

data collected and publicly disseminated by the NASD—to assess the importance of liquidity in the

determination of bond prices. These data have the advantage that they are fairly comprehensive—

all NASD members are required to report their transactions in “TRACE eligible” bonds each day—

allowing us to form a reasonably accurate picture of daily transaction activity on the corporate

bond market. However, the dataset is limited in the time-series dimension because the TRACE

system was not brought on-line until fairly recently. Throughout the discussion, we will make

every attempt to be clear-eyed about the caveats that this limitation induces on our results.

We proxy for the liquidity of a bond on a given day with the absolute daily price change

in a given bond, less the price change in an equivalent maturity Treasury bond, divided by the

dollar volume transacted over the day—a version of the measure of price impact introduced by

Amihud (2002). The difference here is that we subtract the price change in Treasury bonds in

order to remove, at least to a first approximation, the effects of term structure movements on the

illiquidity proxy, a “correction” that reduces the correlation of the variable with some of the other

key variables in our analysis. Since this measure has the advantage that it is based on the widely

used Amihud (2002) measure, thus enhancing the comparability of our results to other studies, we

adopt it as our primary illiquidity proxy. On the other hand, it has the disadvantage that it is based

only on observed transactions, and many bonds transact so infrequently that we cannot construct

the measure. Hence we will also construct some alternative measures of liquidity to assess the

2

robustness of our results to the particular proxy of liquidity that is employed.

We show in the context of a linear APT-style factor model that liquidity is a priced factor in

corporate bond returns; this result appears to be robust to the liquidity proxy that is employed. Our

results indicate that liquidity can account for some, but not all, of the discrepancy between the risk

spreads produced by structural models and observed risk spreads. Moreover, the results add further

support for the hypothesis that liquidity risk is a state variable impacting the returns of all assets.

This paper is organized as follows. In Section 2, we very briefly lay out our empirical method-

ology. Section 3 discusses our data in depth, since the dataset that we employ is relatively new to

the literature. We present our empirical results in Section 4, and Section 5 concludes.

2 Empirical Methodology

In this section, we briefly outline our empirical methodology. Since we employ a standard linear

APT-style model, our discussion is highly abbreviated. For textbook treatments of the techniques

we summarize, one is referred to Cochrane (2000) or Campbell, Lo and MacKinlay (1997).

We define the gross one-period return on bond i as:

Rt+1,i = 1 +Pt+1,i − Pt,i + AIt+1,i

Pt,i

, (1)

where Pt+1,i denotes the price of bond i at time t + 1, and AIt+1,i ≥ 0 denotes the interest accrued

over the period [t, t + 1].

If there are no arbitrage opportunities, then standard arguments show that there exists a stochas-

tic discount factor m such that:1

E [mR] = 1. (2)

Here we take the expectation unconditionally; we do not consider conditioning information though

in principle one can do so. We return to this issue below.

1Since the equality holds for all t and i, we omit subscripts for notational clarity; we will continue to omit subscriptsexcept when they are needed.

3

We assume that the pricing kernel m is a linear combination of M risk factors:

m = δ0 + δ′F, (3)

where δ0 is a scalar, δ is a M ×1 vector of coefficients, and F = [1 ft]′ is a M ×1 vector of factors.

As is well known, the model in equation (3) is equivalent to a model in which the expected total

return on security i is a linear function of the riskless rate, factor risk premia, and security-specific

factor sensitivities:

E [Ri] = λ0 + β′iλ, (4)

where:

λ0 =1

E [m], (5)

λ = −λ0COV [f , f ′] δ, and (6)

βi = COV [f , f ′]−1 COV [f ,Ri] . (7)

Hence to test whether factor j is priced, one tests the null hypothesis H0 : λj = 0; to test whether

factor j is marginally useful in pricing assets, one tests the null H0 : δj = 0. As can be seen

by examining equation (6), these two hypotheses are equivalent only if the factors are orthogonal;

since we will work with factor-mimicking portfolios, our factors will not be orthogonal so we will

carry out both sets of tests.

It is natural to estimate the coefficients δ using the generalized method of moments (GMM),

based on the moment conditions defined by (2). Let Rt denote the N × 1 vector formed by

“stacking” the gross returns; the vector of security-level average pricing errors is given by:

gT (δ) =1

T

T∑t=1

(mtRt − 1) , (8)

where 1 is a N × 1 vector of ones. The GMM estimate of δ minimizes a weighted sum of squares

4

of the pricing errors across assets:

minδ

JT ≡ g′TWgT , (9)

where W is a weight matrix. It is straightforward to solve this equation analytically, producing:

δ̂ = (D′WD)′D′W, (10)

where D = ∂gT

∂δ.

When W is set to the optimal weighting matrix, S−1T = COV [gT , g′

T ]−1, then the variance of

δ̂ is given by:

VAR[δ̂]

=1

T(D′S′D)

−1. (11)

Moreover, we can test over-identifying restrictions on the vector δ with TJT distributed (asymptot-

ically) χ2k, where k is the number of restrictions. Finally, with the estimates δ̂ in hand, it is straight-

forward to compute GMM estimates of the prices of risk, λ̂, using equation (6), and VAR[λ̂

]using

this same equation and the delta method.

Finally, we will use the Hansen and Jagannathan (1997) (HJ) distance measure to assess the

accuracy of our model of bond returns. The HJ distance is given by:

HJ =

√gT (δ)′ (E [RtR

′t])

−1 gT (δ), (12)

and can be interpreted as the least-squares distance between a given pricing kernel and the closest

point in the set of the pricing kernels that can price the base assets correctly. The HJ distance can

also be cast as the maximum mispricing possible per unit of standard deviation. For example, if

the HJ distance is 0.5 and the portfolio has an annualized standard deviation of 20 percent, then

the maximum annualized pricing error is 10 percent.

We estimate the HJ distance using the GMM procedure above with the weighting matrix W =

(E [RtR′t])

−1, the inverse of the covariance matrix of the second moments of asset returns. The

optimal weighting matrix is not useful in this case since it is model-specific: we might fail to

5

reject a model because it is difficult to estimate, rather than because it has small pricing errors. In

contrast, the inverse of the covariance matrix of asset returns is invariant across models.

As noted at the outset, our data sample is relatively short, and this impacts our analysis in a

number of ways. Obviously, a short time series impacts the finite sample properties of our results:

it is likely that the asymptotic distributions used to conduct our inference are poor approximations

of the true finite sample distributions of our test statistics. At least with respect to the HJ statistics,

the evidence to date suggests that finite sample biases tend to lead to over-rejection (Ahn and

Gadarowski (2004)); hence acceptance indicates that one has cleared an even higher hurdle than

that suggested by the asymptotic distribution.

Moreover, we cannot with any confidence examine conditional models where the prices of

fundamental risks vary with variables such as macroeconomic variables chosen to summarize the

state of the business cycle. Finally, we are unable to assess the stability of the model’s parameters;

in particular, we do not have in our sample window a clearly defined “liquidity event” such as the

Russian default and LTCM crises of the late 1990s. These important issues will have to wait until

a longer span of data are available. For now, we learn what we can with the short data sample that

is available.

3 Data

Our data for corporate bond transactions, liquidity, and returns is from the National Association of

Securities Dealers TRACE (Trade Reporting and Compliance Engine) database. The TRACE sys-

tem was implemented as a response to growing pressure to make the corporate bond market more

transparent. Beginning on July 1, 2002, the NASD requires all over the counter bond transactions

to be reported through the TRACE system. Beginning on July 1, 2002, the NASD requires all over

the counter corporate bond transactions in TRACE-eligible securities to be reported through the

TRACE system.2 NASD members were initially required to report corporate bond transactions

2TRACE-eligible securities include all U.S. dollar-denominated debt securities that are depository eligible un-der rule 11310(d). Specifically excluded is debt issued by government-sponsored entities, mortgage or asset-backed

6

within 75 minutes of the trade’s occurrence. On October 1, 2003, this was reduced to 45 minutes.

The required reporting time was further reduced to 30 minutes on October 1, 2004, and is sched-

uled to reach the final goal of 15 minutes on July 1, 2005. Transaction information for bonds on

the public dissemination list is transmitted on a real-time basis to fee-paying subscribers.3

While NASD guidelines require all corporate bond trades to be reported, the public dissemi-

nation of these trade reports has been gradually phased in since the initiation of TRACE. Our data

contains all publicly reported transactions in TRACE-eligible securities for the period from July

1, 2002 through December 31, 2004. That is, our dataset only contains trades for bonds which

are publicly disseminated on the day of the trade. Important dates for the phase-in process are as

follows:

• Phase I (July 1, 2002): Approximately 550 bonds became subject to dissemination. This

included all investment-grade bonds having an original issue size of $1 billion or more, as

well as 50 high yield bonds which were carried over from NASD’s Fixed Income Pricing

System (FIPS).4

• Phase II (March 3, 2003): Approximately 4,200 bonds became subject to dissemination.

This included the original bonds from Phase I, as well as all bonds with an original issue

size of at least $100 million and a credit rating of A or higher. An additional 120 BBB rated

bonds with issue sizes less than $1 billion were added as part of Phase II in April 2003.

• Phase III (October 1, 2004): All corporate bonds (approximately 29,000 total bonds) became

subject to dissemination.

Thus the universe of bonds in our sample expands during the course of the sample period. From

July 2002 through February 2003, an average of 500 unique bonds trade on any given day. From

March 2003 through September 2004, this number grows to around 1,600. And for the last three

securities, collateralized mortgage obligations, and money market instruments.3Trade information is also freely available (with a four hour delay) on the website http://www.nasdbondinfo.com.4FIPS was initiated in April 1994 to improve transparency in the market for high-yield corporate bonds. For more

details on the FIPS 50 see Hotchkiss and Ronen (2002), Alexander, Edwards and Ferri (2000a), Alexander, Edwardsand Ferri (2000b).

7

months of our sample, an average of around 4,000 unique bonds traded at least once on a given

day. These numbers indicate that many of the bonds in phases II and III trade very infrequently.

Table 1 shows statistics concerning the number of unique bonds and the number of trades that

occur during our sample. We run a series of checks on the data to arrive at our final sample. There

are initially over 18,000 unique bonds in the sample. We first remove all trades which are flagged

as canceled, corrected, etc. This deletes approximately 3,400 bonds (1.7 million trades) from the

sample. We then pull descriptive data from Bloomberg for each bond. This includes information

such as coupon rate, maturity date, issue size, credit rating, and flags for special features such as

callable bonds. We are unable to find matching descriptive information for approximately 2,700

bonds, so we ignore the 900,000 trades in these securities. Finally, we remove all trades which

have invalid or missing information on the quantity traded. This final screen only removes about

200 bonds from the sample representing just over 200,000 trades. We are left with a final sample

of 12,376 unique bonds and a total of 4,699,035 trades.

As mentioned, we pull descriptive data for each bond from Bloomberg. Table 2 provides

information regarding a few of these descriptives. Nearly half of the bonds in the sample are

straight fixed-rate securities, while almost as many are callable bonds. Less than seven percent

of the bonds are floating-rate securities, and these are excluded from our analysis. Only a small

fraction have another special feature or some combination of these features. Table 2 also shows

statistics concerning the Standard and Poor’s credit rating for each bond. The vast majority of the

bonds fall somewhere in the BBB to AA range. About seven percent are AAA rated, while less

than five percent of the bonds have a junk level rating of BB or below. Finally, the vast majority

of the bonds in our sample have issue sizes under $500 million, with over half having an issue size

under $50 million.

The first three rows of Table 3 give summary statistics on the average levels of trading activity

for bonds in the sample. One feature that is quickly apparent is that trading activity is heavily

skewed, both in terms of levels of activity and average trade size. While the average bond exhibits

an average dollar volume of $879,000 per day, the average level for the median bond is only

8

$112,000 per day. Similarly, the average trade size for the average bond is $355,000, while the

average trade size for the median bond is only $42,000. Finally, while the average bond trades 386

times over the sample period, the median bond trades only 23 times. 5 The average bond in the

sample is about 3 years old, and has about 8 years left until maturity. These statistics are broadly

consistent with those found by Edwards, Harris and Piwowar (2004) in a sample of TRACE trades

for 2003.

3.1 Illiquidity Measures

The numbers in Table 3 provide a rough estimate of the wide variation in liquidity across the

bonds in our sample. In order to define the liquidity of a bond for the remainder of our analysis,

we use a modified version of Amihud’s (2002) price impact measure. This measure is defined

as the absolute percentage price change divided by the dollar volume of the trade, and Amihud

(2002) shows that this measure calculated daily for equities is strongly related cross-sectionally

to other measures of liquidity calculated using intraday data. We calculate Amihud’s measure for

each trade and then average across all trades in a given week to calculate the illiquidity measure

for a bond in that week. Rather than use the price impact measure exactly as defined by Amihud,

we account for the fact that calculating the measure for bonds may also pick up term structure

movements (in addition to the price impact of a trade). This will lead to an increased correlation of

our liquidity factor with the maturity factor we construct later. To address this issue, we calculate

the percentage return of the bond net of the return on a riskless Treasury security over the same

interval.6 We thus calculate the modified Amihud measure as:

primi,t =1

Ni,t

Ni,t∑j=1

∣∣∣(

Pi,t,j−Pi,t,j−1

Pi,t,j−1

)−

(PTreas,t,j−PTreas,t,j−1

PTreas,t,j−1

)∣∣∣V olumei,t,j

∗ (1, 000) (13)

5Some degree of caution must be used in interpreting these results, due to the phasing in of TRACE public dis-semination. Only approximately 500 bonds have the opportunity to appear every day throughout the entire 2 1/2 yearsample.

6Specifically, we use intraday observations from the GovPx dataset for on-the-run (newly issued) 6 month bills,and 2, 5, and 10 year notes. Bonds with less than 1 year maturity are matched with the 6 month bill; bonds with 1-3years to maturity are matched with the 2 year note; bonds with 3-7 years to maturity are matched with the 5 year note,and bonds with greater than 7 years to maturity are matched with the 10 year note.

9

where Ni,t is the number of trades for bond i in week t, V olumei,t,j is the dollar trading volume

(in thousands) for bond i in the j-th trade in week t, P refers to a transaction price, and the

subscript Treas indicates the benchmark Treasury. We multiply the measure by 1,000 to facilitate

presentation. In calculating the weekly illiquidity metrics, we delete all transactions in which the

price deviates from the price of the last transaction by more than 20%.7

The sixth row of Table 3 shows the distribution of the Amihud illiquidity measure across the

bonds in the sample. We observe a wide range of liquidity across bonds, ranging from an average

price impact of essentially 0 to bonds with an average price impact measure over 11. To put the

price impact numbers in perspective, the average level of 0.55 for the median bond indicates a

price impact of 2.3% for a median-sized trade of $42,000. Figures 1 through 2 illustrate how the

illiquidity measure varies across bonds with different characteristics. Panels A and B of Figure 1

show how bond illiquidity varies based on the time since issuance and the time remaining until

maturity. Bonds with longer maturities tend to be less liquid (higher illiquidity measure) than

short-maturity bonds. This is generally consistent with results from Fleming (2003) and Brandt

and Kavajecz (2004) for the Treasury market, and may be a reflection of investor preferences.

Buy and hold investors such as insurance companies are generally attracted to long-term debt,

potentially decreasing the pool of tradeable securities as the maturity of debt increases. Panel B

shows that bonds become more illiquid as they age, or become more “seasoned”. This is also

consistent with results from the Treasury market, and reflects the fact that bonds gradually become

buried in the portfolios of investors who intend to hold them until maturity.

Figure 2 plots the illiquidity measure as a function of issue size. Average illiquidity for each of

ten size deciles is plotted. It is clear that bonds in the smallest issue size decile are by far the most

illiquid. This decile represents bond issues of approximately $100 million or less. There is no clear

pattern across the remaining size deciles in terms of liquidity. Figure 3 plots the illiquidity measure

as a function of the Standard and Poor’s credit rating of the bond issues. Bonds are separated into

7This eliminates approximately 12,000 transactions (0.2% of all trades) from the sample. Edwards et al. (2004)eliminate all TRACE trades where the price deviates from the daily median by 10%, as well as all trades where theprice deviates from the previous or next transaction by 10%. Our results are qualitatively similar using these filters.

10

five groups: AAA, AA, A, BBB, and Junk. It appears that bonds at the extreme ends of the credit

ladder are the least liquid. Junk bonds in particular appear to be highly illiquid, although this

effect may be partially related to the size effect. Junk bond issues tend to be fairly small, which

likely hurts their liquidity. In addition, given that junk bonds are highly sensitive to firm-specific

information, it is likely that trades will move prices more for low-grade debt than for more secure

debt. Finally, it is worth noting that the relative illiquidity of AAA bond issues may also be related

to the size effect. AAA issues tend to be much smaller than most of the lower rated issues.

To assess the robustness of our results, we also use three alternative measures of liquidity.

These include the dollar volume traded, the ratio of price volatility to volume, and the turnover of

the bond. The ratio of volatility to volume is calculated as

V olatility/V olumei,t =(P max

i,t − P mini,t )/P mean

i,t

V olumei,t

(14)

with P max, P min, and P median representing the maximum, minimum, and median price for bond i

during week t. V olumei,t is the total face value of bond i traded in week t. Turnover is measured

as

Turnoveri,t =

∑Ni,t

j=1 V olumei,j,t

IssueSizei

(15)

where, as before, Ni,t is the number of trades in bond i during week t, V olumei,j,t is the face

value traded in the jth trade of bond i during week t, and IssueSizei is the total face value of

the bond at issuance. The last two rows of Table 3 present the distribution of these alternative

liquidity measures across the bonds in the sample. These measures confirm that liquidity varies

quite substantially across the bonds in the sample.

3.2 Returns

We calculate weekly returns for each bond using equation (1). For each bond, we calculate the

open-to-close return over the week from the first trade to the last trade using transaction prices

from TRACE. We then add on any interest that accrued over the week based on the bond’s coupon

11

rate. The total return for bond i in week t is then given by

Ri,t =Pi,t(last) − Pi,t(first) + AIi,t

Pi,t(first)

(16)

where Pi,t(first) represents the transaction price for the first trade in bond i during week t, Pi,t(last)

is the last trade price from week t, and AIi,t is the accrued interest that accumulated for bond i

during week t. We require that the first trade and last trade be separated by a number of days equal

to one less than the number of trading days in the week. We exclude all weekly return observations

that exceed 100 percent, but this eliminates only five weekly return observations from the sample.

Table 3 displays univariate statistics for the bond returns. The pooled average total return is 0.46

percent, with a median of 0.2 percent (24 percent and 10 percent per annum, respectively).

4 Empirical Results

The assets that we use to test the pricing model are portfolios formed by sorting the bonds in

our sample by maturity, credit rating, and illiquidity; the first subsection focuses on the salient

properties of these test asset portfolios. We then turn to a discussion of portfolios designed to

mimic the risk factors in bond returns that are related to movements in the term structure of risk-

free interest rates, default risk, and illiquidity. Finally, we examine how well the test asset excess

returns are explained by linear combinations of the factor mimicking portfolios, with a focus on

the marginal explanatory power provided by the illiquidity factor-mimicking portfolio.

4.1 Test asset construction: Bond portfolios sorted on maturity, credit rat-

ing and illiquidity

Table 4 displays average weekly returns, standard deviations of these returns, and first-order auto-

correlation coefficients for our basic set of tests assets. Panel A displays the summary statistics for

bond portfolios formed by sorting the sample into quintiles by maturity each week and then calcu-

12

lating the equally weighted return on the bonds in each quintile over the following week. Quintile

one represents the bonds with the shortest time to maturity and quintile five those with the longest

time to maturity. As can be seen, the average weekly return on the portfolios rise almost monoton-

ically as maturity lengthens, reflecting the greater interest-rate sensitivity (duration), default risk,

and perhaps liquidity risk of longer-term bonds. Portfolios with longer maturities also exhibit more

volatile returns, with the standard deviations of the returns rising monotonically from portfolio one

to five. Examining the autocorrelations in the last column, the portfolios with the shortest and

longest time to maturity are somewhat more persistent than the others, but none of the portfolios

exhibit a high degree of persistence in returns. Looking at the extremes, the return difference be-

tween portfolios one and five is 0.21 percent, with a Newey-West t-statistic of approximately two.

The difference in returns is clearly economically significant: a 0.21 percent difference in weekly

returns implies an annualized 11 percent return spread between long and short maturity bonds.

Panel B displays returns, standard deviations, and autocorrelations for five portfolios formed by

sorting the bonds by their S&P long-term credit ratings, with all bonds rated below BBB defined

as junk. Curiously, the AAA-rated portfolio earns an average weekly return of 0.14 percent, while

the AA- and A- rated portfolios earn 0.07 percent and 0.11 percent, respectively. However, recall

from our discussion of Figure 3 that illiquidity varies substantially with credit rating; in particular,

AAA-rated bonds tend to be more illiquid than AA- and A-rated bonds. This could in part help

to explain why the higher rated bonds earn higher returns; we will return to this issue below. The

average weekly returns on the portfolio of bonds rated below BBB is 0.49 percent, substantially

higher than the other four portfolios, as one should expect given their higher default risk and

sharply higher illiquidity. The junk portfolio is also the most volatile, with a standard deviation of

1.70 percent, which corresponds to an annualized standard deviation of 12.27 percent. None of the

portfolios exhibit high degrees of persistence, as evidenced by the autocorrelation coefficients all

being below 0.24. The return difference between the AAA-rated and junk-rated portfolios is 0.35

percent with a t-statistic of two.

Panel C displays the summary statistics for test assets formed by sorting the bonds on the

13

modified Amihud measure of illiquidity. The least illiquid bonds earned slightly higher returns

than portfolios two and three; returns rise monotonically from portfolio two to five. Quintile five

also has the highest standard deviation, similar to those of the longest maturity or lowest credit

rating above. Like the return patterns for the portfolios sorted by credit rating, the determinants of

these return patterns will become clearer when we project the portfolio returns onto the space of

factor-mimicking portfolio returns.

4.1.1 Portfolios sorted on other measures of liquidity

To test the robustness of our results, we consider two natural alternative liquidity measures. Table 5

presents the basic weekly return data for bond portfolios sorted on these other measures of liquidity.

In Panel A, bonds are sorted into five equally-weighted portfolios based on the previous week’s

trading dollar volume (cite here). As can be seen, there are not dramatic return differences between

the portfolios; the difference between the highest dollar volume and lowest dollar volume portfolios

is only -0.1 percent and is borderline statistically significant. However, as noted in Section 3, over

the course of our sample period the NASD phased in the dissemination of many smaller, less

actively traded bonds. If we restrict the sample to trades from March 2003 through December of

2004—the period of full disseminatin—the return difference between the high and low volume

portfolios is -0.16 percent with a t-statistic of -4.74. Hence it appears that dollar volume is a more

effective liquidity proxy over the latter part of the sample.

In Panel B, bonds are sorted into five equally-weighted portfolios based on the volatility-

volume ratio. As expected given the high correlation of this measure with the modified Amihud

measure, the return patterns closely follow those in Panel C of Table 4. The average difference

in returns between the high ratio and low ratio portfolios is 0.16 percent and is statistically sig-

nificant. This compares to a 0.15 percent difference using the modified Amihud measure. The

autocorrelations are also similar in magnitude to the modified Amihud measure.

Eckbo and Norli (2000) employ turnover, defined as the number of shares traded over a period

divided by the number of shares outstanding at the beginning of the period, as a liquidity proxy

14

in a study of liquidity in the equity market. They find the average return difference between the

low turnover portfolio and the high turnover portfolio is significant over the 1973-2000 period.

We compute an analogous turnover measure for the corporate bonds in our sample. The natural

turnover measure for bonds is the total face value traded over a period divided by total face value

outstanding at the beginning of the period. Panel C of Table (5) shows the return spread on port-

folios sorted on this turnover measure. As can be seen, there is scant difference in the returns

between the five portfolios. However, like the volume proxy, the return differences between the

turnover portfolios is somewhat more significant over the period from March 2003 until December

of 2004 (phases II and III). The return difference between the low turnover portfolio and the high

turnover portfolio over this period is 0.06 percent with a t-statistic of 2.23.

Pastor and Stambaugh (2003) associate illiquidity with stronger volume-related return rever-

sals. They construct a liquidity factor by first estimating the sensitivity of individual stock returns

to lagged, signed dollar trading volume. The liquidity factor is the monthly innovations in the

cross-sectional average of these individual sensitivity estimates. While this is an appealing way to

construct a liquidity measure, it is difficult to apply this method in the corporate bond market be-

cause the vast majority of the bonds do not trade on a daily basis. Data limitations also preclude the

construction of bid-ask spreads, indicators of the depth of the trading, and other commonly used

measures that are based on data in the trading book (non-existent in an over-the-counter market)

or on observations of trades segregated into purchases and sales.

4.1.2 Illiquidity after controlling for maturity and rating

As suggested by Figures 1 - 3 and the results in the previous sub-section, illiquidity appears to be

closely related to both a bond’s maturity and its credit rating. In light of these results, it is natural

to question whether the increasing returns to illiquidity that we observed for our basic test asset

portfolios in fact reflect greater interest rate or credit risk. To investigate this, we first sort the bonds

on either maturity or credit rating, and then we sort on illiquidity. We then compute the returns

on each of the portfolios defined by these sequential sorts. Table 6 displays the average weekly

15

returns and other basic statistics for these portfolios, where we have used the modified Amihud

measure of illiquidity.

Panel A shows average returns for the 25 portfolios formed by sequentially sorting on maturity

and then illiquidity. The column labeled “5-1” is the return difference between the most illiquid

portfolio and the least illiquid portfolio. With the exception of the difference at the longest matu-

rity, after conditioning on maturity the return differences are small and statistically insignificant.

However, the return difference for the portfolo with the longest maturity is 0.22 percent with t-

statistic of 2.01, indicating that the illiquidity effect cannot be completely subsumed by maturity.

Moreover, the average return spread on illiquidity over the five maturity quintiles is 0.11 percent

with a t-statistic of 2.24.

In Panel B of Table 6, we display results for portfolios formed by first sorting all of the bonds

by credit rating and then sorting by illiquidity. The return spread on illiquidity is positive for all

of the portfolios except the portfolio of BBB-rated bonds; the AAA-, AA-, and Junk-rated spreads

are statistically significant and sizable. On average, the return difference between portfolio five

and one is 0.24 percent and is statistically significant at five percent level. Again, it is clear that the

liquidity effect cannot be subsumed by credit rating.

4.2 Construction of factor-mimicking portfolios

In the previous sub-section, we saw that portfolios of bonds with high illiquidity measures earn

higher returns than those with low illiquidity, even after controlling for maturity and credit rating.

In this section, we begin to address the question of whether liquidity is a systematic factor that

affects the co-movements of bond returns. The first step in this process is to construct a set of factor

mimicking portfolios that serve as proxies for the underlying interest rate, default, and perhaps

liquidity risks that we believe drive bond returns.

We create 27 portfolios based on sequential sorts on the bonds’ maturity, credit rating, and

illiquidity measures. Because we are conducting sequential sorts along three dimensions, we sort

into a limited number of categories in each dimension so that there are enough bonds in each

16

portfolio to support calculations of returns. First, we sort all of the bonds into three maturity

groups, each containing one third of the bonds that trade in a given week. Within each of these

maturity groups, we next sort the bonds into three credit rating groups. The first credit rating group

contains bonds with S&P credit ratings of AAA and AA; the second credit group contains bonds

carrying an A rating; and the last group contains bonds rated BBB and below. Finally, within each

of the nine groups sorted on maturity and credit rating, we make a final sort into three portfolios

based on the illiquidity measure. This procedure delivers the 27 portfolios that serve as the basic

building blocks for the construction of our factor-mimicking portfolios.

Our first factor portfolio is designed to mimic the underlying risk factor in bond returns related

to the term structure of default-free interest rates. It is natural to form this portfolio from bonds

with different maturities, controlling for credit rating and illiquidity, in hopes that the difference in

returns on long- versus short-maturity bonds reflect movements in default-free interest rates. We

construct this portfolio by computing the average return on the nine portfolios formed from long

positions in the building-block portfolios with long maturity and short positions in the building-

block portfolios with short maturity. We refer to this as the “maturity” factor-mimicking portfolio.

Our second factor portfolio is designed to mimic the underlying default risk factor in bond returns.

We refer to this as the “credit” factor portfolio, computed as the average return from a long position

in the nine portfolios with S&P credit ratings of BBB and below and a short position in the nine

portfolios with S&P credit rating of AAA and AA. The “illiquidity” factor portfolio is constructed

as the average return on long positions in the nine most illiquid portfolios and short positions in the

nine least illiquid portfolios. Finally, we also make use of a “market” factor, defined as the CRSP

value-weighted market return in excess of the overnight general collateral repo rate.

Table 7 displays summary statistics for the factor-mimicking portfolios and the market factor.

The market factor has a mean weekly return of 0.25 percent but is not statistically significant;

the market factor exhibits some left skewness and has fat tails, all well-known properties of stock

returns. The maturity and credit factor earn 0.14 and 0.12 percent per week, respectively, and both

are statistically significant. The maturity factor is left-skewed, while the credit factor exhibits little

17

skew, and both factors have fat-tailed distributions. The illiquidity factor has a mean weekly return

of 0.06 percent, lower than the mean returns on the other factors, but statistically significant. The

liquidity factor is left-skewed and fat-tailed, similar to the maturity and market factors. None of

the factor portfolios exhibit a high degree of autocorrelation; the liquidity factor has the highest

autocorrelation coefficient at 0.25.

As can be seen in the lower panel of Table 7, the maturity factor is largely uncorrelated with the

other factors. The market factor is positively correlated with the credit and illiquidity factors, with

correlation coefficients of 0.32 and 0.25, respectively. The credit and illiqudity factors are most

highly correlated, with a correlation coefficient of 0.52, suggesting that our sequential sorting

procedure is only partially successful in orthogonalizing the factors. As noted in Section 2, this

implies that the hypothesis that the illiquidity factor is not priced is not the same as the hypothesis

that it is not marginally useful in pricing other assets. The relatively high correlation between these

factors also suggests that, with a short time series of data, it might prove difficult to distinguish the

effects of default risk and illiquidity from one another.

4.3 Time series regressions

If the factor-mimicking portfolios successfully mimic the term structure and default risk factors

that drive bond returns, then we should observe that the returns on our test asset portfolios are

well-explained by the returns to the factor-mimicking portfolios. To confirm this, we use equa-

tion (7) from Section 2, that is, we estimate the following time-series regression for each test asset

portfolio:

Rei,t = αi + β0,iMarkett + β1,iMaturityt + β2,iCreditt + εit, (17)

where Rei,t is the excess return on portfolio i at week t. Note that here we are excluding the

illiquidity factor; shortly we will add this factor and assess its marginal contribution to the time-

series fits.

Table 8 displays the estimates of equation (17) for our 15 basic test portfolios. Panel A shows

18

the results for the five portfolios sorted on maturity. The portfolios all have negative loadings on

the market factor: when the market return is positive, these bond portfolios earn lower returns. The

negative loading on the market factor is shared by nearly all of the 15 portfolios that we consider;

with only a few exceptions, the loadings are statistically significant. These results are suggestive

of the “rotation” stories often seen in the popular press: what is good for equities is bad for bonds,

and vice-versa. By construction, the loadings on the maturity factor rise monotonically from -0.08

to 1.16; the loading for the shortest-maturity portfolio is not statistically significant. The loadings

on the credit factor are all positive and highly significant, as expected.

In general, the three-factor model does a reasonably good job of predicting the excess returns

on these test assets. The regression intercepts are close to zero; however, the GRS test statistic

has a p-value of 0.03, indicating rejection of the null hypothesis that the intercepts are jointly zero.

The adjusted-R2 values rise from 0.2 for the short-maturity portfolio to 0.9 for the long-maturity

portfolio.

Panel B displays the results for the five test portfolios formed by sorting on credit ratings. With

the exception of the junk bond portfolio, all of the portfolios load negatively on the market factor;

the junk portfolio loading is 0.02 and it is not statistically significant. In fact, for the junk portfolio,

the only significant loading is on the credit factor, consistent with the notion that default is the

dominant risk factor for these bonds. The investment-grade portfolios have positive and significant

loadings on the maturity factor. The loadings on the credit factor increase monotonically from

AAA to Junk; the credit factor loadings are not statistically significant for the AAA- and AA-rated

bond portfolios.

The three factor model also does a fairly good job of predicting the excess returns of the credit

portfolios. The regression intercepts are again close to zero, and the GRS test statistic has a p-

value of 0.05, indicating borderline joint significance. We note that the individual intercepts are

significant for the AAA portfolio and junk portfolio. The adjusted-R2 values range from 0.52 to

0.81.

Panel C shows the regression results for the five portfolios formed by sorting on the modified

19

Amihud illiquidity measure. As with the other test assets, the illiquidity portfolios have negative

loadings on the market factor. The portfolios all load positively on the maturity factor, with the

loadings rising monotonically from the least illiquid portfolio to the most illiquid portfolio. The

most illiquid portfolio also has the highest loading on the credit factor. The regression results

indicate that the illiquid bonds mostly likely are those with long maturity and low credit ratings.

The GRS test fails to reject the null hypothesis that the intercepts are jointly zero, and the adjusted-

R2 values range from 0.58 to 0.71.

On balance, these results indicate that the market, maturity, and credit factor mimicking port-

folios have substantial power to explain the co-movements of bond returns, at least as summarized

by our test assets. However, the results indicate that exact factor pricing does not hold under this

model, leaving room for an additional factor or set of factors to play a role in explaining returns.

We next consider the model that includes the illiquidity factor-mimicking portfolio:

Rei,t = αi + β0,iMarkett + β1,iMaturityt + β2,iCreditt + β3,iIlliquidityt + εit, (18)

Table 9 reports the regression estimates for equation (18). Comparing the results to those in

Table 8, three points are in order. The most important point is that the intercepts are now all close to

zero, nearly all are individually insignificant, and all three of the joint tests fail to reject the null that

the intercepts are jointly equal to zero. Hence the inclusion of the illiquidity factor produces exact

factor pricing. Second, the presence of the illiquidity factor raises the adjusted-R2 values as much

as 12 percentage points, with the largest increases registered at short maturities, low credit quality,

and high illiquidity—precisely what we expect. Finally, we note that the correlation between the

credit and illiquidity factor is reflected in some loss of precision in the estimates of the loadings

on the credit factor, and the estimates of the credit loadings become smaller, all consistent with the

effects of multicollinearity on multiple regression coefficient estimates. Here again, our short data

sample affects our ability to cut through the correlations in the factors to find their separate effects.

20

4.4 Is Illiquidity risk priced?

In the previous subsection, we demonstrated that the illiquidity factor helps to explain the time-

series variation in bond portfolio returns, and we found that most of the portfolios had significant

loadings on the illiquidity factor. Next we examine whether illiquidity risk is priced in the cross-

section of bond returns. One approach to this question can be based on equation (4): use the

estimated betas from the previous section in a cross-sectional regression in order to estimate λ, and

then test whether the price of risk on the illiquidity factor portfolio is statistically significant. An

alternative approach is to estimate the coefficients of the pricing kernel using the GMM estimator

in equation (10). In general, since GMM is a one-step procedure, it is more efficient than two-step

procedures. Moreover, it is easy to conduct some additional interesting hypothesis tests within the

GMM framework; specifically, we can test whether liquidity enters the pricing kernel and is thus

marginally useful in pricing other assets.

Table 10 displays the GMM estimation results.8 We carry out the estimates using different

sets of test assets in order to check the robustness of our results. In Panel A we report the results

obtained using the 15 portfolios sorted on maturity, credit rating, and illiquidity that, together

with the risk-free rate, are used as our basic set of test assets. All the factors have positive and

statistically significant premia at the five percent significance level. The market factor has the

highest estimated risk premium of 0.86 percent per week; the premia on other risk factors range

from 15 basis point to 19 basis point per week. In particular, the illiqudity factor has an estimated

premium of 0.16 percent with a t-statistic of 3.09, indicating that liquidity is a priced risk-factor

for bonds.

The Hansen’s overidentification test (J-test) does not reject the four factor model; the p-value

is 0.12. The risk premia are jointly significant, with a p-value of 0.0216. Finally, the HJ distance

is 0.47 but not statistically significant from zero.

Does our conclusion on liquidity change if we use different test assets? As we will see, the

8Say here that we ran the beta-lambda version and there was nothing interesting over and above what we havehere?

21

answer to this question is no; however, when confronted with some of the other sets of test assets,

our battery of specification tests suggests that the four factor model may not be the whole story.

In Panel B, we use the 25 portfolios sorted on maturity and then on illiquidity as test assets. The

magnitude of the risk premium on illiquidity is lower, but still statistically significant. The J-test

statistic remains in the acceptance region, and the joint test remains in the rejection region. The HJ

distance increases slightly, and is borderline significant. When we move to Panel C, where we use

the 25 portfolios sorted on credit and then on illiquidity, the risk premium on the illiquidity falls to

0.05 and is only significant at 10 percent level, and now the HJ test soundly rejects the null of zero

distance to the true set of pricing kernels.

Panel D reports the estimation results using 25 portfolios sorted on maturity and credit and

the lastly Panel E shows the GMM results using 75 portfolios used in Panel B, C, and D. As

before, illiquidity is priced, and the J-tests do not reject the model. However, the HJ distances are

significant.

Do our conclusions on the price of liquidity risk change when we consider the alternative

measures? Based on the results in Table 11, we conclude that the answer to this question is also

no. In the table, we display GMM estimation results based on our alternative liquidity measures

and the maturity, credit, and market factor used before. Panel A displays results using dollar

volume as a proxy for liquidity, and Panel B displays results for the volatility-volume ratio. Using

the same 15 test assets, we find that results are consistent with those in Panel A of Table 10. The

illiquidity factor has a positive and significant risk premium, and all if the risk factors are jointly

significant. However, the p-values on the HJ statistics indicate significance.

Finally, we note that all of our results are strengthened if we restrict the sample period to the

post-Phase II period when most or all of the bond trades were disseminated.

22

5 Conclusion

In this paper, we employed the TRACE corporate bond dataset to test whether market liquidity

is a priced factor in bond returns. We conclude from our econometric results that, although the

illiquidity factor is priced, the pricing errors are still large enough that we reject the null of zero

distance between our pricing kernel and the true set of pricing kernels. Exposure to illquidity risk

accounts for a statistically significant portion of the cross-sectional variation in bond returns, but it

is likely that other risk factors are omitted.

An important caveat to our analysis is that the time period we study is relatively short, reflecting

data limitations. This limitation prevents us from analyzing some natural questions about the

robustness of our results. For example, we cannot examine how our results might vary through the

business cycle or during well identified liquidity events. Analysis of issues requiring a long time

series will have to wait until additional data become available through the TRACE system.

A natural direction for future research brings us full circle to one of the motivational issues

raised in the introduction. How can market liquidity be incorporated into structural models of

bond prices? What role does liquidity play in determining corporate bond risk spreads relative to

say, credit risk or interest rate risk? Based on the results in this paper, further theoretical work

along these lines would likely prove fruitful.

23

Figure 1: Illiquidity and Time

Panel A displays the relationship between average illiquidity and years to maturity. The vertical axis shows the

average illiquidity for all of the bonds with the indicated years to maturity. Panel B displays the relationship between

the average illiquidity and the number of years since a bond was issued (bond age). Panel C displays the average

illiquidity measure over the life of a bond. The life of a bond is scaled from 0 to 1 and is computed as the ratio of

bond’s age to its original maturity. In each panel, average illiquidity is computed at a weekly frequency.

0.50

1.00

1.50

2.00

5 10 15 20 25

Ave

rage

Illi

quid

ity

Years to Maturity

Panel A: Average Illiquidity by Years to Maturity

0.50

1.00

1.50

2.00

1 2 3 4 5 6 7 8 9 10

Ave

rage

Illi

quid

ity

Years Since Issuance

Panel B: Average Illiquidity by Years Since Issuance

24

Figure 2: Illiquidity and Issue Size

This plot displays the relationship between the average iliquidity and the size of a bond Issue. Bond issue size is

measured as the log of the par value of the issue; illiquidity is computed at a weekly frequency.

0.50

1.00

1.50

2.00

0 0.5 1 1.5 2 2.5 3

Ave

rage

Illi

quid

ity

Size of Issue ($ Billions)

25

Figure 3: Illiquidity and Credit Rating

The figure displays the average illiquidity for different credit rating groups. The illiquidity averages are computed at a

weekly frequency.

0.50

1.00

1.50

2.00

JunkBBBAAAAAA

Ave

rage

Illi

quid

ity

Credit Rating

26

Table 1: Sample Size

The table displays counts of the number of bonds and number of trades in our sample after we filter forcanceled, corrected, and other types of non-trades (’After filters’), after we merge the raw TRACE data withdescriptive information from other sources (’After merge with descriptives’) and after we throw out bondswith invalid or missing information on quantity traded (’With quantity data’). The remaining number ofunique bonds in our sample is 12,376. These bonds traded a total of 4,699,035 times over the period fromJuly 1, 2002 to December 31, 2004, a total of 631 business days.

Number of Number ofBonds Trades

Total before processing 18,681 7,530,572After filters 15,267 5,815,007After merge with descriptives 12,578 4,917,484With quantity data 12,376 4,699,035

27

Table 2: Qualitative Features of Bonds in Sample

The table displays some qualitative information about the bonds in our sample. The first two columnsbreak the sample down by structure; the ’Combination’ line shows the share of bonds that contain multiplefeatures, such as being both callable and putable; the other categories are mutually exclusive. Columns 3-4display the breakdown of credit ratings; the last two columns display information on issue sizes. ’Large’refers to those bonds with issue size (total face value issued) larger than $500 million, ’Medium’ are thosebetween $50 million and $500 million,and ’Small’ refers to issues smaller than $50 million.

Bond Share of Credit Share of Issue Share ofStructure Sample Rating Sample Size SampleStraight 48.9 AAA 7.1 Large 10.5Callable 39.2 AA 15.0 Medium 36.0Putable 1.1 A 46.3 Small 53.9Floater 6.7 BBB 26.7Sinking fund 0.4 Junk 4.9Zero coupon 0.4Combination 3.3

28

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29

Table 4: Basic Test Assets: Portfolios Sorted on Maturity, Credit Rating, and Illiquidity

The table displays average weekly returns on portfolios formed from sorting the bonds into quintiles on the indicated

factor. The sorts are ordered such that ’1’ always indicates the portfolio formed from bonds in the lowest quintile for

the indicated variable (i.e., shortest maturity, least credit risk, least illiquidity). The autocorrelations of each portfolio

return with its first lag are shown in the column labeled ρ. The ’High-Low’ values indicate the average return one

earns by shorting one dollar of the ’Low’ portfolio and buying one dollar of the ’High’ portfolio.

Panel A: Portfolios sorted on maturity

Sort Std.Order Quintile Mean Dev. ρ

Longer 1 0.0632 0.3555 0.2959Maturity 2 0.0960 0.5181 0.0660

↓ 3 0.1776 0.7129 0.06794 0.1496 0.8418 0.16495 0.2747 0.9860 0.1950

Longest−Shortest 0.2115t-Stat. 2.0430

Panel B: Portfolios sorted on credit rating

S&P Std.Rating Mean Dev. ρAAA 0.1359 0.5479 0.1381AA 0.0704 0.4505 0.1291A 0.1128 0.5458 0.1003

BBB 0.1579 0.7914 0.0837Junk 0.4894 1.7025 0.2319

Junk−AAA 0.3535t-Stat. 2.0247

Panel C: Portfolios sorted on illiquidity

Sort Std.Order Quintile Mean Dev. ρMore 1 0.1361 0.5244 0.1287

Illiquid 2 0.1034 0.4922 0.0917↓ 3 0.1219 0.5676 0.0986

4 0.1602 0.6966 0.17895 0.2813 0.9314 0.1481

High-Low 0.1452t-Stat. 2.1941

30

Table 5: Additional Test Assets: Portfolios Sorted on Alternative Measures of Liquidity

The table displays average weekly returns on portfolios formed from sorting the bonds into quintiles on the indicated

factor. The details about the data displayed in the table are the same as in Table 4 above.

Panel A: Portfolios sorted on dollar volume

Sort Std.Order Quintile Mean Dev. ρ

Greater 1 0.2706 0.7171 0.1991Volume 2 0.1259 0.5936 0.0722

↓ 3 0.1271 0.5700 0.14414 0.1328 0.6350 0.10205 0.1656 0.7070 0.1390

High−Low -0.1049t-Stat. -1.7538

Panel B: Portfolios sorted on volatility-volume ratio

Sort Std.Order Quintile Mean Dev. ρHigher 1 0.1529 0.5190 0.0722Ratios 2 0.1214 0.5339 0.1472

↓ 3 0.1065 0.6046 0.13074 0.1214 0.6405 0.12445 0.3194 0.9319 0.0996

High−Low 0.1665t-Stat. 2.5497

Panel C: Portfolios sorted on turnover

Sort Std.Order Quintile Mean Dev. ρ

Greater 1 0.1909 0.7533 0.1698Turnover 2 0.1535 0.6236 0.1739

↓ 3 0.1083 0.7415 0.12274 0.2083 0.7030 0.15285 0.2037 0.8677 0.0642

High−Low 0.0128t-Stat. 0.2322

31

Table 6: Additional Test Assets: Portfolios Sorted First on Maturity or Credit Rating, Then Illiq-uidity

The table displays average weekly returns on portfolios formed by first sorting the bonds into quintiles on maturity or

credit rating, and then sorting on illiquidity. The details about the data displayed in the table are the same as in Table 4

above.

Panel A: Portfolios sorted first on maturity, then on illiquidity

Sort Greater Illiquidity →Order Quintile 1 2 3 4 5 5−1 t-Stat.

Longer 1 0.0530 0.0297 0.0133 0.0508 0.1691 0.1161 1.0227Maturity 2 0.0962 0.0646 0.0936 0.0159 0.2142 0.1179 1.1520

↓ 3 0.1826 0.1639 0.1222 0.1124 0.3283 0.1457 1.41964 0.1975 0.1679 0.0956 0.1138 0.1792 -0.0184 -0.22525 0.1808 0.2861 0.2334 0.2658 0.4014 0.2206 2.0075

Average 0.1164 2.2427

Panel B: Portfolios Sorted first on credit rating, then on illiquidity

Credit Greater Illiquidity →Quality 1 2 3 4 5 5−1 t-Stat.AAA 0.0858 0.0109 0.0805 0.1843 0.3104 0.2246 2.0584AA 0.0484 0.0327 0.0317 0.0522 0.1870 0.1386 2.1709A 0.1036 0.1016 0.1258 0.0745 0.1496 0.0460 0.9558BBB 0.1763 0.1414 0.1352 0.2117 0.1229 -0.0534 -0.7792Junk 0.3290 0.2719 0.3212 0.3667 1.2125 0.8835 2.5149

Average 0.2479 3.3187

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Table 7: Summary Statistics for Factor-Mimicking Portfolios

This table displays summary statistics for the returns on portfolios designed to mimic the risk factors that we hypothesize are the determinants of

bond returns. The ’Market’ factor is the CRSP value-weighted stock market return index less the rate on overnight general collateral repurchase

agreements (repo). For each week, we first sort all of the bonds into three maturity portfolios; within each maturity portfolio we sort the bonds into

three credit rating portfolios; finally, within the nine maturity and credit rating portfolios, we sort the bonds into three liquidity portfolios, for a total

of 27 basic building-block portfolios. The “Maturity” factor is the average return difference between long maturity and short maturity portfolios,

controlling for credit rating and illiquidity; the “Credit” factor is the average return difference between the nine low credit rating portfolios and nine

high credit rating portfolios; and the liquidity factor is the average return difference between the nine most illiquid portfolios and the nine most

liquid portfolios. The t-statistics are the Newey and West (1987) robust statistics with lag-length equal to three.

Portfolio Mean t-Stat. Skewness Kurtosis ρMarket 0.2450 1.3145 -0.3195 4.4169 0.1560Maturity 0.1369 2.2178 -0.6543 3.5790 0.1840Credit 0.1228 1.8589 0.0931 8.2795 0.1239Iliquidity 0.0630 2.1213 -0.3113 4.9739 0.2547

CorrelationsMaturity Credit Illiquidity

Market 0.0503 0.3151 0.2506Maturity 0.0468 0.0223Credit 0.5189

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Table 8: Time-Series Regessions Excluding Illiquidity FactorThe table reports results for weekly time-series regressions of the indicated test asset returns on the returns of the factor-mimicking portfolios. Theregression specification is given by:

Rei,t = αi + β0,iMarkett + β1,iMaturityt + β2,iCreditt + εit.

The GRS F -stat. is the M. R. Gibbons, Ross and Shanken (1989) test that the alphas are jointly equal to zero.

Panel A: Portfolios sorted on maturity

SortOrder Quintile α Market Maturity Credit Adj.-R2

Longer 1 0.05 -0.04 -0.08 0.24 0.23Maturity 1.66 -1.93 -1.14 3.59

↓ 2 0.03 -0.07 0.32 0.32 0.400.89 -3.28 5.98 4.56

3 0.06 -0.10 0.62 0.51 0.661.56 -4.38 12.85 9.36

4 -0.00 -0.11 0.89 0.47 0.74-0.07 -4.42 12.22 6.17

5 0.06 -0.03 1.16 0.52 0.861.76 -1.17 18.48 6.67

GRS F -stat. = 2.50, p-val. = 0.03

Panel B: Portfolios sorted on credit rating

CreditRating α Market Maturity Credit Adj.-R2

AAA 0.09 -0.04 0.55 -0.12 0.542.43 -1.95 8.24 -1.50

AA 0.02 -0.06 0.44 -0.01 0.520.72 -3.85 8.93 -0.23

A 0.04 -0.07 0.58 0.11 0.611.23 -4.00 9.81 1.82

BBB -0.01 -0.07 0.74 0.66 0.81-0.23 -2.83 18.03 9.33

Junk 0.25 0.02 0.12 1.81 0.662.49 0.35 0.69 10.70

GRS F -stat. = 2.26, p-val. = 0.05

Panel C: Portfolios sorted on illiquidity

SortOrder Quintile α Market Maturity Credit Adj.-R2

More 1 0.04 -0.05 0.48 0.32 0.62Illiquid 1.53 -2.27 10.67 5.63

↓ 2 0.02 -0.06 0.48 0.23 0.610.83 -4.15 10.04 3.94

3 0.04 -0.08 0.54 0.25 0.581.13 -4.78 10.09 4.34

4 0.04 -0.10 0.66 0.47 0.711.19 -4.01 14.02 6.15

5 0.10 -0.05 0.72 0.80 0.701.93 -1.70 8.03 8.53

GRS F -stat. = 1.22, p-val. = 0.30

34

Table 9: Time-Series Regessions Including Illiquidity FactorThe table reports results for weekly time-series regressions of the indicated test asset returns on the returns of the factor-mimicking portfolios. Theregression specification is given by:

Rei,t = αi + β0,iMarkett + β1,iMaturityt + β2,iCreditt + β3,iIlliquidityt + εit.

The GRS F -stat. is the M. R. Gibbons et al. (1989) test that the alphas are jointly equal to zero.

Panel A: Portfolios sorted on maturity

SortOrder Quintile α Market Maturity Credit Illiquidity Adj.-R2

Longer 1 0.03 -0.04 -0.08 0.16 0.37 0.32Maturity 0.79 -2.52 -1.28 2.69 2.58

↓ 2 0.00 -0.08 0.32 0.26 0.29 0.420.13 -3.55 6.27 3.20 2.17

3 0.03 -0.10 0.62 0.46 0.24 0.660.82 -4.10 12.38 6.59 1.14

4 -0.03 -0.12 0.89 0.37 0.43 0.76-0.86 -4.90 13.51 5.28 2.25

5 0.03 -0.03 1.16 0.47 0.21 0.871.06 -1.32 18.76 6.11 1.94

GRS F -stat. = 1.74, p-val. = 0.13

Panel B: Portfolios sorted on credit rating

CreditRating α Market Maturity Credit Illiquidity Adj.-R2

AAA 0.05 -0.05 0.55 -0.23 0.49 0.611.69 -2.52 9.07 -2.79 3.32

AA 0.00 -0.06 0.44 -0.06 0.22 0.540.03 -3.61 9.51 -1.23 1.94

A 0.01 -0.07 0.58 0.05 0.26 0.620.42 -4.27 10.58 0.71 1.82

BBB -0.03 -0.07 0.74 0.64 0.09 0.81-0.92 -2.95 18.22 8.31 0.77

Junk 0.20 0.01 0.12 1.60 0.97 0.692.01 0.11 0.79 8.58 2.09

GRS F -stat. = 1.72, p-val. = 0.14

Panel C: Portfolios sorted on illiquidity

SortOrder Quintile α Market Maturity Credit Illiquidity Adj.-R2

More 1 0.03 -0.05 0.48 0.34 -0.08 0.62Illiquid 1.04 -2.26 10.60 5.81 -0.88

↓ 2 0.00 -0.06 0.48 0.21 0.07 0.610.15 -3.96 10.05 3.46 0.53

3 0.01 -0.08 0.54 0.22 0.17 0.580.41 -4.77 10.43 3.10 1.10

4 0.01 -0.10 0.66 0.38 0.39 0.730.23 -4.66 15.87 5.41 3.22

5 0.05 -0.07 0.72 0.55 1.11 0.821.14 -2.99 10.09 7.27 6.07

GRS F -stat. = 0.64, p-val. = 0.67

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Table 10: GMM Estimation Results

The table displays GMM estimates of the factor prices of risk, λ, and the coefficients of the pricing kernel, δ, for the four factor pricing model. In

each panel, we report results for different sets of test assets. The J-test is Hansen’s (1982) test on the overidentifying restrictions of the model. The

“Joint” test is a Wald test of the joint significance of the risk premia. The J and Joint tests are computed using the optimal weighting matrix of the

GMM estimator. We denote by “HJ” the Hansen-Jagannathan (1997) distance measure, that is, the least-square distance between the given pricing

kernel and the closest point in the set of pricing kernels that price the assets correctly, computed using a weighting matrix of second moments of

asset returns. The p-value of this measure is computed using 10,000 simulations. The calculations are made at a weekly frequency.

Panel A: 15 portfolios sorted on maturity, credit, and illiquidity

Test StatisticsCoefficient Market Maturity Credit Illiquidity J Joint HJλ 0.8614 0.1914 0.1456 0.1634 16.7209 0.0000 0.4717

t-stat. 2.4401 2.1448 1.9788 3.0902 p-val. 0.1164 0.0216 0.1008δ -15.2759 -35.8469 22.8545 -142.7490

t-stat. -1.8805 -2.0493 1.3031 -2.6803

Panel B: 25 portfolios sorted on maturity and then illiquidity

Test StatisticsCoefficient Market Maturity Credit Illiquidity J Joint HJλ 0.1624 0.2270 0.6863 0.0980 19.9547 0.0000 0.5820

t-stat. 0.5444 3.2234 6.3137 3.4054 p-val. 0.5241 0.0000 0.0551δ 9.7976 -38.0862 -133.9629 61.3983

t-stat. 1.3886 -3.0022 -6.3201 2.3033

Panel C: 25 portfolios sorted on credit and then on illiquidity

Test StatisticsCoefficient Market Maturity Credit Illiquidity J Joint HJλ 0.2282 0.1625 0.0677 0.0510 23.6807 0.0000 0.6526

t-stat. 0.7401 2.2029 1.2591 1.7960 p-val 0.3088 0.0438 0.0130δ -3.0939 -31.9604 1.5179 -39.8332

t-stat. -0.4257 -2.2015 0.1096 -1.3938

Panel D: 25 portfolios sorted on credit and then maturity

Test StatisticsCoefficient Market Maturity Credit Illiquidity J Joint HJλ -0.2336 0.1627 0.2020 0.3024 21.7109 0.0000 0.8431

t-stat. -0.6954 2.5546 3.3064 4.7114 p-val 0.4163 0.0000 0.0004δ 17.9125 -38.3638 29.9850 -366.4195

t-stat. 1.8291 -2.6015 1.5864 -4.6217

Panel E: The 75 portfolios of Panels B, C, and D

Test StatisticsCoefficient Market Maturity Credit Illiquidity J Joint HJλ 0.8810 0.5554 0.1988 0.4341 31.5355 0.0000 1.7062

t-stat. 5.8918 14.3500 4.8161 18.4239 p-val 1.0000 0.0000 0.0000δ -14.8905 -185.5877 141.8108 -769.3161t-stat. -2.2072 -14.3713 8.2148 -19.4089

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Table 11: GMM Estimates of Factor Risk Premia: Alternative Illiquidity Measures

The table displays GMM estimates of the factor prices of risk, λ, and the coefficients of the pricing kernel, δ, for the four factor pricing model. In

each panel, we report results for different sets of test assets. The J-test is Hansen’s (1982) test on the overidentifying restrictions of the model. The

“Joint” test is a Wald test of the joint significance of the risk premia. The J and Joint tests are computed using the optimal weighting matrix of the

GMM estimator. We denote by “HJ” the Hansen-Jagannathan (1997) distance measure, that is, the least-square distance between the given pricing

kernel and the closest point in the set of pricing kernels that price the assets correctly, computed using a weighting matrix of second moments of

asset returns. The p-value of this measure is computed using 10,000 simulations. The calculations are made at a weekly frequency.

Panel A: Dollar volume

Test StatisticsCoefficient Market Maturity Credit Illiquidity J Joint HJλ 0.1073 0.1527 0.1857 0.2571 15.2309 0.0000 0.5124

t-stat. 0.2253 1.7193 1.8644 3.0953 p-val. 0.1722 0.0042 0.0250δ 8.7445 -84.7292 -13.5743 -231.7675

t-stat. 0.6329 -2.9933 -0.6116 -2.9976

Panel B: Volatility-volume ratio

Test StatisticsCoefficient Market Maturity Credit Illiquidity J Joint HJλ 0.2683 0.1930 0.1217 0.2075 17.5834 0.0000 0.4961

t-stat. 0.7049 2.0961 1.0941 2.9391 p-val. 0.0918 0.0069 0.0389δ 2.1869 -52.9970 28.5655 -191.5089

t-stat. 0.1985 -2.6376 1.3207 -3.0564

Panel C: Turnover

Test StatisticsCoefficient Market Maturity Credit Illiquidity J Joint HJλ 0.1476 0.0465 0.2308 -0.0783 18.0021 0.0000 0.5240

t-Stat. 0.4926 0.6211 2.5374 -0.9325 p-val 0.0815 0.1155 0.0228δ 0.0622 -26.7466 -31.5968 51.7326

t-stat. 0.0085 -1.4517 -2.1904 1.1151

37

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