Realized Volatility, Liquidity, and CorporateYield Spreads

Marco Rossi∗

December 7, 2009

AbstractThis study revisits the relative role of credit risk and illiquidity as determinants ofcorporate bond prices. Using TRACE transaction data from January 2003 to Decem-ber 2008, I find that high-frequency firm equity volatility measures explain 40% of thevariation of corporate yield spreads and as much as 52% of the variation in a multivari-ate context. In contrast, trading-based illiquidity measures such as the percentage ofzero trading days, the friction measure used by Chen, Lesmond, and Wei (2007), anda new measure derived in this paper explain approximately an incremental 1% of thecross-sectional variation of yield spreads. Unlike other friction models, the proposedmodel includes firm-specific factors in the return generating process and relaxes theassumption of constant liquidity. As a result, liquidity cost estimates are lower anduncorrelated with corporate yield spreads. Overall, my findings suggest that creditrisk is the main determinant of corporate yield spreads and that certain trading-basedilliquidity measures actually capture credit risk.

Keywords: liquidity, credit risk, realized volatility, stochastic friction, TRACE.

∗Smeal College of Business, Department of Finance, Penn State University. Email: marco.rossi@psu.edu;Phone: +1-814-865-2446. This paper has benefited enormously from discussions with Jingzhi Huang (com-mittee chair) and Jean Helwege. I would also like to thank the other members of my committee, namelyJoel Vanden, William Kracaw, and John Liechty, as well as Jack Bao, Peter Ilev, Berardino Palazzo, LukasRoth, Robert Dittmar, Asher Curtis, Lubomir Petrasek, Xuemin (Sterling) Yan, Chunchi Wu, KingsleyFong, and seminar participants at Penn State, the University of Missouri, the University of Kansas, theBank of Canada, Seattle University, HEC Montreal, and the University of New South Wales for their helpfulcomments and suggestions. This paper has benefited also from comments received at the FMA (2009) andNFA(2009) conferences and at the FMA (2009) doctoral consortium.

Introduction

Structural credit risk models cannot adequately explain credit spread levels, credit spread

changes, and observed defaults.1 On one hand, this empirical failure has prompted re-

searchers to reconsider the role of idiosyncratic volatility and jumps to better describe credit

risk. On the other hand, a growing number of studies argues that illiquidity might be an

additional determinant of corporate bond prices. However, in informationally efficient mar-

kets,2 trading decisions are simultaneously affected by investors’ assessment of credit risk

(information side) and by the liquidity of the bond that they wish to trade (friction side).

In order to avoid attributing to illiquidity what is in fact credit risk, I propose a stochastic

friction model of corporate bond returns and liquidity developing the idea that, while system-

atic and idiosyncratic credit risk variables affect expected returns, bond returns are observed

only if there is enough valuable information to justify their liquidity costs. After obtaining

several trading-based liquidity measures, I conduct a cross-sectional regression analysis to

assess the relative importance of illiquidity and credit risk variables in the determination of

corporate yield spreads.

In recent applications of the standard friction model (Rosett (1959)) to stock and bond

liquidity, systematic factors summarize all the relevant information and liquidity is constant

(Lesmond, Ogden, and Trzcinka (1999), and Chen, Lesmond, and Wei (2007)). However,

these assumptions are quite restrictive given that the contingent claim approach to structural

credit risk modeling (e.g. Merton (1974)) assigns an important role to idiosyncratic variables

and given the evidence that corporate bond liquidity is time-varying.

I improve on the standard friction approach in several ways. First, I model bond returns

1See Eom, Helwege, and Huang (2004) on spread levels, Collin-Dufresne, Goldstein, and Martin (2001)on spread changes, and Huang and Huang (2003) on the credit spread puzzle, stating that credit risk modelcannot simultaneously explain observed defaults and average corporate yield spreads.

2Hotchkiss and Ronen (2002) and Ronen and Zhou (2008) conclude that the US corporate bond marketis efficient.

2

as a function of individual factors such as equity returns and realized volatility. The use of

high frequency measures, such as realized volatility, is better able to identify jumps, which

carry more information than historical volatility estimated with daily data (see e.g. Huang

(2007)). Moreover, considering the relation between bond and equity returns of the same firm

allows for a hedging interpretation, rather than a linear asset pricing model interpretation

with all its caveats.3

Next, I use a panel data approach to model liquidity as a function of bond characteristics

such as age and issue size4, whereas in previous friction models bond liquidity measures

were estimated individually as constant parameters. The shrinkage induced by the use of

a hierarchical panel approach has the additional benefit of reducing the variance of the

estimated parameters of both the expected return and liquidity component of the model.

Lastly, I assume heteroscedastic shocks to bond returns. This last modeling approach

makes estimates robust to the presence of outliers, which are typical of tic-by-tic transaction

data (Brownlees and Gallo (2006)). The use of actual transaction data from TRACE is yet

another improvement on studies using Datastream or matrix price data.5

The most important contribution of my study comes from the cross-sectional analysis of

corporate yield spreads and their determinants. Using bond transaction data in the period

from January 2003 to December 2008, I find that credit risk variables account for a substan-

tial portion of the cross-sectional variation of credit spreads. In particular, individual realized

equity volatility measures have as much explanatory power as credit ratings. In contrast,

several trading-based liquidity measures have little explanatory power in a multivariate con-

text that adequately controls for credit risk. For instance, in univariate regressions the LOT

3Schaefer and Strebulaev (2008) show that structural models can produce plausible hedge ratios even ifthey cannot produce correct bond prices.

4Alexander, Edwards, and Ferri (2000), and Hotchkiss and Jostova (2007) show that age and issue sizeare among the main determinants of trading activity.

5See Warga and Welch (1993) on the problems of using matrix-based data for studies involving corporatebonds.

3

measure proposed by Chen, Lesmond, and Wei (2007) is positively associated with credit

spreads, but the strength of this relation goes down substantially once realized volatility is

included in the regression. This happens because firm equity realized volatility and the LOT

measure are significantly positively correlated. This positive correlation is likely to indicate

that this liquidity measure might be capturing a form of mispricing associated to an omitted

variable (volatility) in the return generating process. Overall, this paper establishes the first

order importance of credit risk, as opposed to illiquidity as proxied by certain trading-based

measures, in the determination of credit spreads.

My study has several other notable results. In a univariate analysis, I document that

trading activity is strongly affected by bond characteristics such as age, time to maturity, and

issue size. In particular, I show that small and short-maturity bonds trade very infrequently

relative to young and large issues. I corroborate these findings with the friction model

estimates which show that round-trip liquidity costs are decreasing with issue size and time

to maturity, and increasing with age and the eurodollar-treasury (TED) spread, which is

meant to pick up flight-to-liquidity effects. Using year dummies to capture calendar time

effects, I find that liquidity cost spiked in 2007 (credit-crisis) and went down again in 2008

without reaching, however, pre-crises levels.

The remainder of the paper is structured as follows. Section 1 presents a literature review.

Section 2 describes the derivation of the proposed illiquidity measure. Section 3 presents the

data. Section 4 applies the model to the data and presents estimation results. In section

5, I conduct a regression analysis of the determinants of corporate yield spreads. Section 6

concludes the study.

4

1 Related Literature

This paper relates to the study of Chen, Lesmond, and Wei (2007) who propose a limited

dependent variable (LDV) approach to derive a bond-specific measure of liquidity (also

known as the LOT measure). Applied on Datastream data, this liquidity measure explains

7% of the cross-sectional variation in yield spreads for investment grade bonds, and up to

22% of the variation for speculative grade bonds. The LDV approach posits the existence of a

linear model of bond returns which are observed only when they are beyond a threshold value.

These thresholds can be seen as an estimate of transaction costs. In the LOT measure, the

pricing factors are given by the daily change in the 10-year risk-free interest rate (systematic

bond factor), and the returns on the Standard & Poor’s 500 index (systematic equity factor).

The thresholds are simply two constants: one for negative returns (sell-side cost), and one

for positive returns (buy-side cost). However, being contingent claims on the firm’s assets,

debt and equity should depend on idiosyncratic as well as systematic factors. Furthermore,

there is evidence of substantial time variation in recently proposed liquidity measures (see

e.g. Longstaff, Mithal, and Neis (2005), and Bao, Pan, and Wang (2008)).

My paper is also closely related to studies by Campbell and Taksler (2003) and Zhang,

Zhou, and Zhu (2009) who use equity and realized equity volatility measures to explain

credit and CDS spreads.6 Campbell and Taksler (2003) find that the correlation between

equity volatility and the spread of an index of A-rated bonds over treasuries is 0.7 in the

sample period from 1965 to 1999. Zhang, Zhou, and Zhu (2009) take this approach one step

further by using diffusive and jump volatility measures from high frequency equity data as

explanatory variables in CDS spread regressions. They find that volatility alone explains

roughly 50% of the variation in CDS spreads. I use the information from equity volatility

in two ways. First, similarly to Zhang, Zhou, and Zhu (2009), I extract equity jumps from

6Goyal and Santa-Clara (2003) conclude that idiosyncratic risk matters, even in the context of corporatebond pricing.

5

equity realized variance and use both jumps and the resulting diffusive volatility as factors in

my bond return generating process. The idea is that jumps typically represent news entering

investors’ information sets and, therefore, are likely to be associated with trading activity in

the bond market. Second, given the documented importance of jumps,7 I investigate which

component of equity volatility impacts credit spreads the most.

More generally, my study is related to a series of papers dealing with liquidity, credit

risk, and their interaction. Longstaff, Mithal, and Neis (2005) use CDS data to extract the

default component of credit spreads and suggest that taxes and illiquidity in the bond mar-

ket explain the non-default component. Bao and Pan (2008) relate excess bond volatility (at

short horizons) to corporate bond liquidity. Bao, Pan, and Wang (2008) use the negative

of the auto-covariance of bond prices as a measure of liquidity. They document substantial

commonality across individual measures and correlation with market volatility (VIX). Ma-

hanti, Nashikkar, Subrahmanyam, Chacko, and Mallik (2008) propose a measure of liquidity

defined as the “weighted average turnover of investors who hold a particular bond, where the

weights are the fractional holdings of the amount outstanding of the bond”. The intuition

behind this measure is that investors with high turnover prefer to hold bonds with lower

transaction costs, and they further improve the liquidity of these bonds by trading them.

Finally, Ambrose, Cai, and Helwege (2008) and Ambrose, Cai, and Helwege (2009) consider

the confounding effects of credit risk and selling pressure for fallen angels, and argue that

most of the price variation taking place with a downgrade to junk status is due to credit risk

(information) rather than selling pressure (friction).8

7See also Tauchen and Zhou (2006), who show that the volatility of realized market jumps is able toexplain more than 60% of the variation of Moody’s AAA and BAA credit spread monthly indices, andCremers, Driessen, and Maenhout (2008) on the importance of firm-specific jumps.

8See also Driessen (2005), Houweling, Mentink, and Vorst (2003), and Kalimipalli and Nayak (2009).

6

2 Model

The idea of a friction model of liquidity is that while true returns depend on several stochastic

factors, observed returns will reflect changes in the underlying factors only if the information

value of the marginal trader is sufficient to cover the liquidity cost incurred upon trading

(Chen, Lesmond, and Wei (2007)). Unlike the LOT model, the empirical model that I pro-

pose allows for liquidity costs to vary over time,9 with bond characteristics, and with macro

variables.10 I also include a bond-specific effect to account for those effects not captured by

the variables included in the model. Another important difference between my model and

the LOT model is that I use a hierarchical panel data approach to simultaneously estimate

all the bonds’ parameters, whereas Chen, Lesmond, and Wei (2007) carry out individual es-

timations. The authors recognize that their modeling approach prevents them from reliably

estimating their illiquidity measures for bonds with a censoring in excess of 85% in any given

year. These “extreme” bonds, however, are likely to carry precious information regarding

liquidity in the corporate bond market. Therefore, the additional benefit of the proposed

methodology is to include bonds that trade very infrequently.

2.1 Evolution of Bond Returns and Liquidity Costs

I characterize the effect of liquidity on observed bond returns as Rit = R∗it − Ljit, j = {s, b}.

The latent true bond return R∗it, and the sell-side and buy-side liquidity costs Ljit, indexed

9To be precise, Chen, Lesmond, and Wei (2007) estimate their models once a year for every bond in orderto preserve some time variation. and with observable bond characteristics.

10In a recent paper, Omori and Miyawaki (2009) independently derive a tobit model with covariate depen-dent thresholds and homoscedastic errors. However, their thresholds are linear functions of the covariates,which complicates and slows down their sampling scheme in order to attain non-negative thresholds, anddepend on individuals only, but not on time.

7

by s and b respectively, are given by

R∗it = β′ixit + εit, εit ∼ N(0, σ2it) (2.1)

Lsit = αs

i (zi)× exp{γ′zit}, αsi < 0 (2.2)

Lbit = αb

i(zi)× exp{γ′zit}, αbi > 0, (2.3)

where xit is a vector containing both systematic and idiosyncratic risk factors, zi is a vector

of time-invariant bond characteristics, and zit is a vector of time varying variables.11 The

choice of the exponential in Equations (2.2) and (2.3) guaranties positivity of liquidity costs.

Returns are observed only when they are large enough (in absolute value) to justify

transaction costs. The observation rule of bond returns that I propose is a generalization of

the friction model originally proposed by Rosett (1959)

Rit =

R∗it − Ls

it, R∗it < Lsit

0, Lsit ≤ R∗it ≤ Lb

it

R∗it − Lbit, R∗it > Lb

it

(2.4)

Round trip liquidity costs are obtained as

Costit ≡ Lbit − Ls

it = [αbi(zi)− αs

i (zi)]× exp{γ′zit}, (2.5)

which reduce to the LOT measure when zi = 1 and zit = ∅.11See Section 4 for the exact specification of the proposed models.)

8

2.2 Observed and Augmented Likelihood

Given the parameters, the latent variables, and the friction model, the likelihood of every

observation is given by

p(Rit|Lsit, L

bit, βi, σ

2it, xit) =

[1

σit

φ

(Rit + Ls

it − β′ixit

σit

)]1{Rit<0}

×[1

σit

φ

(Rit + Lb

it − β′ixit

σit

)]1{Rit>0}

×[Φ

(Lb

it − β′ixit

σit

)− Φ

(Ls

it − β′ixit

σit

)]1{Rit=0}

(2.6)

For purpose of estimation of the parameters entering Equation 2.1, it is better to work with

the augmented likelihood function (see e.g. Chib (1992))

p(R∗it|Rit, Lsit, L

bit, βi, σ

2, xit) =1

σit

φ

(R∗it − β′ixit

σit

), (2.7)

where the rule for obtaining the latent variable R∗it is given by

p(R∗it|Rit, Lsit, L

bit, βi, σ

2it, xit) =

Rit + Ls

it, Rit < 0

∼ TN(Lsit,L

bit)

(β′ixit, σ2it), Rit = 0

Rit + Lbit, Rit > 0

(2.8)

2.3 Prior Distributions and Hierarchical Structure

In order to reduce the variance of the estimates and preserve enough heterogeneity across

bonds, I propose a hierarchical bayesian panel regression model with random coefficients.12

12See Tsionas (2002) for a similar approach in the context of stochastic frontier models, and Greene (2005)for a frequentist treatment of heterogeneity in panel data estimators of the stochastic frontier model.

9

Formally, factor loadings and bond-specific average liquidity are distributed as

βi|β,∆ ∼ N(β,∆) (2.9)

−αsi |αs, zi, σ

2s ∼ LogN(α′szi, σ

2s) (2.10)

αbi |αb, zi, σ

2b ∼ LogN(α′bzi, σ

2b ), (2.11)

where zi is a vector of time-invariant bond characteristics.

I simplify the variance structure of the error term in equation (2.1) as in Geweke (1993)

by using the decomposition σ2it = σ2 × vit where

σ2 ∼ IG(sh, sc) (2.12)

r/vit ∼ χ2(r). (2.13)

The degrees-of-freedom parameter r captures the extent of heteroscedasticity in the data.

Low values of r reflect the prior beliefs that the data might contain several large outliers, while

large values of r are consistent with homoscedastic error terms. The model in equation (2.1)

may be represented (for every bond i) in matrix notation as R∗i = Xiβi +εi, εi ∼ N (0, σ2Vi),

where Vi = diag(vi1, vi2 . . . , viTi).

To complete the model, a few more prior distributions need to be specified. I impose flat

priors for the parameters β, αs, αb, and γ, and diffuse Inverse Wishart and Inverse Gamma

priors for the remaining parameters:

∆ ∼ IW (∆0, N0) (2.14)

σ2s , σ

2b ∼ IG(sh, sc). (2.15)

10

2.4 Posterior Distributions

Bayesian estimation of the model parameters and latent variables requires the combination of

the likelihood of the model, and the use of prior information on the parameters. To simplify

notation, collect the parameters of the data generating process into ΘR, and those of the

liquidity processes into Θl, and define Θ ≡ [ΘR,Θl] and the prior over these parameters as

p(Θ). We obtain the posterior distribution of the parameters and the latent variables, given

the data, as

p(Θ, Ls, Lb|R,X,Z, Z) ∝ p(R|Ls, Lb, X,ΘR)× p(Ls, Lb|Z, Z,Θl)× p(Θ). (2.16)

For the estimation of βi and σit, it is convenient to work with the augmented likelihood:

p(Θ, Ls, Lb, R∗|R,X,Z, Z) ∝ p(R∗|Ls, Lb, R,X,ΘR)× p(Ls, Lb|Z, Z,Θl)× p(Θ). (2.17)

Sampling directly from the joint posterior distribution of the parameters is not feasible.

However, the parameters can be estimated using a Markov Chain Monte Carlo (MCMC)

algorithm (see Appendix D), which is an iterative scheme to draw from the conditional

distributions of blocks of parameters of the vector Θ. Conditional posterior distributions of

these blocks of parameters are derived in Appendix A.

3 Data

The data in this study come from five sources. The fixed investment securities data base

(FISD) provides bond characteristics; the trades reporting and compliance engine (TRACE)

contains the actual bond transaction data; CRSP contains stock price data; COMPUSTAT

contains balance sheet data; the Dow Jones trades and quotes (TAQ) database is used to

11

construct realized volatility, and its diffusive and jump components. In Appendix B, I provide

details on these databases, and on the filters used to determine the final sample.

3.1 Corporate Bond Transaction Data (TRACE)

Under the pressure from several government bodies and buy-side traders, on July 1, 2002,

the National Association of Securities Dealers (NASD) started a three-phase dissemination

process of corporate bond transactions through its trades reporting and compliance engine

(TRACE).13 This process progressively increased the pool of bonds subject to dissemination

resulting, after 2004, in approximately 95% coverage of US corporate bonds. The only bond

transactions not reported to TRACE are those that take place in exchanges, e.g. NYSE’s

automated bond system (ABS). Although the role of TRACE is to increase transparency

in the corporate bond market (see Bessembinder, Maxwell, and Venkataraman (2006) and

Edwards, Harris, and Piwowar (2007)), not all information is released after each transaction.

For instance, until recently the side of the transaction was unknown, and the size of the trade

is top-coded to one and five million dollars for junk and investment-grade bonds respectively.

I use bond transaction data from January 2003 to December 2008. Bond characteristics

are obtained from the fixed investment securities database (FISD) compiled by Mergent Inc.

I only include corporate bonds (medium term notes and debentures) with no optionality and

no credit enhancement which leads to a final sample with senior unsecured bonds without

call, put, and conversion options. Because utilities and financial firms are heavily regulated,

and their leverage might not be representative of the true credit situation of the firm, I only

include industrial firms in my final sample. Implementing these filters, I obtain a starting

sample of 3099 corporate bonds which I then merge with TRACE to obtain transaction data.

The merge with TRACE results in a sample of 1601 corporate bonds.

I use information from the TRACE record file to filter out irregular and special trades, and

13The body that oversees TRACE now is the Financial Industry Regulatory Authority (FINRA)

12

trades that include any dealer commission. To minimize the impact of unusual observations,

I keep price observations that pass the following screening:14

|p−med(p, k)| ≤ 5 ∗MAD(p, k) + g, (3.1)

where g is a granularity parameter which I set equal to $1, and med(p, k), and MAD(p, k)

are respectively the centered rolling median, and median absolute deviations of the price

p using k observations (I set k = 20).15 After implementing this screening, I keep bonds

that are traded on at least 20 distinct days. After imposing these filters, and merging the

resulting bond data with the CRSP, COMPUSTAT, and TAQ databases, I obtain a final

sample of 984 bonds.

Table 1 reports descriptive statistics, grouped by year, on bond characteristics and trans-

actions. The final sample includes 984 bonds issued by 181 entities (identified using the

parent id in FISD). Panel A of the table provides information on bond characteristics such

as issue size, coupon rate, and time to maturity at issuance. As can be seen, the time to

maturity at issuance has increased over time, while the average coupon rate has remained

quite stable ranging from 6.59% to 7.09%. Panel B reports descriptive statistics on trading

activity. There is a total of 1,916,412 trades spread over 6 years. Although this seems quite a

large number, there are in fact on average approximately 325 (≈ 1916412/984/6) trades per

bond per year, or 6.25 trades per week. Furthermore, there are substantial cross-sectional

differences in trading activity depending on bond characteristics (see Table 2). This is the

first indication of the scarce liquidity of the corporate bond market relative to the equity

market as the equity of a typical issuer in my sample often trades more than 10,000 times

in a single day. The bonds’ age at the time of trade averages 6.63 years, and is lower in

14Brownlees and Gallo (2006) propose a similar algorithm, based on rolling trimmed statistics, to filterTAQ data.

15I also try to eliminate return reversals as in Bessembinder, Maxwell, and Venkataraman (2006) but thisprocedure leaves too many observations that are clearly outliers.

13

the early TRACE years. Finally, the price and size percentiles of the distribution show that

approximately half of the trade prices are within 5% of par, and that a significant proportion

of trade sizes (50%) is below $25,000 indicating an active presence of retail investors in the

corporate bond market.

Table 2 reports the number of bond transactions grouped by issue size (in one dimension)

and by two measures of bond seasoning (in the other dimension): Panel A considers a

classification by age; Panel B considers a classification by time to maturity. The number

in parenthesis represent the number of bonds in each category. This table shows that large

issues trade much more frequently. Bonds with an issue size smaller than 50 million (80

in total) trade 15568 times, while bonds with an issue size in excess of 250 million (353

in total) trade almost 1.5 million times during the same period. There is also substantial

variation in trading volume depending on age and time maturity, with large issues trading

less frequently as they age, and smaller issues doing just the opposite. Finally, trading

activity declines rapidly when bonds approach maturity.

3.2 Bond Returns and Credit Spreads

Bond returns are defined as

Rt =Pt + AIt + Ct

Pt−1 + AIt−1

,

where Pt is the clean price of the bond, AIt is the accrued interest over one period, and Ct is

the coupon payment whenever it is paid (in which case AIt = 0). This definition presupposes

the existence of consecutive bond price observations. However, many bonds can go without

trading for weeks, or even months. Consistently with my model, I set Rt = 0 if no trading

occurs. Care must be taken, however, on the first day of trading after a period of stale prices,

as two consecutive genuine price observations are not available. To compute this return, I

use the last available stale price. I also conduct my analysis using a linearly interpolated

14

price16 as the price that goes in the denominator, and the results are virtually identical.

Table 3 reports the first four moments of the distribution of bond returns across credit

ratings and time to maturity. The last two rows in each panel report the percentage of

days in which bonds in a given category trade, and the total number of days in which these

bonds could have traded. Two things are worth noting in this table. First, the observed

bond return distribution is highly non-normal as can be seen from the the excess kurtosis

(fat tails). Second, there is a negative relation between ratings and trading for short- and

medium-term bonds, which disappears for long-maturity bonds. This lack of monotonicity

for long-term bonds is most likely due, however, to the presence of the heavily traded General

Motors and Ford bonds in the (long term) BB/B/CCC categories.

I compute credit yield spreads as the difference between the daily yield on the corporate

bond (obtained by averaging the available yields on a given day) and the yield on the treasury

benchmark with the same time to maturity. The constant maturity benchmark yields are

from Datastream and are for the following yearly maturities: 1/12, 1/4, 1/2, 1, 2, 3, 5,

7, 10, 20, 30. I use linear interpolation to get the yield of intermediate maturities. Table

4 presents average credit spreads categorized by rating in one dimension and by time to

maturity (Panel A) or by year (Panel B) in the other dimension. With the exception of the

medium-term CCC-D category, which contains few observations, it can be seen that credit

spreads are increasing in time to maturity and in credit risk. The break down by year (Panel

B) reveals the effect of the credit crisis on credit spreads taking place in 2007 and 2008.

3.3 High-Frequency Equity Data (TAQ)

Given the documented importance of equity volatility, I use high-frequency equity data from

the NYSE trades and quotes (TAQ) database to compute equity realized volatility and to

16For example, suppose there is a trade at t = 0 for $100, and no trade at t = 1, but there is a trade at t= 2 for $102. The return for t = 1 is 0 and the return for t = 2 is 1%. In case of stale prices the return att=2 would be equal to 2%.

15

disentangle its diffusive and jump components (see Figure 1 for an example). These two

components, as well as individual equity returns, are used as explanatory variables of bond

returns, and yield spreads. Before processing the data, I impose the filter presented in (3.1)

with k = 50 and g = 0.05. The choice of these parameters reflects the fact that stock

tit-by-tic data are much more numerous and closer in time than bond transactions.

To screen out jumps, I use a nonparametric approach developed by Barndorff-Nielsen and

Shephard (2004) which relies on the concepts of realized variance and bipower variation.17 In

Appendix C, I explain briefly the methodology to recover jumps from high frequency data,

and provide references for its exact implementation.

As an example, in Figure 1, I present the realized volatility (top graph) of Ford Motor

Company which is one of the most important players in the corporate bond market. The

middle and bottom graphs represent the diffusive and jump component of volatility which

sum to realized volatility. As can be seen, volatility has increased substantially since Jan

2007, a pattern shared with the equity return volatility of most firms in the sample.

4 Estimation by Markov Chain Monte Carlo (MCMC)

In this section, I estimate the parameters of the bond return generating process, and the two

liquidity thresholds. I implement a Gibbs sampler whenever the posterior distribution of a

given block is a standard one, and a Metropolis-Hastings algorithm otherwise. In Appendix

D, I provide a pseudo-code which describes the estimation algorithm in detail.

17See Huang and Tauchen (2005), Barndorff-Nielsen and Shephard (2006) and Huang (2007) for an appli-cation of this approach.

16

4.1 Friction Model Specifications

I consider 3 different specifications for the bond return generating process in equation (2.1):

for the LOT measure I use the changes in the long-term default-free rate (systematic bond

factor), and market equity returns (systematic equity factor) as in Chen, Lesmond, and Wei

(2007); in the second specification, in addition to the bond factor, I use firm equity returns,

and realized equity return volatility; in the last specification I substitute realized volatility

with its jump and diffusive components. Following Chen, Lesmond, and Wei (2007), I

interact all the factors with the duration of the bond.

The liquidity covariates in the threshold component of the model, i.e. equations (2.2)

and (2.3), are: issue size (in log), coupon rate, time to maturity at issuance (in log), age (6

year interval dummies), calendar time (6 yearly dummies), and the difference between the

30-day eurodollar rate and the 30-day treasure rate (TED spread). I use the log of issue size

and time to maturity because the distribution of the level of these variables is very skewed18.

4.2 LOT Measure with TRACE Data

Table 5 reports average estimates of the friction model proposed by Chen, Lesmond, and Wei

(2007) grouped by median time to maturity and median rating, where the median is taken

over the estimation period. Although I have estimated this measure for every bond, to be

consistent with Chen, Lesmond, and Wei (2007), I report results only for bonds that trade

on at least 15% of trading days. The results reported in the table are comparable to those

reported in Table 2 (page 129) of Chen, Lesmond, and Wei (2007) and are characterized by

a mostly negative relation between the LOT measure and credit quality. With regard to the

factor loadings, it can be seen that the loadings on the bond factors are mostly negative,

but not increasing with credit risk as one would expect from theory. The idea is that highly

18Histograms of these distributions are available upon request

17

rated bonds behave more like treasuries and their value varies with interest rates rather than

with the equity market factor. With regard to the loading on the systematic equity factor,

the table shows that this loading is often positive, but there are some exceptions especially

for short-term bonds.

Figure 2 reports the distribution of the parameter estimates of the LOT model: the first

row reports the distribution of the factor loadings; the second row reports the distribution of

the sell- and buy-side transaction costs; the last row reports the distributions of the estimated

round-trip liquidity costs. The most important feature of these graphs is the dispersion in

the parameter estimates. While the bond factor loadings are negative on average, it can

be seen that there are estimates as high as almost 10. The same observation can be made

for the equity factor loading: it’s slightly positive on average, but it can get quite negative.

Finally the LOT measure can be as high as 45%, making this measure hard to attribute only

to transaction costs. Moreover, when I estimate the LOT measure on bonds that trade very

infrequently, this measure can be as high as 80%. The top panel of Figure 4 shows a scatter

plot of the LOT measure against the percentage of zero trading days. As can be seen, for

bonds that trade less than 15% of the time (when the x axis approaches 1) estimation of the

LOT measure becomes very unreliable.

4.3 Idiosyncratic Factors and Time-Varying Liquidity

Table 6 presents average estimation results for the parameters in the return component of

the model (Equation (2.1)). The results are categorized by (median) rating and time to

maturity. The difference between the two models in the table is that in the second one

realized volatility is replaced by its diffusive and jump components. At daily frequency, the

results are very similar and, therefore, the following comments are good for both models.

Bond factor loading are mostly negative, and are increasing with credit rating. The average

18

firm-specific equity factor loading is always positive, which means that good news for equity

holders are typically good news for bond holders given that debt and equity are both positive

claims on the assets of the firm. The average volatility factor loading is always negative which

means, assuming that leverage is not too variable, that an increase in business risk typically

causes a deterioration in the value of debt. Contrary to what I find for the LOT measure,

on average the time-invariance component of round-trip liquidity costs (given by the first

part of the expression in Equation (2.5)) is on average well below 5% and does not display

a systematic relation with credit ratings (hence credit spreads).

The hierarchical panel approach, and the resulting shrinkage of individual parameters

toward a common parameter, provides a more robust estimation which avoids the occurrence

of very extreme estimates. In Figure 3, I provide graphical evidence of the effect of shrinkage

in the estimation of the factor loadings. As we would expect from theory, systematic bond

factor loadings are mostly negative. Similarly, the equity and volatility factor loading are

mostly positive and negative respectively. Overall, joint estimation of the parameters avoids

the occurrence of very extreme estimates. The bottom plot of Figure 4 shows graphically

the “stability” of the proposed liquidity measure, relatively to the LOT measure, for bonds

that trade very infrequently.

Table 7 reports estimates of the sensitivity of liquidity costs to several bond character-

istics and to the Eurodollar-Treasury (TED) spread (see Equations (2.2) and (2.3)). The

difference between the two models in the table is in the return component only: in the second

model realized volatility is replaced by its diffusive and jump components. The first two sets

of estimates (Panel A and B) refer to time-invariant regressors and account for asymmetric

responses depending on the sign of observed returns. As can be seen, on average, liquidity

costs are quite symmetric. Panel C refers to those regressors that change over time.19 For

19For numerical reasons I do not allow for asymmetric responses as this would slow the estimation downas the γ vector is most expensive parameter to estimate in terms of time/iterations.

19

each model, the first two columns provide the mean and standard deviation of the poste-

rior distribution; the next two columns report the 1st and 99th percentiles of the posterior

distribution and can be seen as a bayesian confidence interval for the estimated coefficients.

Except for the first three age brackets, all the variables are statistically significant in the

sense that the posterior means are at least two standard deviations away from zero.

Age and issue size are strong determinants of trading activity and round-trip transaction

costs (see Hotchkiss and Jostova (2007)). In particular, a 1% increase in issue size is asso-

ciated with a 0.55% decrease in liquidity cost. The coefficients on the age dummies reveal

that for young bonds age has not much of an effect on liquidity, but, after approximately 9

years, age starts having a bigger and bigger impact. The negative coefficient on the log of

maturity means that the effect of ageing is more pronounced for bonds that are close to ma-

turity. The coupon rate is supposed to capture the effect of taxes (Elton, Gruber, Agrawal,

and Mann (2001)) and the coefficient on this variable reveals that one point increase in the

coupon rate increases liquidity costs by approximately 0.13%. The TED spread, which is

meant to capture flight-to liquidity effects, has a positive effect on liquidity costs. Lastly, the

coefficients on the year dummies reveal clearly the anatomy of the financial crisis. Relative

to 2003, liquidity costs have been going up every year until 2007 (when they picked), and

have come down in 2008, but not to pre-crisis levels.

4.4 A Brief Discussion on the Role of Jump Variation

Model 2 estimates in Table 6 and 7 reveal that, at daily frequencies, disentangling the dif-

fusive and jump components of realized volatility does not change the estimated coefficients

very much. However, although the changes (going from Model1 to Model2) in the “LOT”

estimates in Table 6 are not very big, it is evident that by controlling for more risk factors

the estimated liquidity costs are smaller.

20

The inability of jump variation to reveal its impact on the estimates is also likely due

to the fact that jumps are rare events and many days can go by without observing any

jumps. As a result, the jump variation variable has many more zeros elements than non-zero

ones and this might make it difficult to appreciate its impact. This zero-element problem

is exacerbated at daily frequency and could be substantially mitigated using weekly data. I

leave the exploration of other lower frequencies for future research.

Finally, it is important to observe that although disentangling jumps from diffusive move-

ments does not add explanatory power at a daily frequency, this does not mean that jumps

are not important. It just means that, for the purposes of this study, the impact of jumps

is best appreciated using realized volatility, which is the sum of both jump and diffusive

volatility.

5 Volatility, Liquidity, and Yield Spreads

In this section, I examine the determinants of credit spreads with the objective of establishing

whether volatility (information) or liquidity (friction) plays a dominant role. The general

specification of the regressions is given by

Y ield Spreadit = α + β′1Illiquidityit + β′2V olatiltiyit

+β′3Rating Dummiesit + β′4Accounting V ariablesit

+β′5Macro V ariablest + β′6Other V ariablesit + εit,

and is similar to those estimated by Campbell and Taksler (2003) and Chen, Lesmond, and

Wei (2007).

21

5.1 Yield Spread Determinants: Univariate Analysis

A preliminary graphical analysis reveals that realized volatility plays a very important role

in explaining credit spreads. Figure 5 shows that equity realized volatility (bottom graphs)

is at least as important as credit ratings (upper graphs) in explaining credit spreads. In the

plots I use logs to deal with the heteroscedastic nature of bond yields, the variance of which

increases with credit risk. Another way to deal with heteroscedasticity, as it is often done

in a regression context (e.g. Campbell and Taksler (2003)), is to use a trimmed sample in

which the top and bottom percentiles of yield spreads are eliminated.

Table 8 reports regression estimates of both level and log of yield spreads on several

liquidity measures, realized volatility, and credit ratings. Consistently with the graphical

evidence, the table shows that realized volatility has at least as much explanatory power

as credit ratings. Specifications that include both variables show that rating and realized

volatility alone can explain as much as 60% of the variation in corporate yield spreads.

With regard to the explanatory power of the three trading-based illiquidity measures

(model 1 through 3) considered in this study, it can be seen that the LOT measure is

positively associated with credit spreads, which is to be expected given the results of Table

5, and can explain as much as 14% of the cross sectional variation. In contrast to the LOT

measure, the other two measures of illiquidity are (unconditionally) uncorrelated with yield

spreads. Two facts are worth noting about this result. First, in unreported results I find that

when I regress yield spreads on the LOT measure using the full sample (including infrequently

traded bonds), the R2 falls to zero. Secondly, the explanatory power of the LOT measure in

a univariate context seems to stem from the positive correlation (approximately 12%) that

this measure has with realized volatility.20 The other two measures are uncorrelated with

realized volatility.

20This correlation is to be considered as a lower bound since, unlike realized volatility, the LOT measurevaries only over bonds and not over time.

22

5.2 Yield Spread Determinants: Multivariate Analysis

In Table 9, I estimate several regression models similar to those estimated by Campbell and

Taksler (2003) and Chen, Lesmond, and Wei (2007). The main message of this table is that

liquidity variables contribute minimally to the determination of credit spreads, while credit

risk variables such as volatility and leverage have significant economic impact.21 In the first

three models of the table I include one liquidity measure at the time, while model 4 excludes

them. Although the liquidity variables are characterized by a positive and significant coeffi-

cient, their exclusion has practically no impact on the model fit (R2) and on the magnitude

and sign of the remaining coefficients.

To appreciate the economic impact of credit risk variables, note that a one-standard

deviation change (approximately 0.18) in realized volatility is associated with a variation

in credit spreads of 82 basis points (≈ 4.58% × 0.18) and a one-standard deviation change

(approximately 0.12) in market leverage is associated with a positive variation in credit

spreads of approximately 43 basis points (≈ 3.61%× 0.12).

The ability of credit risk variables to explain credit spreads so well relatively to other

studies is likely due to two reasons. First, realized volatility carries more information than

equity volatility estimated with historical data up to 180 days before the date of interest.

The realized volatility measure that I propose are updated daily, and therefore incorporate

information as it happens and are not smoothed like more conventional historical measures.

Secondly, in the sample period of my study, which includes the recent credit crisis, credit

risk has manifested itself pervasively confirming, ex-post, the expectations that investors

formed, ex-ante, when requiring such high premia to hold corporate bonds. The subsection

on robustness checks further develops this last point.

Regressions in which I substitute realized volatility with its diffusive and jump compo-

21Using different data, and a different methodology, Kalimipalli and Nayak (2009) reach a similar conclu-sion.

23

nents are similar to those reported in Table 9. The coefficients on the diffusive component

are approximately the same as those on realized volatility, while the coefficients on the

jump component are half the size. Given the similarity of the results, I do not report these

regressions, but make them available on request.

5.3 Robustness Checks

In the remainder of the paper, I conduct some further analysis to assess the robustness of

the results to the use of more general standard errors and to several partitions of the sample.

5.3.1 Clustered Standard Errors

As can be seen from Tables 8 and 9, the t-statistics associated to the estimated coefficients

are rather large, which is likely due to the assumption of spherical disturbances underlying

OLS standard errors. It is reasonable to assume that observations relative to bonds issued

by the same issuer are not mutually independent. To account for the dependence induced by

issuer-specific effects, I rerun the multivariate regressions of Table 9 using standard errors

clustered by issuer.22

Table 10 reports the multivariate regressions of Tables 9 with clustered-adjusted stan-

dard errors. As can be seen, while the t-statistics have shrunk substantially making some

variables no longer significant, they are still high enough to make the variables of interest,

e.g. realized volatility, strongly significant in a statistical sense. Finally, the large reduction

in the t-statistics signals the relevance of individual effects and the inappropriateness of the

assumption of independent observations implied by the OLS procedure for deriving standard

errors.

22See Petersen (2009) and Gow, Ormazabal, and Taylor (2009) for applications of these standard errors infinance and accounting. I obtained the Matlab code for running regressions with clustered standard errorsat http://www.stanford.edu/~djtaylor/research/.

24

5.3.2 Investment Grade Vs Junk Bonds

Table 11 reports several yield spread regressions for two sub-samples: one sub-sample in-

cludes only investment grade bonds (models 1 and 2); the other sub-sample includes junk

bonds (models 3 and 4). Table 11 differs from Table 9 in two ways. First, the highest two

interest coverage dummies have been merged because there were no junk bonds with high

enough interest coverage. Secondly, the regressions do not include rating dummies as the

sample is already partitioned using rating information. Notice, that, in the interest of space,

I have only included the two most important liquidity measures.

The regressions in Table 11 reveal that both volatility and (to some extent) illiquidity play

a bigger role in the sub-sample of junk bonds. While volatility is still strongly economically

and statistically significant, the proposed illiquidity measure loses its significance in the junk

bond sample and the size of the coefficient remains practically unchanged. The standard LDV

measure of Chen, Lesmond, and Wei (2007) is characterized by a much higher coefficient in

the junk bond sample, which is consistent with the finding of their paper. This phenomenon is

further evidence that this illiquidity measure is more correlated to credit spreads in situations

in which credit risk is more relevant.

5.3.3 Pre-Crisis Vs Crisis Yield Spreads

Table 12 reports several yield spread regressions for two sub-samples: one sub-sample in-

cludes the period January 2003-December 2006 (models 1 and 2); the other sub-sample

includes the period January 2007-December 2008 (models 3 and 4). The pattern of the

change in coefficients is similar to the case in which the partition is done according to credit

rating. For instance, the coefficient on realized volatility is much higher now. The stan-

dard deviation of volatility is also higher in the crisis sub-sample which makes the economic

impact of volatility even higher.

25

The most striking difference between the two sets of regressions is in their explained

variation. Approximately 15% in the pre-crisis sample, the R2 jumps to 57% in the crisis

sample. This finding is consistent with the manifestation of credit risk particularly in the

last two years of data.

6 Conclusion

In order to remedy the inability of structural credit risk models to adequately fit corporate

bond data, the literature has focused on illiquidity as a concurring determinant of corporate

yield spreads. Several recent papers have concluded that liquidity is an important factor in

the determination of credit spreads, especially for junk bonds. In this paper, I argue that

certain illiquidity measures appear to do especially well in explaining yield spreads because

they are in fact picking up credit risk.

Using transaction data from TRACE, and high-frequency volatility measures from TAQ,

I find that a substantial portion of yield spreads is explained by equity volatility, indicating

that investors seek reward for bearing credit risk. In particular, I find that equity realized

volatility explains as much yield spread variation as that explained by credit ratings. The

explanatory power of credit risk variables, and their ability to explain away well known

illiquidity measures, suggests that the liquidity component of yield spreads is less important

than previously thought.

26

A Conditional Posterior Distributions

A.1 Conditional Distribution of βi and σ2it

To the derive the conditional posterior distribution of βi and σ2it, it is more convenient to

work with the augmented likelihood in (2.7). To see this, notice that, once we augment

that data with the auxiliary variable R∗it, the likelihood function is the standard likelihood

function of a linear regression model, and the standard conditional distributions apply.

Using vector notation on the time observations, and defining Vi = diag(vi1, vi2 . . . , viTi),

we can multiply the likelihood in (2.17) and the prior distribution in (2.9) to obtain

p(βi|β,∆, σ2, Vi, R∗i , Xi) ∼ N(Bi, Vi), i = 1, . . . , N (A.1)

where Bi = Vi ×(X ′iV

−1i R∗i /σ

2 + ∆−1β)

and Vi =(X ′iV

−1i Xi/σ

2 + ∆−1)−1

.

The posterior distribution for σ2 is given by

p(σ2|{β}Ni=1,∆, R∗i , Xi) ∼ IG(

N∑i=1

Ti/2 + sh,N∑

i=1

SSRi/2 + sc), (A.2)

where SSRi ≡ (R∗i −Xiβi)′V −1

i (R∗i −Xiβi) and Ti is equal to the number of censored and

uncensored observations available for bond i.

Finally, the distribution of the time-varying component of the variance has been shown

by Geweke (1993) to be implicitly given by

e2it/σ

2 + r

vit

|βi, σ2 ∼ χ2(r + 1), (A.3)

where e2it is the squared residual of observation it.

27

A.2 Conditional Distribution of αsi and αbi

To obtain the posterior distribution for this parameter, I need to combine the observed

likelihood in (2.6) with the expressions in (2.2), and (2.3). Defining l = log(L), The posterior

distribution of αsi is given by

p(αsi |βi, σ

2it, Rit, xit, L

sit, L

bit) ∼ p(Rit|Ls

it, Lbit, βi, σ2, xit)× p(lsit|γsi, σ

2s , zit)

∝ exp

{−1

2

(Rit + Ls

it − β′ixit

σit

)2}1{Rit<0}

×[Φ

(Lb

it − β′ixit

σit

)− Φ

(Ls

it − β′ixit

σit

)]1{Rit=0}

×

exp

{−1

2

(αs

i − α′szi

σs

)2}. (A.4)

The posterior distribution of the buy-side liquidity costs is given by

p(αbi |βi, σ

2it, Rit, xit, L

sit, L

bit) ∝ exp

{−1

2

(Rit + Lb

it − β′ixit

σit

)2}1{Rit>0}

×[Φ

(Lb

it − β′ixit

σit

)− Φ

(Ls

it − β′ixit

σit

)]1{Rit=0}

×

exp

{−1

2

(αb

i − α′bzi

σb

)2}. (A.5)

The expressions in (A.4) and (A.5) do not resemble the kernels of any well known distribution.

Therefore, I implement a Metropolis-Hastings algorithm to sample from these unknown

target distributions.

28

A.3 Conditional Distribution of αs and αb

Given the flat prior, the distribution of αs is given by

p(αs|{αsi}Ni=1, σ

2s , zi) ∼ N(α, V ), (A.6)

where α = V × (z′αs/σ2s) and Vi = σ2

s (z′z)−1. Note that the posterior parameters of the dis-

tribution are just the OLS slope and its covariance matrix. A similar posterior distributions

can be obtained for αb.

A.4 Conditional Distribution of β and ∆

Combining the distributions in which it appears, β can be shown to have the following

posterior conditional distribution:

p(β|{β}Ni=1,∆) ∼N∏

i=1

N(βi,∆)

∼ N

(N∑

i=1

βi/N,∆/N

). (A.7)

Using the the linearity and cyclic property of the trace operator, the posterior conditional

distribution for ∆ is given by

p(β|{β}Ni=1,∆) ∼ IW (∆0, N0)×N∏

i=1

N(βi,∆)

∝ |∆|−N+N0+K+1

2 exp{−tr(∆−1(∆0 + ∆1))}

∼ IW (∆0 + ∆1, N +N0), (A.8)

where ∆1 ≡∑N

i=1(βi − β)(βi − β)′.

29

A.5 Conditional Distribution of γ

To obtain the posterior distribution for this parameter, I need to combine the observed

likelihood in (2.6) with the expressions in (2.2), and (2.3). The posterior distribution of γ is

given by

p(αsi |βi, σ

2it, Rit, xit, L

sit, L

bit) ∼ p(Rit|Ls

it, Lbit, βi, σ2, xit)× p(lsit|γsi, σ

2s , zit)

∝ exp

{−1

2

(Rit + Ls

it − β′ixit

σit

)2}1{Rit<0}

×[Φ

(Lb

it − β′ixit

σit

)− Φ

(Ls

it − β′ixit

σit

)]1{Rit=0}

×

exp

{−1

2

(Rit + Lb

it − β′ixit

σit

)2}1{Rit>0}

. (A.9)

The expressions in (A.9) does not resemble the kernel of any well known distribution. There-

fore, I implement a Metropolis-Hastings algorithm to sample from this unknown target dis-

tribution.

A.6 Conditional Distribution of σ2s and σ2

b

The derivation of the posterior conditional distribution for σ2j , j = {s, b} is given by

p(σ2j |{α

ji , zi}Ni=1) ∼ IG (N/2 + sh, SSR/2 + sc) , j = {s, b} (A.10)

where SSRi ≡ (αj − zαj)′(αj − zαj) and N is equal to the number of bonds.

B Databases and Merging

To conduct my analysis, I use 5 databases. Below, I briefly describe the databases and the

filters used for the sample selection.

30

1. FISD to obtain bond characteristics (e.g. rating, issue size) and identify the sample

of bonds to include in the study; the unique identifier for each issue in this database is

the bond 9-letter cusip; I impose several filters to define the initial sample (before the

merge with TRACE) with the following self-explanatory SAS commands (in a data

step):

where security_level = ’SEN’ and

convertible = ’N’ and

putable = ’N’ and

redeemable = ’N’ and

exchangeable = ’N’ and

bond_type in (’CDEB’,’CMTN’) and

coupon_type = ’F’ and

foreign_currency = ’N’ ;

The above commands exclude bonds that are convertible, putable, callable, and ex-

changeable and keeps bonds that are either corporate debentures or medium term

notes, and fixed-rate bonds denominated in US dollars;

2. TRACE to obtain transaction prices; note that most of the bonds in TRACE are

covered by FISD; the unique identifier for each issue in this database is the bond 9-

letter cusip; I use the following SAS commands (in an SQL step) to impose filters on

TRACE data:

where [...]

&StartDate <= a.trd_exctn_dt <= &EndDate and /*time filter*/

a.sale_cndtn_cd = ’@’ and /*regular sale condition*/

31

a.spcl_trd_fl ne ’Y’ and/*get rid of special sales and prices*/

a.wis_fl = ’N’ and /*regular, no when-issued basis*/

a.cmsn_trd = ’N’ and /*price excludes commission*/

a.spcl_trd_fl = ’’ and /*non-special price trade*/

a.asof_cd = ’’ ; /*regular trade, e.g. no reversal*/

In addition to imposing the above filters, I also make sure to exclude or correct misre-

ported transactions as documented by TRACE and to eliminate repeated inter-dealer

trades;

3. CRSP to obtain stock returns of the company currently backing a given bond; the

unique identifier for each firm and securitiy in this database are the PERMCO and

PERMNO numbers respectively;

4. COMPUSTAT to obtain balance sheet information on the company backing a given

bond; firms are identified by their GVKEY number;

5. TAQ to obtain 5-minute returns to construct the realized variance measures used in

the specification of the bond return generating process; securities are identified by their

TICKER symbol (which varies over time and is not a unique identifier).

The link between these databases is straightforward in some cases and quite complicated in

others. FISD and TRACE are easily linked through the 9-letter cusip. Once a preliminary

sample of bonds is formed, to see whether a firm with public equity is backing them, I match

the six-letter cusip (which identifies the firm at issuance in the FISD database) with the

historical cusip (NCUSIP) in the CRSP “stocknames” table. During this merge I obtain the

historical tickers, and PERMCO and PERMNO numbers associated with firms’ CUSIPs,

which I then use to get data from COMPUSTAT and TAQ.

32

C Extracting Jumps from Realized Variance

To screen out jumps, I use a nonparametric approach developed by Barndorff-Nielsen and

Shephard (2004) which relies on the concepts of realized variance and bipower variation.23

The idea is that, as we sample price data at very high frequency, the limiting behaviors of

the return realized variance and bipower variation capture different aspects of the return

process. More formally, given a log asset price p(t), we can define the instantaneous return

of the associated jump-diffusion process as

dp(t) = µ(t)dt+ σ(t)dW (t) + k(t)dq(t), 0 ≤ t ≤ T, (C.1)

where µ(t) and σ(t) are the drift and the diffusion of the process, W (t) is a standard Brownian

motion, q(t) is a counting process which controls the arrival of jumps, and k(t) is the size

of the jumps upon arrival. I refer to Andersen, Bollerslev, and Diebold (2007) for a precise

description of the parameters of the process and their properties. Given a sample of high-

frequency price data in a given day, one can create ∆-period returns, where ∆ is a fraction

of the day, as rt,∆ ≡ p(t)− p(t−∆). Setting the time interval to unity, we get 1/∆ intervals

in a day, and we also have rt+1 ≡ rt+1,1. It can be shown that the realized variance converges

uniformly in probability to the quadratic variation of the process:

RVt+1(∆) ≡1/∆∑j=1

r2t+j∆,∆ −→

∫ t+1

t

σ2(s)ds+∑

t<s≤t+1

k2(s), (C.2)

23See Huang and Tauchen (2005), Barndorff-Nielsen and Shephard (2006) and Huang (2007) for an appli-cation of this approach.

33

for ∆ −→ 0. The other object of interest, the bipower variation, converges to just the

diffusive component of the quadratic variation of the process:

BVt+1(∆) ≡ µ−1

1/∆∑j=2

|rt+j∆,∆||rt+(j−1)∆,∆| −→∫ t+1

t

σ2(s)ds, (C.3)

for ∆ −→ 0, where µ−1 ≡√

2/π.

It can be shown (see Barndorff-Nielsen and Shephard (2004) and Andersen, Bollerslev,

and Diebold (2007)) that the difference between the quantities in expressions (C.2) and (C.3)

converges to∑

t<s≤t+1 k2(s). In most applications (e.g. Andersen, Bollerslev, Diebold, and

Ebens (2001)), including mine, 5-minute returns are typically used to obtain daily measures

of realized variance and bipower variation, i.e. ∆ is small but not zero, so this difference is

not even guarantied to be positive. To deal with this issue, Barndorff-Nielsen and Shephard

(2004) propose a statistical procedure to determine whether price variation is due to jumps

or diffusive movements based on the test statistics RJit ≡ RVit−BVit

RVit, which, appropriately

scaled, converges to a standard normal distribution. I implement this methodology exactly

as in the appendix (p. 35) of Zhang, Zhou, and Zhu (2009).

D Estimation Algorithm

Given the conditional posterior densities derived in the previous section, I implement the

Gibs sampler, and Hasting-Metropolis algorithm it, as follows.

1. Initialize the chain by assigning {R∗i , β1, . . . , βN , α1, . . . , αN , σ, Vi, β,∆, σs, σb, γ, αs, αb}0;

2. Move the Markov chain one step forward by drawing parameters from the posterior

densities derived in the previous sections. In particular, we obtain updated values (not

necessarily in this order), for j > 0, as follows:

34

• R∗i j|Ri, Lsij−1, Lb

ij−1, i = 1, . . . , N : use equation (2.8);

• βji |σj−1, V j−1

i , βj−1,∆j−1, i = 1, . . . , N : use equation (A.1);

• σj|{β1 . . . βN}j, {V1 . . . VN}j−1: use equation (A.2);

• V ji |β

ji , σ

j, i = 1, . . . , N : use equation (A.3);

• βj|{β1 . . . βN}j,∆j−1: use equation (A.7);

• ∆j−1|βj, {β1 . . . βN}j: use equation (A.8);

• σjs|{Ls

1 . . . LsN}j−1: use equation (A.10); similarly for σb;

• αjs|{Ls

1 . . . LsN}j−1, {Lb

1 . . . LbN}j−1: use equation (A.6); similarly for αj

b;

• αsi , α

bi , γ

j: the posterior densities of interest are proportional to the expressions in

(A.4) , (A.5), and (A.9) respectively, and a Metropolis-Hastings (within-Gibbs-

sampler) algorithm is required to sample from this non-standard distributions.

The procedure for generating a generic sample θ from one of these distributions

works as follows:

– given the current sample previously drawn, θc, generate a new sample θp from

the proposal distribution q(θp|θc); the proposal and target density should have

the same support;

– evaluate the acceptance probability as

α(θp|θc) = min

(1,p(θp)q(θc|θn)

p(θc)q(θp|θc)

)

– accept the proposed value θp with probability α(θp|θc), i.e.

{θ}j+1 =

θp, with prob α(θp|θc)

θc, with prob 1− α(θp|θc)

35

The proposal density for αs and αb is a truncated normal, i.e. q(θp|θc) ∼ TN(0, ν2),

where ν2 is a perturbation parameter; the proposal for γ is a normal distribution;

3. Repeat step 2 J times, where J is large enough to ensure convergence of the chain;

4. Discard the first K

36

References

Alexander, G. J., A. K. Edwards, and M. G. Ferri, 2000, “The Determinants of Trading

Volume of High-Yield Corporate Bonds,” Journal of Financial Markets, 3(2), 177 – 204.

Ambrose, B. W., N. Cai, and J. Helwege, 2008, “Forced Selling of Fallen Angels,” The

Journal of Fixed Income, 18(1), 72 – 85.

, 2009, “Fallen Angels and Price Pressure,” Penn State Working Paper.

Andersen, T. G., T. Bollerslev, and F. X. Diebold, 2007, “Roughing It Up: Including Jump

Components in the Measurement, Modeling, and Forecasting of Return Volatility,” The

Review of Economics and Statistics, 89(4), 701720.

Andersen, T. G., T. Bollerslev, F. X. Diebold, and H. Ebens, 2001, “The distribution of

realized stock return volatility,” Journal of Financial Economics, 61(1), 43 – 76.

Bao, J., and J. Pan, 2008, “Excess Volatility of Corporate Bonds,” SSRN eLibrary.

Bao, J., J. Pan, and J. Wang, 2008, “Liquidity of Corporate Bonds,” SSRN eLibrary.

Barndorff-Nielsen, O. E., and N. Shephard, 2004, “Power and Bipower Variation with

Stochastic Volatility and Jumps,” Journal of Financial Econometrics, 2(1), 1–37.

, 2006, “Econometrics of Testing for Jumps in Financial Economics Using Bipower

Variation,” Journal of Financial Econometrics, 4(1), 1–30.

Bessembinder, H., W. Maxwell, and K. Venkataraman, 2006, “Market Transparency, Liq-

uidity Externalities, and Institutional Trading Costs in Corporate Bonds,” Journal of

Financial Economics, 82(2), 251–288.

37

Brownlees, C., and G. Gallo, 2006, “Financial Econometric Analysis at Ultra-High Fre-

quency: Data Handling Concerns,” Computational Statistics & Data Analysis, 51(4), 2232

– 2245.

Campbell, J. Y., and G. B. Taksler, 2003, “Equity Volatility and Corporate Bond Yields,”

The Journal of Finance, 58(6), 2321–2349.

Chen, L., D. A. Lesmond, and J. Wei, 2007, “Corporate Yield Spreads and Bond Liquidity,”

The Journal of Finance, 62(1), 119–149.

Chib, S., 1992, “Bayes Inference in the Tobit Censored Regression Model,” Journal of Econo-

metrics, 51(1-2), 79 – 99.

Collin-Dufresne, P., R. S. Goldstein, and J. S. Martin, 2001, “The Determinants of Credit

Spread Changes,” The Journal of Finance, 56(6), 2177–2207.

Cremers, K. M., J. Driessen, and P. Maenhout, 2008, “Explaining the Level of Credit Spreads:

Option-Implied Jump Risk Premia in a Firm Value Model,” Review of Financial Studies,

21(5), 2209–2242.

Driessen, J., 2005, “Is Default Event Risk Priced in Corporate Bonds?,” Review of Financial

Studies, 18(1), 165–195.

Edwards, A. K., L. E. Harris, and M. S. Piwowar, 2007, “Corporate Bond Market Transaction

Costs and Transparency,” The Journal of Finance, 62(3), 1421–1451.

Elton, E. J., M. J. Gruber, D. Agrawal, and C. Mann, 2001, “Explaining the Rate Spread

on Corporate Bonds,” The Journal of Finance, 56(1), 247–277.

Eom, Y. H., J. Helwege, and J.-Z. Huang, 2004, “Structural Models of Corporate Bond

Pricing: An Empirical Analysis,” Rev. Financ. Stud., 17(2), 499–544.

38

Geweke, J., 1993, “Bayesian Treatment of the Independent Student-t Linear Model,” Journal

of Applied Econometrics, 8, S19–S40.

Gow, I. D., G. Ormazabal, and D. Taylor, 2009, “Correcting for Cross-Sectional and Time-

Series Dependence in Accounting Research,” Accounting Review.

Goyal, A., and P. Santa-Clara, 2003, “Idiosyncratic Risk Matters!,” The Journal of Finance,

58(3), 975–1007.

Greene, W., 2005, “Reconsidering Heterogeneity in Panel Data Estimators of the Stochastic

Frontier Model,” Journal of Econometrics, 126(2), 269 – 303.

Hotchkiss, E. S., and G. Jostova, 2007, “Determinants of Corporate Bond Trading: A Com-

prehensive Analysis,” SSRN eLibrary.

Hotchkiss, E. S., and T. Ronen, 2002, “The Informational Efficiency of the Corporate Bond

Market: An Intraday Analysis,” Review of Financial Studies, 15(5), 1325–1354.

Houweling, P., A. Mentink, and T. Vorst, 2003, “How to Measure Corporate Bond Liquid-

ity?,” Tinbergen Institute Discussion Papers 03-030/2, Tinbergen Institute.

Huang, J., and M. Huang, 2003, “How Much of the Corporate-Treasury Yield Spread is Due

to Credit Risk?,” http://ssrn.com/abstract=307360.

Huang, X., 2007, “Macroeconomic News Announcements, Financial Market Volatility and

Jumps,” Working Paper - Duke University.

Huang, X., and G. Tauchen, 2005, “The Relative Contribution of Jumps to Total Price

Variance,” Journal of Financial Econometrics, 3(4), 456–499.

Kalimipalli, M., and S. Nayak, 2009, “Idiosyncratic Volatility Vs. Liquidity? Evidence from

the US Corporate Bond Market,” SSRN eLibrary.

39

Lesmond, D. A., J. P. Ogden, and C. A. Trzcinka, 1999, “A New Estimate of Transaction

Costs,” The Review of Financial Studies, 12(5), 1113–1141.

Longstaff, F. A., S. Mithal, and E. Neis, 2005, “Corporate Yield Spreads: Default Risk or

Liquidity? New Evidence from the Credit Default Swap Market,” The Journal of Finance,

60(5), 2213–2253.

Mahanti, S., A. Nashikkar, M. G. Subrahmanyam, G. Chacko, and G. Mallik, 2008, “Latent

Liquidity: A New Measure of Liquidity, with an Application to Corporate Bonds,” Journal

of Financial Economics.

Merton, R. C., 1974, “On the Pricing of Corporate Debt: The Risk Structure of Interest

Rates,” The Journal of Finance, 29(2), 449–470.

Omori, Y., and K. Miyawaki, 2009, “Tobit Model with Covariate Dependent Thresholds,”

Computational Statistics & Data Analysis, In Press, –.

Petersen, M. A., 2009, “Estimating Standard Errors in Finance Panel Data Sets: Comparing

Approaches,” Rev. Financ. Stud., 22(1), 435–480.

Ronen, T., and X. Zhou, 2008, “Where Did All the Information Go? Trade in the Corporate

Bond Market,” SSRN eLibrary.

Rosett, R. N., 1959, “A Statistical Model of Friction in Economics,” Econometrica, 27(2),

263–267.

Schaefer, S. M., and I. A. Strebulaev, 2008, “Structural Models of Credit Risk are Useful:

Evidence from Hedge Ratios,” The Journal of Financial Economics.

Tauchen, G. E., and H. Zhou, 2006, “Realized Jumps on Financial Markets and Predicting

Credit Spreads,” SSRN eLibrary.

40

Tsionas, E. G., 2002, “Stochastic Frontier Models with Random Coefficients,” Journal of

Applied Econometrics, 17(2), 127–147.

Warga, A., and I. Welch, 1993, “Bondholder losses in leveraged buyouts,” Review of Financial

Studies, 6, 959–982.

Zhang, B. Y., H. Zhou, and H. Zhu, 2009, “Explaining Credit Default Swap Spreads with

the Equity Volatility and Jump Risks of Individual Firms,” Review of Financial Studies.

41

Table 1: Summary Statistics of Corporate Bond Transactions

This table presents summary statistics, categorized by year, of bond characteristics (Panel A) and bondtransactions (Panel B). The bonds described in this table must have data on FISD, TRACE, CRSP, COM-PUSTAT, and TAQ. Moreover, bonds that trade on less than 20 trading days are excluded from the sample.Issues is the number of bonds. Issuers is the number of issuers at the parent company level according tothe FISD database. Issue Size is the average issue size. Coupon is the average fixed coupon rate. TTMat Issuance is the time to maturity at issuance. Trades is the total number of trades of the bonds in thesample. Age is age measured in years, or as a percentage of the life span of the bond, at the time of trade.Trade Price is the price, as a percentage of par, of the bond (several quantiles are provided). Trade Size isthe dollar size of the transaction (several quantiles are provided).

2003 2004 2005 2006 2007 2008 2003-2008Panel A: Bond Characteristics

Issues 480 816 887 789 696 607 984Issuers 104 161 176 164 151 136 181Issue Size (× $1,000,000) 663.43 439.46 406.91 414.34 451.63 465.80 441.77Coupon 6.59 6.73 6.87 6.90 6.98 7.09 6.83TMM at Issuance 16.89 17.37 17.65 18.47 19.98 21.36 16.68

Panel B: Transactions CharacteristicsTrades 257,157 290,672 427,881 359,588 265,618 315,496 1,916,412Bond Age as of Trade

Years 4.70 5.30 6.70 7.43 7.89 7.36 6.63Pct Life 0.39 0.40 0.48 0.54 0.53 0.52 0.48

Trade Price (pct of par )Minimum 39.50 19.00 10.00 18.95 10.50 0.01 0.01First Quartile 99.78 99.75 92.95 91.00 96.33 96.00 96.30Median 105.15 103.75 100.30 99.39 99.90 100.27 100.64Third Quartile 110.40 108.59 104.25 101.87 102.96 102.90 105.10Maximum 158.91 151.07 159.85 148.66 144.45 151.47 159.85

Trade Size (× $1,000)Minimum 1 1 1 1 1 1 1First Quartile 10 10 10 10 10 10 10Median 30 25 25 25 25 20 25Third Quartile 175 100 100 95 90 50 100Maximum 5000 5000 5000 5000 5000 5000 5000

42

Table 2: Trading Activity by Bond Characteristics

This table presents summary statistics of corporate bond transactions categorized by age and issue size(Panel A), and time to maturity and issue size (Panel B). The bonds described in this table must have dataon FISD, TRACE, CRSP, COMPUSTAT, and TAQ. Age and Time to Maturity are measured in years atthe time of trade. Size is the dollar issue size of the bond. The numbers in the table represent the tradeoccurrences with a given size-age or size-TTM combination. The number in parenthesis represent the numberof bonds falling in each category. Notice that for the number in parentheses the marginal distribution is notobtained by summing summing numbers in the joint table as the a given bond might be in more than onecategory during its life.

Issue Size (×$1,000,000)≤ 50 (50− 100] (100− 250] (250− 500] > 500

PANEL A: by Age (in years)≤ 3 705 3,200 22,793 116,676 269,446 412,820

( 1) ( 5) ( 39) ( 80) ( 55) ( 180)(3− 5] 1,404 3,048 17,280 73,834 230,130 325,696

( 3) ( 19) ( 83) ( 81) ( 58) ( 244)(5− 7] 2,351 7,830 50,974 80,643 190,461 332,259

( 18) ( 37) ( 150) ( 89) ( 48) ( 342)(7− 10] 6,854 22,231 139,183 203,068 122,685 494,021

( 52) ( 85) ( 246) ( 112) ( 46) ( 541)> 10 5,254 23,702 130,511 139,360 52,789 351,616

( 56) ( 73) ( 237) ( 102) ( 30) ( 498)16,568 60,011 360,741 613,581 865,511 1,916,412( 80) ( 134) ( 417) ( 243) ( 110) ( 984)

PANEL B: by Time to Maturity (in years)≤ 1 1,611 8,057 45,707 75,362 55,135 185,872

( 42) ( 74) ( 208) ( 131) ( 50) ( 180)(1− 3] 5,472 15,411 85,534 156,583 125,382 388,382

( 55) ( 75) ( 200) ( 131) ( 52) ( 244)(3− 5] 3,822 9,705 54,543 105,007 144,106 317,183

( 50) ( 50) ( 140) ( 97) ( 41) ( 342)(5− 10] 3,483 10,177 47,375 89,355 252,125 402,515

( 27) ( 38) ( 95) ( 71) ( 36) ( 541)> 10 2,180 16,661 127,582 187,274 288,763 622,460

( 13) ( 43) ( 162) ( 74) ( 45) ( 498)16,568 60,011 360,741 613,581 865,511 1,916,412( 80) ( 134) ( 417) ( 243) ( 110) ( 984)

43

Table 3: Summary Statistics of Corporate Bond Returns

This table presents summary statistics of bond returns grouped by rating and maturity, with short-maturitybonds in Panel A, medium-maturity bonds in Panel B, and long-maturity bonds in Panel C. The first fourrows of each panel report the average across bonds of the first four moments of returns. The mean is reportedin basis points. Pct Trading is the percentage of days in which the bonds in a given category traded, i.e.days with trading over trading days. The last row reports the total number of days in which the bonds ina given category could have traded. The statistics are calculated using data from trading days only, thusignoring zero returns originating from zero-trading days.

AAA AA A BBB BB B CCC-DPanel A: Short Maturity (0-2 years)

Mean (bp) 0.673 0.274 -0.423 -0.351 -0.784 0.754 -0.865St. Dev. 0.005 0.005 0.007 0.007 0.013 0.018 0.025Skewness -0.110 -0.385 0.012 -0.502 0.055 0.172 -0.772Kurtosis 8.604 9.112 21.945 20.835 15.733 172.947 113.249Pct. Trading 0.809 0.633 0.441 0.349 0.292 0.396 0.371Trading Days 2472 25935 93286 69803 13617 9077 4680

Panel B: Medium Maturity (3-10 years)Mean (bp) 1.355 1.140 0.477 -1.312 -2.911 -2.648 -0.267St. Dev. 0.012 0.010 0.013 0.018 0.022 0.035 0.053Skewness 0.063 0.025 0.019 1.796 -3.009 5.870 -0.065Kurtosis 6.496 11.313 14.673 174.692 71.886 288.178 22.081Pct. Trading 0.792 0.624 0.449 0.366 0.382 0.356 0.436Trading Days 9192 41482 142264 103432 17567 18053 12699

Panel C: Long Maturity (more than 10 years)Mean (bp) 4.514 3.515 2.026 -1.102 -5.895 -1.669 1.482St. Dev. 0.024 0.026 0.026 0.028 0.032 0.033 0.065Skewness 0.100 -0.053 -0.038 -0.087 -0.125 -0.280 0.892Kurtosis 5.600 8.008 7.198 16.828 9.866 23.624 16.866Pct. Trading 0.431 0.245 0.259 0.290 0.498 0.394 0.464Trading Days 7541 32968 138939 117910 17718 19006 20906

44

Table 4: Average Corporate Credit Spreads

This table presents average credit spreads categorized by S&P rating, and subsequently by time to maturity(Panel A), and by year (Panel B). Credit spreads are defined as the difference between daily yield spreads(obtained by averaging the available yields on a given day) and the yields on the treasury benchmark withthe same time to maturity. The constant maturity benchmark yields are from Datastream and are for thefollowing yearly maturities: 1/12, 1/4, 1/2, 1, 2, 3, 5, 7, 10, 20, 30. I use linear interpolation to get the yieldof intermediate maturities. Transactions for which the spread is negative are not included in the sample.

AAA AA A BBB BB B CCC-DPanel A: Breakdown by Time to Maturity (in years)

Short (0-2) 42 72 107 126 331 443 900Medium (3-10) 63 61 90 169 331 497 1292Long (> 10) 62 108 142 225 444 590 904

Panel B: Breakdown by Year2003-2008 60 73 109 179 391 531 10022003 36 62 92 202 200 553 8192004 60 53 75 137 205 231 5612005 40 47 76 159 385 286 11842006 46 57 84 123 411 413 7442007 65 81 109 148 270 399 5272008 152 163 270 382 591 1247 1679

45

Table 5: LOT Measure with TRACE Data

This table presents estimation results for the parameters of the bond return generating process underlyingthe modified LOT measure as described in the paper by Chen, Lesmond, and Wei (2007). One difference withthe original implementation of this measure is that I use all the observations for each bond in the estimationinstead partitioning bond data by year. Another difference is in the estimation procedure: the original LOTmeasure is estimated with maximum likelihood, while I use Markov Chain Monte Carlo (MCMC) techniques.The estimates are averaged across ratings and and time to maturity. The table represents bonds that tradeat lease on 15% of the available trading days. The betas are the average estimated factor loadings. Thealphas are the average estimates of the positive and negative thresholds. The lot measure is the differencebetween the two thresholds, i.e. the sum of the buy-side and sell-side transaction costs. The table alsoreports the number of bonds in each rating-maturity category.

AAA AA A BBB BB B CCC-DPanel A: Short Maturity (1-2 years)

Bond Factor (β1) -0.8173 -0.1906 -0.5428 -1.0091 0.5507 -1.1752 1.3262Equity Factor (β2) -0.0068 0.0368 0.0116 -0.0166 -0.7638 -0.3175 0.1451Sell-Side Cost (αs) -13 -60 -105 -116 -266 -301 -412Buy-Side Cost (αb) 12 55 102 111 206 218 369LOT = αb − αs 25 115 207 227 472 519 781Num. Bonds 5 39 113 99 14 12 4

Panel B: Medium Maturity (3-10 years)Bond Factor (β1) -0.8532 -0.8301 -0.7423 -0.7964 -0.1801 -0.2858 -0.5793Equity Factor (β2) -0.0002 0.0070 0.0009 0.0103 -0.0136 0.0475 0.0618Sell-Side Cost (αs) -28 -73 -125 -199 -321 -502 -915Buy-Side Cost (αb) 26 69 117 189 307 475 856LOT = αb − αs 53 142 242 389 629 976 1771Num. Bonds 7 36 106 57 12 15 11

Panel C: Long Maturity (more than 10 years)Bond Factor (β1) -0.5387 -0.6741 -0.5358 -0.6047 -0.2790 0.0155 0.0132Equity Factor (β2) 0.0051 0.0135 0.0049 0.0093 0.0215 0.0297 0.0043Sell-Side Cost (αs) -225 -257 -357 -399 -357 -370 -1159Buy-Side Cost (αb) 204 246 343 389 330 347 1075LOT = αb − αs 429 503 700 789 687 717 2233Num. Bonds 4 8 54 50 17 27 7

46

Table 6: Stochastic Friction Model: Idiosyncratic Factors

This table presents average estimation results for the parameters in the return component (Equation (2.1))of the model proposed in this paper. The results are categorized by (median) rating and (median) timeto maturity. In addition to the systematic bond market factors, a firm equity return and equity realizedvolatility factor are included in the specification (Model 1). In the bottom panel (Model 2), I provideestimates for a specification in which volatility is divided in its diffusive and jump component. The last rowof each panel, i.e. “LOT”, is the time invariant component of the estimated round-trip liquidity costs (firstpart of the expression in Equation (2.5)).

AAA AA A BBB BB B CCC-D

Model 1: Realized VolatilityShort Maturity (1-2 years)

Bond Market -0.605 -0.484 -0.482 -0.475 -0.420 -0.430 -0.249Firm Equity 0.002 0.003 0.003 0.004 0.004 0.007 0.023Realized Volatility -0.001 -0.001 -0.001 -0.001 -0.001 -0.001 0.001“LOT” = αb − αs 103 268 393 414 451 383 382

Medium Maturity (3-10 years)Bond Market -0.762 -0.677 -0.624 -0.549 -0.328 -0.300 -0.255Firm Equity 0.003 0.002 0.002 0.003 0.006 0.007 0.014Realized Volatility -0.001 -0.001 -0.001 -0.001 -0.002 -0.002 0.001“LOT” = αb − αs 146 211 311 398 409 350 373

Long Maturity (more than 10 years)Bond Market -0.353 -0.320 -0.323 -0.333 -0.240 -0.081 -0.188Firm Equity 0.002 0.005 0.002 0.003 0.005 0.010 0.005Realized Volatility -0.001 -0.001 -0.001 -0.001 -0.002 -0.001 -0.002“LOT” = αb − αs 355 495 473 466 253 258 467

Model 2: Diffusive and Jump VariationShort Maturity (1-2 years)

Bond Market -0.609 -0.485 -0.483 -0.476 -0.420 -0.429 -0.245Firm Equity 0.002 0.003 0.003 0.004 0.004 0.007 0.023Diffusive Variation -0.001 -0.001 -0.001 -0.001 -0.001 -0.001 0.001Jump Variation 0.002 0.002 0.002 0.002 0.002 0.002 0.002“LOT” = αb − αs 103 265 390 411 447 380 380

Medium Maturity (3-10 years)Bond Market -0.762 -0.679 -0.625 -0.550 -0.330 -0.300 -0.255Firm Equity 0.003 0.002 0.002 0.003 0.006 0.007 0.014Diffusive Variation -0.001 -0.001 -0.001 -0.001 -0.002 -0.002 0.000Jump Variation 0.002 0.002 0.002 0.002 -0.000 0.002 0.004“LOT” = αb − αs 145 210 308 395 406 347 369

Long Maturity (more than 10 years)Bond Market -0.351 -0.320 -0.323 -0.333 -0.241 -0.081 -0.188Firm Equity 0.002 0.005 0.002 0.003 0.005 0.010 0.005Diffusive Variation -0.002 -0.001 -0.001 -0.002 -0.002 -0.002 -0.002Jump Variation 0.001 0.001 0.001 0.001 0.002 0.002 0.001“LOT” = αb − αs 352 490 468 462 251 256 462

47

Table 7: Stochastic Friction Model: Liquidity Costs

This table presents estimation results for the parameters of the threshold components (Equations (2.2) and(2.3)) of the model proposed in this paper. The estimates are categorized by (median) rating and (median)time to maturity. The time-invariant individual effects (Panel A) are allowed to respond asymmetricallydepending on whether observed reruns are negative (first group of estimates) or positive (second group ofestimates). Panel B reports the sensitivity of liquidity cost to the time varying covariates, which includecalendar time (year dummies), age (year interval dummies), and the TED spread (defined as the differencebetween the 30-day treasury and eurodollar rates).

Model 1 Model 2Posterior Percentiles Posterior Percentiles

Mean St Dev 1st 99th Mean St Dev 1st 99th

Panel A: Time-invariant CovariatesSell Side (αs)

Intercept 1.638 0.304 0.940 2.349 1.630 0.304 0.913 2.343Log Issue Size -0.541 0.022 -0.592 -0.490 -0.540 0.022 -0.591 -0.489Coupon 0.126 0.017 0.087 0.166 0.125 0.017 0.085 0.165Log Maturity -0.052 0.037 -0.136 0.036 -0.053 0.036 -0.136 0.032

Buy Side (αb)Intercept 1.718 0.315 1.004 2.456 1.708 0.311 0.979 2.431LogIssue Size -0.555 0.023 -0.609 -0.502 -0.554 0.023 -0.606 -0.502Coupon 0.138 0.018 0.097 0.180 0.137 0.018 0.096 0.178Log Maturity -0.060 0.038 -0.149 0.030 -0.062 0.038 -0.147 0.025

Panel B: Time-varying Covariates (γ)Calendar Year Effects (in terms of 2003)

Year 2004 0.080 0.005 0.069 0.092 0.079 0.005 0.067 0.089Year 2005 0.097 0.006 0.086 0.112 0.095 0.005 0.084 0.105Year 2006 0.111 0.006 0.099 0.126 0.108 0.005 0.095 0.118Year 2007 0.171 0.007 0.157 0.190 0.167 0.005 0.154 0.179Year 2008 0.142 0.008 0.126 0.161 0.137 0.006 0.124 0.154

Age Brackets Effects (in terms of [0, 3))[3, 6) -0.004 0.008 -0.021 0.016 0.003 0.006 -0.011 0.017[6, 9) -0.005 0.011 -0.029 0.020 0.005 0.006 -0.008 0.019[9, 12) 0.014 0.012 -0.014 0.043 0.026 0.006 0.013 0.040[12, 15) 0.034 0.015 -0.002 0.066 0.048 0.008 0.032 0.064≥ 15 0.069 0.017 0.027 0.107 0.085 0.009 0.064 0.104

Ted Spread 0.017 0.002 0.012 0.021 0.017 0.002 0.012 0.021

48

Table 8: Yield Spreads Determinants: Univariate Analysis

Using bond data from TRACE from January 2003 to December 2008, I regress end-of-month credit spreadson several liquidity measures, realized volatility, and credit ratings. The first three models use three liquiditymeasures: the percentage of zero trading days (over total trading days), the LOT measure, and the measureproposed in this model (Rossi). In model 4 I regress credit spreads on realized equity volatility. In model 5 Iregress credit spreads on credit rating codes. In the last model I include both ratings and realized volatility.In Panel A the regressors (except for ratings) and the regressand are in logs, while in Panel B they are inlevels. For the specifications in levels the sample is trimmed and excludes the credit spreads below and abovethe bottom and top percentiles respectively. The table reports OLS estimates and t-statistics in parenthesis.

Model (1) (2) (3) (4) (5) (6)

Panel A: Log Specifications - Whole Sample

Intercept 0.12 -0.01 0.19 1.90 -1.15 0.22( 11.84) ( -2.03) ( 24.78) (144.74) (-135.05) ( 13.20)

Pct Zeros 0.13( 7.38)

Log LOT 0.27( 62.12)

Log Rossi -0.00( -0.78)

Log Realized Volatility 1.27 0.77(138.27) ( 94.03)

Rating 0.17 0.13(175.03) (134.18)

R2 0.00 0.13 0.00 0.38 0.50 0.61N 31173 26811 31173 31173 31173 31173

Panel B: Level Specification - Trimmed Sample

Intercept 0.10 -0.11 0.10 -0.59 -1.29 -2.20( 10.99) (-17.20) ( 11.27) (-30.83) (-53.42) (-103.67)

Pct Zeros 0.159.08

LOT 0.07( 64.03)

Rossi 0.02( 9.36)

Realized Volatility 8.43 6.22(147.45) (121.13)

Rating 0.41 0.29(142.54) (115.94)

R2 0.00 0.14 0.00 0.42 0.40 0.59N 30577 26249 30577 30577 30577 30577

49

Table 9: Yield Spreads Determinants: Multivariate Analysis

Using data from January 2003 to December 2008, I regress end-of-month credit spread levels on severaldeterminants. The sample is trimmed and excludes credit spreads below and above the bottom and toppercentiles respectively. The table reports OLS estimates and t-statistics in parenthesis.

Regression Models1 2 3 4

Liquidity and Volatility MeasuresPct Zeros 0.33

( 10.70)Log LOT 0.13

( 14.77)Log Rossi 0.10

( 8.44)Realized Volatity 4.59 4.88 4.58 4.58

( 75.29) ( 74.00) ( 75.07) ( 74.89)Credit Ratings (relative to AAA-AA)

A 0.05 0.01 0.05 0.03( 0.87) ( 0.15) ( 0.93) ( 0.56)

BBB 0.45 0.38 0.45 0.41( 6.83) ( 5.28) ( 6.83) ( 6.30)

BB 1.44 1.37 1.43 1.40( 18.73) ( 16.41) ( 18.64) ( 18.20)

B or worse 2.15 2.06 2.14 2.11( 26.19) ( 23.07) ( 26.00) ( 25.59)

Firm-specific/Accounting VariablesPre Tax Int Cov [5,10) 0.12 0.12 0.12 0.12

( 3.68) ( 3.40) ( 3.75) ( 3.78)Pre Tax Int Cov [10,20) 0.20 0.25 0.20 0.20

( 4.35) ( 4.93) ( 4.31) ( 4.36)Pre Tax Int Cov [20,∞) 0.21 0.28 0.21 0.22

( 3.55) ( 4.26) ( 3.46) ( 3.70)Inc to Sale -2.53 -3.42 -2.48 -2.40

(-11.97) (-13.53) (-11.76) (-11.39)LT Debt to Assets -1.36 -1.35 -1.40 -1.29

( -7.59) ( -7.07) ( -7.76) ( -7.16)Market Leverage 3.69 3.75 3.66 3.61

( 20.45) ( 19.70) ( 20.32) ( 19.98)Macroeconomic and Other Variables

Treasury (1y) -0.54 -0.50 -0.53 -0.53(-20.25) (-17.56) (-20.16) (-20.19)

Term Spread (10y-2y) -0.47 -0.40 -0.46 -0.47(-10.68) ( -8.42) (-10.49) (-10.58)

TED Spread (30-day) -0.05 -0.07 -0.06 -0.05( -3.66) ( -4.70) ( -3.77) ( -3.60)

Time to Maturity 0.01 0.01 0.01 0.01( 17.17) ( 7.54) ( 18.02) ( 19.65)

R2 0.51 0.52 0.51 0.51N 30370 26105 30370 30370

50

Table 10: Yield Spreads Determinants: Clustered Standard Errors

Using data from January 2003 to December 2008, I regress end-of-month credit spread levels on severaldeterminants. The sample is trimmed and excludes credit spreads below and above the bottom and toppercentiles respectively. The table reports OLS estimates with t-statistics clustered by issuer.

Regression Models1 2 3 4

Liquidity and Volatility MeasuresPct Zeros 0.33

( 5.41)Log LOT 0.13

( 7.35)Log Rossi 0.10

( 4.20)Realized Volatity 4.59 4.88 4.58 4.58

( 7.91) ( 7.30) ( 7.88) ( 7.87)Credit Ratings (relative to AAA-AA)

A 0.05 0.01 0.05 0.03( 0.51) ( 0.08) ( 0.58) ( 0.37)

BBB 0.45 0.38 0.45 0.41( 3.39) ( 2.46) ( 3.50) ( 3.22)

BB 1.44 1.37 1.43 1.40( 4.22) ( 3.51) ( 4.24) ( 4.14)

B or worse 2.15 2.06 2.14 2.11( 7.03) ( 6.53) ( 7.06) ( 6.95)

Firm-specific/Accounting VariablesPre Tax Int Cov [5,10) 0.12 0.12 0.12 0.12

( 1.26) ( 1.18) ( 1.30) ( 1.30)Pre Tax Int Cov [10,20) 0.20 0.25 0.20 0.20

( 1.07) ( 1.24) ( 1.06) ( 1.07)Pre Tax Int Cov [20,∞) 0.21 0.28 0.21 0.22

( 0.99) ( 1.21) ( 0.97) ( 1.03)Inc to Sale -2.53 -3.42 -2.48 -2.40

( -1.98) ( -2.21) ( -1.96) ( -1.89)LT Debt to Assets -1.36 -1.35 -1.40 -1.29

( -1.38) ( -1.27) ( -1.42) ( -1.28)Market Leveage 3.69 3.75 3.66 3.61

( 3.46) ( 3.41) ( 3.45) ( 3.39)Macroeconomic and Other Variables

Treasury (1y) -0.54 -0.50 -0.53 -0.53( -6.28) ( -5.10) ( -6.24) ( -6.27)

Term Spread (10y-2y) -0.47 -0.40 -0.46 -0.47( -5.76) ( -4.40) ( -5.68) ( -5.75)

TED Spread (30-day) -0.05 -0.07 -0.06 -0.05( -1.62) ( -2.03) ( -1.68) ( -1.60)

Time to Maturity 0.01 0.01 0.01 0.01( 7.57) ( 3.26) ( 7.66) ( 9.69)

R2 0.51 0.52 0.51 0.51N 30370 26105 30370 30370

51

Table 11: Yield Spreads Determinants: Investment Grade Vs Junk Bonds

Using data from January 2003 to December 2008, I regress end-of-month credit spread levels on severaldeterminants. The sample is trimmed and excludes credit spreads below and above the bottom and toppercentiles respectively. The first two regressions use the sub-sample of investment-grade bonds while thelast two regressions use only junk bonds. The table reports OLS estimates with t-statistics clustered byissuer.

Investment Grade Junk1 2 3 4

Liquidity and Volatility MeasuresLog LOT 0.12 0.33

( 10.78) ( 4.14)Log Rossi 0.09 0.10

( 6.23) ( 0.61)Realized Volatity 3.62 3.50 6.92 6.57

( 12.46) ( 12.97) ( 9.09) ( 8.69)Firm-specific/Accounting Variables

Pre Tax Int Cov [5,10) 0.10 0.08 0.33 0.52( 1.07) ( 0.90) ( 0.85) ( 1.22)

Pre Tax Int Cov [10,∞) 0.06 -0.02 0.37 0.82( 0.38) ( -0.11) ( 0.63) ( 1.36)

Inc to Sale -1.98 -0.93 -4.96 -5.71( -1.47) ( -0.92) ( -1.44) ( -1.60)

LT Debt to Assets -1.74 -1.95 -6.68 -5.71( -3.05) ( -3.42) ( -2.21) ( -2.04)

Market Leveage 3.11 2.97 8.10 7.74( 2.67) ( 2.70) ( 3.02) ( 2.83)

Macroeconomic and Other VariablesTreasury (1y) -0.45 -0.46 -1.30 -1.38

( -9.36) (-10.43) ( -3.53) ( -3.28)Term Spread (10y-2y) -0.65 -0.67 -1.62 -1.71

( -8.50) ( -9.24) ( -2.76) ( -2.65)TED Spread (30-day) 0.29 0.30 0.24 0.26

( 8.39) ( 9.24) ( 2.14) ( 2.37)Time to Maturity 0.01 0.01 0.00 0.01

( 4.21) ( 8.56) ( 0.15) ( 2.05)R2 0.47 0.46 0.57 0.55N 22298 26086 3807 4284

52

Table 12: Yield Spreads Determinants: Pre-Crisis Vs Crisis

Using data from January 2003 to December 2008, I regress end-of-month credit spread levels on severaldeterminants. The sample is trimmed and excludes credit spreads below and above the bottom and toppercentiles respectively. The first two regressions use the sample period going from January 2003 to December2006 while the last two regressions use sample period from January 2007 to December 2008. The table reportsOLS estimates with t-statistics clustered by issuer.

Pre-Crisis Crisis1 2 3 4

Liquidity and Volatility MeasuresLog LOT 0.12 0.12

( 7.31) ( 3.55)Log Rossi 0.07 0.14

( 3.29) ( 3.77)Realized Volatity 2.62 2.31 4.88 4.67

( 4.47) ( 4.07) ( 6.68) ( 6.88)Credit Ratings (relative to AAA-AA)

A 0.02 0.07 -0.85 -0.81( 0.32) ( 1.19) ( -3.42) ( -3.34)

BBB 0.14 0.26 0.03 -0.06( 1.02) ( 2.52) ( 0.06) ( -0.12)

BB 1.36 1.41 0.76 0.84( 3.02) ( 3.22) ( 2.04) ( 2.27)

B or worse 1.77 1.88 3.37 3.29( 7.02) ( 7.59) ( 4.17) ( 4.19)

Firm-specific/Accounting VariablesPre Tax Int Cov [5,10) 0.22 0.16 -0.24 -0.00

( 2.38) ( 1.83) ( -0.27) ( -0.00)Pre Tax Int Cov [10,20) 0.41 0.24 -0.66 -0.36

( 2.21) ( 1.55) ( -0.68) ( -0.41)Pre Tax Int Cov [20,∞) 0.43 0.23 -0.04 0.33

( 1.92) ( 1.27) ( -0.03) ( 0.29)Inc to Sale -3.76 -2.16 -0.04 0.23

( -2.22) ( -1.55) ( -0.01) ( 0.04)LT Debt to Assets -0.59 -0.73 -1.03 -1.50

( -0.72) ( -0.92) ( -0.41) ( -0.62)Market Leveage 3.61 3.60 4.85 4.96

( 2.98) ( 2.72) ( 2.04) ( 2.21)Macroeconomic and Other Variables

Treasury (1y) -0.09 -0.10 -0.94 -0.97( -1.42) ( -1.33) ( -7.82) ( -9.15)

Term Spread (10y-2y) -0.19 -0.21 -1.23 -1.28( -2.05) ( -1.96) ( -6.16) ( -7.08)

TED Spread (30-day) -0.12 -0.11 0.06 0.07( -1.90) ( -1.90) ( 2.38) ( 2.44)

Time to Maturity 0.01 0.02 -0.00 -0.00( 5.13) ( 9.29) ( -1.18) ( -0.03)

R2 0.17 0.14 0.57 0.56N 18363 21361 7742 9009

53

Figure 1: Realized Volatility of Ford Motor Company

This figure shows the realized volatility (top panel) of the equity returns of Ford Motor Company. Themiddle and bottom panel show the diffusive and jump component respectively of the realized volatility. Todisentangle the diffusive and jump variation, I use the ratio statistics RJit ≡ RVit−BVit

RVit, which converges to

a normal distribution under the null hypothesis of no jumps.

54

Figure 2: LOT Model: Estimates Distribution

The upper row of the figure shows the distribution of the estimated factor loading (bond factor on theleft and equity factor on the right). The middle row presents the distribution of the negative and positivethresholds (the negative of αs on the left and αb on the right). The last row report the distribution of theestimates of round-trip transaction costs, given by αb − αs.

55

Figure 3: Idiosyncratic Volatility and Distribution of Factor Loadings

This figure reports the distribution of the estimated factor loadings of the friction model with idiosyncraticrisk and time-varying liquidity. The factors are (respectively from top to bottom) the market bond factor,the individual equity return factor, and the individual equity return realized volatility factor.

56

Figure 4: Liquidity Measurers and Trading Activity

This figure reports scatter plots of the estimated liquidity cost measures from the LOT model and the modelthat I propose versus the percentage of zero trading days, which is obtained dividing the number of days onwhich a bond trades by the total available number of trading days.

57

Figure 5: Explanatory Power of Ratings and Realized Volatility

This figure reports several scatter plots of the log yield spreads on credit ratings (top) and realized volatility(bottom). A linear and quadratic fit (and their relative equations) are also included in the bottom panel.Yield spreads are calculated as the difference between corporate and treasury yields with similar character-istics. Realized volatility is calculated (on a daily basis) with high-frequency data by squaring and adding5-minute returns.

58