Keywords:
Abstract:
We estimate the nondefault component of corporate bond yield spreads and examine its relationship with bond liquidity. We measure bond liquidity using intraday transactions data and estimate the default component using the term structure of credit default swaps (CDS) spreads. With swap rate as the risk free rate, the estimated nondefault component is generally moderate but statistically significant for AA, A, and BBBrated bonds and increasing in this order. With Treasury rate as the risk free rate, the estimated nondefault component is the largest in basis points for BBBrated bonds but, as a fraction of yield spreads, it is the largest for AAArated bonds. Controlling for the unobservable firm heterogeneity, we find a positive and significant relationship between the nondefault component and illiquidity for investmentgrade bonds but no significant relationship for speculativegrade bonds. We also find that the nondefault component comoves with macroeconomic conditionsnegatively with the Treasury term structure and positively with the stock market implied volatility.
JEL Classifications: G12, G13, G14
Key words: Corporate bond yield spreads, credit default swaps, liquidity
We estimate the nondefault component of corporate bond yield spreads and examine its relationship with bond liquidity. We measure bond liquidity using intraday transactions data and estimate the default component using the term structure of credit default swaps (CDS) spreads. With swap rate as the risk free rate, the estimated nondefault component is generally moderate but statistically significant for AA, A, and BBBrated bonds and increasing in this order. With Treasury rate as the risk free rate, the estimated nondefault component is the largest in basis points for BBBrated bonds but, as a fraction of yield spreads, it is the largest for AAArated bonds. Controlling for the unobservable firm heterogeneity, we find a positive and significant relationship between the nondefault component and illiquidity for investmentgrade bonds but no significant relationship for speculativegrade bonds. We also find that the nondefault component comoves with macroeconomic conditionsnegatively with the Treasury term structure and positively with the stock market implied volatility.
JEL Classifications: G12, G13, G14
Key words: Corporate bond yield spreads, credit default swaps, liquidity
To what extent do corporate bond yield spreads reflect default risk? How is the nondefault component of yield spreads, if it exists, associated with bond liquidity? These are fundamental issues to understanding how financial markets value corporate bonds and thus important for corporate financing, risk management, and monetary policy (Kohn; 2007). Early studies compared observed yield spreads to the estimates based on bond pricing models fit to historical data on corproate bond defaults and found mixed results (e.g., Jones et al. (1984), Longstaff and Schwartz (1995), Duffie and Singleton (1997), Duffee (1999), Elton et al. (2001), CollinDufresne et al. (2001), Delianedis and Geske (2001), Huang and Huang (2003), Eom et al. (2004)). For example, Elton et al. (2001) suggested that, when taking into account both expected credit loss and associated risk premiums, most of yield spreads are attributable to default risk. In contrast, Huang and Huang (2003) suggested that the nondefault component accounts for the majority of yield spreads, especially so for highrated investmentgrade bonds. These conflicting results may be due largely to data limitations and model sensitivity in estimating the default component (Huang and Huang; 2003; Delianedis and Geske; 2001; Eom et al.; 2004).
To address these issues, recent studies examine the determinants of corporate bond yield spreads using data on credit default swap (CDS) spreads (e.g., Longstaff et al. (2005), Nashikkar and Subrahmanyam (2006), Ericsson et al. (2007)). They generally find that the majority of corporate yield spreads are due to default risk. To understand the advantage of using CDS data, a brief description of CDS is useful. A CDS is like an insurance contract on credit risk, where a protection seller promises to buy the reference bond at its par value when a predefined credit event occurs. In return, a protection buyer makes periodic payments to the seller until the maturity date of the contract or until a credit event occurs. This periodic payment, usually expressed as a percentage of the notional value of protection, is called the "CDS spread". Since default risk is traded through CDS separately from other factors, such as embedded options, that may affect the bond price, the CDS spread allows for a reasonable estimate for the default component of yield spread without explicitly estimating expected credit loss and associated risk premium.
In this paper we also use CDS spreads to estimate the default component of corporate bond yield spreads and examine the link between the nondefault component and liquidity. Utilizing a dataset far richer than those in existing studies, our comprehensive analysis contributes to the literature in three dimensions. First, we develop a new method to estimate the default component by deriving a firmspecific discount rate curve from the term structure of its CDS spreads. We use the discount rate curve to price each of the firm's senior unsecured straight bonds and compute the implied yield as our estimate for the default component of the observed yield. Because the CDSimplied yield and the observed yield are based on identical cash flow, we are able to match exactly each bond's maturity and fully correct any coupon effects. In contrast, most existing studies used only 5year CDS spreads and thus had to estimate a hypothetical 5year bond yield using a set of existing bonds. As a result, liquidity and bond characteristics, such as bond age and cash flow, of this hypothetical bond are not directly observable, limiting the scope of crosssectional analysis and the ability to correct the coupon effect.^{3}
Second, we improve the analysis of the effect of liquidity on the nondefault component of yield spreads by using intraday transactions data to measure bond liquidity. Previous studies suggested that liquidity may manifest through the price impact of trades or market depth (e.g., Kyle (1985)), transaction costs (e.g., Acharya and Pedersen (2005)), or trading frequency (e.g., Vayanos (1998) and Lo et al. (2004)). We explore a number of measures to capture each of these aspects of bond liquidity.^{4}Importantly, our liquidity measures vary both across bonds and over time. By contrast, most existing studies used bond characteristics, such as coupon, size, maturity, and age, as proxies for bond liquidity (Fisher; 1959; Ericsson et al.; 2007; Houweling et al.; 2005; Longstaff et al.; 2005; Perraudin and Taylor; 2003).^{5}Interpreting the relation between bond spreads and these proxies may be complicated by the possible correlations between the proxies and the issuer's credit risk. More importantly, while these proxies may vary across bonds, they are either constant or changing deterministically with the passage of time. Thus, they may not identify the effects of stochastic variation in bond liquidity on the nondefault component of yield spreads.
Third, our methodology allows us to better control for the unobservable firm heterogeneity that may have biased previous estimates. The nondefault component of yield spreads may be affected by firmspecific factors, such as clientele effects, that are correlated with our liquidity measures. To the extent that these factors are unobservable, an omitted variable bias occurs in the regression estimation. Since our estimation method allows for multiple bonds by each firm at any time, we have enough degrees of freedom to apply a fixedeffects approach to control for the crossfirm variation attributable to the unobservable firm characteristics (Chen et al.; 2007).
Our main results are based on swap rate as the risk free rate, as swap rate is widely believed to be closer to the risk free rate benchmark used by market participants in pricing corporate debt and its derivatives (e.g., Hull et al. (2004) and Ericsson et al. (2007)). We find that the estimated nondefault component of yield spreads is statistically significant for only AA, A, and BBBrated bonds and increasing in this order both in basis points and as a fraction of yield spreads. For speculativegrade bonds, the estimated nondefault components are generally insignificant. Among those statistically significant, the sizes of the estimated nondefault components are in general moderateranging from 3 basis points or 13 percent of yield spreads for AArated bonds to 24 basis points or 22 percent of yield spreads for BBBrated bonds. Even so, our point estimates appear to be larger than those in existing studies, in particularly for BBBrated bonds. For example, Longstaff et al. (2005) found the nondefault components are statistically significant for A and BBBrated bonds, accounting respectively for about 10 and 6 percent of their yield spreads.
We also find that with Treasury rate as the risk free rate, the nondefault components are statistically significant for all investmentgrade bonds (i.e., those rated AAA, AA, A, and BBB) and BBrated bonds. In basis points, the nondefault component is the largest for BBBrated bonds, about 60 basis points, and the smallest for AAArated bonds, about 32 basis points. As a fraction of yield spreads, the nondefault components are decreasing in bond rating, that is, the highest for AAArated bonds, 77 percent, and the lowest for BBrated bonds, 17 percent. The nondefault components account for more than half of yield spreads for A and higherrated bonds, opposite to the empirical results in Elton et al. (2001), Longstaff et al. (2005) but consistent with the calibration results in Huang and Huang (2003).
In our regression analysis, we link the the nondefault component to our liquidity measures constructed from intraday transactions data. We find a positive and statistically significant relationship between the nondefault component of yield spreads and illiquidity for investmentgrade bonds (i.e., those rated AA, A, and BBB) but no significant relationship for speculativegrade bonds. This result contrasts to Chen et al. (2007) who suggested the liquidity effects are stronger for speculativegrade bonds.^{6} Our point estimates suggest that relative to total yield spreads, the liquidity effects decrease in ratingthe strongest for AArated bonds and the weakest for BBBrated bonds. Specifically, when one of our liquidity measures deteriorates by the magnitude of its interquartile range, the increase in the nondefault component can be as high as 10 percent of total yield spreads for AArated bonds, 7 percent for Arated bonds, and 4 percent for BBBrated bonds. While previous studies such as Longstaff et al. (2005) and Nashikkar and Subrahmanyam (2006) also suggested the nondefault component is positively related to illiquidity, they generally did not distinguish the liquidity effects by rating groups.^{7}
We also find that the nondefault component of bond spreads comoves with macroeconomic conditionsnegatively with the Treasury term structure and positively with the stock market implied volatility (VIX). This result is consistent with previous studies suggesting that corporate yield spreads are associated with marketwide liquidity factors (Longstaff; 2004; Liu et al.; 2006; Delianedis and Geske; 2001; CollinDufresne et al.; 2001; Duffie and Singleton; 1997). In addition, controlling for conventional liquidity proxies affects little the statistical significance of our transactionbased liquidity measures, suggesting our measures identify a unique part of the variation in the nondefault component of yield spreads. Finally, the estimated effects of our transactionbased liquidity measures are largely robust to a number of alternative model specifications and data samplings, such as exclusing newsdriven trades and using Treasury rate as the risk free rate.
The rest of the paper is organized as follows: Section 2 describes data sources and sampling schemes; Section 3 presents our methodology estimating the nondefault component of yield spreads and examines its crosssectional and timeseries properties; Section 4 reports our regression results on the effects of liquidity on the nondefault component; and Section 5 concludes.
Our overall sample consists of bonds with data available on both bond prices and associated CDS spreads from January 1, 2001 to April 30, 2007. We use this sample to examine the crosssectional and timeseries properties of the nondefault component of yield spreads. To analyze the effect of liquidity on the nondefault component, we further merge the overall sample with intraday bond transactions data from NASD's TRACE (Trading Reporting and Compliance Engine) system, resulting in a smaller "regression sample." Throughout this paper, we conduct our analysis at the monthly frequency, where, unless noted otherwise, the monthly value of a timevarying variable is the average of its corresponding daily values. The rest of this section provides details on our data and sampling method.
The data on daily bond yields are from Merrill Lynch's Corporate Bond Index Database ("the ML Database").^{8} The ML Database also contains information on some bond characteristics, including the amount of face value outstanding and a composite rating based on S&P and Moody's ratings. Additional bond descriptive information is obtained from both Bloomberg and Moody's DRS databases.^{9} We retain only senior unsecured U.S. dollardenominated bonds issued by U.S. firms that pay fixed semiannual or zero coupons with remaining maturity less than 15 years. We also delete bonds that are callable, puttable, convertible, or have sinking fund features.^{10}
We use issuer ticker to merge the bond yield data with the CDS spread data provided by Markit Partners. Issuer tickers are manually checked and adjusted to ensure the merge accuracy. The Markit's data contain daily composite spread quotes on CDS contracts with maturities at 6 month, 1, 2, 3, 5, 7, 10, 20, and 30 years.^{11} Following the common practice, we use quotes corresponding to the modified restructuring clause for U.S. dollardenominated notional values. In addition, a reference entity is included on any day only if its CDS quotes are nonmissing at 1 and 10year and at additional two or more of the four maturities in between.
As shown in Panel A of Table 1 (memo item), the overall sample consists of 1263 unique bonds from 328 firms (identified by unique issuer ticker), with on average nearly 4 bonds per firm. The numbers of bonds and firms vary significantly by bond rating. Slightly over three quarters of the sample are investmentgrade bonds, somewhat more than the proportion in the overall corporate bond universe. Also, in term of number of bonds, A and BBBrated bonds are by far the most available; AA and BBrated bonds come next; and bonds in both tails of the rating distribution (i.e., AAA and CCC/below) are the fewest. In addition, excluding the tails of the rating distribution, the average number of bonds per firm increases in rating, from slightly over 2 for Brated bonds to about 10 for AArated bonds.
We use intraday transactions data provided by NASD's TRACE to compute measures for corporate bond liquidity. TRACE started to disseminate to the public intraday transactions data on July 1, 2002 for a small number of selected corporate bonds; but the dissemination expanded gradually and began to cover most of the corporate bonds traded over the counter on October 1, 2004 (see Appendix A for more details on the TRACE data). The data contain trading information such as transaction price, trading size, settlement date and time. Following the practice in the existing studies using the TRACE data, we remove observations with "data errors" (e.g., Edwards et al. (2007)).^{12}
We first estimate daily liquidity measures and then compute their monthly average values, which in turn are merged with our overall sample using bond CUSIPs. The resultant "regression sample" is significantly smaller than the overall sample due mainly to the limited coverage of TRACE data before the full dissemination phase. As shown in Panel B of Table 1 (memo item), the regression sample consists of 808 unique bonds from 242 firms, with on average slightly over 3 bonds per firm. Even so, the distribution of the number of bonds by rating is similar to that in the overall sample. First, about 80 percent of the regression sample are investmentgrade bonds. Second, most of investmentgrade bonds are A or BBBrated, and most of speculativegrade bonds are BBrated. Third, excluding the tails of rating categories, the average number of bonds per firm increases in rating, from close to 2 for Brated bonds to about 7 for AArated bonds.
Our analysis focuses on the results with swap rate as the risk free rate. It is now widely believed that swap rate is closer to the risk free rate benchmark used by market participants in pricing corporate debt and its derivatives, in part because swaps face similar tax and regulatory treatments as corporate credits do (see, e.g., Hull et al. (2004); Houweling and Vorst (2005); Longstaff et al. (2005); Blanco, Brennan and Marsh (2005); Zhu (2006)). In contrast, although Treasury securities are almost truly default free, Treasury yields may be affected by other factors, such as the specialness of Treasury securities and taxation benefits.^{13}
Nonetheless, we also contrast our main results with those using Treasury yields as the risk free rate, not only because some existing studies used Treasury yields but also because swap rate is not completely risk free due to the counterparty credit risk in the swap contract and the credit risk in the LIBOR rate.
We use the following conventional variables to measure macroeconomic conditions: the level and the slope of Treasury term structure, the return, historical and implied volatilities on the S&P 500 index, and Treasury 10year ontherun premiums. These variables are collected from Bloomberg and the Federal Reserve Board.
In this section, we first describe, with an example, our method of using the CDS termstructure to estimate the nondefault component of corporate bond yield spreads. We then examine the properties of the estimated nondefault component in both cross section and time series.
The key issue of estimating the nondefault component of corporate bond yield spreads is to estimate appropriately the default component. Broadly speaking, there are two approaches to estimating the default component: one based on corporate bond pricing models, and the other based on CDS spreads. Typically, the former approach first calibrates a corporate bond pricing model to match historical data on corporate bond default frequency and loss given default, then uses the yield spread implied by the model as the estimate for the default component of the observed yield spread (e.g., Huang and Huang (2003)). This approach has two main drawbacks: one, the estimates are sensitive to the model assumptions on both default process and risk premium (Huang and Huang; 2003; Delianedis and Geske; 2001; Eom et al.; 2004); two, it is difficult, if not impossible, to estimate expected credit loss on individual bonds with reasonable precision. Estimations using aggregate default data ignore completely the heterogeneous risk profiles among different bonds and may have significant statistical errors because historical default events are sparse and clustered in a small number of recession periods.
The CDSbased approach avoids these potential problems because CDS spreads reflect market expectations on both default probability and loss given default and the associated risk premiums. As shown in Duffie (1999), under certain conditions, CDS spreads are equal to the yield spread on a bond with the same credit risk exposure. Due to data limitations, most existing studies use only 5year CDS spread data (e.g., Longstaff et al. (2005); Blanco, Brennan and Marsh (2005); Zhu (2006); and Nashikkar and Subrahmanyam (2006)). Of course, it is rare for a reference entity to have a bond maturing in exact 5 years on any given day. As a result, researchers rely on pricing information on the bonds straddling the 5year maturity to estimate the yield spread on a hypothetical bond at the 5year maturity. This may induce an estimation error because the reference entity might have issued a 5year bond with different terms and the price on the 5year hypothetical bond might have been different if it were actually traded. In addition, it is hard to fully address the coupon effect in bond yield computations, partly because the cash flow of the hypothetical bond is not well defined. Also, because there are no observable data on the hypothetical bond for either liquidity proxies or transactions data, statistical analysis on the liquidity effect has to be done using the bonds in the bracket (see footnote 1).
We also use CDS data to estimate the default component of yield spreads, and our approach avoids constructing any hypothetical bonds and addresses the issues of both maturity mismatch and coupon effect.^{14} Our estimation has three steps. First, for each firm on each day, we estimate a CDSimplied par yield curve by adding swap rates to CDS spreads at observed maturity points and interpolating across maturities using the piecewise cubic Hermite interpolating polynomial (PCHIP) algorithm.^{15}Under certain conditions laid out in Duffie (1999) and assuming swap rate is the appropriate measure of risk free rate, the resulting curve equals the par yield curve for floatingrate bonds with the same credit profile as the reference entity. Duffie and Liu (2001) further show that par yields on floatingrate and fixedrate bonds by the same issuer would differ only a bit for the usual range of interest rate term structures and term to maturities (see also Longstaff et al. (2005) and Nashikkar and Subrahmanyam (2006)). Thus, we use the resulting curve as a reasonable approximation for the paryield curve for fixedrate bonds with the same credit profiles.^{16}
Second, from a firm's CDSimplied par yield curve, we compute zero yield curve and discount rate curve using the standard bootstrap method. Finally, we use the estimated discount rate curve to discount the cash flow of each bond and obtain an estimate of the bond price implied by the firm's CDS term structure. We call the yield computed from the resulting bond price "the CDSimplied yield". Importantly, the actual bond yield and the CDSimplied yield have identical cash flows, so we remove both maturity mismatch and coupon effect. Moreover, our approach implies that on any given period when a firm has multiple bonds meeting our sampling criteria, they are all kept in our final sample. As discussed later, these extra degrees of freedom allow us to apply a fixedeffects approach to control for the unobservable firm heterogeneity, which effectively identifies the liquidity effect using variation across bonds by the same issuer.
Our estimate for the nondefault component of yield spreads is simply the difference between the actual bond yield and the CDSimplied yield, and the default component of yield spreads is simply the difference between yield spread and the nondefault component.
In Figure 1, we show an example of our estimation for CocaCola Inc.. On April 30, 2007, the firm has 7 Arated bonds outstanding, with their remaining maturities ranging from 2.4 years to 14.8 years. Quotes on CDS spreads are available at maturities from 6 month to 15 years. As shown in the top panel, our first step is to add up CDS spread and swap rate, marked as "O", and use PCHIP algorithm to fit the CDSimplied par yield curve, the solid line. Typical during this period, CocaCola's CDSimplied par yield curve is inverted at the short end of its maturity range. From this par yield curve, we use the standard bootstrap method to derive a zero yield curve, the dashdotted line in the top panel, and then compute the corresponding discount rate curve, shown in the middle panel. Finally, we use this discount rate curve to price each of CocaCola's bonds and compute their corresponding CDSimplied yields. In the bottom panel, we contrast these CDSimplied yields, marked as "O", to the actual yields, marked as "X". Clearly, the nondefault components of yield spreads, as measured by the difference between "X" and "O" marks, vary across bonds. Below we examine the statistical properties of such variation with better controlled samples in both crosssection and timeseries.
We examine the crosssectional characteristics of the components of yield spreads for a sample of bonds with relatively stable risk profile during the period. Specifically, we remove bonds whose ratings ever changed by one or more whole rating letter and bonds that appear in less than three months over the period.^{17} For each bond, we then compute its average yield spread and average default and nondefault components over the entire period. This results in a pure crosssectional sample, consisting of 743 investmentgrade bonds and 111 speculativegrade bonds.
Table 2 reports average values of yield spread and its components by bond rating. Column (1) shows the average spread of bond yield over comparablematurity swap rate. Columns (2) and (3) show, respectively, the default and nondefault components of the spread. Column (4) calculates the nondefault component as a fraction of yield spreads. Several patterns emerge from the table. First, not surprisingly, both yield spread and the default component increase with worse rating, from under 10 basis points for AAArated bonds to over 10 percent for CCrated bonds. Second, the nondefault component, both in basis points and as a fraction of yield spreads, is statistically significantly different from zero for all investmentgrade bonds except AAArated ones, with their sizes increasing with worse rating. In term of economic maganitude, the nondefault component is moderate in general, ranging 3 basis points and 13 percent of yield spreads for AArated bonds to 24 basis points and 22 percent of yield spreads for BBBrated bonds.^{18}Even so, they are still notably larger than those in Longstaff et al. (2005), which, in contrast, found that nondefault components are insignificant for AAA/AArated bonds and decrease with worse rating (in particular, only 6 percent for BBBrated bonds). Third, the nondefault components are statistically insignificantly different from zero for all speculativegrade categories except Brated bonds. Notably, except for BBrated bonds, these nondefault components are all negative. Fourth, for all investmentgrade bonds together, the nondefault component averages 12 basis points and accounts for about 20 percent of yield spreads, while for speculativegrade bonds, the nondefault component is not significantly different from zero.
Columns (5)(8) repeat the same exercises with Treasuryrate as the risk free rate measure. The results contrast to those with swap rate in several aspects. First, the nondefault components, both in basis points and as a fraction of yield spreads are statistically significantly different from zero for all investmentgrade rating categories and, as a fraction of yield spreads, decrease with worse ratings. In particular, the nondefault components account for more than half of yield spreads for A or betterrated bonds, and just over 40 percent of yield spreads for BBBrated bonds. This contrasts to the result in Longstaff et al. (2005), which found that the nondefault components are less than half of yield spreads for all investmentgrade bonds when using Treasury rate as the risk free rate. Second, the nondefault components are statistically significant for BBrated bonds, accounting for 17 percent of yield spreads, but insignificant for other speculativegrade bonds. The results for BBrated bonds are close to those found in Huang and Huang (2003) and Longstaff et al. (2005). Third, for all investmentgrade bonds together, the nondefault component accounts for nearly half of spreads; while for speculativegrade bonds, the nondefault component is less than 10 percent of yield spreads. Both averages are statistically different from zero.
It is interesting to note that the choice of different risk free rate does not have much impact on the default component estimates (i.e., Columns (2) and (6)). That is, the different patterns of the nondefault components with alternative risk free rates reflect mostly the differences in yield spreads due to the factors causing the divergence between Treasury and swap rates, such as Treasury specialness and tax benefits. To the extent that these factors do not vary with corporate bond ratings, their effects account for a bigger part of yield spreads for higherrated investmentgrade bonds because their yield spreads are already low.
After having examined the means, Figure 2 plots by bond rating the histograms of the average nondefault component for each bond in the crosssectional sample with swap rate as the risk free rate measure. We group all speculativegrade bonds except the CCrated bond into a single category and don't show AAArated bonds due to their small sample sizes. A striking pattern of these histograms is that for each rating category, the density of the the nondefault component all peaks at nearly zero basis point. In addition, while the distributions are fairly narrow for AA and Arated bonds with right skewness, they are rather flat and fattailed for BBBrated and, especially, speculativegrade bonds. Previous studies suggest that the variation in bond liquidity attribute to these crosssectional variation in the nondefault component, a hypothesis we will test in the next section.
Figure 3 plots by bond rating the median values of the monthly nondefault component for the bonds in the overall sample.^{19} The top panel uses swap rate as the risk free rate. Several points are worth to note. First, as we have seen in the crosssectional analysis, the nondefault component for BBBrated bonds, dotted line, was almost always the highest among all rating categories. In addition, it declined notably from about 30 basis points in 2001 to about zero in early 2004 and then trended slightly up since 2006. Second, before 2004, the nondefault component for Arated bonds, averaging 10 basis points, was generally higher than that for AArated bonds, averaging just below zero. However, since 2004, the two series became statistically indifferent; and both trended slightly up since 2006. Third, the nondefault component for speculativegrade bonds appeared to be volatile before 2003, due mainly to the small number of bonds in the early period (from about 10 bonds in early 2001 to about 60 bonds at the end of 2002). Since 2003, it had fluctuated around zero and fallen below zero in 2007.
The time series of nondefault component with Treasury rate, plotted in the lower panel of Figure 3, show similar patterns to those with swap rate, but with two notable differences. First, all series shifted upward; Second, we see more clearly a secular decline in the nondefault components for all investmentgrade bonds from 2001 to 2004 and a gradual pickup since 2005.
One of our goals in the following analysis is to understand to what extent the observed timeseries variation in the nondefault component are attributable to the stochastic variation in bond liquidity.
In this section, we first describe the construction of our liquidity measures using corporate bond intraday transactions data. Then we report the regression results on the effects of liquidity on the nondefault component of yield spreads. We find a statistically significant positive relationship between the nondefault component and bond illiquidity for investmentgrade bonds. Our analysis also suggests that our liquidity measures identify a unique portion of the time variation in the nondefault component and that the nondefault component comoves with macroeconomic conditions.
Using intraday transactions data for corporate bonds reported in TRACE, we compute one measure for each of the following three types of bond liquidity definitions: price impact of trades, transaction cost, and trading frequency.^{20} Considering these multiple measures is important because different aspects of the liquidity concept may manifest itself in different fashions in the intraday trading statistics. We also discuss bond characteristics that are used in the literature as proxies for bond liquidity, and examine their relationship with our tradingbased liquidity measures. Table 3 reports descriptive statistics for these liquidity measures.
Bond liquidity may manifest through the price impact of trades or market depth (Kyle; 1985). We adopt one of the most frequentlyused price impact measures, proposed by Amihud (2002), by defining the Amihud measure as the ratio of the absolute percentage change in bond price to the dollar size of a trade (in million dollars). That is, for each day and bond , we define
The Amihud measure indicates illiquidity in that a larger value implies that a trade of a given size would move the price more, suggesting the bond is more illiquid. By construction, daily Amihud measures are nonmissing for only bonds traded at least twice on the day.
As shown on Line 1 of Table 3, for all rating categories together, the median Amihud measure is 0.34, suggesting that a median trade, at about $ (Line 10), would move price by roughly 1 percent. By rating, the median Amihud measure is the highest for speculativegrade bonds, at 0.42, which is only modestly higher than those for other rating categories, all at about 0.32.
Liquidity is also often defined by transaction costs (e.g., Amihud and Mendelson (1986), Acharya and Pedersen (2005)). A commonlyused measure for transaction costs is bidask spread. Unfortunately, our data do not have information on bidask quotes or on the side initiating a tradewhich potentially could be used to trace out effective bidask spreads. Instead, we estimate bidask spreads using the wellknown Roll (1984) model. Under certain assumptions, Roll showed that the effective bidask spread equals to the square root of the negative covariance between price changes in adjacent trades. That is,
The intuition of the Roll model is the following. Assuming informational efficiency and no news on a bond's fundamental values, bond prices should bounce up and down within the band formed by bidask quotes, generating a negative correlation between price changes in adjacent trades. The extent of this negative correlation depends on the the width of the band. By construction, daily bidask spread estimates are nonmissing for only bonds traded at least three times on the day.
As shown on Line 2 of Table 3, for all rating categories together, the median estimated bidask spread is 0.91 percent of price, rather costly comparing to trading stocks and Treasury securities (Hasbrouck; 2005; Chakravarty and Sarkar; 2003; Fleming; 2003). By rating, the median estimated bidask spreads increase with worse ratings, with the lowest at percent of price for AArated bonds and the highest at 1.3 percent of price for for speculativegrade bonds.
Bond liquidity may also be reflected in trading frequency. Intuitively, all else equal, bonds that are more illiquid would trade less frequently. Trading frequency measures have been widely used as indicators for asset liquidity (see, e.g., Vayanos (1998), Lo et al. (2004), and Chen et al. (2007)). We consider monthly turnover rate as our trading frequency measure, which is the ratio of total trading volume in a month to the amount of face value outstanding.
As shown on Line 3 of Table 3, for all rating categories together, the median monthly turnover rate is merely 0.04, meaning that for the average bond in our sample, it takes about 25 months to turn over once. That corporate bonds are traded sparsely is also evident by other measures: the median number of traded days, Line 8, is days, the median number of trades in a month, Line 9, is 44, and the median monthly trading volume, Line 11, is about $15 million.
There is no apparent difference by rating in the median turnover rate. While betterrated bonds tend to have higher median numbers of trades or traded days in a month, they are also generally larger in face values outstanding. For example, the median number of trades for AArated bonds is 100 times a month, notably larger than 35 times a month for speculativegrade bonds (Line 9); but the median size of AArated bonds is $ 800 million, also notably larger than just under $ 300 million for speculativegrade bonds (Line 7).
Table 4 shows pairwise correlations among the above three liquidity measures within each rating category. The correlations vary widely and are generally not particularly strong. Specifically, the correlations between the Amihud measure and bidask spread, are positive as expected, but they are less than 50 percent for all rating groups. The correlations between the Amihud measure and turnover rate are negative as expected, but they range from statistical insignificance for BBBrated and speculativegrade bonds to only 8 percent for AArated bonds. The correlations between the bidask spread and turnover rate also vary widely, ranging from 4 percent for Arated bonds to 8 percent for speculativegrade bonds.
The large variation in the correlations among these liquidity measures may reflect the multifaceted nature of the liquidity concept, suggesting that each of these measures may have captured only some aspects of bond liquidity. Thus, it would be helpful to combine these measures in our analysis to exploit their potential complimentary features.
Lacking of intraday transactions data, previous studies often use bond characteristics as proxies for bond liquidity, such as coupon rate, bond age, remaining maturity, and bond size. To save space, we don't recite the various hypotheses that are proposed in the literature on why these proxies may be reasonable. See, for example, Longstaff et al. (2005) for a reference.
Average bond characteristics are shown on Lines 4 to 7 of Table 3. For the entire regression sample, the median bond in a typical month has a coupon rate of 6.4 percent, is close to 4 years since issuance, has slightly over 4 years of remaining maturity, and has $ 400 million dollars outstanding. Not surprisingly, the median coupon rate increases in bond rating. In addition, speculativegrade bonds tend to be smaller and notably older, but the remaining maturity is the longest for BBBrated bonds and the shortest for Arated bonds .
Figure 4 shows the distributions of bond age, remaining maturity, and maturity at issuance for the regression sample. The number of bonds decreases quickly for those older than 9 years (top panel) or those with more than 10 years of remaining maturity (middle panel). These distributions suggest that in interpreting results related to age and remaining maturity, we have to be cautious about the reliability over the range greater than 10 years. In addition, while there are wide variation in the maturity at issuance (bottom panel), about half of the bonds were issued at 10 years, with other mass points at 3, 5, 7, 15, 20, and 30 years.
As argued earlier, bond characteristics used as proxies for liquidity are either constant or deterministic. So we cannot use them to identify timevarying liquidity effects from other stochastic shocks in the nondefault component. To help assess later to what extent our transactionbased liquidity measures contribute to our understanding of the stochastic variation in the nondefault component, we use a regression approach to analyze the relationship between our liquidity measures and bond liquidity proxies. It is worth to point out that our results on the Amihud and bidask spread measures are new to the literature and that those on the turnover rate measure are in general consistent with the evidence in the existing literature (Downing et al.; 2005; Hotchkiss and Ronen; 2002; Alexander et al.; 2004; Edwards et al.; 2007).
Table 5 presents the regression results. Note that to allow for more flexible and potentially nonlinear functional forms, we use a 4th order polynomials for bond age and remaining maturity.^{21} We also include firm and time fixedeffects to account for unobservable firm heterogeneity and macroeconomic effects. The following findings are worth mentioning. First, our transactionbased liquidity measures are weakly related to bond characteristics, especially for lower rated bonds. Specifically, s are modest, from 11 to 36 percent, and generally decreasing with lower ratings. The weak correlation suggests that our liquidity measures and bond characteristics may have captured different aspects of bond liquidity, especially for the lower rated bonds. Second, relationships between different transactionbased liquidity measures and bond characteristics don't necessarily follow the same directions. For example, bonds with larger coupon or smaller size are more liquid by the Amihud measure but less liquid by the turnover rate measure. Again, this points to the multifaceted nature of bond liquidity. Third, as for bond age and remaining maturity, the coefficients on their polynomials are jointly statistically significant at the 95 percent confidence level in all specifications. Their functional forms, plotted in Figure 5, suggest that bonds that are older or have longer remaining maturities are generally more illiquid. The only exception is that turnover rate increases with termtomaturity for speculativegrade bonds.
Before presenting our main regression results, we examine how the nondefault components are related unconditionally to our liquidity measures. Table 6 shows pairwise correlations between nondefault components with swap rate and liquidity measures for each rating group. The results with Treasury rate, not shown, are similar.
For AA and Arated bonds, the correlations between the nondefault component and transactionbased measures are statistically significant and have expected signs. Specifically, the correlations are positive with the Amihud and bidask measures and negative with turnover rate. For BBBrated and speculativegrade bonds, the correlations with turnover rate are significant and negative, as expected, but statistically insignificant with the Amihud measure and negative with the bidask measure. Overall, all correlations are generally low, with the strongest (in absolute values) being those between the nondefault component and turnover rate, 36 percent, for BBBrated bonds.
The correlations between the nondefault component and bond characteristics are more consistent across bond ratings. The nondefault component tends to be larger for bonds with higher coupon, older age, longer remaining maturity, or smaller size. Again, all correlations are moderate with the strongest (in absolute values) being those between the nondefault component and bond size, 39 percent, for both BBBrated and speculativegrade bonds.
We now report regression results on the effects of bond liquidity on the nondefault component of yield spreads. First, we demonstrate the importance of controlling for unobservable firm heterogeneity in identifying the liquidity effect. Second, we show that controlling for CDS liquidity and bond market informational efficiency increases significantly both the model fit and the economic significance of liquidity effects. Third, we test whether our key results are affected by controlling for conventional liquidity proxies. Finally, we present results from a number of analyses for robustness, including explicitly controlling for macroeconomic conditions and using Treasury rate as the risk free rate. Note that, unless specified otherwise, the risk free rate used in the nondefault component estimation is swap rate. In addition, to reduce the impact of outliers, we windsorize the sample at 5 percent of both the nondefault component and liquidity measures used in each regression. We also use log scale for our liquidity measures in all regressions.
Table 7 reports the results from OLS regressions of the nondefault component for four broad rating categories. For each sample, we first regress the nondefault component on each of our three transactionbased liquidity measures, and then on all three measures together. For each regression, we include dummy variables indicating the month of each observation as controls for macroeconomic conditions. Standard errors of the estimated coefficients are computed using the Huber/White robust method assuming that regression residual terms may be correlated across bonds issued by the same firm but uncorrelated across firms.
The results lend some support for the liquidity effect. Specifically, consistent with the common view, the coefficients on turnover rates are all negative, and statistically significant at the 95 percent confidence level for six out of eight regressions. The coefficients on the Amihud illiquidity measure and bidask spread are positive for only AA and Arated bonds and statistical significance in only some regressions (Columns 1 and 4 for the Amihud measure, Columns 2 and 6 for bidask spread). However, the coefficients on the Amihud illiquidity and bidask spread measures are all negative for BBBrated and speculatedgrade bonds, although none is statistically significant. The statistics for all regressions appear to be modest: when all three liquidity measures are included at the same time, ranges from 10 percent for speculativegrade bonds to 36 for BBBrated bonds.
A potential issue with the above OLS regressions is that the nondefault component may be affected by unobservable firm characteristics correlated with our liquidity measures, in which case an omitted variable bias occurs and the direction of biase is unpredictable (Chen et al.; 2007). An example of such unobservable heterogeneity is the "clientele effect". That is, institutional investors may form their bond portfolios based on certain firm characteristics that may be correlated with either credit risk or liquidity. Transactions by these investors in turn may generate liquidity impacts on yield spreads or on the nondefault component (see, e.g., Chacko (2006), Mahanti et al. (2006), and Nashikkar and Subrahmanyam (2006)). To address this issue, we add firm fixedeffects to each of the above models, where a firm is represented by a unique Merrill Lynch ticker. With the fixedeffects model, we now effectively identify the liquidity effect using the variation across bonds issued by the same firm. The richness of our data, especially the full term structure of CDS spreads allowing for multiple bonds by the same firm, gives us enough degrees of freedom to estimate these fixedeffects models.
As shown in Table 8, overall, controlling for the unobservable firm heterogeneity leads to stronger support for the liquidity effect on the nondefault component, especially for investmentgrade bonds. Specifically, comparing to Table 7, the main change is that the coefficients on the Amihud illiquidity and bidask spread measures become positive and statistically significant at the 95 percent confidence level for AA and Arated bonds. In addition, results on turnover rate now show significant liquidity effects in all regressions. But the signs of the coefficients on the Amihud illiquidity and bidask spread measures remain mostly negative for both BBBrated and speculativegrade bonds and even become statistically significant.
The reliability of using CDS spreads to estimate the default component of yield spreads depends on two critical assumptions. First, CDS spreads reflect solely credit risk and the associated risk premium. In particular, this requires that the CDS market is perfectly liquid. While the CDS market may be arguably more liquid than the cash market, partly due to the absence of shortsale constraints and its unfunded nature, (Hull et al.; 2004; Longstaff et al.; 2005), it is still evolving and its liquidity may have been varying over time. Indeed, some recent studies suggest that the effect of CDS illiquidity on CDS spreads may be positive and statistically significant (Tang and Yan; 2007; Nashikkar and Subrahmanyam; 2006). Thus, in the presence of CDS illiquidity, our CDSbased method may have underestimated the nondefault component of yield spreads. Put it differently, our estimated nondefault component would be negatively (positively) correlated with a CDS illiquidity (liquidity) measure. Empirically, it implies that all else equal, if liquidity conditions in bond and CDS markets are (positively) correlated, not controlling for CDS illiquidity results in (downward) biased estimates on the effect of bond illiquidity on the nondefault component of yield spreads.
Second, we assume that both the CDS and bond markets are similarly informational efficient in the sense that bond prices react to the news on credit risk as quickly as CDS spreads do. Recent studies suggest that bond markets may lag behind CDS in price discovery, possibly caused by, among other things, the shortselling constraint or higher transaction costs on corporate bonds (Blanco, Brennan and Marsh; 2005; Zhu; 2006). Specifically, when the issuer's credit quality deteriorates (improves), bond markets may have priced too little (much) spreads relative to CDS spreads, resulting in underestimation (overestimation) of the nondefault component. Empirically, this suggests that without controlling for the less informational efficiency in the bond markets, our estimated nondefault component would have a bias that is increasing in the issuer's credit quality.
To address the above issues, we should control for CDS liquidity and the difference in the informational efficiency between the bond and CDS markets. First, in the absence of direct CDS liquidity measures, e.g., CDS bidask spreads, we use the number of quotes on 5year CDS contracts to control for the CDS liquidity effect. Presumably, a larger number of quotes indicates more dealers making the market, thus improving the CDS liquidity. Thus, our discussion above implies the coefficient on the number of quotes is expected to be positive. Second, instead of trying to measure directly the difference in the informational efficiency between the two markets, we include the oneperiod lagged CDS spread as a measure for the issuer's credit condition to control directly for the potential bias. This variable is read at the corresponding bond's maturity from the CDS term structure fitted using the PCHIP algorithm described above. Our discussions above suggest that all else equal, the coefficients on the lagged CDS spread are expected to be negative.
The results with these two additional controls are shown in Table 9. Overall, controlling for CDS liquidity results in firmer support for the liquidity effect, in terms of coefficient signs, statistical significance, and model fit, especially for investmentgrade bonds. First, more coefficients on the liquidity measures for BBBrated bonds now have expected signs and statistically significant at the 95 percent confidence level. Second, except for AArated bonds, all coefficients on the lagged CDS spread are negative as expected and mostly statistically significant. This suggests that all else equal, the nondefault component of yield spreads increases with the improvement in the issuer's credit quality, consistent with the less informational efficiency in the bond markets. Third, except for AArated bonds, all coefficients of the number of CDS quotes are positive as expected but only statistically significant for the Arated and some BBBrated regressions, generally consistent with the existence of CDS illiquidity. Fourth, notably, the statistics increase significantly across all specifications but most dramatically for the speculativegrade bonds.
To examine the economic magnitude of the liquidity effect, we use the point estimates in Table 9 to calculate how the nondefault components change when each of the liquidity measures changes from its 25th to 75th percentile. We only report those estimates being statistically significant. The results are stated in Table 10. Overall, in basis points, turnover rate has the largest impact, ranging from 1.5 to 2.6 basis points; bidask spread comes the second, about 1 to 2 bps; and the Amihud measure is slightly smaller, about 1 to 1.5 bps. Relative to the median yield spreads for the regression samples, the liquidity effects range from 4 to 10 percent (in absolute values). These calculations suggest that the liquidity effects appear to be quantitatively moderate but nontrivial both relative to the nearzero nondefault components and even to their full yield spreads.
We now examine the significance of our transactionbased liquidity measures after controlling for conventional liquidity proxies. The results are shown in Table 11. Comparing to our benchmark results in Table 9, the point estimates on our transactionbased liquidity measures become somewhat smaller in absolute values, but their statistical significances remain largely unchanged (except column 2). These changes are consistent with the moderate correlations we find above between the transactionbased liquidity measures and bond characteristics. Coefficients on the number of CDS quotes and lagged CDS spreads are largely unchanged. These findings suggest that our transactionbased liquidity measures identify a unique portion of the variation in the nondefault component that is orthogonal to the conventional liquidity proxies.
As for the liquidity proxies, the nondefault components are positively associated with coupon rate but uncorrelated with bond size for all rating groups. Interpreting these coefficients is difficult since both coupon rate and bond size may be correlated with the issuer's credit risk. Nondefault components are also statistically significantly related to bond age and remaining maturity. As plotted in the top panel of Figure 6, for investmentgrade bonds, nondefault components are marginally lower for younger bonds; but for speculativegrade bonds, nondefault components first decrease as bonds get older within about the first four years but then increase in age. As shown in the bottom panel, for investmentgrade bonds, nondefault components are higher for the first couple of years of remaining maturity and then remain roughly flat; but for speculativegrade bonds, nondefault components decrease more precipitously in remaining maturity. Our findings on remaining maturity are consistent with previous studies suggesting that a large fraction of investmentgrade bond yield spreads, especially at the short end of the maturity range, cannot be accounted for by credit risk (e.g., Huang and Huang 2003).
It is worth pointing out that some of our results are opposite to what have been found in the literature, for example, Longstaff et al. (2005) found nondefault components were found to be negatively related to bond size and positively with remaining maturity. Besides that our sample is much more representative, another possible reason for these differences may be due to our control for unobservable firm heterogeneity. In particular, previous studies may have picked up the correlation between bond characteristics and nondefault components effectively by comparing, say, large or longterm bonds issued by one firm to, respectively, small or shortterm bonds issued by another firm. If credit quality and unobservable firm heterogeneity are not well controlled for, those findings may just reflect the correlation between bond size or maturity and credit risk.
While using time dummy variables may control for macroeconomic conditions, their coefficients may not be easily interpreted. To get a sense how the nondefault component is associated with macroeconomic conditions, we replace the time dummies with the following commonlyused macroeconomic variables as explicit controls: 6month Tbill rate and term spread between 10year Treasury rate and 6month Tbill rate; monthly returns, historical volatilities, and implied volatilities on the S&P 500 index; and the ontherun spread for 10year Treasury securities.
The results are shown in Table 12. Comparing to Table 11, the results on our transactionbased liquidity measures are largely unchanged (with somewhat lower significance level), so are those on CDS liquidity proxies and bond characteristics (not shown). On the macroeconomic variables, nondefault components are negatively associated with short rate and term spread. Since Treasury term structures often increase on stronger outlook for economic growth, this result suggests that nondefault components decrease on better economic perspectives. This is consistent with the negative correlation between nondefault components and S&P 500 stock returns (when they are statistically significant). However, this interpretation should be taken with a grain of salt, considering that the recent behavior in the Treasury term structure, especially its inverting yield curve, is still not well understood. Finally, nondefault components are found to increase in S&P implied volatility but decrease in the historical volatility, possibly because implied volatility is forward looking. Results on 10year Treasury ontherun premium are only positively significant for AArated bonds, as they may be closer substitutes for Treasury securities.
This section presents a number of exercises that check for the robustness of our results. These include: (1) constructing our transactionbased liquidity measures using trades that occurred in the time window less subjected to news; (2) using Treasury rate as the risk free rate measure in estimating the nondefault component; and (3) using the nondefault component estimated without adjusting for coupon effects. Overall, our results are robust to these alternative model specifications, estimation methods, and samplings.
Since transaction price, trade size, and trading frequency may be affected by both bond liquidity and valuations, changes in our transactionbased liquidity measures may also reflect changes in firm fundamentals, especially when news arrives. To mitigate the potential impact of news, we now use only transactions occurring between 10:30AM and 3:30PM each day to exclude possibly newsdriven trades. We choose this time window because company news usually arrives in the aftermarket hours and major economic data are generally released no later than 10AM.
The results, shown in Table 13, suggest that excluding newsdriven trades in general leads to more moderate liquidity effects. Comparing to Table 11, the results on Arated bonds are roughly unchanged. But for AA and BBBrated bonds, most coefficients become statistically insignificant, although they continue to have the expected signs. Coefficients for speculativegrade bonds remain statistically insignificant. To the extent that bond liquidity may vary when news arrives, the above results also suggest that news helps to identify the dynamic liquidity effect on the nondefault component of yield spreads.
Swap rate has been regarded as the appropriate risk free rate for studying the effects of liquidity on the nondefault component, as it offers a better control for tax effects and is arguably closer to dealers' funding cost. Nonetheless, as mentioned early, using swap rate has its own drawbacks. For example, swap rate may have a component compensating for counterparty default risks, and the benchmark LIBOR rate also has a credit risk component. For robustness, we follow the literature to repeat our regressions with the nondefault component estimated using Treasury rate as the risk free rate.
The results are shown in Table 14. Comparing to Table 11, the results are roughly unchanged for both investmentgrade and speculativegrade bonds. These suggest that the difference in the estimated nondefault components resulting from using alternative risk free rates is largely uncorrelated with our transactionbased liquidity measures.
Among other regressors, notable changes occur to the coefficients on coupon rate: They become slightly smaller for investmentgrade bonds but slightly larger for speculativegrade bonds. On a related note, Longstaff et al. (2005) argued that one can use the difference in the estimated coefficients on coupon rate between using Treasury rate and using swap rate as an estimate for the tax effect on corporate bond yield spread. Based on our estimates, this would result in a negative tax effect for investmentgrade bonds but a positive tax effect for speculativegrade bonds! Our results thus suggest that their method of identifying tax effect at best may not be robust to the controlling for transactionbased liquidity effect or for unobservable firm heterogeneity. Clearly, more research questions remain regarding the tax effect.
We have argued that we improve the estimation of the nondefault component of yield spreads by fully correcting coupon effect. What happens if we don't adjust for coupon effect? We reestimate our models with the nondefault component equal to bond spreads minus the CDS spread that is read directly at the comparable maturity from the CDS term structure (i.e., Line 3 in Table 3).
The results with swap rate as the risk free rate are shown in Table 15. Comparing to Table 11, the results on our liquidity measures are roughly unchanged, suggesting that the coupon effects are largely orthogonal to our transactionbased liquidity measures, although they may affect the estimated levels of the nondefault component.
Not surprisingly, failing to adjust the coupon effect has significant impacts on the coefficients on coupon rates. Indeed, for investmentgrade bonds they decrease by about 0.4 on average, implying that all else equal, for each percentage of coupon rate, one would underestimate the nondefault component by 0.4 basis points if the coupon effects were not removed. The impact for speculativegrade bonds is more modest.
In this paper we estimate the nondefault component of corporate bond yield spreads and examine its relationship with bond liquidity. We construct three types of bond liquidity measures, including price impact of trades, transaction costs, and trading frequency variables, using newly available intraday transactions data. In addition, we control for the default component of bond spreads using the term structure of CDS spreads, addressing both maturity mismatch and coupon effect that may have biased existing estimations. Importantly, in doing so, our methodology allows us to have enough degrees of freedom to use fixedeffects models to control for the unobservable firm heterogeneity that may otherwise bias the regression analysis.
Using swap rate as the risk free rate, the estimated nondefault component of yield spread is in general moderate and statistically significant for only AA, A, and BBBrated bonds and increasing in this order both in basis points and as a fraction of yield spreads. With Treasury rate as the risk free rate, the estimated nondefault component is statistically significant for all investmentgrade bonds (i.e., those rated AAA, AA, A, and BBB) and BBrated bonds. In basis points, the nondefault component is the largest for BBBrated bonds; but as a fraction of yield spreads, the nondefault component is decreasing in bond rating, that is, the highest for AAArated bonds. In addition, the nondefault component accounts more than half of yield spreads for A and higherrated bonds.
We find a positive and significant relationship between the nondefault component and bond illiquidity for investmentgrade bonds (i.e., those rated AA, A, and BBB) but no significant relationship for speculativegrade bonds. We demonstrate that such estimated relationship would appear weaker if the unobservable firm heterogeneity were not well controlled for. We also find that the nondefault component of bond spreads comoves with macroeconomic conditionsnegatively with the Treasury term structure and positively with the stock market implied volatility (VIX). In addition, controlling for conventional liquidity proxies does not affect the statistical significance of our transactionbased liquidity measures, suggesting our measures identify a unique portion of the nondefault component associated with the stochastic variation in bond liquidity. Finally, the estimated effects of our transactionbased liquidity measures are robust to a number of alternative model specifications and samplings, such as excluding newsdriven trades and using Treasury rate as the risk free rate.
For future research, the strong statistical evidence of the positive relationship between the nondefault component of yield spreads and bond illiquidity suggests that it is important to incorporate liquidity factors into the bond pricing models. In addition, our results call for careful reevaluations on the effects of tax on corporate yield spreads.
We construct corporate bond liquidity measures using the intraday transactions data from the NASD's Trading Reporting and Compliance Engine, or TRACE, reporting system. Under the pressure from both regulators and investors to increase the transparency of the corporate bond market, the NASD now requires its members to report to the NASD through TRACE all overthecounter secondary market transactions for a list of eligible fixed income securities. The NASD updates the eligible list daily before the market opens. Specifically, the NASD adopted three phases to incrementally disseminate these trade reports to the public.
More details on TRACE rules can be found in NASD (2004). We obtain the publicly disseminated intraday transactions data through MarketAccess. The data include bond CUSIP, NASD composite ratings, transaction price (including the effect of any dealer commission), trade size, settlement time, and other trade related variables. Our data, however, do not have some critical transaction information such as whether the trade was initiated by the buyer or the seller. An additional limitation is that the trade size available in our data is capped at $1 million for highyield bonds and $5 million for investmentgrade bonds for those trades with quantities greater than these limits.
Our overall sample is constructed by merging Merrill Lynch's Corporate Bond Index Database and Markit Partner's CDS Database for the period from January 1, 2001 to April 30, 2007. We retain only senior unsecured U.S. dollardenominated bonds issued by U.S. firms that pay fixed semiannual coupons with remaining maturity less than 15 years. We also delete bonds that are callable, puttable, convertible, or have sink fund features. In addition, to include a reference entity, we require its CDS quotes be nonmissing at 1 and 10year maturities and nonmissing at additional two of the four maturities in between (i.e., 2, 3, 5, and 7year). We merge this overall sample with the TRACE data to obtain our regression sample. The sampling period is from July 1, 2002 to April 30, 2007. In addition, for bond transaction data, we remove trades with "data errors" as in Edwards et al. (2007). The figures shown in Panel B reflect the sample of the bonds with at least one nonmissing trading liquidity measure for any month (without winsorizing). Note that we conduct our analysis at the monthly frequency, where monthly values of all timevarying variables are the average of their corresponding daily values. 



(A) To construct the crosssectional sample, we first remove bonds that were ever either upgraded or downgraded (in terms of changing whole rating letter) in the overall sample. We also remove bonds that appear in less than three months over the sample period.^{1} For each bond, we then compute means of the relevant variables over the sample period. For this resulting crosssectional of bonds, we report means of bond spreads, default and nondefault components of the spreads with either Treasury or swap rate as the riskfree rate. (B) * indicates statistically significance at the 95 percent confidence level of a test of the null hypothesis that the nondefault component (in basis point in Columns (3) and (7), and in fraction in Columns in (4) and (8)) is zero. 



Bond Ratings All
(N. of Obs.)* (15270) Variable Mean 
Bond Ratings All
(N. of Obs.)* (15270) Variable P5 
Bond Ratings All
(N. of Obs.)* (15270) Variable P25 
Bond Ratings All
(N. of Obs.)* (15270) Variable P50 
Bond Ratings All
(N. of Obs.)* (15270) Variable P75 
Bond Ratings All
(N. of Obs.)* (15270) Variable P95 
Bond Ratings AA
(N. of Obs.)* (2332) Variable P25 
Bond Ratings AA
(N. of Obs.)* (2332) Variable P50 
Bond Ratings AA
(N. of Obs.)* (2332) Variable P75 
Bond Ratings A
(N. of Obs.)* (7615) Variable P25 
Bond Ratings A
(N. of Obs.)* (7615) Variable P50 
Bond Ratings A
(N. of Obs.)* (7615) Variable P75 
Bond Ratings BBB
(N. of Obs.)* (2927) Variable P25 
Bond Ratings BBB
(N. of Obs.)* (2927) Variable P50 
Bond Ratings BBB
(N. of Obs.)* (2927) Variable P75 
Bond Ratings Highyield
(N. of Obs.)* (2396) Variable P25 
Bond Ratings Highyield
(N. of Obs.)* (2396) Variable P50 
Bond Ratings Highyield
(N. of Obs.)* (2396) Variable P75 


Price impact of trades: 1. Amihud illiq. (abs(ret)/$M)  0.55  0.00  0.14  0.34  0.65  1.61  0.18  0.32  0.56  0.15  0.33  0.65  0.08  0.32  0.66  0.11  0.42  0.78 
Transaction costs: 2. Estimated bidask spread (%)  1.11  0.21  0.55  0.91  1.42  2.57  0.55  0.80  1.16  0.54  0.87  1.35  0.50  0.97  1.50  0.72  1.28  1.92 
Trading frequency: 3. Turnover rate  0.05  0.00  0.01  0.04  0.07  0.17  0.02  0.04  0.06  0.01  0.03  0.06  0.01  0.04  0.09  0.01  0.03  0.07 
Bond characteristics: 4. Coupon (%)  6.24  3.60  5.25  6.38  7.20  8.75  4.63  5.45  6.63  5.00  6.15  7.05  5.50  6.40  7.20  6.63  7.20  7.90 
Bond characteristics: 5. Age (year)  4.88  0.32  1.69  3.73  7.45  12.72  1.43  3.16  6.23  1.63  3.72  7.10  1.48  3.24  7.51  2.82  5.97  8.45 
Bond characteristics: 6. Termtomaturity (year)  5.13  1.28  2.42  4.21  7.38  11.79  2.42  4.13  6.59  2.37  4.04  6.91  2.54  4.87  8.04  2.50  4.37  8.01 
Bond characteristics: 7. Bond size ($100mm)  6.30  1.50  2.50  4.00  8.00  20.00  3.00  8.00  13.00  2.50  4.00  7.50  2.50  3.50  7.50  1.99  2.91  5.00 
Memo items: 8. Number of traded days  13.91  3  9  15  20  22  13  19  21  10  16  20  5  10  19  8  13  18 
Memo items: 9. Number of trades  118.88  4  17  44  133  450  33  100  224  20  48  119  9  23  127  15  35  90 
Memo items: 10. Median trade size ($MM)  0.20  0.01  0.02  0.03  0.08  1.00  0.03  0.03  0.05  0.02  0.03  0.05  0.02  0.04  0.26  0.02  0.04  0.35 
Memo items: 11. Monthly trading vol ($MM)  43.83  0.33  4.01  14.82  47.22  170.70  6.09  28.00  69.52  3.93  14.13  43.72  4.06  15.68  63.46  2.92  10.21  27.90 
* The numbers of observations for the liquidity measures may be smaller than stated on this line. Data sources: Merill Lynch, Markit, TRACE, Federal Reserve Board, from July 2002 to April 2007.
This table shows the pairwise correlations of transactionbased liquidity measures for each rating group. * indicates the correlation coefficient is statistically significant at the 95 percent confidence level.
AA, AA, AA+
(1) Amihud 
AA, AA, AA
(2) Bidask 
AA, AA, AA
(3) Turnover 
A, A, A+
(4) Amihud 
A, A, A
(5) Bidask 
A, A, A
(6) Turnover 
BBB, BBB, BBB+
(7) Amihud 
BBB, BBB, BBB+
(8) Bidask 
BBB, BBB, BBB+
(9) Turnover 
Speculativegrade
(10) Amihud 
Speculativegrade
(11) Bidask 
Speculativegrade
(12) Turnover 


Coupon  0.082**  0.040**  0.044*  0.056**  0.0035  0.066**  0.035  0.0029  0.078**  0.29**  0.028  0.072* 
Coupon: (Standard error)  (0.02)  (0.01)  (0.02)  (0.02)  (0.008)  (0.01)  (0.05)  (0.03)  (0.04)  (0.07)  (0.04)  (0.04) 
Log (Bond size)  0.20**  0.068**  0.28**  0.27**  0.059**  0.35**  0.0044  0.092**  0.26**  0.43**  0.099  0.064 
Log (bond size): (Standard error)  (0.03)  (0.02)  (0.04)  (0.03)  (0.02)  (0.02)  (0.08)  (0.03)  (0.06)  (0.2)  (0.06)  (0.08) 
Bond age/10  2.91**  0.96**  4.21**  2.88**  0.32*  3.43**  4.83**  1.66**  3.22**  5.75**  2.43**  0.98 
Bond age/10: (Standard error)  (0.4)  (0.2)  (0.4)  (0.4)  (0.2)  (0.3)  (0.9)  (0.5)  (0.6)  (1.6)  (0.7)  (0.7) 
(Bond age/10)  2.47**  0.72  5.69**  2.11**  0.52  4.92**  5.75**  2.57**  4.13**  5.66*  3.19**  0.23 
(Bond age/10): (Standard error)  (0.8)  (0.5)  (0.8)  (0.7)  (0.4)  (0.6)  (1.9)  (1.1)  (1.3)  (3.0)  (1.3)  (1.4) 
(Bond age/10)  1.06**  0.17  3.18**  0.72  0.62**  2.90**  3.40**  1.70**  2.24**  3.19  1.94**  0.32 
(Bond age/10): (Standard error)  (0.5)  (0.3)  (0.6)  (0.5)  (0.3)  (0.5)  (1.3)  (0.7)  (0.9)  (2.0)  (0.9)  (1.0) 
(Bond age/10)  0.20*  0.019  0.59**  0.12  0.13**  0.54**  0.68**  0.34**  0.41**  0.69*  0.40**  0.097 
(Bond age/10): (Standard error)  (0.1)  (0.07)  (0.1)  (0.09)  (0.06)  (0.09)  (0.3)  (0.1)  (0.2)  (0.4)  (0.2)  (0.2) 
Termtomat/10  2.65*  3.76**  4.78**  1.97**  2.16**  5.35**  6.98**  3.57**  1.04  1.15  3.63**  4.81** 
Termtomat/10: (Standard error)  (1.4)  (0.9)  (1.5)  (1.0)  (0.5)  (0.8)  (2.9)  (1.3)  (1.8)  (3.4)  (1.8)  (1.9) 
(TTM/10)  1.95  5.83**  11.7**  1.91  0.93  12.4**  11.6  4.22  3.14  6.31  3.81  7.14 
(TTM/10): (Standard error)  (3.8)  (2.4)  (4.3)  (2.6)  (1.4)  (2.2)  (8.0)  (3.6)  (4.8)  (9.0)  (4.8)  (5.0) 
(TTM/10)  1.08  5.38**  11.8**  4.97*  0.58  11.2**  9.03  2.15  2.76  12.7  0.45  5.14 
(TTM/10): Standard error)  (4.0)  (2.6)  (4.7)  (2.7)  (1.5)  (2.3)  (8.8)  (3.9)  (5.0)  (9.2)  (4.8)  (5.0) 
(TTM/10)  0.37  1.88**  3.94**  2.16**  0.40  3.40**  2.70  0.38  0.61  5.60*  0.54  1.41 
(TTM/10): (Standard error)  (1.4)  (0.9)  (1.7)  (0.9)  (0.5)  (0.8)  (3.2)  (1.4)  (1.8)  (3.1)  (1.6)  (1.7) 
Constant  3.17**  0.026  3.41**  3.50**  0.041  3.56**  2.36**  0.18  3.03**  3.49**  1.10**  4.37** 
Constant: (Standard error)  (0.3)  (0.2)  (0.3)  (0.3)  (0.1)  (0.2)  (0.7)  (0.3)  (0.5)  (1.3)  (0.5)  (0.6) 
Observations  2185  2050  2138  6497  5751  6689  2135  1603  2214  1266  916  1366 
Number of firms  20  19  19  82  77  81  95  83  99  61  59  66 
0.22  0.36  0.26  0.13  0.26  0.19  0.12  0.20  0.11  0.12  0.13  0.16 
Corr(nondef. comp. with swap rate, liquidity measure)  

Correlation with 
AA

A

BBB

Highyield

Transactionbased measures Amihud  0.17*  0.07*  0.03  0.08* 
Transactionbased measures Bidask  0.19*  0.11*  0.18*  0.19* 
Transactionbased measures Turnover  0.13*  0.13*  0.36*  0.25* 
Bond char. as proxies Coupon  0.30*  0.28*  0.07*  0.07* 
Bond char. as proxies Bond size  0.08*  0.10*  0.39*  0.39* 
Bond char. as proxies Age  0.22*  0.20*  0.19*  0.05* 
Bond char. as proxies Termtomaturity  0.12*  0.07*  0.01  0.03 
AA, AA, AA+ Independent var. (1) 
AA, AA, AA Independent var. (2) 
AA, AA, AA+ Independent var. (3) 
AA, AA, AA Independent var. (4) 
A, A, A+ Independent var. (5) 
A, A, A+ Independent var. (6) 
A, A, A+ Independent var. (7) 
A, A, A+ Independent var. (8) 
BBB, BBB, BBB+ Independent var. (9) 
BBB, BBB, BBB+ Independent var. (10) 
BBB, BBB, BBB+ Independent var. (11) 
BBB, BBB, BBB+ Independent var. (12) 
Speculativegrade Independent var. (13) 
Speculativegrade Independent var. (14) 
Speculativegrade Independent var. (15) 
Speculativegrade Independent var. (16) 


Log(Amihud illiq.)  1.13**  1.36**  0.53  0.65  0.51  0.36  1.01  0.13  
Log(Amihud illiq.): (Standard error)  (0.4)  (0.3)  (0.4)  (0.4)  (1.0)  (1.0)  (0.7)  (0.9)  
Log(Bidask spreads)  2.39*  1.11  1.94**  1.01*  3.19  3.56  1.80  1.38  
Log(Bidask spreads): (Standard error)  (1.3)  (1.3)  (0.8)  (0.6)  (3.1)  (2.2)  (2.4)  (2.5)  
Log(Turnover rate)  1.61**  1.38**  1.30**  1.11*  3.61**  5.39**  1.85  4.24**  
Log(Turnover rate): (Standard error)  (0.6)  (0.5)  (0.5)  (0.6)  (1.7)  (2.1)  (2.2)  (1.6)  
Constant  2.41  3.60  7.02*  6.19*  17.9  20.3*  15.4  21.8*  9.85  9.54  18.3*  24.5**  29.2**  53.3**  24.8**  61.3** 
Constant: (Standard error)  (3.3)  (3.0)  (3.7)  (3.0)  (12)  (11)  (13)  (13)  (9.6)  (9.2)  (10)  (12)  (0.1)  (2.6)  (6.8)  (5.8) 
Observations  2185  2050  2138  1914  6497  5751  6689  5182  2135  1603  2214  1320  1266  916  1366  824 
0.10  0.11  0.13  0.14  0.14  0.15  0.13  0.16  0.26  0.29  0.28  0.36  0.06  0.08  0.05  0.10 
AA, AA, AA+ Independent var. (1) 
AA, AA, AA Independent var. (2) 
AA, AA, AA+ Independent var. (3) 
AA, AA, AA Independent var. (4) 
A, A, A+ Independent var. (5) 
A, A, A+ Independent var. (6) 
A, A, A+ Independent var. (7) 
A, A, A+ Independent var. (8) 
BBB, BBB, BBB+ Independent var. (9) 
BBB, BBB, BBB+ Independent var. (10) 
BBB, BBB, BBB+ Independent var. (11) 
BBB, BBB, BBB+ Independent var. (12) 
Speculativegrade Independent var. (13) 
Speculativegrade Independent var. (14) 
Speculativegrade Independent var. (15) 
Speculativegrade Independent var. (16) 


Log(Amihud illiq.)  0.79**  0.74**  0.76**  0.65**  0.11  0.16  0.96**  0.63  
Log(Amihud illiq.): (Standard error)  (0.2)  (0.3)  (0.1)  (0.2)  (0.2)  (0.4)  (0.4)  (0.7)  
Log(Bidask spreads)  1.80**  0.96**  1.74**  0.89**  0.98*  2.31**  5.17**  4.31**  
Log(Bidask spreads): (Standard error)  (0.4)  (0.5)  (0.3)  (0.3)  (0.6)  (0.7)  (1.2)  (1.6)  
Log(Turnover rate)  1.37**  1.28**  1.09**  0.90**  1.38**  1.98**  3.34**  4.07**  
Log(Turnover rate): (Standard error)  (0.2)  (0.3)  (0.2)  (0.2)  (0.3)  (0.4)  (0.7)  (1.1)  
Constant  3.28  4.48  7.59*  7.89*  3.60  7.47*  2.45  6.36*  3.46  6.63**  3.33  9.98**  12.3  6.83  37.8**  13.6** 
Constant: (Standard error)  (3.8)  (3.6)  (4.2)  (4.3)  (3.6)  (3.9)  (3.4)  (3.7)  (2.7)  (2.7)  (3.3)  (2.7)  (20)  (4.2)  (8.2)  (6.7) 
Observations  2185  2050  2138  1914  6497  5751  6689  5182  2135  1603  2214  1320  1266  916  1366  824 
Number of firms  20  19  19  18  82  77  81  75  95  83  99  75  61  59  66  58 
0.11  0.11  0.13  0.14  0.13  0.13  0.13  0.15  0.10  0.10  0.12  0.12  0.06  0.09  0.07  0.11 
AA, AA, AA+ Independent var. (1) 
AA, AA, AA Independent var. (2) 
AA, AA, AA+ Independent var. (3) 
AA, AA, AA Independent var. (4) 
A, A, A+ Independent var. (5) 
A, A, A+ Independent var. (6) 
A, A, A+ Independent var. (7) 
A, A, A+ Independent var. (8) 
BBB, BBB, BBB+ Independent var. (9) 
BBB, BBB, BBB+ Independent var. (10) 
BBB, BBB, BBB+ Independent var. (11) 
BBB, BBB, BBB+ Independent var. (12) 
Speculativegrade Independent var. (13) 
Speculativegrade Independent var. (14) 
Speculativegrade Independent var. (15) 
Speculativegrade Independent var. (16) 


Log(Amihud illiq.)  0.44  0.52  0.70**  0.56**  0.69**  0.58  0.17  0.0025  
Log(Amihud illiq.): (Standard error)  (0.3)  (0.3)  (0.1)  (0.2)  (0.2)  (0.4)  (0.4)  (0.6)  
Log(Bidask spreads)  1.30**  0.57  1.89**  1.09**  1.39**  0.47  0.56  0.075  
Log(Bidask spreads): (Standard error)  (0.5)  (0.6)  (0.3)  (0.3)  (0.5)  (0.6)  (1.2)  (1.4)  
Log(Turnover rate)  1.35**  1.27**  1.00**  0.75**  1.20**  1.30**  0.54  0.78  
Log(Turnover rate): (Standard error)  (0.2)  (0.2)  (0.2)  (0.2)  (0.3)  (0.4)  (0.8)  (1.1)  
N. of CDS quotes  0.15  0.15  0.037  0.10  0.41**  0.36**  0.40**  0.40**  0.040  0.33*  0.038  0.56**  0.22  0.030  0.34  0.23 
N. of CDS quotes: (Standard error)  (0.09)  (0.10)  (0.09)  (0.10)  (0.07)  (0.08)  (0.07)  (0.08)  (0.1)  (0.2)  (0.1)  (0.2)  (0.3)  (0.3)  (0.3)  (0.4) 
Lagged CDS spread  0.08**  0.07**  0.09**  0.08**  0.03**  0.03**  0.02  0.03**  0.17**  0.18**  0.16**  0.17**  0.13**  0.12**  0.13**  0.11** 
Lagged CDS spread: (Standard error)  (0.03)  (0.03)  (0.03)  (0.03)  (0.01)  (0.01)  (0.01)  (0.01)  (0.01)  (0.01)  (0.01)  (0.01)  (0.01)  (0.01)  (0.01)  (0.01) 
Constant  1.36  4.91  8.64*  9.69*  8.06*  7.76*  1.86  14.8**  28.2**  28.7**  25.2**  21.5**  317**  80.0**  23.4**  22.4** 
Constant: (Standard error)  (4.1)  (5.8)  (4.5)  (5.5)  (4.7)  (4.4)  (4.3)  (3.6)  (2.1)  (2.7)  (2.0)  (2.9)  (26)  (14)  (7.6)  (10.0) 
Observations  1987  1868  1949  1759  6068  5401  6259  4911  1940  1468  2007  1225  1059  756  1139  684 
Number of firms  19  18  18  18  77  76  77  73  86  76  87  69  57  52  59  52 
0.12  0.13  0.14  0.15  0.14  0.15  0.14  0.16  0.25  0.27  0.25  0.29  0.32  0.30  0.30  0.30 
Changes in nondefault component AA 
Changes in nondefault component A 
Changes in nondefault component BBB 
Changes as percent of total spreads AA 
Changes as percent of total spreads A 
Changes as percent of total spreads BBB 


1. Amihud illiquidity  1.03  1.46  4.1  2.4  
Amihud illiquidity: (Standard error)  [0.74, 1.31]  [0.63, 2.28]  [3.0, 5.2]  [1.0, 3.7]  
2. Bidask spread  0.97  1.73  1.53  6.5  6.9  2.5 
Bidask spread: (Standard error)  [0.24, 1.70]  [1.20, 2.27]  [0.46, 2.60]  [1.6, 11.3]  [4.8, 9.1]  [0.7, 4.3] 
3. Turnover rate  1.48  1.79  2.64  9.9  7.2  4.3 
Turnover rate: (Standard error)  [1.91, 1.05]  [2.49, 1.09]  [3.92, 1.35]  [12.7, 7.0]  [10.0, 4.4]  [6.4, 2.2] 
4. Yield spread  15  25  61  
5. Nondefault comp.  1.25  1.37  5.00 
Yield spread bond yield  swap rate.
Nondefault comp. bond yield  CDS implied yield with swap rate as riskfree rate.
AA, AA, AA+
Independent var. (1) 
AA, AA, AA+
Independent var. (2) 
AA, AA, AA+
Independent var. (3) 
AA, AA, AA
Independent var. (4)+ 
A, A, A+
Independent var. (5) 
A, A, A+
Independent var. (6) 
A, A, A+
Independent var. (7) 
A, A, A+
Independent var. (8) 
BBB, BBB, BBB+
Independent var. (9) 
BBB, BBB, BBB+
Independent var. (10) 
BBB, BBB, BBB+
Independent var. (11) 
BBB, BBB, BBB+
Independent var. (12) 
Speculativegrade
Independent var. (13) 
Speculativegrade
Independent var. (14) 
Speculativegrade
Independent var. (15) 
Speculativegrade
Independent var. (16) 


Log(Amihud illiq.)  0.024  0.100  0.40**  0.18  0.45**  0.23  0.40  0.28  
Log(Amihud illiq.): (Standard error)  (0.2)  (0.3)  (0.1)  (0.2)  (0.2)  (0.3)  (0.4)  (0.6)  
Log(Bidask spreads)  0.66  0.37  1.17**  0.93**  1.14**  0.26  0.77  0.32  
Log(Bidask spreads: (Standard error)  (0.5)  (0.5)  (0.3)  (0.3)  (0.5)  (0.6)  (1.2)  (1.4)  
Log(Turnover rate)  0.68**  0.52**  0.36**  0.23  0.61**  0.68*  0.56  0.33  
Log(Turnover rate): (Standard error)  (0.2)  (0.3)  (0.2)  (0.2)  (0.3)  (0.4)  (0.8)  (1.1)  
N. of CDS quotes  0.17**  0.20**  0.055  0.16*  0.36**  0.35**  0.35**  0.36**  0.08  0.41**  0.11  0.61**  0.21  0.10  0.31  0.32 
N. of CDS quotes: (Standard error)  (0.08)  (0.09)  (0.08)  (0.09)  (0.07)  (0.07)  (0.07)  (0.07)  (0.1)  (0.2)  (0.1)  (0.2)  (0.3)  (0.3)  (0.3)  (0.4) 
Lagged CDS spread  0.08  0.08  0.10  0.08  0.06**  0.05**  0.04**  0.05**  0.14**  0.16**  0.14**  0.16**  0.10**  0.09**  0.10**  0.09** 
Lagged CDS spread: (Standard error)  (0.07)  (0.08)  (0.07)  (0.08)  (0.01)  (0.01)  (0.01)  (0.01)  (0.01)  (0.02)  (0.02)  (0.02)  (0.01)  (0.01)  (0.01)  (0.01) 
Coupon  1.14**  1.04**  1.03**  0.92**  2.01**  1.89**  1.91**  1.90**  2.37**  2.69**  2.21**  2.14**  4.16**  4.48**  3.62**  4.65** 
Coupon: (Standard error)  (0.3)  (0.3)  (0.3)  (0.3)  (0.2)  (0.2)  (0.2)  (0.2)  (0.5)  (0.5)  (0.5)  (0.6)  (1.1)  (1.3)  (1.0)  (1.4) 
Log(Bond size)  0.60*  0.49  0.31  0.27  0.030  0.082  0.024  0.28  0.31  0.16  0.40  0.87  3.62  2.92  2.46  3.20 
Log(Bond size): (Standard error)  (0.3)  (0.4)  (0.3)  (0.4)  (0.3)  (0.3)  (0.3)  (0.3)  (1.0)  (1.1)  (1.0)  (1.1)  (2.2)  (2.7)  (2.0)  (2.9) 
Constant  16.2**  11.4**  7.11  8.70  3.33  14.0**  13.2**  7.12  42.4**  45.1**  38.6**  38.8**  253**  19.4  23.7  14.7 
Constant: (Stanard error)  (5.3)  (5.5)  (5.2)  (5.5)  (4.3)  (5.1)  (5.0)  (4.7)  (9.5)  (11)  (9.4)  (12)  (28)  (21)  (15)  (23) 
Bond age polyn. (4)  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes 
TTM polyn. (4)  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes 
Observations  1987  1868  1949  1759  6068  5401  6259  4911  1940  1468  2007  1225  1059  756  1139  684 
Number of firms  19  18  18  18  77  76  77  73  86  76  87  69  57  52  59  52 
0.29  0.31  0.30  0.32  0.25  0.25  0.24  0.26  0.37  0.40  0.37  0.41  0.37  0.36  0.35  0.36 
AA, AA, AA+ (1) 
AA, AA, AA+ (2) 
AA, AA, AA+ (3) 
AA, AA, AA+ (4) 
A, A, A+ (5) 
A, A, A+ (6) 
A, A, A+ (7) 
A, A, A+ (8) 
BBB, BBB, BBB+ (9) 
BBB, BBB, BBB+ (10) 
BBB, BBB, BBB+ (11) 
BBB, BBB, BBB+ (12) 
Speculativegrade (13) 
Speculativegrade (14) 
Speculativegrade (15) 
Speculativegrade (16) 


Log(Amihud illiq.)  0.041  0.066  0.48**  0.24  0.33*  0.23  0.37  0.18  
Log(Amihud illiq.): (Standard error)  (0.2)  (0.3)  (0.1)  (0.2)  (0.2)  (0.3)  (0.4)  (0.6)  
Log(Bidask spreads)  0.66  0.41  1.24**  0.96**  1.17**  0.38  1.42  0.34  
Log(Bidask spreads): (Standard error)  (0.4)  (0.5)  (0.3)  (0.3)  (0.5)  (0.6)  (1.2)  (1.4)  
Log(Turnover rate)  0.55**  0.38  0.28*  0.17  0.39  0.23  0.81  1.14  
Log(Turnover rate): (Standard error)  (0.2)  (0.3)  (0.2)  (0.2)  (0.3)  (0.4)  (0.7)  (1.2)  
6Month Tbill  4.52**  4.37**  4.81**  4.61**  5.62**  5.61**  5.51**  5.48**  6.20**  4.83**  6.14**  5.25**  6.69**  4.34  4.37  2.40 
6Month Tbill: (Standard error)  (0.7)  (0.8)  (0.8)  (0.8)  (0.5)  (0.5)  (0.5)  (0.5)  (1.1)  (1.2)  (1.0)  (1.3)  (3.2)  (3.8)  (3.0)  (4.0) 
Treas term spread  6.72**  6.82**  7.18**  7.04**  8.12**  8.14**  8.16**  8.23**  7.60**  5.96**  6.94**  6.62**  7.20*  3.56  4.01  2.17 
Treas term spread: (Standard error)  (0.9)  (1.0)  (0.9)  (1.0)  (0.6)  (0.7)  (0.6)  (0.7)  (1.4)  (1.5)  (1.3)  (1.6)  (4.2)  (4.8)  (3.9)  (5.1) 
S&P 500 Return  5.80*  4.84  7.46**  6.27*  7.29**  8.17**  4.75**  7.71**  2.21  2.74  3.91  6.97  34.8  31.1  21.1  11.2 
S&P 500 Return: (Standard error)  (3.1)  (3.3)  (3.2)  (3.4)  (2.3)  (2.4)  (2.3)  (2.5)  (5.6)  (5.9)  (5.5)  (6.4)  (24)  (32)  (24)  (32) 
S&P500 real. vol.  9.14**  9.32**  9.17**  8.99**  6.22**  5.43*  5.87**  5.15*  17.4**  13.1**  19.2**  17.4**  0.72  1.34  9.55  3.72 
S&P500 real. vol: (Standard error)  (2.9)  (2.9)  (3.0)  (3.1)  (2.5)  (2.8)  (2.6)  (2.8)  (4.6)  (5.4)  (4.5)  (6.0)  (10)  (11)  (8.5)  (11) 
S&P impl. vol.  0.44**  0.40**  0.51**  0.43**  0.24*  0.09  0.26**  0.16  0.70**  0.76**  0.67**  0.89**  0.24  0.26  0.14  0.14 
S&P impl. vol: (Standard error)  (0.2)  (0.2)  (0.2)  (0.2)  (0.1)  (0.1)  (0.1)  (0.1)  (0.3)  (0.3)  (0.3)  (0.3)  (0.6)  (0.7)  (0.5)  (0.7) 
Treas. liquidity  0.22**  0.24**  0.26**  0.27**  0.092  0.11  0.096  0.14  0.12  0.12  0.052  0.11  0.61  0.80  1.11**  0.67 
Treas. liquidity: (Standard error)  (0.1)  (0.1)  (0.1)  (0.1)  (0.1)  (0.1)  (0.1)  (0.1)  (0.2)  (0.2)  (0.2)  (0.2)  (0.4)  (0.5)  (0.4)  (0.6) 
Constant  38.2**  39.8**  35.7**  37.8**  44.2**  44.3**  41.0**  42.4**  62.7**  54.8**  59.8**  50.7**  41.7**  21.8  31.9*  17.3 
Constant: (Standard error)  (4.9)  (5.1)  (4.8)  (5.4)  (4.0)  (4.2)  (3.9)  (4.4)  (11)  (12)  (10)  (14)  (19)  (24)  (18)  (26) 
Bond char.  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes 
CDS liq. proxies  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes 
Observations  1987  1868  1949  1759  6068  5401  6259  4911  1940  1468  2007  1225  1059  756  1139  684 
Number of firms  19  18  18  18  77  76  77  73  86  76  87  69  57  52  59  52 
0.24  0.25  0.24  0.26  0.17  0.17  0.17  0.18  0.26  0.27  0.26  0.27  0.24  0.25  0.27  0.26 
AA, AA, AA+
Indepentdent var. (1) 
AA, AA, AA+
Indepentdent var. (2) 
AA, AA, AA+
Indepentdent var. (3) 
AA, AA, AA+
Indepentdent var. (4) 
A, A, A+
Independent var. (5) 
A, A, A+
Independent var. (6) 
A, A, A+
Independent var. (7) 
A, A, A+
Independent var. (8) 
BBB, BBB, BBB+
Independent var. (9) 
BBB, BBB, BBB+
Independent var. (10) 
BBB, BBB, BBB+
Independent var. (11) 
BBB, BBB, BBB+
Independent var. (12) 
Speculativegrade
Independent var. (13) 
Speculativegrade
Independent var. (14) 
Speculativegrade
Independent var. (15) 
Speculativegrade
Independent var. (16) 


Log(Amihud illiq.)  0.00  0.13  0.31**  0.10  0.23  0.26  0.06  0.34  
Log(Amihud illiq.): (Standard error)  (0.21)  (0.28)  (0.13)  (0.20)  (0.17)  (0.37)  (0.32)  (0.69)  
Log(Bidask spreads)  0.18  0.04  1.24**  1.10**  0.44  0.18  0.12  0.49  
Log(Bidask spreads): (Standard error)  (0.37)  (0.40)  (0.28)  (0.32)  (0.58)  (0.68)  (1.14)  (1.29)  
Log(Turnover rate)  0.26  0.41  0.42**  0.26  0.41*  0.61  0.91  1.31  
Log(Turnover rate): (Standard error)  (0.22)  (0.26)  (0.14)  (0.19)  (0.22)  (0.38)  (0.63)  (1.13)  
Bond char.  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes 
CDS liq. proxies  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes 
Observations  1964  1771  1955  1682  5947  4837  6229  4406  1860  1259  1985  1066  1033  638  1135  585 
Number of firms  18  17  18  17  77  74  77  72  84  69  85  62  57  52  59  51 
0.29  0.32  0.29  0.32  0.25  0.26  0.24  0.27  0.37  0.41  0.38  0.41  0.36  0.38  0.34  0.38 
AA, AA, AA+
Independent var. (1) 
AA, AA, AA+
Independent var. (2) 
AA, AA, AA+
Independent var. (3) 
AA, AA, AA+
Independent var. (4) 
A, A, A+
Independent var. (5) 
A, A, A+
Independent var. (6) 
A, A, A+
Independent var. (7) 
A, A, A+
Independent var. (8) 
BBB, BBB, BBB+
Independent var. (9) 
BBB, BBB, BBB+
Independent var. (10) 
BBB, BBB, BBB+
Independent var. (11) 
BBB, BBB, BBB+
Independent var. (12) 
Speculativegrade
Independent var. (13) 
Speculativegrade
Independent var. (14) 
Speculativegrade
Independent var. (15) 
Speculativegrade
Independent var. (16) 


Log(Amihud illiq.)  0.13  0.20  0.51**  0.31  0.38*  0.19  0.46  0.52  
Log(Amihud illiq.): (Standard error)  (0.27)  (0.33)  (0.15)  (0.22)  (0.20)  (0.35)  (0.39)  (0.65)  
Log(Bidask spreads)  0.76  0.69  1.07**  0.68*  0.69  0.25  0.01  1.05  
Log(Bidask spreads): (Standard error)  (0.50)  (0.54)  (0.31)  (0.35)  (0.51)  (0.62)  (1.15)  (1.44)  
Log(Turnover rate)  0.93**  0.71**  0.37**  0.23  0.51**  0.54  0.51  0.81  
Log(Turnover rate): (Standard error)  (0.27)  (0.29)  (0.18)  (0.21)  (0.25)  (0.39)  (0.79)  (1.23)  
Coupon  0.95**  0.90**  0.93**  0.79**  1.93**  1.76**  1.85**  1.72**  2.01**  2.47**  2.01**  2.01**  5.04**  5.53**  4.32**  5.72** 
Coupon: (Standard error)  (0.28)  (0.29)  (0.29)  (0.30)  (0.17)  (0.18)  (0.17)  (0.18)  (0.47)  (0.50)  (0.46)  (0.59)  (1.07)  (1.30)  (0.97)  (1.43) 
Bond char.  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes 
CDS liq. proxies  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes 
Observations  1986  1868  1949  1759  6173  5487  6389  4982  1891  1412  1955  1174  1003  722  1078  652 
Number of firms  19  18  18  18  79  77  79  74  87  76  88  69  57  52  59  52 
0.46  0.49  0.48  0.50  0.32  0.32  0.31  0.33  0.35  0.37  0.33  0.35  0.35  0.37  0.33  0.37 
AA, AA, AA+
Independent var. (1) 
AA, AA, AA+
Independent var. (2) 
AA, AA, AA+
Independent var. (3) 
AA, AA, AA+
Independent var. (4) 
A, A, A+
Independent var. (5) 
A, A, A+
Independent var. (6) 
A, A, A+
Independent var. (7) 
A, A, A+
Independent var. (8) 
BBB, BBB, BBB+
Independent var. (9) 
BBB, BBB, BBB+
Independent var. (10) 
BBB, BBB, BBB+
Independent var. (11) 
BBB, BBB, BBB+
Independent var. (12) 
Speculativegrade
Independent var. (13) 
Speculativegrade
Independent var. (14) 
Speculativegrade
Independent var. (15) 
Speculativegrade
Independent var. (16) 


Log(Amihud Illiquidity)  0.06  0.17  0.45**  0.12  0.40**  0.21  0.34  0.28  
Log(Amihud Illiquidity): (Standard error)  (0.24)  (0.29)  (0.14)  (0.19)  (0.20)  (0.34)  (0.36)  (0.62)  
Log(Bidask spreads)  1.01**  0.62  1.35**  1.13**  0.95*  0.14  0.23  0.76  
Log(Bidask spreads): (Standard error)  (0.44)  (0.49)  (0.27)  (0.32)  (0.50)  (0.59)  (1.19)  (1.37)  
Log(Turnover rate)  0.80**  0.63**  0.34**  0.25  0.55**  0.82**  0.68  0.38  
Log(Turnover rate): (Standard error)  (0.23)  (0.25)  (0.16)  (0.19)  (0.25)  (0.38)  (0.74)  (1.06)  
Coupon  0.79**  0.72**  0.70**  0.59**  1.62**  1.47**  1.56**  1.48**  2.04**  2.33**  1.90**  1.86**  3.80**  4.50**  3.24**  4.80** 
Coupon : (Standard error)  (0.25)  (0.25)  (0.25)  (0.27)  (0.16)  (0.16)  (0.16)  (0.17)  (0.48)  (0.52)  (0.47)  (0.60)  (1.01)  (1.27)  (0.92)  (1.38) 
Bond char.  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes 
CDS liq. proxies  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes 
Observations  1984  1865  1947  1757  6106  5426  6303  4931  1914  1446  1976  1203  1021  729  1103  658 
Number of firms  19  18  18  18  77  76  77  73  85  75  86  68  56  52  59  51 
0.27  0.29  0.28  0.31  0.25  0.25  0.24  0.26  0.36  0.39  0.36  0.40  0.37  0.37  0.35  0.37 