Charles A. Dice Center for

Research in Financial Economics

Comovement of Corporate Bonds and Equities

Jack Bao, The Ohio State University

Kewei Hou, The Ohio State University

Dice Center WP 2013-11 Fisher College of Business WP 2013-03-11

Original: July 2013

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Fisher College of Business

Working Paper Series

Comovement of Corporate Bonds and Equities

Jack Bao and Kewei Hou∗

July 7, 2013

Abstract

We study heterogeneity in the comovement of corporate bonds and equities, bothat the bond level and at the firm level. Using an extended Merton model, weillustrate that corporate bonds that mature late relative to the rest of the bonds inits issuer’s maturity structure should have stronger comovement with equities. Incontrast, endogenous default models suggest that a bond’s position in its issuer’smaturity structure has little relation with the strength of the comovement betweenbonds and equities. Empirically, we find results consistent with the prediction ofthe extended Merton model. In addition, we find that comovement between bondsand equities is stronger for firms with higher credit risk as proxied by the book-to-market ratio and distance-to-default even after controlling for ratings. Our evidencesuggests that market participants are able to assess credit quality at a more granularlevel than ratings.

∗Bao is at the Fisher College of Business, Ohio State University, bao 40@fisher.osu.edu. Hou is at theFisher College of Business, Ohio State University and CAFR, hou.28@osu.edu. We have benefited fromhelpful comments from and discussions with Sergey Chernenko, Eric Powers, and Rene Stulz and seminarparticipants at Ohio State and the South Carolina Fixed Income Conference. All remaining errors areour own.

1

1 Introduction

Corporate bonds and equities issued by the same firm are different contingent claims on

the same cash flows. The comovement between them is of important interest as it has

implications for understanding the relative prices in the two markets as well as for hedging

common exposures across markets. Previous research has found that corporate bonds

and equities typically have positively correlated contemporaneous returns. Furthermore,

hedge ratios, the ratio of corporate bond to equity returns, are larger for bonds with

poorer ratings1 and their magnitudes are in-line with what the Merton (1974) model

predicts.2 While the existing literature largely focuses on the comovement at the rating

level, in this paper we study the heterogeneity in comovement at both the bond level and

the firm level within a rating.

We first investigate the differences in comovement at the bond level. Extending the

Merton model to allow for both junior and senior bond issues suggests that junior bonds

should have a greater sensitivity to underlying firm value than senior bonds and thus

should have stronger comovement with equities than senior bonds. However, there is a

paucity of significant seniority structure among public debt, making this dimension of

heterogeneity difficult to examine empirically using corporate bond data.3 Instead, the

existing capital structure of firms provides an interesting set of maturity structures of

corporate bonds. According to Brunnermeier and Oehmke (2013), a bond that matures

after most of the other bonds issued by the same firm is de facto junior. The intuition

arises from the fact that a firm in financial trouble may still be able to repay bonds that

1See Kwan (1996).2Schaefer and Strebulaev (2008) investigate the Merton model implied hedge ratio within a rating

class and find that they cannot reject the hypothesis that the model implied and empirical hedge ratiosare equal.

3Bank debt is normally more senior than public debt. However, firms with significant amounts ofpublic debt often do not have significant amounts of bank debt. Rauh and Sufi (2010) report that only48.6% of the firms in their sample with significant amounts of public debt also have at least 10% of theirdebt in bank debt. Our sample, which is at the bond-month level, is even more extreme, with a medianof 2.53% of debt in bank debt.

2

are due early in its maturity structure, but not bonds that are due later. This creates

the effect that bond issues that mature later are more sensitive to firm value and are

effectively junior. We formalize this intuition in an extension of the Merton model and

present relative hedge ratios in a numerical exercise which shows that bonds that are de

facto junior in its firm’s maturity structure have higher hedge ratios.

Alternatively, consider the endogenous default model of Geske (1977), in which equi-

tyholders must pay in to retire outstanding bonds and continue the firm. In this case, the

relation between the maturity structure of bonds and hedge ratios is unclear as equity-

holders will take the full maturity structure of bonds into account when making optimal

decisions on continuation of the firm, rather than solely considering the bond due today.

In fact, we illustrate that for reasonable parameter values, the Geske (1977) model pre-

dicts that a bond that matures relatively late in its firm’s maturity structure could have

a lower hedge ratio. This contrasts the Brunnermeier and Oehmke (2013) intuition and

also the prediction of the extended Merton model.

To empirically test the predictions of the two models, we measure a bond’s place in its

firm’s maturity structure using the proportion of debt due prior to the bond and relate it

to the hedge ratio of the bond. Our primary empirical finding is that bonds that are due

relatively late in their firms’ maturity structure have higher hedge ratios and thus stronger

comovement with equities even after controlling for ratings. For example, if we increase

the proportion of debt due prior for a Baa bond with seven years to maturity from 10%

to 80%, its hedge ratio will more than double from 5.98% to 12.34%. This confirms the

prediction of the extended Merton model and is consistent with the Brunnermeier and

Oehmke (2013) intuition of the effect of de facto seniority on the credit risk of corporate

bonds.

Next, we study the firm-level heterogeneity in the comovement between bonds and

equities. Using book-to-market ratio, a reduced form measure of credit risk, and distance-

to-default, a structural model-based measure of credit worthiness, we examine whether

3

the comovement of corporate bonds and equities is related to market perceptions of credit

risk that are finer than credit ratings. We find that this is indeed the case, as high book-

to-market and low distance-to-default firms (firms that are perceived as having higher

credit risk) have stronger comovement between bonds and equities within ratings. For

example, a one standard deviation increase in book-to-market ratio (distance-to-default)

above the mean is associated with an increase (decrease) in the hedge ratio for a seven-

year Baa bond from 7% to 9.52% (from 6.94% to 3.87%). Our results suggest that at least

in the corporate bond market, market participants recognize more granular differences in

credit risk than what is provided by ratings, and this is reflected in the joint dynamics of

corporate bond and equity returns.

The results that hedge ratios are related to both book-to-market and distance-to-

default within the same rating group provides additional commentary on the recent lit-

erature that examines how dependent the market is on ratings. In the context of MBS,

Adelino (2009) finds that the market was unable to distinguish between different Aaa-

rated securities, but was able to distinguish between different securities within lower rating

groups. As the vast majority of MBS are Aaa-rated, this suggests a heavy dependence by

the market on ratings. In contrast, our results suggest that at least for corporate bonds,

the market is capable of evaluating credit quality at a finer level and does not naively rely

on credit ratings.

The rest of the paper is organized as follows. In Section 2, we discuss the Merton model,

an extension to the Merton model to include maturity structures of debt, the Geske model,

and other measures of credit risk. In Section 3, we discuss the data, summary statistics,

and the construction of some key variables. In Section 4, we discuss our empirical results.

Finally, we conclude in Section 5.

4

2 Hedge Ratios

The theoretical relation between corporate bond and equity returns is formalized in struc-

tural models of default, the earliest of which is the Merton (1974) model. Structural

models posit processes for firm value and other state variables and all securities which

are claims on the firm can be priced from these processes. This ties all securities together

through their sensitivities to state variables and provides important intuition as to why

corporate bond and equity returns should be related. Though Huang and Huang (2003)

find that structural models of default perform quite poorly in explaining the levels of

corporate bond prices, Schaefer and Strebulaev (2008) find that even a structural model

as simple as the Merton model does well in characterizing the average relative returns of

corporate bonds and equities. This points to trying to understand relative returns rather

than relative prices as the more promising direction to explore structural models.4

2.1 Merton Model

In the Merton model, the only state variable is the value of the firm, which is assumed to

follow a Geometric Brownian Motion under the risk-neutral probability measure

d lnVt =

(r − 1

2σ2v

)dt+ σvdW

Qt . (1)

The firm has a single zero-coupon bond issue with face value K and equity is a call option

on the firm. The single bond issue is equivalent to a risk-free bond short a put option on

4Note that many structural models including the Merton model do not take a stand as to what isthe correct pricing kernel. Instead, the models are based on no arbitrage and the consistent pricing ofdifferent securities.

5

the firm. Its value is

B = V (1−N(d1)) +Ke−rTN(d2), (2)

where d1 =ln(VK

)+(r + 1

2σ2v

)T

σv√T

, d2 = d1 − σv√T .

As illustrated by Schaefer and Strebulaev (2008), the relative returns of the bond and

equity (the hedge ratio) under the Merton model is

hE ≡∂ lnB

∂ lnE=

(1

N(d1)− 1

)(V

B− 1

). (3)

The Merton model formalizes the important insight that corporate bonds and equities

are linked through their exposures to the underlying firm value.5 In addition, the Merton

model-implied hedge ratio is increasing in both asset volatility and leverage as higher asset

volatility and leverage are associated with greater likelihood of default. Consistent with

this prediction, Kwan (1996) finds that the correlation between changes in bond yields

and stock returns is stronger for speculative grade bonds than investment grade bonds.

In other words, high grade bonds behave similarly to Treasuries and low grade bonds are

more sensitive to underlying firm value and behave like equities.

In Panel A of Table 1, we compute Merton model-implied hedge ratios for various

leverage-asset volatility combinations.6 The reported hedge ratios bear out the intuition

that bonds issued by safer firms have lower hedge ratios.7 For a firm with a market

leverage of 10% and asset volatility of 20%, the model hedge ratio is 0.0018%. That is, a

5Note that an important source of variation in bond prices, stochastic interest rates, is not modeledhere as the tie between bonds and equities through interest rates is indirect and less important than the tiethrough firm value. See Shimko, Tejima, and van Deventer (1993) for an extension of the Merton modelto stochastic interest rates and Schaefer and Strebulaev (2008) and Bao and Pan (2013) for applicationsof this model.

6The maturity T is set to seven years.7The numerical examples here are designed to provide some intuition on the effects of leverage and

asset volatility on hedge ratios. They should not be expected to match perfectly the empirical hedgeratios in the data.

6

10% equity return corresponds to only a 0.00018% corporate bond return, an economically

insignificant return. Such a small relative return for the corporate bond reflects the fact

that at 10% leverage and 20% asset volatility, the likelihood of default is extremely low.

As a result, the bond has similar dynamics as a Treasury bond and its returns are very

weakly tied to the returns of the equity. In contrast, the relative returns of corporate

bonds to equities are much more important at higher leverages and asset volatility. For

example, at 70% market leverage and 20% asset volatility, the hedge ratio is 14.39%.

Thus, a 10% equity return corresponds to a 1.439% bond return. Or consider 10% market

leverage and 50% asset volatility, the hedge ratio is 8.49%, which means a 10% equity

return is associated with a 0.849% bond return. Finally, at 70% market leverage and 50%

asset volatility, the hedge ratio is 37.93%. In this case, a 10% equity return is associated

with close to 4% bond return. The effects of leverage and asset volatility on hedge ratios

can also be seen in Figure 1(a) where we plot the Merton model-implied hedge ratios

against market leverage and asset volatility.

2.2 Extending the Merton Model

A number of authors have derived extensions of the Merton model that involve changing

the underlying state variables, state variable processes, default triggers, and capital struc-

ture.8 Here we will focus our attention on the capital structure. The simplest extension

to the capital structure in the Merton model is to allow for both junior and senior bonds.9

Such an extension is straightforward. Equity and the senior bond are priced in the same

way as in the standard Merton model, but the junior bond is the difference between two

call options. If the recovery rate in default states is sufficiently high, the senior bond is

very safe and has dynamics that are similar to Treasuries. The junior bond, which bears

8For example, see Black and Cox (1976), Leland (1994), Longstaff and Schwartz (1995), Leland andToft (1996), Collin-Dufresne and Goldstein (2001), Duffie and Lando (2001), Chen, Collin-Dufresne, andGolstein (2009), and Collin-Dufresne, Goldstein, and Helwege (2010), among others.

9For example, see Lando (2004). We also provide further discussion in Appendix A.

7

the first loss, is riskier and has a higher hedge ratio than the senior bond.

Our extension is similarly based on having multiple bond issues, but in the dimension

of the maturity structure of the firm. We analyze how the position of a bond in a firm’s

maturity structure affects its hedge ratio. As discussed by Brunnermeier and Oehmke

(2013), shorter maturity bonds are de facto senior to longer maturity bonds. The intuition

is that a firm that is in financial distress may remain solvent long enough to repay bond

issues that mature early. However, the firm is then likely to suffer solvency issues when

the bonds due later eventually mature.

To formally analyze the effect of maturity structure, we extend the Merton model to

include three zero-coupon bond issues, which have the same explicit seniority, but mature

at different times. The three bond issues have face values Ki and maturities Ti where

T1 < T2 < T3. At Ti, a firm remains solvent if Vi > Ki. Otherwise the firm defaults, Vi is

discounted by a proportional bankruptcy cost (L), and the remaining bond issues share

the remaining firm value in proportion to their face values.10

This model, while clearly a simplification of the complex capital structure typical in

large U.S. corporations, allows us to examine the sensitivity of a firm’s bond returns to

its equity returns while varying the de facto seniority of a bond. By including three

bonds, we are able to focus on the bond with the middle maturity, T2, and adjust the

relative amounts of debt due before and after it.11 This changes the de facto seniority of

the intermediate maturity bond, allowing us to gauge the impact of de facto seniority on

hedge ratios.

In this extended Merton model, equity remains a call option on the firm, but has

10We do not formally model the decision to rollover debt, but the states where Vi is barely larger thanKi are the states where the rollover of debt would be particularly expensive, if not impossible. These arealso the states that are primarily responsible for driving the wedge between bond issues that creates defacto seniority. See He and Xiong (2012) and He and Milbradt (2013) for recent papers that examine theissue of rollover risk.

11Our set-up essentially maps all bonds maturing prior to T2 into a single bond that matures at T1 andall bonds maturing after T2 into a single bond that matures at T3.

8

intermediate monitoring points at each bond maturity

E = EQ0

[e−rT31V1>K11V2>K21V3>K3 (V3 −K3)

]. (4)

Though there is no closed-form solution for equity value, it can be simplified to a function

of normal CDFs and integrals by noting that firm value returns are lognormal. Similarly,

the value of the bond maturing at T2 is

B = EQ0

[e−rT21V1>K11V2>K2K2

]+ EQ

0

[e−rT11V1<K1(1− L)V1

K2

K1 +K2 +K3

](5)

+ EQ0

[e−rT21V1>K11V2<K2(1− L)V2

K2

K2 +K3

],

where the three terms represent solvency at T2, default at T1, and default at T2, respec-

tively. The hedge ratio (∂ lnB∂ lnE

) can then be calculated using numerical procedures. See

Appendix B for further details.

To examine the implications of the extended Merton model, we consider two scenarios.

In both scenarios, we fix the value of the bond due at T2 to be 10% of the total market

value of debt. In the first scenario, 80% of the firm’s market value of debt is from the

bond due at T3 and only 10% of the market value of debt is from the bond due at T1. In

this scenario, the intermediate maturity bond, due at T2, is de facto senior to most of the

debt in the firm. In the second scenario, we flip the proportions of the market value of

debt due to the bonds maturing at T1 and T3, making the intermediate maturity bond de

facto junior to most of the firm’s debt. We set the risk-free rate to be 3%, the maturities

of the three bonds to be T1 = 5, T2 = 7, and T3 = 10, and the bankruptcy costs to be

L = 0.5. We present the hedge ratios of the intermediate bond for the two scenarios in

Table 2 and also plot them in Figure 1(b).

Table 2 and Figure 1(b) show that the hedge ratio for the intermediate bond is higher

when a larger proportion of the firm’s debt is due prior to the bond (the bond is de

9

facto junior), especially for bonds issued by firms with significant leverage and/or asset

volatility. For example, a bond that is de facto senior has a hedge ratio of 0.0048%

when the firm has a market leverage of 70% and an asset volatility of 20%. This is an

economically small hedge ratio as it implies that a 10% equity return is only associated

with a 0.00048% bond return. This is despite the fact that the firm has very high leverage.

In contrast, when the bond is de facto junior, the hedge ratio is substantially higher at

26.52%. A 10% equity return now corresponds to a 2.652% bond return. The intuition

follows from the sequential repayment of debt. If a firm is in financial distress, it may

still be able to repay the principal on bonds that mature early in the firm’s maturity

structure. It is much less likely that the firm will be able to remain solvent long enough

to repay the principal on the bonds that are late in the firm’s maturity structure. Thus, a

bond that is early in a firm’s maturity structure (de facto senior) can be very safe even if

the firm itself is in some financial distress. These bonds will have little sensitivity to the

underlying firm value, the main tie between corporate bonds and equities. Bonds that

are late in the maturity structure of a risky firm will be particularly risky, behaving more

like equities, and consequently should have stronger comovement with equities.

We can also compare the hedge ratios of de facto senior and de facto junior bonds to

the original Merton case with no maturity structure that is presented in Table 1. For the

case of 70% market leverage and 20% asset volatility discussed above, the Merton model

hedge ratio is 14.39%, which is significantly greater than the de facto senior hedge ratio

of 0.0048% but is considerably lower than the de facto junior hedge ratio of 26.52%. A

comparison of other market leverage-asset volatility combinations between Tables 1 and 2

shows a similar pattern. For most cases, the hedge ratio given by the Merton model falls

between the hedge ratios of de facto senior and de facto junior bonds. This suggests that if

de facto seniority is not taken into account, the Merton model will produce model-implied

hedge ratios that are too high for bonds that mature early in a firm’s maturity structure

and too low for bonds that mature late in the firm’s maturity structure.

10

2.3 Geske Model

The extended Merton model in Section 2.2 is based on the assumption that upon bond

maturity, the face value is paid out of the value of the firm. There is no strategic decision

for the manager or equityholders to make. Another class of structural models of default,

endogenous default models, force equityholders to directly bear the cost of payouts (either

coupons, face value payments, or both) to bondholders.12 Thus, at each moment that

there is a payoff, equityholders make an decision by trading-off the value they must pay

to continue the firm against their claim to the value of the firm as a going concern.

Here, we consider the Geske (1977) model.13 In the Geske model, equityholders are

residual claimants to the firm. That is, if the firm survives to the very last promised

payment to bondholders at time T , equityholders receive max(0, VT −KT ), where VT is

the value of the firm at T and KT is the last promised payment. At any prior time t

with a payment to bondholders of Kt, equityholders must decide whether to pay Kt and

allow the firm to continue or to force the firm to default and put the firm to bondholders.

Effectively, equityholders are making an optimal decision based on the market value of

max(0, VT −KT ) at t compared to the payment Kt.

Similar to our extension of the Merton model, we include three zero coupon bonds

with maturities T1 < T2 < T3 and focus on the intermediate maturity bond while varying

the proportion of the firm’s debt value due to the first and third bonds. We again consider

two scenarios, the first where the bond maturing at T3 accounts for 80% of the firm’s debt

value and the second where the bond maturing at T1 accounts for 80% of the firm’s debt

value. This allows us to compare the case where most of the firm’s debt is due after the

intermediate maturity bond and the case where most of the firm’s debt is due before the

intermediate maturity bond.

12Masulis and Korwar (1986) find that in a sample of 1,406 announcements of stock offerings, 372offerings are planned exclusively to refund outstanding debt. This suggests that there is a realistic rolefor equityholders in paying off debt.

13See also Leland (1994) and Leland and Toft (1996) for models that incorporate optimal decisions byequityholders.

11

As our paper implements rather than extends the Geske model, we do not provide the

full set of Geske pricing formulas for general cases, but instead focus on the intuition of the

Geske model and the special case that we implement, which are best understood through

a recursive framework. In the Geske model, firm value follows a Geometric Brownian

motion, just like in the Merton model. Also as in the Merton model, if the firm survives

until there is only one bond outstanding (the bond with maturity T3 in our example),

equity is a residual claim and receives a payoff of max(0, V3 −K3) at T3.

Consider the value of equity and the remaining bond at T2 and suppose that equity-

holders have just decided to pay the face value of the intermediate maturity bond, K2,

to continue the firm. This becomes the standard Merton model as equityholders have no

remaining decisions to make. The value of equity is now

E2 = V2N(d1)−K3e−r(T3−T2)N(d2), (6)

d1 =ln(V2K3

)+(r + 1

2σ2v

)(T3 − T2)

σv√T3 − T2

, d2 = d1 − σv√T3 − T2.

That is, equity is a call option on the firm and the long maturity bond is a risk-free bond

short a put option on the firm.

We can now consider the optimal decision of equityholders at T2. The decision that

equityholders make is simple. If the value of the firm as a going concern given in equation

(6) is greater than K2, they pay K2 and continue the firm. Otherwise, equityholders force

the firm to go bankrupt. Equityholders’ decision at T1 is similar, though the trade-off

that equityholders make is between the payment of the face value of the short maturity

bond, K1, and the value of equity as a going concern which is a compound call option.

As noted by Eom, Helwege, and Huang (2004), the Geske formula is not straight-

forward to implement accurately, particularly for long maturity bonds. Thus, we follow

Huang (1997) and Eom, Helwege, and Huang (2004) and implement the Geske model

using a Cox, Ross, and Rubinstein (1979) binomial tree, and the equity and bond values,

12

as well as the hedge ratio, are determined numerically. We also verify the accuracy of this

numerical implementation by running simulations.

Table 3 and Figure 1(c) present the hedge ratios for the intermediate bond for the

scenario where (1) most of the debt is due after the intermediate bond and (2) most of

the debt is due prior to the intermediate bond. Recall that for the extended Merton

model, the hedge ratio is generally lower in the first scenario. The intuition follows from

the fact that if most of the debt is due after T2, the firm would very likely have enough

value to pay bondholders at T2 directly from firm value. Thus, the intermediate bond has

little sensitivity to firm value and a low hedge ratio. In the Geske model, the intuition is

less clear. Even if most of the firm’s debt is due at T3, equityholders may still choose to

default at T2 as their payoff at T3 contingent on continuation, max(0, V3−K3), may have

a low expected value due to the large amount of debt due at T3. In fact, we find that

for the same set of parameters as the extended Merton model, the Geske model generates

slightly higher hedge ratios for scenario 1 than scenario 2 for almost all leverage-asset

volatility combinations. For example, consider again the firm with a 70% market leverage

and a 20% asset volatility, the Geske model predicts a hedge ratio of 30.43% for the

intermediate bond for scenario 1 versus a hedge ratio of 27.79% for scenario 2. Thus,

a 10% equity return is associated with a 3.043% return for the intermediate bond when

most debt is due after the bond compared to a 2.779% return when most debt is due

before the intermediate bond. These results contrast with the extended Merton model

which predicts a hedge ratio of 0.0048% for scenario 1 and 26.52% for scenario 2. This

contrast between the extended Merton model and Geske model can clearly be seen in

Figure 1(d) where we plot the differences in hedge ratio between the two scenarios for

both models against leverage and asset volatility.

13

2.4 Other Measures of Credit Risk

The simple Merton model suggests that firms with greater credit risk have higher hedge

ratios for their bonds. The intuition follows from the fact that as credit risk increases,

the value of bonds become more sensitive to firm value. Indeed, both Kwan (1996)

and Schaefer and Strebulaev (2008) find that hedge ratios are larger as ratings decline.

Ratings, however, are both coarse and slowly updated measures of credit risk. If market

participants are able to assess credit quality at a more granular level than ratings, we

would expect to see larger hedge ratios for firms with greater credit risk within ratings.

On the other hand, recent evidence from the MBS market suggests that it is unclear

whether the market makes its own assessment of credit quality above and beyond what is

provided by credit ratings. Adelino (2009) finds evidence that yield spreads at issuance do

not predict future downgrades and defaults within Aaa-rated MBS, but do predict future

downgrades and defaults for all other ratings. As Aaa-rated MBS constitute roughly 90%

of MBS issuance, this suggests that the vast majority of the MBS market do rely primarily

on ratings.

We consider two measures of credit risk that have been used extensively in the aca-

demic literature and are also prominent in credit risk assessment in practice, book-to-

market ratio and distance-to-default. Book-to-market ratio, the ratio of the book value of

equity to the market value of equity, is a reduced form proxy of a firm’s default likelihood

and has been found to be related to the relative strength of a firm’s economic funda-

mentals (Lakonishok, Shleifer, and Vishny (1994) and Fama and French (1995)). Though

there is much debate as to whether the return premium associated with book-to-market

ratio is due to risk or mispricing (see e.g., Fama and French (1993, 1996), Lakonishok,

Shleifer, and Vishny (1994), Haugen (1995), MacKinlay (1995), and Daniel and Titman

(1997)), our use of the book-to-market ratio is much more basic. We simply take it as a

indicator of the market’s assessment of the default prospects of a firm.

14

Distance-to-default is a structural-based proxy for default risk that is predicated on the

Merton model. It is widely used in both the academic literature (e.g., Hillegeist, Keating,

Cram, and Lundstedt (2004), Vassalou and Xing (2004), Campbell, Hilscher, and Szilagyi

(2008), and Bharath and Shumway (2008)) and investment management practice (e.g.,

KMV) to measure how far a firm is from its default boundary.14

3 Data and Summary Statistics

3.1 Data

The corporate bond dataset used in this paper is the Financial Industry Regulatory

Authority’s (FINRA) TRACE. This dataset is the result of a regulatory initiative to

increase the price transparency in the corporate bond market and was implemented in

three phases. On July 1, 2002, the first phase was implemented, requiring transaction

information to be disseminated for investment grade bonds with an issue size of $1 billion

or greater and 50 legacy high-yield bonds from the Fixed Income Pricing System (FIPS).

Approximately 520 bonds were included in Phase I. Phase II, which was fully implemented

on April 14, 2003, increased the coverage to approximately 4,650 bonds and included

smaller investment grade issues. Phase III, which was fully implemented on February

7, 2005, covered approximately 99% of secondary bond market transactions. In 2010,

FINRA added coverage of U.S. Agency bonds and in 2011, asset-backed and mortgage-

backed securities. In our study, we use TRACE data on corporate bonds from July 2002

to December 2010.

Bond transaction data from TRACE is merged with bond characteristics data from

Mergent FISD, which allows us to determine bond characteristics such as age, maturity,

14KMV, which was acquired by Moody’s in 2002, uses distance-to-default to assess expected defaultfrequency (EDF). Though Moody’s puts out both credit ratings and EDFs, EDFs are not used explicitlyin assigning ratings. Moody’s publishes guides for their ratings methodologies by industry. The fourbroad criteria that are typically used are scale and diversification, profitability, leverage and coverage,and financial policy.

15

amount outstanding, and rating. This data is then merged with equity return data from

CRSP and accounting information from Compustat. Treasury data is obtained from the

Constant Maturity Treasury (CMT) series. Mergent FISD is used to eliminate bonds that

are putable, convertible, or have a fixed-price call option.15 CRSP is used to eliminate

bonds issued by financial firms.16

3.2 Bond Returns and Summary Statistics

While equity returns can be directly obtained from CRSP, both corporate bond and

Treasury returns need to be calculated. To calculate monthly corporate bond returns, we

use the last trade for a bond in a month and calculate log returns as

rt+1 = ln

(Pt+1 + AIt+1 + Ct+1

Pt + AIt

), (7)

where Pt+1 is a clean price, AIt+1 is the accrued interest, and Ct+1 is the coupon paid

during the month (if applicable).17

Unlike equities, many corporate bonds do not trade every day. Thus, the returns that

we calculate for a bond in a particular month may not necessarily be based exactly on

month-end to month-end prices and may not encompass exactly 21 trading days. To

account for this, we make two adjustments. First, we match Treasury and equity returns

to the exact dates that are used to calculate bond returns. This eliminates any issues

of non-synchronization as our goal is to compare contemporaneous bond, equity, and

15Bonds where the only provision is a make whole call provision, which constitutes a significant portionof non-financial bonds, are kept. Make whole calls involve redemption at the maximum of par and thepresent value of all future cash flows using a discount rate of a comparable Treasury plus X basis points,where the most common values of X are 20, 25, and 50. Historically, the average AAA yield spreadhas been greater than 50 basis points, leaving make whole calls with little chance of being in-the-money.Powers and Tsyplakov (2008) show that theoretically, make whole calls have little effect on bond prices.They also find that by 2002-2004, the empirical effect of make whole calls on bond prices is extremelysmall.

16Though some corporate bond studies, primarily those on corporate bond illiquidity, retain financials,studies on structural models of default typically drop financials due to differences of the implications ofleverage for financials.

17We apply standard filters for corrected and canceled trades and eliminate obvious errors.

16

Treasury returns. Second, we scale all log returns to make them monthly. For example, if

a log bond return is calculated over 25 business days, we scale the return by 2125

to make

it comparable to a return that actually occurs over exactly a one month period (≈ 21

business days).

An additional complication that arises in calculating corporate bond returns is the se-

vere illiquidity of the corporate bond market. As shown in Edwards, Harris, and Piwowar

(2007) and Bao, Pan, and Wang (2011), the effective bid-ask spreads of corporate bonds

are very large. This creates the problem that observed bond returns can be non-zero even

absent changes to fundamentals. One way to address this problem is to average prices

over the last few days of the month to obtain prices where the bid-ask spread is washed

out by averaging over a number of trades at both bid and ask. However, this could create

a mismatch between the timing of corporate bond returns and that of equity and Trea-

sury returns. We choose not to make any adjustments for illiquidity as we note that the

illiquidity simply adds noise to bond returns,18 and as long as bond returns are used as

dependent variables, our empirical analysis will still provide unbiased inferences.

To calculate Treasury returns, we use yields from the US Department of Treasury’s

Constant Maturity Treasury (CMT) series. The provided yields are par yields for on-the-

run (and hence, liquid) bonds. Because the yields are par yields, we can calculate the

return for a bond that trades at par at the start of a month by discounting cash flows

using yields at the end of the month. Only discrete maturities are provided in the CMT

series. Thus, we interpolate between maturities to calculate the return for a Treasury

with the same maturity as the corporate bond being priced.

Table 4 reports summary statistics for our corporate bond sample. The majority of

the sample are rated A or Baa by Moody’s, with these ratings covering 73.5% of the

bond-months of the sample. Across ratings, the average time-to-maturity ranges from 6.8

18Indeed, Bao and Pan (2013) find that corporate bond returns are too volatile compared to equityreturns.

17

to 9.5 years and the average age from 4.5 to 6.3 years. To measure a bond’s position in

its firm’s maturity structure, we calculate the proportion of the firm’s total face value of

current outstanding debt from FISD that has a shorter remaining maturity than the bond

of interest. Its average value varies from 33% to 50% across ratings. On the dimensions of

amount outstanding, coupons, and yield spreads, we observe more systematic differences

across ratings. Poorer rated bonds tend to have smaller amount outstanding, higher

coupon rates, and higher yield spreads. The smaller amounts outstanding reflect the fact

that firms with poorer ratings tend to be smaller and the higher coupon rates reflect the

fact that most corporate bonds are issued at par. The average yield spread for our Aaa

& Aa sample is 113 basis points compared to 573 basis points for B. There is also a large

jump at the investment grade/junk cut-off as Baa bonds have an average yield spread of

217 basis points compared to 426 basis points for Ba.

In comparison to yield spreads, the log realized returns for bonds in our sample show

a less pronounced relation with ratings. This can be attributed primarily to two reasons.

First, realized bond returns for a period of 8.5 years do not necessarily reflect ex-ante ex-

pected returns.19 Second, while yield spreads may be monotonically increasing as ratings

become poorer, this is not necessarily the case for expected returns. Yield spreads are

promised returns and reflect both expected returns and the probability of default. As the

probability of default has historically been higher for bonds of poorer ratings, we would

expect to see higher yield spreads for poorer ratings even if expected returns do not vary

across ratings.

3.3 Other Default Proxies and Firm Summary Statistics

We construct the book-to-market ratio (B/M) similar to Asness and Frazzini (2011), who

argue that it is important to use more timely stock price data relative to book data in

19As shown in Hou and van Dijk (2012), conclusions can vastly differ when ex-ante expected returnsare used rather than ex-post realized returns in asset pricing tests.

18

order to better forecast future book-to-market ratio.20 For the book value of equity, we use

the book value of common equity (CEQ) from Compustat and assume that it is publicly

available three months after the fiscal year end. For the market value of equity, we use

common shares outstanding (CSHO) times price at the end of the fiscal year (PRCC F),

but adjust this by the returns without dividends from CRSP. For example, to calculate

the B/M ratio for a firm in May 2010 whose last fiscal year ended in December 2009, we

take the book equity and market equity (CSHO × PRCC F) from Compustat as of the

end of the 2009 fiscal year and scale the market equity by (1 + r) where r is the return of

the firm’s equity without dividends from December 2009 to May 2010.21

In our sample, firms with poorer ratings tend to have higher B/M ratios. Reported in

Table 5, the average B/M for bonds rated Aaa & Aa and A are 0.39 and 0.41, respectively.

In comparison, the average B/M are 0.61, 0.91, and 0.79 for Baa, Ba, and B-rated bonds,

respectively. This is consistent with the notion that high B/M is associated with relative

distress, though our main focus will be on using B/M as a measure of perceived distress

risk within rather than across credit ratings.

Following Campbell, Hilscher, and Szilagyi (2008), we calculate the distance-to-default

as

DD =ln(VK

)+ 0.06 + rf − 1

2σ2v

σv. (8)

As in the previous literature, we measure K as debt in current liabilities (DLC) plus one-

half times long-term debt (DLTT). The primary complication that arises in solving for

DD is that the firm value, V, and asset volatility, σv, are unobservable. As in Hillegeist,

Keating, Cram, and Lundstedt (2004) and Campbell, Hilscher, and Szilagyi (2008), we

20This is particularly critical for our purposes as one of the drawbacks to using credit ratings in creditassessment is that they are updated slowly.

21Directly using market equity from CRSP for May 2010 creates errors if the firm had significantissuance or repurchases between December 2009 and May 2010 which would be reflected in the marketequity for May 2010, but not in the book equity for December 2009.

19

solve for V and σv by simultaneously solving the Merton model equation for equity value

and equity volatility

E = V N(d1)−Ke−rTN(d2), (9)

and

σE = N(d1)V

Eσv. (10)

Table 5 reports the average distance-to-default estimates for different ratings. The

distance-to-default decreases monotonically from a high of 10.24 for Aaa & Aa bonds to a

low of 4.55 for B bonds. Since differences in distance-to-default should reflect differences

across firms in volatility and leverage, Table 5 also reports the equity volatility and a

measure of the firm’s leverage (labeled DD Leverage), defined as the face value of a firm’s

debt (debt in current liabilities plus one-half times long-term debt) divided by the sum

of the face value of debt and the market value of equity. In addition, we also report

a more traditional measure of pseudo-leverage, with the face value of debt defined as

debt in current liabilities plus long-term debt (rather than one-half of long-term debt).

Both equity volatility and leverage are higher for junk rated bonds than investment grade

bonds. In particular, the average equity volatility are 26.86%, 27.89%, and 32.55% for

Aaa & Aa, A, and Baa bonds respectively compared to 40.01% and 43.17% for Ba and B

bonds respectively, whereas the average pseudo-leverage are 39.73%, 30.94%, and 36.16%

for Aaa &Aa, A, and Baa bonds respectively compared to 53.76% and 63.13% for Ba and

B bonds respectively.

20

4 Empirical Results

4.1 Empirical Specification

To examine the empirical relation between corporate bond returns and equity returns, we

run the following panel regression rating-by-rating

rD = β0 + β1rE × Z + βErE + βT rT + β2rE × T + β3rE × T 2 + β4Z + β5T + β6T2 + ε,

(11)

where Z is any variable (for example, proportion of debt due prior) that is hypothesized

to be related to equity-corporate bond comovement (the hedge ratio), and β1 measures

its effect on the comovement.

We include returns of Treasuries with similar maturity, rT , as a control. Though the

models described in Section 2 have constant interest rates for parsimony, interest rates

display important variation in reality and can affect corporate bond returns. This is

particularly important for high grade corporate bonds, which have low default rates and

are primarily sensitive to interest rate movements rather than firm value.

We also include interaction terms between equity returns and time-to-maturity, T , and

time-to-maturity-squared, T 2. Under standard structural models of default, the hedge ra-

tio becomes larger as the time-to-maturity increases.22 This suggests that we should allow

the corporate bond-equity relation to vary with the maturity of a bond. To determine the

order of maturity that needs to be included in a regression framework, we appeal to the

relation between Merton model-implied hedge ratios and maturity. For asset volatility

ranging from 10% to 50% and market leverage ranging from 10% to 70%, we calculate

Merton model-implied hedge ratios and for each asset volatility-market leverage combi-

nation for each integer time-to-maturity from 1 to 30 years. Each asset volatility-market

22In a recent paper, Chen, Xu, and Yang (2012) find that firms with more systematic volatility chooselonger debt maturity.

21

leverage combination represents a different level of creditworthiness and for each combina-

tion, we regress model hedge ratios on T and T 2. The R2 of these regressions are reported

in Panel B of Table 1. With the exception of the 10% asset volatility-10% leverage com-

bination, the R2 of all of these regressions are above 95%, suggesting that a quadratic

function accounts for the vast majority of the theoretical effect of maturity on the hedge

ratio.23

Prior to estimating our main specification (11), we first run the regression

rD = β0 + βErE + βT rT + ε. (12)

This regression allows us to gauge the typical relation between corporate bond returns and

equity and Treasury returns, providing a baseline comparison for our results on proportion

of debt due prior, book-to-market, and distance-to-default. Table 6 reports the results.

We see that, consistent with the results in Kwan (1996) and Schaefer and Strebulaev

(2008), there is a positive and significant relation between corporate bond returns and

equity returns for all ratings, and this relation strengthens steadily as ratings become

poorer with an empirical hedge ratio of 0.0489 (t-stat = 9.70) for Aaa & Aa bonds, 0.0886

(t-stat = 4.73) for Baa bonds, and 0.1260 (t-stat = 8.87) for B bonds. In contrast, the

relation between corporate bond returns and Treasury returns weakens monotonically as

ratings become poorer. The coefficient on Treasury returns is a highly statistically and

economically significant 0.6632 (t-stat = 15.62) for Aaa & Aa bonds, but falls to a still

significant 0.4022 (t-stat = 4.94) for Baa bonds. For junk-rated Ba and B bonds, the

relation between corporate bond and Treasury returns is negative and insignificant.

The variation in the relations between corporate bond and equity and Treasury returns

across ratings reinforces some important intuition about corporate bonds. High grade

corporate bonds have low default probabilities and little sensitivity to underlying firm

23Adding a cubic term for maturity has little effect on our results.

22

value. Instead, they are very sensitive to interest rates, which are also the main driver

of Treasury bond prices. Thus, high grade corporate bonds will tend to comove with

Treasuries. Low grade corporate bonds are more sensitive to underlying firm value, the

primary driver of equity prices. Thus, low grade corporate bonds comove more with

equities.

4.2 Maturity Structure

In our empirical analysis, we first consider the position of a bond in its issuer’s maturity

structure. As discussed in Section 2, our extended Merton model demonstrates that a

bond that matures after most of the other bonds issued by a firm is de facto junior and

thus, should have a higher hedge ratio. However, this prediction is not borne out in

the Geske (1977) model. Our regressions follow from equation (11) with Z equal to the

proportion of the issuer’s debt due prior to a bond. The extended Merton model predicts

that the coefficient on the interaction term between equity returns and the proportion

due prior should be positive. Alternatively, the Geske (1977) model predicts that the

coefficient should be close to zero or slightly negative.

Table 7 shows that the interaction term is positive for all ratings and is statistically

highly significant in four out of five cases (except for B). A shift from being the most

de facto senior bond to the most de facto junior bond is associated with an increase in

the hedge ratio of 0.0854 (t-stat = 2.45), 0.0468 (t-stat = 2.95), 0.0908 (t-stat = 2.60),

0.1621 (t-stat = 4.40), and 0.0348 (t-stat = 1.03) for Aaa & Aa, A, Baa, Ba, and B bonds,

respectively. Our results are thus consistent with the prediction of the extended Merton

model. To better gauge the economic impact of de facto seniority, consider for example the

marginal relation between corporate bond and equity returns for Baa bonds. Fixing the

time-to-maturity at seven years and the proportion of debt due prior at 10%, the marginal

effect of equity returns on corporate bond returns is 0.0598.24 Increasing the proportion

24The marginal effect is calculated as βE+β1×Z+β2×T+β3×T 2 = 0.0355+0.0908(0.1)+0.00226(7)−

23

of debt due prior to 80% and thus making the bond de facto junior, the marginal effect

more than doubles to 0.1234, an increase of 0.0636.25 The corresponding increases in

the marginal effect for Aaa & Aa, A, Ba, and B are 0.0598, 0.0328, 0.1135, and 0.0244,

respectively. Thus, for bonds with the same credit rating but different de facto seniority,

there may be economically significant differences in their comovement with equities.

Fully understanding why hedge ratios can be different for bonds with different de facto

seniorities even after controlling for ratings requires a systematic examination of ratings

that goes beyond the scope of this paper. However, a cursory examination of ratings-

related documents published by Moody’s suggests that ratings are based on assessment

of a firm’s profitability, leverage, and other competitive factors. While whether a bond

is secured or is explicitly junior to other bonds is considered, the full maturity structure

of a firm’s debt is not. Our results are consistent with the market recognizing the effect

of de facto seniority on a bond’s credit risk even though ratings agencies do not use it in

credit assessments.

4.3 Book-to-Market

As shown in Table 6 and also in Kwan (1996) and Schaefer and Strebulaev (2008), there

is both empirical and theoretical evidence that bonds issued by firms with better credit

ratings comove less with equities than bonds issued by firms with poorer credit ratings.

This implies that the sensitivity of bond returns to equity returns is higher for poorer

rated bonds. Further, Section 4.2 shows that bonds that are de facto junior have higher

hedge ratios than bonds that are de facto senior controlling for ratings. Thus, the market

perceives de facto junior bonds to be riskier even though credit rating agencies do not

assign different ratings on this dimension. Even more broadly, if what the market perceives

0.000013(72) = 0.0598.25To put this increase in the marginal effect into perspective, if we keep the proportion of debt due

prior fixed at 10% but instead decrease the rating to Ba, the marginal effect will only increase by 0.0124to 0.0722.

24

as credit risk26 is more granular than the ratings provided by credit ratings agencies, we

should see higher hedge ratios for the particular set of firms that the market perceives to

be riskier within ratings.

We first use a firm’s book-to-market ratio (B/M) as a proxy for the market’s perception

of a firm’s credit risk. To study the effect of B/M on the comovement between corporate

bonds and equities within ratings, we use regression equation (11) and the B/M ratio

normalized period-by-period across ratings as Z.27 Table 8 shows that the effect of B/M

on the comovement between corporate bonds and equities is positive for all ratings and

is statistically significant for all ratings except B. Specifically, a one standard deviation

increase in B/M is associated with an increase in the hedge ratio of 0.0543 (t-stat = 5.78),

0.0114 (t-stat = 6.10), 0.0252 (t-stat = 4.22), 0.0168 (t-stat = 3.03), and 0.0062 (t-stat =

1.00) for Aaa & Aa, A, Baa, Ba, and B bonds, respectively. Given that high B/M firms

are perceived to have higher default likelihood, this suggests that even after controlling for

ratings, firms that are perceived to have greater default risk have higher hedge ratios.28

To gauge the economic importance of the effect of B/M on the relation between cor-

porate bond and equity returns, consider for example a Baa bond with seven years to

maturity and B/M ratio at the period mean, the marginal effect of equity returns on

corporate bond returns is 0.0700. If we keep rating unchanged but increase B/M to one

standard deviation above the period mean, the marginal effect of equity returns on cor-

porate bond returns increases to 0.0952, which is only slightly lower than the marginal

effect of 0.0999 if we keep B/M fixed at the period mean but instead reduce the rating to

Ba.

26In this context, risk is not necessarily systematic default risk and can include firm-specific risk.27We normalize B/M each period by subtracting the cross-sectional mean and dividing by the cross-

sectional standard deviation. Unlike the proportion of debt due prior, the distribution of B/M canvary considerably over time with macroeconomic conditions. To avoid our results being driven solely bymacroeconomic factors, we normalize B/M.

28Adding more granular controls for ratings (e.g. adding rE × 1Baa1 and rE × 1Baa3 into the Baaregression) does not change our conclusions.

25

4.4 Distance-to-Default

In addition to considering B/M, a reduced form proxy for credit risk, we also examine

the distance-to-default, a structural model-based measure of credit risk.29 Our analysis

of distance-to-default is similar to that using B/M, and as with B/M, we normalize the

distance-to-default each period across ratings.

Table 9 report the effect of distance-to-default on the relation between corporate bond

and equity returns for each rating. It shows that for all ratings, a higher distance-to-

default is associated with a lower hedge ratio and the relation is statistically significant

except for B bonds. These results are economically meaningful as a one standard deviation

increase in distance-to-default is associated with a decrease in the hedge ratio by 0.0164

(t-stat = 5.26) for Aaa & Aa bonds, 0.0155 (t-stat = 3.08) for A bonds, 0.0307 (t-stat =

3.44) for Baa bonds, 0.0502 (t-stat = 3.71) for Ba bonds, and 0.0292 (t-stat = 1.21) for B

bonds. Again, consider a seven-year Baa bond, if we increase the distance-to-default from

the period mean to one standard deviation above the mean, the marginal effect of equity

returns on corporate bond returns decreases from 0.0694 to 0.0387, which is smaller than

the marginal effect of 0.0495 for an A bond with distance-to-default at the period mean.

Overall, the findings in Table 9 reinforce the findings in Table 8. They show that

firms with higher credit risk measured by either high B/M or low distance-to-default are

associated with a higher sensitivity of corporate bond returns to equity returns than firms

with lower credit risk, even after controlling for credit ratings.

5 Conclusion

In this paper, we use hedge ratios, the relative returns of corporate bonds and equities,

to examine the comovement of securities issued by the same firm. Structural models of

29A high distance-to-default is a sign of financial health while a low distance-to-default is a sign offinancial distress.

26

default, beginning with Merton (1974), imply that corporate bonds and equities should

comove due to their common sensitivity to the underlying firm value, and that hedge

ratios should be larger for riskier bonds. Our focus is on the heterogeneity of the hedge

ratios both at the bond level as well as at the firm level.

Within firm, there is potentially heterogeneity across bonds in both the seniority and

maturity structure. While it is clear from the Merton model that junior bonds should

have higher hedge ratios than senior bonds, the lack of priority structure in public debt

data makes it difficult to test the Merton prediction empirically. Instead, we focus on

the maturity structure of public debt and illustrate that a bond’s position is in its firms

maturity structure should affect its hedge ratio in an extended Merton model but not

in the Geske model. Empirically, we find that bonds that are later its firms maturity

structure have higher hedge ratios, consistent with the prediction of the extended Merton

model. The intuition is similar to Brunnermeier and Oehmke (2013), who argue that

bonds that are due early in its firms maturity structure are de facto senior. This arises

from the possibility that a firm that is in financial trouble can remain solvent enough to

repay short-term bonds, but not long enough to repay long-term bonds.

At the firm level, we proxy for credit risk using book-to-market ratio and distance-

to-default, and find that hedge ratios are larger for firms with greater credit risk within

ratings. This suggests that in the corporate bond market, market participants are able

to assess credit risk above and beyond what is provided by credit ratings agencies.

27

Tables & Figures

Table 1: Merton Model Hedge Ratios

Panel A: Merton Model Hedge Ratiosσv

0.1 0.15 0.2 0.25 0.3 0.4 0.50.1 0 8.9e-07 0.0018 0.0621 0.4412 3.2324 8.48780.2 1.0e-07 0.0042 0.1898 1.1643 3.2411 9.6727 17.10890.3 0.0003 0.1437 1.3279 3.9790 7.5999 15.7851 23.6740

Market Leverage 0.4 0.0243 0.9330 3.7700 7.8166 12.2781 21.0071 28.73550.5 0.2987 2.8144 7.1155 11.9300 16.6855 25.3360 32.67550.6 1.3750 5.6615 10.8029 15.8607 20.5773 28.8424 35.71180.7 3.5755 9.0054 14.3865 19.3400 23.8277 31.5563 37.9315

Panel B: R2 of regressions of Merton Model Hedge Ratios on Time-to-Maturityσv

0.1 0.15 0.2 0.25 0.3 0.4 0.50.1 0.7954 0.9643 0.9980 0.9965 0.9925 0.9928 0.99680.2 0.9507 0.9989 0.9944 0.9924 0.9942 0.9983 0.99760.3 0.9954 0.9949 0.9932 0.9959 0.9983 0.9977 0.9916

Market Leverage 0.4 0.9978 0.9935 0.9967 0.9990 0.9987 0.9927 0.98410.5 0.9939 0.9966 0.9992 0.9983 0.9951 0.9866 0.97810.6 0.9955 0.9993 0.9979 0.9942 0.9899 0.9818 0.97460.7 0.9991 0.9978 0.9935 0.9893 0.9856 0.9790 0.9732

Panel A of this table reports Merton model-implied hedge ratios, ∂ lnB∂ lnE

, scaled by 100. TheMerton hedge ratio follows from equation (3) and is calculated for different levels of marketleverage and asset volatility. In Panel B, we report the R2 from regressions of Merton modelhedge ratios on T and T 2, where T is the maturity of the bond. For each market leverage-asset volatility combination, we calculate model hedge ratios for maturities ranging from 1year to 30 years. We then estimate the regression for each market leverage-asset volatilitycombination.

28

Table 2: Extended Merton Model Hedge Ratios

σv0.1 0.15 0.2 0.25 0.3 0.4 0.5

0.1 0.0000 0.0000 0.0000 0.0000 0.0002 0.0695 1.04610.0000 0.0000 0.0001 0.0174 0.2339 2.7979 8.4882

0.2 0.0000 0.0000 0.0000 0.0001 0.0099 0.5905 3.58710.0000 0.0004 0.0917 0.9799 3.3836 11.3312 20.1151

0.3 0.0000 0.0000 0.0000 0.0028 0.0696 1.5494 6.19450.0000 0.0679 1.3394 4.9732 10.0149 20.4587 29.3097

Market Leverage 0.4 0.0000 0.0000 0.0001 0.0159 0.2121 2.6408 8.33680.0047 0.9530 5.3511 11.6261 17.8715 28.2927 36.0544

0.5 0.0000 0.0000 0.0008 0.0467 0.4153 3.6005 9.86210.1952 4.3125 11.9132 19.2059 25.2896 34.3312 40.6166

0.6 0.0000 0.0000 0.0026 0.0888 0.6138 4.2703 10.75851.8473 10.6608 19.5015 26.2566 31.3300 38.3678 43.1563

0.7 0.0000 0.0000 0.0048 0.1226 0.7392 4.5833 11.06697.0904 18.6201 26.5155 31.6959 35.2889 40.1597 43.7222

This table reports the hedge ratio for the intermediate maturity bond priced in equation (5)of the extended Merton model. The top number for each market leverage-asset volatilitycombination is the hedge ratio that corresponds to the case where most of a firm’s debtis due after the intermediate bond (so it is de facto senior). The bottom number is thehedge ratio that corresponds to the case where most of a firm’s debt is due prior to theintermediate bond (so it is de facto junior). A firm is assumed to have three zero-couponbonds with maturities T1 = 5, T2 = 7, and T3 = 10. The risk-free rate, r, is 3%. For eachmarket leverage-asset volatility combination, the face values of the three bond issues K1,K2, and K3, are chosen to match the market leverage. The top number corresponds to 10%of the market value being attributable to the short maturity bond, 10% to the intermediatematurity bond, and 80% to the long maturity bond. The bottom number corresponds to80% attributable to the short maturity bond, 10% to the intermediate maturity bond, and10% to the long maturity bond.

29

Table 3: Geske Model Hedge Ratios

σv0.1 0.15 0.2 0.25 0.3 0.4 0.5

0.1 0.0000 0.0000 0.0007 0.0522 0.3962 3.2189 8.57350.0000 0.0000 0.0000 0.0163 0.2500 2.7136 7.9962

0.2 0.0000 0.0000 0.1940 1.3455 3.7954 12.0600 21.93610.0000 0.0000 0.1068 1.0227 3.5251 11.2047 19.0319

0.3 0.0000 0.1476 1.9184 5.9669 11.2312 21.9087 31.71110.0000 0.0619 1.5626 5.3673 10.4166 20.1260 28.9815

Market Leverage 0.4 0.0000 1.6304 6.8316 13.1491 19.2157 30.7761 41.21950.0000 1.1566 6.2016 12.8181 18.6349 28.4030 34.8844

0.5 0.2455 6.3485 14.3616 21.7644 28.5736 37.2128 46.10860.0583 5.7181 13.9096 20.4583 25.8839 33.8802 40.3855

0.6 4.0958 14.5166 22.8674 29.7942 34.3876 43.2369 50.54313.3421 13.0815 21.6618 27.8414 31.0798 38.1453 43.2457

0.7 13.2835 23.3547 30.4281 35.6507 40.2505 46.1002 50.188912.3024 21.5210 27.7943 31.4527 35.6294 39.6380 43.4748

This table reports the hedge ratio for the intermediate maturity bond priced in a Geske(1977) framework. The top number for each market leverage-asset volatility combination isthe hedge ratio that corresponds to the case where most of a firm’s debt is due after theintermediate bond. The bottom number is the hedge ratio that corresponds to the casewhere most of a firm’s debt is due prior to the intermediate bond. A firm is assumed tohave three zero-coupon bonds with maturities T1 = 5, T2 = 7, and T3 = 10. The risk-freerate, r, is 3%. For each market leverage-asset volatility combination, the face values ofthe three bond issues K1, K2, and K3, are chosen to match the market leverage. The topnumber corresponds to 10% of the market value being attributable to the short maturitybond, 10% to the intermediate maturity bond, and 80% to the long maturity bond. Thebottom number corresponds to 80% attributable to the short maturity bond, 10% to theintermediate maturity bond, and 10% to the long maturity bond.

30

Table 4: Bond Summary StatisticsAaa & Aa A Baa Ba B

Bonds 1,103 4,024 4,341 2,039 717Bond-months 25,959 82,708 91,221 25,220 11,499

Time-to-maturity mean 6.80 8.28 9.50 6.97 6.82med 4.50 4.34 6.19 3.96 3.71std 7.65 10.28 9.77 9.00 8.46

Age mean 4.45 4.72 4.86 5.09 6.32med 3.30 3.74 3.92 4.11 5.58std 3.94 3.81 3.82 3.78 3.90

Proportion due prior mean 0.5003 0.4002 0.3913 0.3502 0.3332med 0.5444 0.3623 0.3608 0.2886 0.2441std 0.2758 0.2795 0.2824 0.2777 0.3016

Amount outstanding mean 527 420 465 291 297med 250 296 350 225 200std 860 516 446 336 330

Coupon mean 5.21 5.76 6.38 6.69 7.16med 5.00 5.75 6.38 6.85 7.30std 1.55 1.37 1.29 1.39 1.45

Return mean 0.0045 0.0043 0.0052 0.0067 0.0048med 0.0037 0.0038 0.0045 0.0062 0.0064std 0.0264 0.0282 0.0359 0.0506 0.0702

Yield spread mean 0.0113 0.0137 0.0217 0.0426 0.0573med 0.0082 0.0100 0.0169 0.0361 0.0391std 0.0106 0.0140 0.0177 0.0374 0.1022

This table reports summary statistics for the bonds in our sample by rating.Time-to-maturity is reported in years. Age is the number of years since issuance.Amount outstanding is the face value outstanding in $mm. Proportion due prioris the proportion of a firm’s face value of debt due prior to a particular bondand is measured in decimals. Coupon is the coupon rate in %. Return is themonthly log return of a bond. Yield spread is the difference between the yieldof a bond and a Treasury with similar maturity from the Constant MaturityTreasury (CMT) series.

31

Table 5: Firm Summary StatisticsAaa & Aa A Baa Ba B

Firms 50 225 468 228 116Firm-months 2,289 10,761 18,200 5,391 2,456Observations 25,959 82,708 91,221 25,220 11,499

Equity market cap mean 204.96 45.51 23.11 10.32 8.14med 173.27 34.32 15.73 10.13 5.98std 121.71 42.89 27.37 7.90 8.46

Equity volatility mean 0.2686 0.2789 0.3255 0.4001 0.4317med 0.2035 0.2459 0.2737 0.3460 0.3647std 0.1614 0.1255 0.1659 0.1849 0.1912

DD leverage mean 0.3315 0.2319 0.2638 0.4417 0.5213med 0.3966 0.1553 0.2092 0.3468 0.4756std 0.2370 0.2043 0.2083 0.3044 0.2522

Pseudo-leverage mean 0.3973 0.3094 0.3616 0.5376 0.6313med 0.4816 0.2408 0.3257 0.4969 0.6327std 0.2583 0.2131 0.2116 0.2829 0.2224

Book-to-market mean 0.39 0.41 0.61 0.91 0.79med 0.31 0.35 0.52 0.73 0.70std 0.22 0.27 0.36 0.62 0.54

Distance-to-default mean 10.24 9.63 8.01 5.87 4.55med 10.80 9.07 7.60 5.09 4.30std 5.44 4.37 3.77 3.00 2.24

Equity Return mean 0.0020 0.0072 0.0059 -0.0018 -0.0090med 0.0053 0.0121 0.0128 0.0080 0.0029std 0.0837 0.0805 0.0994 0.1362 0.1665

This table reports firm characteristics for our sample. Equity market cap is theequity market capitalization in $bn. Equity volatility is the volatility of a firm’sequity using daily log returns over the past year, annualized. DD Leverage is theface value of debt divided by the sum of the face value of debt and the marketvalue of equity. Following Campbell, Hilscher, and Szilagyi (2008), the face valueof debt is defined as Debt in Current Liabilities plus one-half times Long-TermDebt. Pseudo-leverage is defined similarly as DD Leverage, but uses the sum ofDebt in Current Liabilities and Long-Term Debt as the face value of debt. Book-to-market is the book value of equity divided by the market value of equity as inAsness and Frazzini (2011). Distance-to-default is calculated following Campbell,Hilscher, and Szilagyi (2008). Equity Return is the monthly log equity return.

32

Table 6: Co-movement between Corporate Bonds and EquitiesAaa & Aa A Baa Ba B

rE 0.0489 0.0470 0.0886 0.1161 0.1260[9.70] [2.99] [4.73] [7.18] [8.87]

rT 0.6632 0.5779 0.4022 -0.0249 -0.1488[15.62] [8.80] [4.94] [-0.24] [-1.32]

Constant 0.0025 0.0024 0.0034 0.0071 0.0066[4.24] [2.45] [2.19] [4.33] [3.50]

R2 0.2386 0.2128 0.1203 0.1335 0.1695Obs 25,959 82,708 91,221 25,220 11,499

This table reports a regression of rD = β0 + βErE + βT rT . rD, rE, andrT are log returns (in decimals) for corporate bonds, equities, and Trea-suries, respectively. t-statistics are reported in brackets using standarderrors that are clustered by firm and month.

33

Table 7: Co-movement and Maturity StructureAaa & Aa A Baa Ba B

rE × prop due prior 0.0854 0.0468 0.0908 0.1621 0.0348[2.45] [2.95] [2.60] [4.40] [1.03]

rE -0.0044 0.0120 0.0355 0.0786 0.0956[-1.27] [1.99] [4.92] [7.49] [8.04]

rT 0.6683 0.5842 0.4164 -0.0195 -0.1154[16.07] [8.78] [5.12] [-0.18] [-1.07]

rE × T 0.0019 0.0029 0.0023 -0.0034 0.0047[1.98] [1.34] [1.02] [-1.08] [2.64]

rE × T 2 -0.0000 -0.0000 -0.0000 0.0000 -0.0001[-1.32] [-1.49] [-0.60] [0.78] [-3.60]

prop due prior 0.0020 -0.0001 -0.0006 0.0029 0.0000[1.29] [-0.17] [-0.40] [1.27] [0.01]

T -0.0000 0.0000 0.0001 -0.0002 -0.0000[-0.30] [0.18] [0.44] [-1.00] [-0.15]

T 2 0.0000 0.0000 -0.0000 0.0000 0.0000[0.66] [0.06] [-0.68] [1.08] [1.90]

Constant 0.0015 0.0022 0.0031 0.0070 0.0063[7.16] [6.25] [4.85] [8.22] [4.27]

R2 0.2471 0.2192 0.1336 0.1448 0.1776Obs 25,959 82,708 91,221 25,220 11,499

This table reports a regression of rD = β0 + β1rE × prop due prior + βErE +βT rT +β2rE×T +β3rE×T 2 +β4prop due prior+β5T +β6T

2. prop due prioris the proportion of a firm’s face value of debt due prior to a particular bondand is measured in decimals. rD, rE, and rT are log returns (in decimals) forcorporate bonds, equities, and Treasuries, respectively. T is the maturity ofa bond in years. t-statistics are reported in brackets using standard errorsthat are clustered by firm and month.

34

Table 8: Co-movement and Book-to-MarketAaa & Aa A Baa Ba B

rE ×B/M 0.0543 0.0114 0.0252 0.0168 0.0062[5.78] [6.10] [4.22] [3.03] [1.00]

rE 0.0248 0.0238 0.0302 0.0747 0.0958[4.24] [2.91] [3.11] [4.99] [6.88]

rT 0.6731 0.5842 0.4143 -0.0212 -0.1239[16.90] [8.79] [5.14] [-0.20] [-1.02]

rE × T 0.0063 0.0047 0.0060 0.0039 0.0063[3.95] [2.32] [3.64] [1.55] [3.42]

rE × T 2 -0.0001 -0.0001 -0.0000 -0.0000 -0.0001[-2.86] [-2.35] [-2.57] [-1.46] [-2.69]

B/M 0.0008 -0.0003 -0.0003 0.0012 0.0007[1.15] [-1.20] [-0.64] [3.18] [1.25]

T 0.0001 0.0000 0.0000 -0.0000 -0.0000[0.93] [0.10] [0.26] [-0.11] [-0.37]

T 2 0.0000 0.0000 -0.0000 0.0000 0.0000[0.17] [0.10] [-0.56] [0.40] [0.54]

Constant 0.0022 0.0021 0.0033 0.0063 0.0061[14.70] [4.39] [4.82] [7.15] [3.35]

R2 0.2518 0.2208 0.1385 0.1449 0.1830Obs 25,959 81,720 89,896 24,456 8,955

This table reports a regression of rD = β0 + β1rE × B/M + βErE +βT rT +β2rE×T +β3rE×T 2 +β4B/M +β5T +β6T

2. B/M is the book-to-market ratio of a firm, winsorized at 1% of each tail and normalizedby the cross-sectional mean and standard deviation each month. rD,rE, and rT are log returns (in decimals) for corporate bonds, equities,and Treasuries, respectively. T is the maturity of a bond in years.t-statistics are reported in brackets using standard errors that are clus-tered by firm and month.

35

Table 9: Co-movement and Distance-to-DefaultAaa & Aa A Baa Ba B

rE ×DD -0.0164 -0.0155 -0.0307 -0.0502 -0.0292[-5.26] [-3.08] [-3.44] [-3.71] [-1.21]

rE 0.0133 0.0178 0.0326 0.0444 0.0594[1.38] [2.63] [3.86] [2.33] [1.86]

rT 0.6683 0.5824 0.4152 -0.0190 -0.1193[16.04] [8.79] [5.15] [-0.18] [-1.09]

rE × T 0.0063 0.0049 0.0055 0.0038 0.0064[3.46] [2.50] [3.28] [1.52] [4.05]

rE × T 2 -0.0001 -0.0001 -0.0000 -0.0000 -0.0001[-2.69] [-2.41] [-2.32] [-1.34] [-4.10]

DD -0.0006 0.0000 -0.0001 -0.0023 0.0014[-1.63] [0.21] [-0.13] [-3.20] [0.87]

T 0.0001 0.0000 0.0000 -0.0000 -0.0000[1.10] [0.13] [0.32] [-0.21] [-0.34]

T 2 0.0000 0.0000 -0.0000 0.0000 0.0000[0.01] [0.11] [-0.61] [0.47] [1.11]

Constant 0.0023 0.0022 0.0031 0.0054 0.0083[4.01] [5.03] [4.29] [5.96] [2.72]

R2 0.2477 0.2200 0.1356 0.1424 0.1784Obs 25,959 82,457 90,361 25,139 11,300

This table reports a regression of rD = β0+β1rE×DD+βErE+βT rT +β2rE×T+β3rE×T 2+β4DD+β5T+β6T

2. DD is the distance-to-defaultof a firm, winsorized at 1% of each tail and normalized by the cross-sectional mean and standard deviation each month. rD, rE, and rT arelog returns (in decimals) for corporate bonds, equities, and Treasuries,respectively. T is the maturity of a bond in years. t-statistics arereported in brackets using standard errors that are clustered by firmand month.

36

Fig

ure

1:H

edge

Rat

ios

for

the

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ton,

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nded

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ton,

and

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kem

odel

s.F

orpan

els

(b)

and

(c),

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aces

repre

sent

the

case

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ea

bon

dis

late

init

sfirm

’sm

aturi

tyst

ruct

ure

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aces

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the

case

wher

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bon

dis

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firm

’sm

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tyst

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.

37

Appendix

A Merton Model with Seniority

As discussed by Lando (2004) among others, for a firm with a zero-coupon junior bond

with a face value of KJ and a zero-coupon senior bond with a face value of KS (both

maturing at T ), the values of the junior and senior bonds are

BJ = V N(d1,S)−KSe−rTN(d2,s)− V N(d1) +Ke−rTN(d2), (13)

BS = V − V N(d1,S) +KSe−rTN(d2,S),

where d1,S =ln(

VKS

)+(r + 1

2σ2v

)T

σv√T

, d2,S = d1,S − σv√T ,

and d1 =ln(

VKS+KJ

)+(r + 1

2σ2v

)T

σv√T

, d2 = d1 − σv√T .

It can be shown that

∂ lnBJ

∂ lnV− ∂ lnBS

∂ lnV=

V

BS ×BJ

[BS (N(d1,S)−N(d1))−BJ (1−N(d1,S))] . (14)

Examining the above equation for T ranging from 0.5 to 30, r ranging from 0.01 to

0.2, σV ranging from 0.01 to 1, KS and KJ each ranging from 1 to 200, and V fixed at

100, we find that ∂ lnBJ

∂ lnV− ∂ lnBS

∂ lnV> 0, indicating that the junior bond is more sensitive to

underlying asset value and thus, has a higher hedge ratio than the senior bond.

38

B Merton Model Extension to Include Maturity Struc-

ture

In our extension of the Merton model, the firm has three zero coupon bond issues with

face values K1, K2, and K3, and maturities T1 < T2 < T3. All of the bonds share the

same seniority. For example, suppose that the firm defaults prior to T1. The proportion

of the total recovered value of the firm that is paid to bond 1 is K1

K1+K2+K3. In addition,

we also allow for a proportion L of the residual firm value to be lost in bankruptcy. That

is, if the value of the firm at default is Vdefault, the proportion payable to bondholders is

(1− L)Vdefault.

B.1 Equity Value

E = EQ0

[e−rT31V1>K11V2>K21V3>K3 (V3 −K3)

]The firm value process is lognormal

V1 = V0 exp

[(r − 1

2σ2v

)T1 + σv

√T1w1

], (15)

where w1 ∼ N(0, 1).

39

Similarly,

V2 = (V1 −K1) exp

[(r − 1

2σ2v

)(T2 − T1) + σv

√T2 − T1w2

]if V1 > K1

and

V3 = (V2 −K2) exp

[(r − 1

2σ2v

)(T3 − T2) + σv

√T3 − T2w3

]if V2 > K2

After some algebra, it can be shown that

E =e−rT2

2π

∫ ∞−d2

∫ ∞−d̃2

exp

(−1

2z22 −

1

2z21

)(V2 −K2)N

(˜̃d1

)dz2dz1 (16)

−K3−e−rT3

2π

∫ ∞−d2

∫ ∞−d̃2

exp

(−1

2z22 −

1

2z21

)N(

˜̃d2

)dz2dz1,

where ˜̃d2 =ln(V2−K2

K3

)+(r − 1

2σ2v

)(T3 − T2)

σv√T3 − T2

, ˜̃d1 = ˜̃d2 + σv√T3 − T2

and d̃2 =ln(V1−K1

K2

)+(r − 1

2σ2v

)(T2 − T1)

σv√T2 − T1

.

B.2 Bond Value

Our focus is on the intermediate bond with maturity T2 and face value K2. The price

of this bond can be written as the sum of three expectations, which correspond to (1)

solvency at T2, (2) default at T1, and (3) default at T2. The value of each of these pieces

corresponds to

1. EQ0

[e−rT21V1>K11V2>K2K2

];

2. EQ0

[e−rT21V1<K1(1− L)V1

K2

K1+K2+K3

]; and

3. EQ0

[e−rT21V1>K11V2<K2(1− L)V2

K2

K2+K3

].

40

After some algebra, it can be shown that the three pieces are

1. K2e−rT2√

2π

∫ ∞−d2

exp(−1

2z21)N(d̃2

)dz1;

2. V0(1− L) K2

K1+K2+K3N(−d1); and

3. K2(1−L)e−rT1

(K2+K3)√2π

∫ ∞−d2

exp(−1

2z21)

(V1 −K1)N(−d̃1

)dz1.

B.3 Hedge Ratios

In principle, numerical partial derivatives can be calculated once we have bond prices.

However, in our implementation of the extended Merton model, we find that numerical

partial derivatives calculated for prices that are determined though numerical integration

can become quite imprecise. Thus, we apply Leibniz’s rule to first calculate partial deriva-

tives as functions of integrals (much like the bond prices themselves) before numerically

integrating. In particular, the three pieces of the partial derivative of the bond price (from

B.2) with respect to firm value (∂B/∂V ) are

1. K2e−rT2

∫ ∞−d2

1√2πexp

(−1

2z21)n(d̃2

)V1V0

1(V1−K1)σv

√T2−T1

dz1;

2. (1−L)K2N(−d1)K1+K2+K3

− (1−L)K2n(−d1)(K1+K2+K3)σv

√T1

; and

3. (1−L)K2e−rT1

K2+K3

[∫ ∞−d2

1√2πexp

(−1

2z21)V1V0N(−d̃1

)dz1 −

∫ ∞−d2

1√2πexp

(−1

2z21) (V1−K1)n(−d̃1)V1σv√T2−T1(V1−K1)V0

dz1

].

The hedge ratio can then be calculated by taking ∂B∂V

VB

and dividing by the numerically

calculated ∂E∂V

VE

.

41

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