Comovement of Corporate Bonds and Equities
Jack Bao and Kewei Hou∗
January 29, 2014
Abstract
We study heterogeneity in the comovement of corporate bonds and equities, both at the
bond level and at the firm level. To formalize empirical predictions, we use an extended
Merton model to illustrate that corporate bonds that mature late relative to the rest of
the bonds in their issuers’ maturity structure should have stronger comovement with
equities. In contrast, endogenous default models suggest that a bond’s position in its
issuer’s maturity structure has little relation with the strength of the comovement.
In the data, we find that bonds that are later in their issuers’ maturity structure
comove more strongly with equities, consistent with the prediction of the extended
Merton model. In addition, we find that the comovement between bonds and equities
is stronger for firms with higher credit risk as proxied by higher book-to-market ratios
and lower distance-to-default even after controlling for ratings. Our results highlight
the important effects of bond and firm level characteristics on the relative returns of
corporate bonds and equities.
∗
Bao is at the Fisher College of Business, Ohio State University, bao [email protected]. Hou is at the
Fisher College of Business, Ohio State University and CAFR, [email protected]. We have benefited from
helpful comments from and discussions with Manuel Adelino, Markus Brunnermeier, Sergey Chernenko,
Jean Helwege, Jing-zhi Huang, Rainer Jankowitsch, Martin Oehmke, Jun Pan, Eric Powers, Ilya Strebulaev,
Rene Stulz, Yuhang Xing, and Wei Xiong and seminar participants at Baruch, Boston University, the 2013
EFA Meetings, the Federal Reserve Board of Governors, Inquire UK, Ohio State, Rutgers, South Carolina,
the South Carolina Fixed Income Conference, and Texas A&M. All remaining errors are our 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 has important 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 In this paper we empirically study the
hedge ratios at both the bond level and the firm level to highlight the heterogeneity in the
corporate bond-equity comovement.
We first investigate differences in comovement at the bond level. Most firms issue bonds
at different times with different maturities, providing a rich maturity structure. A bond that
matures after most of the other bonds issued by the same firm is potentially de facto junior.
This arises from the fact that a firm in financial trouble may remain solvent long enough to
repay bonds that are due early in its maturity structure, but not bonds that are due later.
The effect is that bond issues that mature later are more sensitive to underlying firm value.
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 late in their firms’ maturity
structure have higher hedge ratios even if they have the same explicit seniority as other
bonds issued by the same firms.3
1
See Kwan (1996).
Schaefer 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 ratios are equal.
Huang and Shi (2013) confirm this result and reconcile it with the Collin-Dufresne, Goldstein, and Martin
(2001) finding that most of the variation in credit spread changes cannot be explained by the Merton model.
3
It is important to note that a de facto junior bond is not explicitly junior. While it is straightforward
to extend the Merton model to allow for explicitly junior and senior bonds, it is difficult to empirically
examine the effect of explicit seniority on hedge ratios because most corporate bond issues in the United
States are senior unsecured. Furthermore, though bank debt is normally senior to corporate bonds, firms
with significant amounts of corporate bond outstanding often do not have significant bank debt. Rauh and
Sufi (2010) report that more than 50% of the firms in their sample with significant amounts of corporate
bonds have less than 10% of their debt in bank debt. Our sample, which is at the bond-month level, is even
more extreme, with a median of 2.53% of debt in bank debt.
2
2
As an alternative to the extended Merton model, we consider the endogenous default
model of Geske (1977), in which equityholders 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 equityholders 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. Equityholders may choose to default on early debt issues
even if current firm value is much larger than the face value of debt due early because they
take into account the large amount of debt due in the future that will have to be paid
before they receive a payout as residual claimants. In fact, we illustrate that for reasonable
parameter values, the Geske (1977) model predicts that a bond that matures relatively late
in its firm’s maturity structure could have a lower hedge ratio, constrasting the extended
Merton model.
To empirically test the predictions of the two models, we regress corporate bond returns
on equity and Treasury bond returns to calculate empirical hedge ratios. We measure a
bond’s place in its firm’s maturity structure using the proportion of debt due prior to the
bond and interact it with equity returns to determine its effect on 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 de facto seniority affecting corporate bond-equity comovement.
Next, we study firm-level heterogeneity in the comovement between bonds and equities.
Using a simple Merton model, it can be illustrated that firms with greater credit risk have
higher hedge ratios. To the extent that credit ratings capture credit risk well, this should be
reflected in higher average hedge ratios for poorer credit ratings. Furthermore, if the market
is able to distinguish between the credit quality of bonds issued by firms with similar credit
3
ratings, we should see significant within-rating variation in hedge ratios. However, the recent
subprime mortgage crisis has cast doubt on the quality of credit ratings and suggested that
market participants have relied too heavily on ratings. 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 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.
Our paper is primarily related to three strands of literature. At the most fundamental
level, our paper is related to the literature that tries to empirically evaluate the pricing
of corporate bonds through the lens of structural models of default. Huang and Huang
(2003) is the seminal paper, showing that a large part of credit spreads cannot be explained
by structural models of default. In contrast, Schaefer and Strebulaev (2008) argue that
even a first generation model like the Merton model can reasonably characterize the average
magnitude of hedge ratios. Bao and Pan (2013) find that the corporate bond market is
excessively volatile compared to equities when a Merton model is used, but attribute this
largely to illiquidity in the corporate bond market.
Our focus on the maturity structure of bonds is most closely related to the theoretical
work of Brunnermeier and Oehmke (2013) and Chen, Xu, and Yang (2012) who construct
equilibrium models of maturity choice. The primary contribution in Brunnermeier and
Oehmke (2013) is in using the intuition of de facto seniority and showing that in equilib-
4
rium, financing is inefficiently short-term. Chen, Xu, and Yang (2012) instead consider the
interaction between rollover and systematic risk, finding that firms with more systematic
volatility will choose longer bond maturities. In our paper, we do not aim to explain why
a firm has chosen a particular maturity structure, instead taking the outcome of maturity
choice as given and focusing on the empirical effects of maturity structure outcomes on the
observed comovement of corporate bonds and equities. Importantly, we note that our results
are robust to controlling for the average firm-level maturity (the outcome of the endogenous
choices discussed in other papers) and also the maturity of the bond. Furthermore, corporate bond issuers will have both de facto senior and de facto junior bonds regardless of the
equilibrium choice to largely issue short-term or long-term debt as de facto priority is defined
relative to bonds from the same issuer, not based on the absolute maturity of a bond.
Finally, our paper is related to the recent literature 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 formalize empirical predictions for hedge ratios by using structural models of default. In Section 3, we discuss the data,
summary statistics, and the construction of some key variables. In Section 4, we discuss our
empirical results. We conclude in Section 5.
2
Credit Risk and Hedge Ratios
The theoretical relation between corporate bond and equity returns is formalized in structural
models of default, the earliest of which is the Merton (1974) model. Structural models posit
5
processes for firm value and other state variables and all securities that are claims on the firm
can be priced from these processes. This ties corporate bonds and equities together through
their sensitivities to state variables and provides important intuition as to why their returns
should be related. Though Huang and Huang (2003) find that structural models of default,
when calibrated using historical default rates and the equity premium, 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
1 2
d ln Vt = r − σv dt + σv dWtQ .
2
(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
the firm. Its value is
B = V (1 − N (d1 )) + Ke−rT N (d2 ),
V
√
ln K
+ r + 21 σv2 T
√
where d1 =
, d2 = d1 − σv T .
σv T
4
(2)
Note that many structural models including the Merton model do not take a stand as to what is the
correct pricing kernel. Instead, the models are based on no arbitrage and the consistent pricing of different
securities.
6
As illustrated by Schaefer and Strebulaev (2008), the relative returns of the bond and equity
(the hedge ratio) under the Merton model is
∂ ln B
=
hE ≡
∂ ln E
1
V
−1
−1 .
N (d1 )
B
(3)
The Merton model formalizes the important insight that corporate bonds and equities
are linked through their exposures to the underlying firm value.5 Furthermore, 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. To see this, in
Panel A of Table 1, we report 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. For a firm with a market leverage of 10% and
asset volatility of 20%, the model hedge ratio is 0.0018%. That is, a 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 risk-free bond and its returns are essentially unrelated to the returns
of 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
5
Note that an important source of variation in bond prices, stochastic interest rates, is not modeled here
as the tie between bonds and equities through interest rates is indirect and less important than the tie
through firm value. See Shimko, Tejima, and van Deventer (1993) for an extension of the Merton model to
stochastic interest rates and Schaefer and Strebulaev (2008) and Bao and Pan (2013) for applications of this
model.
6
The maturity T is set to seven years to be in line with the average remaining maturity of outstanding
U.S. corporate bonds.
7
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 structure.7
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.8 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 a risk-free bond; the junior bond, which bears the first loss, is
riskier and has a higher hedge ratio than the senior bond.9
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
7
For example, see Black and Cox (1976), Leland (1994), Longstaff and Schwartz (1995), Leland and Toft
(1996), Collin-Dufresne and Goldstein (2001), Duffie and Lando (2001), Chen, Collin-Dufresne, and Golstein
(2009), and Collin-Dufresne, Goldstein, and Helwege (2010), among others.
8
See Black and Cox (1976) and Lando (2004). We also provide further discussion in Appendix A.
9
However, as discussed in the introduction of this paper, this is difficult to examine empirically due to
the paucity of significant explicit seniority structure among corporate bonds.
8
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, holding all else constant, 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 intermediate monitoring points at each bond maturity
E = E0Q e−rT3 1V1 >K1 1V2 >K2 1V3 >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
−rT2
K2
Q
−rT1
B=
e
1V1 >K1 1V2 >K2 K2 + E0 e 1V1 <K1 (1 − L)V1
K1 + K2 + K3
K2
+ E0Q e−rT2 1V1 >K1 1V2 <K2 (1 − L)V2
,
K2 + K3
E0Q
(5)
where the three terms represent solvency at T2 , default at T1 , and default at T2 , respectively.
B
The hedge ratio ( ∂∂ ln
) can then be calculated using numerical procedures. See Appendix B
ln E
for further details.
10
We do not formally model the decision to rollover debt, but the states where Vi is barely larger than
Ki are the states where the rollover of debt would be particularly expensive, if not impossible. These are
also the states that are primarily responsible for driving the wedge between bond issues that creates de facto
seniority. See He and Xiong (2012) and He and Milbradt (2013) for recent papers that examine the issue of
rollover risk.
11
Our set-up essentially maps all bonds maturing prior to T2 into a single bond that matures at T1 and
all bonds maturing after T2 into a single bond that matures at T3 .
9
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 cost 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 the bond is de facto junior (a larger proportion of the firm’s debt is due prior to the
bond) than when it is de facto senior, especially for 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, bonds that are 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
10
will be particularly risky, 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. For the case of 70% market leverage
and 20% asset volatility discussed above, the Merton model hedge ratio reported in Table
1 is 14.39%, which is significantly higher 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.
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, as in most structural
models of default, 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
12
Masulis and Korwar (1986) find that in a sample of 1,406 announcements of stock offerings, 372 offerings are planned exclusively to refund outstanding debt. This suggests that there is a realistic role for
equityholders in paying off debt.
13
See also Leland (1994) and Leland and Toft (1996) for models that incorporate optimal decisions by
equityholders.
11
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.
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
as 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 equityholders
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
12
decisions to make. The value of equity is now
E2 = V2 N (d1 ) − K3 e−r(T3 −T2 ) N (d2 ),
V2
1 2
+
r
+
(T3 − T2 )
ln K
σ
p
v
2
3
√
d1 =
, d2 = d1 − σv T3 − T2 .
σv T3 − T2
(6)
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. Because
equityholders take into account the full maturity structure of the firm’s debt, even bonds that
mature relatively early can have significant exposure to underlying firm value. Intuitively,
even if the debt due early has a low face value, equityholders know that they will only
receive a non-zero payoff in the future if the firm value is high enough to cover all face value
payments in the future. Having large face values due in the future would be equivalent to
buying an out-of-the-money call option.
As noted by Eom, Helwege, and Huang (2004), the Geske formula is not straightforward
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, 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 scenarios where (1) most of the debt is due after the intermediate bond and (2) most of the debt is
13
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 consequently 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.14
For example, consider again the firm with a 70% market leverage and a 20% asset volatility,
the Geske model produces 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) which shows that the difference in hedge ratios is close to
0 for the Geske model whereas the difference in hedge ratios is economically significant for
the extended Merton model.
2.4
Other Measures of Credit Risk
In the simple Merton model, credit risk is increasing in both leverage and asset volatility.
As illustrated earlier in Table 1, hedge ratios also increase with credit risk. The intuition
follows from the fact that as credit risk increases, the value of bonds become more sensitive
14
The slightly lower hedge ratios for the case where the bond is relatively late in the maturity structure is
attributable to the fact that we model bonds as zero coupon bonds and the recovery as being in proportion
to the face value. Thus, bonds late in the maturity structure receive a slightly disproportionate recovery
relative to market values.
14
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 academic
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 fundamentals (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.
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)
15
to measure how far a firm is from its default boundary.15
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, 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.16 CRSP is used to eliminate bonds issued by
15
KMV, which was acquired by Moody’s in 2002, uses distance-to-default to assess expected default
frequency (EDF). See Crosbie and Bohn (2003). Though Moody’s puts out both credit ratings and EDFs,
EDFs are not used explicitly in assigning ratings. Moody’s publishes guides for their ratings methodologies
by industry. The four broad criteria that are typically used are scale and diversification, profitability, leverage
and coverage, and financial policy. If a firm has explicit seniority structure, Moody’s will then notch the
bonds. Secured bonds receive a one notch upgrade from senior unsecured bonds and subordinated bonds
receive a one notch downgrade.
16
Bonds where the only provision is a make whole call provision, which constitutes a significant portion of
16
financial firms.17
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).18
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 monthend to month-end prices. 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 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
21
25
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 severe
illiquidity of the corporate bond market. This illiquidity creates two potential problems in
non-financial bonds, are kept. Make whole calls involve redemption at the maximum of par and the present
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 spread has 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 extremely small.
17
Though some corporate bond studies, primarily those on corporate bond illiquidity, retain financials,
studies on structural models of default typically drop financials due to different implications of leverage for
financials.
18
We apply standard filters for corrected and canceled trades and eliminate obvious errors.
17
our analysis. First, by using transaction prices, returns can be non-zero simply because a
trade at bid follows a trade at ask or vice versa.19 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 introduces additional noise in our measurement of bond returns, making
it more difficult to find statistically significant results. However, it should not bias point
estimates as long as bond returns are used as a regressor.20
A second potential concern is that illiquidity is priced in the corporate bond market. The
past literature has found a significant relation between yield spreads and different proxies
for illiquidity.21 Thus, we might expect that the average level of corporate bond returns is
positively related to illiquidity and that some of the variation in bond returns is explained
by changes in illiquidity. However, there does not appear to be a clear ex-ante theoretical
reason for illiquidity to affect the hedge ratio. Therefore, we do not make an adjustment for
illiquidity in the results reported in the paper. Nevertheless, we also re-run our main tests
to allow for both the level of returns and hedge ratios to vary with bond illiquidity and find
that our primary conclusions are unchanged.
To calculate Treasury returns, we use yields from the U.S. Department of Treasury’s
Constant Maturity Treasury (CMT) series. The provided yields are par yields for on-therun (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.
19
One way to address this issue would be to average prices over the last few days of the month to obtain
prices where the bid-ask spread is (partially) washed out by averaging over a number of trades at both bid
and ask. However, this creates a mismatch between the timing of corporate bond returns and that of equity
and Treasury returns as corporate bond prices would be an average over multiple days.
20
Indeed, both Kwan (1996) and Schaefer and Strebulaev (2008) regress bond returns on equity returns
rather than equity returns on bond returns to avoid the attenuation bias arising from imprecisely measured
regressors.
21
See Chen, Lesmond, and Wei (2007), Bao, Pan, and Wang (2011), and Dick-Nielsen, Feldhutter, and
Lando (2012), among others.
18
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 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. Note
that this includes all debt in Mergent FISD, not just bonds traded in TRACE. 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 expected
returns.22 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.
22
As shown in Hou and van Dijk (2012), conclusions can vastly differ when ex-ante expected returns are
used rather than ex-post realized returns in asset pricing tests.
19
3.3
Other Default Proxies and Firm Summary Statistics
We construct the book-to-market ratio (B/M) in a similar manner to Asness and Frazzini
(2011), who argue that it is important to use more timely stock price data relative to book
data in order to better forecast future book-to-market ratio.23 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 using 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 at 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 2010.24
Table 5 shows that firms with poorer ratings tend to have higher B/M ratios. 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.
Following Campbell, Hilscher, and Szilagyi (2008), we calculate the distance-to-default
as
DD =
ln
V
K
+ 0.06 + rf − 12 σv2
.
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,
23
This is particularly critical for our purposes as one of the drawbacks to using credit ratings in credit
assessment is that they are updated infrequently.
24
Directly using market equity from CRSP for May 2010 introduces errors if the firm had significant
issuance or repurchases between December 2009 and May 2010 which would be reflected in the market
equity for May 2010, but not in the book equity for December 2009.
20
and Lundstedt (2004) and Campbell, Hilscher, and Szilagyi (2008), we solve for V and σv
by simultaneously solving the Merton model equation for equity value and equity volatility
E = V N (d1 ) − Ke−rT N (d2 ),
(9)
V
σE = N (d1 ) σv .
E
(10)
and
Table 5 reports the average distance-to-default estimates, which 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 longterm 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 volatilities 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-leverages 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.
21
4
Empirical Results
4.1
Empirical Specification
To empirically examine the relation between corporate bond returns and equity returns, we
run the following panel regression rating-by-rating
rD = β0 + β1 rE × Z + βE rE + βT rT + β2 rE × T + β3 rE × T 2 + β4 Z + β5 T + β6 T 2 + 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 the 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 in the regression. Under standard structural models of default,
the hedge ratio becomes larger as the time-to-maturity increases.25 This suggests that we
should allow the empirical relation between corporate bond and equity returns to vary with
the maturity of a bond. To determine the order of maturity that needs to be included in
the regression, we appeal to the relation between Merton model-implied hedge ratios and
maturity. For asset volatility from 10% to 50% and market leverage from 10% to 70%, we
calculate Merton model-implied hedge ratios and for each asset volatility-market leverage
combination and each integer time-to-maturity from 1 to 30 years. Each asset volatility25
In a recent paper, Chen, Xu, and Yang (2012) find that firms with more systematic volatility choose
longer bond maturity. We find that our results are unchanged when controlling for the average bond maturity
of a firm. See Appendix C.1 for more details.
22
market leverage combination represents a different level of creditworthiness and for each
combination, 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
combination, 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.26
Prior to estimating our main specification (11), we first run the regression
rD = β0 + βE rE + 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. More importantly, this relation strengthens steadily as ratings become
poorer, with the empirical hedge ratio increasing from 0.0489 (t-stat = 9.70) for Aaa & Aa
bonds to 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
economically and statistically significant 0.6632 (t-stat = 15.62) for Aaa & Aa bonds, but
falls to 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 value.
Instead, they are very sensitive to interest rates, which are also the main driver of Treasury
26
Adding a cubic term for maturity has little effect on our results.
23
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. These results
are consistent with both Schaefer and Strebulaev (2008) and Huang and Shi (2013).
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, the extended Merton model demonstrates that a bond
that matures after most of the other bonds issued by a firm is de facto junior and 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 highly statistically
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.27 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
27
For robustness, we re-run the regressions and include the Amihud (2002) measure and the Amihud
measure interacted with equity returns as additional controls. This allows for both the level of returns and
the hedge ratio to be related to illiquidity. We find that the coefficient on rE × prop due prior variable are
now 0.0675, 0.0565, 0.0994, 0.1419, and 0.0038 for the five ratings classes, with similar levels of statistical
significance as before. Results are similar if we use the IRC measure from Feldhutter (2012) to measure
illiquidity. See Appendix C.2 fore more details.
24
returns on corporate bond returns is 0.0598.28 Increasing the proportion 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 0.0636.29 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 are 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 strong evidence that bonds issued by firms with poorer credit ratings have higher hedge
ratios than bonds issued by firms with better credit ratings. 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 as credit risk30 is more granular than the ratings
provided by credit ratings agencies, we should see higher hedge ratios for the particular set
28
The marginal effect is calculated as βE + β1 × Z + β2 × T + β3 × T 2 = 0.0355 + 0.0908(0.1) + 0.00226(7) −
0.000013(72 ) = 0.0598.
29
To 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 0.0124 to 0.0722.
30
In this context, risk is not necessarily systematic default risk and can include firm-specific risk.
25
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.31 Table 8 shows that the effect of B/M on the relation
between corporate bond and equity returns 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 is an indicator of higher default
likelihood, this suggests that even after controlling for ratings, firms that are perceived to
have greater default risk have higher hedge ratios.32
To gauge the economic importance of the effect of B/M on the corporate bond-equity
relation, 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 this bond’s 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 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.
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. Our analysis of
distance-to-default is similar to that using B/M, and as with B/M, we normalize the distance31
We normalize B/M each period by subtracting the cross-sectional mean and dividing by the crosssectional standard deviation. Unlike the proportion of debt due prior, the distribution of B/M can vary
considerably over time with macroeconomic conditions. To avoid our results being driven solely by macroeconomic factors, we normalize B/M.
32
Adding more granular controls for ratings (e.g. adding rE × 1Baa1 and rE × 1Baa3 into the Baa regression)
does not change our conclusions.
26
to-default each period across ratings.
Table 9 reports 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 effect is statistically significant except for B
bonds. These results are also economically significant 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 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
empirically examine the comovement between them. Structural models of 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 at the bond level as well
as at the firm level.
Within firm, there is potential heterogeneity across bonds in both the seniority and
maturity structure. While it is clear from the Merton model that junior bonds should have
27
higher hedge ratios than senior bonds, the lack of priority structure in corporate bonds
makes it difficult to test the Merton seniority prediction empirically. Instead, we focus on
the maturity structure of corporate bonds and illustrate that a bond’s position in its firm’s
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 in their 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 their 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-todefault, 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. Overall, our
results suggest that there are interesting differences in the comovement of corporate bonds
and equities when conditioning on both bond- and firm-level characteristics.
28
Tables & Figures
Table 1: Merton Model Hedge Ratios
Panel A: Merton Model Hedge Ratios
σv
0.1
0.15
0.2
0.25
0.3
0.4
0.5
0.1
0 8.9e-07 0.0018 0.0621 0.4412 3.2324 8.4878
0.2 1.0e-07 0.0042 0.1898 1.1643 3.2411 9.6727 17.1089
0.3 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 28.7355
0.5 0.2987 2.8144 7.1155 11.9300 16.6855 25.3360 32.6755
0.6 1.3750 5.6615 10.8029 15.8607 20.5773 28.8424 35.7118
0.7 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-Maturity
σv
0.1
0.15
0.2
0.25
0.3
0.4
0.5
0.1 0.7954 0.9643 0.9980 0.9965 0.9925 0.9928 0.9968
0.2 0.9507 0.9989 0.9944 0.9924 0.9942 0.9983 0.9976
0.3 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 0.9841
0.5 0.9939 0.9966 0.9992 0.9983 0.9951 0.9866 0.9781
0.6 0.9955 0.9993 0.9979 0.9942 0.9899 0.9818 0.9746
0.7 0.9991 0.9978 0.9935 0.9893 0.9856 0.9790 0.9732
B
Panel A of this table reports Merton model-implied hedge ratios, ∂∂ ln
, scaled by 100. The
ln E
Merton hedge ratio follows from equation (3) and is calculated for different levels of market
leverage and asset volatility. In Panel B, we report the R2 from regressions of Merton model
hedge ratios on T and T 2 , where T is the maturity of the bond. For each market leverageasset volatility combination, we calculate model hedge ratios for maturities ranging from 1
year to 30 years. We then estimate the regression for each market leverage-asset volatility
combination.
29
Table 2: Extended Merton Model Hedge Ratios
Market Leverage
0.1
0.1 0.0000
0.0000
0.15
0.0000
0.0000
0.2
0.0000
0.0001
σv
0.25
0.0000
0.0174
0.2 0.0000
0.0000
0.0000
0.0004
0.0000
0.0917
0.0001
0.9799
0.3 0.0000
0.0000
0.0000
0.0679
0.0000
1.3394
0.0028
4.9732
0.4 0.0000
0.0047
0.0000
0.9530
0.0001
5.3511
0.0159
11.6261
0.5 0.0000
0.1952
0.0000
4.3125
0.6 0.0000
1.8473
0.0000
10.6608
0.0026
19.5015
0.0888 0.6138 4.2703 10.7585
26.2566 31.3300 38.3678 43.1563
0.7 0.0000
7.0904
0.0000
18.6201
0.0048
26.5155
0.1226 0.7392 4.5833 11.0669
31.6959 35.2889 40.1597 43.7222
0.0008 0.0467
11.9132 19.2059
0.3
0.0002
0.2339
0.0099
3.3836
0.4
0.0695
2.7979
0.5
1.0461
8.4882
0.5905 3.5871
11.3312 20.1151
0.0696 1.5494
10.0149 20.4587
6.1945
29.3097
0.2121 2.6408 8.3368
17.8715 28.2927 36.0544
0.4153 3.6005
25.2896 34.3312
9.8621
40.6166
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 volatility
combination is the hedge ratio that corresponds to the case where most of a firm’s debt
is due after the intermediate bond (so it is de facto senior). The bottom number is the
hedge ratio that corresponds to the case where most of a firm’s debt is due prior to the
intermediate bond (so it is de facto junior). A firm is assumed to have three zero-coupon
bonds with maturities T1 = 5, T2 = 7, and T3 = 10. The risk-free rate, r, is 3%. For each
market 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 intermediate
maturity bond, and 80% to the long maturity bond. The bottom number corresponds to
80% attributable to the short maturity bond, 10% to the intermediate maturity bond, and
10% to the long maturity bond.
30
Table 3: Geske Model Hedge Ratios
0.1
0.0000
0.0000
0.15
0.0000
0.0000
σv
0.2
0.0007
0.0000
0.2
0.0000
0.0000
0.0000
0.0000
0.1940
0.1068
1.3455
1.0227
3.7954
3.5251
0.3
0.0000
0.0000
0.1476
0.0619
1.9184
1.5626
5.9669
5.3673
11.2312
10.4166
21.9087
20.1260
31.7111
28.9815
0.4
0.0000
0.0000
1.6304
1.1566
6.8316
6.2016
13.1491
12.8181
19.2157
18.6349
30.7761
28.4030
41.2195
34.8844
0.5
0.2455
0.0583
6.3485
5.7181
14.3616
13.9096
21.7644
20.4583
28.5736
25.8839
37.2128
33.8802
46.1086
40.3855
0.6
4.0958
3.3421
14.5166
13.0815
22.8674
21.6618
29.7942
27.8414
34.3876
31.0798
43.2369
38.1453
50.5431
43.2457
0.7 13.2835
12.3024
23.3547
21.5210
30.4281
27.7943
35.6507
31.4527
40.2505
35.6294
46.1002
39.6380
50.1889
43.4748
0.1
Market Leverage
0.25
0.0522
0.0163
0.3
0.3962
0.2500
0.4
3.2189
2.7136
0.5
8.5735
7.9962
12.0600 21.9361
11.2047 19.0319
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 is
the hedge ratio that corresponds to the case where most of a firm’s debt is due after the
intermediate bond. The bottom number is the hedge ratio that corresponds to the case
where most of a firm’s debt is due prior to the intermediate bond. A firm is assumed to
have three zero-coupon bonds with maturities T1 = 5, T2 = 7, and T3 = 10. The risk-free
rate, r, is 3%. For each market 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 intermediate maturity bond, and 80% to the long maturity bond. The
bottom number corresponds to 80% attributable to the short maturity bond, 10% to the
intermediate maturity bond, and 10% to the long maturity bond.
31
Table 4: Bond Summary Statistics
Aaa & Aa
A
Baa
Bonds
1,103 4,024 4,341
Bond-months
25,959 82,708 91,221
Time-to-maturity
mean
6.80
8.28
9.50
med
4.50
4.34
6.19
std
7.65 10.28
9.77
Age
mean
4.45
4.72
4.86
med
3.30
3.74
3.92
std
3.94
3.81
3.82
Proportion due prior mean
0.5003 0.4002 0.3913
med
0.5444 0.3623 0.3608
std
0.2758 0.2795 0.2824
Amount outstanding mean
527
420
465
med
250
296
350
std
860
516
446
Coupon
mean
5.21
5.76
6.38
med
5.00
5.75
6.38
std
1.55
1.37
1.29
Return
mean
0.0045 0.0043 0.0052
med
0.0037 0.0038 0.0045
std
0.0264 0.0282 0.0359
Yield spread
mean
0.0113 0.0137 0.0217
med
0.0082 0.0100 0.0169
std
0.0106 0.0140 0.0177
Ba
2,039
25,220
6.97
3.96
9.00
5.09
4.11
3.78
0.3502
0.2886
0.2777
291
225
336
6.69
6.85
1.39
0.0067
0.0062
0.0506
0.0426
0.0361
0.0374
B
717
11,499
6.82
3.71
8.46
6.32
5.58
3.90
0.3332
0.2441
0.3016
297
200
330
7.16
7.30
1.45
0.0048
0.0064
0.0702
0.0573
0.0391
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 prior
is the proportion of a firm’s face value of debt due prior to a particular bond
and is measured in decimals. Coupon is the coupon rate in %. Return is the
monthly log return of a bond. Yield spread is the difference between the yield
of a bond and a Treasury with similar maturity from the Constant Maturity
Treasury (CMT) series.
32
Table 5: Firm Summary Statistics
Aaa & Aa
A
Baa
Ba
B
Firms
50
225
468
228
116
Firm-months
2,289 10,761 18,200
5,391
2,456
Observations
25,959 82,708 91,221 25,220 11,499
Equity market cap mean
204.96 45.51 23.11
10.32
8.14
med
173.27 34.32 15.73
10.13
5.98
std
121.71 42.89 27.37
7.90
8.46
Equity volatility
mean
0.2686 0.2789 0.3255 0.4001 0.4317
med
0.2035 0.2459 0.2737 0.3460 0.3647
std
0.1614 0.1255 0.1659 0.1849 0.1912
DD leverage
mean
0.3315 0.2319 0.2638 0.4417 0.5213
med
0.3966 0.1553 0.2092 0.3468 0.4756
std
0.2370 0.2043 0.2083 0.3044 0.2522
Pseudo-leverage
mean
0.3973 0.3094 0.3616 0.5376 0.6313
med
0.4816 0.2408 0.3257 0.4969 0.6327
std
0.2583 0.2131 0.2116 0.2829 0.2224
Book-to-market
mean
0.39
0.41
0.61
0.91
0.79
med
0.31
0.35
0.52
0.73
0.70
std
0.22
0.27
0.36
0.62
0.54
Distance-to-default mean
10.24
9.63
8.01
5.87
4.55
med
10.80
9.07
7.60
5.09
4.30
std
5.44
4.37
3.77
3.00
2.24
Equity Return
mean
0.0020 0.0072 0.0059 -0.0018 -0.0090
med
0.0053 0.0121 0.0128 0.0080 0.0029
std
0.0837 0.0805 0.0994 0.1362 0.1665
This table reports firm characteristics for our sample. Equity market cap is the
equity market capitalization in $bn. Equity volatility is the volatility of a firm’s
equity using daily log returns over the past year, annualized. DD Leverage is the
face value of debt divided by the sum of the face value of debt and the market
value of equity. Following Campbell, Hilscher, and Szilagyi (2008), the face value
of debt is defined as Debt in Current Liabilities plus one-half times Long-Term
Debt. Pseudo-leverage is defined similarly as DD Leverage, but uses the sum of
Debt in Current Liabilities and Long-Term Debt as the face value of debt. Bookto-market is the book value of equity divided by the market value of equity as in
Asness and Frazzini (2011). Distance-to-default is calculated following Campbell,
Hilscher, and Szilagyi (2008). Equity Return is the monthly log equity return.
33
Table 6: Co-movement between Corporate Bonds
Aaa & Aa
A
Baa
Ba
rE
0.0489 0.0470 0.0886 0.1161
[9.70] [2.99] [4.73]
[7.18]
rT
0.6632 0.5779 0.4022 -0.0249
[15.62] [8.80] [4.94] [-0.24]
Constant
0.0025 0.0024 0.0034 0.0071
[4.24] [2.45] [2.19]
[4.33]
2
R
0.2386 0.2128 0.1203 0.1335
Obs
25,959 82,708 91,221 25,220
and Equities
B
0.1260
[8.87]
-0.1488
[-1.32]
0.0066
[3.50]
0.1695
11,499
This table reports a regression of rD = β0 + βE rE + βT rT . rD , rE , and
rT are log returns (in decimals) for corporate bonds, equities, and Treasuries, respectively. t-statistics are reported in brackets using standard
errors that are clustered by firm and month.
34
Table 7: Co-movement
Aaa & Aa
rE × prop due prior
0.0854
[2.45]
rE
-0.0044
[-1.27]
rT
0.6683
[16.07]
rE × T
0.0019
[1.98]
2
rE × T
-0.0000
[-1.32]
prop due prior
0.0020
[1.29]
T
-0.0000
[-0.30]
2
T
0.0000
[0.66]
Constant
0.0015
[7.16]
R2
0.2471
Obs
25,959
and Maturity Structure
A
Baa
Ba
0.0468 0.0908 0.1621
[2.95]
[2.60]
[4.40]
0.0120 0.0355 0.0786
[1.99]
[4.92]
[7.49]
0.5842 0.4164 -0.0195
[8.78]
[5.12] [-0.18]
0.0029 0.0023 -0.0034
[1.34]
[1.02] [-1.08]
-0.0000 -0.0000 0.0000
[-1.49] [-0.60]
[0.78]
-0.0001 -0.0006 0.0029
[-0.17] [-0.40]
[1.27]
0.0000 0.0001 -0.0002
[0.18]
[0.44] [-1.00]
0.0000 -0.0000 0.0000
[0.06] [-0.68]
[1.08]
0.0022 0.0031 0.0070
[6.25]
[4.85]
[8.22]
0.2192 0.1336 0.1448
82,708 91,221 25,220
B
0.0348
[1.03]
0.0956
[8.04]
-0.1154
[-1.07]
0.0047
[2.64]
-0.0001
[-3.60]
0.0000
[0.01]
-0.0000
[-0.15]
0.0000
[1.90]
0.0063
[4.27]
0.1776
11,499
This table reports a regression of rD = β0 + β1 rE × prop due prior + βE rE +
βT rT + β2 rE × T + β3 rE × T 2 + β4 prop due prior + β5 T + β6 T 2 . prop due prior
is the proportion of a firm’s face value of debt due prior to a particular bond
and is measured in decimals. 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 clustered by firm and month.
35
Table 8: Co-movement and Book-to-Market
Aaa & 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]
T2
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]
2
R
0.2518 0.2208 0.1385 0.1449 0.1830
Obs
25,959 81,720 89,896 24,456
8,955
This table reports a regression of rD = β0 + β1 rE × B/M + βE rE +
βT rT + β2 rE × T + β3 rE × T 2 + β4 B/M + β5 T + β6 T 2 . B/M is the bookto-market ratio of a firm, winsorized at 1% of each tail and normalized
by 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 clustered by firm and month.
36
Table 9: Co-movement and Distance-to-Default
Aaa & 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]
T2
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]
2
R
0.2477 0.2200 0.1356 0.1424 0.1784
Obs
25,959 82,457 90,361 25,139 11,300
This table reports a regression of rD = β0 +β1 rE ×DD +βE rE +βT rT +
β2 rE ×T +β3 rE ×T 2 +β4 DD+β5 T +β6 T 2 . DD is the distance-to-default
of a firm, winsorized at 1% of each tail and normalized by the crosssectional 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 clustered by firm
and month.
37
38
Figure 1: Hedge Ratios for the Merton, Extended Merton, and Geske models. For panels (b) and (c), the cyan surfaces represent
the case where a bond is late in its firm’s maturity structure and the red surfaces with point markers represent the case where
a bond is early in its firm’s maturity structure.
Appendix
A
Merton Model with Seniority
As discussed by Black and Cox (1976) and 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 ) − KS e−rT N (d2,s ) − V N (d1 ) + Ke−rT N (d2 ),
(13)
BS = V − V N (d1,S ) + KS e−rT N (d2,S ),
ln KVS + r + 12 σv2 T
√
√
where d1,S =
, d2,S = d1,S − σv T ,
σ T
v
ln KS V+KJ + r + 12 σv2 T
√
√
and d1 =
, d2 = d1 − σv T .
σv T
It can be shown that
∂ ln BJ
∂ ln BS
V
−
=
[BS (N (d1,S ) − N (d1 )) − BJ (1 − N (d1,S ))] > 0.
∂ ln V
∂ ln V
BS × BJ
B
(14)
Merton Model Extension to Include Maturity Structure
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 Vdef ault , the proportion payable to bondholders is (1−L)Vdef ault .
39
B.1
Equity Value
E = E0Q e−rT3 1V1 >K1 1V2 >K2 1V3 >K3 (V3 − K3 )
The firm value process is lognormal
p
1 2
V1 = V0 exp r − σv T1 + σv T1 w1 ,
2
(15)
where w1 ∼ N (0, 1).
Similarly,
p
1 2
V2 = (V1 − K1 ) exp r − σv (T2 − T1 ) + σv T2 − T1 w2
2
if V1 > K1
and
p
1 2
V3 = (V2 − K2 ) exp r − σv (T3 − T2 ) + σv T3 − T2 w3
2
if V2 > K2
After some algebra, it can be shown that
Z Z
e−rT2 ∞ ∞
1 2 1 2
E=
exp − z2 − z1 (V2 − K2 )N d˜˜1 dz2 dz1
2π −d2 −d˜2
2
2
−rT3 Z ∞ Z ∞
1 2 1 2
−e
− K3
exp − z2 − z1 N d˜˜2 dz2 dz1 ,
2π
2
2
˜
2
−d2 −d
2
+ r − 21 σv2 (T3 − T2 )
ln V2K−K
p
3
˜
√
where d˜2 =
, d˜˜1 = d˜˜2 + σv T3 − T2
σ T −T
v 3 2
1
ln V1K−K
+ r − 12 σv2 (T2 − T1 )
2
˜
√
and d2 =
.
σv T2 − T1
40
(16)
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. E0Q e−rT2 1V1 >K1 1V2 >K2 K2 ;
2.
E0Q
h
−rT2
e
1V1 <K1 (1 −
K2
L)V1 K1 +K
2 +K3
i
; and
h
i
2
.
3. E0Q e−rT2 1V1 >K1 1V2 <K2 (1 − L)V2 K2K+K
3
After some algebra, it can be shown that the three pieces are
Z ∞
−rT2
e√
1. K2 2π
exp − 12 z12 N d˜2 dz1 ;
−d2
K2
2. V0 (1 − L) K1 +K
N (−d1 ); and
2 +K3
Z ∞
K2 (1−L)e−rT1
1 2
˜
√
3. (K +K ) 2π
exp − 2 z1 (V1 − K1 ) N −d1 dz1 .
2
B.3
3
−d2
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 derivatives 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
Z ∞
V
1 √
−rT2
˜2 1
√1 exp − 1 z 2 n d
dz1 ;
1. K2 e
2 1
V0 (V1 −K1 )σv T2 −T1
2π
−d2
2.
(1−L)K2 N (−d1 )
K1 +K2 +K3
3.
(1−L)K2 e−rT1
K2 +K3
−
Z
(1−L)K2 n(−d1 )
√ ;
(K1 +K2 +K3 )σv T1
and
∞
−d2
√1 exp
2π
− 21 z12
V
1
V0
Z
˜
N −d1 dz1 −
∞
−d2
41
√1 exp
2π
− 12 z12
(V1 −K1 )n(−d˜1 )V1
√
dz1
σv T2 −T1 (V1 −K1 )V0
.
The hedge ratio can then be calculated by taking
calculated
C
C.1
∂B V
∂V B
and dividing by the numerically
∂E V
.
∂V E
Robustness Checks
Firm-level Maturity
Recent theoretical literature has examined the equilibrium choice of debt maturity by firms.
Brunnermeier and Oehmke (2013) solve a model where the equilibrium is a maturity rate race
and the outcome is an inefficiently large amount of short-term debt. Chen, Xu, and Yang
(2012) solve a model that shows that firms with higher systematic volatility choose longerterm debt and find empirical support for this prediction. We note that these studies are
about aggregate firm-level maturity choice, while our focus is on how the relative maturity
of a bond compared to the rest of the bonds issued by the same issuer. Nevertheless,
we repeat our analysis after controlling for average firm-level maturity in Table A.1. We
find that our de facto seniority results from Table 7 are largely unchanged. In particular,
there are few differences in the economic and statistical significance for the coefficient on
rE × prop due prior.
C.2
Liquidity Measures
Though the structural models of default considered here do not allow for a natural role
for liquidity to affect hedge ratios, we acknowledge that numerous papers have shown that
bond illiquidity affects corporate bond pricing and pricing dynamics. Thus, we control
for the Amihud measure in Table A.2. We find that our baseline results are unchanged.
Interestingly, we also find that hedge ratios are larger for more illiquid bonds. This can
potentially be explained by the He and Xiong (2012) and He and Milbradt (2013) findings
that illiquidity can drive additional credit risk. This additional credit risk drives additional
exposure to the underlying firm value, increasing hedge ratios.
42
Table A.1: Firm-level Maturity
Aaa & Aa
A
Baa
rE × prop due prior
0.0871 0.0540 0.1148
[2.02]
[2.14]
[2.99]
rE × avg firm mat
0.0002 0.0006 0.0019
[0.15]
[0.64]
[1.83]
rE
-0.0066 0.0055 0.0141
[-0.49]
[0.61]
[1.08]
rT
0.6681 0.5842 0.4170
[16.12]
[8.78]
[5.13]
rE × T
0.0018 0.0025 0.0011
[2.11]
[1.05]
[0.46]
2
rE × T
-0.0000 -0.0000 -0.0000
[-1.31] [-1.26] [-0.16]
2
R
0.2472 0.2193 0.1339
Obs
25,959 82,708 91,221
Ba
B
0.1791 0.0599
[3.63]
[2.74]
0.0024 0.0039
[0.81]
[1.96]
0.0565 0.0603
[2.14]
[2.41]
-0.0175 -0.1136
[-0.17] [-1.05]
-0.0045 0.0034
[-1.15]
[3.58]
0.0000 -0.0001
[0.89] [-4.31]
0.1456 0.1791
25,220 11,499
This table reports regressions of corporate bond returns on equity returns and other controls.
avg firm mat is the average remaining time-to-maturity for a bond issuer, weighted by
amount outstanding. Other variables are as defined in Table 7. Additional unreported
controls are prop due prior, avg firm mat, T , T 2 , and a constant term. t-statistics are
reported in brackets using standard errors that are clustered by firm and month.
43
Table A.2: Liquidity
Aaa & Aa
A
Baa
Ba
B
rE × prop due prior
0.0630 0.0548 0.0999 0.1438 0.0226
[2.14]
[3.61]
[3.03]
[4.32]
[0.57]
rE × Amihud
0.0175 0.0092 0.0242 0.0308 0.0120
[2.84]
[1.73]
[3.04]
[2.36]
[1.22]
rE
-0.0084 0.0087 0.0266 0.0632 0.0878
[-2.11]
[1.53]
[4.09]
[6.70]
[7.48]
rT
0.6637 0.5797 0.4126 -0.0323 -0.1113
[15.11]
[8.33]
[5.20] [-0.31] [-1.06]
rE × T
0.0027 0.0022 0.0019 -0.0024 0.0058
[2.68]
[1.03]
[0.81] [-0.80]
[2.97]
rE × T 2
-0.0000 -0.0000 -0.0000 0.0000 -0.0001
[-2.01] [-0.93] [-0.42]
[0.36] [-3.94]
2
R
0.2548 0.2261 0.1425 0.1551 0.1855
Obs
23,667 74,068 82,831 23,502 10,758
This table reports regressions of corporate bond returns on equity returns and other controls.
Amihud is the Amihud (2002) measure. Other variables are as defined in Table 7. Additional
unreported controls are prop due prior, Amihud, T , T 2 , and a constant term. t-statistics are
reported in brackets using standard errors that are clustered by firm and month.
44
References
Adelino, M. (2009). Do Investors Rely Only on Ratings? The Case of Mortgage-Backed
Securities. Working Paper, Duke.
Amihud, Y. (2002). Illiquidity and stock returns: cross-section and time-series effects.
Journal of Financial Markets 5, 31–56.
Asness, C. and A. Frazzini (2011). The Devil in HML’s Details. Working Paper, AQR
Capital Management.
Bao, J. and J. Pan (2013). Bond Illiquidity and Excess Volatility. Review of Financial
Studies 26, 3068–3103.
Bao, J., J. Pan, and J. Wang (2011). The Illiquidity of Corporate Bonds. Journal of
Finance 66, 911–946.
Bharath, S. T. and T. Shumway (2008). Forecasting Default with the Merton Distance to
Default Model. Review of Financial Studies 21, 1339–1369.
Black, F. and J. C. Cox (1976). Valuing Corporate Securities: Some Effects of Bond
Indenture Provisions. Journal of Finance 31, 351–367.
Brunnermeier, M. K. and M. Oehmke (2013). The Maturity Rat Race. Journal of Finance 68, 483–521.
Campbell, J. Y., J. Hilscher, and J. Szilagyi (2008). In Search of Distress Risk. Journal of
Finance 63, 2899–2939.
Chen, H., Y. Xu, and J. Yang (2012). Systematic Risk, Debt Maturity, and the Term
Structure of Credit Spreads. Working Paper, MIT and Bank of Canada.
Chen, L., P. Collin-Dufresne, and R. S. Golstein (2009). On the Relation Between the
Credit Spread Puzzle and the Equity Premium Puzzle. Review of Financial Studies 22,
3367–3409.
Chen, L., D. Lesmond, and J. Wei (2007). Corporate Yield Spreads and Bond Liquidity.
Journal of Finance 62, 119–149.
Collin-Dufresne, P. and R. S. Goldstein (2001). Do Credit Spreads Reflect Stationary
Leverage Ratios? Journal of Finance 56, 1929–1957.
Collin-Dufresne, P., R. S. Goldstein, and J. Helwege (2010). Is Credit Event Risk Priced?
Modeling Contagion via the Updating of Beliefs. Working Paper, Columbia, Minnesota,
and South Carolina.
Collin-Dufresne, P., R. S. Goldstein, and S. Martin (2001). The Determinants of Credit
Spread Changes. Journal of Finance 56, 2177–2207.
Cox, J. C., S. A. Ross, and M. Rubinstein (1979). Option Pricing: A Simplified Approach.
Journal of Financial Economics 7, 229–263.
Crosbie, P. and J. Bohn (2003). Modeling Default Risk. Working Paper, Moody’s KMV.
Daniel, K. and S. Titman (1997). Evidence on the Characteristics of Cross Sectional
Variation in Stock Returns. Journal of Finance 52, 1–33.
45
Dick-Nielsen, J., P. Feldhutter, and D. Lando (2012). Corporate bond liquidity before and
after the onset of the subprime crisis. Journal of Financial Economics 103, 471–492.
Duffie, D. and D. Lando (2001). Term Structures of Credit Spreads with Incomplete
Accounting Information. Econometrica 69, 633–664.
Edwards, A. K., L. E. Harris, and M. S. Piwowar (2007). Corporate Bond Market Transaction Costs and Transparency. Journal of Finance 62, 1421–1451.
Eom, Y. H., J. Helwege, and J. Z. Huang (2004). Structural Models of Corporate Bond
Pricing: An Empirical Analysis. Review of Financial Studies 17, 499–544.
Fama, E. F. and K. R. French (1993). Common risk factors in the returns on stocks and
bonds. Journal of Financial Economics 33, 3–56.
Fama, E. F. and K. R. French (1995). Size and Book-to-Market Factors in Earnings and
Returns. Journal of Finance 50, 131–155.
Fama, E. F. and K. R. French (1996). Multifactor Explanations of Asset Pricing Anomalies.
Journal of Finance 51, 55–84.
Feldhutter, P. (2012). The same bond at different prices: identifying search frictions and
selling pressures. Review of Financial Studies 25, 1155–1206.
Geske, R. (1977). The Valuation of Corporate Liabilities as Compound Options. Journal
of Financial and Quantitative Analysis 12, 541–552.
Haugen, R. A. (1995). The new finance – The case against efficient markets. Prentice-Hall.
He, Z. and K. Milbradt (2013). Endogenous Liquidity and Defaultable Bonds. Econometrica, forthcoming.
He, Z. and W. Xiong (2012). Rollover Risk and Credit Risk. Journal of Finance 67, 391–
429.
Hillegeist, S. A., E. Keating, D. Cram, and K. Lundstedt (2004). Assessing the Probability
of Bankruptcy. Review of Accounting Studies 9, 5–34.
Hou, K. and M. A. van Dijk (2012). Profitability shocks and the size effect in the crosssection of expected stock returns. Working Paper, Ohio State and Erasmus.
Huang, J. Z. (1997). Default Risk, Renegotiation, and the Valuation of Corporate Claims.
Ph. D. thesis, NYU.
Huang, J. Z. and M. Huang (2003). How Much of the Corporate-Treasury Yield Spread is
Due to Credit Risk. Working Paper, Penn State and Stanford.
Huang, J. Z. and Z. Shi (2013). A Revisit to the Equity-Credit Market Integration
Anomaly. Working Paper, Penn State.
Kwan, S. H. (1996). Firm Specific Information and the Correlation Between Individual
Stocks and Bonds. Journal of Financial Economics 40, 63–80.
Lakonishok, J., A. Shleifer, and R. W. Vishny (1994). Contrarian Investment, Extrapolation, and Risk. Journal of Finance 49, 1541–1578.
Lando, D. (2004). Credit Risk Modeling. Princeton University Press.
46
Leland, H. E. (1994). Corporate Debt Value, Bond Covenants, and Optimal Capital Structure. Journal of Finance 49, 157–196.
Leland, H. E. and K. Toft (1996). Optimal Capital Structure, Endogenous Bankruptcy,
and the Term Structure of Credit Spreads. Journal of Finance 51, 987–1019.
Longstaff, F. A. and E. S. Schwartz (1995). A Simple Approach to Valuing Risky Fixed
and Floating Rate Debt. Journal of Finance 50, 789–819.
MacKinlay, A. C. (1995). Multifactor models do not explain deviations from the CAPM.
Journal of Financial Economics 38, 3–28.
Masulis, R. W. and A. N. Korwar (1986). Seasoned Equity Offerings: An Empirical Investigation. Journal of Financial Economics 15, 91–118.
Merton, R. C. (1974). On the Pricing of Corporate Debt: The Risk Structure of Interest
Rates. Journal of Finance 29, 449–470.
Powers, E. A. and S. Tsyplakov (2008). What is the Cost of Financial Flexibility? Theory
and Evidence from Make-whole Call Provisions. Financial Management 37, 485–512.
Rauh, J. D. and A. Sufi (2010). Capital Structure and Debt Structure. Review of Financial
Studies 23, 4242–4280.
Schaefer, S. M. and I. A. Strebulaev (2008). Structural Models of Credit Risk are Useful:
Evidence from Hedge Ratios on Corporate Bonds. Journal of Financial Economics 90,
1–19.
Shimko, D. C., N. Tejima, and D. R. van Deventer (1993). The Pricing of Risky Debt
when Interest Rates are Stochastic. Journal of Fixed Income 3, 58–65.
Vassalou, M. and Y. Xing (2004). Default Risk in Equity Returns. Journal of Finance 59,
831–868.
47