Download Structural Models of Credit Risk are Useful: Evidence

Structural Models of Credit Risk are Useful:
Evidence from Hedge Ratios on Corporate Bonds∗
Stephen M. Schaefer
Ilya A. Strebulaev
London Business School
This version: 10 November 2003
ABSTRACT
It is well known that structural models of credit risk provide poor predictions of bond
prices. We show that they may perform much better as a predictor of debt return
sensitivities to equity. This is important since it gives us an opportunity to identify
much better the reasons for model failure. The main result of this paper is that
even the simplest of the structural models (Merton (1974)) produces hedge ratios
that are in line with those observed empirically. As well as providing insight into
the determinants of corporate bond prices our results are also useful to practitioners
who wish to hedge their positions in corporate debt. The paper also shows that
corporate bond prices are sensitive to some variables – e.g., VIX – in a way that
appears unrelated to credit risk.
Keywords: Credit risk, structural models, hedge ratios, credit spreads
JEL Classification Numbers: G12, G13
∗
We would like to thank Crispin Southgate and Joseph Nehoraj from Merrill Lynch and European Credit Management
for help with the data. We are responsible for all remaining errors. Address for correspondence: Institute of Finance and
Accounting, London Business School, Regent’s Park, London NW1 4SA, UK. E-mail: [email protected].
Structural Models of Credit Risk are Useful:
Evidence from Hedge Ratios on Corporate Bonds
Abstract
It is well known that structural models of credit risk provide poor predictions of bond prices. We show that they may perform much better as
a predictor of debt return sensitivities to equity. This is important since
it gives us an opportunity to identify much better the reasons for model
failure. The main result of this paper is that even the simplest of the
structural models (Merton (1974)) produces hedge ratios that are in line
with those observed empirically. As well as providing insight into the
determinants of corporate bond prices our results are also useful to practitioners who wish to hedge their positions in corporate debt. The paper
also shows that corporate bond prices are sensitive to some variables –
e.g., VIX – in a way that appears unrelated to credit risk.
Keywords: Credit risk, structural models, hedge ratios, credit spreads.
I
Introduction
It is commonly agreed that structural models of credit risk over-value corporate bonds.1
Structural models employ the contingent claims approach to value the put option inherent
in the contract between lenders and equityholders. Contingent claims models are, of
course, widely used in practice and are seen as one of the major successes of financial
theory. The failure of such models to explain satisfactorily actual corporate debt prices
and spreads is therefore surprising. The poor performance of these models in this area
has been recognized for many years but their failure continues to surprise.
This paper makes two simple but important points. First, while structural models
provide a poor prediction of prices and returns, they may perform much better as a
predictor of the sensitivity – or hedge ratio – of debt to equity. This is important because
hedge ratios determine the composition of the replicating portfolio which, according to
the theory, determines the price. Thus, if we find that a model provides a good prediction
of hedge ratios but a poor prediction of the price, we are better able to identify the reasons
for model failure. In fact, we find that even the simplest structural model (Merton (1974))
predicts hedge ratios that are in line with those observed empirically. This leads us to
reconsider the possible explanations for the failure of structural models to explain better
the level of prices and yields.
At present, there are two main explanations for this failure. First, that structural
models fail to predict accurately the probability of default and/or the recovery rate.
This explanation has weight because all current models violate at least some known
facts about capital structure and/or the circumstances of corporate default. Consequently, the development of much of the theory has been in pursuit of improved ways to
model credit events. As a result, we possess an arsenal of models that include stochastic
default boundaries, dynamic capital structure and opportunistic behavior on the part
1
For early empirical investigation of the Merton model see Jones, Mason, and Rosenfeld (1984). For
a more recent analysis see Eom, Helwege and Huang (2003) who study the empirical performance of a
number of structural models.
2
of claimholders (e.g., Anderson and Sundaresan (1996), Collin-Dufresne and Goldstein
(2001)). Empirical tests have showed, however, that these modifications do not substantially improve the ability of the models to explain the level of corporate bond prices
(Eom, Helwege and Huang (2002), Huang and Huang (20002)).
A second explanation is that the pricing of corporate bonds is influenced by factors
that are not related to credit risk, and are therefore not included in structural models.
It could even be the case that structural models account very well for the credit risk
component of bond prices and returns while, at the same time, credit risk is actually
responsible for only a part, perhaps not even a very large part, of spreads and returns.
Some recent results tend to support this view. Elton, Gruber, Agrawal, and Mann (2001)
find that differences in taxation account for about a third of credit spreads. Huang and
Huang (2002) estimate that credit risk accounts only for a small fraction of the observed
credit spread. Collin-Dufresne, Goldstein, and Martin (2001) find that the variables
present in structural models can not explain changes in spreads.
On the positive side, a recent paper by Leland (2002) shows that the default probability prediction of structural models are indeed roughly consistent with observed default
frequency.2,3
Our paper contributes to this on-going debate but, unlike many previous authors
(e.g., Huang and Huang (2002)), we do not focus on the level of prices or the size of
the spread. Instead we investigate the ability of structural models to explain rates of
return and we ask is whether these models can be used to hedge corporate bond returns.
Using data on monthly returns for a large sample of U.S. corporate bonds over a five-year
period, we find that the variables present in structural models explain a large fraction
of the returns on investment grade bonds and a smaller but significant fraction for high
yield bonds. This result is in itself not surprising, since it has been known for some
2
This finding means that those models which try to find the ways to make predicted spreads closer
to the actual ones, can substantially overpredict default frequency.
3
This result is also in line with the eponymous KMV method that used a Merton-based approach to
predict “distance to default”.
3
time that investment grade and government bonds have similar returns (Campbell and
Ammer, 1993). We also find that debt returns are significantly related to returns on the
underlying asset and that the pattern of sensitivities is broadly consistent with the level
of credit exposure.
In other words, this paper focusses attention on the second-moment predictions of
the model. In structural models, any change in the value of a credit risky bond credit is
a result of a change in the value of the assets that collateralize the debt or a change in
riskless rates. In our empirical analysis we employ the change in the value of the equity
together with descriptors of the change in riskless rates and ask whether the sensitivities
of corporate bond returns to equity and riskless bond returns are consistent with the
model. Our main result is that even the Merton (1974) model produces equity sensitivities that are roughly in line with those observed empirically. Our test is supportive of
the view that structural models account well for the credit risk component of corporate
bond returns. This positive result is also consistent with Leland’s favorable and recent
findings on the default probability predictions of structural models.
A number of authors have found that the returns on corporate bonds are also related
to a number of factors that are not present in structural models. These include the FamaFrench SMB and HML factors (Elton et. al. (2001)), returns on a broad index of equity
prices and implied volatility from options on equity indices, “VIX” (Collin-Dufresne et.
al. (2001)). In the second half of the paper we include these variables in our analysis of
hedged corporate bond returns.
The results represent the second main finding of this paper. We find that returns
on corporate bonds are significantly related to changes in the VIX implied volatility
index but that the sensitivities are not related to any of the standard measures of credit
exposure such as rating, leverage or asset volatility. Our results on the Fama-French
SMB factor are similar. Thus VIX and SMB have significant effects on the prices of
corporate bonds but are not related to their credit risk..
Our results also have potential interest for practitioners who wish to hedge corporate
4
debt positions. Our findings suggest that structural models are in fact more useful for
this purpose than might appear from their performance in explaining the size of credit
spreads.
Other authors have previously studied these issues. Blume, Keim, and Patel (1991)
also study the behavior of corporate bond returns but not within the framework of structural models. Huang and Huang (2002) use a variety of models to determine whether
structural models explain the average level of yield spreads but do not study hedge ratios. Collin-Dufresne, Goldstein, and Martin (2001) analyze changes in yield spreads in
a regression framework where the choice of regressors is motivated by structural models. They do not, however, examine whether the size (as distinct from the sign) of the
estimated coefficients is consistent with the theory.
The closest paper to ours in spirit is Leland (2002) in sense that we investigate the
predictions of structural models that do not directly touch on the level of bond prices
(or yield spreads). While Leland (2002) studies whether structural models are able to
predict the presence of corporate default, we study their implications for hedge ratios
and so attempt to define more narrowly the reasons why these models fail to explain the
level of spreads.
Our paper proceeds as follows. Section II provides a description of the data set and
the sample selection procedure and also gives descriptive statistics. Section III describes
some preliminary regression analysis of ability of structural models to explain returns
on corporate bonds. In section IV the procedure is refined and the regressions take into
account the effect of changes in asset values, volatility and leverage on the hedge ratios
predicted by the structural model. Section V examines the sensitivity of bond returns
to other variables such as VIX and the Fama-French SMB and HML factors. Section VI
concludes.
5
II
Data, Sample Selection and Descriptive Statistics
II.1
Data
We use monthly prices on corporate bonds that are included either in the Merrill Lynch
Corporate Master index or the Merrill Lynch Corporate High Yield index. These indices
include most rated U.S. publicly issued corporate bonds. The data covers the period from
December 1996 to September 2002. Table I provides descriptive statistics on the bonds
in the data set. The data set contains more than 323000 bond-month observations, with
about 2900 issuers and 9000 issues. Matching with CRSP and COMPUSTAT allows us
to use about 50% of the total number of observations and all rating categories (from
AAA to CCC) are represented. As we would expect, as we move down the ratings the
average time-to-maturity decreases and the average coupon rate increases. The median
size at issuance is $200 million dollars. Detailed information on each bond is obtained
from the Fixed Income Securities Database (FISD) as provided by LJS Global Services)
and equity and treasury bond returns are from CRSP.
II.2
Sample selection
The specific bonds included in the analysis satisfy the following criteria: (1) the bond
is not convertible, exchangeable, callable or putable; (2) the bond is issued by a U.S.
company and denominated in $U.S.;4 (3) it is possible to match unambiguously the
bond issuer with a company in CRSP using the CUSIP; (4) the bond is issued by a nonfinancial corporation; (5) the bond has an initial maturity of at least four years and (6)
the bond has at least 25 consecutive monthly price observations. Table I gives summary
statistics for the entire sample and Table II for the remaining sample of 1362 bonds.
II.3
Descriptive statistics on returns
For each bond j we calculate the return between months t and t − 1 as follows:
4
More specific, the company is of the U.S. origin according to the FISD definition. In particular, its
headquarters should be located in the U.S. and it is subject to the U.S. legal practice.
6
rj,t =
Pj,t + AIj,t + Ij,t Cj /Nj
,
Pj,t−1
where Pj,t is the price of bond j at the end of month t and AIj,t is the change in
the accrued interest between t − 1 and t. Since the calculation of the accrued interest
restarts with each coupon payment, if the coupon date falls between t − 1 and t, Cj /Nj is
added to the price, where Cj is the annual coupon rate and Nj is the coupon frequency
per annum of bond j. Ij,t is an indicator function taking the value of 1 if the coupon is
due between t − 1 and t. The excess return is then calculated as
rj,t = rj,t − rf1m,t ,
where rf1m,t is the return on one-month Treasury bills up to month t.
Table III provides summary statistics on individual raw returns and excess returns
for the whole data set and by rating (where rating, as provided by the S&P, is defined as
the rating on the first date that a bond is present in the dataset). The realized average
monthly return on corporate bonds is 0.50% and the average excess return is 0.11%.5
Over this period the low-grade bonds had a lower average return than investment-grade
bond and a substantially higher standard deviation of rate of return. for the period
1977-1989.
III
Structural models and returns
In this section we study the determinants of corporate bond returns implied by structural
credit risk models. We take initially a conservative view and include in the the regressions
only those factors which are explicitly stochastic in the Merton model. Later other
variables are included – specifically VIX and the Fama-French factors – and the sensitivity
of debt returns to equity re-examined.
5
We include these statistics on mean returns only for comparison. The period covered by our data is
too short to be able to identify means with any precision. The focus of our study, however, is on second
moments (e.g. hedge ratios) and for this purpose the data is adequate.
7
The value of the firm, V , is the driving state variable in most structural models;
indeed the presence of V is the distinguishing feature of the structural approach. In this
section we estimate the sensitivity of debt values to changes in the value of the firm by
regressing excess rates of return on corporate bonds against the excess return on the
equity of the issuing firm.6 In a one-factor model the elasticity of the value of debt to
equity is related to the delta of the debt against V by:
∂D E
=
∂E D
Ã
∂D
∂V
∂E
∂V
!
E
=
D
Ã
1
∂E
∂V
!
−1
E
,
D
(1)
Clearly, this elasticity is a function of both V and interest rates and therefore varies
over time. These regressions are, therefore, simply a first step. Later in the paper we use
the Merton model to account for time variation in the elasticity. In our regressions we
also control for changes in the riskless term structure by including returns on a ten-year
constant maturity Treasury bond. While in the first generation of structural models
the risk-free rate was held constant, more recent versions include a stochastic interest
rate (e.g., Kim, Ramaswamy, and Sundaresan (1993), Longstaff and Schwartz (1995)).
An increase in the risk-free rate has two opposing effects on the price of debt. First,
it decreases the value of debt by decreasing the present value of all future cash flows.
Second, it leads to increase in the risk-neutral drift of the value or the firm, V , and so
increases the value of debt by decreasing the likelihood of default. The second effect will
be relatively more important for bonds with a higher likelihood of default.
Thus we regress the excess return on each bond, rj,t , on the excess return on the
issuing firm’s equity, rE,t , and return on riskless bonds, rf 10y,t :
rj,t = αj,0 + αj,E rE,t + αj,rf rf 10y,t ,
6
(2)
An increase in E causes an increase in D in case this increase is caused by an increase in V , i.e. an
increase in the total pie to be divided between equityholders and debtholders. It also can be the case that
an increase in E is a result of a wealth transfer from debtholders to equityholders (with V kept constant).
The latter issue has been investigated in the structural framework by the strategic debt service models
such as Anderson and Sundaresan (1996) and Mella-Barral and Perraudin (1997).
8
Table IV reports the average value of the coefficients and their t-statistics calculated
from their cross-sectional standard deviation.7
A number of points are worth noting. Both factors are significant for the whole
sample and for most of the rating categories. For the whole sample, a one percent
return on the riskless bond leads to 0.45% increase in the corporate debt price. The
standard deviation of treasury bond returns is about 1.5% per month and so the average
impact on one-month returns on corporate bonds of a one standard deviation return on
government bonds is about 0.75%. The impact of the riskless rate becomes smaller for
lower credit rating categories: the loading is significantly positive for investment-grade
bonds and negative but insignificant for B and CCC grade bonds. This suggests, quite
interestingly, that for very low quality bonds the effect of an increase in the riskless rate
– and thus the risk-neutral drift of V – in reducing credit exposure may outweigh the
resulting decline in the present value of firm’s liabilities.
Ignoring the effects of credit risk and assuming parallel shifts in the riskless term
structure, the coefficient on the return on Treasuries would equal the ratio of the duration
of the corporate bond to the duration of the Treasury. As the average maturity of our
sample of corporate bonds is just under 13.5 years (see Table II) and the Treasuries have
a constant maturity of 10 years, the actual coefficient of 0.75 is clearly lower than the
ratio of the durations. This is consistent with a negative correlation between changes in
the yield spreads on corporate debt and changes in Treasury yields.
For the full sample, the return impact of a 1% return on equity is 0.042%. If we
assume that the standard deviation of monthly equity returns is 12%, then a one standard deviation return on equity increases a bond’s return by 0.4% on average. This is
smaller than, but of the same order of magnitude as the effect of the risk-free rate. The
sensitivities of bond returns to equity are convex in the credit rating: a one percent
increase in the stock price increases returns by 1-2 basis points (bp) for AA-A bonds, 4
7
Similar procedure was employed by Collin-Dufresne, Goldstein, and Martin (2001) in the context of
studying the changes in credit spreads.
9
bp for BBB bonds, and 7-11 bp for BB-B bonds. The average coefficient on equity is
significant for each of the ratings (apart from AAA).8
In univariate regressions (not reported here) returns on the riskless bond are the
major determinant of returns on high-grade bonds, while for low-grade bonds equity
is the more important. The two factors combined explain about half of a variation
in returns on AAA-A-grade bonds and 20% for low-grade bonds. We show later that,
even when other regressors are included and the results are computed for subperiods,, the
sensitivity of corporate bond returns to both equity and riskless bonds remain significant
and the coefficients exhibit the same relationship with credit ratings.
IV
IV.1
Debt sensitivity to equity
Sensitivity in structural models
Having established that the sensitivity of corporate debt returns to the underlying equity
and riskless debt are significant, we now ask whether the magnitude of these sensitivities is consistent with the predictions of structural models. In this paper we employ
the Merton (1974) model. This may be a surprising choice because, not only are its
assumptions regarding capital structure clearly oversimplified, but it is also well known
to underestimate credit spreads by a wide margin (Jones, Mason, and Rosenfeld, 1984).
However, it remains an open question whether this simple model is able to explain the
sensitivity of debt returns to equity and this is the question we now address.
In the Merton model the value of equity is simply the value of a European call option
on the assets of the firm with exercise price equal to the face value of the debt. Using
equation (1) we may therefore write the sensitivity of the return on a credit risky bond
to the return on equity as:
8
This is a different result from Collin-Dufresne, Goldstein, and Martin (2001) who find that changes
in equity and quasi-market leverage do not have a significant impact on changes in credit spreads.
10
µ
hE =
1
−1
∆E
¶µ
¶
1
−1 ,
L
(3)
where L is market leverage (defined as market value of debt as a percentage of the
market value of the firm) and ∆E is the delta of a European call option on the value
of the firm (equity). In the Merton model the parameters of interest are book leverage,
B, the volatility of the firm’s assets, σA , time to maturity, T − t, and the risk-free rate,
r. The table below shows the comparative statics of hE (assuming that the option to
default is out-of-the-money).
Parameter
hE
B
+
σA
+
T −t
+/−
r
−
For short maturity bonds the effect of an increase in time to maturity on the dispersion
of asset values at maturity is greater than the effect on the risk-neutral expected value
of the firm’s assets and the hedge ratio increases. For long maturity bonds the second
effect dominates and an increase in time to maturity decreases the hedge ratio. Our
calculations suggest that, for reasonable parameter values, the actual times-to-maturity
in our sample are too small for the second effect to dominate.
Table V shows the values of hE for the Merton model for asset volatilities between
10% and 50% and values of L between 10% and 70%. The risk-free rate is 5 %per annum
and the time-to-maturity is 10 years. These values have been chosen to reflect those
encountered in our sample. The table shows that the sensitivity varies significantly from
zero (less than 0.01), when both leverage and volatility are low, to about 0.3 when both
leverage and volatility are high.
While in the remainder of the paper we focus on the Merton model, it would be
straightforward to carry out the same analysis using other debt pricing models. To
11
illustrate, we consider two other models of corporate debt pricing. In the Leland (1994)
model, where debt is assumed to be perpetual and to pay a constant coupon, the bond
price is given by:
"
µ ¶−x #
µ ¶−x
Vt
cF
Vt
∗
1−
Pt =
+ max(wV − K, 0)
,
∗
r
V
V∗
where c is the coupon rate, F is the principal, K and w are fixed and proportional
liquidation (bankruptcy) costs, V ∗ is the time-invariant default boundary,
µ
∗
V =
cF
r
¶
1
,
1 + x−1
and x is constant and a function of other parameters, x = x(σ, δ, r), where δ is the
firm’s payout ratio.
Taking the derivative of Pt with respect to Vt , we find that
∂Pt
=
∂Vt
µ
1
V∗
¶−x
·
Vt−x−1 x
¸
cF
∗
− max(wV − K, 0) .
r
Substituting the above formulas into equation (3) we obtain the corresponding value of
hE .
In the strategic debt service models of Anderson and Sundaresan (1996) and MellaBarral and Perraudin(1997), and if we also assume that the bond is a constant coupon
perpetuity, the result is similar to Leland’s, the only difference being the value of the
boundary:
V∗ =
IV.2
cF/r + K
w (1 + x−1 )
Preliminary comparison of sensitivity in the Merton model and
actual data
As a first step in investigating the relationship between the Merton model sensitivities
and those estimated empirically using regressions we perform a simulation. The object of
this simulation is to calculate the mean value of the sensitivity of bond returns to equity
in our data when (a) the Merton model holds and (b) the sensitivities are estimated
12
using linear regression on monthly data that has the same characteristics as our sample.
For a given rating class, the difference between the mean sensitivity we obtain in this
way and the sensitivity calculated from Merton model using the mean values by rating
class, leverage, volatility, and time to maturity can be attributed to (a) non-linearity and
(b) the discrete time interval between observations.
We aggregate the seven principal rating classes into three: “AAA-A” (including AAA,
AA, and A bonds), “BBB”, and “Junk” class (BB, B, and CCC bonds). For each of
the three classes we find the 5% and 95% quantiles of leverage,9 volatility, and time to
maturity and we take these to be the minimum and maximum points of unform distributions from which we draw values in the simulation.10 The table below gives the upper
and lower limits for the three variables for each rating class.
Parameter
Leverage
AAA-As
BBB
Junk
min
max
0.15
0.35
0.2
0.45
0.3
0.7
min
max
0.1
0.3
0.15
0.35
0.2
0.5
min
max
5
20
0.48
5
20
0.35
5
20
0.17
Asset volatility
Time to maturity
Frequency
We now generate 2000 time series of “bond returns” as follows. First, we assign a
rating class to each time series according to the proportions found in the actual data.
Second, again for each time series, we randomly draw values for leverage, volatility, and
time to maturity from the distributions for the relevant credit class and generate a time
series of asset values. Using the Merton model we then calculate monthly equity and
9
10
The ratio of the face value of the firm’s debt to the market value of assets.
We also tried to account better for the distribution patterns of leverage and volatility within the
rating class; it did not produces any significant changes.
13
bond prices and, from these, monthly returns. Finally, we estimate the hedge ratio, hE ,
by running a regression of the simulated bond returns on simulated equity returns (i.e.,
a regression similar to (2) but excluding the return on the riskless bond.11 )
Table VI reports the mean values of hE and R̄2 for these regressions. Comparing
these results and those in Table IV we see that the sensitivities are surprisingly similar.
For high quality (AAA-A) bonds, the average sensitivity is found to be about 0.01 for
simulations and 0.0003–0.02 for the actual the data; for BBB bonds both for the model
and the actual data produce a sensitivity of 0.04 and for junk bonds we find 0.15 for the
simulations and 0.07–0.11 bp for the actual data. The mean values of the simulated and
empirical hedge ratios are not significantly different at a 5% confidence for the entire
sample and the “AAA-A” and “BBB” subsamples and at the 10% level for the “Junk”
subsample. These results are surprising since, for the same simulated data, the model
underestimates the observed level of credit spreads by more than 80%, or by more than
50bp in absolute terms.
IV.3
Preliminary Analysis of Hedge Ratios
The results in the previous section raise the possibility that, although the Merton model
leads to poor predictions of credit spreads, it may perform better as a predictor of hedge
ratios. This result, if substantiated, would be important because, in contingent claims
pricing theory, the hedge ratios define the composition of the replicating portfolio which,
in turn, defines price of the contingent claim. Thus, if we find that the model provides
good predictions of hedge ratios but poor predictions of the bond price, we are better
able to identify the reasons for model failure.
To address this issue, we next test the second-moment prediction of the model in a
more rigorous manner. Note that if the model is correct then in equation (2) αj,E is
11
We also run regressions on the actual data using only equity returns to be consistent. The results
are broadly similar with the equity sensitivities slightly larger.
14
equal to hE and we can therefore rewrite the regression as
rj,t = αj,0 + βj,E hE,j,t rE,t + αj,rf rf 10y,t ,
(4)
where hE,j,t is the model hedge ratio for firm j at time t and under the null hypothesis
that the Merton model holds, βj,E is equal to one. To implement (2) the following
parameters need to be estimated for each firm: the ratio of the book value of debt to the
market value of assets, B/V , the volatility of assets, σA , time to maturity, T − t, and
the riskless rate.
To estimate
B
V,
we take the ratio of the book value of debt (sum of COMPUSTAT
items 9 (long-term debt) and 34 (debt due within a year)) to the quasi-market value
of assets (sum of COMPUSTAT items 9 and 34 plus the number of shares outstanding
times stock price (both CRSP)). The COMPUSTAT data are taken at the date of the
last annual accounting report and the CRSP data are taken on the date of observation.
The estimation of asset volatility is a challenging task and here we consider a number
of alternatives. First, we compute upper and lower bounds on asset volatility as follows.
The maximum equates asset and equity volatility, σE , i.e., assuming zero leverage. The
minimum is calculated as σE (1 − L), where L is the market leverage, i.e., assuming that
the debt bears no asset risk. In this case, the theoretical hedge ratio is zero.
A more realistic estimate of asset volatility recognizes that debt bears some asset risk
and that equity and debt covary. For firm j at time t we have:
2 2
2 2
2
σd
Ajt = (1 − Ljt ) σEjt + Ljt σDjt + 2Ljt (1 − Ljt )σED,jt ,
(5)
where σDjt is the time t volatility of firm j’s debt, and σED,jt is the time t covariance
between returns on firm j’s debt and equity.12 We could estimate firm j’s debt volatility
using the returns on each of firm j’s bonds but this approach has two drawbacks. First,
it assumes that all of a firm’s outstanding debt has the same volatility as its publicly
traded debt. Second, much corporate debt is relatively illiquid and some of the observed
volatility may be spurious.
12
This calculation assumes again that leverage is measured instantaneously.
15
We therefore estimate equation (5) as follows. Firm j’s equity volatility at time t is
estimated as the time series volatility of returns on firm j’s equity using three years of
data up to month t. For the volatility of returns on firm j’s debt we first calculate the
average volatility of debt returns by credit rating. Thus, for rating category BBB and
firm j, for example, we take the returns on firm j’s debt for each month that the debt
was rated BBB at the start of the month. If the bond in question is rated BBB in at
least 15 months we then compute the time series volatility. Averaging these volatilities
over all firms we obtain the average volatility for BBB debt. The volatility of firm j’s
debt in month t is then set equal to the average volatility for the rating category of
firm j at month t.13 . The covariance of equity and debt returns, σED,jt , is estimated as
ρED,jt σD,jt σE,jt where ρED,jt is estimated in a similar way to σD,jt .
Finally, we take time to maturity as equal to the median time to maturity for each
rating class and the riskless rate equal 5%.
Table VII reports summary statistics for these estimates. As expected, leverage is
higher for lower rating categories and, similarly, so is equity volatility. The average values
of equity volatility and quasi-market leverage are broadly consistent with similar results
reported in other studies. Deleveraging equity volatility using L but taking no account
of the asset risk borne by debtholders (the third panel of Table VII) results in estimates
of asset volatility that are relatively constant across the rating categories.
The fourth panel of Table VII gives estimates of σ
d
At using equation (5) and the
method described above. Here the mean values of asset volatility are quite similar for
investment grade bonds (22% to 24%) but noticeably higher for junk bonds: 27% for
BB, 29% for B and 30% for C. The range of values, as measured by the 5% and 95%
quantiles, is also much wider for the lower rated bonds, e.g.,12-33% for AA vs. 14-56%
for B.
Table VIII shows estimates of the hedge ratio hE (σE ). The first two panels set σA
13
One can think of this as a form of “switching regime” where the volatility for firm j switches between
the volatilities of the different rating categories
16
equal to σE and (1 − L)σE respectively. The final panel shows estimates of the hedge
ratio using the estimates of σA derived using equation (5). As expected, these rise
monotonically as the rating category declines.
IV.4
Testing Merton Model Predictions of Hedge Ratios
We now use our estimates of asset volatility to test more formally whether hedge ratios
from the Merton model are consistent with the empirical relation between equity and
corporate bond returns.
For firm j we take the estimate of time t asset volatility described above and use this
as an input to the calculation of the time t hedge ratio, hE,jt (c
σA ).14 We then estimate
the following regression for each firm, j:
rj,t = αj,0 + βj,E hE,jt (c
σA )rE,t + αj,rf rf 10y,t .
(6)
Under the null hypothesis that the Merton model correctly estimates the sensitivity
of returns on firm j’s debt to firm j’s equity, the coefficient βj,E should be unity.
The results are given in Table IX. For the entire sample the mean estimate of βj,E
is 1.206. The t-statistic against unity, the value of βj,E under the null is 1.156. For the
six rating categories the mean value of βj,E is different from unity in only two cases: BB
where the mean value if 2.498 and CCC where the mean is 0.415. For the other four
categories the mean value ranges from 0.55 (AA) to 1.54 (B) and none is significantly
different from one.
These results are supportive of the structural approach, and the Merton model in
particular, in a way that previous analyses of the level of prices or credit spreads have
not been. They are also complementary to the results recently obtained by Leland (2002)
who shows that the default frequency predictions of structural models are also broadly
consistent with the data.
Apart from the size of the yield spread, there is a further prediction of structural
14
The other inputs – book leverage, time to maturity and the riskless rate – are as described earlier.
17
models that is inconsistent with the results in Table IX. This is the R̄2 in the regression
which, if the Black-Scholes conditions supporting the structural approach apply, should
be much higher. In our simulations reported in Table VI the R̄2 varied from 0.65% for
AAA-A to 0.936% for BB-CCC.15 In Table IX the R̄2 are much lower (11% for CCC and
37% for AA).
The low variation in the fraction of rate of return volatility accounted for by equity
returns and interest rates has a number of possible explanations. One is simply that it
reflects noise in the bond return data (or, possibly in the equity or riskless bond data).
This almost certainly accounts for some of the unexplained variability of corporate debt
returns. Another is that the model is mis-specified and that either the functional form of
the hedge ratio is incorrect or that other variables are necessary to account for the credit
exposure of corporate debt. For example, the Merton could hold except that volatility
is stochastic and other variables are necessary to predict volatility. Finally, returns on
corporate debt could be related to other variables in a way that is not directly related
to credit risk. For example, returns on corporate debt might be related to variables that
proxy for changes in liquidity.
V
Other Determinants of Returns on Corporate Bonds
In this section we consider the impact of variables that other authors have found to be
significant explanators of returns on corporate bond. These include (i) changes in the
10-year minus 2-year yield spread on US Treasuries, (ii) the return on the S&P 500 index,
(iii) changes in the VIX index of implied volatility of options on the S&P 100 index and
(iv & v) the Fama-French SMB and HML factors. All five factors are included in the
recent study by Collin-Dufresne et. al. (2001) and the Fama-French factors are included
by Elton et. al. (2001).
The results are shown in Table X. The mean coefficients on 10-year Treasury returns
are very similar to those in Table IV and those on equity are also similar but little lower.
15
Note that in this procedure interest rate is held constant.
18
The mean coefficients on the S&P are not significant except for the A credit category.
The mean coefficients on HML are significant for the whole sample and for three out of
the six rating subsamples while those on SMB are significant for the whole sample and
all the subsamples except for CCC.
Perhaps the most interesting results, however, are those for changes in the VIX
volatility index. Collin-Dufresne et. al. (2001) had previously found this variable to be
significant in regressions of changes in yield spreads on a very similar list of regressors.
The mean coefficients for ∆(V IX) are significant in every case except for CCC; in fact,
the t-statistics are higher than those on changes in the firm’s own equity for all the
investment grade subsamples. A natural interpretation of the role of ∆(V IX) in the
regression is as a proxy for changes in the volatility of equity. In this interpretation VIX
is related to a bond’s credit exposure via its effect on the default put. However, when we
examine the magnitude of the coefficients for the different rating subcategories we see
that they are essentially the same. For example, the mean coefficient for AAA and AA
are -0.043 and -0.060 respectively and for BBB and BB are -0.056 and -0.054 respectively.
If ∆(V IX) were acting as a proxy for changes in equity volatility the sensitivity for the
lower credit categories, as in the case of the coefficients on equity returns, would be much
larger.16
We find, therefore, that the change in the VIX index is a variable that, while strongly
related to returns returns on credit risky bonds, is apparently unrelated to credit risk.
The R̄2 in Table X are significantly higher than in Table IV but particularly so for the
non-investment grade bonds: the average R̄2 for BB and B increases to 28% from 20%.
The precise rôle of ∆(V IX) and the other variables is, at this point, unclear but it seems
highly unlikely that is connected with credit exposure.
16
We have also carried out cross-sectional regressions of the individual coefficients on ∆(V IX) on
simple descriptors of credit risk, such as equity volatility and leverage, and found no relation.
19
VI
Conlusion
This paper studies the ability of structural models to explain excess returns on corporate
bonds and the main question we ask is whether these models provide accurate predictions
of hedge ratios. Using data on monthly returns for a large sample of U.S. corporate bonds
over a five-year period, we find those variables included in structural models – returns
on the issuing firm’s equity and on riskless bonds – explain a large fraction of returns
on investment grade bonds and a smaller but significant fraction for high yield bonds.
Further, and this is the main result of the paper, we find that, for most rating categories,
the equity ratios predicted by the Merton model are not rejected by time series data.
The next step is to account for other factors. We include in our regression variables
that in previous studies have been shown to influence corporate bond prices. The variables we use are: (i) changes in the 10-year minus 2-year yield spread on US Treasuries,
(ii) the return on the S&P 500 index, (iii) changes in the VIX index of implied volatility
of options on the S&P 100 index and (iv & v) the Fama-French SMB and HML factors.
We find that none of the included variables undermines the significance of either the
risk-free rate or equity. Our main result here, and the second main result of the paper,
is that changes in the VIX index have an impact on corporate bond returns that is both
significant and apparently unrelated to a bonds exposure to credit risk. It seems clear,
therefore, that returns on credit risky bonds are systematically related to at least one
factor that lies outside standard measures of “credit risk”. Whether there are other
factors, and the precise role of ∆(V IX) in the determination of risky bond prices, is a
question for further research.
20
References
[1] Anderson Ronald W., and Suresh M. Sundaresan, 1996, “The Design and Valuation
of Debt Contracts”, Review of Financial Studies, 9, 37-68.
[2] Blume, M. E., D. B. Keim, and S. Patel, 1991, “Returns and Volatility of Low-Grade
Bonds: 1977–1989”, Journal of Finance, 46.
[3] Campbell, John Y., and John Ammer, 1993,“What Moves the Stock and Bond
Markets? A Variance Decomposition for Long-Term Asset Returns,” Journal of
Finance, 48, 3–37.
[4] Collin-Dufresne Pierre, and Robert S. Goldstein, 2001, “Do Credit Spreads Reflect
Stationary Leverage Ratios?”, Journal of Finance, 56, 1929-1957.
[5] Collin-Dufresne Pierre, Robert S. Goldstein, and J.Spencer Martin, 2001, “The
Determinants of Credit Spread Changes”, Journal of Finance, 56, 2177-2207.
[6] Elton Edwin J., Martin J. Gruber, Deepak Agrawal, and Christopher Mann, 2001,
“Explaining the Rate Spread on Corporate Bonds”, Journal of Finance, 56, 247-277.
[7] Eom Young Ho, Jean Helwege, and Jing-zhi Huang, 2003, “Structural Models of
Corporate Bond Pricing: An Empirical Analysis”, Review of Financial Studies,
forthcoming.
[8] Huang, Jing-zhi, and Ming Huang, 2002, “How Much of the Corporate-Treasury
Yield Spread is Due to Credit Risk?”, working paper, Pennsylvania State University.
[9] Jones P.E., S.P. Mason, and E. Rosenfeld, 1984, “Contigent Claims Analysis of
Corporate Capital Structures: An Empirical Analysis”, Journal of Finance, 39,
611-25.
[10] Kim, J., Ramaswamy, and S. Sundaresan, 1993, “Does Default Risk in Coupons
Affect the Valuation of Corporate Bonds?: A Contingent Claims Model”, Financial
Management, 117–131.
21
[11] Leland, Hayne E., 2002, “Predictions of Expected Default Frequencies in Structural
Models of Debt”, Venice Conference on Credit Risk, Sept 2002
[12] Longstaff Francis A., and Eduardo S. Schwartz, 1995, “A Simple Approach to Valuing Risky Fixed and Floating Rate Debt”, Journal of Finance, 50, 789–819.
[13] Mella-Barral Pierre, and William R.M. Perraudin, 1997, “Strategic Debt Service”,
Journal of Finance, 52, 531–556.
[14] Merton Robert C., 1974, “On the Pricing of Corporate Debt: The Risk Structure
of Interest Rates”, Journal of Finance, 29, 449-470.
22
Table I
Summary statistics for the entire data sample
This table reports summary statistics for the entire sample of corporate debt returns over
period 12.1996–07.2002. The number of observations is given in thousands. T − t is the time
to maturity remaining on the date of each observation. The coupon rate is in %. Volume is
in million $US dollars. % in CRSP reports the fraction of observations that are matched with
CRSP and COMPUSTAT.
Observations
Issuers
Issues
Mean T − t
Median T − t
Mean Coupon
Median Coupon
Mean Volume
Median Volume
% in CRSP
All
323.53
2894
9049
10.60
7.04
7.88
7.50
281.52
200
49.96
AAA
7.23
77
301
11.90
7.34
7.37
7.20
338.62
250
13.80
AA
32.20
240
1154
11.13
6.50
7.07
6.88
337.90
200
39.52
23
A
117.17
824
3811
10.94
6.80
7.31
7.13
293.11
200
50.41
BBB
93.38
1035
3332
12.10
7.88
7.58
7.38
269.72
200
55.92
BB
26.93
667
1515
9.36
6.67
8.42
8.36
247.53
200
52.26
B
35.18
1084
1738
6.94
6.97
10.09
10
246.87
179.80
49.97
CCC
11.41
568
879
6.74
6.30
10.69
10.63
250.92
200
43.50
Table II
Summary statistics for the final sample
This table reports summary statistics for the final sample (the selection procedure is described in
the paper) of corporate debt returns over period 12.1996–07.2002. The number of observations
is given in thousands. T − t is the time to maturity remaining on the date of each observation.
The coupon rate is in %. Volume is in million $US dollars. % in CRSP reports the fraction of
observations that are matched with CRSP and COMPUSTAT.
Observations
Issuers
Issues
Mean T − t
Median T − t
Mean Coupon
Median Coupon
Mean Volume
Median Volume
% in CRSP
All
58.49
493
1362
13.46
8.49
7.76
7.40
259.10
200
100
AAA
0.27
4
6
20.53
24.16
7.26
7.15
321.88
300
100
AA
4.03
30
110
13.87
8.63
6.97
6.75
321.03
250
100
A
18.73
149
560
16.10
12.80
7.41
7.13
266.06
200
100
BBB
23.07
226
687
14.15
8.96
7.49
7.25
258.68
200
100
BB
5.03
121
265
8.71
6.79
8.27
8
247.08
200
100
B
6.20
148
229
7.13
7.13
9.56
9.50
213.59
175
100
CCC
1.16
57
83
8.38
6.88
9.86
9.75
220.90
200
100
Table III
Summary statistics on returns
This table reports summary statistics for excess and raw returns for the final sample (the
selection procedure is described in the paper) of corporate bond returns over period 12.1996–
07.2002. We calculate for each corporate bond j the return between months t and t − 1 as
follows:
Pj,t + AIj,t + Ij,t Cj /Nj
rj,t =
,
Pj,t−1
where Pj,t is the price of bond j at the end of month t and AIj,t is the accrued interest between
t − 1 and t. Since the calculation of the accrued interest restarts with each coupon payment, if
the coupon falls between t − 1 and t, Cj /N is added to the price, where Cj is the annual coupon
rate and Nj is the coupon frequency per annum of bond j. Ij,t is an indicator function taking
the value of 1 if the coupon is due between t − 1 and t. The excess return is then calculated as
rj,t = rj,t − rf1m,t ,
where rf1m,t is the return on one-month Treasury bills. Returns are first calculated for each
individual bond and then averaged across bonds. Excess returns are given in parentheses. N
is the number of bonds.
Mean
5%
95%
std
N
All
0.50
( 0.11)
-0.18
(-0.57)
0.90
( 0.51)
2.87
( 2.87)
1362
AAA
0.59
( 0.20)
0.38
( 0.00)
0.73
( 0.34)
1.72
( 1.70)
6
AA
0.59
( 0.20)
0.38
(-0.00)
0.75
( 0.36)
1.85
( 1.84)
102
A
0.56
( 0.16)
0.20
(-0.19)
0.76
( 0.37)
2.15
( 2.14)
475
24
BBB
0.49
( 0.10)
0.04
(-0.34)
0.83
( 0.44)
2.72
( 2.72)
473
BB
0.48
( 0.09)
-1.40
(-1.80)
0.97
( 0.59)
3.36
( 3.36)
124
B
0.34
(-0.05)
-2.51
(-2.91)
1.29
( 0.91)
5.16
( 5.16)
172
CCC
-0.22
(-0.62)
-4.07
(-4.49)
3.14
( 2.73)
9.73
( 9.74)
10
Table IV
Regressions of Excess Returns
This table reports regressions of excess returns on corporate bonds over period 12.1996–07.2002.
ret
rf10y
is the excess return on the 10-year constant maturity U.S. Treasury bond. E ret is the
excess return on the issuer’s equity and N is the number of observations.
Intercept
ret
rf10y
E ret
R̄2
N
All
-0.001
(-4.158)
0.451
(36.655)
0.042
(18.796)
0.375
42.943
(1362)
AAA
-0.001
(-3.630)
0.748
(6.125)
0.003
(0.685)
0.658
51.500
(6)
AA
-0.001
(-8.244)
0.690
(28.087)
0.016
(4.635)
0.553
49.039
(102)
A
-0.001
(-13.142)
0.611
(49.690)
0.021
(10.322)
0.471
43.137
(475)
BBB
-0.001
(-4.951)
0.507
(27.826)
0.039
(11.038)
0.352
42.655
(473)
BB
0.002
(2.876)
0.201
(3.816)
0.074
(6.725)
0.254
41.968
(124)
B
0.001
(1.961)
-0.066
(-1.845)
0.100
(10.895)
0.153
40.244
(172)
CCC
-0.006
(-1.073)
-0.380
(-1.475)
0.109
(2.794)
0.216
38.500
(10)
Table V
Sensitivities of bond returns to equity in the Merton (1974)
model
This table reports the sensitivities of bond returns to equity in the Merton (1974) model for
different values of leverage and asset volatility. Leverage is the ratio of the face value of debt
to the market value of assets. σA is asset volatility. Leverage and asset volatility are given in
%. The riskless rate is assumed to be equal to 5%, time to maturity 10 years. Sensitivities are
given in 0.01%. The procedure is described in the paper. t-statistics are in parentheses.
σA
Leverage
10
20
30
40
50
60
70
10
15
20
25
30
40
50
0
(18.40)
0
( 4.39)
0
( 9.66)
0
(10.74)
0.01
( 7.57)
0.04
(15.08)
0.20
(12.61)
0
( 2.42)
0
( 7.93)
0.02
(12.67)
0.15
(10.28)
0.55
(15.88)
1.33
(17.82)
2.41
(23.32)
0
( 7.90)
0.07
(13.12)
0.42
(18.21)
1.35
(20.91)
2.96
(23.96)
4.44
(27.88)
6.33
(34.45)
0.05
( 9.88)
0.66
(13.67)
2.35
(17.17)
4.31
(26.88)
6.23
(32.97)
8.63
(35.39)
10.31
(41.62)
0.37
(16.33)
2.44
(18.82)
5.19
(25.23)
7.72
(33)
10.31
(39.25)
12.70
(42.87)
14.75
(46.37)
3.66
(20.55)
7.91
(27.80)
12.02
(32.86)
15.57
(40.93)
18.16
(48.93)
20.36
(54.62)
21.53
(56.57)
8.70
(27.54)
14.18
(38.65)
20.10
(50.08)
22.64
(53.12)
24.83
(58.37)
26.64
(61.85)
28.72
(74.88)
25
Table VI
Sensitivities of debt returns to equity implied by the Merton
(1974) model for the actual data set
This table reports sensitivities of bond returns to equity in the Merton (1974) model. The
sensitivities were obtained using a simulation, as described in the paper. Sensitivities are given
in 0.01%. t-statistics are in parentheses.
hE
R̄2
All
0.046
(29.50)
0.74
AAA-A
0.011
(17.95)
0.65
BBB
0.040
(22.87)
0.78
BB-CCC
0.153
(32.54)
0.93
Table VII
Leverage and Volatilities
This table reports results of the first step of a more extensive analysis of hedge ratios. σE is
the historical equity volatility. L is market leverage. σc
A is the estimated asset volatility, hE
is the hedge ratio calculated for three different estimates of asset volatility. Volatilities and
leverage are given in % and hedge ratios are given in 0.01%. N is the number of observations.
The details of the estimation procedure are given in the paper.
L
Mean
Median
Std. Dev.
5% quantile
95% quantile
σE
Mean
Median
Std. Dev.
5% quantile
95% quantile
(1 − L)σE
Mean
Median
Std. Dev.
5% quantile
95% quantile
σcA
Mean
Median
Std. Dev.
5% quantile
95% quantile
N
All
AAA
AA
A
BBB
BB
B
CCC
0.346
0.301
0.232
0.035
0.804
0.047
0.026
0.143
0.014
0.042
0.162
0.110
0.169
0.028
0.458
0.239
0.208
0.165
0.036
0.564
0.364
0.342
0.204
0.057
0.754
0.439
0.449
0.222
0.064
0.802
0.587
0.614
0.219
0.192
0.886
0.751
0.817
0.209
0.343
0.975
0.361
0.326
0.165
0.186
0.669
0.251
0.241
0.045
0.188
0.346
0.268
0.258
0.074
0.156
0.402
0.291
0.277
0.092
0.177
0.468
0.348
0.328
0.117
0.193
0.585
0.438
0.421
0.142
0.240
0.716
0.543
0.517
0.193
0.285
0.905
0.803
0.753
0.376
0.504
1.205
0.221
0.204
0.110
0.080
0.420
0.240
0.236
0.056
0.183
0.332
0.222
0.222
0.071
0.111
0.328
0.219
0.207
0.081
0.117
0.365
0.219
0.200
0.110
0.086
0.448
0.246
0.217
0.138
0.077
0.524
0.222
0.189
0.148
0.060
0.524
0.194
0.135
0.193
0.020
0.549
0.239
0.219
0.106
0.118
0.440
58488
0.240
0.237
0.055
0.184
0.333
272
0.224
0.223
0.069
0.116
0.329
4027
0.223
0.211
0.080
0.125
0.366
18729
0.230
0.211
0.106
0.108
0.453
23068
0.270
0.244
0.128
0.117
0.530
5031
0.287
0.258
0.133
0.136
0.557
6203
0.300
0.262
0.184
0.095
0.622
1158
26
Table VIII
Hedge Ratios
This table reports results of the first step of a more extensive analysis of hedge ratios. σE is
the historical equity volatility. L is market leverage. σc
A is the estimated asset volatility, hE
is the hedge ratio calculated for three different estimates of asset volatility. Volatilities and
leverage are given in % and hedge ratios are given in 0.01%. N is the number of observations.
The details of the estimation procedure are given in the paper.
hE (σE )
Mean
Median
Std. Dev.
5% quantile
95% quantile
hE ((1 − L)σE )
Mean
Median
Std. Dev.
5% quantile
95% quantile
hE (σcA )
Mean
Median
Std. Dev.
5% quantile
95% quantile
N
All
AAA
AA
A
BBB
BB
B
CCC
0.095
0.053
0.105
0
0.306
0.006
0
0.022
0
0.029
0.023
0.001
0.055
0
0.129
0.047
0.011
0.071
0
0.213
0.092
0.064
0.089
0
0.265
0.138
0.133
0.098
0.001
0.312
0.221
0.230
0.097
0.045
0.367
0.334
0.343
0.074
0.208
0.438
0.020
0.003
0.037
0
0.095
0.002
0
0.006
0
0.017
0.005
0
0.014
0
0.034
0.012
0.001
0.026
0
0.066
0.018
0.005
0.032
0
0.083
0.030
0.013
0.042
0
0.118
0.045
0.023
0.056
0
0.163
0.053
0.021
0.073
0
0.196
0.028
0.007
0.044
0
0.121
58488
0.002
0
0.006
0
0.017
272
0.006
0
0.014
0
0.036
4027
0.013
0.001
0.027
0
0.070
18729
0.021
0.008
0.033
0
0.091
23068
0.040
0.027
0.045
0
0.135
5031
0.083
0.071
0.055
0.011
0.191
6203
0.123
0.116
0.073
0.016
0.250
1158
Table IX
Hedge ratio regressions
This table reports results of a regression analysis of hedge ratios. The regression is
rj,t = αj,0 + βj,E hE,j,t rE,t + αj,rf rf 10y,t ,
where hE,j,t is the hedge ratio for firm j at time t as implied by the model. Under the null
ret
is the excess
hypothesis that the Merton model holds, βj,E is equal to one (100 bp). rf10y
return on the 10-year constant maturity U.S. Treasury bond. The details of the estimation
procedure are given in the paper. N is the number of observations.
Intercept
βE
ret
rf10y
R̄2
N
All
-0.001
(-4.554)
1.206
(1.156)
0.369
(18.180)
0.266
42.076
(686)
AAA
(—)
(—)
(—)
(—)
(—)
(—)
(—)
35
(1)
AA
-0.002
(-6.764)
0.552
(-1.304)
0.815
(25.734)
0.372
48.536
(28)
27
A
-0.002
(-8.601)
1.173
(0.608)
0.688
(34.809)
0.371
42.902
(143)
BBB
-0.001
(-4.978)
0.787
(-0.629)
0.479
(22.968)
0.289
41.569
(288)
BB
0.001
(1.312)
2.498
(2.968)
0.140
(2.172)
0.203
42.976
(83)
B
-0.001
(-1.207)
1.540
(1.762)
-0.116
(-2.165)
0.128
40.471
(136)
CCC
-0.007
(-0.854)
0.415
(-2.159)
-0.408
(-1.386)
0.108
41.714
(7)
Table X
Regression of Excess Returns
This table reports results of regression analysis of excess returns on corporate bonds over period
ret
12.1996-07.2002. rf10y
is the excess return on the 10-year constant maturity U.S. Treasury
ret
bond. E
is the excess return on the issuer’s equity. ∆(rf10y − rf2y ) is the change in the
slope of the term structure (the difference between the yield on ten-year and two-year constantmaturity U.S. Treasury bonds). S&P ret is the return on the S&P index. VIX is the implied
option volatility on the S&P100 index options. SMB and HML are the Fama-French Small
minus Big and High minus Low factors. N is the number of observations. t-statistics are given
in parentheses.
Int
ret
rf10y
∆(Spd)
E ret
S&P ret
∆(V ix)
SMB
HML
R̄2
N
All
-0.001
(-8.628)
0.509
(41.557)
-0.015
(-11.239)
0.030
(13.196)
0.008
(1.691)
-0.053
(-13.073)
0.104
(19.429)
0.023
(5.558)
0.463
42.943
(1362)
AAA
-0.001
(-4.049)
0.796
(6.137)
-0.006
(-0.958)
0.002
(0.426)
0.023
(1.487)
-0.043
(-4.414)
0.052
(2.518)
0.021
(1.532)
0.700
51.500
(6)
AA
-0.002
(-10.237)
0.743
(27.578)
-0.008
(-3.419)
0.009
(2.423)
-0.002
(-0.253)
-0.060
(-9.083)
0.077
(8.165)
0.017
(4.154)
0.631
49.039
(102)
28
A
-0.002
(-17)
0.659
(53.560)
-0.012
(-7.504)
0.008
(4.249)
0.019
(4.157)
-0.045
(-10.712)
0.092
(15.424)
0.012
(3.241)
0.553
43.137
(475)
BBB
-0.002
(-7.561)
0.573
(31.885)
-0.009
(-4.369)
0.028
(7.377)
-0.002
(-0.178)
-0.056
(-8.282)
0.104
(13.536)
0.008
(1.145)
0.455
42.655
(473)
BB
0
(1.077)
0.291
(6.223)
-0.027
(-6.261)
0.054
(5.304)
0.033
(1.913)
-0.054
(-3.314)
0.182
(6.929)
0.080
(5.400)
0.347
41.968
(124)
B
0.001
(0.905)
-0.018
(-0.468)
-0.027
(-4.526)
0.091
(9.198)
-0.014
(-0.593)
-0.059
(-3.071)
0.100
(3.986)
0.040
(1.923)
0.221
40.244
(172)
CCC
-0.004
(-0.931)
-0.416
(-1.805)
-0.131
(-3.941)
0.110
(2.502)
0.146
(0.892)
-0.095
(-1.035)
0.124
(1.020)
0.303
(4.098)
0.247
38.500
(10)