Using Yield Spreads to Estimate
Expected Returns on Debt and Equity
Ian A. Cooper∗
Sergei A. Davydenko
London Business School
This version: 27 February 2003
ABSTRACT
This paper proposes a method of extracting expected returns on debt and equity
from corporate bond spreads. It is based on an easily implementable calibration of
the Merton (1974) model to market debt spreads and other observable variables. For
rating classes, the approach generates robust expected default loss estimates very
similar to historical default data. It also provides forward-looking estimates for individual firms unavailable from historical data. The method can be used to adjust
the cost of the firm’s debt for the probability of default, which is essential for lowrated firms. The approach can also be applied to provide independent estimates of
expected equity premia consistent with historical default experiences. These equity
risk premium estimates vary from three percent for typical investment-grade firms to
over eight percent for the average junk bond issuer.
Keywords: cost of debt, equity premium, credit spreads, expected default
JEL Classification Numbers: G12, G32, G33
∗ Corresponding author. Please address correspondence to: London Business School, Sussex Place, Regent’s
Park, London NW1 4SA. E-mail: icooper@london.edu. Tel: +44 020 7262 5050 Fax: +44 020 7724 3317. This is
a revised version of our earlier paper “The Cost of Debt”. We thank Ilya Strebulaev for helpful comments.
I
Introduction
Corporate bond yields reflect a variety of factors, including liquidity, taxes, risk premia, and
expected losses from default.1 In many uses, such as cost of capital estimation, lending decisions,
portfolio allocation, performance measurement, and bank regulation, estimates of expected returns on risky debt are required. These are equal to the promised debt yield minus the part
of the yield that reflects expected default. To obtain expected returns, therefore, estimates of
expected losses due to default are required.
Most methods of estimating the expected default loss are based either on historical default
data, or accounting and equity market information (see Lao (2000), Elton et al. (2001), Crosbie
and Bohn (2002)).2 However, such estimates ignore the most relevant variable incorporating the
market consensus expectation of default: the debt yield itself. In this paper we propose a method
of estimating the expected default loss and expected debt returns using individual companies’
bond yields. It is based on calibrating the simplest structural model of corporate debt pricing,
Merton (1974), to observed debt yield spreads. It allows to estimate how much of the observed
market spread for individual bonds is due to expected default. From these, expected returns on
the bonds can be calculated. The procedure is simple and uses only easily observed variables.
The proposed method has important advantages over alternative methods based on historical
default experiences as proxies for future default incidence (for example, Altman (1989), and
Blume, Keim and Patel (1991)). Elton et al. (2001) use historical data on rating migrations
and recovery rates to estimate the expected default spread. Their approach uses data on ratings
classes. Thus, it does not provide estimates for individual bonds unless they are typical of a
ratings class. Our method, on the other hand, recovers the part of the spread due to expected
default for individual bonds. Another disadvantage of the historical approach is that it does
not provide forward-looking estimates. Asquith, Mullins and Wolff (1989) argue that historical
default frequencies may differ from future probabilities, because available historical data do not
cover all likely future economic and market conditions (see also Waldman et al. (1998)). In
contrast, our method uses the observed market yield, which should reflect expectations of the
economic and market conditions for the period to which it refers. The estimates that we derive
are, on average, consistent with historical default data for ratings classes. Where they differ, our
estimates appear to be better behaved than those based on historical averages.3
The model that we use to split the yield spread between expected default and other components is the Merton (1974) model. The variables that we calibrate to are the yield spread,
leverage and the equity volatility. The Merton model makes a number of simplifying assumptions about capital structure and bankruptcy procedures. Many papers, including Black and
1 They
may also reflect option features, such as call provisions, but these are assumed away in our analysis.
also Driessen (2002) for a different approach to decomposing the credit spread.
3 An entirely different approach to estimating the expected return on debt is to apply an asset pricing model
such as the CAPM to risky debt (Blume and Keim (1987)). However, this approach requires debt transaction
price series to estimate debt betas, which are often unavailable. Moreover, applying this method is complicated by
the fact that debt betas change significantly with changes in capital structure and over time. Also, it depends on
the CAPM being the correct equilibrium model, and using a correct estimate of the market risk premium. Taking
into account these difficulties, this approach is hard to implement.
2 See
2
Cox(1976), Leland (1994), Longstaff and Schwartz (1995), and Collin-Dufresne and Goldstein
(2001) have extended the basic framework to incorporate more realistic assumptions about corporate bond markets. These models improve the fit to the general level of yields, but none gives
a generally good cross-sectional fit to bond prices (see Eom et al. 2002). Despite the large variety
of structural models available, Huang and Huang (2002) show that very different models predict
similar debt spreads when they are calibrated to fit observed default and recovery rates. In this
sense, the choice of observed variables for calibration appears to be more important than the
particular model structure. Moreover, since our goal is not to predict the total spread, but rather
determine the fraction of the observed market spread which is due to the expected default loss,
the importance of the choice of the model is likely to be reduced further. For these reasons, we
choose the simplest model for our calibration. If the Merton framework picks up first-order effects
relevant to the relative valuation of risky debt and equity, then our estimates of the part of the
spread due to expected default should not be overly sensitive to the model assumptions. We test
robustness of our estimates by varying the structure of the model and the parametrization. We
find that the procedure, though simple, is robust in estimating the default loss component of the
spread, confirming the result of Huang and Huang that predictions of these types of models do
not vary much when calibrated to the same variables.
Various other calibrations of structural models have been proposed.4 For individual firms,
KMV (described in Crosbie and Bohn (2002)) calibrate a version of the Merton model to the
face value and maturity of debt and a time series of equity values. They recover asset value and
volatility and use this to calculate a ’distance to default’. This is then used in conjunction with
KMV’s proprietary default database to estimate the probability of default. Huang and Huang
(2002) calibrate several structural models to historical default probabilities for rating classes,
using average leverage, equity premia and debt maturity. They solve for the implies volatility
of assets and use it to calculate the sum of the expected default loss and risk premium due
to default. They conclude that different models have similar performance when calibrated to
historical default data, and also that default cannot account for much of the spread for high
grade bonds. Delianedis and Geske (2001) calibrate a version of the Merton model to debt face
value, maturity, equity value and equity volatility. Like Huang and Huang, they recover the part
of the spread caused by default risk. They also show that this cannot explain the entire spread
for high grade firms. An important difference from our approach is that these papers focus on
the total spread due to default risk, including the associated risk premium. We, on the other
hand, use the total market spread adjusted for non-default factors as an input, and split out the
single component due to the expected default loss.
We make three innovations in the calibration procedure. First, we calibrate the model to debt
yield spreads. None of the above papers uses yield spreads for calibration.5 Of all capital market
variables, bond yields should contain the most relevant information about consensus predictions
of default. Thus, estimates of default rates that do not use yields as inputs may be inconsistent
with market expectations, and the resulting inferences about default may be misleading. In
4 An alternative to structural models of risky debt is reduced-form models. These models are concerned only
with pricing under the risk-adjusted probability measure and so cannot assist in the estimation of the actual
default probability. See Madan (2000) for a review of these models.
5 Delianedis and Geske (2001) mention this as a possibility but do not implement it.
3
contrast, our approach assumes that bond are fairly priced, and backs out expected default
adjustments that are consistent with observed spreads, leverage and volatility. This procedure
allows to estimate the parameters of the firm asset value distribution implied by observed market
prices. This is then used to determine the part of the yield due to expected default, given the
expected return on equity.
Second, we control for factors other than default risk by measuring spreads relative to the
AAA rate rather than the treasury rate. We base this adjustment on two observations. The
first is that AAA debt has a very low chance of default. The second is that the components
of the spread unrelated to default, such as taxes and bond-market risk factors, appear to be
relatively constant across ratings categories. For the tax spread component discussed by Elton et
al., we derive an explicit formula that demonstrates this independence. The adjustment for the
AAA spread results in expected default spread estimates which are similar in magnitude to those
predicted from historical default data. It appears to overcome, for the purposes of calibration,
the commonly observed inability of structural debt models to explain spreads on investment
grade debt (Jones, Mason and Rosenfeld (1984), Delianedis and Geske (2001), Huang and Huang
(2002), Eom et al. (2002)).6
Third, to make the model more flexible, we ”endogenize” the time to maturity of the debt. The
Merton model assumes a single class of zero-coupon debt. Because of omitted factors, including
coupons, default before maturity, strategic actions, and complex capital structures, the Merton
model is too simple to reflect reality. Therefore, the choice of maturity when implementing the
model is difficult. For instance, Huang and Huang use actual maturity, Delianedis and Geske use
duration, and KMV use a procedure that mainly depends on liabilities due within one year. To
avoid this issue and give the model enough flexibility to fit actual yield spreads, we simply solve
for the value of maturity which makes the model consistent with observed spreads adjusted for
non-default factors.
There are many possible applications of the equilibrium expected default spreads that we
recover. We illustrate these with the use in estimating a firm’s expected cost of debt for use
in its cost of capital. The cost of capital is used in valuation, capital budgeting, goal-setting,
performance measurement and regulation, and is perhaps the most important number in corporate finance. Its key inputs are the cost of equity and the cost of debt. Yet while the cost
of equity has been the subject of extensive debate, little attention has thus far been focused on
estimating the cost of debt. Two common approaches are to use either the yield on the debt or
the riskless interest rate as proxies. Neither is correct when part of the yield spread is due to
expected default. The errors are most significant when the debt is risky. As Brealey and Myers
(2000) say: ‘This is the bad news: There is no easy or tractable way of estimating the rate of
return on most junk debt issues’ (p. 548). Our method helps to overcome this problem.
Another application of the approach is related to equity premia estimates. In the form that
we use it for most of the paper, the procedure derives the expected default rate on debt con6 As discussed in the section on adjusting for factors other than default, we do not claim that this procedure
helps to explain investment grade spreads. It simply enables a calibration that is consistent with the data.
4
ditional on the expected equity return. Alternatively, if expected default rates are known from
an independent source, such as historical default data, the procedure can be reversed to give
estimates of the expected return on equity consistent with observed debt yields. We use this
approach to provide a new set of equity risk premium estimates, based on data hitherto unused
for this purpose. This method, based on bond yields, should contain information about consensus expectations of risk premia. An important advantage of our approach is that the resulting
estimates are largely based on a forward-looking capital market variable.7 It may provide a useful benchmark comparison to estimates of equity premia derived in more standard ways. Other
estimates are typically based on the CAPM, APT or variants of the dividend growth model.8
All these methods generate large standard errors. Our method, based on different information,
provides new insights into the equity premium.
Our main results are as follows. The technique we propose for estimating the equilibrium
expected default component of the spread appears quite robust. It recovers estimates which
are not very different from those obtained from historical average default and recovery data
for ratings categories. Our estimates also appear to be better behaved than estimates based on
historical default frequencies in the following sense: The historical method gives expected returns
on debt for our sample that are not monotonic as debt rating changes, whereas our procedure
gives monotonic estimates. It also generates sensible estimates of the asset volatilities of firms,
which are consistent with measures obtained in other ways.
In line with historical default rates, we find that only a small fraction of the spread for highgrade debt is due to expected default loss. For lower-grade debt, this component is larger, and
our approach provides a method for adjusting yields to give expected debt returns. We find that
the expected default component of the spread varies significantly within ratings categories, so
using average figures for ratings categories for individual companies may be misleading. The
estimates of equity risk premia that we obtain using the technique are well-behaved. They are
consistent with asset risk premia of about three percent and equity risk premia of between three
and nine percent.
The balance of the paper is organized as follows. The next section presents the calibration
approach for the Merton (1974) model and the relationship between the cost of debt and equity.
Section III discusses adjusting the yield spread for factors other than the risk of default. Section
IV provides a description of our data set. Section V discusses the calibration method and examines
its robustness. Section VI provides estimates of expected returns on debt. Section VII presents
estimates of equity risk premia. The following sections discuss various applications and extensions
of the method. Section X summarizes. Technical details are given in Appendices.
7 The ’DCF’ method of relating equity prices to earnings forecasts relies on consensus expectations reflected in
the share price, but requires earnings forecasts which are obtained from surveys.
8 See Welch (2000) for a survey of existing practices.
5
II
The Merton Model
The Merton model is the simplest equilibrium model of corporate debt. It assumes that the
value of the firm’s assets follows a geometric Brownian motion:
dV
= µdt + σdWt
V
(1)
where V is the value of the firm’s assets, µ and σ are constants, and Wt is a standard Wiener
process.9 The model further assumes that the firm has a single class of zero coupon risky debt
of maturity T . Other assumptions include a constant flat risk-free yield curve and a very simple
bankruptcy procedure.
Merton applies the Black-Scholes option pricing analysis to value equity as a call option on
firm’s assets. Merton’s formula can be written in a form that gives a relationship between the
firm’s leverage w, the maturity of the debt T, the volatility of the assets of the firm σ, and the
promised yield spread s (see Appendix A):
N (−d1 )/w + esT N (d2 ) = 1
(2)
where N (·) is the cumulative normal distribution function and
d1
d2
√
[− ln w − (s − σ 2 /2)T ]/σ T
√
= d1 − σ T
=
(3)
(4)
Of the variables in Equation (2), leverage and the spread are observable, and σ and T are
generally unobservable. Another implication of the model is that the observable equity volatility10
σ E satisfies:
σ E = σN (d1 )/(1 − w)
(5)
We now have three inputs: w, s and σ E , and two unknowns: σ and T . We solve equations (2)
and (5) simultaneously to find values of σ and T that are consistent with the observed values of
w, s and σ E .11 Thus, σ is computed as the implied volatility of the firm’s assets when the equity
is viewed as a call option on the assets.
Once the values of σ and T are known, the relationship between the expected return on assets,
equity and debt are related as follows. Since equity in this model is a call option on the assets
and therefore has the same underlying source of risk, the risk premia on assets π and equity π E
9 The
drift must be adjusted for cash distributions.
contrast to the asset volatility, the short-term equity volatility is easily observable either from option-implied
volatilities or from historical returns data.
11 The system of equations is well-behaved, and we generally had no difficulties solving it applying standard
numerical methods. To assure a starting point for which standard algorithms quickly yield a solution, one can
solve equations (2) and (5) separately for σ for a few fixed values of T (or vice versa). This procedure always
converged for any reasonable starting points. The intersection of the solution curves σ(T ) from equations (2) and
(5) can then be used as a starting point for the system of these equations.
10 In
6
are related as:
π
µ−r
σ
≡
=
πE
µE − r
σE
(6)
π = π E σ/σ E
(7)
or:
Now the expected return premium on debt over the maturity period ΠD can be calculated as
(see Appendix A):12
ΠD = s +
i
√
√
1 h (π−s)T
ln e
N (−d1 − π T /σ) /w + N (d2 + π T /σ)
T
(8)
and the spread which is due to expected default δ ≡ s − ΠD which should be excluded from the
expected return on debt is thus:
δ=−
i
√
√
1 h (π−s)T )
ln e
N (−d1 − π T /σ) /w + N (d2 + π T /σ)
T
(9)
The right-hand side of this expression is positive, and the expected return on debt is lower than
the promised yield. Note also that the probability of default predicted by the model is:
√
P = N (−d2 − π E T /σ E )
(10)
If, on the other hand, the expected default loss on debt δ is known, then the expected equity
premium can be estimated. Equation (9) can be solved for π and combined with (7) to find π e
consistent with the expected default.13
III
Adjusting for factors other than default risk
Before we apply the Merton model, we adjust for factors other than default risk by subtracting
the AAA spread. There is growing evidence that corporate yield spreads cannot be entirely due
to the risk of default. Huang and Huang (2002) and Delianedis and Geske (2001) measure the
part of the spread that is due to default risk and find that, for AAA bonds, very little of it can be
explained by default. Table I presents their estimates of the proportion of the AAA spread that
is due to default risk. All estimates suggest that very little of the AAA spread can be explained
by default risk.14 Thus, high grade spreads must be almost entirely due to other factors.15 Elton
et al. (2001) argue that a part of the spread for U.S. corporate bonds is due to the state tax on
corporate bond coupons which is not paid on government coupons. Collin-Dufresne et al. (2001)
12 Note that, unlike the return on assets and equity, the calculated return on debt is an annualized compounded
return rather than an instantaneous return.
13 If the probability of default is known, then it can also be used to estimate the equity premium.
14 Huang and Huang report that models such as Leland and Toft (1996) can explain up to half of the ten-year
spread. However, these models are for infinite maturity debt, so the comparison with ten year yields is, as Huang
and Huang state, not very informative.
15 We also tested the influence of default risk on AAA spread by regressing them on fundamental determinants of
default risk, including the equity volatility and leverage of the issuer, and found that these factors were statistically
insignificant determinants of AAA spreads.
7
demonstrate the presence of a systematic factor in credit spreads which appears to be unrelated
to equity markets. Lower liquidity of corporate bonds relative to government bonds is also likely
to be responsible for a part of the spread.
INSERT TABLE I HERE
The Merton model does not include factors such as tax, liquidity, and bond-market risk
factors unrelated to the equity market. Thus, the calibration of the model to credit spreads must
exclude the part of the spread unrelated to default risk. The magnitude of the tax, liquidity
and bond-specific risk components of the spread are hard to estimate. For instance, Elton et
al. estimate the component of the spread caused by differential state taxes on corporate and
government bonds. Their estimates range from 29 to 50 basis points for the tax component of
the AA spread. Even this large range might be questioned, as any positive tax spread could
be subject to arbitrage by institutions that are exempt from state taxes, such as pension funds.
Uncertainties at least as great affect estimates of the bond-specific risk component. Elton et al.
claim that equity-related risk factors can explain almost all of the spread unexplained by default
and taxes, whereas Collin-Dufresne et al. identify a substantial bond-market risk factor.
For these reasons, direct estimation of the non-default components of the spread does not
appear practical. We therefore need a variable that contains these components to adjust spreads
so that they reflect only default-related factors. The evidence that the AAA spread does not
contain a significant default risk component suggests that it reflects only non-default factors. So
we could use the AAA spread to proxy these other factors as long as they are cross-sectionally
constant. For the tax and bond-market risk components, there is evidence that this is the case.
The tax-induced spread is modelled by Elton et al. (2001). In the US, coupon payments on
corporate bonds are subject to state income taxes, while government bonds are not. Elton et al.
(2001) do not give an explicit formula for the part of the spread induced by this tax effect. In
Appendix B we derive such a simple formula. The yield spread due to tax is given by:
∆y
tax
=
1
tM
1−τ
ln
1 − τ e−rtM tM
(11)
where: ∆y tax is the spread due to tax, tM is the time to maturity, τ is the applicable tax rate
and rtM is the riskless interest rate. This formula holds for any bond, as long as capital gains
and losses are treated symmetrically and the capital gain tax rate is the same as the income tax
on coupons.16 Table II shows estimates of the tax-induced spread based on the above formula
16 In
a more general case, when the income tax rate τ I atwhich coupons are taxed does not coincide with the
1−τ I
1
ln
. This is similar
−r
t
tM
e /B ])
1−e tM M (τ C +(τ I −τ C )E ∗ [F
to (11) when rtM tM is high or the risk-neutral expectation of the principal repayment is not very different from
the purchasing price: E ∗ [F ] ≈ B. Another nuance is that in reality taxation rules for bonds originally sold
significantly below par (called original-discount bonds), such as zero-coupon bonds, are different from our model.
Capital gains on such bonds are appreciated for tax purposes gradually throughout the life of the bond, so that
only a small part of the tax is paid at maturity. For such bonds formula (11 will also be an approximation.
capital gains tax τ C , formula (11) becomes ∆y tax =
8
for the average tax rates estimated in Elton et al. (2001) and maturity tM =10 years.
INSERT TABLE II HERE
The key feature for our application is that the tax-induced spread does not depend on the riskiness
of the bond or the coupon rate. So this factor will influence the AAA yield by exactly the same
amount as lower grade yields. Even if we do not know the appropriate tax rate to use for the
adjustment, subtracting the AAA yield will eliminate this factor.
A similar result regarding the equality across ratings categories of non-default factors is given
in Collin-Dufresne et al. (2001). They report that there is a systematic factor in bond returns that
appears to be unrelated to equity markets. The loadings of bonds from different rating categories
on this factor are very similar. They do not give a price of risk for this factor. However, if
the factor loadings are constant across ratings categories, then subtracting the AAA yield will
eliminate the effect of this factor, regardless of the price of risk.
The final component of yields that is not related to default is liquidity. The difference in
liquidity between the corporate bond and government bonds may be responsible for part of the
bond spread. Subtracting the corporate AAA spread will adjust for the difference in liquidity
between corporate AAA and government bonds. It will not adjust for differences between the
liquidity of the bonds we are analyzing and the AAA bonds we use as benchmarks. In our
estimations, this difference is limited by the data we use, which is for transactions involving
insurance companies. The fact that the prices we use represent actual transactions ensures some
liquidity in the bonds we analyze, and limits the difference between their liquidity and the liquidity
of the AAA benchmark bonds we use. Therefore, we assume that relative liquidity effects are of
second-order importance across our sample.
IV
Data
We use bond trade data supplied by the National Association of Insurance Companies (NAIC),
which records all transactions in fixed-income securities by insurance companies in the US. The
original dataset includes trade prices for more than six hundred thousand transactions in the
period 1994-1999. We exclude all bonds other than senior unsecured fixed-coupon straight US
industrial corporate bonds without call/put/sinking fund provisions and other optionalities. We
also exclude bonds for which we were able to unambiguously identify the promised cash flow
stream and Moody’s or Standard and Poor’s rating at the date of trade, as well as find issuing
company’s accounting data in Compustat for the fiscal year immediately preceding the date of
trade, and a 2-year history of its stock prices in CRSP. We use only senior unsecured debt, as
this is the type of debt on which Moody’s company ratings are based. To improve the matching
of the inputs, we retain in the sample only bonds with remaining maturity between 7.5 and 10
years, where the trade is executed within three months after the fiscal year end of the issuing
company. We use this maturity subsample because yield spreads vary with maturity, so we do
9
not want to mix very different maturities with different term premia. We use a relatively long
maturity because we are interested in expected returns for relatively long horizons. We use the
subsample of trades shortly after the year end to better match the accounting information to the
trade data. Thus constructed, the final sample includes 2632 trades on 553 bonds of 292 issuers.
We estimate spreads on these bonds using data on U.S. Treasury STRIPS (risk-free zerocoupon securities) using the procedure suggested in Davydenko and Strebulaev (2002). We first
compute the yield on each bond trade from the transaction price recorded by NAIC. We then
calculate the yield on a risk-free bond with the same promised cash flows using Treasury STRIPS
prices as of the settlement date of trade.17 We subtract the estimated cash-flow matched risk-free
rate from the yield on the bond to obtain the yield spread for the trade.
We measure leverage as a ratio of the Compustat-recorded book value of debt to the sum of
the book value of debt and total market value of equity obtained from CRSP for the last business
day before the trade. We measure total debt by the book value of short term and long term
debt. Some other authors do not use observed leverage in structural debt models because the
book value of debt may not proxy for its market value. An alternative is to use the book value of
debt to proxy for the face value of debt (Crosbie and Bohn (2002), Delianedis and Geske (2001)).
This approach is also subject to criticism unless the structural model used is one that explicitly
deals with the coupon flows on the bonds. We follow Huang and Huang (2002) in using leverage
based on the book value of debt, but later find that our results are insensitive to the precise
measurement of leverage. We estimate equity volatility as the volatility of daily equity returns
as recorded in CRSP over the two years prior to the bond trade. We also tried six months and
1 and 3 years, with no significant change in the results. We use equity risk premium estimates
for different ratings classes from Huang and Huang (2002). These are based on the empirical
relationship between leverage and equity returns in Bhandari (1988).
Table III shows summary statistics for spreads and other fundamental variables for our sample. All the variables behave as one would expect, with spread, equity volatility, and leverage
increasing on average as the rating deteriorates. There is, however, significant variation of these
variables within ratings classes. The table also shows maturity and duration, which are similar
across ratings classes for the subsample. Finally, it gives the asset volatility calculated using the
KMV method described in Crosbie and Bohn (2002) and also employed in Vassalou and Xing
(2002), which we later use to benchmark our asset volatility. One interesting feature of this
variable is that its average is relatively constant across ratings categories, apart from the single-B
category. The asset volatility estimate recovered from the KMV procedure for this class is double
that for the other classes.
INSERT TABLE III HERE
17 For
the majority of trades there are 4 annual STRIPS returns available. We use linear approximation of the
STRIPS yield curve to discount the corporate bond coupon payment which occurs between maturity dates on two
STRIPS.
10
To benchmark our procedure and to estimate equity premia, we need default loss estimates
based on historical data. Elton et al. provide such estimates for different bond classes over
different maturities conditional on starting from a particular rating class. We use these as one
benchmark. However, the usefulness of these estimates for our purposes is limited. They assume
that recovery is a function of ratings class, whereas Altman and Kishore (1996) document that
recovery rates are highly dependent on industry and bond seniority, but not on initial rating after
controlling for these variables. Moreover, Elton et al. also assume that annual ratings transititions
are Markovian and use these to estimate cumulative default probabilities for different horizons.
Instead, we produce our own estimates of the yield equivalent of expected default using Moody’s
historical default frequencies reported in Keenan et al., also used by Huang and Huang. These
are direct estimates of cumulative default probabilities. We complement these with a recovery
rate for senior unsecured bonds of 48.2% from Altman and Kishore (1996). This recovery rate is
close to the 51.3% recovery from Moody’s used by Huang and Huang (2002).
As a benchmark, we compute the expected default loss spread using historical data on defaults
and recoveries. This spread, called in Elton et al. the risk-neutral spread, is the coupon with
which the bond would trade at par in a risk-neutral world. This coupon, C, is defined implicitly
by the relationship:
c
X (Ptc − Pt−1
)R + (1 − Ptc )C
(1 +
r)t
+
1 − PTc
=1
(1 + r)T
where R is the recovery rate and Ptc is the cumulative probability of defaulting over t years. The
spread due to expected default risk is C − r. The resulting estimates are discussed in the next
section.
V
Calibrating the model
In the form that we use it, the Merton model gives two equations relating five variables:
leverage, yield spread, equity volatility, debt maturity and asset volatility. In addition, there is
another equation relating the expected default loss on debt and the expected equity premium under the real probability measure. We calibrate the model using leverage, yield spread, and equity
volatility as observable inputs, and solve the first two equations simultaneously to give implied
debt maturity and asset volatility. Asset volatility cannot be observed directly, so imputing its
value from a model using equity volatility as an input is standard.18 Once these parameters are
found, we use the formula relating the expected return on debt to the expected return on equity.
18 See,
for instance, Vasicek (1984).
11
V.1
Calibration for rating classes
Before we use the model to estimate the expected default for individual firms, we test the
calibration primarily based on aggregate data for ratings used in Huang and Huang (2002). Table
IV shows in columns (1) to (3) the values of spreads, leverage and equity expected return given
by Huang and Huang for six ratings groups for bonds of ten years maturity. These are drawn by
them from a variety of sources, not all for the same period of time. We also need an estimate
of equity volatility, which Huang and Huang do not use. As the Huang and Huang data are for
mixed periods, and we use them only to test the general properties of the calibration, we simply
obtain the volatility from our dataset. This is given in column (4).
Table IV shows three calibrations using different spread adjustments. The first, called ’none’,
uses the spread against treasuries as the input variable. The second, called ’tax’, deducts 20
basis points for the estimate of the tax-induced spread. This is calculated using formula (11)
with the estimate of the relevant tax rate of 4.875% from Elton et al., an interest rate of 5%, and
a maturity of 10 years. The third adjustment, called ’AAA’, deducts the average AAA spread of
63 basis points to control for the non-default component not accounted for in the Merton (1974)
model.
The calibrated model parameters T, σ, and π are given in columns (5) to (7). The asset risk
premium, π, is generally about 4.5%, and constant across ratings groups. This suggests that our
procedure is not generating any systematic bias in the relationship between equity and asset risk.
The calibrated values of asset volatility, σ, also appear reasonable. Although they should not
necessarily be equal across ratings classes, they are sufficiently similar to suggest that most of the
variation in equity volatility is coming from differences in leverage between ratings classes rather
than differences in the nature of the assets. We later test whether our procedure recovers asset
volatilities that are similar to those produced by the KMV procedure, which is rather different
and uses time series of equity prices.
The maturity parameter, T , that we recover is given in column (5). This reflects not only the
actual ten year maturity of the debt, but also any other factors such as complex capital structures,
distress costs, strategic behavior by equity holders, and other complications not included in the
Merton model but reflected in the adjusted spread s. The values of T found are typically higher
than the true debt maturity. We later test whether this is a problem for the procedure by also
calibrating a variant of the strategic debt service model of Anderson and Sundaresan (1996),
which has liquidation costs in bankruptcy as another free parameter that we can use to match
the actual maturity of the debt.
INSERT TABLE IV HERE
12
V.2
Estimated default loss for rating classes
The estimated values of the default loss spread are given in column (8) of Table IV. Column
(9) shows them as a proportion of the spread relative to treasuries, and column (10) gives the
predicted probability of default from the fitted model. Interestingly, the estimated spread due to
default is not highly sensitive to the deduction of the AAA spread. However, we believe that the
reasons for making this adjustment are so compelling that we use it throughout the rest of the
paper.
The estimated spreads due to default for AAA, AA, A, BBB, BB and B ratings categories
are 4, 4, 9, 21, 78, and 237 basis points out of total spreads of 63, 91, 123, 194, 320, and 470
b.p., respectively. The corresponding estimates based on historical data, reported in column (11),
are 4, 5, 8, 24, 132, and 353 b.p. Apart from the non-investment grades, our model generates
estimates of default spreads which are very close to those estimated with historical default and
recovery rates. For the junk grades, uncertainties about the estimates based on historical data
are quite large, so the correspondence between the fitted and historical default risk components
appears reasonable. Indeed, it is unlikely that a B spread of 470 basis points can be consistent
with the expected default spread of 353 basis points computed on the basis of historical data.
The historical estimate would leave only 117 basis points for liquidity, tax, bond-specific risk
premia and the default risk premium. This is lower than the corresponding quantity for the BBB
spread, which is difficult to justify. So our estimate based on calibration appears more reasonable
in the cases where it differs from the estimate based on historical data.
Our estimates of the proportion of the spread due to expected default share one important
property with the historical estimates. This is that the proportion is increasing as the debt
quality deteriorates. This means that using a constant value for the default risk spread within
a ratings category, or even a constant proportion of the spread, is unlikely to be correct, and
a procedure such as ours is necessary to estimate the default spread accurately, even within a
category.
Elton et al. also provide historical estimates of the spread due to default risk for AA, A and
BBB categories. These are given in column (14) and are 5, 14, and 41 b.p. respectively. These
differ somewhat from our calibrated estimates of 4, 9 and 21 b.p., but they also differ from our
historical estimates. For reasons given above, we believe that our historical estimates are more
appropriate in this context.
As well as the spread due to default risk, which reflects the probability of default and the
recovery rate, we also report the predicted and historical probability of default. The fitted value
of this variable is given in column (10). This differs markedly from the historical estimate, given
in column (13). Even though our procedure matches the expected spread due to default well, it
gives a generally higher probability of default than the historical data. This is because continuous
models of the Merton type tend to have a high probability of small losses in default. It is difficult
to have a high loss rate in conjunction with a low probability of default in such models. So our
calibration of the model fits the spread due to expected default but not the default probability.
13
V.3
Sensitivity to inputs and model specifications
Table V presents sensitivity analysis and robustness checks for the calibration procedure. We
use AA and BB ratings categories and vary the input parameters. These are given in columns
(1) to (5). The parameters are each varied individually up or down by 10 percent of their base
value. The results from calibrating the basic version of the model are given in columns (6) to
(10).
For both ratings categories, the estimated values of asset volatility and asset risk premia are
very robust with respect to variation in the inputs and the structure of the model. Even when
equity volatility is varied, the asset volatility estimates remain stable. The expected default
spread, δ, is also quite robust. Table IV showed that it is not sensitive to the subtraction of
the AAA spread. Equation (9) shows that it does not depend at all on the risk-free interest
rate. The sensitivity analysis in Table V also shows that it is insensitive to leverage. Thus, the
relatively crude measure of leverage that we use is unlikely to be an issue. The value of δ is
somewhat sensitive to the estimate of the equity risk premium. We use this sensitivity below
when we invert the procedure to estimate equity premia. The one variable to which the expected
default spread shows high elasticity is equity volatility, σ E . Second moments such as σ E can
be estimated quite accurately for equity returns. Thus, the procedure has the merit of giving a
result that is sensitive only to a parameter that can be observed relatively accurately.19
For AA bonds, the estimates of the expected default spread, δ, are all between two and
seven basis points out of a total spread of 91 basis points. Thus, the expected default is a very
small proportion of the spread regardless of the parameters or the form of the model. For BB
debt, across all the parameter values and different models, the expected default component of
the spread is between 50 and 120 basis points out of a total spread of 320 basis points. So
the expected default component is between 16% and 38% of the spread. Although this is quite
substantial variation, it reflects a very wide range of parameter values and model structures. So
the split of the spread between expected default and other components is quite robust.
To test robustness to model specifications, we also use a simple variation of the Merton model
which allows for liquidation costs and strategic debt service of the Anderson and Sundaresan
(1996) type. This is described in Appendix C. It requires another parameter θ, which may
be thought of as the proportional deadweight loss in liquidation. When liquidation costs and
strategic debt service are introduced into the model, the implied value of maturity T inversely
depends on the assumed liquidation costs. In particular, one can solve for the value of Altering the
model in this way results in a reduction in the implied value of T . The amount of the reduction
depends on the assumed value of bankruptcy costs. In particular, we can solve for the level of
bankruptcy costs θ which make the implied value of T to equal the actual ten year maturity of
the debt.
19 Note that this suggests that measures of default probability derived models of the Merton type are very
sensitive to volatility. These measures have recently been shown by Vassalou and Xing (2002) to explain some
important factor risk premia in the equity market. Because of the close relationship between volatility and the
default measure, however, it may be difficult in practice to distinguish between the effect of volatility per se and
the effect of volatility through the default measure.
14
Columns (11) to (15) of Table V give the results of these estimates, first using bankruptcy cost
of 5% of the debt face value suggested by Anderson and Sundaresan, and then solving for θ using
the actual 10 year maturity for T . In most cases, these adjustments result in 15-30 basis point
changes in the estimated default loss for the BB bond out of total spread of 320 basis points, and
much smaller changes for the AA bond. These are small proportions of the total spread, and so
the results are not highly sensitive to the choice of model.
INSERT TABLE V HERE
V.4
Sensitivity to dividend payouts
The version of the Merton model that we use does not include distributions in the form of
dividends or coupons on debt. We deal with the debt structure by allowing the maturity of debt
to be endogenous. To test for sensitivity to dividends, we amend the standard Merton model by
assuming that the firm pays continuous dividends that are a constant proportion γ of the value
of the firm V . This version of the model is given in Appendix B. We set the level of γ from the
instantaneous dividend yield g by the relationship γ = (1 − w)g.
Table VI shows the sensitivity of our estimates to the level of dividend yields. The first
column shows the dividend yield varying from zero to three percent. The final column shows
the impact of this on the proportion of the spread due to expected default. In all cases other
than BB debt, the change from zero to three percent dividend yield changes the expected default
estimate by less than four percent of the spread. In the case of BB bonds, the impact is less than
seven percent of the spread. So substantial variation in the dividend yield has little impact on
the proportion of the spread that is due to expected default.
INSERT TABLE VI HERE
The dividend yield does, however, have a major impact on the implied maturity of the debt,
T. It brings this down from the high levels shown in Tables IV and V to levels that match those of
the bonds used for the observed spreads. This means that the procedure can match the implied
maturity of the debt to the actual maturity. However, since the inclusion of dividends does
not significantly affect the expected default that we recover, we use the standard version of the
Merton model in the rest of the paper.
VI
Estimates of the expected return on debt
To estimate the expected default loss for individual transactions, for each trade in our database
we estimate the bond spread and match it with leverage information from Compustat and historical equity volatility from CRSP. We use the equity risk premia for ratings categories described
15
in the previous section. We first adjust for the AAA spread in each year to account for varying
systematic factors and factors other than expected default. Spreads vary with maturity and
over time. Therefore, we make the adjustment using a AAA spread that is matched on these
characteristics. We then solve the model to determine the level of the expected default spread.20
The calibrations produce well-behaved estimates of asset volatility and risk premia. In particular, the asset volatility estimates are similar across ratings classes, and do not exhibit the
very high estimate for the single-B class that is given by the KMV estimate shown in Table III.
This suggests that our procedure is incorporating different information from that of the KMV
procedure. To check this, we measured the correlation of the two measures, and it was 0.65 across
the whole sample, confirming that the two measures are somewhat different.
The estimates of δ show large variation within ratings classes. The coefficients of variation are
greater than one for all but the B class. For the B class, the standard deviation of δ within the
class is 124 basis points, which is large in absolute magnitude but lower than the mean spread.
These results indicate that average expected default for a ratings class is misleading if applied
to the individual bonds in the class. Because of the large variation within each ratings class,
methods which largely rely on ratings to estimate the expected default of bonds are unreliable.
A slightly better assumption is that the ratio of δ to the spread is constant within a ratings
class. This combines the information on rating with the spread in a simple way. It has a lower
coefficient of variation than the absolute value of δ for all ratings classes. However, it also exhibits
significant variation within ratings classes.
Interestingly, the averages of the ratio of δ to the spread for the ratings classes in Table VII are
very similar to those obtained in Table IV. Table VII is based on individual bonds for the NAIC
database, whereas Table IV is based on aggregate data from Huang and Huang for a different
period. The similarity of the results is another indication of the robustness of our technique.
INSERT TABLE VII HERE
VII
Equity risk premia estimates
In this section we use our method to estimate equity risk premia from debt yields. The equity
risk premium is one of the most important parameters in corporate finance and valuation, and its
value is the subject of great controversy (Welch (2000)). Even small differences in estimates can
have a major effect on valuations, and estimates ranging from 0% to more than 9% have been
advocated. The reason for the disagreement is that all estimates are potentially subject to large
errors.21 For instance, historical returns data are often used as the basis of the estimate. This
leads to large standard errors because equity returns data are very noisy and the risk premium
20 We also computed the results with an adjustment for the average AAA spread over the period rather than in
each year, and the results were very similar.
21 Estimates of equity risk premia used in valuation typically do not have standard errors attached. A notable
exception is Fama and French (1997).
16
may be time-varying. This has led to extensive criticism of the use of historical data as the basis
of equity risk premia estimates, and many other estimation methods have been proposed, each
of which has its own drawbacks.
As one alternative, some advocate the use of expectational or “ex-ante” methods of estimation. These rely on the observation of expectational data from surveys or models of expectations
formation (see Harris and Marston (1992) for a good example). In these methods, errors arise
because the relevant expectations are only indirectly observable, the relevant weighting for different agents’ expectations are unknown, and the appropriate method to convert expectations to
a risk premium estimate is debatable.
The value of our approach is that it uses the corporate debt spread, a market variable that
aggregates agents’ expectations and is observable, therefore avoiding the measurement problems
of other expectational variables. And since the debt spread incorporates a risk premium for the
future rather than the past, it avoids the problems associated with the use of historical returns to
estimate the equity risk premia. The equity risk premium is such an important and controversial
parameter that any extra information that assists in its estimation has value. Our method
provides new information that is based on data (bond yields) which have not been employed for
this purpose before. Furthermore, these data are current capital market data that should reflect
market consensus expectations. No other methods commonly used to estimate equity risk premia
are based on such data.
Our method of estimating the equity premium relies on obtaining an estimate of expected
default from historical data. This is set equal to the Merton model estimate to impute the equity
risk premium. Historical data can be used only for ratings classes, so we use the method to
estimate equity risk premia for ratings classes. To estimate equity premia we use the data for
individual trades, as described in the previous section. We calibrate the model in the same way,
but then solve the equation relating the equity premium and the expected default spread found
from the historical default and recovery data to determine the level of the equity premium. Table
VIII shows the results.
The mean equity premium across our sample of firms is 4.8% and the asset premium 3.3%.
These estimates are lower than estimates based upon an unadjusted use of historical averages.
For instance, Brealey and Myers report an average premium for equities relative to treasury bills
of 9.1% based on the period 1926-2000. Our estimates are more consistent with those of Dimson
et al. (2001), who use a different historical period and make various adjustments to the raw
historical averages. For ratings classes, the equity premia range from 3.1% for AA companies to
8.5% for B companies, reflecting the different leverage in the different ratings classes. The asset
risk premia are similar across classes, indicating that differences in risk premia for different rating
classes are primarily driven by differences in leverage .
INSERT TABLE VIII HERE
These estimates depend on the validity of the historical data for ratings classes as predictions
17
of the future, and also the robustness of the procedure used to impute from them equity risk
premia. Our tests suggest that the procedure is robust. So the main issue is the use of historical
data to estimate the average expected default for ratings classes. This is justifiable if past data
can be expected to give a good indication of expected default over our horizon, which is eight
years.
VIII
Applications
The expected return on debt plays a central role in many applications. Here we discuss three:
applications in corporate finance, banking, and portfolio management.
The corporate finance application is adjusting the cost of debt for use in the cost of capital.
Standard methods of cost of debt estimation assume that the cost of debt is equal to either the
promised yield on newly-issued debt of the firm (Erhardt, 1994), or the risk-free rate (Bodie,
Kane and Marcus, 1993). Both approaches fail to make a proper adjustment for the possibility
of default. Using them may result in serious errors in the cost of capital and value estimates.
As a simple example, consider a company with 60% leverage and a promised yield spread
of 4%. Suppose that half of the spread is due to default. Suppose that real interest rate is 2%,
and the equity risk premium for this firm is 6%.22 Using the riskless rate or the yield into the
standard weighted-average cost of capital formula, excluding taxes, gives:
W ACCriskless
W ACCpromised
rate
= 0.6 × 2% + 0.4 × (2% + 6%) = 4.4%
yield
= 0.6 × 6% + 0.4 × (2% + 6%) = 6.8%
The true WACC, using the promised real yield of 6% minus the expected default of 2% as
the cost of debt, is:
W ACCtrue = 0.6 × (6% − 2%) + 0.4 × (2% + 6%) = 5.2%
These differences can have a material impact on valuations. The multipliers for a real perpetuity growing at 2.5% are 53 times, 23 times and 37 times, for the WACC estimated with the
riskless rate, the debt yield and the true expected return on debt, respectively.
An important banking application of the spread due to expected default loss is to control
the risk and measure the performance of lending. In some uses, such as making the initial
lending decision, it may be desirable to speculate on a bank’s private view regarding future
default. For risk measurement and performance measurement, however, it is often useful to have
a non-speculative market-based benchmark against which to measure outcomes. For instance,
22 These
illustrative numbers correspond roughly to the figures for B debt in Tables IV and VII.
18
measuring whether a portfolio of low grade debt has a higher or lower incidence of default than
could have been expected on the basis of our measure of expected default would be an indicator
of whether there is predictive ability with respect to default.
A similar issue of using market consensus beliefs as a neutral expectation arises in portfolio
management. In deciding how much risky debt to hold in a portfolio, the expected return plays
a key role. Our method enables one to extract the market consensus expected return, which
should assist in determining the neutral holding of each bond. Holdings based on speculative
views will be deviations from this neutral holding and should be based on the difference between
the speculative view and the market consensus view.
These three applications illustrate the importance of a technique to estimate the consensus
expected default and expected return on debt. In addition, we have provided a new method of
estimating the expected return on equity, which is the central number in many corporate finance
and valuation applications.
IX
Extensions
The method we use in this paper could be extended and applied in many ways. For instance,
we calibrate to the yield spread using a novel method which assumes that leverage is observable
and generates an implied value of maturity. We could, alternatively, calibrate to the yield spread
using the procedure of Delianedis and Geske (2001), which assumes that the market value of debt
is unobservable and generates an implied firm value. As another example of possible extensions,
our procedure for estimating the equity premium uses an historical estimate of the spread due
to default as an input. It could also be implemented using the historical default frequency as an
input, in a way similar to Huang and Huang (2002).
The analysis indicates that equity volatility is an important input to the estimation. In fact,
equity volatility plays two roles: One is that it enters the equation relating the instantaneous
volatilities of equity and assets. The other is that it enters the equation determining the total
volatility of the asset value over the life of the bond. It may be an improvement in the implementation of this type of model to use separate estimates for these two volatilities. They could
be related to each other by one of the standard stochastic volatility models. Although this would
not be strictly consistent with the use of a simple contingent claim model such as Merton, it
might improve the performance of the model, and could be justified as a first approximation to
a more complex model.
Our analysis is based on using firm-specific variables as the basis of the expected default
spread estimate. An alternative would be to use these to determine the relative characteristics
of different firms and rely on average historical results for ratings classes for the typical firm in
a class. This would give a way of using the evidence on expected default for ratings classes in
conjunction with a relative adjustment based on our method.
19
The method proposed in this paper produces expected default probabilities that are consistent
with yield spreads. Delianedis and Geske (1998) present evidence that structural models can help
to predict ratings transitions, even in their risk-neutral form, without the use of the information in
yield spreads. An obvious extension of their work would be to see if the incremental information
provided by using information in yields in the way proposed here can add to this predictive
ability.
X
Summary
This paper uses a new calibration approach to calculate the expected default loss on corporate bonds using the observed market debt spread. It first adjust the spread for factors other
than default risk by using an explicit formula for the tax spread, or subtracting the yield on
a matched AAA-rated corporate bond under the assumption that it is largely default-free. By
”endogenizing” the value of the bond maturity, the method allows to apply the Merton model
to complex capital structures and implicitly to account for many real-life characteristics of the
bond market ignored in the Merton model.
The adjustment for the AAA spread enables us to recover expected spreads due to default
that are similar in magnitude to observed frequencies. It appears to overcome, for the purposes
of calibration, the commonly observed inability of structural debt models to explain spreads
on investment grade debt. We test robustness by varying the structure of the model and the
parametrization, and find that the procedure, though simple, is robust in estimating the default
loss component of the spread. We believe that this robustness comes from the fact that all models
of risky debt must preserve the basic structure of debt and equity: that debt is senior to equity.
This makes the choice of the particular structural model secondary when the goal is splitting the
observed market spread into default and non-default components.
Our estimates suggest that only a small fraction of the spread for a high-grade firm is due
to expected default. For lower-grade firms, this component is larger, and our approach provides
a method for adjusting yields to give expected debt returns. The expected default component
of the spread varies significantly within ratings categories, so using average figures for ratings
categories for individual firms could be misleading.
There are many possible applications of the equilibrium expected default spreads that we
recover. We illustrate these with the use in determining a firm’s expected cost of debt for use
in its cost of capital. We also use the procedure to provide an entirely new set of equity risk
premium estimates, based on data hitherto unused for this purpose. They are consistent with
asset risk premia of about three percent and equity risk premia of between three and nine percent.
20
XI
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21
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22
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23
Appendix A
The Merton Model and the Expected
Return on Debt
The standard form of the Merton model is:
B = V N (−d1 ) − F e−rT N (d2 )
(A1)
where
d1
=
d2
=
√
[ln[V /F ] + (r + σ 2 /2)T ]/σ T
√
d1 − σ T
(A2)
(A3)
V is the value of the assets of the firm, B is the value of the debt, and F is the promised debt payment
(the face value of the debt for a zero-coupon bond). The other variables are as in the main text.
If the continuously compounded promised yield on debt is y then:
F = BeyT
(A4)
Promised debt spread and financial leverage are defined by:
s
=
y−r
(A5)
w
=
B/V
(A6)
Substitution of (A4)-(A6) into (A1)-(A3) yields equations (2)-(4) in the main text.
It follows from (1) that the asset returns are log-normally distributed:
ln(VT /V ) ∼ N (µ − σ 2 /2)T, σ 2 T
where VT is the value of the firm’s assets at date T . Therefore,
VT = V e(µ−
σ2
2
)T +σ
√
TZ
(A7)
where Z is a standard normal variable.
The compound return on debt over T , denoted by ΠD , is determined by:
ΠD T
Be
VZ
T =F
=
Z∞
VT dN (Z) +
−∞
F dN (Z)
VT =F
Evaluating the integrals and taking into account (A6), we obtain:
wV eΠD T = V eµT N (−K1 ) + F N (K2 )
where F is given by (A4) and:
K1
=
K2
=
√
ln [V /F ] + (µ + σ 2 /2)T /σ T
√
K1 − σ T
Rearranging terms in (A8) and noting that (see (3), (4) and (6)):
√
√
K1 = d1 + (µ − r) T /σ = d1 + π T /σ
√
K2 = d2 + π T /σ
we obtain equation (8) of the main text.
24
(A8)
Appendix B
Tax and the yield spread
Elton et al. (2001) claim that a significant proportion of yield spreads is due to tax. Their focus is
the fact that corporate bonds are subject to state taxes in the US, whereas government bonds are not.
They employ a data-fitting procedure to estimate the contribution to spread due to tax. We derive a
simple expression for the tax-induced spread using assumptions very similar to theirs.
Consider a security with maturity tM , at which a random ‘principal’ Fe is repaid. The security may
et at intermediate moments 0 < ti < tM . Let rt denote the
also deliver random ‘coupon’ payments C
i
risk-free discount rate from time 0 to t. Then before-tax value of such security is the expectation of its
before-tax cash flows taken under the equivalent martingale measure:
"
#
h i
X −r t
∗
ti i e
(B1)
B̄ = E
e
Ct + e−rtM tM E ∗ Fe
i
i
Assume that income tax is paid on ‘coupons’, and that the capital gain tax must be paid on the ‘principal’
at maturity, with a symmetric treatment of capital gains and losses. Further, assume that relevant income
and capital gain tax rates are the same and equal to τ .23 Then the after-tax value of the bond is:
"
#
h
i
X −r t
∗
ti i
e
B=E
e
(1 − τ ) Ct + e−rtM tM E ∗ B + (1 − τ ) (Fe − B)
i
i
Substituting for E ∗
B
hP
i
et from (B1):
e−rti ti C
=
h i
h i
(1 − τ ) B̄ − e−rtM tM E ∗ Fe + e−rtM tM τ B + (1 − τ ) e−rtM tM E ∗ Fe =
=
(1 − τ ) B̄ + e−rtM tM τ B
from which it follows that
1−τ
B̄
1 − τ e−rtM tM
Thus, the yield spread due to tax can be computed from the simple formula:
1
1−τ
∆y tax =
ln
tM
1 − τ e−rtM tM
B=
(B2)
(B3)
This depends on the level of interest rates, so we cannot use the Elton et al. results with our data, which
refer to a different period. Also, we have slightly different assumptions, in that they assume principal
acceleration upon early bond default, and deductibility of the principal amount, whereas we have no
acceleration and deductibility of the purchase price. Their estimated tax spreads are almost constant
across ratings, the variations being within 1-2 basis points.
Appendix C
Bankruptcy costs and strategic default in
the Merton model
A simple way to introduce bankruptcy costs strategic bankruptcy into the Merton model is to use
a one-period specification of the Anderson-Sundaresan (1996) model. In this model bankruptcy results
in a fixed cost, H. However, both borrowers and lenders act strategically, and bankruptcy does not
automatically happen when the borrower fails to repay the debt face value, F, to bondholders at maturity.
Instead, the borrower offers the bondholders a payoff which makes them indifferent between accepting
the payment and filing for bankruptcy when F is not repaid in full. In this model costly bankruptcy
23 The relevant tax rate for the tax spread between corporate and government bonds is τ = t (1 − t ) where t
s
g
s
and tg are the state and federal tax rate, respectively.
25
never occurs in equilibrium. The payoff to debt at maturity is:
BT = min{F, max{VT − H, 0}}
The value of such a security is:
B
=
=
Call(H) − Call(F + H) =
h
i
h
i
F +H
+H
V N (dH
) + F e−rT (1 + Θ)N (dF
) − ΘN (dH
1 ) − N (d1
2 )
2
where Θ = H/F is the bankruptcy costs expressed as a fraction of the debt par value, and
√
√
dH
=
− ln w − (s − σ 2 /2)T /σ T − ln Θ/σ T
1
√
√
+H
dF
=
− ln w − (s − σ 2 /2)T /σ T − ln(1 + Θ)/σ T
1
√
di2 = di1 − σ T ,
i = H, F + H
Dividing by B yields an equation for T and σ in terms of w and s:
h
i
h
i
+H
F +H
1 = N (dH
) /w + esT (1 + Θ)N (dF
) − ΘN (dH
1 ) − N (d1
2 )
2
(C1)
It follows that:
h
i
∂E
F +H
V /E = 1 − N (dH
) /(1 − w)
(C2)
1 ) + N (d1
∂V
For a given value of Θ we solve system (C1)-(C2) to determine T and σ which are consistent with the
observed firm’s characteristics w, σ E , and s. The asset return premium on debt is:
σ E /σ =
π = π E σ/σ E
(C3)
The derivation of the expected default loss on the debt is similar to that described in II. The expected
basis points loss over the maturity period is:
δ=−
i
1 h (π−s)T N (K1H ) − N (K1F +H )
ln e
+ (1 + Θ)N (K2F +H ) − ΘN (K2H )
T
w
√
i
i
Kj = dj − π T /σ,
i = H, F + H, j = 1, 2
(C4)
which is the analog of formula (9) for this case.
Appendix D
Dividends in the Merton model
To take into account dividend payments, we amend the Merton model by assuming that the firm
pays continuous dividends that are a constant proportion γ of the value of the firm V .
Repeating the analysis of similar to described in Appendix Aunder this assumption, equations (2)–(9)
become:
e−γT N (−d1 )/w + esT N (d2 ) = 1
(D1)
where
d1
=
d2
=
√
[− ln w − (s + γ − σ 2 /2)T ]/σ T
√
d1 − σ T
(1 − w)σ E = σ[1 − e−γT N (−d1 )]
26
(D2)
(D3)
(D4)
Asset and equity return premia (including distributions) π ≡ µ − r and π E ≡ µE − r are related as
before:
π
σ
=
(D5)
πE
σE
The spread which is due to expected default and should be excluded from the expected return on debt
is thus:
i
√
√
1 h
δ ≡ s − ΠD = − ln e(π−γ−s)T N −d1 − (π − γ) T /σ /w + N d2 + (π − γ) T /σ
(D6)
T
27
Table I. The Default Risk Component of AAA Spreads
The table shows the average AAA spread and the part of the spread due to default.
Estimates are from Huang and Huang (2002) and Delianedis and Geske (2001).
Source
Model
Maturity
Years
Spread
b.p.
Default
b.p.
Default
%
Huang and Huang
Huang and Huang
Huang and Huang
Delianedis and Geske
Longstaff/Schwartz I
Longstaff/Schwartz II
Collin-Dilfresne/Goldstein
Merton/Geske
10
10
10
1-10
63
63
63
36
10
6
11
2
16
10
18
5
Table II. Corporate bond tax spread
The table gives estimates of spreads which is due to differential
taxation of corporate and government bond coupons on the state
level. T is the maturity of the bond in years. rTm is the riskless interest rate over the life of the bond, and τ is the average
applicable state tax rate. Assumed bond maturity is 10 years.
r%
τ = 4%
τ = 4.875%
τ = 6.7%
3.0
4.0
5.0
6.0
7.0
10.7
13.6
16.3
18.6
20.8
13.2
16.8
20.0
22.9
25.5
18.4
23.4
27.9
31.9
35.5
28
Table III. Summary Statistics on Credit Risk Variables
This table reports summary statistics for the final studied sample, which includes
trades on bonds between 7.5 and 10 years to maturity, executed within 3 calendar
months after the last fiscal year end. Leverage is the ratio of book value of debt to
the book value of debt plus the market value of equity on the last business day before
the trade date. Equity volatility is the volatility of daily share price returns over 252
business days before the trade date. Asset volatility is the volatility of asset returns
implied by the Merton (1974) model estimated using the KMV procedure described
in Crosbie and Bohn (2002). Volatilities and spreads are annualized.
AAA
AA
A
BBB
BB
B
All
Mean
Median
Std. Dev.
0.28
0.28
0.10
0.45
0.47
0.21
0.69
0.66
0.39
1.07
0.95
0.50
2.08
1.82
1.11
3.96
3.69
1.19
1.01
0.81
0.80
Mean
Median
Std. Dev.
0.06
0.06
0.03
0.13
0.12
0.07
0.25
0.24
0.14
0.35
0.35
0.14
0.49
0.44
0.20
0.63
0.61
0.15
0.31
0.29
0.17
Equity volatility
Mean
Median
Std. Dev.
0.33
0.33
0.17
0.30
0.28
0.10
0.33
0.29
0.12
0.33
0.30
0.11
0.39
0.38
0.12
0.62
0.62
0.18
0.34
0.30
0.12
Maturity
Mean
Median
Std. Dev.
8.53
8.53
0.21
8.81
8.81
0.75
9.00
9.12
0.75
8.86
8.88
0.74
8.91
9.15
0.69
8.87
9.01
0.87
8.92
8.99
0.74
Duration
Mean
Median
Std. Dev.
6.56
6.56
0.16
6.69
6.71
0.48
6.76
6.79
0.52
6.57
6.58
0.47
6.47
6.48
0.56
6.14
5.99
0.54
6.64
6.65
0.52
Mean
Median
Std. Dev.
0.24
0.24
0.07
0.25
0.25
0.06
0.25
0.22
0.09
0.25
0.23
0.10
0.26
0.25
0.10
0.49
0.39
0.30
0.25
0.23
0.10
2
231
1088
1003
266
42
2632
Spread
Leverage
KMV asset
volatility
N
29
30
None
Tax
None
Tax
AAA
None
Tax
AAA
None
Tax
AAA
None
Tax
AAA
None
Tax
AAA
AAA
AA
A
BBB
BB
B
Rating
Spread
adjustment
470.00
450.04
407.00
320.00
300.04
257.00
194.00
174.04
131.00
123.00
103.04
60.00
91.00
71.04
28.00
63.00
43.04
s
b.p.
(1)
0.66
0.54
0.43
0.32
0.21
0.13
8.76
7.30
6.55
5.99
5.60
5.38
Model
inputs
w
πE
%
(2)
(3)
0.57
0.38
0.31
0.29
0.28
0.27
(4)
σE
9.87
9.34
8.27
43.05
38.05
29.25
65.38
53.75
35.22
49.86
39.63
22.97
51.97
39.81
19.60
51.08
37.22
0.30
0.29
0.28
0.27
0.26
0.25
0.23
0.22
0.21
0.23
0.22
0.21
0.23
0.23
0.22
0.24
0.24
4.54
4.48
4.33
5.07
4.96
4.73
4.91
4.78
4.51
4.68
4.58
4.37
4.74
4.67
4.51
4.83
4.78
Model
parameters
T
σ
π
yrs.
%
(5)
(6)
(7)
271.34
260.40
236.70
93.02
88.44
78.15
26.85
25.32
21.28
15.58
13.92
9.46
9.45
7.98
3.87
5.36
4.01
δ
b.p.
(8)
0.58
0.55
0.50
0.29
0.28
0.24
0.14
0.13
0.11
0.13
0.11
0.08
0.10
0.09
0.04
0.09
0.06
(9)
Model
δ/s
51.08
48.96
44.37
51.94
47.78
39.37
28.40
24.51
17.24
15.75
12.65
6.82
10.37
7.67
2.71
6.11
3.90
P
%
(10)
352.87
132.24
24.40
8.33
5.41
3.95
0.75
0.41
0.13
0.07
0.06
0.06
43.91
20.63
4.39
1.55
0.99
0.77
Default loss
Historical
δh
δ h /s
Ph
b.p.
%
(11)
(12)
(13)
40.90
14.00
4.80
EG
δ EG
b.p.
(14)
The table gives model-predicted expected default losses and probabilities for generic bonds in 4 rating groups. Three methods,
indicated in the second column, are used to adjust the observed spread for non-default factors: no adjustment, subtracting a tax
spread, and subtracting the average spread on AAA bonds. The inputs are mean values in each rating class of the promised spread
on debt s, leverage w, and assumed equity risk premium, π E , as used in Huang and Huang (2002); and the median volatility of
equity σ E from our studied sample. T , σ and π are model-implied maturity, volatility of assets and return on assets. The outputs
are the expected percentage default loss on debt δ and the expected probability of default P . Average historical default losses and
default frequencies δ h and Ph are reported for comparison.
Table IV. Estimated expected default losses
31
28
18.9
37.1
257
225
289
91.0
81.9
100.1
320
288
352
s
b.p.
(1)
0.48
0.59
0.54
0.19
0.23
0.21
Model
inputs
s adj.
w
b.p.
(2)
(3)
6.57
8.03
7.30
5.04
6.16
5.60
πE
%
(4)
0.34
0.42
0.38
0.25
0.30
29.25
24.03
35.56
30.23
28.71
29.25
29.25
52.26
18.35
19.60
15.80
23.44
20.88
18.51
19.60
19.60
27.61
14.53
(6)
(5)
0.28
T
σE
0.25
0.24
0.26
0.26
0.23
0.25
0.25
0.24
0.26
0.22
0.22
0.22
0.23
0.22
0.22
0.22
0.20
0.24
4.73
4.56
4.90
5.01
4.44
4.26
5.20
5.06
4.49
4.51
4.48
4.55
4.62
4.40
4.06
4.96
4.54
4.49
δ/s
%
(10)
4.25
3.38
4.86
3.99
4.49
5.33
3.37
2.30
6.40
17.64
14.34
20.92
18.90
16.57
17.64
17.64
24.70
13.13
0.22
0.22
0.22
0.23
0.22
0.22
0.22
0.20
0.24
4.51
4.48
4.54
4.62
4.40
4.06
4.96
4.53
4.49
78.15
70.10
85.86
75.96
80.10
90.00
67.47
51.14
101.86
24.42
24.34
24.39
23.74
25.03
28.13
21.08
15.98
31.83
16.94
15.04
18.74
18.53
15.51
16.94
16.94
24.94
11.80
0.24
0.23
0.25
0.26
0.22
0.24
0.24
0.23
0.25
4.60
4.45
4.74
4.91
4.27
4.14
5.06
4.86
4.39
Panel B: Typical BB-rated bonds
3.87
2.77
4.87
3.63
4.09
4.85
3.06
2.09
5.83
92.71
81.59
104.14
88.37
97.28
104.13
82.24
69.45
113.78
4.16
2.97
5.26
3.90
4.41
5.16
3.33
2.32
6.15
Bankruptcy
θ = 0.05, Endogenous T
T
σ
π
δ
%
b.p.
(11) (12) (13)
(14)
Panel A: Typical AA-rated bonds
Basic model
θ=0
σ
π
δ
%
b.p.
(7)
(8)
(9)
28.97
28.33
29.59
27.62
30.40
32.54
25.70
21.70
35.56
4.57
3.62
5.26
4.28
4.85
5.68
3.66
2.55
6.76
0.12
0.11
0.13
0.15
0.10
0.12
0.12
0.17
0.07
0.36
0.25
0.43
0.42
0.30
0.36
0.36
0.54
0.19
0.23
0.23
0.24
0.25
0.22
0.23
0.23
0.22
0.25
0.22
0.22
0.22
0.23
0.22
0.22
0.22
0.20
0.24
4.45
4.35
4.55
4.76
4.14
4.01
4.90
4.57
4.35
4.49
4.47
4.52
4.61
4.38
4.05
4.94
4.50
4.49
108.31
92.45
124.55
106.12
110.63
119.18
98.22
97.06
118.46
5.99
3.77
8.37
5.91
6.07
7.09
5.04
4.87
7.09
33.85
32.10
35.38
33.16
34.57
37.24
30.69
30.33
37.02
6.58
4.60
8.36
6.49
6.67
7.80
5.54
5.36
7.79
costs and strategic default
T = 10, Endogenous θ
δ/s
θ
σ
π
δ
δ/s
%
%
b.p.
%
(15)
(16) (17) (18)
(19)
(20)
The table reports the sensitivity of the model-predicted expected default losses and probabilities to inputs and model assumptions, for typical
A and BB bonds. Observed spreads are adjusted by subtracting the AAA spread of 63 basis points. Reported are results for the basic model,
as well as the Anderson-Sundaresan procedure for two levels of bankruptcy costs. The inputs are sample median values in each rating class of
the promised spread on debt s, leverage w, assumed equity risk premium, π E , and the volatility of equity σ E . T , σ and π are model-implied
maturity, volatility of assets and return on assets. The outputs are the expected percentage default loss on debt δ and the expected probability
of default P .
Table V. Sensitivity and Robustness of Estimated Default Loss
Table VI. Sensitivity of Estimated Default Loss to Dividend Yield
The table reports the sensitivity of the model-predicted expected default losses for
generic bonds in 4 rating groups to the assumed instantaneous equity dividend yield
g. Mean values of the promised spread on debt s in each rating class, adjusted by
subtracting the median AAA spread of 63 b.p., are used as inputs. Other inputs are
mean leverage w and assumed equity risk premium π E used in Huang and Huang
(2002), and the median volatility of equity σ E from our sample. γ, T , σ and π are the
model-implied asset payout ratio, maturity, volatility of assets and return on assets.
The outputs are the expected basis point loss in debt yield due to default δ, and the
proportion of the promised spread which is due to default, δ/s.
g
%
γ
%
T
σ
π
%
δ
b.p.
δ/s
%
Panel A: Typical AA-rated bonds
s=91 bp, s adj.=28 bp, w = 0.21, π E = 5.60%, σ E = 0.28
0.00
0.00 19.60 0.22 4.51
3.87
4.25
1.00
0.79 16.26 0.22 4.51
4.08
4.48
2.00
1.58 14.02 0.22 4.50
3.02
3.32
3.00
2.36 12.40 0.22 4.50
0.42
0.46
Panel A: Typical A-rated bonds
s=123 bp, s adj.=60 bp, w = 0.32, π E = 5.99%, σ E = 0.29
0.00
1.00
2.00
3.00
0.00
0.68
1.36
2.04
22.39
17.37
14.41
12.41
0.21
0.21
0.21
0.21
4.36
4.34
4.32
4.31
9.78
11.03
10.22
7.38
7.95
8.97
8.31
6.00
Panel C: Typical BBB-rated bonds
s=194 bp, s adj.=131 bp, w = 0.43, π E = 6.55%, σ E = 0.31
0.00
1.00
2.00
3.00
0.00
0.57
1.14
1.71
33.28
22.14
17.07
14.07
0.21
0.21
0.21
0.21
4.51
4.42
4.37
4.33
22.68
28.22
29.68
28.31
11.69
14.54
15.30
14.59
Panel D: Typical BB-rated bonds
s=320 bp, s adj.=257 bp, w = 0.54, π E = 7.30%, σ E = 0.38
0.00
1.00
2.00
3.00
0.00
0.46
0.93
1.40
29.25
19.36
14.91
12.28
0.25
0.24
0.24
0.23
4.73
4.60
4.51
4.45
78.15
90.40
96.71
99.78
24.42
28.25
30.22
31.18
Panel E: Typical B-rated bonds
s=470 bp, s adj.=407 bp, w = 0.66, π E = 8.76%, σ E = 0.57
0.00
1.00
2.00
3.00
0.00
0.34
0.68
1.02
8.22
7.03
6.20
5.57
0.28
0.28
0.27
0.27
4.30
4.24
4.19
4.15
32
237.36
243.80
248.48
251.90
50.50
51.87
52.87
53.60
33
Mean
Median
Std. Dev.
N
Mean
Median
Std. Dev.
N
Mean
Median
Std. Dev.
N
Mean
Median
Std. Dev.
N
Mean
Median
Std. Dev.
N
Mean
Median
Std. Dev.
N
AA
A
BBB
BB
B
All
112.79
88.25
78.63
2161
395.65
368.59
119.35
42
208.13
182.16
110.75
255
110.13
96.14
45.63
949
78.06
70.40
29.36
825.00
s
b.p.
(1)
57.06
53.24
12.08
90.00
61.55
38.00
76.24
335.46
315.65
121.24
155.07
130.42
109.85
58.56
47.14
43.48
27.63
21.79
25.06
0.34
0.32
0.17
0.63
0.61
0.15
0.49
0.45
0.19
0.36
0.35
0.14
0.27
0.26
0.14
6.43
6.55
0.55
8.76
8.76
0.00
7.30
7.30
0.00
6.55
6.55
0.00
5.99
5.99
0.00
Model inputs
s adj.
w
πE
b.p.
%
(2)
(3)
(4)
11.08 0.17 5.60
7.90
0.17 5.60
9.99
0.08 0.00
0.33
0.30
0.12
0.62
0.62
0.18
0.39
0.37
0.12
0.33
0.30
0.10
0.31
0.28
0.11
24.49
14.67
48.44
16.33
4.44
23.64
28.57
19.01
33.26
28.22
15.03
68.19
19.87
14.10
18.81
0.23
0.21
0.09
0.29
0.29
0.09
0.23
0.21
0.08
0.23
0.21
0.09
0.23
0.21
0.10
4.52
4.58
0.84
4.35
4.10
1.37
4.49
4.85
1.20
4.55
4.58
0.76
4.49
4.52
0.79
(8)
4.71
4.68
0.41
(6)
19.67
14.24
16.87
(5)
0.28
0.27
0.09
(7)
0.24
0.22
0.08
Model parameters
T
σ
π
σE
18.88
5.67
44.04
195.74
157.15
124.44
55.58
31.75
69.17
13.83
7.03
20.79
6.19
2.31
10.39
10.78
6.44
12.50
47.28
48.90
18.36
22.04
20.04
16.18
10.70
7.56
10.23
6.40
3.41
7.53
Model estimates
δ
δ/s
b.p.
%
(9)
(10)
1.78
2.73
0.54
1.19
2.98
3.97
The table gives model-predicted expected default losses and probabilities for bonds in the sample by ratings.
The sample consists of senior unsecured bonds with 7.5-10 years to maturity. The observed spread was adjusted
for non-default factor by subtracting the sample median spread on AAA bonds for the year of trade.
Table VII. Expected losses for individual bonds in the sample
34
Mean
Median
Std. Dev.
N
Mean
Median
Std. Dev.
N
Mean
Median
Std. Dev.
N
Mean
Median
Std. Dev.
N
Mean
Median
Std. Dev.
N
Mean
Median
Std. Dev.
N
AA
A
BBB
BB
B
All
115.74
91.45
80.15
1702
534.86
496.50
96.30
14
280.53
244.47
125.31
119.00
116.68
103.32
45.83
815
82.58
74.14
29.51
696
61.13
57.98
11.96
58
s
b.p.
(1)
64.75
41.40
77.64
475.14
438.34
90.08
226.46
185.09
125.20
65.53
52.13
43.03
32.01
25.30
24.92
15.84
13.02
9.46
0.34
0.32
0.17
0.61
0.60
0.14
0.55
0.50
0.21
0.37
0.37
0.14
0.28
0.28
0.14
0.18
0.20
0.08
0.33
0.30
0.11
0.73
0.82
0.22
0.42
0.41
0.11
0.33
0.30
0.10
0.31
0.28
0.10
0.28
0.27
0.09
Model inputs
s adj.
w
σE
b.p.
(2)
(3)
(4)
27.07
22.38
44.05
367.05
368.47
4.50
134.93
134.48
1.40
23.52
23.58
0.80
7.74
7.78
0.43
5.45
5.46
0.07
δ
b.p.
(5)
27.12
16.38
53.73
14.91
2.35
21.72
35.60
19.22
43.71
31.04
16.59
73.08
21.68
15.32
19.64
22.80
15.54
18.51
0.23
0.21
0.09
0.35
0.37
0.08
0.24
0.23
0.09
0.23
0.20
0.08
0.23
0.20
0.09
0.24
0.22
0.08
3.27
2.20
3.48
3.83
2.91
3.47
2.31
1.38
2.72
3.02
2.03
3.22
3.78
2.61
3.88
2.51
2.09
2.28
Model parameters
T
σ
π
%
(6)
(7)
(8)
4.79
3.18
5.12
8.53
6.36
8.34
4.30
2.44
4.31
4.49
2.98
4.87
5.30
3.42
5.51
3.06
2.68
2.95
Model
estimate
πE , %
(9)
The table gives cost of equity values consistent with Moody’s historical default data and an average historical
recovery for senior unsecured bonds of 48%. The sample consists of senior unsecured bonds with 7.5-10 years
to maturity.The observed spread was adjusted for non-default factor by subtracting the sample median spread
on AAA bonds for the year of trade.
Table VIII. Estimated cost of equity for individual bonds in the sample