Unformatted text preview: THE JOURNAL OF FINANCE • VOL. LIX, NO. 2 • APRIL 2004 Default Risk in Equity Returns
MARIA VASSALOU and YUHANG XING∗
This is the first study that uses Merton’s (1974) option pricing model to compute
default measures for individual firms and assess the effect of default risk on equity
returns. The size effect is a default effect, and this is also largely true for the bookto-market (BM) effect. Both exist only in segments of the market with high default
risk. Default risk is systematic risk. The Fama–French (FF) factors SMB and HML
contain some default-related information, but this is not the main reason that the FF
model can explain the cross section of equity returns. A FIRM DEFAULTS WHEN IT FAILS to service its debt obligations. Therefore, default
risk induces lenders to require from borrowers a spread over the risk-free rate
of interest. This spread is an increasing function of the probability of default of
the individual firm.
Although considerable research effort has been put toward modeling default
risk for the purpose of valuing corporate debt and derivative products written on
it, little attention has been paid to the effects of default risk on equity returns.1
The effect that default risk may have on equity returns is not obvious, since
equity holders are the residual claimants on a firm’s cash f lows and there is no
promised nominal return in equities.
Previous studies that examine the effect of default risk on equities focus
on the ability of the default spread to explain or predict returns. The default
spread is usually defined as the yield or return differential between long-term
BAA corporate bonds and long-term AAA or U.S. Treasury bonds.2 However,
∗ Vassalou is at Columbia University and Xing is at Rice University. This paper was presented
at the 2002 Western Finance Association Meetings in Park City, Utah; at London School of Economics; Norwegian School of Management; Copenhagen Business School; Ohio State University;
Dartmouth College; Harvard University (Economics Department); the 2003 NBER Asset Pricing
Meeting in Chicago; and the Federal Reserve Bank of New York. We would like to thank John
Campbell, John Cochrane, Long Chen (WFA discussant), Ken French, David Hirshleifer, Ravi
Jagannathan (NBER discussant), David Lando, Lars Tyge Nielsen, Lubos Pastor, Jay Ritter, Jay
Shanken, and Jeremy Stein for useful comments. Special thanks are due to Rick Green and an
anonymous referee for insightful comments and suggestions that greatly improved the quality and
presentation of our paper. We are responsible for any errors.
For papers that model default risk see for instance, Madan and Unal (1994), Duffie and Singleton (1995, 1997), Jarrow and Turnbull (1995), Longstaff and Schwartz (1995), Zhou (1997), Lando
(1998), and Duffee (1999), among others.
For instance, many studies have shown that the yield spread between BAA and AAA corporate
bond spread can predict expected returns in stocks and bonds. Such studies include those of Fama
and Schwert (1977), Keim and Stambaugh (1986), Campbell (1987), and Fama and French (1989), 831 832 The Journal of Finance as Elton et al. (2001) show, much of the information in the default spread is
unrelated to default risk. In fact, as much as 85 percent of the spread can be
explained as reward for bearing systematic risk, unrelated to default. Furthermore, differential taxes seem to have a more important inf luence on spreads
than expected loss from default. These results lead us to conclude that, independent of whether the default spread can explain, predict, or otherwise relate
to equity returns, such a relation cannot be attributed to the effects that default
risk may have on equities. In other words, we still know very little about how
default risk affects equity returns.
The purpose of this paper is to address precisely this question. Instead of
relying on information about default obtained from the bonds market, we estimate default likelihood indicators (DLI) for individual firms using equity data.
These DLI are nonlinear functions of the default probabilities of the individual
firms. They are calculated using the contingent claims methodology of Black
and Scholes (1973) (BS) and Merton (1974). Consistent with the Elton et al.
(2001) study, we find that our measure of default risk contains very different
information from the commonly used aggregate default spreads. This occurs
despite the fact that our DLI can indeed predict actual defaults.
We find that default risk is intimately related to the size and book-to-market
(BM) characteristics of a firm. Our results point to the conclusion that both the
size and BM effects can be viewed as default effects. This is particularly the
case for the size effect.
The size effect exists only within the quintile with the highest default risk.
In that segment of the market, the return difference between small and big
firms is of the order of 45 percent per annum (p.a.). The small stocks in the
high-default-risk quintile are typically among the smallest of the small firms
and have the highest BM ratios. Furthermore, even within the high-defaultrisk quintile, small firms have much higher default risk than big firms, and
default risk decreases monotonically as size increases.
A similar result is obtained for the BM effect. The BM effect exists only
in the two quintiles with the highest default risk. Within the highest default
risk quintile, the return difference between value (high BM) and growth (low
BM) stocks is around 30 percent p.a., and goes down to 12.7 percent for the
stocks in the second highest default risk quintile. There is no BM effect in the
remaining stocks of the market. Again, the value stocks in these categories have
the highest BMs of all stocks in the market, and the smallest size. Value stocks
have much higher default risk than growth stocks, and there is a monotonic
relation between BM and default risk.
We also find that high-default-risk firms earn higher returns than low default
risk firms, only to the extent that they are small in size and high BM. If these
firm characteristics are not met, they do not earn higher returns than low
default risk firms, even if their risk of default is actually very high.
among others. In addition, Chen, Roll, and Ross (1986), Fama and French (1993), Jagannathan and
Wang (1996), and Hahn and Lee (2001) consider variations of the default spread in asset-pricing
tests. Default Risk in Equity Returns 833 We finally examine whether default risk is systematic. We find that it is
indeed systematic and therefore priced in the cross section of equity returns.
Fama and French (1996) argue that the SMB and HML factors of the Fama
and French (1993) (FF) model proxy for financial distress. Our asset-pricing
results show that, although SMB and HML contain default-related information,
this is not the reason that the FF model can explain the cross section. SMB and
HML appear to contain important priced information, unrelated to default risk.
Several studies in the corporate finance literature examine whether default
risk is systematic, but their results are often conf licting. Denis and Denis
(1995), for example, show that default risk is related to macroeconomic factors
and that it varies with the business cycle. This result is consistent with ours
since our measure of default risk also varies with the business cycle. Opler and
Titman (1994) and Asquith, Gertner, and Sharfstein (1994), on the other hand,
find that bankruptcy is related to idiosyncratic factors and therefore does not
represent systematic risk. The asset-pricing results of the current study show
that default risk is systematic.
Contrary to the current study, previous research has used either accounting
models or bond market information to estimate a firm’s default risk and in some
cases has produced different results from ours.
Examples of papers that use accounting models include those of Dichev (1998)
and Griffin and Lemmon (2002). Dichev examines the relation between
bankruptcy risk and systematic risk. Using Altman’s (1968) Z-score model and
Ohlson’s (1980) conditional logit model, he computes measures of financial distress and finds that bankruptcy risk is not rewarded by higher returns. He
concludes that the size and BM effects are unlikely to proxy for a distress factor related to bankruptcy. A similar conclusion is reached in the case of the BM
effect by Griffin and Lemmon (2002), who use Olson’s model and conclude that
the BM effect must be due to mispricing.
There are several concerns about the use of accounting models in estimating
the default risk of equities. Accounting models use information derived from
financial statements. Such information is inherently backward looking, since
financial statements aim to report a firm’s past performance, rather than its
future prospects. In contrast, Merton’s (1974) model uses the market value of a
firm’s equity in calculating its default risk. It also estimates its market value
of debt, rather than using the book value of debt, as the accounting models do.
Market prices ref lect investors’ expectations about a firm’s future performance.
As a result, they contain forward-looking information, which is better suited
for calculating the likelihood that a firm may default in the future.
In addition, and most importantly, accounting models do not take into account
the volatility of a firm’s assets in estimating its risk of default. Accounting models imply that firms with similar financial ratios will have similar likelihoods
of default. This is not the case in Merton’s model, where firms may have similar
levels of equity and debt, but very different likelihoods to default, if the volatilities of their assets differ. Clearly, the volatility of a firm’s assets provides crucial information about the firm’s probability to default. Campbell et al. (2001)
demonstrate that firm level volatility has trended upward since the mid-1970s. 834 The Journal of Finance Furthermore, using data from 1995 to 1999, Campbell and Taksler (2003) show
that firm level volatility and credit ratings can explain equally well the crosssectional variation in corporate bond yields. Clearly, a firm’s volatility is a key
input in the Black–Scholes option-pricing formula.
As mentioned, an alternative source of information for calculating default
risk measures is the bonds market. One may use bond ratings or individual
spreads between a firm’s debt issues and an aggregate yield measure to deduce
the firm’s risk of default. When a study uses bond downgrades and upgrades as a
measure of default risk, it usually relies implicitly on the following assumptions:
that all assets within a rating category share the same default risk and that
this default risk is equal to the historical average default risk. Furthermore,
it assumes that it is impossible for a firm to experience a change in its default
probability, also without experiencing a rating change.3
Typically, however, a firm experiences a substantial change in its default risk
prior to its rating change. This change in its probability of default is observed
only with a lag, and measured crudely through the rating change. Bond ratings
may also represent a relatively noisy estimate of a firm’s likelihood to default
because equity and bond markets may not be perfectly integrated, and because
the corporate bond market is much less liquid than the equity market.4 Merton’s
model does not require any assumptions about the integration of bond and
equity markets or their efficiencies, since all the information needed to calculate
the default risk measures is obtained from equities.
Examples of studies that use bond ratings to examine the effect of upgrades
and downgrades on equity returns include those of Holthausen and Leftwich
(1986), Hand, Holthausen, and Leftwich (1992), and Dichev and Piotroski
(2001), among others. The general finding is that bond downgrades are followed by negative equity returns. The effect of an increase in default risk on
the subsequent equity returns is not examined in the current study.
The remainder of the paper is organized as follows. Section I discusses the
methodology used to compute DLI for individual firms. Section II describes
the data and provides summary statistics. Section III examines the ability of
the DLI to predict actual defaults. In Section IV we report results on the performance of portfolios constructed on the basis of default-risk information. In
Section V, we provide asset-pricing tests that examine whether default risk is
priced. We conclude in Section VI with a summary of our results.
I. Measuring Default Risk
A. Theoretical Model
In Merton’s (1974) model, the equity of a firm is viewed as a call option on
the firm’s assets. The reason is that equity holders are residual claimants on
3 See also, Kealhofer, Kwok, and Weng (1998).
For instance, Kwan (1996) shows that lagged stock returns can predict current bond yield
changes. However, Hotchkiss and Ronen (2001) find that although the correlation between bond
and stock returns is positive and significant, there is no causal relation between the two markets.
4 Default Risk in Equity Returns 835 the firm’s assets after all other obligations have been met. The strike price of
the call option is the book value of the firm’s liabilities. When the value of the
firm’s assets is less than the strike price, the value of equity is zero.
Our approach in calculating default risk measures using Merton’s model is
very similar to the one used by KMV and outlined in Crosbie (1999).5 We assume
that the capital structure of the firm includes both equity and debt. The market
value of a firm’s underlying assets follows a geometric Brownian motion (GBM)
of the form:
dV A = µV A d t + σ A V A dW , (1) where VA is the firm’s assets value, with an instantaneous drift µ, and an
instantaneous volatility σA . A standard Wiener process is W .
We denote by Xt the book value of the debt at time t, that has maturity equal
to T . As noted earlier, Xt plays the role of the strike price of the call, since the
market value of equity can be thought of as a call option on VA with time to
expiration equal to T . The market value of equity, VE , will then be given by the
Black and Scholes (1973) formula for call options:
VE = V A N (d 1 ) − X e−rT N (d 2 ), (2) where
ln(V A / X ) + r + σ A T
σA T √
d2 = d1 − σ A T , (3) r is the risk-free rate, and N is the cumulative density function of the standard
To calculate σA we adopt an iterative procedure. We use daily data from the
past 12 months to obtain an estimate of the volatility of equity σE , which is
then used as an initial value for the estimation of σA . Using the Black–Scholes
formula, and for each trading day of the past 12 months, we compute VA using
VE as the market value of equity of that day. In this manner, we obtain daily
values for VA . We then compute the standard deviation of those VA ’s, which is
used as the value of σA , for the next iteration. This procedure is repeated until
the values of σA from two consecutive iterations converge. Our tolerance level
for convergence is 10E-4. For most firms, it takes only a few iterations for σA to
converge. Once the converged value of σA is obtained, we use it to back out VA
through equation (2).
There are two main differences between our approach and the one used by KMV. They use a
more complicated method to assess the asset volatility than we do, which incorporates Bayesian
adjustments for the country, industry, and size of the firm. They also allow for convertibles and
preferred stocks in the capital structure of the firm, whereas we allow only equity, as well as shortand long-term debt. 836 The Journal of Finance The above process is repeated at the end of every month, resulting in the
estimation of monthly values of σA . The estimation window is always kept equal
to 12 months. The risk-free rate used for each monthly iterative process is the
1-year T -bill rate observed at the end of the month.
Once daily values of VA are estimated, we can compute the drift µ, by calculating the mean of the change in lnVA .
The default probability is the probability that the firm’s assets will be less
than the book value of the firm’s liabilities. In other words,
Pdef , t = Prob V A, t +T ≤ X t |V A, t = Prob ln V A, t +T ≤ ln ( X t ) |V A, t . (4) Since the value of the assets follows the GBM of equation (1), the value of the
assets at any time t is given by:
T + σ A T εt +T , (5) and εt +T ∼ N (0, 1). (6) ln V A, t +T = ln V A, t + µ −
εt +T = W (t + T ) − W (t )
2 Therefore we can rewrite the default probability as follows:
Pdef , t = Prob ln V A, t − ln ( X t ) + µ − 2
2 Pdef , t σ2
V A, t
+ µ− A
2 = Prob −
√ σA T √
T + σ A T εt + T ≤ 0 T ≥ εt + T . (7) We can then define the distance to default (DD) as follows:
ln(V A, t / X t ) + µ − σ A T
D Dt =
σA T (8) Default occurs when the ratio of the value of assets to debt is less than 1, or
its log is negative. The DD tells us by how many standard deviations the log
of this ratio needs to deviate from its mean in order for default to occur. Notice
that although the value of the call option in (2) does not depend on µ, DD does.
This is because DD depends on the future value of assets which is given in
We use the theoretical distribution implied by Merton’s model, which is the
normal distribution. In that case, the theoretical probability of default will be
given by: Default Risk in Equity Returns 837 Pdef 12
ln(V A, t / X t ) + µ − σ A T 2
= N (−DD) = N −
√ σA T (9) Strictly speaking, Pdef is not a default probability because it does not correspond
to the true probability of default in large samples. In contrast, the default
probabilities calculated by KMV are indeed default probabilities because they
are calculated using the empirical distribution of defaults. For instance, in the
KMV database, the number of companies times the years of data is over 100,000,
and includes more than 2,000 incidents of default. We have a much more limited
database. For that reason, we do not call our measure default probability, but
It is important to note that the difference between our measure of default risk
and that produced by KMV is not material for the purpose of our study. The DLI
of a firm is a positive nonlinear function of its default probability. Since we use
our measure of default risk to examine the relation between default risk and
equity returns rather than price debt or credit risk derivatives, this nonlinear
transformation cannot affect the substance of our results.
II. Data and Summary Statistics
We use the COMPUSTAT annual files to get the firm’s “Debt in One Year”
and “Long-Term Debt” series for all companies. COMPUSTAT starts reporting
annual financial data in 1963. However, prior to 1971, only a few hundred firms
have debt data available. Therefore, we start our analysis in 1971.
As book value of debt we use the “Debt in One Year” plus half the “LongTerm Debt.” It is important to include long-term debt in our calculations for
two reasons. First, firms need to service their long-term debt, and these interest payments are part of their short-term liabilities. Second, the size of the
long-term debt affects the ability of a firm to roll over its short-term debt, and
therefore reduce its risk of default. How much of the long-term debt should enter our calculations is arbitrary, since we do not observe the coupon payments
of the individual firms. KMV uses 50 percent and argues that this choice is
sensible, and captures adequately the financing constraints of firms.7 We do
Our procedure also differs from the one used in KMV with respect to the way we calculate the
distance to default. Whereas we use the formula that follows from the Black-Scholes model, KMV
uses the one below: DD = (Market value of Assets − Default Point)/(Market value of Assets × Asset
To obtain an idea of how sensitive our results would be to our choice about the proportion of
long-term debt included in our calculations of DLI, we performed the following test. We examined
the variation of the ratio of long-term debt to total debt across size and BM quintiles. If there is
no substantial variation, our results should not be influenced by the choice we make. We find that
there is virtually no variation across BM portfolios. There is a small variation across size portfolios,
with the small firms having a somewhat smaller ratio than the big firms. However, the small firms
have also a larger standard deviation than the big firms. Overall, the difference in the ratios is not
deemed large enough to alter the qualitative results of the paper. The Journal of Finance Default Likelihood Indicator 838 0.1802
n7 1 0.0002 Time
Figure 1. Aggregate default likelihood indicator. The aggregate DLI is defined as the simple
average of the DLI of all firms. The shaded areas denote recession periods, as defined by NBER. We use annual data for the book value of debt. To avoid problems related to
reporting delays, we do not use the book value of debt of the new fiscal year,
until 4 months have elapsed from the end of the previous fiscal year.8 This
is done in order to ensure that all information used to calculate our default
measures was available to the investors at the time of the calculation.
We get the daily market values for firms from the CRSP daily files. The book
value of equity information is extracted from COMPUSTAT. Each month, the
BM ratio of a firm is the 6-month prior book value of equity divided by the
current month’s market value of equity. Firms with negative BM ratios are
excluded from our sample.
As risk-free rate for the computation of DLI, we use monthly observations
of the 1-year Treasury Bill rate obtained from the Federal Reserve Board
Table I reports the number of firms per year for which DLI could be calculated,
as well as the number of firms that filed for bankruptcy (Chapter 11) or were
The aggregate default likelihood measure P(D) is defined as a simple average
of the DLI of all firms. A graph of the P(D) is provided in Figure 1 for the whole
sample period (January 1971 to December 1999). The shaded areas represent
recession periods as defined by the NBER. The graph shows that default probabilities vary greatly with the business cycle and increase substantially during
8 The SEC requires firms to report 10K within three months after the end of the fiscal year, but
a small percentage of firms report it with a longer delay. Default Risk in Equity Returns 839 Table I Firm Data
The second column of the table reports the number of firms each year for which DLI could be
calculated. The third column reports the number of firms that filed for bankruptcy (Chapter 11),
while the fourth reports the number of liquidations.
Year No. of Stocks in Sample No. of Bankruptcy No. of Liquidations 1971
13 We define the aggregate survival rate, SV as 1 − P(D). The change in aggregate survival rate (SV) at time t is given by SV t − SV t−1 . Summary statistics
for SV and (SV) are presented in Panel A of Table II.
The default return spread is from Ibbotson Associates, and it is defined as the
return difference between BAA Moody’s rated bonds and AAA Moody’s rated
bonds. Similarly, the default yield spread is defined as the yield difference between Moody’s BAA bonds and Moody’s AAA bonds. The series is obtained from
the Federal Reserve Bank of St. Louis. The change in spread (spread) is obtained from Hahn and Lee (2001). The spread in Hahn and Lee is defined as the
difference in the yields between Moody’s BAA bonds and 10-year government
bonds. (spread) is the change in that spread.
Panel B of Table II provides the correlation coefficients between the abovedefined default spreads and (SV). The correlations are very low and reveal 840 The Journal of Finance
Table II Summary Statistics
In this table, SV denotes the survival rate and it is equal to 1 minus the aggregate DLI. The
variable (SV) is the change in the survival rate. Mean, Std, Skew, Kurt, and Auto refer to the
mean, standard deviation, skewness, kurtosis and autocorrelation at lag one, respectively. The
variable RDEF is the return difference between Moody’s BAA corporate bonds and AAA corporate bonds. The variable YDEF is the yield difference between Moody’s BAA bonds and Moody
AAA corporate bonds. The variable (spread) is the default measure used in Hahn and Lee
(2001) which is defined as: (spread) = (yBAA − yTB ) − (yBAA − yTB ), where yBAA is the yield of
the Moody’s BAA corporate bonds, and yTB is yield on 10-year government bonds. The variable
EMKT denotes the value-weighted excess return on the stock market portfolio over the risk-free
rate; SMB and HML are the Fama and French (1993) factors. Size denotes the firm’s market capitalization and BM its book-to-market ratio. DLI is the firm’s DLI. T -values are calculated from
Newey and West (1987) standard errors, which are corrected for heteroskedasticity and serial
correlation up to three lags. The R2 ’s are adjusted for degrees of freedom. In Panel F, SMB and
HML are the Fama–French factors. When the expression (within sample) appears next to SMB
and HML, it means that these factors are calculated using the data in the current study and
following exactly the same methodology as in Fama and French. “Auto” refers to the first-order
Panel A: Summary Statistics on Aggregate Survival Indicator (SV)
(SV) Std Skew Kurt Auto 0.9579
0.1657 Panel B: Correlation between (SV) and Other Default Measures (SV)
(Spread) RDEF YDEF 1
−0.113 Panel C: Correlation between
−0.1709 EMKT EMKT SMB HML 1
−0.1422 1 SMB
t-value 1 (SV) and Other Factors Panel D: Time-Series Regression of Fama–French Factors on
Factor (Spread) 0.0064
0.0264 Default Risk in Equity Returns 841 Table II—Continued
Panel E: Firm Characteristic and Default Risk
SIZE BM DLI Average cross-sectional correlation between firm characteristics
0.4332 1 Average time-series correlation between firm characteristics
−0.4119 1 1
0.432 Panel F: SMB and HML within Sample
Mean t-value Std Auto SMB
SMB within Sample 0.0864
HML within Sample 0.3076
0.2000 that the information captured by our measure is markedly different from that
captured by the commonly used default spreads. This is consistent with the
findings in Elton et al. (2001).
The Fama–French factors HML and SMB, and the market factor EMKT are
obtained from Kenneth French’s web page.9 From the same web page we also
obtain data for the 1-month T-bill rate used in our asset-pricing tests. Panel C
of Table II reports the correlation coefficients between (SV) and the Fama–
French factors. The correlations of (SV) with EMKT and SMB are positive and
of the order of 0.5, whereas that with HML is negative and equal to −0.18. This
suggests that EMKT and SMB contain potentially significant default-related
information. The regressions of Panel D in Table II show that (SV) can explain
a substantial portion of the time-variation in EMKT and SMB. This does not
mean, however, that the priced information in EMKT and SMB is related to
default risk. The default-related content of the priced information in SMB and
HML will be examined in Section V.
Finally, given that the need to compute DLI for each stock constrains us to use
only a subset of the U.S. equity market as presented in Table I, it is important
to verify that our results are representative of the U.S. market as a whole. To
this end, we construct the Fama–French factors HML and SMB within our
sample, and compare them with those constructed by Fama and French using
a much larger cross section of U.S. equities. The results are reported in Panel E
We thank Ken French for making the data available. Details about the data, as well as the
actual data series, can be obtained from http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ 842 The Journal of Finance of Table II. The distributional characteristics of the HML and SMB factors
constructed within our sample are similar to those of the HML and SMB factors
provided by Fama and French. Furthermore, their correlations are quite large
and of the order of 0.95 for SMB and 0.86 for HML. The above comparisons
reveal that the subsample we use in our study is largely representative of the
U.S. equity samples used in other studies of equity returns.
III. Measuring Model Accuracy
In this section, we evaluate the ability of our default measure to capture
default risk. To do that, we employ Moody’s Accuracy Ratio. In addition, we
compare the DLI of actually defaulted firms with those of a control group that
did not default.
A. Accuracy Ratio
The accuracy ratio (AR) proposed by Moody’s reveals the ability of a model
to predict actual defaults over a 5-year horizon.10
Let us suppose a model that ranks the firms according to some measure of
default risk. Suppose there are N firms in total in our sample and M of those
actually default in the next 5 years. Let θ = M be the percentage of firms that
default. For every integer λ between 0 and 100, we look at how many firms
actually defaulted within the λ percent of firms with the highest default risk. Of
course, this number of defaults cannot be more than M . We divide the number
of firms that actually defaulted within the first λ percent of firms by M and
denote the result by f (λ). Then f (λ) takes values between 0 and 1, and is an
increasing function of λ. Moreover, f (0) = 0 and f (100) = 1.
Suppose we had the “perfect measure” of future default likelihood, and we
were ranking stocks according to that. We would then have been able to capture
all defaults for each integer λ, and f (λ)would be given by
f (λ) = λ
for λ < θ
θ and f (λ) = 1 for λ ≥ θ. (10) Suppose we also calculate the average f (λ) for all months covered by the sample.
The graph of this function of average f (λ) is shown as the kinked line in Figure 2,
At the other extreme, suppose we had zero information about future default
likelihoods, and we were ranking the stocks randomly. If we did that a large
number of times, f (λ) would be equal to λ. Graphically, the average f (λ) would
correspond to the 45◦ line in the graphs of Figure 2.
We measure the amount of information in a model by how far the graph of
the average f (λ) function lies above the 45◦ line. Specifically, we measure it by
See, “Rating Methodology: Moody’s Public Firm Risk Model: A Hybrid Approach to Modeling
Short Term Default Risk,” Moody’s Investors Service, March 2000. The AC ratio is somewhat
related to the Kolmogorov-Smirnov test. Default Risk in Equity Returns 843 Figure 2. Accuracy ratio. Accuracy Ratio = 0.59231 (defined as the ratio of Area A over
Area B). the area between the 45◦ line and the graph of average f (λ). The accuracy ratio
of a model is then defined as the ratio between the area associated with that
model’s average f (λ) function and the one associated with the “perfect” model’s
average f (λ) function. Under this definition, the “perfect” model has accuracy
ratio of 1, and the zero-information model has an accuracy ratio of 0.
The measure implied by Merton’s model is the distance-to-default (DD).
Therefore, if we rank stocks according to DD, the accuracy ratio we obtain is
equal to 0.592. This means that our measure contains substantial information
about future defaults.
By construction, our measure of default risk is related to size. It is therefore
tempting to conclude that it contains virtually the same information as the
market value of equity. This is not the case, however. If we rank stocks on the
basis of their market value of equity and compute the corresponding accuracy
ratio, this will be equal to only 0.089. Therefore, DD contains much more information than that conveyed by the size of the firms. This is an important point,
since part of our analysis in Section IV provides an interpretation of the size
effect, based on the information contained in DLI. 844 The Journal of Finance Finally, an important parameter in the DD measure is the volatility of assets.
Therefore, one may conjecture that what we capture with our default measure
is simply the volatility of assets. This is again not the case. If we rank stocks on
the basis of their volatility of assets, the accuracy ratio we obtain is 0.290, which
is much lower than that based on DD (0.592). In other words, our measure of
default risk captures important default information beyond what is conveyed
by the market value of equity or the volatility of the firm’s assets alone.
B. Comparison between Defaulted Firms and Non-defaulted Firms 0.6
0.1 Bankrupt firms 8 16 24 32 40 48 56 64 72 80 88 96 104 112 0
120 Average Default Likelihood Indicator As a further test of the ability of our measure to capture default risk, we
compare the DLI of firms that actually defaulted with those of a control group
of firms that did not default. Similar comparisons have been performed in the
past in Altman (1968) and Aharony, Jones, and Swary (1980). To make the comparison meaningful, we choose firms in the control group that have similar size
and industry characteristics as those in the experimental group. In particular,
for every firm that defaults, we select a firm with a market capitalization similar to that of the firm in the experimental group before it defaulted. In addition,
the firm in the control group shares the same two-digit industry code as the
one in the experimental group.
We compute the average DLI for each group. Figure 3 presents the results.
We find that the average DLI of the experimental group goes up sharply in the
5 years prior to default. In contrast, the average DLI of the control group stays
at the same level throughout the 5-year period. Note that in the graph, t = 0
corresponds to about 2 to 3 years prior to default, since the database does not Control group Figure 3. Average default likelihood indicators of bankrupt firms and firms in a control
group. The control group contains firms with the same size and industry characteristics as those
in the experimental group that did not default. Firms are delisted 2 to 3 years prior to bankruptcy.
Numbers in x-axis denote months prior to delisting, and not prior to the actual default. Default Risk in Equity Returns 845 provide data up to the date of default. Therefore, an average DLI of 0.57 for
the experimental group can be considered high. The results of this test provide
further assurance that our DLIs do indeed capture default risk.
IV. Default Risk and Variation in Equity Returns
We start our analysis of the relation between default risk and equity returns
by examining whether portfolios with different default risk characteristics provide significantly different returns. A significant difference in the returns would
indicate that default risk may be important for the pricing of equities.
Table III reports simple sorts of stocks based on their DLI. At the end of each
month from December 1970 to November 1999, we use the most recent monthly
default probability for each firm to sort all stocks into portfolios. We first sort
stocks into five portfolios. We examine their returns when the portfolios are
equally weighted or value-weighted and report the average DLI for each one of
them. Evidently, the lower the average DLI, the lower the risk of default.
Table III Portfolios Sorted on the Basis of DLI
From December 1970 to November 1999, at each month end, we use the most recent monthly DLI
of each firm to sort all portfolios into quintiles and deciles. We then compute the equally and valueweighted returns over the next month. “Return” denotes the average portfolio return and “ADLI”
the average portfolio DLI. Portfolio 1 is the portfolio with the highest default risk and portfolio 10 is
the portfolio with the lowest default risk. When stocks are sorted in quintiles, Portfolio 5 contains
the stocks with the lowest default risk. “High–Low” is the difference in the returns between the
high and low default risk portfolios. T -values are calculated from Newey–West standard errors.
The value of the truncation parameter q was selected in each case to be equal to the number of
autocorrelations in returns that are significant at the 5 percent level.
1 2 3 4 5 6 7 8 9 Low
10 High–Low t-value Equally weighted
1.72 1.29 1.41 1.38 1.19
19.38 1.61 0.24 0.04 0.01 0.53
19.37 (1.96) Value-weighted
1.26 1.27 1.28 1.36 1.12
14.92 1.38 0.21 0.03 0.03 0.14
14.89 (0.46) Equally weighted
2.12 1.32 1.25 1.32 1.44 1.39 1.37 1.39 1.24 1.14
31.74 7.25 2.35 0.86 0.34 0.14 0.06 0.03 0.01 0.01 0.98
31.73 (2.71) Value-weighted
1.20 1.21 1.19 1.30 1.19 1.37 1.29 1.41 1.31 1.04
29.18 6.44 2.12 0.86 0.33 0.11 0.06 0.02 0.01 0.04 0.16
29.15 (0.44) Average size
Average BM Average size
Average BM 2.56 3.52 4.24 4.89 5.59
1.64 0.99 0.82 0.74 0.64 2.24 2.87 3.32 3.71 4.08 4.40 4.73 5.06 5.40 5.78
2.01 1.27 1.05 0.92 0.84 0.79 0.75 0.72 0.68 0.61 846 The Journal of Finance Note that in calculating the returns of portfolios in Section IV, we use the
following procedure. Every time a stock gets delisted due to default, we set
the return of the portion of the portfolio invested in that stock equal to −100
percent. In other words, we assume that the recovery rate for equity holders is
zero. In this way, we fully take into account the cost of default in our calculations
of average portfolio returns. In fact, the returns we report may be considered as
the lower bounds of returns (before transaction costs) earned by equity-holders.
The reason is that often, the recovery rate is not zero.
The t-values of all tests in Section IV are computed from Newey and West
(1987) standard errors. In particular, they are corrected for White (1980) heteroskedasticity and serial correlation up to the number of lags that are statistically significant at the 5 percent level.
The return difference between the equally weighted high-default-risk portfolio and low-default-risk portfolio is 53 basis points (bps) per month or 6.36
percent per annum (p.a.). The difference is statistically significant at the 5 percent level. This is not the case for the value-weighted portfolios whose difference
in returns is only 14 bps per month.
When we sort stocks into 10 portfolios, the results we obtain are similar.
The difference in returns between the high-default-risk portfolio and the lowdefault-risk portfolio is statistically significant for the equally weighted portfolios but not for the value-weighted portfolios. The return differential for the
equally weighted portfolios is 98 bps per month or 11.76 percent p.a.
Notice though that the aggregate default measure for the equally weighted
portfolios assumes bigger values than it does for the value-weighted portfolios. It appears that small-capitalization stocks have on average higher default
risk, and as a result, they earn higher returns than big-capitalization stocks
do. In addition, both in the case of default quintiles and deciles, the average
market capitalization of a portfolio (size) and its BM ratio vary monotonically
with the average default risk of the portfolio. In particular, the average size
increases as the default risk of the portfolio decreases, whereas the opposite is
true for BM. These results suggest that the size and BM effects may be linked
to the default risk of stocks. Recall that both effects are considered stock market anomalies according to the literature of the Capital Asset-Pricing Model
(CAPM). The reason for their existence remains unknown. The remainder of
the paper investigates further the possible link between default risk and those
effects. Our analysis will focus on equally weighted portfolios, since this is the
weighting scheme typically employed in studies that consider the size and BM
effects.11 However, all the results of the paper remain qualitatively the same
when portfolios are value-weighted.
A. Size, BM, and Default Risk
To examine the extent to which the size and BM effects can be interpreted as
default effects, we perform two-way sorts and examine each of the two effects
within different default risk portfolios.
11 For recent references, see for instance Chan, Hamao, and Lakonishok (1991) and Fama and
French (1992). Default Risk in Equity Returns 847 Table IV Size Effect Controlled by Default Risk
From January 1971 to December 1999, at the beginning of each month, stocks are sorted into
five portfolios on the basis of their DLI in the previous month. Within each portfolio, stocks are
then sorted into five size portfolios, based on their past month’s market capitalization. The equally
weighted average returns of the portfolios are reported in percentage terms. “Small–Big” is the
return difference between the smallest and biggest size portfolios within each default quintile. BM
stands for book-to-market ratio. The rows labeled “Whole Sample” report results using all stocks in
our sample. T -values are calculated from Newey–West standard errors. The value of the truncation
parameter q was selected in each case to be equal to the number of autocorrelations in returns that
are significant at the 5 percent level.
1 2 3 4 Big
5 Small–Big t-stat 0.8048
(3.2146) Panel A: Average Return
High DLI 1
Low DLI 5
Whole sample 4.6256
1.2837 Panel B: Average Size
High DLI 1
Low DLI 5
Whole sample 0.6883
7.0886 Panel C: Average DLI
High DLI 1
Low DLI 5
Whole sample 27.4500
0.6141 Panel D: Average BM
High DLI 1
Low DLI 5
Whole sample 2.2378
0.6646 A.1. The Size Effect
Table IV presents results from sequential sorts. Stocks are first sorted into
five quintiles according to their default risk. Subsequently, the stocks within
each default quintile are sorted into five size portfolios. This procedure produces 848 The Journal of Finance 25 portfolios in total. In what follows, we examine whether the size effect exists
in all default risk quintiles, as well as in the whole sample.
The results of Panel A show that the size effect is present only within the
quintile that contains the stocks with the highest default risk (DLI 1). The
effect is very strong with an average return difference between small and big
firms of 3.82 percent per month or a staggering 45.84 percent p.a. Notice that
the difference in returns drops to close to zero for the remaining default-sorted
portfolios. There is a statistically significant size effect in the whole sample,
but the return difference between small and big firms is more than four times
smaller than in DLI 1.
The results of Panel A suggest that the size effect exists only within the
segment of the market that contains the stocks with the highest default risk.
To what extent, however, are we truly capturing the size effect? Is there really
substantial variation in the market capitalizations of stocks within the DLI 1
portfolio? Panel B addresses this question. We see that there is indeed large
variation in the market caps of stocks within the highest default risk portfolio.
But in terms of the average market caps for the size quintiles formed using the
whole sample, the biggest firms in DLI 1 are rather medium to large firms. On
the other hand, the DLI 1-Small portfolio contains the smallest of the small
firms compared to the small size quintile formed on the basis of the whole
sample. These results imply that the size effect is concentrated in the smallest
of the small firms, which also happen to be among those with the highest default
How much riskier are the stocks in DLI 1 compared to the other default risk
quintiles? Panel C of Table IV shows that they are a lot riskier. The small firms
in DLI 1 are almost 14 times riskier in terms of likelihood of default than the
small firms in DLI 2. They are also on average more than twice as risky in terms
of default than the stocks in the small size quintile constructed using the whole
sample. Therefore, the large average returns that small high-default stocks
earn compared to the rest of the market can be considered to be compensation
for the large default risk they have.
To see that, notice also that in the high DLI quintile, DLI decreases monotonically as size increases. In other words, the large difference in returns between
small and big stocks in the DLI 1 quintile can be explained by the large difference in the default risk of those portfolios. In the remaining default quintiles
where there is no evidence of a size effect, the difference in default risk between
small and big stocks is also very small.
Panel D reports the average BM of the default- and size-sorted portfolios.
These results are useful in order to understand the extent to which size, default
risk, and BM are interrelated. Panel D shows that the average BMs in the sizesorted portfolios of DLI 1 are the highest in the sample. The BM decreases
monotonically with DLI, which suggests that the BM effect may also be related
to default risk.
The conclusion that emerges from Table IV is that the size effect is in fact a
default effect. There is a size effect only in the segment of the market with the
highest default risk. Within that segment, the difference in returns between Default Risk in Equity Returns 849 small and big stocks can be explained by the difference in their default risk. In
the remaining stocks in the market, where there is no significant size effect, the
difference in default risk between small and big stocks is minimal. BM seems
also to be related to default risk and size, and we will examine these relations
in the following section.
A.2. The BM Effect
Table V presents results from portfolio sortings in the same spirit as those of
Table IV. Stocks are first sorted into five default risk quintiles, and then each
of the five default quintiles is sorted into five BM portfolios. In what follows,
we will examine the BM effect within each default quintile, as well as for the
market as a whole.
Panel A shows that the BM effect is prominent only in the two quintiles with
the highest default risk, with the return differential between value (high BM)
and growth (low BM) stocks being almost two and a half times bigger in DLI
1 than in DLI 2. There is a BM effect in the whole sample, but the return
differential is about half as big as that found in DLI 1.
Notice that within DLI 1, the average DLI is much higher for value stocks
than it is for growth stocks. In DLI 2, where the BM effect is weaker, the difference in default risk between value and growth stocks is also small. These
results imply that, similar to the size effect, the BM effect seems to be due to
default risk. The only difference is that the BM effect is significant within the
two-fifths of the stocks with the highest default risk, whereas the size effect is
present only in the one-fifth of stocks with the highest default risk. In other
words, the interrelation between size and default risk seems to be a bit tighter.
This is confirmed in Section IV.C using regression analysis.
There is a lot of dispersion in the average BM ratios within the DLI portfolios. This is particularly true for DLI 1 and 2, which means that indeed the
return differential we examine captures a BM effect. In fact, the average BM
ratio varies more across portfolios in DLI 1 than it does across BM portfolios
formed using the whole sample. In DLI 1 and 2 where default risk is higher
than in the other quintiles and the market as a whole, the average BM ratios
of the BM-sorted portfolios are also higher. This result underlines again the
interrelation between BM and default risk discussed above. Furthermore, the
average DLIs in Panel C exhibit a monotonic relation with BM only in the DLI
1 and 2 quintiles, that is, the two quintiles with the highest default risk, where
the BM effect is significant. For the rest of the sample, the relation between default risk and BM ratios does not appear to be linear. A similar result emerges
from Table IV, Panel C. Default risk varies monotonically with size only within
the two highest default risk quintiles. It seems that there are linear relations
between default risk and size, and default risk and BM, only to the extent
that default risk is sizeable. When the risk of default of a company is very
small, the linearity in the relation between default and size and default and
BM disappears, probably because defaults are very unlikely to occur in those
cases. 850 The Journal of Finance
Table V BM Effect Controlled by Default Risk
From January 1971 to December 1999, at the beginning of each month, stocks are sorted into five
portfolios on the basis of their DLI in the previous month. Within each portfolio, stocks are then
sorted into five BM portfolios, based on their past month’s BM ratio. The equally weighted average
returns of the portfolios are reported in percentage terms. “High–Low” is the return difference
between the highest BM and lowest BM portfolios within each default quintile. The rows labeled
“Whole Sample” report results using all stocks in our sample. T -values are calculated from Newey–
West standard errors. The value of the truncation parameter q was selected in each case to be equal
to the number of autocorrelations in returns that are significant at the 5 percent level.
1 2 3 4 Low BM
5 High–Low t-stat 0.8170
(4.5879) Panel A: Average Returns
High DLI 1
Low DLI 5
Whole sample 3.3636
1.0963 Panel B: Average BM
High DLI 1
Low DLI 5
Whole sample 3.7233
1.1258 High DLI 1
Low DLI 5
Whole sample 30.9210
0.2472 Panel C: Average DLI
1.5062 Panel D: Average Size
High DLI 1
Low DLI 5
Whole sample 2.0112
4.7197 Panel D shows again that DLI 1 contains mainly small firms. However, size
does not vary monotonically with BM, except within the two highest default
risk quintiles. The same conclusion can be reached from Panel D of Table IV.
The average BM ratios vary monotonically with size only within the two highest
default risk quintiles. In both cases the variation is small. Default Risk in Equity Returns 851 It seems that size and BM proxy to some extent for each other only within
the segment of the market with the highest default risk. This implies that they
are not identical phenomena. Furthermore, the return premium of small firms
over big firms is more than 1 percent larger than that of high BM stocks over
low BM stocks. In addition, the size effect is present in a subset of the segment
of the market in which the BM effect exists. Both are linked, however, to a
common risk measure, which is default risk.
B. The Default Effect
Tables IV and V show that size and BM are intimately related to default risk.
But does this mean that there is also a default risk in the data? And if there
is, is it confined only within certain size and BM quintiles? In other words, is
default risk rewarded differently depending on the size and BM characteristics
of the stock? These are the questions we address in this section. We define the
default effect as a positive average return differential between high and low
default risk firms.
B.1. The Default Effect in Size-sorted Portfolios
Table VI examines whether there is a default effect in size-sorted portfolios by reversing the sorting procedure of Table IV. In particular, we first sort
stocks into five size quintiles, and then sort each size quintile into five default
portfolios. As we will see below, this exercise also allows us to obtain a better
understanding of small firms as an asset class.
Panel A shows that there is a statistically significant default effect only within
the small size quintile. The average monthly return is 2.2 percent or 26.4 percent p.a. In most of the remaining size quintiles, the difference in returns between high and low default risk portfolios is in fact negative. This means that
high-default-risk firms earn a higher return than low default risk firms, only
if they are also small in size.
To verify this point, see Panel B of Table VI. All high-DLI portfolios have
substantial default risk, independent of the market capitalization of the stocks.
Similarly, all low-DLI portfolios have virtually no default risk. However, only
small high-default-risk stocks earn higher returns than low default risk
This result may indicate that firms differ in their ability to re-emerge from
Chapter 11, depending on their size. If small firms, for instance, are less likely to
emerge from the restructuring process as public firms, investors may require a
bigger risk premium to hold them, compared to what they require for bigger size
high-default-risk firms. This will induce the average returns of small high-DLI
firms to be higher than those of bigger high-DLI firms.12 Empirical evidence
This interpretation assumes that default risk is systematic, and therefore, not diversifiable.
In Section V we test whether default risk is priced in the cross section of equity returns. Our results
show that default risk is indeed priced, and therefore, it constitutes a systematic source of risk. 852 The Journal of Finance
Table VI Default Effect Controlled by Size
From January 1971 to December 1999, at the beginning of each month, stocks are sorted into
five portfolios on the basis of their market capitalization (size) in the previous month. Within
each portfolio, stocks are then sorted into five portfolios, based on past month’s DLI. Equally
weighted average portfolio returns are reported in percentage terms. “HDLI-LDLI” is the return
difference between the highest and lowest default risk portfolios within each size quintile. T -values
are calculated from Newey–West standard errors. The value of the truncation parameter q was
selected in each case to be equal to the number of autocorrelations in returns that are significant
at the 5 percent level.
1 2 3 4 Low DLI
5 High–Low t-stat 2.2295
(1.7074) Panel A: Average Returns
Big 5 3.7315
1.0428 Panel B: Average DLI
Big 5 41.5360
0.1200 Small 1
Big 5 1.2008
0.0063 Panel C: Average Size
7.4129 Panel D: Average BM
Big 5 2.4472
0.4538 from the corporate bankruptcy literature shows that indeed large firms are
more likely to survive Chapter 11 than small firms.13
Panel B of Table VI also provides insights into the profile of small firms as an
asset class. Notice that within the small size quintile, DLI varies between 41.53
percent and 0.09 percent. This implies that small firms can differ a lot with
respect to their (default) risk characteristics. They can also differ significantly
with respect to their returns, as Panel A reveals. These results suggest that
small firms do not constitute a homogenous asset class, as is commonly believed.
13 See for instance, Moulton and Thomas (1993) and Hotchkiss (1995). Default Risk in Equity Returns 853 Finally, Panel B shows that default risk decreases monotonically as size increases, confirming the close relation between size and default risk observed in
Table IV. Panels C and D show that the small–high DLI portfolio contains the
smallest of the small stocks and those with the highest BM ratio.
Two important conclusions emerge from this table. First, default risk is rewarded only in small value stocks. Firms that have high default risk, but are
not categorized as small and high BM, will not earn higher returns than firms
with low default risk and similar size and BM characteristics. This result further underlines the close link among size, default risk, and BM. Second, small
firms are not made equal. They differ substantially in terms of both their return and (default) risk characteristics. This result reveals that small firms do
not constitute a homogeneous asset class.
B.2. The Default Effect in BM-sorted Portfolios
To further examine the link between default risk and BM, Table VII examines
the presence of a default effect in BM-sorted portfolios. Assets are first sorted
in five BM quintiles, and subsequently, each BM-sorted quintile is subdivided
into five default-sorted portfolios.
Panel A reveals that the default effect is again present only within the high
BM quintile. This result is consistent with that of Table VI. Since the smallest
high-DLI firms are also typically the highest BM firms, the same interpretation
applies here. Specifically, default risk is rewarded only for small, value stocks,
and not for any other stocks in the market, independently of their risk of default.
This is confirmed in Panels C and D.
Once again, Panel B shows that value stocks can differ a lot with respect to
their default risk characteristics. Given that they also differ significantly in
terms of their returns, Panels A and B suggest that, similar to small firms,
value stocks do not constitute a homogeneous asset class either.
The results of Table VII are consistent and analogous to those of Table VI.
High-default-risk stocks earn a higher return than low default risk stocks, only
to the extent that they are small and high BM. If the size and BM criteria are
not fulfilled, they will not earn higher returns than low default risk stocks, even
if their default risk is very high. Furthermore, our analysis implies that small
firms and value stocks do not constitute homogeneous asset classes.14
C. Examining the Interaction of Size and Default, and BM and Default
Using Regression Analysis
In this section, we summarize and quantify the degree of interaction between
size and default and BM and default using regression analysis. Two different methodologies are employed. The first one is a portfolio-based regression
The results presented in Section IV based on sequential sorts hold also when independent sorts
are performed. To conserve space, we do not report those results here. The main insight offered by
the independent sorts is that most small stocks are also high-DLI stocks, whereas most big stocks
are low-DLI stocks. Similarly, most value stocks are high default risk stocks, whereas most growth
stocks have low risk of default. 854 The Journal of Finance
Table VII Default Effect Controlled by BM
From January 1971 to December 1999, at the beginning of each month, stocks are sorted into five
portfolios on the basis of their BM ratio in the previous month. Within each portfolio, stocks are then
sorted into five portfolios, based on past month’s DLI. Equally weighted average portfolio returns
are reported in percentage terms. “HDLI-LDLI” is the return difference between the highest and
lowest default risk portfolios within each size quintile. T -values are calculated from Newey–West
standard errors. The value of the truncation parameter q was selected in each case to be equal to
the number of autocorrelations in returns that are significant at the 5 percent level.
1 2 3 4 Low DLI
5 High–Low t-stat 1.6042
(−0.8369) Panel A: Average Returns
High BM 1
Low BM 5 3.2285
0.9711 Panel B: Average DLI
High BM 1
Low BM 5 42.0930
0.0063 Panel C: Average BM
High BM 1
Low BM 5 3.1427
0.2453 Panel D: Average Size
High BM 1
Low BM 5 1.9664
6.2550 approach developed in Nijman, Swinkels, and Verbeek (2002). The second one
uses the Fama and MacBeth (1973) methodology on individual stock returns. C.1. The Portfolio-based Regression Approach
The regression methodology in Nijman, Swinkels, and Verbeek (2002) is an
extension of the methodology in Heston and Rouwenhorst (1994), which allows
for the presence of interaction terms between the variables of interest. In the
current application, we analyze average returns of portfolios grouped on the Default Risk in Equity Returns 855 basis of DLI, size, and BM, and examine the relative magnitudes of the individual effects, as well as their interactions.
Similar to Daniel and Titman (1997), Nijman, Swinkels, and Verbeek (2002)
assume that the conditional expected return of a stock can be decomposed into
several effects. In other words,
Na Nb Et Ri, t +1 = αa, b X i, t (a, b) (11) a=1 b=1 where Et (·) denotes the expectation conditional on the information available at
time t, Ri, t+1 is the return of the stock at time t + 1, Xi, t (a, b) is a dummy variable
that indicates the membership of the stock in a particular portfolio, and αa, b
the expected return of a stock with characteristics a and b. In our application,
a and b are either size and default risk or BM and default risk. Therefore,
equation (7) simply states the conditional expected return of a stock, given its
size/BM and default risk characteristics that grant it membership to a particular portfolio.15
The conditional expected return on a portfolio p of N stocks with weights wi, t ,
can then be written as:
Na p E t R t +1 =
p Nb p αa, b X i, t (a, b) , (12) a=1 b=1 p where Xi, t (a, b) = wi, t Xi, t (a, b). Since the portfolios we use for our tests are all
equally weighted and sorted on the basis of the characteristics a and b, we can
simplify the above equation as follows:
Na p E t R t +1 = Nb αa, b X t (a, b) . (13) a=1 b=1 The regression equation implied can be written as:
p R t +1 = Na Nb αa, b X t (a, b) + εt +1 , (14) a=1 b=1 p P
where εt+1 ≡ Rt+1 − Et (Rt+1 ), which is by construction orthogonal to the regressors. The only assumption made is that the cross-autocorrelation structure is
zero, that is, E(εt+h εt ) = 0. However, equation (10) can be written in a more parsimonious way by imposing an additive structure similar to that in Roll (1992) 15
Note that, in principle, we could examine all three effects simultaneously, that is the size, BM,
and default effects. This, however, would increase the parameters to be estimated considerably, at
the expense of efficiency. For that reason, we concentrate on two effects at a time. 856 The Journal of Finance and Heston and Rouwenhorst (1994). In that case, the conditional expected
return of portfolio p will be given by:
p Et Rt +1 = α1,1 + Na Nb φa X t (a, ·) +
a=2 Na Nb φb X t (·, b) +
b=2 αa, b X t (a, b) , (15) a=2 b=2 where Xt (., b), for instance, denotes that only the argument b is considered. In
that case, all stocks in group b are considered, irrespectively of their a characteristic. The constant α1,1 denotes the return on the reference portfolio. The
reference portfolio is arbitrarily chosen and is used to avoid the dummy trap.
When we examine the interaction of size and default effects, the reference portfolio we use is the portfolio that contains big cap and low-DLI stocks. In the
tests of the interaction of BM and default effects, the reference portfolio is the
one that contains stocks with low BM and low DLI.
The estimated coefficients φ can be interpreted as the difference in return
between portfolio p and the reference portfolio attributed to a particular effect.
Similarly, the coefficients α denote the additional expected return for portfolio
p due to the interaction of two effects. The total expected return on portfolio
p is given by the sum of the returns of the reference portfolio, the individual
effects, and the interaction effects.
Each set of estimations uses 15 left-hand-side portfolios. In the case of the
size-default effects test, they are comprised of three size portfolios, three defaultsorted (DLI) portfolios, and nine portfolios created from the intersection of two
independent sorts on three size and three DLI portfolios. In the case of the BMdefault effects test, the portfolios include three BM portfolios, three defaultsorted portfolios, and nine portfolios from the intersection of two independent
sorts on three BM portfolios and three DLI portfolios. In both sets of tests, there
are eight parameters to be estimated.
The results are reported in Table VIII. The first panel refers to the tests of
the size-default effects, whereas the second panel contains the results for the
Panel A shows that the economically and statistically most important coefficients for the individual effects are for small size and high DLI. In addition,
the strongest interaction effect refers to the interaction of small size and high
DLI. In other words, a portfolio will earn higher return, the smaller its market
cap, and the higher its default risk. It will also earn an additional return from
the interaction of high default risk and small size. This additional return is
zero if the small firms have medium default risk. These results are consistent
with our earlier finding that the size effect exists only among high-default-risk
stocks. Note also that the coefficient on the interaction term between high DLI
and medium size is negative and statistically insignificant. This is again in line
with our previous result that the default effect exists only within small firms.
The return on the reference portfolio (big firms, low DLI) is 1.1363 percent
per month (p.m.). This means that a portfolio of small firms with high DLI will
earn 2.37(1.136 + 0.4287 + 0.50 + 0.31) percent per month, compared to 1.79 Default Risk in Equity Returns 857 Table VIII A Decomposition of Returns in Size, BM, and DLI Portfolios Using
Panel A provides results using 15 size- and DLI-sorted portfolios. Out of these 15 portfolios, 3
are sorted on the basis of size, 3 on the basis of DLI, and 9 portfolios are created from the intersection of two independent sorts on three size and three DLI portfolios. The reference portfolio
contains big firms with low DLI. Its average return is 1.1363 percent per month. Panel B provides
results based on 15 BM- and DLI-sorted portfolios. The portfolios are constructed in an analogous
fashion to that of the portfolios of Panel A. The reference portfolio contains now low BM and low
DLI firms, and has an average return of 1.0529 percent per month. The results presented are
from Fama–MacBeth regressions. T -values are computed from standard errors corrected for White
(1980) heteroskedasticity and serial correlation up to three lags using the Newey–West estimator.
The Wald test examines the hypothesis that the coefficients of each individual effect are jointly
Panel A: Size Effect and Default Effect
Size (M) Wald test
p-value DLI (M) DLI (H) Size (M)
DLI (M) Size (M)
DLI (H) Size (S)
DLI (M) Size (S)
DLI (H) 0.1069
2.3895 Size Coefficient
t-value Size (S) DLI 5.9102
Panel B: BM Effect and Default Effect BM (M) Wald test
p-value DLI (M) DLI (H) BM (M)
DLI (M) BM (H)
DLI (M) BM (M)
DLI (H) BM (H)
DLI (H) 0.1533
1.9991 BM Coefficient
t-value BM (H) DLI 48.9252
0.0097 percent p.m. that a portfolio of small firms of medium DLI will earn. Similarly, a
portfolio of medium firms of high DLI will earn 1.62 percent per month, whereas
a portfolio of medium size firms of medium DLI will earn only 1.37 percent per
Notice that the returns above are smaller than those in Tables IV and VI. The
reason is that stocks here are classified into tertiles of size and DLI portfolios
rather than quintiles as in Table IV to VII. The pattern of returns and the
conclusions remain the same: the highest returns are earned by stocks with
the highest DLI and smallest size.
Similar conclusions emerge for the BM-DLI portfolios in Panel B. The stocks
that earn the highest returns are stocks that are both high BM and high DLI.
The return of the reference portfolio here (low BM, low DLI) is 1.05 percent
p.m. Therefore, the high BM, high DLI portfolio will earn a total return of 858 The Journal of Finance
Table IX Fama–MacBeth Regressions on the Relative Importance of Size, BM,
and DLI Characteristics for Subsequent Equity Returns
The Fama–MacBeth regression tests are performed on individual equity returns. The variables size
and BM are rendered orthogonal to DLI. The regressions relate individual stock returns to their past
month’s size, BM, and DLI characteristics. Size2, BM2, DLI2 denote the characteristics squared,
whereas SizeDLI and BMDLI denote the products of the respective variables. Those products aim
to capture the interaction effects of each pair of variables.
Constant DLI DLI2 Size Size2 BM BM2 SizeDLI BMDLI Coef
4.4352 −4.8980 17.8748 −0.0030
0.0000 0.5710 −0.0293 −0.6800
−2.7120 4.3832 −0.5061 −0.2406 5.5091 −1.5762 −3.8740 Coef
4.3906 −6.2470 19.7108 −0.0072
−3.3818 4.6873 −1.1187 Coef
5.7721 −2.4581 0.1345
2.1236 2.38 percent p.m. as opposed to the 1.84 percent earned by the high BM, medium
DLI portfolio. Medium default risk firms earn an extra return for default risk,
but it is smaller than that earned by high-default-risk firms. In addition, the
only positive and statistically significant interaction coefficient is the one referring to high BM and high DLI stocks. By the same token, a portfolio of medium
BM and high DLI stocks will earn 1.58 percent per month compared to 1.45 percent per month earned by a portfolio of medium BM and medium DLI firms. In
both cases, the interaction term is economically and statistically equal to zero. C.2. The Fama–MacBeth Regression Approach on Individual Stock Returns
Table IX presents results from Fama–MacBeth regressions of individual
stocks on their past month’s size, BM, and DLI characteristics. The regressions consider both a linear relation between stock returns and characteristics,
as well as a nonlinear relation by including the characteristics squared (size2,
BM2, DLI2). In addition, there are interaction terms proxied by the product of
size with DLI (sizeDLI) and BM with DLI (BMDLI). We render size and BM orthogonal to DLI before performing the tests, in order to avoid possible problems
in the interpretation of the results.
The results show that what explains next month’s equity returns is the current default risk of securities, their BM, and the interaction of default risk and
size. Size per se does not appear to play any role. This is confirmed in tests
where only the DLI and size variables are considered. Indeed, only DLI, DLI2,
and sizeDLI are important for explaining the next period’s equity returns. In
contrast, BM seems to contain incremental information about next period’s returns, over and above that contained in DLI. The regressions that consider
only DLI and BM variables show that the BM variables and DLI2, in addition to the interaction term, are important for explaining next period’s equity Default Risk in Equity Returns 859 returns. The regression results in Table IX also highlight the importance of the
squared terms, and therefore, the nonlinearity in the relations between equity
returns, DLI, and BM.
The bottom line from these tests is the following. The observed relation in
the literature between size and equity returns is completely due to default risk.
Size proxies for default risk and this is why small caps earn higher returns than
big caps. They do so because small caps have higher default risk than big caps.
BM also proxies partially for default risk. Default risk is not however all the
information included in BM.
D. Conclusions About the Size, BM, and Default Effects
The results in Section IV point to the following conclusions. The size effect
is a default effect as it exists only within the quintile of firms with the highest
default risk. The BM effect is also related to default risk, but it exists among
firms with both high and medium default risk. Default risk is rewarded only to
the extent that high-default-risk firms are also small and high BM and in no
other case. In other words, default risk and size share a nonlinear relation, and
the same is true for default risk and BM. The exact functional form of these
relations is not completely mapped out here. Rather, we highlight some of the
principal characteristics of these relations. It is clear that the highest returns
are earned by stocks that are either both small in size and high DLI, or both
high DLI and high BM. It is also clear though that default risk is a variable
worth considering above and beyond size and BM, and the asset-pricing tests
of the following section confirm that.
V. The Pricing of Default Risk
The results of the previous section imply that the size and BM effects are
compensations for the high default risk that small and high BM stocks exhibit.
But does this mean that default risk is systematic? The answer to this question
is not obvious, since defaults are rare events and seem to affect only a small
number of firms. However, the default of a firm may have ripple effects on other
firms, which may give rise to a systematic component in default risk.
The purpose of this section is to investigate through asset-pricing tests,
whether default risk is systematic, and therefore whether it is priced in the
cross section of equity returns.
A. The Tested Hypotheses
Two hypotheses are examined as part of our asset-pricing tests. First, we
test whether default risk is priced. To do so, we need to consider a plausible
empirical asset-pricing specification in which default risk appears as a factor.
It is clear that an asset-pricing model that includes only default as a risk
factor would certainly be mis-specified, since even if default risk is priced, it is
unlikely to be the only risk factor that affects equity returns. For that reason, 860 The Journal of Finance we consider an asset-pricing model that includes as factors the excess return
on the market portfolio (EMKT) and the aggregate survival measure (SV).
The empirical asset-pricing specification is given below.
Rt = a + bEMKTt + d (SV)t + εt (16) where Rt represents the return at time t of a stock in excess of the risk-free
Such a model can be understood in the context of an Intertemporal Capital
Asset-Pricing Model (ICAPM) as in Merton (1974). One can postulate a version of ICAPM where default risk affects the investment opportunity set, and
therefore, investors want to hedge against this source of risk.
The second hypothesis examined is whether the FF factors SMB and HML
proxy for default risk. Recall that the FF model is empirical in nature, and
includes apart from the market factor, a factor related to size (SMB) and a factor
related to BM (HML). Fama and French (1996) argue that SMB and HML proxy
for financial distress. We test this hypothesis here, by including (SV) in the
FF model. In other words, we test the following empirical specification:
Rt = a + bEMKTt + sSMBt + hHMLt + d (SV)t + εt . (17) If indeed all the priced information in SMB and HML is related to financial
distress, we would expect to find that in the presence of (SV), SMB and HML
lose all their ability to explain equity returns.
To get a sense of the performance of the two empirical specifications examined, we also present results from tests of the CAPM and FF model. These two
models act as benchmarks for comparison purposes.
B. The Test Assets
As previously mentioned, two hypotheses are examined in our asset-pricing
tests. First, whether default risk is priced, and second, whether SMB and HML
proxy for default risk. This implies that there are three variables against which
the test assets have to exhibit maximum dispersion: (SV), size, and BM. By
test assets we mean the portfolios whose returns the asset-pricing models will
be called upon to explain.
To obtain maximum dispersion against all three variables, we perform a
three-way independent sort. All equities in our sample are sorted in three portfolios according to (SV). They are also sorted in three portfolios according
to size. Finally, they are sorted in three portfolios according to BM. Twentyseven equally weighted portfolios are formed from the intersection of the three
independent sorts. Summary statistics of the 27 portfolios are provided in
C. Empirical Methodology of the Asset-Pricing Tests
To test the asset-pricing models of Section V.A, we use the Generalized Methods of Moments (GMM) methodology of Hansen (1982), and employ the
asymptotically optimal weighting matrix. For each model considered, we also Default Risk in Equity Returns 861 Table X Summary Statistics on the 27 Size, BM, and DLI Sorted Portfolios
The 27 portfolios are constructed from the intersection of three independent sorts of all stocks into
three size, three BM, and three default risk portfolios. Default risk is measured by the DLI. The
second, third, and fourth columns describe the characteristics of each portfolio in terms of its size,
BM, and DLI. Size refers to the market value of equity. Equally weighted average returns are
reported in percentage terms.
SIZE DLI Average Return Size BM DLI Small
0.0150 compute Hansen’s J -statistic on its overidentifying restrictions. In addition, we
report a Wald test (Wald(b)) on the joint significance of the coefficients of the
pricing kernel implied by each model.
To compare the alternative models, we use the Hansen and Jagannathan
(1997) (HJ) distance measure. To calculate the p-value of the HJ -distance, we
simulate the weighted sum of n − k χ 2 (1) random variables 100,000 times,
where n is the number of test assets, and k is the number of factors in the
D. Asset-Pricing Results
The results from the asset-pricing tests are reported in Table XI. The rows
labeled “coefficient” refer to the coefficient(s) of the factor(s) in the pricing
16 See Jagannathan and Wang (1996). Table XI Premium
(39.2795) Constant 0.0079
(0.8804) EMKT SMB 0.0043
(SV) Model HML
Panel A: The EMKT + Statistic
p-value Test 63.6054
(0.0000) J -test (0.0001) Wald(b) 0.8678
(0.0000) HJ The GMM estimations use Hansen’s (1982) optimal weighting matrix. The tests are performed on the excess returns of the 27 equally weighted
portfolios of Table IX. EMKT refers to the excess return on the stock market portfolio over the risk-free rate. (SV) is the change in the survival rate,
which is defined as 1 minus the aggregate DLI. HML is a zero-investment portfolio, which is long on high BM stocks and short on low BM stocks.
Similarly, SMB is a zero-investment portfolio, which is long on small market capitalization (size) stocks and short on big size stocks. The J -test is
Hansen’s test on the overidentifying restrictions of the model. The Wald(b) test is a joint significance test of the b coefficients in the pricing kernel. The
J -test and Wald(b) tests are computed in GMM estimations that use the optimal weighting matrix. We denote by HJ the Hansen–Jagannathan (1997)
distance measure. It refers to the least-square distance between the given pricing kernel and the closest point in the set of pricing kernels that price
the assets correctly. The p-value of the measure is obtained from 100,000 simulations. The estimation period is from January 1971 to December 1999.
In Panel A we test the hypothesis that default risk is priced in the context of a model that includes the EMKT along with a measure of default risk
( (SV)). Panels B and C present results from tests of the CAPM and Fama–French models, which are used as benchmarks for comparison purposes.
Finally, in Panel D we test the hypothesis that the Fama–French factors SMB and HML include default-related information, by including in the
Fama–French model our aggregate measure of default risk. Optimal GMM Estimation of Competing Asset-pricing Models 862
The Journal of Finance Premium
(−4.7691) Panel D: The Fama–French Model Augmented by 0.0009
(0.0270) Panel C: The Fama–French Model Panel B: The CAPM Statistic
p-value (SV) Statistic
(0.0000) (0.0000) (0.0001) (0.0425) 0.8032
(0.0000) Default Risk in Equity Returns
863 864 The Journal of Finance kernel, whereas the rows labeled “premium” refer to the risk premium(s) implied for the factor(s).
The first panel shows the results of the model that includes the market and
(SV ) as factors. We see that (SV ) commands a positive and statistically
significant risk premium. This implies that default risk is systematic and it
is priced in the cross section of equity returns. As expected, the J -test, and
the HJ -distance measure have both very small p-values, which means that the
model cannot price assets correctly. Even though both the EMKT and (SV)
are priced, it appears that there are other factors that may be important for
explaining the cross-sectional variation in equity returns, and which are not
considered here. Despite this implication, the model considered has a smaller
HJ distance than both the CAPM (Panel B) and the FF model (Panel C). This
means that, any mis-specification present in this model translates into at least
as small an annualized pricing error as those resulting from the two standard
asset-pricing models in the literature, the CAPM and FF model.17
Panel D reports the results from testing the hypothesis that SMB and HML
proxy for default risk. In particular, we test the model of equation (8). The
results show that (SV) continues to receive a positive and statistically significant risk premium, even when it is considered part of the augmented model.
HML is also priced again, as in Panel C, and SMB is not priced either in Panel
C or Panel D.
Notice, however, that the coefficients of SMB and HML are very different in
Panel D than they are in Panel C, and this is particularly the case for SMB.
The fact that the coefficients of SMB and HML change in the presence of (SV)
suggests that SMB and HML share some common information with (SV).
The dramatic change in the coefficient of SMB between Panels C and D is
an indication that SMB shares more common information with (SV) than
HML does. In general, we expect the coefficients to change when the factors in
the pricing kernel are not orthogonal. Table II shows that (SV) is positively
and highly correlated with EMKT and SMB, but has a small and negative
correlation with HML.
Recall that statistically significant coefficients in the pricing kernel imply
that the corresponding factors help price the test assets, whereas a statistically
significant premium means that the corresponding factor is priced.18 The results in Panel D show that although all factors help price the test assets, SMB
is not a priced factor.
Notice also that the coefficient on SMB is not statistically significant in Panel
C, whereas it is in Panel D. This may be the case if the FF model is more misspecified than the model in Panel D. It seems that SMB needs the presence
of (SV) in the pricing kernel in order for its coefficient to become significant.
The fact that the coefficient of SMB becomes significant in this case further
shows that although there is some common information between SMB and
For an interpretation of the HJ -distance as the maximum annualized pricing error, see Campbell and Cochrane (2000).
See Cochrane (2001), Section 13.5. Default Risk in Equity Returns 865 5
0 27 25 23 21 19 17 15 13 11 9 7 5 3 1 -1
FF3+D(SV) Figure 4. Loadings of the 27 portfolios on ∆(SV). This graph shows the loadings of the 27
portfolios of Table IX on the survival indicator (SV). The portfolios are ordered in the same way
as in Table IX. EMKT + (SV ) labels the loadings on (SV) from the model that includes the
market factor and (SV) in the pricing kernel. Similarly, FF3 + (SV ) labels the loadings on
(SV) from the augmented Fama–French (1993) model by the (SV) factor. The sample period is
from January 1971 to December 1999. (SV), there is also residual information in both factors, which is important for
pricing the test assets.
This interpretation is also supported by the values of the HJ -distance measures for the models of Panels C and D. The HJ distance for the FF model is
larger in value than that of the model in Panel D. This suggests that the FF
model may be more mis-specified than the model in Panel D. An implication of
this result is that although there is some common information between (SV)
and the FF factors, there is also a lot of additional important information in
SMB and HML which helps explain the test assets, but is unrelated to default
Figure 4 plots the loadings of the 27 portfolios on (SV) from the models
of Panels A and D. The portfolios are ordered in the same way as in Table X.
It is interesting to note that the loadings on (SV) for the model of Panel A
are equal or larger than 1, for 20 of the 27 portfolios. This means that default
risk is important for a large segment of the cross section that includes not just
small firms but also medium-sized and big firms. In other words, the pricing of
default risk is not driven by only a handful of portfolios.
Once SMB and HML are included in the pricing kernel, the loadings of (SV)
are reduced substantially for all portfolios. For the two portfolios that include
small, high BM stocks with high or medium level of default risk, the loadings
are also significantly reduced, but they remain around 1. The fact that the
Vassalou (2003) shows, for instance, that a model which includes the market factor along with
news about future GDP growth absorbs most of the priced information in SMB and HML. In the
presence of news about future GDP growth in the pricing kernel, SMB and HML lose virtually all
their ability to explain the cross section. Furthermore, Li, Vassalou, and Xing (2000) show that
the investment component of GDP growth can price equity returns very well, and can completely
explain the priced information in the Fama–French factors. 866 The Journal of Finance loadings of (SV) are so drastically reduced for all 27 portfolios suggests again
that SMB and HML include important default-related information.
The conclusion that emerges from the asset-pricing tests is that default risk
is priced, and it is priced even when (SV) is included in the FF model. SMB
and HML contain some default-related information. However, this information
does not appear to be the reason that the FF model is able to explain the cross
section of equity returns.
This paper uses for the first time the Merton (1974) model to compute monthly
DLI for individual firms, and examine the effect that default risk may have on
Our analysis provides a risk-based interpretation for the size and BM effects.
It shows that both effects are intimately related to default risk. Small firms earn
higher returns than big firms, only if they also have high default risk. Similarly,
value stocks earn higher returns than growth stocks, if their risk of default is
high. In addition, high-default-risk firms earn higher returns than low default
risk firms, only if they are small in size and/or high BM. In all other cases,
there is no significant difference in the returns of high and low default risk
We also examine through asset-pricing tests whether default risk is systematic, and we find that it is indeed. Fama and French (1996) argue that their
factors SMB and HML proxy for default risk. Our results show that, although
SMB and HML contain some default-related information, this is not the reason that the Fama–French model is able to explain the cross section of equity
returns. SMB and HML appear to contain other significant price information,
unrelated to default risk. Risk-based explanations for this information are provided in Vassalou (2003) and Li, Vassalou, and Xing (2000). Our results show
that default is a variable worth considering in asset-pricing tests, above and
beyond size and BM.
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