ï»¿ WPS5319
Policy Research Working Paper 5319
Is There a Distress Risk Anomaly?
Pricing of Systematic Default Risk in the Cross Section
of Equity Returns
Deniz Anginer
Ã‡elim YÄ±ldÄ±zhan
The World Bank
Development Research Group
Finance and Private Sector Development Team
May 2010
Policy Research Working Paper 5319
Abstract
The standard measures of distress risk ignore the fact premia, that is stocks with higher systematic default
that firm defaults are correlated and that some defaults risk exposures, have higher expected equity returns.
are more likely to occur in bad times. The paper uses Consistent with structural models of default, they show
risk premium computed from corporate credit spreads that the premium to a high-minus-low systematic default
to measure a firmâ€™s exposure to systematic variation in risk hedge portfolio is largely explained by the market
default risk. Unlike previously used measures that proxy factor. The authors confirm the robustness of these results
for a firmâ€™s physical probability of default, credit spreads by using an alternative systematic default risk factor for
proxy for a risk-adjusted default probability and thereby firms that do not have bonds outstanding. The results
explicitly account for the non-diversifiable component of show no evidence of firms with high systematic default
distress risk. In contrast to prior findings in the literature, risk exposure delivering anomalously low returns.
the authors find that stocks that have higher credit risk
This paperâ€”a product of the Finance and Private Sector Development Team, Development Research Groupâ€”is part of
a larger effort in the department to understand the asset pricing implications of systematic credit risk.. Policy Research
Working Papers are also posted on the Web at http://econ.worldbank.org. The author may be contacted at danginer@
worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
Is there a Distress Risk Anomaly?
Pricing of Systematic Default Risk in the Cross Section of Equity Returns
Deniz Anginer and Ã‡elim YÄ±ldÄ±zhan 1
JEL Classifications: G11, G12, G13, G14.
Keywords: Default risk, systematic default risk, credit risk, distress risk, bankruptcy,
credit spread, asset-pricing anomalies, pricing of default risk, corporate bonds
1
Deniz Anginer can be reached at Pamplin School of Business, Virginia Tech, Blacksburg, VA, 24061,
Email:danginer@vt.edu. Ã‡elim YÄ±ldÄ±zhan can be reached at Terry College of Business, University of
Georgia, Athens, GA, 30602, E-mail: celim@uga.edu. We would like to thank Alexander Barinov, Tobias
Berg, Sugato Bhattacharyya, Dennis Capozza, Ilia Dichev, Stu Gillan, Jack He, Jens Hilscher, Sara
Holland, Alex Hsu, Paul Irvine, Haitao Li, Jim Linck, Russell Lundholm, Harold Mulherin, Jeff Netter,
Shawn Park, Paolo Pasquariello, Bradley Paye, Annette Poulsen, Amiyatosh Purnanandam, Uday Rajan,
Nejat Seyhun, Tao Shu, Tyler Shumway, Jeff Smith, Ralph Steuer, Ginger Wu, Julie Wu, Lu Zhang, and
seminar participants at the University of Michigan, University of Georgia, Virginia Tech, World Bank,
University of Delaware, BlackRock, Wilfrid Laurier, University of Connecticut, CFTC, Cornerstone
Research, Ozyegin Universitesi, Sabanci Universitesi, 37th EFA Annual Meeting for helpful discussion and
guidance.
1. Introduction
A fundamental tenet of asset pricing is that investors should be compensated with higher
returns for bearing systematic risk that cannot be diversified. As default risk remains a
major source of potential large losses to equity investors, a number of recent papers have
examined whether default risk is a systematic risk and whether it is priced in the cross
section of equity returns. From a theoretical perspective, default risk can be a priced
factor if a firmâ€™s capital asset pricing model (CAPM) beta does not fully capture default-
related risk. Empirical work has focused on determining the probability of firms failing
to meet their financial obligations using accounting and market-based variables and
testing to see if estimated default probabilities are related to future realized returns. The
existing empirical evidence contradicts theoretical expectations and suggests that firms
with high default risk earn significantly lower average returns. 2
The low returns on stocks with high default risk cannot be explained by Fama-French
(1993) risk factors. Stocks with high distress risk tend to have higher market betas and
load more heavily on size and value factors. This leads to significantly negative alphas
for the high-minus-low default risk hedge portfolio and makes the anomaly even larger in
magnitude. These empirical results provide a challenge to the standard risk-reward trade-
off in financial markets and to the contention that small firms and value firms earn high
average returns because they are financially distressed (Chan and Chen 1991; Fama and
French 1996; Kapadia 2011).
We argue that the anomalous results documented in the literature are due to
incorrectly measuring systematic default risk. Figure 1, which plots the historical default
2
See for example Dichev (1998) and Campbell, Hilscher, and Szilagyi (2008) for a discussion of this
anomaly.
2
rates on Moodyâ€™s rated corporate issuers, suggests that default rates are highly dependent
on the stage of the business cycle. This casual analysis of the historical data suggests that
there is an important systematic component of default risk and that the incidence of
ï¬?nancial distress is correlated with macroeconomic shocks such as major recessions.
Previous papers measure financial distress by determining firmsâ€™ expected probabilities
of default inferred from historical default data. This calculation ignores the fact that firm
defaults are correlated and that some defaults are more likely to occur in bad times, and
therefore fails to appropriately account for the systematic nature of default risk.
Investors, however, will take into account the covariance of default losses from a
company with the rest of the assets in their portfolio when pricing distress risk.
We use credit risk premium computed from corporate credit spreads to proxy for a
firmâ€™s exposure to the non-diversifiable portion of default risk. The fixed-income
literature provides evidence of a significant risk premium component in corporate credit
spreads, justifying our use of this measure as a proxy for firm exposure to systematic
default risk. 3 It has been well-documented (Almeida and Philippon 2007; Berndt, Duffie,
Ferguson and Schranz 2005; Hull, Predescu, and White 2004) that there is a substantial
difference between the risk-adjusted (or risk-neutral, as commonly designated in
contingent claim pricing) and physical probabilities of default. Ranking stocks based on
their physical default probabilities inferred from historical default dataâ€”as done in
Dichev (1998), Campbell, Hilscher, and Szilagyi (2008), and others in this literatureâ€”
implicitly assumes that stocks with high physical probabilities of default also have high
3
The spread between corporate bond yields and maturity-matched treasury rates is too high to be fully
captured by expected default and has been shown to contain a large risk premium for systematic default
risk. See, for detailed analysis, Elton et al. (2001), Huang and Huang (2003), Longstaff et al. (2005),
Driessen (2005), and Berndt et al. (2005).
3
exposures to systematic variation in default risk. George and Hwang (2010) show that a
firmâ€™s physical probability of default does not necessarily reflect its exposure to
systematic default risk. In fact, George and Hwang (2010) show that firms with higher
sensitivities to systematic default risk make capital structure choices that reduce their
physical probabilities of distress. It is therefore not correct to rank firms based on their
physical default probabilities when pricing financial distress, because such a ranking does
not properly reflect firmsâ€™ exposures to systematic default risk, the only type of default
risk that should be rewarded with a premium.
Moreover, previous papers have shown that three stock characteristicsâ€”high
idiosyncratic volatility, high leverage, and low profitabilityâ€”are associated with high
historical default rates. However, these are the same characteristics that are known to be
associated with low expected future returns. Within the q-theory framework (Cochrane
1991; Liu, Whited and Zhang 2009), low profitability (more likely to default) firms have
low expected future returns. Similarly, firms with high leverage (more likely to default)
and high idiosyncratic volatility (more likely to default) have low expected future stock
returns (Korteweg 2010; Dimitrov and Jain 2008; Penman, Richardson and Tuna 2007;
Ang, Hodrick, Xing and Zhang 2009). It is not clear if the distress anomaly is at least
partially attributable to one or more of these previously documented return relationships. 4
4
There is a strong relationship between distress risk and the three stock characteristics. When we form
quintile portfolios sorted on historical probabilities of default -computed using coefficients from Column 1
of Table 2-, idiosyncratic volatility increases monotonically from 2.5% for the lowest distress group to
4.5% for the highest distress group. Leverage increases from 0.22 for the lowest distress group to 0.61 for
the highest distress group. Similarly, profitability for the lowest distress group is 1.2% and decreases
monotonically to -1.1% for the highest distress group. The 3-factor alpha for the zero cost portfolio formed
by going long high distress stocks and shorting low distress stocks is -1.078% per month, yet this premium
decreases to -0.36% after controlling for leverage. When we control for idiosyncratic volatility, the return
spread between high and low distress stocks reduces to -0.29%. Finally, controlling for profitability
reduces the spread to -0.29% per month, making it statistically insignificant.
4
We take a different approach and use a market-based measure, credit risk premium
computed from corporate credit spreads, to proxy for systematic default risk exposure.
We compute credit spreads as the difference between the bond yield of the firm and the
corresponding maturity-matched treasury rate. We then compute credit risk premium by
taking into account expected losses, taxes, and liquidity effects (Elton, Gruber, Agrawal
and Mann 2001; Chen, Lesmond, and Wei 2007; Driessen and de Jong 2007) and using
only the fraction of the spread that is due to systematic default risk exposure. This
measure offers two distinct advantages over others that have been used in the literature.
First, unlike stock characteristics used to measure default risk, which may reflect
information about future returns unrelated to distress risk, credit spreads reflect the
market consensus view of the credit risk of the underlying firm. Second, credit spreads
contain risk premium for systematic default risk, and are a proxy for the market-implied
risk-adjusted probability of default. Using credit risk premia sorted portfolios, we find
that firms with higher exposures to systematic default risk have higher expected equity
returns. This premium is subsumed by the market factor, as predicted by structural
models of default and rational asset pricing theory, and is further reduced economically
and statistically by the Fama-French risk factors.
Our measure of systematic default risk exposure, calculated from credit spreads,
limits the sample of firms to those that have issued corporate bonds. To ensure the
robustness of our results, we show that when firms are ranked based on their physical
default probabilities, as previously done in the literature, the distress anomaly is also
observed in the Bond sample. To further alleviate sample selection issues, we extend the
analysis to the full CRSP-COMPUSTAT sample. We compute a measure of systematic
5
default risk exposure for all firms regardless of whether they have bonds outstanding.
Following Hilscher and Wilson (2010), we assume a single factor structure for default
risk and measure a firmâ€™s systematic default risk exposure as the sensitivity of its default
probability to the common factor. We refer to the common factor as the systematic
default risk factor, and the sensitivity of a firmâ€™s default probability to the common factor
as its systematic default risk beta. First, we verify that systematic default risk beta is
significantly priced in the cross section of corporate bond risk premia, justifying our use
of corporate bond risk premium as a measure of systematic default risk exposure. This
relationship is robust to controlling for bond ratings, physical default probabilities,
accounting variables, market variables, and structural model parameters. Second, and
differently from Hilscher and Wilson (2010), we form decile portfolios by sorting all
equities in the CRSP-COMPUSTAT sample based on their systematic default risk betas.
Consistent with the bond sample results, we find that the portfolio with the highest
systematic default risk exposure has higher equity returns than the lowest systematic
default risk exposure portfolio. Moreover, we find that once we control for the market
factor, the difference in returns between the highest and lowest systematic default risk
portfolios becomes insignificant.
In our analyses of the sample of firms with bonds outstanding and of the full CRSP-
COMPUSTAT sample, we find no evidence of firms with high systematic default risk
exposure delivering anomalously low equity returns. These results are consistent with
the basic structural models of default in which aggregate risk factors drive default
probabilities as well as the returns on bonds and equities (Merton 1974; Campello, Chen
and Zhang 2008).
6
Ours is not the first paper to study the relationship between default risk and equity
returns. Dichev (1998) uses Altmanâ€™s z-score and Ohlsonâ€™s o-score to measure financial
distress. He finds a negative relationship between default risk and equity returns for the
1981â€“1995 time period. In a related study, Griffin and Lemmon (2002), using the o-
score to measure default risk, find that growth stocks with high probabilities of default
have low returns. Using a comprehensive set of accounting and market-based measures,
Campbell, Hilscher, and Szilagyi (2008, hereafter CHS) show that stocks with high risk
of default deliver anomalously low returns. Garlappi, Shu, and Yan (2008), who obtain
default risk measures from Moodyâ€™s KMV, find results similar to those of Dichev (1998)
and CHS (2008). They attribute their findings to the violation of the absolute priority
rule. Vassalou and Xing (2004) find some evidence that distressed stocks, mainly in the
small value group, earn higher returns. 5
George and Hwang (2010) suggest that firms with higher sensitivities to systematic
default risk make capital structure choices that reduce their overall physical probabilities
of default. They show that when in distress, low leverage firms suffer greater losses and
have greater exposures to systematic risk compared to high leverage firms. Avramov,
Jostova, and Philipov (2007) show that the negative return for high default risk stocks is
concentrated around rating downgrades. Chava and Purnanandam (2010) argue that the
poor performance of high distress stocks is limited to the post-1980 period, when
investors were positively surprised by defaults. When they use implied cost of capital
estimates from analysts' forecasts to proxy for ex-ante expected returns, they find a
positive relationship between default risk and expected returns. Kapadia (2011) creates a
5
Da and Gao (2010) argue that Vassalou and Xingâ€™s results are driven by one-month returns on stocks in
the highest default likelihood group that trade at very low prices. They show that returns are contaminated
by microstructure noise and that the positive one-month return is compensation for increased liquidity risk.
7
portfolio that tracks changes in aggregate firm failure rate in the U.S. He uses the return
to the tracking portfolio as an asset pricing factor along with the market risk premium to
explain size and value premiums. Our paper contributes to the literature by constructing
a default risk measure that ranks equities explicitly based on their exposures to systematic
default risk rather than ranking firms based on their physical probabilities of default.
The rest of the paper is organized as follows. Section 2 describes the data. Section 3
describes the physical default probability measure used in this study. Section 4 describes
the use of credit spreads as a proxy for systematic default risk exposure. Section 5
contains asset pricing tests, in which equities are ranked based on their physical default
probabilities and systematic default risk exposures. Section 6 describes the construction
and use of an alternative systematic default risk factor and extends the equity return
analyses to the full CRSP-COMPUSTAT sample. Finally, Section 7 concludes.
2. Data
Corporate bond data used to compute the credit risk-premium in this study comes
from three separate databases: the Lehman Brothers Fixed Income Database (Lehman)
for the period 1974 to 1997, the National Association of Insurance Commissioners
Database (NAIC) for the period 1994 to 2006, and the Trade Reporting and Compliance
Engine (TRACE) system dataset for the period 2003 to 2010. We also use the Fixed
Income Securities Database (FISD) for bond descriptions. Due to the small number of
observations prior to 1980, we include only the period 1980 to 2010 in the analyses that
follow. We match the bond information with firm-level accounting and price information
obtained from COMPUSTAT and CRSP for the same time period. We exclude financial
8
firms (SIC codes 6000â€“6999) from the sample. To avoid the influence of microstructure
noise, we also exclude firms priced less than one dollar.
Our sample includes all U.S. corporate bonds listed in the above datasets that satisfy a
set of selection criteria commonly used in the corporate bond literature. 6 We exclude all
bonds that are matrix-priced (rather than market-priced) from the sample. We remove all
bonds with equity or derivative features (i.e., callable, puttable, and convertible bonds),
bonds with warrants, and bonds with floating interest rates. Finally, we eliminate all
bonds that have less than one year to maturity.
For all selected bonds, we extract beginning of month credit spreads, calculated as the
difference between the corporate bond yield and the corresponding maturity-matched
treasury rate. There are a number of extreme observations for the variables constructed
from the different bond datasets. To ensure that statistical results are not heavily
influenced by outliers, we set all observations higher than the 99th percentile value of a
given variable to the 99th percentile value. All values lower than the first percentile of
each variable are winsorized in the same manner. Using credit spreads we compute
credit risk premia (CRP) as described in the next section. For each firm, we then
compute a value-weighted average of that firmâ€™s CRP, using market values of the bonds
as weights. There are 121,714 firm-months and 1,071 unique firms with CRP and
corresponding firm-level accounting and market data. There is no potential survivorship
bias in our sample as we do not exclude bonds of firms that have gone bankrupt or bonds
that have matured.
6
See for instance Duffee (1999), Collin-Dufresne, Goldstein, and Martin (2001), and Avramov et al.
(2007).
9
We use hazard regressions using historical defaults to compute physical default
probabilities. Corporate defaults between 1981 and 2010 are identified from the
Moodyâ€™s Default Risk Servicesâ€™ Corporate Default database, SDC Platinumâ€™s Corporate
Restructurings Database, Lynn M. LoPucki's Bankruptcy Research Database, and
Shumwayâ€™s (2001) list of defaults. We choose 1981 as the earliest year for identifying
defaults because the Bankruptcy Reform Act of 1978 is likely to have caused the
associations between accounting variables and the probability of default to change.
Furthermore, we have little corporate bond yield information prior to 1980. In all, we
obtain a total of 1,290 firm defaults covering the period 1981â€“2010. We have complete
accounting-based measures for 728 of these failures. Of these 728 failures, 118 also have
corresponding corporate bond information. For the full CRSP-COMPUSTAT sample as
well as for the subsample of firms that have bonds outstanding we use accounting and
market-based variables used by CHS (2008) when predicting defaults. The variables we
use are the following: NIMTAAVG is a geometrically declining average of past values of
the ratio of net income to the market value of total assets; TLMTA is the ratio of total
liabilities to the market value of total assets; EXRETAVG is a geometrically declining
average of monthly log excess stock returns relative to the S&P 500 index; SIGMA is the
standard deviation of daily stock returns over the previous three months; RSIZE is the log
ratio of market capitalization to the market value of the S&P 500 index; CASHMTA is the
ratio of cash to the market value of total assets; MB is the market-to-book ratio, PRICE is
the log price per share truncated at $15 for shares priced above $15; DD is the Merton
(1974) â€œdistance-to-defaultâ€? measure, which is the difference between the asset value of
10
the firm and the face value of its debt, scaled by the standard deviation of the firmâ€™s asset
value. These variables are described in detail in the Appendix.
The bond sample covers a small portion of the total number of companies, but a
substantial portion in terms of total market capitalization. For instance, in the year 1997,
the number of firms with active bonds in our sample constitutes about 4% of all the firms
in the market. However, in terms of market capitalization, the dataset captures about
40% of aggregate equity market value in 1997. We compute summary statistics for
default measures and financial characteristics of the companies in our bond sample and
for all companies in CRSP. These results are summarized in Table 1. As not all
companies issue bonds, it is important to discuss the limitations of our bond dataset. Not
surprisingly, companies in the bond sample are larger and show a slight value tilt. They
also have higher profitability, more leverage, and higher equity returns; they hold less
cash and are less likely to default. There is, however, significant dispersion in size,
market-to-book ratio, default probability, and credit spread values of firms in the bond
sample. To ensure that our results are not driven by sample selection, in Section 5, we
show that when firms are ranked based on physical default probabilities the distress
anomaly is observed in the Bond sample. In Section 6, we extend the analyses to the
CRSP/COMPUSTAT sample.
3. Physical Default Probabilities
There is a vast literature on modeling the probability of default. In this paper, we utilize
dynamic models of default prediction (Shumway 2001; Chava and Jarrow 2004; CHS
2008), that avoid biases of static models by adjusting for potential duration dependence
11
issues. 7 We compute physical default probabilities by estimating a hazard regression
using the set of defaults described in the previous section. We use information available at
the end of the calendar month to predict defaults 12 months ahead. Specifically, we
assume that the probability of default in 12 months, conditional on survival in the
dataset for 11 months, is given by:
1
PDti - 1(Yti- 1+ 12 = 1|Yti- 2+ 12 = 0)= (1)
1 + exp (- a 12 - b12X ti - 1 )
where Yti- 1+ 12 is an indicator that equals one if the firm defaults in 12 months
conditional on survival for 11 months. X ti - 1 is a vector of explanatory variables
available at the time of prediction. We use accounting and market-based variables used
in CHS (2008) when predicting defaults. In addition we use Mertonâ€™s distance to default
measure that has been utilized in a number of previous studies. 8 All the variables
included in the hazard regressions are described in detail in the Appendix. We use
quarterly accounting variables lagged by two months and market variables lagged by
one month to ensure that this information is available at the time of default prediction.
We run two sets of hazard regressions, one using the sample of firms in the Bond
sample, and the other using all firms in the CRSP-COMPUSTAT sample. As mentioned
earlier, to ensure that our results are not driven by sample selection, we construct physical
default probabilities for the Bond sample using coefficients obtained from hazard
7
Altman (1968) and Ohlson (1980) are examples of such static models.
8
Mertonâ€™s (1974) structural default model treats the equity value of a company as a call option on the
companyâ€™s assets. The probability of default is based on the â€œdistance-to-defaultâ€? measure, which is the
difference between the asset value of the firm and the face value of its debt, scaled by the standard
deviation of the firmâ€™s asset value. There are a number of different approaches to calculating the distance-
to-default measure. We follow CHS (2008) and Hillegeist et al. (2004) in constructing this measure, the
details of which are provided in the appendix.
12
regressions that use only the firms in the Bond sample. This ensures that the distress
anomaly documented by the prior literature exists for the subset of firms that have bonds
outstanding.
Table 2 reports the results from the hazard regressions. In the first column, we use
the same covariates (NIMTAAVG, TLMTA, EXRETAVG, SIGMA, RSIZE, CASHMTA, MB
and PRICE) used in CHS (2008) to predict corporate defaults. The sample includes all
CRSP-COMPUSTAT firms for the 1980 to 2010 time period. As a comparison, we
report the estimates from the CHS (2008) study in column 2. The coefficient estimates
from these two regressions are very similar, suggesting that our default dataset, although
smaller than the CHS (2008) default dataset, captures a significant portion of the
variation in firm defaults. In column 3, we limit the sample to firms with only bonds
outstanding. Relative value (MB), liquidity position (CASHMTA), and share price
(PRICE) are no longer statistically significant predictors of failure. In the bond sample,
relatively larger firms are less likely to default, consistent with the full CRSP-
COMPUSTAT sample. We also use Mertonâ€™s distance to default (DD) measure as a
predictor of defaults in the bond sample (reported in column 6). We obtain qualitatively
similar results to those in the full CRSP-COMPUSTAT sample using our own set of
defaults (reported in column 4) as well as when compared to CHS (2008) results
(reported in column 5).
4. Corporate Spread as a Measure of Systematic Default Risk Exposure
In this section, we describe our use of corporate bond risk premia to measure systematic
distress risk exposure.
13
There is now a significant body of research that shows that compensation for default
risk constitutes a considerable portion of credit spreads. Huang and Huang (2003), using
the Longstaff-Schwartz (1995) model, find that distress risk accounts for 39%, 34%,
41%, 73%, and 93% of the corporate bond spread, respectively, for bonds rated AA, A,
BAA, BA, and B. Longstaff, Mithal, and Neis (2005) use the information in credit
default swaps (CDS) to obtain direct measures of the size of the default and non-default
components in corporate spreads. They find that the default component represents 51%
of the spread for AAA/AA-rated bonds, 56% for A-rated bonds, 71% for BBB-rated
bonds, and 83% for BB-rated bonds. Blanco, Brennan, and Marsh (2005) and Zhu (2006)
show significant similarity in the information content of CDS spreads and bond credit
spreads with respect to default. They confirm, through co-integration tests, that the
theoretical parity relationship between these two credit spreads holds as a long run
equilibrium condition. 9
As mentioned earlier, our focus in this paper is on measuring compensation for
systematic default risk exposure. We create this measure by extracting the credit risk
premium component from the credit spreads. Although credit risk makes up a significant
portion of corporate spreads, liquidity risk and taxes have also been shown to be
important (Elton et al. 2001; Chen, Lesmond, and Wei 2007; Driessen and de Jong 2007).
In computing the credit risk premium, we take into account expected losses, taxes, and
liquidity effects, and use only the fraction of the spread that is likely to be due to
systematic default risk exposure. We follow Driessen and de Jong (2007), Elton et al.
9
In this study we have chosen to use bond spreads instead of CDS spreads because bond data is available
for a substantially larger number of companies and is available for a much longer time period.
14
(2001), and Campello, Chen, and Zhang (2008) and compute the credit risk premium
(CRP) for each firm i and month t as:
í µí°¶í µí±…í µí±ƒí µí±–,í µí±¡ = ï¿½í µí±ƒí µí°·í µí±–,í µí±¡ Ã— ï¿½1 âˆ’ í µí°¿í µí±– ,í µí±¡ ï¿½ + ï¿½1 âˆ’ í µí±ƒí µí°·í µí±– ,í µí±¡ ï¿½ï¿½ Ã— ï¿½1 + í µí°¶í µí±Œí µí±–,í µí±¡ ï¿½
(2)
âˆ’ï¿½1 + í µí±Œí µí°ºí µí±– ,í µí±¡ ï¿½ âˆ’ í µí±‡í µí±‹í µí±–,í µí±¡ âˆ’ í µí°¿í µí±„í µí±–,í µí±¡ .
In Equation (2), PD is the physical probability of default computed from hazard
regressions described in Section 3. 10 L is the loss rate in the event of default. We follow
Elton et al. (2001) and Driessen and de Jong (2007) and use historical loss rates reported
in Altman and Kishore (1998) by rating category. The loss rates vary from 32% for
AAA-rated firms to 62% for CCC-rated firms. CY is the corporate bond yield, and YG is
the corresponding maturity-matched treasury yield. The equation assumes that all losses
are incurred at maturity.
Because bond investors have to pay state and local taxes on bond coupons whereas
treasury bond investors do not, we also remove this tax differential from the corporate
yields. Expected tax costs, TX, are computed as:
ï¿½ï¿½1 âˆ’ í µí±ƒí µí°·í µí±– ,í µí±¡ ï¿½ Ã— í µí°¶í µí±œí µí±¢í µí±?í µí±œí µí±›í µí±– ,í µí±¡ + í µí±ƒí µí°·í µí±– ,í µí±¡ Ã— ï¿½1 âˆ’ í µí°¿í µí±–,í µí±¡ ï¿½ï¿½ Ã— í µí¼?. (3)
The first part of Equation (3) captures the coupon rate, Coupon, conditional on no
default. The second part captures the tax refund in the event of default. í µí¼? is the effective
tax rate and following Elton et al. (2001) is set to 4.875%.
10
We compute default probabilities using coefficients obtained from column 3 of Table 2. In computing
default probabilities, we use quarterly accounting variables lagged by two months and market variables
lagged by one month to ensure that this information is available at the beginning of the month over which
default probabilities are measured.
15
The recent literature emphasizes the role of liquidity risk in the pricing of corporate
bonds (Driessen and de Jong 2007; Lin, Wang and Wu 2011; Downing, Underwood and
Xing 2005). We explicitly account for the liquidity effect in credit spreads by computing
liquidity risk premium for each bond in our dataset. The analysis follows Driessen and
de Jong (2007) and is based on a linear multifactor asset pricing model in which expected
corporate bond returns are explained by their exposure to market risk and liquidity risk
factors. 11 We consider two types of liquidity risk, one originating from the equity market
and one from the treasury bond market. For the stock market, we use the liquidity
innovations of Pastor and Stambaugh (2003); for the treasury market, we use changes in
quoted bid-ask spreads on long-term treasury bonds. 12 We compute expected bond
returns for 11 rating-maturity groups using equation (2), and use a cross-sectional
regression to compute risk premium associated with liquidity innovations in the stock and
treasury markets. 13 We then subtract the computed liquidity premium, LQ, from the
corporate bond spreads with the corresponding rating and maturity. Table 3 summarizes
the computations for different rating-maturity groups.
Our results are in line with the findings in the literature (Driessen and de Jong 2007;
Elton et al. 2001; Campello, Chen and Zhang 2008). Figure 2 plots the computed
expected losses, taxes, and liquidity premium against corporate spreads. In the rest of
this paper, we use the portion of credit spreads that compensates for systematic default
risk exposure, net of expected losses, taxes, and liquidity premium. We call this variable
CRP (Credit Risk Premium).
11
As in Driessen and de Jong (2007) we also included changes in implied market volatility orthogonalized
by market returns as an additional factor, and we obtained similar results.
12
We thank Alex Hsu for providing the data on treasury bond bid-ask quotes.
13
We refer to bonds with maturity greater than seven years as having â€œlong maturityâ€? and with maturity
less than seven years as having â€œshort maturity.â€?
16
5. Pricing of Distress Risk
5.1. Physical PDâ€™s and Equity Returns
In this section, we analyze the relationship between physical default probabilities and
future stock returns using the firms in the CRSP-COMPUSTAT sample and using the
firms that have bonds outstanding in the Bond sample. For the CRSP-COMPUSTAT
sample we compute default probabilities using coefficients obtained from column 1 of
Table 2. 14 For the Bond sample we compute default probabilities using coefficients
obtained from column 3 of Table 2. In computing these default probabilities, we use
quarterly accounting variables lagged by two months and market variables lagged by one
month to ensure that this information is available at the beginning of the month over
which default probabilities are measured. We sort stocks in the full CRSP-
COMPUSTAT sample into deciles each month from 1981 through 2010 according to
their physical default probabilities, and compute value-weighted returns for each
portfolio. If a delisting return is available, we use the delisting return; otherwise, we use
the last available return in CRSP.
We repeat the same analyses for stocks that have bonds outstanding. We construct
physical default probabilities in the Bond sample using coefficients obtained from hazard
regressions using the bond sample. This analysis ensures that the distress risk anomaly
observed in the full CRSP-COMPUSTAT sample also exists for the bond sample when
firms are ranked using physical default probabilities. To save space, we report returns for
only the top and bottom deciles, and the difference between the top and bottom deciles.
14
We obtain similar results using CHS coefficients computed on a rolling basis (we thank Jens Hilscher for
providing this data), Mertonâ€™s distance-to-default measure, Ohlsonâ€™s o-score and Altmanâ€™s z-score, which
are not reported to save space.
17
We compute value-weighted returns for these decile portfolios on a monthly basis
and regress the portfolio return in excess of the risk-free rate on the market (MKT), size
(SMB), value (HML), and momentum (MOM) factors:
i
rti = a i + bMKT i
MK Tt + bSMB i
SMBt + bHML i
HMLt + bMOM MOM t + eti . (4)
In Panel A of Table 4, we report portfolio return results for the CRSP-
COMPUSTAT sample. Our results are consistent with those obtained in previous
studies. Stocks in the highest default risk portfolio have significantly lower returns. The
difference in returns between the highest and lowest default risk portfolios is -1.184% per
month. The alphas from the market and the 3- and 4-factor models are economically and
statistically significant. The monthly 4-factor alpha for the zero cost portfolio formed by
going long on stocks in the highest default risk decile, and short on stocks in the lowest
default risk decile is -0.83% per month. Portfolio return analyses that utilize historical
default probabilities calculated using coefficients from the bond sample are reported in
Panel B of Table 4. The results are weaker for the bond sample, but still economically
and statistically significant. Using firms that have credit spread information, the monthly
4-factor alpha for the zero cost portfolio formed by going long on stocks in the highest
default risk decile and short on stocks in the lowest default risk decile is -0.49%.
Distressed stocks load positively on the size and value factors. The negative loading on
the momentum factor is consistent with the intuition that distressed stocks tend to have
low returns prior to portfolio formation.
As a robustness check, we also compute risk adjusted returns per unit of distress risk
for the bond sample as well as for the CRSP-COMPUSTAT sample. One reason that the
18
distress anomaly is smaller in the bond sample is that the companies in the highest
distress decile in the CRSP-COMPUSTAT sample have higher default probabilities than
the stocks in the highest distress decile in the bond sample. To take into account the
differences in default probabilities, we follow CHS (2008) and regress the return of each
long-short portfolio onto the differences in log default probabilities including no intercept
in the regression. The coefficients from this regression would provide us with a distress
premium per unit of log default probability. We use long-short distress portfolio returns
adjusted for the Famaâ€“French three-factor model. The coefficient estimate on the log
default probability is 6.492 (t-stat = 5.02) for the CRSP-COMPUSTAT sample and 5.657
(t-stat = 3.24) for the bond sample, suggesting that per unit of log default probability, the
distress effect is similar in the CRSP-COMPUSTAT and Bond samples.
The analyses in this section show that using physical default probabilities computed
in the Bond sample and the CRSP-COMPUSTAT sample produces results similar to
those of CHS (2008) and others in the literature. The distress anomaly persists in our
Bond sample when we use physical probabilities of default to rank firms.
5.2 Credit Risk Premium and Equity Returns
In this section, we examine how CRPs (credit risk premia) are related to future realized
equity returns. We sort stocks into deciles from 1981 to 2010, using CRPs in the
previous month. We compute value-weighted returns for each portfolio and update the
portfolios each month. As before, if a delisting return is available we use the delisting
return; otherwise we use the last available return in CRSP. To save space, we only report
returns for the top and bottom decile portfolios, and the return difference between the top
and bottom deciles in Table 5.
19
Our results challenge those obtained in the previous studies. Using CRPâ€™s as a
measure of systematic default risk exposure, the difference in raw returns between the
highest and lowest default risk portfolios is 0.521% per month and statistically
significant. The 4-factor monthly alpha for a portfolio formed by going long on stocks in
the highest default risk exposure portfolio and short on stocks in the lowest default risk
exposure portfolio is -0.005% and statistically insignificant when we use CRP as our
measure of systematic default risk exposure.
There is a positive relationship between CRP and raw equity returns, and the return
of the high-minus-low excess spread portfolio is statistically significant. CAPM and
multi-factor regressions show that alphas are subsumed in all CRP portfolios, suggesting
that variation in systematic default risk exposure is captured mainly by the market factor
and partly by the size and value factors. The size and value factors have statistically
significant positive loadings for the high minus low CRP portfolio suggesting that these
factors are intimately related to systematic default risk exposure. These results are
consistent with structural models of default in which aggregate risk factors drive default
probabilities as well as the returns on bonds and equities (Merton 1974; Campello, Chen
and Zhang 2008).
Ranking stocks on their physical default probabilities inferred from historical data, as
done in Dichev (1998), CHS (2008), and others, implicitly assumes that high default
probability stocks also have high exposures to the systematic component of default risk.
Using CRP, we explicitly rank firms based on their exposures to the systematic
component of default risk and we find no evidence of systematic default risk being
negatively priced.
20
6. Alternative Measure of Systematic Default Risk
We now extend the analysis of Section 5.2 to the full CRSP-COMPUSTAT sample
to ensure the robustness of our results. In particular, we follow Hilscher and Wilson
(2010) and identify a measure of systematic default risk that can be calculated for all
firms regardless of whether they have bonds outstanding. We form decile portfolios by
sorting all equities in the CRSP-COMPUSTAT sample based on their systematic default
risk betas and investigate the pricing of systematic default risk in the cross section of
equity returns.
We measure a firmâ€™s systematic default risk exposure as the sensitivity of its default
probability to the median default probability of all firms in the CRSP-COMPUSTAT
sample. We refer to this measure as systematic default risk beta. We find that portfolios
with high systematic default risk betas, on average, have higher returns than portfolios
with low systematic default risk betas, verifying our results in Section 5.2. We also show
that systematic default risk beta is significantly priced in the cross-section of credit risk
premia validating the use of CRP as a measure of systematic default risk exposure.
6.1 Measuring Systematic Default Beta
We assume that historical default probabilities have a single common factor and use the
median cross-sectional default probability to proxy for this common factor. The
assumption of a single factor is a good approximation as we find that the first principal
component explains 74.7% of variation in default probabilities.15 The first principal
15
Extracting principal components in the standard way from the full panel of CRSP-COMPUSTAT firms is
problematic because the cross-section is much larger than the time series. We therefore first shrink the size
of the cross-section by assigning each firm-month to a given rating-month and calculating equal-weighted
average 12-month cumulative default probabilities as done by Hilscher and Wilson (2010). We group all
21
component and the median default probability have a correlation of 0.96 and are
significantly higher during and after recessions. This is consistent with economic theory
that suggests that systematic risk (discount rate) is higher during recessions.
To compute each firmâ€™s sensitivity to the systematic default factor, we estimate the
following regression for each firm over 48-month rolling windows:
í µí±– í µí±– í µí±–
í µí±ƒí µí°·í µí±¡ = í µí»¼í µí¼? + í µí±†í µí±Œí µí±†í µí°·í µí°¸í µí°¹í µí°µí µí°¸í µí±‡í µí°´í µí±–
í µí¼? Ã— í µí±€í µí±ƒí µí°·í µí±¡ + í µí¼€í µí±¡ . (5)
í µí±–
í µí±ƒí µí°·í µí±¡ is the 12-month annualized physical default probability for firm i in month t. It is
computed each month using coefficients from column 1 in Table 2. As before, in
computing physical default probabilities, we use quarterly accounting variables lagged by
two months and market variables lagged by one month to ensure that this information is
available at the beginning of the month over which default probabilities are measured.
í µí±€í µí±ƒí µí°·í µí±¡ is the cross-sectional median physical default probability across all firms.16
í µí±†í µí±Œí µí±†í µí°·í µí°¸í µí°¹í µí°µí µí°¸í µí±‡í µí°´í µí±–
í µí¼? is exposure to systematic default risk in month Ï„, obtained from rolling
regressions using the past 48 months of data.
6.2 Physical PDâ€™s, Systematic Default Risk Exposures and Firm Characteristics
We examine how systematic default risk exposures are related to physical default
probabilities and firm characteristics. Each month, from January 1981 through December
2010, we rank and put into decile portfolios companies in the CRSP-COMPUSTAT and
Bond samples based on their systematic default risk exposures in the previous month.
firms with ratings of CCC+ and below together. This leaves us with a panel of 17 ratings groups with 360
months of data. Forming industry groups rather than ratings groups yields similar results..
16
The results are similar if we instead use the first principal component.
22
For the CRSP-COMPUSTAT sample we use SYSDEFBETA as our measure of systematic
default risk exposure and for the Bond Sample we use CRP as our measure of systematic
default risk exposure. We calculate average market-to-book ratio (MB), market equity
(ME), physical default probability (PD), and Mertonâ€™s distance to default (DD) values for
all the companies in a given systematic default risk exposure decile portfolio for the two
samples. The results are reported in Table 6. Panel A of Table 6 reports results for the
CRSP-COMPUSTAT sample while Panel B of Table 6 reports results for the Bond
sample.
Panel A shows that there is not a monotonic relationship between physical default
probabilities and systematic default risk exposures. For example, while the average
physical default probability of the lowest SYSDEFBETA portfolio is 0.130%, the average
physical default probabilities of the next eight larger SYSDEFBETA decile portfolios are
lower than 0.130%. Panel B yields similar results to Panel A. The average physical
default probability of the lowest í µí°¶í µí±…í µí±ƒ portfolio is 0.052%. This default probability is
larger than the average physical default probabilities of the next seven CRP decile
portfolios. The relationship between physical default probabilities and systematic default
risk exposures is U-shaped both in the CRSP-COMPUSTAT and Bond samples. Firms
with very high and very low physical default probabilities command greater credit risk
premium. This result is consistent with prior work reporting that asset correlations
implied from historical defaults are similarly U-shaped (Chernih, Vanduffel and Henrard
2006), and it highlights our main point that a firmâ€™s expected probability of default does
not necessarily reflect the firmâ€™s exposure to systematic default risk.
23
6.3 Default Risk Beta and Credit Spreads
In this section, we analyze the relationship between our measure of credit risk premium
calculated in Section 4 and systematic default risk beta. We show that systematic default
risk beta (í µí±†í µí±Œí µí±†í µí°·í µí°¸í µí°¹í µí°µí µí°¸í µí±‡í µí°´) can explain the cross-sectional variation in credit risk premia
in corporate bonds. This finding provides further evidence that SYSDEFBETA is a good
measure of systematic default risk exposure, and that investors demand compensation for
this exposure. This result also validates our use of CRPs to measure firmsâ€™ exposures to
systematic default risk.
Table 7 summarizes Fama-MacBeth cross-sectional regression results when monthly
credit risk premium (in %) are regressed on lagged systematic default risk beta
(í µí±†í µí±Œí µí±†í µí°·í µí°¸í µí°¹í µí°µí µí°¸í µí±‡í µí°´ as calculated in equation 5) and firm characteristics that are related to
credit risk. In the regression, we control for the CAPM beta (BETACAPM), return
volatility (SIGMA), profitability (NIMTAAVG), leverage (TLMTA), amount of liquid
assets (CASHMTA), market-to-book ratio (MB), and relative size of the firm (RSIZE).
We also control for two bond characteristics: average issue amount (OAMT) and average
time to maturity (TTM) of a firmâ€™s outstanding bonds. As alternative credit risk
measures, we include Mertonâ€™s distance to default (DD), physical default probability
(PD), and the Standard & Poorâ€™s (S&P) rating (RATING). The t-statistics for the slopes
are based on the time series variability of the estimates, incorporating a Newey-West
(1987) correction with four lags to account for possible autocorrelation in the estimates.
In column 1, we control for stock characteristics that have been shown to be important
determinants of credit risk by CHS (2008) as well as time to maturity and the offering
amount of the firmâ€™s outstanding bonds. In column 2 we control for rating and Mertonâ€™s
24
distance to default, in addition to time to maturity and bond offering amount. In column
3 we control for time to maturity, offering amount of the bond, Mertonâ€™s distance to
default and the physical probability of default. In column 4 we control for all the CHS
(2008) variables, firm rating, Mertonâ€™s distance to default, and the physical probability of
default. In all specifications the loading on systematic default risk beta, í µí±†í µí±Œí µí±†í µí°·í µí°¸í µí°¹í µí°µí µí°¸í µí±‡í µí°´,
is positive and statistically significant.
The impact of í µí±†í µí±Œí µí±†í µí°·í µí°¸í µí°¹í µí°µí µí°¸í µí±‡í µí°´ on spreads is also economically significant. Results in
column 4 of Table 7 suggest that moving from the 75th percentile systematic default risk
beta firm (í µí±†í µí±Œí µí±†í µí°·í µí°¸í µí°¹í µí°µí µí°¸í µí±‡í µí°´ = 0.156) to the 95th percentile firm (í µí±†í µí±Œí µí±†í µí°·í µí°¸í µí°¹í µí°µí µí°¸í µí±‡í µí°´ = 0.954)
leads to an increase of 45 basis points in bond risk premium after controlling for all
parameters known to influence credit spreads.
The results suggest that systematic default risk beta is an important driver of the
credit risk premium in corporate bond spreads. CRP, our measure of exposure to
systematic default risk computed from corporate bond spreads, and systematic default
risk beta (í µí±†í µí±Œí µí±†í µí°·í µí°¸í µí°¹í µí°µí µí°¸í µí±‡í µí°´) are comparable proxies for exposure to systematic default risk.
In the next section we use systematic default risk beta (í µí±†í µí±Œí µí±†í µí°·í µí°¸í µí°¹í µí°µí µí°¸í µí±‡í µí°´) to examine the
pricing of systematic default risk in the cross section of equity returns in the CRSP-
COMPUSTAT sample.
6.4 Pricing of Systematic Default Risk in the CRSP-COMPUSTAT Sample
The systematic default risk beta described in the previous section allows us to test
whether systematic default risk is priced in the larger CRSP-COMPUSTAT sample. In
25
Section 5.2, our analysis was confined to firms that have outstanding bonds because we
used the bond credit risk premium as our proxy for systematic default risk compensation.
We use the same portfolio approach described in Section 5. In particular, we sort
stocks into deciles each month from January 1981 through December 2010 according to
their systematic default risk betas obtained at the beginning of the previous month. We
then calculate the value-weighted decile portfolio returns for all stocks in the CRSP-
COMPUSTAT sample on a monthly basis and regress the portfolio return in excess of the
risk-free rate on the market (MKTRF), size (SMB), value (HML), and momentum (MOM)
factors. In Table 8, we report regression results for only the top and bottom decile
portfolios along with the top decile minus bottom decile hedge portfolio to save space.
Results in Table 8, which are obtained from the CRSP-COMPUSTAT sample, are
similar to those reported in Table 5, which are obtained using the bond sample. Table 5
shows that the highest CRP decile portfolio earns on average 52 basis points more per
month compared to the lowest CRP decile portfolio. Similarly, Table 8 shows that the
highest systematic default risk beta decile portfolio in the full CRSP-COMPUSTAT
sample earns 46 basis points more per month compared to the lowest systematic default
risk beta decile portfolio. This result is significant at the 10% level. Once we control for
the market factor, the statistical significance of the hedge portfolio return disappears,
suggesting a strong link between systematic default risk and market risk. Controlling for
Fama-French size and value factors further reduces the economic and statistical
significance of the systematic default risk premium, supporting the Fama and French
(1992) conjecture that size and value premiums may be related to systematic distress risk.
26
Overall, the results in this section verify the robustness of using credit spreads as a proxy
for systematic default risk exposure and confirm our results in Section 5.
7. Conclusion
We argue that the distress risk anomaly documented in the cross section of equity returns
is due to mismeasuring systematic default risk. Previous papers measure financial
distress by computing firmsâ€™ expected probabilities of default inferred from historical
default data. This calculation ignores the fact that firm defaults are correlated and that
some defaults are more likely to occur in bad times, thus failing to appropriately account
for the systematic nature of default risk. We use credit risk premium obtained from
corporate credit spreads to proxy for a firmâ€™s exposure to systematic default risk. Unlike
previously used measures that proxy for physical probabilities of default, credit risk
premia proxy for risk-adjusted default probabilities, reflecting the risk premium for the
non-diversifiable component of distress risk. We find that stocks that have higher credit
risk premium have higher expected equity returns. Consistent with structural models of
default, we also show that the premium to a high minus low systematic default risk hedge
portfolio is largely explained by the market factor, suggesting that CAPM beta captures
most of the variation in systematic default risk exposure.
To show that our results are robust to sample biases, we conduct two analyses. First,
we show that when firms in our Bond sample are ranked according to traditional
measures of default risk used in the literature, the default risk anomaly exists in the bond
sample. Second, we construct an alternative proxy to measure systematic default risk
exposure (í µí±†í µí±Œí µí±†í µí°·í µí°¸í µí°¹í µí°µí µí°¸í µí±‡í µí°´) and extend the analyses to the full CRSP sample. We obtain
27
results similar to what we find using the bond sample. These results are consistent with
the basic structural models of default in which aggregate risk factors drive the default
probability as well as the returns on bonds and equities.
28
APPENDIX
Here we explain the details of the variables used to compute the physical probability of default
(PD) and the Merton distance-to-default (DD) measure. We use quarterly accounting data from
COMPUSTAT and monthly market data from CRSP. MB is the market-to-book ratio. Book
equity, BE is defined as in Davis, Fama, and French (2000). To deal with outliers, we adjust total
value of assets, TA (COMPUSTAT quarterly data item: ATQ) by the difference between the
market equity (ME) and book equity (BE):
MT Ai ,t = T Ai ,t + 0.1(MEi ,t - BEi ,t )
(A.1)
.
NIMTAAVG is a geometrically declining average of past values of the ratio of net income
(data item: NIQ) to adjusted total assets:
1- f 2
NI MT AAVGt - 1,t - 12 = (NI MT At - 1,t - 3 + ... + NI MT At - 10,t - 12 ) (A.2)
1 - f 12 .
EXRETAVG is a geometrically declining average of monthly log excess stock returns
relative to the S&P 500 index:
1- f
EXRET AVGt - 1,t - 12 = (EXRETt - 1 + ... + f 11EXRETt - 12 ) (A.3)
1 - f 12 .
The weighting coefficient is set to Ï† = 2âˆ’1/3, such that the weight is halved each quarter.
TLMTA is the ratio of total liabilities (data item: NIQ) to adjusted total assets. SIGMA is
the standard deviation of daily stock returns over the previous three months. SIGMA is
coded as missing if there are fewer than five observations. RSIZE is the log ratio of
market capitalization to the market value of the S&P 500 index. CASHMTA is the ratio
of the value of cash and short-term investments (data item: CHEQ) to the value of
29
adjusted total assets. PRICE is the log price per share truncated from above at $15. All
variables are winsorized using a 1/99 percentile interval in order to eliminate outliers.
We follow CHS (2008) and Hillegeist, Keating, Cram, and Lunstedt (2004) to
calculate Mertonâ€™s distance-to-default measure. The market equity value of a company is
modeled as a call option on the companyâ€™s assets:
V E = V Ae- Â¶T N (d1) - Xe- rT N (d2 ) + (1 - e- Â¶T )V A
2
log(V A / X ) + (r - Â¶ - (s A / 2))T
d1 = (A.4)
sA T
d2 = d1 - s A T .
V E is the market value of a firm. V A is the value of the firmâ€™s assets. X is the face value
of debt maturing at time T. r is the risk-free rate, and Â¶ is the dividend rate expressed in
terms of V A . s A is the volatility of the value of assets, which is related to equity
volatility through the following equation:
s E = (V Ae- Â¶ T N (d1)s A ) / V E (A.5)
.
We simultaneously solve the above two equations to find the values of V A and s A . We
use the market value of equity for V E and short-term plus one-half long-term book debt to
proxy for the face value of debt X (data items: DLCQ+1/2*DLTTQ). s E is the standard
deviation of daily equity returns over the past three months. T equals one year, and r is
the one-year treasury bill rate. The dividend rate, d, is the sum of the prior yearâ€™s
30
common and preferred dividends, obtained from COMPUSTAT Annual, (data items:
DVP+DVC) divided by the market value of assets. We use the Newton method to
simultaneously solve the two equations above. For starting values for the unknown
variables we use, V A = V E + X , and s A = s EV E (V E + X ) . Once we determine asset
values, V A , we then compute asset returns as in Hillegeist et al. (2004):
V + Dividends - V A,t - 1 Ã¹
Ã©
m = max Ãª A,t ,r Ãº (A.6)
t Ãª V Ãº
Ãª
Ã« A ,t - 1 Ãº.
Ã»
Because expected returns cannot be negative, if asset returns are below zero, they are set
to the risk-free rate. 17 Mertonâ€™s distance to default is finally computed as:
log (V A / X ) + ( m- Â¶ - (s A
2
/ 2 ))T
DD = - (A.7)
sA T .
17
We obtain similar results if we use a 6% equity premium instead of asset returns as in CHS (2008).
31
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37
Table 1: Summary Statistics
Table 1 reports summary statistics for firm characteristics and distress measures for companies in the CRSP sample (left panel) and the bond sample (right panel). MB is the
market-to-book ratio, and ME is market capitalization in millions of dollars. CASHMTA is the ratio of cash to the market value of total assets. EXRETAVG is a geometrically
declining average of monthly log excess stock returns relative to the S&P 500 index. NIMTAAVG is a geometrically declining average of past values of the ratio of net income to
the market value of total assets. TLMTA is the ratio of total liabilities to the market value of total assets, and RSIZE is the log ratio of market capitalization to the market value of
the S&P 500 index. IDIOVOL is the standard deviation of regression errors obtained from regressing daily excess returns on the Fama and French (1993) factors. TOTVOL is the
standard deviation of daily stock returns over the previous twelve months. PRICE is the log price per share truncated at $15. PD is the physical probability of default reported as a
percentage. DD is the Merton distance to default measure. The Appendix describes how these variables are calculated. P25, P50, and P75 represent 25th, 50th, and 75th percentiles,
respectively. Statistical significance at the 10%, 5%, and 1% levels is denoted by *, **, and ***, respectively.
CRSP Sample Bond Sample
Variables Mean STD P25 P50 P75 Mean STD P25 P50 P75 Difference
MB 1.983 1.466 0.900 1.533 2.644 1.794 1.131 0.999 1.486 2.268 0.189***
ME 1,273.8 5,713.0 20.7 91.8 271.6 5,327.7 17,251.1 417.5 1,297.2 3,811.6 -4,053.4***
CASHMTA 0.091 0.091 0.024 0.070 0.114 0.050 0.058 0.010 0.028 0.070 0.041***
EXRETAVG -0.010 0.043 -0.034 -0.006 0.018 -0.001 0.030 -0.017 0.000 0.016 -0.008***
NIMTAAVG 0.003 0.015 -0.001 0.005 0.012 0.008 0.008 0.003 0.008 0.012 -0.005***
TLMTA 0.413 0.282 0.159 0.374 0.643 0.536 0.229 0.360 0.535 0.708 -0.123***
RSIZE -10.708 1.604 -11.907 -10.790 -9.617 -8.031 1.160 -8.724 -7.701 -7.113 -2.677***
IDIOVOL 0.035 0.027 0.018 0.028 0.044 0.018 0.010 0.012 0.015 0.020 0.018***
TOTVOL 0.037 0.028 0.020 0.030 0.046 0.020 0.010 0.014 0.018 0.023 0.017***
PRICE 2.116 0.705 1.646 2.431 2.708 2.635 0.263 2.708 2.708 2.708 -0.519***
PD * 100 0.081 0.155 0.021 0.039 0.078 0.043 0.067 0.020 0.031 0.048 3.762***
DD 7.094 39.000 2.906 5.024 8.177 8.384 5.856 5.063 7.518 10.643 -1.290***
Table 2: Default Prediction
Table 2 reports results from hazard regressions of the default indicator on the predictor variables. The data are constructed
such that all of the predictor variables are observable 12 months before the default event. NIMTAAVG is a geometrically
declining average of past values of the ratio of net income to the market value of total assets. TLMTA is the ratio of total
liabilities to the market value of total assets. EXRETAVG is a geometrically declining average of monthly log excess stock
returns relative to the S&P 500 index. SIGMA is the standard deviation of daily stock returns over the previous three
months. RSIZE is the log ratio of market capitalization to the market value of the S&P 500 index. CASHMTA is the ratio of
cash to the market value of total assets. MB is the market-to-book ratio; PRICE is the log price per share truncated at $15,
and DD is Mertonâ€™s distance to default. These variables are described in detail in the Appendix. Results under â€œAll Firmsâ€?
are estimates computed using the full CRSP-COMPUSTAT sample of defaults with available accounting information.
Results under â€œCHS Sampleâ€? show the estimates CHS (2008) report in their paper. Results under â€œFirms with Bondsâ€? are
estimates computed using the sample of defaults from companies that have issued bonds with available accounting
information. Absolute values of z-statistics are reported in parentheses below coefficient estimates. McFadden pseudo R2
values are reported for each regression. Statistical significance at the 10%, 5%, and 1% levels is denoted by *, **, and ***,
respectively.
(1) (2) (3) (4) (5) (6)
Sample Period: 1981â€“2010 1963â€“2003 1981â€“2010 1981â€“2010 1981â€“2010 1981â€“2010
Lag (Months) 12 12 12 12 12 12
NIMTAAVG -21.989*** -20.260*** -18.308***
(10.33) (18.09) (2.74)
TLMTA 2.188*** 1.420*** 1.503***
(16.84) (16.23) (2.76)
EXRETAVG -7.871*** -7.13*** -6.241**
(10.28) (14.15) (2.13)
SIGMA 1.461*** 1.410*** 1.774***
(11.19) (16.49) (5.17)
RSIZE -0.063*** -0.045** -0.614***
(4.21) (2.09) (7.28)
CASHMTA -1.516*** -2.130*** -1.064
(7.85) (8.53) (1.21)
MB 0.085*** 0.075*** 0.127
(2.63) (6.33) (0.91)
PRICE -0.167* -0.058 -0.017
(1.74) (1.40) (0.95)
DD -0.356*** -0.345*** -0.460***
(17.18) (33.73) (8.07)
CONSTANT -9.718*** -9.160*** -13.844*** -3.401*** Not -2.634***
(18.12)
(30.89) (8.90) (48.52) Reported (11.10)
Observations 993,560 1,565,634 54,551 993,560 1,565,634 54,551
Defaults 728 1968 118 728 1968 118
Pseudo R2 0.134 0.114 0.156 0.083 0.066 0.129
All Firms in All Firms in
Firms with CHS Firms with
Sample Type CRSP- CHS Sample CRSP-
Bonds Sample Bonds
COMPUSTAT COMPUSTAT
Table 3: Expected Losses, Taxes, and Liquidity Premia in Credit Spreads
In Table 3, we report average credit spreads, spreads in excess of expected losses and taxes and liquidity premium for various
rating-maturity groups. Column (1) reports corporate bond yields minus maturity-matched government treasuries; column (2)
reports spreads in excess of expected losses and taxes; and column (3) reports the liquidity premium for each corresponding
rating/maturity portfolio. The estimation of these components is described in Section 4.1. Bonds with maturity greater than
seven years are referred to as having â€œlong maturity,â€? and bonds with maturity less than seven years are referred to as having
â€œshort maturity.â€?
Spread in Excess of
Expected Losses and
Portfolio Spread Taxes Liquidity Premium
AAA short-mat 0.97% 0.62% 0.13%
AAA long-mat 0.95% 0.62% 0.23%
AA short-mat 1.04% 0.56% 0.24%
AA long-mat 1.26% 0.84% 0.35%
A short-mat 1.32% 0.81% 0.33%
A long-mat 1.28% 0.81% 0.41%
BBB short-mat 1.99% 1.20% 0.50%
BBB long-mat 2.06% 1.32% 0.73%
BB 3.78% 2.09% 0.88%
B 5.28% 2.10% 1.30%
CCC 10.36% 4.75% 1.40%
40
Table 4: Distress Portfolio Returns Sorted on Physical Default Probabilities
Table 4 reports time series averages, CAPM, 3-factor and 4-factor regression results for distress risk portfolios. We sort
stocks into deciles each month from January 1981 to December 2010 according to their physical default probabilities,
obtained at the beginning of the previous month, calculated using the hazard coefficients computed using the CRSP-
COMPUSTAT sample (Panel A) and using the bond sample (Panel B). We compute the value-weighted returns for these
decile portfolios on a monthly basis and regress the portfolio return in excess of the risk-free rate on the market (MKT), size
(SMB), value (HML), and momentum (MOM) factors. The factors are obtained from Ken Frenchâ€™s website We report
regression results for only the top and bottom decile portfolios as well as the high-minus-low distress risk hedge portfolio to
save space. Absolute values of t-statistics are reported in parentheses below their respective coefficient estimates. Statistical
significance at the 10%, 5%, and 1% levels is denoted by *, **, and ***, respectively.
Panel A: Monthly Equity Returns For Default Risk Portfolios in the full CRSP-COMPUSTAT sample sample
Physical PDâ€™s constructed with coefficients from Column (1) of Table 2
Alpha * 100 MKT SMB HML MOM
10th 0.608**
(2.01)
0.166 1.041***
(0.99) (28.01)
0.433*** 0.879*** 0.109** -0.462***
(2.86) (23.63) (2.17) (8.05)
0.096 0.949*** 0.083* -0.37*** 0.337***
(0.72) (29.23) (1.94) (7.42) (11.05)
Alpha * 100 MKT SMB HML MOM
90th -0.576
(1.19)
-1.216*** 1.507***
(3.87) (21.46)
-1.509*** 1.511*** 0.923*** 0.43***
(5.29) (21.63) ( 9.82) (3.99)
-0.736*** 1.351*** 0.981*** 0.219*** -0.772***
(3.24) (24.48) (13.45) (2.58) (14.89)
Alpha * 100 MKT SMB HML MOM
90th - 10th -1.184**
(2.34)
-1.382*** 0.466***
(2.96) (4.28)
-1.942*** 0.632*** 0.814*** 0.892***
(4.68) (6.04) ( 6.73) (6.02)
-0.832*** 0.402*** 0.898*** 0.589*** -1.109***
(2.64) (5.69) (10.96) (6.25) (18.14)
41
Panel B: Monthly Equity Returns For Default Risk Portfolios in the Bond sample
Physical PDâ€™s constructed with coefficients from Column (3) of Table 2
Alpha * 100 MKT SMB HML MOM
10th 0.825***
(3.05)
0.382** 0.847***
(2.29) (22.64)
0.385** 0.891*** -0.274*** 0.003
(2.36) (22.27) (5.18) (0.05)
0.271* 0.913*** -0.283*** 0.031 0.114***
(1.65) (22.76) (5.41) (0.51) (3.07)
Alpha * 100 MKT SMB HML MOM
90th 0.318
(0.82)
-0.323 1.224***
(1.36) (22.92)
-0.694*** 1.437*** 0.009 0.685***
(3.19) (26.89) (0.13) (8.39)
-0.217 1.345*** 0.047 0.566*** -0.475***
(1.15) (29.42) (0.79) (8.14) (11.24)
Alpha * 100 MKT SMB HML MOM
90th - 10th -0.507*
(1.66)
-0.705*** 0.378***
(2.60) (5.74)
-1.079*** 0.546*** 0.284*** 0.682***
(3.83) (7.89) (3.10) (6.45)
-0.487** 0.432*** 0.330*** 0.535*** -0.589***
(1.97) (7.17) (4.20) (5.84) (10.58)
42
Table 5: Monthly Equity Returns for Credit Risk Premium Portfolios
In Table 5, we report time series averages, CAPM, 3-factor, and 4-factor regression results for distress risk portfolios. Each
month from January 1981 through December 2010, we sort stocks into 10 portfolios based on their credit risk premia (CRP) at
the beginning of the previous month. We compute the value-weighted return for these decile portfolios on a monthly basis and
regress the portfolio return in excess of the risk-free rate on the market (MKT), size (SMB), value (HML), and momentum
(MOM) factors. The factors are obtained from Ken Frenchâ€™s website. We report regression results for only the top and bottom
decile portfolios to save space. Absolute values of t-statistics are reported in parentheses below coefficient estimates. Statistical
significance at the 10%, 5%, and 1% levels is denoted by *, **, and ***, respectively.
Equity Returns in Credit Risk Premia
P tf li
Alpha * 100 MKT SMB HML MOM
10th 0.463*
(1.65)
-0.074 0.826***
(0.52) (23.63)
-0.021 0.890*** -0.319*** 0.020
(0.17) (27.51) (9.29) (0.47)
0.01 0.878*** -0.314*** 0.013 -0.03
(0.08) (26.00) (9.07) (0.29) (1.20)
Alpha * 100 MKT SMB HML HML
th
90 0.984***
(2.58)
0.325 1.014***
(1.33) (17.12)
-0.193 1.28*** 0.157*** 0.715***
(0.93) (22.83) (2.63) (9.62)
0.005 1.205*** 0.191*** 0.668*** -0.193***
(0.02) (21.65) (3.34) (9.37) (4.64)
Alpha * 100 MKT SMB HML HML
90th - 10th 0.521**
(1.98)
0.399 0.188***
(1.50) (2.91)
-0.172 0.391*** 0.476*** 0.695***
(0.75) (6.32) (7.25) (8.49)
-0.005 0.327*** 0.505*** 0.656*** -0.163***
(0.02) (5.21) (7.84) (8.15) (3.48)
43
Table 6: Stock Characteristics and Systematic Default Risk Exposure
In Table 6 we report summary statistics of stock characteristics for firms belonging to systematic default risk decile portfolios.
Default risk is measured in two alternative ways: once using physical default probabilities and once using systematic default
risk exposures. For the full CRSP-COMPUSTAT sample, each month from January 1981 through December 2010, we sort
stocks into 10 portfolios based on their systematic default risk betas obtained at the beginning of the previous month. For the
Bond sample, each month from January 1981 through December 2010, we sort stocks into 10 portfolios based on their value-
weighted credit risk premia (CRP). We then compute cross-sectional average values for various stock characteristics in each
group. Market-to-book ratio (MB) is calculated as the ratio of market equity in the previous calendar month to book equity in
the previous month. ME is the market value of equity in millions of dollars. PD is the physical probability of default reported as
a percentage. DD is Mertonâ€™s distance to default measure. For the full CRSP-COMPUSTAT sample PDâ€™s are calculated using
coefficients from column (1) of Table 2, whereas for the Bond sample PDâ€™s are calculated using coefficients from column (3)
of Table 2.
Panel A: Systematic Default Beta Portfolios, CRSP Sample
Portfolio MB ME PD*100 DD
Low 2.17 1269 0.130 5.006
2 2.06 2,444 0.071 7.296
3 2.13 3,342 0.048 8.954
4 2.11 3,359 0.041 9.630
5 2.07 2,837 0.045 9.240
6 2.02 2,327 0.052 8.207
7 1.97 1,651 0.064 7.854
8 1.95 823 0.085 5.997
9 1.97 482 0.118 6.063
High 2.18 266 0.175 4.393
Panel B: Credit Risk Premium Portfolios, Bond Sample
Portfolio MB ME PD*100 DD
Low 2.03 9,442 0.052 7.938
2 2.10 12,902 0.042 8.763
3 2.24 13,086 0.040 8.905
4 2.18 13,170 0.038 9.551
5 2.14 15,638 0.041 8.713
6 2.12 12,881 0.044 8.748
7 2.03 10,544 0.046 8.531
8 1.86 9,095 0.051 8.030
9 1.82 8,764 0.060 7.714
High 1.74 6,065 0.098 6.109
44
Table 7: Pricing of Systematic Default Risk Beta in the Cross Section of Credit Spreads
In Table 7, we run monthly Fama-MacBeth (1973) regressions of credit risk premium (in %) on default risk prediction variables
used in CHS 2008, firm rating, market beta, and systematic default risk beta. Our sample period covers January 1981 to
December 2010. We report Fama-MacBeth regression coefficients as well as their corresponding Newey-West (1987) corrected
t-statistics in parentheses. Credit risk premium are calculated in month t+1 as the difference between the corporate bond yield
and the corresponding maturity-matched treasury rate minus expected losses, liquidity compensation, and tax compensation.
BETACAPM is the firmâ€™s CAPM beta at time t and is calculated using rolling regressions over the t-48 to t-1 time frame.
SYSDEFBETA is the firmâ€™s systematic default risk beta (failure beta) at time t and is calculated as the sensitivity of its default
probability to the median default probability. SYSDEFBETA is also calculated over the t-48 to t-1 time frame on a rolling
basis. SIGMA, NIMTAAVG, TLMTA , CASHMTA, MB, RSIZE, RATING, and DD are all calculated at time t. These variables are
described in detail in Table 2. OAMT is the market value of debt at the time of its issuance in millions of dollars, and TTM is the
time to maturity of debt in years. PD is the physical probability of default reported as a percentage. Absolute values of t-
statistics are reported in parentheses below coefficient estimates. Statistical significance at the 10%, 5%, and 1% levels is
denoted by *, **, and ***, respectively.
(1) (2) (3) (4)
Credit Risk Credit Risk Credit Risk Credit Risk
VARIABLES Premium Premium Premium Premium
BETACAPM 0.072*** 0.187** 0.189*** 0.082***
(2.64) (4.54) (5.18) (2.90)
SYSDEFBETA 0.555*** 1.424*** 1.408*** 0.567***
(3.74) (7.08) (6.93) (4.38)
SIGMA 3.556*** 3.320***
(16.23) (13.18)
NIMTAAVG -41.575*** -29.324***
(10.29) (8.75)
TLMTA 0.442*** 0.411***
(5.75) (4.50)
CASHMTA -1.296*** -0.661***
(5.16) (2.80)
OAMT -0.098*** -0.103* -0.375*** 0.023
(4.39) (1.89) (10.28) (1.01)
TTM 0.009*** 0.012*** 0.012*** 0.009***
(4.54) (7.13) (6.88) (4.30)
MB -0.019 -0.009
(1.10) (0.70)
RSIZE -0.569*** -0.428***
(18.00) (13.46)
RATING 0.123*** 0.086***
(16.00) (18.19)
DD -0.099*** -0.108*** 0.023*
(9.20) (9.82) (1.80)
PD*106 29.028*** 10.969***
(6.66) (3.73)
Constant -3.715*** 0.889*** 1.828*** -3.843***
(16.49) (5.68) (14.42) (15.85)
Observations 83,202 83,020 83,124 83,020
R-squared 0.501 0.459 0.370 0.601
45
Table 8: Equity Returns for Systematic Default Risk Beta Portfolios
In Table 8, we report the time series averages, CAPM, 3-factor, and 4-factor regression results for distress risk portfolios. We
sort stocks into deciles each month from January 1981 through December 2010 according to their systematic default risk
betasâ€”SYSDEFBETAsâ€”obtained at the beginning of the previous month. We calculate the value-weighted decile portfolio
returns for all stocks in the CRSP-COMPUSTAT sample on a monthly basis and regress the portfolio return in excess of the
risk-free rate on the market-rf (MKTRF), size (SMB), value (HML), and momentum (MOM) factors. The factors are obtained
from Ken Frenchâ€™s website. We report regression results for only the top and bottom decile portfolios along with the top decile
minus bottom decile hedge portfolio to save space. Absolute values of t-statistics are reported in parentheses below coefficient
estimates. Statistical significance at the 10%, 5%, and 1% levels is denoted by *, **, and ***, respectively.
Equity Returns for SYSDEFBETA Portfolios in CRSP
Alpha*100 MKT SMB HML MOM
10th 1.187***
(2.74)
0.214 1.199***
(0.81) (19.82)
0.204 1.069*** 0.897*** 0.096
(0.72) (21.3) (13.01) (1.29)
0.501** 0.962*** 0.910*** -0.031 -0.320***
(2.16) (20.35) (14.63) (0.45) (7.88)
Alpha * 100 MKT SMB HML MOM
90th 1.644***
(3.13)
0.612* 1.313***
(1.66) (15.93)
0.502 1.172*** 1.250*** 0.322***
(1.52) (17.18) (13.33) (3.18)
0.909*** 1.024*** 1.270*** 0.144 -0.450***
(3.08) (16.03) (15.09) (1.55) (8.14)
Alpha * 100 MKT SMB HML MOM
90th - 10th 0.457*
(1.70)
0.398 0.114**
(1.48) (1.97)
0.298 0.104* 0.353*** 0.226***
(1.13) (1.75) (4.32) (2.59)
0.408 0.062 0.360*** 0.175** -0.130***
(1.54) (1.00) (4.43) (1.95) (2.61)
46
Figure 1: Historical Corporate Default Rates
This figure plots the historical default rates on Moodyâ€™s rated corporate issuers. The data is from Moodyâ€™s Investor
Services. Grey areas indicate NBER recessions.
6.0%
5.0%
4.0%
3.0%
2.0%
1.0%
0.0%
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
47
Figure 2: Components of Corporate Spreads
This figure plots the expected losses, taxes, and liquidity premium components of corporate spreads. The estimation
of these components is described in Section 4.1. Bonds with maturity greater than seven years are referred to as
having â€œlong maturityâ€? and bonds with maturity less than seven years are referred to as having â€œshort maturity.â€?
12.00% 12.00%
10.00% 10.00%
Liquidity Premium
Taxes
8.00% 8.00%
Expected Loss
Spread
6.00% 6.00%
4.00% 4.00%
2.00% 2.00%
0.00% 0.00%
AAA short-mat
AA short-mat
A short-mat
BBB short-mat
AAA long-mat
AA long-mat
A long-mat
BBB long-mat
B
CCC
BB
48