Decomposing CDS Spreads and Their Variation∗
Antje Berndt†
April 2014
Abstract
I decompose CDS spreads into an expected loss component, a credit risk premium component
and a residual component. Based on data for all actively traded U.S. public-firm CDS contracts
from 2001 to 2010, expected losses account, at the median, for less than 25% of the level of credit
spreads. Less than 30% of the variation in credit spreads—both over time and across firms—
corresponds to variation in expected losses. Using a popular pricing kernel that is successful
in explaining equity data I find that expected losses and credit risk premia combined account
for less than 45% of the level of credit spreads. In that sense, the data are consistent with a
sizable residual component. To reconcile standard CDS pricing models which imply a residual
component of zero with the data, CDS market investors would have to be more risk averse
(γ = 3.80) than equity market investors (γ = 2.45). In the absence of market segmentation,
however, two thirds of the temporal variation and more than half of the cross-sectional variation
in CDS spreads reflects variation in residual spreads. Much of the variation in residual spreads
is statistically explained by variation in firm-specific and market-wide equity-option-implied
volatility. While expected losses are calibrated to forward-looking firm-specific probabilities
of default (PDs), I document the pitfalls of using backward-looking rating-based PDs instead:
Estimates of the size of the residual component, and its cross-sectional covariation with credit
spreads, would be substantially smaller.
JEL Classifications: G12, G13, G22, G24
Keywords: CDS spreads; Expected losses; Credit risk premia; Variance decomposition
∗
I am grateful to Darrell Duffie for extensive discussions and helpful advice, and thank Jean Helwege and Marliese
Uhrig-Homburg for useful comments. I thank Moody’s and the Risk Management Institute at NUS for various
data bearing on expected default losses, and Richard Cantor, Duan Jin-Chuan, Ashish Das, Kenneth Emery, David
Hamilton, David Kreisman, Albert Metz, Sharon Ou, Li Pei, Steffen Sorensen and Roger Stein for their assistance
with the data. I thank Mark Mitchell for data on the CDS-bond basis.
†
Associate Professor of Finance, Poole College of Management, NC State University. Phone: 919-515-4539. Email:
[email protected].
1.
Introduction
The prices that investors charge for bearing corporate default risk reflect their information about
expected future losses, the market value of their systematic credit risk exposure and, potentially,
the supply of and demand for risk bearing capital. The goal of this paper is to uncover investors’
information. At first, I focus on the information that credit spreads reveal about expected losses.
In particular, what fraction of credit spreads reflects compensation for expected losses? And to
what extent do changes in credit spreads correspond to changes in expected losses?
The standard in the literature is to link credit spreads to expected losses through a structural
or reduced-form model, or by fitting a statistical regression-based model.1 In contrast, I propose
a model-free decomposition of at-market credit default swap (CDS) spreads, C, into an expected
loss component, ExpL, and an excess spread component, C ex :
Cti = ExpLit + Ctex,i .
(1)
For a given reference firm i, ExpLit is computed as the time-t discounted expected loss in notional
due to default of firm i, scaled by a survival-conditional annuity factor, and Ctex,i is defined as the
difference between the firm’s CDS spread, Cti , and the expected loss component, ExpLit .
Since I am interested in investors’ information about expected losses rather than the implications
of any particular credit risk model, I compute expected losses using probability of default and loss
given default measures that are broadly disseminated and widely used by market participants.
Specifically, benchmark expected losses are based on daily, firm-by-firm estimates of conditional
default probabilities disseminated by Moody’s Analytics in form of their expected default frequency
(EDF) measure, and on constant loss given default chosen to match historical market-wide averages.
Using daily Markit 5-year CDS data for 240 firms representative of the active U.S. public-firm
default swap market between 2001 and 2010, I show that the median ExpL-to-C ratio—over time
and across firms—is 0.23. This implies that, in general, less than one quarter of the level of CDS
spreads reflects compensation for expected losses. The ExpL-to-C ratio is usually somewhat higher
for investment-grade than for high-yield firms. It peaks in 2003, shortly after the 2001-02 credit
crunch, and in 2010, following the 2007-09 credit crunch.
Modifications of the credit spread decomposition in Equation (1) can be found in the literature,
although they usually rely on an approximation of ExpLit by the annualized default probability
1
Structural models go back to the seminal work of Black and Scholes (1973) and Merton (1974) and include,
among others, Geske (1977), Longstaff and Schwartz (1995), Anderson and Sundaresan (1996), Anderson, Sundaresan, and Tychon (1996), Mella-Barral and Perraudin (1997), Leland (1994), Leland and Toft (1996), and CollinDufresne and Goldstein (2001). Empirical implementations of structural models that are calibrated to expected
losses include Huang and Huang (2012) and Chen, Collin-Dufresne, and Goldstein (2009). An overview over popular
reduced-form models is given in Duffie and Singleton (2003) and Schönbucher (2003), and empirical applications that
calibrate these models to expected losses and credit spreads include Driessen (2005), Berndt, Douglas, Duffie, Ferguson, and Schranz (2005), Saita (2006) and Bai, Collin-Dufresne, Goldstein, and Helwege (2012). Regression-based
models include Berndt, Douglas, Duffie, Ferguson, and Schranz (2005), Anderson (2011), Raunig (2011), Bai and Wu
(2012) and Diaz, Groba, and Serrano (2013), among many others.
1
times loss given default, which is accurate only for a flat term structure of default probabilities at
low levels, and are used for purposes other than computing ExpL-to-C ratios.2 Instead, I explicitly
take into account the term structure of default probabilities and quantify the fraction of credit
spreads that corresponds to expected losses. A similar approach, but for corporate bonds rather
than default swaps, is taken in Elton, Gruber, Agrawal, and Mann (2001), who show that over the
time period 1987-1996, expected losses do not account for more than 25% of corporate bond yield
spreads. The fact that my estimates for ExpL-to-C ratios are close to theirs was initially surprising
to me, given that CDS spreads tend to be substantially lower than associated corporate bond yield
spreads, and that loss given default in my study is somewhat higher than in their study.3 The
findings can be reconciled, however, by observing that the forward-looking firm-specific default
probabilities used here tend to be lower than their backward-looking rating-based counterparts,
which are the basis for the analysis in Elton, Gruber, Agrawal, and Mann (2001).
Equation (1) implies that a change in credit spreads indicates a change in investors’ information
about expected losses or excess spreads. To uncover slices of investors’ information from price
movements, I decompose credit spread variation into the covariation of credit spreads with expected
losses and the covariation of credit spreads with excess spreads:
var(Cti ) = cov(Cti , ExpLit ) + cov(Cti , Ctex,i ).
(2)
For each firm i, I estimate the fraction of the time-series variation in credit spreads that corresponds
ts,i
= cov(Cti , ExpLit ))/var(Cti ). For the median firm that fraction
to variation in expected losses, βExpL
is 0.24, implying that less than one quarter of the temporal credit spread variation reflects expected
ts,i
estimate is somewhat higher for the median high-yield firm (0.26) than
loss variation. The βExpL
for the median investment-grade firm (0.22).
cs,t
of cross-sectional credit spread variation
Similarly, for each day t, I estimate the fraction βExpL
cs,t
estimate is 0.27, meaning that on
that corresponds to expected loss variation. The median βExpL
most days less than 30% of the cross-sectional credit spread variation signals expected loss variation.
cs,t
The βExpL
estimates vary dramatically over time, from a low of 0.16 in 2005 to a high of 0.55 in
2010. They tend to be higher towards the end and shortly after the 2001-02 and the 2007-09 credit
crunch, and lower for the years in between. Overall, however, changes in credit spreads—both over
time and across firms—signal, to a large extent, changes in excess spreads.
I decompose excess spreads into credit risk premia, RP , and a residual component, Res:
Ctex,i = RPti + Resit .
(3)
The credit risk premium is defined as the market value of the systematic credit risk that the
protection seller is exposed to, scaled by a survival-conditional annuity factor. It consists of two
2
See, for example, Remolona, Scatigna, and Wu (2008), Raunig (2011) and Berg (2013).
For a firm-by-firm comparison of CDS and corporate bond yield spreads, see Table 2 in Longstaff, Mithal, and
Neis (2005).
3
2
components: The jump-to-default risk premium, JtD, and the survival-conditional mark-to-market
credit risk premium, M tM : RPti = JtDti + M tMti . JtDti denotes the scaled market value of
bearing the systematic component of the risk associated with default of the reference firm in the
first year, whereas M tMti denotes the scaled market value of bearing the systematic credit risk for
the remainder of the life of the default swap, conditional on survival for the first year.
I use the Chen, Collin-Dufresne, and Goldstein (2009) adaptation of the Campbell and Cochrane
(1999) pricing kernel—a popular pricing kernel that is successful in explaining equity market data—
to obtain a direct measure of JtDti and M tMti . While a parametric specification of the pricing kernel
cannot be avoided, given such a specification the estimates for JtDti and M tMti rely primarily on
the statistical comovement between realized default rates at the rating cohort level and the pricing
kernel. I find that the median fraction of credit spreads due to jump-to-default risk premia is
minuscule at less than 0.01, and that the median fraction due to survival-conditional mark-tomarket credit risk premia is 0.17. Taken together, expected losses and credit risk premia generally
account for 44% of the level of credit spreads. In that sense, the data are consistent with a sizable
residual component Resit .
I also find pronounced positive comovement between credit spreads and residual spreads. The
median fraction of the time-series credit spread variation that corresponds to expected loss or credit
ts,i
= cov(Cti , ExpLit + RPti )/var(Cti ), is only 0.33, implying that
risk premium variation, βExpL+RP
67% of the temporal variation in CDS spreads signals variation in residual spreads. Similarly, the
median fraction of the cross-sectional credit spread variation that corresponds to expected loss or
cs,t
, is 0.44, meaning 56% of the cross-sectional credit spread
credit risk premium variation, βExpL+RP
variation signals residual spread variation.
The residual component Resit —which captures excess spreads above and beyond credit risk
premia—may be due to risk-bearing-capital supply and demand effects or, more generally, CDS
market liquidity risk or liquidity risk premia, to clientele effects or to measurement noise. In
standard CDS pricing models, it is set equal to zero: Resit = 0.4 The empirical evidence of sizable
residual spreads that I present gives rise to a credit spread level puzzle in the sense that actual
spreads are too high to be explained by expected losses plus credit risk premia, as stipulated by
standard models.5 Similarly, the positive covariation between credit spreads and residual spreads
points to a credit spread volatility puzzle in the sense that actual credit spread variation is too
high to be explained by expected loss and credit risk premium variation.6 Both the level and the
temporal volatility puzzle are somewhat more pronounced for investment-grade (IG) firms than for
4
For standard CDS pricing models, see Duffie and Singleton (2003), Schönbucher (2003), Berndt, Douglas, Duffie,
Ferguson, and Schranz (2005), and the references cited therein.
5
The term “credit spread (level) puzzle” is frequently used in papers that analyze the ability of structural credit risk
models to explain corporate bond yield spreads. There, it usually refers to the observation that yield spreads between
Baa- and Aaa-rated bonds are too high to be explained by standard structural models. See, for example, Huang and
Huang (2012), Eom, Helwege, and Huang (2004), Cremers, Driessen, and Maenhout (2008), Chen, Collin-Dufresne,
and Goldstein (2009), Feldhütter and Schaefer (2014), and the references cited therein.
6
For corporate bonds, the phrase “credit spread volatility puzzle” has been coined in Chen, Collin-Dufresne, and
Goldstein (2009).
3
high-yield (HY) firms.
Since any mismeasurement of expected losses or credit risk premia would be absorbed into
Resit , I re-estimate ExpLit and RPti using two alternative probability of default (PD) measures:
(i) monthly, firm-by-firm PD estimates provided by the Risk Management Institute (RMI) at the
National University of Singapore, and (ii) a refined credit-rating-based PD measure. Both EDFs
and RMI PDs are forward-looking firm-specific estimates of conditional default probabilities. In
contrast, rating-based PDs are backward-looking and treat firms within a given rating category as
homogeneous.
Interestingly, I find that rating-based PDs imply a median (ExpL + RP )-to-C ratio that at 0.67
is much higher than that obtained for EDFs (0.44) or RMI PDs (0.43). In other words, I show that
the credit spread level puzzle tends to be much more pronounced when expected losses and credit
risk premia are calibrated to forward-looking firm-by-firm PDs than when they are calibrated to
historical rating-based PDs. As long as model spreads ExpL + RP are calibrated to rating-based
PDs, my findings are consistent with the evidence for structural-model-based corporate bond yield
spreads in Eom, Helwege, and Huang (2004), Ericsson, Reneby, and Wang (2007) and Feldhütter
and Schaefer (2014), in that model spreads tend to be lower than actual spreads for IG firms, but
higher than actual spreads for HY firms. EDF- and RMI-based model spreads, on the other hand,
tend to underpredict actual spreads for both IG and HY firms. The reason for the discrepancy
between rating- and EDF/RMI-based model spreads is that rating-based PDs are substantially
higher than EDFs and RMI PDs during the sample period, especially for HY firms and between
2004 and 2007.
The time-series variation of credit spreads is more aligned with the variation of EDF-based
model spreads than it is with that of RMI- or rating-based model spreads, for whom the median
ts,i
βExpL+RP
ratio is only 0.17. Cross-sectional credit spread variation, on the other hand, is much
cs,t
closer aligned with rating-based model spread variation—the median βExpL+RP
ratio is 0.79—than
with variation in their EDF-based (0.44) or RMI-based (0.26) counterparts. One interpretation of
these results is that ratings do a better job in aligning the cross-sectional ranking of actual and
model spreads, whereas EDFs do a better job in aligning the time-series ranking.
My findings highlight the importance of the choice of the PD measure for the decompositions
in Equations (1) and (3). I further elaborate on this point by exploring the notion of market
segmentation. Specifically, I allow equity and CDS markets to be populated by investors with
different levels of risk aversion and quantify that level of risk aversion of CDS market investors, γ c ,
that reconciles standard CDS pricing models with the data. I find γ c = 3.8 for EDFs, γ c = 3.75
for RMI-based PDs and γ c = 3.3 for rating-based PDs. While standard CDS pricing models can
be reconciled with the data only if CDS market investors are more risk averse than equity market
investors, the implied discrepancy between risk aversion in equity markets (γ e = 2.45) and credit
markets is more pronounced when model spreads are calibrated to forward-looking firm-by-firm
PDs than when they are calibrated to historical rating-based PDs.
I show that my findings are qualitatively robust to a wide range of alternative loss given default
4
specifications. While expected loss measures often ignore the covariation between future loss given
default and the default indicator, I propose and implement a method for computing these covariance
terms explicitly and show that my results are robust to these covariance effects. Lastly, I build a
prediction model for residual spreads and show that much of their variation—both over time and
across firms—can be statistically explained by firm-specific and market-wide equity-option-implied
volatility. After controlling for benchmark expected losses, I also find evidence of permanent ratingbased clientele and temporary price pressure effects in CDS markets.
2.
Decomposing CDS Spreads
For a given firm i, Cti denotes the time-t annualized at-market T -year CDS spread. I define Nti
as
T
−1
∆
Nti
= Et
X
i
i
πt,(k+1)∆ ,
) − Lit+k∆,∆ Dt+k∆,∆
∆Cti (1 − Dt,k∆
(4)
k=0
i is the indicator of default of
where ∆ = 1/4 is the time in years between coupon payments, Dt,y
firm i in period (t, t + y], and Lit,y is the loss given default, as a fraction of notional, if firm i were to
default in period (t, t + y]. I use πt,y to denote the relative state price density for period (t, t + y].
In standard CDS pricing models, Nti measures the time-t value of the net payments received by
the protection seller in a T -year CDS contract on $1 notional and is set equal to zero.7 In the
presence of CDS market liquidity risk, liquidity risk premia, clientele effects, measurement noise or
misspecification of the state price density, however, Nti may deviate from zero.8 I do not impose
any constraints on Nti , and let the data speak for themselves.
Let δt,y = Et πt,y denote the time-t price of a default-free zero-coupon bond with y years to
maturity. The discounted expected future loss in $1 notional due to default of firm i at or before
time t + T is given by
T
−1
∆
Vti
=
X
i
δt,(k+1)∆ Et Lit+k∆,∆ Dt+k∆,∆
.
(5)
k=0
7
For standard CDS pricing models, see Duffie and Singleton (2003), Schönbucher (2003), and Berndt, Douglas,
Duffie, Ferguson, and Schranz (2005), and the references cited therein. These models generally view Cti as the atmarket CDS spread for a fresh T -year CDS contract initiated at time t. In practice, however, standardized CDS
contracts expire either March, June, September or December 20th. Hence a 5-year CDS spread quoted on April 20th
is for a contract with four years and 11 months to maturity. If t falls between standardized maturity dates, Nti may
differ slightly from the time-t value of the net payments received by the protection seller due to accrued interest
conventions. The latter may also be true if default occurs within a payment period rather than at the end of one.
8
Since Arora, Gandhi, and Longstaff (2012) argue that they are vanishingly small, I abstract from the effects of
counterparty risk on CDS spreads.
5
The market value of the systematic credit risk that the protection seller is exposed to is
T
−1
∆
Wti
=
X
T
−1
∆
i
covt (Lit+k∆,∆ Dt+k∆,∆
, πt,(k+1)∆ )
+
∆Cti
k=0
X
i
covt (Dt,k∆
, πt,(k+1)∆ ).
(6)
k=0
I combine Equations (5) and (6) with Equation (4) and obtain
Cti Ait = Vti + Wti + Nti ,
T
P∆
−1
i
where Ait = ∆ k=0
δt,(k+1)∆ (1 − Et Dt,k∆
) is a survival-conditional annuity factor. This allows
me to express CDS spreads as the sum of ExpLit = Vti /Ait , RPti = Wti /Ait and Resit = Nti /Ait ,
Cti = ExpLit + RPti + Resit .
(7)
I refer to ExpLit as the expected loss component, RPti as the credit risk premium component and
Resit as the residual component of CDS spreads. Excess spreads Ctex,i are defined via Equation (3)
as the sum of credit risk premia and residual spreads.
3.
Data Sources
The decomposition of CDS spreads in Equation (1) relies, first and foremost, on CDS data
and on data pertaining to expected losses, meaning probability of default and loss given default
estimates.
3.1
CDS data
I consider the time period from January 2001 to June 2010 and identify all public U.S. firms
that had been a constituent of one of the major North American CDS indices at some point during
that period. The CDS indices considered are the Markit CDX investment-grade index (CDX.NA.IG
Series 1 through 13), the Markit CDX high-yield index (CDX.NA.HY Series 1 through 13), and
the Markit CDX cross-over index (CDX.NA.XO Series 5 through 11). Since Markit forms the CDX
indices by selecting the most actively traded names, the firms in the sample generally represent the
most liquid segment of the U.S. public-firm CDS market.9 I only retain firms whose identifiers can
be matched unambiguously across all sources of data used in this study, and for which I observe at
least one year worth of CDS data. The cleaned sample includes 240 firms in six industry sectors.10
9
For details on the index formation, see markit.com/Product/CDX.
The industry sectors are Mining, Utilities and Construction (identified as firms with NAICS codes between 210000
and 239999), Manufacturing (NAICS codes from 310000 to 339999), Trade and Transportation (NAICS codes from
420000 to 429999, 440000 to 459999, and 480000 to 499999), Information (NAICS codes from 510000 to 519999),
Finance, Insurance and Real Estate (NAICS codes from 520000 to 539999), and Others. The latter include, among
others, 12 Services firms (NAICS codes from 540000 to 569999 and 810000 to 819999), six Education and Healthcare
firms (NAICS codes from 610000 to 629999), and 12 Entertainment and Accommodation firms (NAICS codes from
710000 to 729999).
10
6
The firms’ distribution across sectors may be judged from Table 1, which indicates a concentration of Manufacturing firms. Table 1 also reports the distribution of credit quality, as measured by
Moody’s senior unsecured issuer rating, across all firms and within sectors. While the majority of
firms are of medium credit quality, Finance, Insurance and Real Estate firms tend to have a rating
of A or above whereas Mining, Utilities and Construction companies tend to have a rating of Ba
or below.
Table 1: Distribution of firms across sectors and by credit quality The table reports the distribution of
firms across sectors and by median Moody’s senior unsecured issuer ratings. The data include 414,816 firm-day
observations for 240 firms and cover the period from January 2001 to June 2010.
Sector
Finance, Insurance, Real Estate
Information
Manufacturing
Mining, Utilities, Construction
Trade, Transportation
Others
All
Aaa-Aa
A
Baa
Ba
B
3
8
5
19
1
8
21
9
7
15
2
1
14
10
3
5
61
35
3
4
3
6
9
34
15
21
7
7
39
92
1
Caa-C
All
2
1
20
26
89
34
40
31
6
240
3
For each firm, I obtain daily 5-year Markit CDS quotes for senior unsecured, U.S.-dollardenominated, at-the-money default swaps. I only include CDS contracts for which default is
triggered by bankruptcy or a missed payment, and exclude those that also cover restructuring
events.11 This restriction keeps the degree of heterogeneity over the definition of a default event to
a minimum, and ensures that there is less recovery-value heterogeneity. Indeed, both bankruptcy
or failure to pay normally trigger cross-acceleration covenants that cause debt of equal seniority
to convert to immediate obligations that are pari passu, that is, of equal legal priority (see, for
example, Berndt, Jarrow, and Kang (2007)). The spreads provided by Markit are composite CDS
quotes. They are computed based on quotes obtained by two or more anonymous sources, including
investment banks and default swap brokers. The distribution of the number of sources is shown in
Figure A.1 in Appendix A. The number of contributors ranges from two to 27, with a mean and
median composite depth of seven.
Credit spreads vary over time and across firms. Figure 1 reveals that median spreads are
substantially higher during the credit crunch of 2001-02 and 2007-09 than during the years in
between. At a given point in time, credit spreads are generally higher for high-yield firms and
lower for investment-grade firms, as shown in Figure A.2 in the appendix. Along industry groups,
median credit spreads are often highest for Mining, Utility and Construction firms and lowest for
Finance, Insurance and Real Estate firms, which is consistent with the within-sector distribution
of credit quality reported in Table 1.
11
Since the Big Bang protocol in April 2009, restructuring events are excluded from the default definition for
standard CDS contracts. Prior to that, CDS traded with restructuring and without.
7
400
Citigroup
bailout
350
300
Bear Stearns
bailout
Basis points
250
200
Sep 11
150
Bear Stearns
hedge funds
Ford and GM
downgrade
100
Lehman
default
50
0
Worldcom
default
Dec01
Dec02
Dec03
Dec04
Dec05
Dec06
Dec07
Dec08
Dec09
Figure 1: Median CDS spreads The figure shows the daily times series of median 5-year CDS spreads. Only days
for which CDS quotes are available for 30 or more firms are shown. The data include 414,816 firm-day observations
for 240 firms and cover the period from January 2001 to June 2010.
3.2
Probability of default data
For each firm in the sample, Moody’s Analytics provides daily firm-by-firm estimates of conditional 1- and 5-year default probabilities. For a given firm and time horizon, their expected default
frequency (EDF) measure is a non-parametric fit from the historical default frequency of other firms
that had the same estimated “distance to default” as the target firm.12 The distance to default
of a given firm is a leverage measure adjusted for asset volatility. Roughly speaking, distance to
default is the number of standard deviations of annual asset growth by which the firm’s expected
assets at a given maturity exceed a measure of book liabilities. The liability measure is, in the
current implementation of the EDF model, the firm’s short-term book liabilities plus one half of
its long-term book liabilities. Estimates of current assets and the current standard deviation of
asset growth (“volatility”) are calibrated from historical observations of the firm’s equity-market
capitalization and of the liability measure.
The calibration, explained for example in Duffie, Saita, and Wang (2007), is based on the model
of Black and Scholes (1973) and Merton (1974), by which the price of a firm’s equity may be viewed
12
Moody’s definition of default includes, in addition to bankruptcy and failure to pay, debt restructurings that
are materially adverse to the interests of creditors. While the probability of default widens as more events are
covered, Berndt, Jarrow, and Kang (2007) show that the likelihood of a restructuring event is substantially smaller
than that of a bankruptcy or missed payment. EDFs therefore assign a slightly higher probability to default than
if only the events covered by no-restructuring CDS were considered. While this may bias the results towards higher
expected loss estimates, the effect is minimal.
8
as the price of an option on assets struck at the level of liabilities. Bharath and Shumway (2008)
show that the fitting procedure is relatively robust. For most years in the sample, the incidence
of defaults is not especially “surprising” relative to the EDF-predicted number of defaults, and I
am not aware that marginal investors in corporate debt had access to better default probability
estimates than those supplied by Moody’s Analytics, but of course this is hard to verify.13
While I use EDFs as benchmark PD estimates, I also consider two alternatives: (i) PD estimates
provided by the Risk Management Institute (RMI) at the National University of Singapore, and
(ii) a refined credit-rating-based PD measure. RMI provides, on a monthly and firm-by-firm basis,
conditional default probabilities for different horizons. The model and estimation methodology is
based on Duan, Sun, and Wang (2012), and is an extension of the hazard rate approach in Duffie,
Saita, and Wang (2007) and Lando and Nielsen (2010).14 On any given day, I assign each firm the
RMI PDs reported for that firm at the end of the previous month.
Every year, Moody’s reports average cumulative issuer-weighted global default rates by alphanumeric senior unsecured issuer rating and maturity horizon, using data dating back to 1983.15 To
obtain rating-based PDs, on any given day I first assign each firm a refined rating that raises the
firm’s alphanumeric rating by one notch (i.e., from Aa2 to Aa1) if the firm is on positive outlook and
by two notches if the firm is on the upgrade watch list. Similarly, I decrease the firm’s alphanumeric
rating by one (two) notches if the firm is on negative outlook (the downgrade watch list). Second,
I set a firm’s y-year PD estimate equal to the most recent y-year default rate reported by Moody’s
for the firm’s refined rating category. As a result, in any given year, a change in rating-based PDs
may occur only if the firm’s refined rating changes. Figure A.3 in the appendix shows that the
average annual frequency of refined rating changes per firm fluctuates between a low of 0.46 in
2001 and a high of 1.03 in 2006. The figure highlights that PDs based on refined ratings exhibit
substantially more time-series variation than PDs based on raw alphanumeric ratings. Indeed, the
average annual frequency of alphanumeric rating changes per firm is only 0.25 in 2001 and 0.52 in
2006. In the sample, downgrades seem to outweigh upgrades, especially in 2002-03 and in 2008, i.e.
shortly after the onset of a credit crunch.
i and E D i . PD estimates for other horizons y are
For each PD measure, I observe Et Dt,1
t t,5
13
Korablev and Dwyer (2007) estimate that the model that produces Moody’s Analytics EDFs would have placed
a probability of 49.5% on the event that there would have been as few or fewer defaults in 2001 by firms in their
sample than the actual number of defaults. Similarly, the p-values for 2002, 2003, 2004, 2005 and 2006 are 52.0%,
21.3%, 51.4%, 74.5% and 45.1%, respectively. A low p-value is observed only for 2003, when, according to Korablev
and Dwyer (2007), EDFs predicted “too many” defaults. Crossen, Qu, and Zhang (2011) also argue that the EDF
model performs consistently over time and in different credit cycles. They provide accuracy ratios for EDFs, ranging
from 78.7% to 93.9% between 2001 and 2006, and from 82.9% to 87.6% between 2007 and 2009.
14
Specifically, Duan, Sun, and Wang (2012) propose a new reduced-form approach based on forward intensities to
forecast a firm’s PD. Their method produces a term structure of default probabilities without explicitly modeling
and estimating the state vector process.
15
These statistics are reported in Moody’s annual “Corporate Default and Recovery Rates” studies, available at
moodys.com.
9
obtained via interpolation:
i
Et Dt,y
= 1 − exp(−λit,y y),
i = 1 − exp(−λi y) for y = 1, 5, λi = λi for y < 1, and λi is linearly interpolated
where Et Dt,y
t,y
t,y
t,y
t,1
in between λit,1 and λit,5 .
3.3
Loss given default data
The benchmark specification of loss given default (LGD) stipulates that at any time t, the
expected fractional loss of notional given default at a future date is 0.63, independent of the firm
or the timing of default. The fraction 0.63 is chosen to reflect one minus the average recovery
rate measured by post-default trading prices for senior unsecured bonds between 1982 and 2010,
as reported by Moody’s Investors Service (2011). In Section 9, I perform robustness checks using
a range of alternative LGD measures, including (i) a specification that updates average historical
LGD on an annual basis, (ii) Markit LGD quotes, (iii) rating-based LGD estimates, and (iv)
regression-based LGD estimates. Unlike benchmark LGD, these alternative specifications allow
for time variation in LGD, for firm-specific LGD (Markit and rating-based LGD), and take into
account the timing of default (rating- and regression-based LGD).
“Historical LGD” in any given year is set equal to one minus the historical average senior unsecured recovery rate between 1982 and the end of the previous year, as reported in Moody’s annual
“Corporate Default and Recovery Rates” studies. “Markit LGD” relies on quotes of composite recovery rates, provided by Markit on a daily basis for each firm, seniority and restructuring clause.
My understanding is that these quotes are based on estimates of expected future recovery rates
provided to Markit by market makers. At time t, the “rating-based LGD” specification sets the yyears-ahead LGD for investment-grade firms equal to one minus Moody’s average senior unsecured
recovery rate between 1982 and the end of the previous year for firms with IG status y years prior
to default. For high-yield firms, historical average recovery rates for firms with HY status y years
prior to default are used. “Regression-based LGD” estimates exploit the close contemporaneous
link between realized default and recovery rates: Expected future LGD is computed as a linear
function of expected future aggregate default rates.
3.4
Summary statistics
Summary statistics for CDS spreads and alternative PD measures are reported in Table 2. They
reveal at least some amount of comovement between credit spreads and PDs, both over time and
across firms. Years with high median CDS spreads generally correspond to years with high median
PDs, and firms with larger credit spreads usually have higher default probabilities.
Also noticeable is the fact that rating-based PDs are substantially higher than EDFs and RMI
PDs, especially between 2004 and 2007 and for HY firms.16 In Table A.1 in Appendix A I show
16
For HY firms, the median rating-based PD for 2001 is substantially lower than that for the remaining years. The
10
Table 2: Descriptive statistics for CDS spreads and PDs The table reports median 5-year CDS spreads in basis
points (columns marked “CDS”) and median annualized 5-year EDFs (columns marked “EDF”), RMI PDs (columns
marked “RMI”) and rating-based PDs (columns marked “Rtg”). The data include 414,816 firm-day observations for
240 firms and cover the period from January 2001 to June 2010. They are stratified by year and credit quality.
All firms
IG firms
HY firms
CDS
EDF
RMI
Rtg
CDS
EDF
RMI
Rtg
CDS
EDF
RMI
All
95
35
37
60
54
21
28
37
284
91
64
Rtg
439
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
128
110
82
68
65
55
58
177
187
138
73
58
56
33
20
14
11
26
128
94
48
36
45
33
25
29
29
34
64
70
38
44
45
78
73
72
65
60
59
62
100
88
53
41
38
31
35
94
90
79
66
50
44
22
13
9
7
15
57
47
47
32
40
24
17
18
20
26
45
50
24
31
36
44
43
28
38
38
37
41
398
454
365
225
199
181
204
424
498
394
127
132
123
76
48
42
26
112
412
268
51
54
63
53
44
61
54
62
97
101
168
461
453
439
416
548
509
354
425
438
that for HY firms, rating-based PDs tend to overpredict realized default rates, and dramatically so
between 2004 and 2007. The table highlights that historical default rates are not necessarily a good
predictor of future default rates. Korablev and Dwyer (2007) and Crossen, Qu, and Zhang (2011),
on the other hand, perform an ex-post analysis of the predictive power of EDFs. They argue that
the EDF model performs well over time, with the only exception being 2003 (see Footnote 13). By
reporting results for alternative PD measures as well as for EDFs as the benchmark PD measure, I
demonstrate the importance of the choice of the PD measure for the decomposition of CDS spreads.
The median expected one-year-ahead loss given default is 0.63 for the benchmark and historical
LGD specification, 0.60 when Markit quotes are used, 0.62 for rating-based LGDs, and 0.50 for
regression-based LGDs. Additional details are provided in Section 9, where the alternative LGD
specifications are more formally introduced.
4.
The Expected Loss Component of CDS Spreads
For constant loss given default Lit+k∆,∆ = L and a flat term structure of default at low levels,
i . More
ExpLit is a close approximation of annualized expected losses, that is, ExpLit ≈ L Et Dt,1
generally, however, ExpLit is given by
T
P∆
−1
ExpLit
=
k=0
T
P∆
−1
i
i
δt,(k+1)∆ Et Lit+k∆,∆ Et Dt+k∆,∆
+ k=0
δt,(k+1)∆ covt (Lit+k∆,∆ , Dt+k∆,∆
)
. (8)
T
P ∆ −1
i
∆ k=0 δt,(k+1)∆ (1 − Et Dt,k∆ )
reason is that in 2001—unlike later in the sample—the majority of HY firms has a refined credit rating of Ba1, the
highest non-IG rating.
11
As long as at time t, the value of Lit+k∆,∆ is set as a function of variables observable at time t,
the covariance terms in Equation (8) are equal to zero and ExpLit is a simple function of defaultfree discount factors, expected LGD and PDs. This is the case for the benchmark specification of
LGD, which assumes Lit+k∆,∆ = 0.63. Section 9 offers a range of robustness checks for alternative
LGD specifications, including a specification where the conditional comovement between loss given
default and the default indicator is taken into consideration.
Table 3 reports summary statistics for ExpLit across all firms in the sample.17 For EDFbased expected losses, the median value of ExpLit —over time and across firms—is 22 basis points,
compared to 95 basis points for Cti (see Table 2). The median fraction of CDS spreads due to
expected losses, ExpLit /Cti , is 0.23. It is highest in 2003 (following the 2001-02 credit crunch) and
in 2009-10 (towards the end and shortly after the 2007-09 financial crisis). The latter episode in
particular is consistent with an increase in EDFs during the crisis that is delayed relative to that in
CDS spreads, and a decrease in EDFs towards the end of the crisis that is slower than that in credit
spreads. Overall, the expected-loss-to-CDS ratio tends to be somewhat higher for IG firms (0.25)
than for HY firms (0.20), but that relation is reversed between 2008 and 2010. These patterns are
confirmed in Figure 2, which displays the expected loss component of 5-year CDS spreads for one
particular IG firm, Target Corporation, and one HY firm, Goodyear Tire & Rubber.
Target
Goodyear
300
2000
1750
250
1500
200
Basis points
1250
150
1000
750
100
500
50
250
0
Dec01
Dec03
Dec05
Dec07
0
Dec09
Dec01
Dec03
Dec05
Dec07
Dec09
Figure 2: CDS spreads and their expected loss component The figure shows daily 5-year CDS spreads,
for Target Corporation (left panel) and The Goodyear Tire & Rubber Company (right panel). The shaded red
area identifies the expected loss component of the credit spreads. The shaded blue area identifies the excess spread
component. Expected losses are based on EDFs and benchmark loss given default.
At 23 basis points, the median level of RMI-based expected losses is nearly the same as that
for EDF-based expected losses. A closer look reveals that RMI-based expected losses tend to be
somewhat higher than their EDF-based counterparts for IG firms and somewhat lower for HY
17
Data on default-free discount factors δt,(k+1)∆ are obtained from Datastream.
12
Table 3: Median expected loss component and expected-loss-to-CDS ratio The table reports the median
expected loss component of 5-year CDS spreads in basis points, and the median expected-loss-to-CDS ratio. Expected
losses are based on EDFs (columns marked “EDF”), RMI PDs (columns marked “RMI”) or rating-based PDs (columns
marked “Rtg”), and on benchmark loss given default. The data include 414,816 firm-day observations for 240 firms
and cover the period from January 2001 to June 2010. They are stratified by year and credit quality.
EDF
All firms
RMI Rtg
EDF
IG firms
RMI Rtg
All
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
22
45
36
35
21
13
9
7
16
78
58
23
30
22
28
20
15
18
18
21
40
43
37
24
27
28
48
45
44
40
37
37
39
13
41
31
27
14
8
6
4
10
35
29
ExpL
17
29
20
25
15
11
11
13
16
28
31
All
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
0.23
0.37
0.30
0.39
0.28
0.18
0.16
0.12
0.12
0.41
0.43
0.21
0.23
0.17
0.29
0.23
0.20
0.27
0.26
0.13
0.21
0.26
0.46
0.18
0.22
0.44
0.66
0.67
0.76
0.65
0.26
0.29
0.40
0.25
0.41
0.33
0.46
0.32
0.20
0.17
0.13
0.11
0.38
0.38
ExpL/C
0.29
0.26
0.20
0.41
0.32
0.27
0.37
0.34
0.16
0.28
0.36
EDF
HY firms
RMI Rtg
23
15
19
23
27
26
17
24
24
23
25
56
78
81
76
47
30
26
16
69
240
160
40
32
34
39
33
27
38
34
39
60
62
249
101
260
256
249
237
304
285
205
242
249
0.35
0.16
0.19
0.36
0.53
0.52
0.59
0.51
0.18
0.20
0.25
0.20
0.20
0.18
0.21
0.22
0.16
0.15
0.10
0.15
0.45
0.47
0.13
0.09
0.08
0.11
0.15
0.13
0.19
0.15
0.08
0.13
0.17
0.76
0.39
0.38
0.69
1.09
1.11
1.24
1.03
0.43
0.46
0.66
firms, consistent with the evidence in Table 2. For HY firms, the differences between EDF- and
RMI-based expected losses are particularly pronounced in 2001-03 and 2008-10, when EDF-based
expected losses are substantially larger. A similar picture emerges for EDF- versus RMI-based
expected-loss-to-CDS ratios.
Not surprisingly, given the evidence in Table 2, rating-based expected losses are substantially
higher than their EDF- and RMI-based counterparts: The median expected loss is 37 basis points,
compared to 22 and 23 basis points for EDF- and RMI-based expected losses, and the median
ExpL-to-C ratio is 0.46, compared to 0.23 for EDFs and 0.21 for RMI PDs. The reason for the
discrepancy between rating- and EDF/RMI-based expected losses is that rating-based PDs are
substantially higher than EDFs/RMI PDs, especially between 2004 and 2007 and for HY firms, as
discussed at the end of Section 3. By reporting results for EDFs as well as alternative PD measures,
I demonstrate the importance of the choice of the PD measure for the estimation of the expected
loss component of CDS spreads.
13
5.
The Credit Risk Premium Component of CDS Spreads
The credit risk premium RPti is defined in Section 2 as the market value of the systematic
credit risk that the protection seller is exposed to, scaled by a survival-conditional annuity factor.
It is equal to the sum of the jump-to-default risk premium, JtDti , and the survival-conditional
mark-to-market credit risk premium, M tMti :
RPti = JtDti + M tMti .
The jump-to-default risk premium is the scaled market value of bearing the systematic risk
associated with default in the first year. It is computed as JtDti = WtJ,i /Ait , where
1
−1
∆
WtJ,i =
X
1
−1
∆
i
covt (Lit+k∆,∆ Dt+k∆,∆
, πt,(k+1)∆ ) + ∆Cti
k=0
≈
X
i
covt (Dt,k∆
, πt,(k+1)∆ )
k=0
i
covt (Lit,1 Dt,1
, πt,1 ).
(9)
The approximation in Equation (9) holds with equality if (i) default, should it occur in year (t, t+1],
occurs towards the end of the year, and (ii) at time t, loss given default one year ahead is assumed
to be equal to Lit,1 . The latter condition is automatically satisfied for benchmark LGD and all
alternative LGD specifications.18
The survival-conditional mark-to-market credit risk premium is the scaled market value of
bearing the systematic credit risk for the remainder of the life of the default swap, conditional on
survival for the first year. It is computed as M tMti = WtM,i /Ait , where
T
−1
∆
T
−1
∆
WtM,i =
X
i
, πt,(k+1)∆ ) + ∆Cti
covt (Lit+k∆,∆ Dt+k∆,∆
1
k= ∆
≈
T
−1
X
≈
i
, πt,(k+1)∆ )
covt (Dt,k∆
1
k= ∆
T
−1
∆
i
covt (Lit+y,1 Dt+y,1
, πt,y+1 ) + ∆Cti
y=1
T
−1
X
X
X
i
, πt,(k+1)∆ )
covt (Dt,k∆
1
k= ∆
i
covt (Lit+y,1 Dt+y,1
, πt,y+1 )
+
y=1
Cti
T
−1
X
i
covt (Dt,y
, πt,y+1 ).
(10)
y=1
The first approximation holds with equality if (i) default, should it occur in year (t + y, t + y + 1],
occurs towards the end of the year, and (ii) at time t, loss given default y +1 years ahead is assumed
P(y+1)/∆−1
i
to be equal to Lit+y,1 . The second approximation replaces ∆ k=y/∆
covt (Dt,k∆
, πt,(k+1)∆ ) by
i
19
covt (Dt,y , πt,y+1 ), for y = 1, . . . , T − 1.
18
Indeed, for all practical purposes, I do not distinguish between Lit+k∆,∆ and Lit+y,1 , for y ≤ k∆ < y + 1.
P(y+1)/∆−1
i
I have verified that results remain nearly unchanged if ∆ k=y/∆
covt (Dt,k∆
, πt,(k+1)∆ ) is replaced by
i
i
covt (Dt,y , πt,y ) instead of covt (Dt,y , πt,y+1 ).
19
14
For the benchmark LGD specification or, more generally, as long as at time t, the value of
Lit+y,1 is set as a function of variables observable at time t, Lit+y,1 = Lit (y), Equations (9) and (10)
simplify to
i
WtJ,i ≈ Lit (0) covt (Dt,1
, πt,1 ),
WtM,i ≈
T
−1
X
(11)
i
Lit (y) covt (Dt+y,1
, πt,y+1 ) + Cti
y=1
5.1
T
−1
X
i
covt (Dt,y
, πt,y+1 ).
(12)
y=1
Towards quantifying the credit risk premia components
The goal going forward is to quantify WtJ,i and WtM,i , and hence JtDti and M tMti . As a
i
i ,π
first step, I express the conditional covariance terms covt (Dt+y,1
, πt,y+1 ) and covt (Dt,y
t,y+1 ) in
et
Equations (11) and (12) in a form that better lends itself to estimation. To that effect, let E
denote expectations conditional on all time-t information plus aggregate information—including
the realizations of the state price density and of aggregate default and recovery rates—up to time
t + T . Then, by the tower property of conditional expectations,
i
et Di
covt (Dt+y,1
, πt,y+1 ) = covt E
t+y,1 , πt,y+1 ,
i
covt (Dt,y
, πt,y+1 )
=
y−1
X
et Di
covt E
t+k,1 , πt,y+1 .
k=0
et Di
E
t+y,1 is the time-t probability of firm i’s default y + 1 years from now, conditional on
the augmented information set. I assume that as long as realized aggregate default rates for the
i
et Di
firm’s rating cohort match their time-t expectations, E
t+y,1 = Et Dt+y,1 . But if realized aggregate
default rates for the firm’s rating cohort turn out to be higher (lower) than their time-t expectations,
then firm-specific conditional default probabilities are adjusted upward (downward):
o
n
i
i
et Dt+y,1
e R(i,t) − D
e R(i,t) − Et D
e R(i,t) − D
e R(i,t) .
E
= Et Dt+y,1
+ D
t,y
t,y
t,y+1
t,y+1
(13)
R is the realized cumulative y-year default
e t,y
Here, R(i, t) denotes the rating of firm i at time t and D
R , which is part of the augmented information set, in
e t,y
rate for firms rated R at time t. I observe D
form of the y-year cumulative issuer-weighted global default rates by annual rating cohort reported
in Moody’s annual “Corporate Default and Recovery Rates” studies.
a denote the aggregate realized cumulative y-year default rate across all firms that are
e t,y
Let D
rated as of time t. Then,
a
a
e t,y+1
e t,y
D
−D
=
X
R
R
e t,y+1
e t,y
wtR D
−D
,
(14)
R
where wtR denotes the number of firms in rating cohort R at time t, as a fraction of all rated firms.
R statistics. Figure A.4 in the appendix
e t,y
The weights wtR are published by Moody’s alongside the D
15
reveals that weights may change, both over time and across ratings. For example, the cohort of
Baa = 0.21), and the
Baa-rated firms is the third-largest cohort at the beginning of the sample (w2001
Baa = 0.25). Overall, however, during the 2001-10 sample
largest cohort at the end of the sample (w2010
period the majority of Moody’s rated firms has an A, Baa or B rating, with each of these three
categories generally accounting for more than 20% of all rated firms.
e R measures the realized number of defaults in period (t, t + y]
For each rating cohort R, wtR D
t,y
for firms rated R at time t, as a fraction of all rated firms. Equation (14) simply states that the
total number of defaults in any given period must be equal to the sum of defaults out of each rating
cohort. The variation in aggregate (y + 1)-year-ahead default rates that corresponds to variation in
the fraction of defaults out of rating cohort R is measured as the δ R,y+1 coefficient in the regression
a
a
R
R
e t,y+1
e t,y
e t,y+1
e t,y
−D
+ R,y+1
.
−D
= δ0R,y+1 + δ R,y+1 D
wtR D
t
(15)
The estimates for δ R,y+1 are reported in Table 4.20,21 They reveal that much of the variation
in aggregate default rates corresponds to variation in the fraction of defaults of high-yield firms,
P
particularly firms out of the B rating cohort. Per Equation (14), R δ R,y+1 = 1.
Table 4: The link between rating cohort and aggregate default rates The table reports the estimates for
R
e t,y
δ R,y+1 in regression (15). The values for D
are observed as the realized cumulative y-year issuer-weighted default
a
a
e
e t,y
rates by annual letter rating cohort, and Dt,y+1
−D
is computed according to Equation (14). The data include
annual observations from 1991 to 2010 and are obtained from Moody’s annual “Corporate Default and Recovery
Rates” studies. T-statistics are reported in parentheses.
Aaa-Aa
A
Baa
Ba
B
Caa or lower
y=0
y=1
y=2
y=3
y=4
0.001
(0.248)
0.010
(2.296)
0.035
(4.687)
0.063
(2.512)
0.345
(5.520)
0.547
(7.628)
0.000
(0.097)
0.024
(3.191)
0.038
(4.988)
0.090
(5.134)
0.538
(11.162)
0.309
(6.231)
0.000
(-0.053)
0.022
(2.896)
0.055
(8.063)
0.110
(3.968)
0.554
(17.785)
0.260
(8.172)
-0.002
(-0.502)
0.025
(2.390)
0.058
(6.707)
0.160
(5.942)
0.539
(14.536)
0.220
(6.741)
-0.004
(-0.724)
0.030
(1.648)
0.083
(7.072)
0.140
(4.760)
0.530
(16.099)
0.222
(5.920)
I take the simple view that R,y+1
are i.i.d. disturbances that are conditionally independent of
t
20
I use data from 1991 onwards since aggregate PD estimates, which will be introduced later in this section, are
available only from 1991 onwards.
21
I have explored an alternative specification of the link between rating cohort and aggregate default rates that
R
R
R
R
e t,y+1
e t,y
e t,y+1
e t,y
replaces the dependent variable in regression (15) by D
−D
. D
−D
is the fraction of firms rated R
at time t that defaults in period (t, t + y]. Table A.2 in the appendix reports the R2 s for both specifications. It
documents that, with the exception of the Ba cohort, the specification in (15) is a better description of the data.
Moreover, the specification in (15) has the added benefit that given an increase in the number of defaults among all
rated firms, the additional defaults out of each rating cohort add up to the overall increase in defaults.
16
ea
ea
future aggregate default rates, D
t,y+1 − Dt,y , and the state price density process. It implies that
eR
eR
all conditional comovement between D
t,y+1 − Dt,y and the state price density process is captured
through comovement between aggregate default rates and the density process. This, together with
Equations (13) and (15), implies
i
covt (Dt+y,1
, πt,y+1 ) =
i
covt (Dt,y
, πt,y+1 )
=
δ R(i,t),y+1
R(i,t)
wt
a
a
e t,y+1
e t,y
−D
, πt,y+1 ,
covt D
y−1 R(i,t),k+1
X
δ
R(i,t)
k=0
wt
ea
ea
covt D
t,k+1 − Dt,k , πt,y+1 .
(16)
(17)
According to Equations (16) and (17), firm-specific information enters into the calculation
of the conditional covariance between a firm’s default indicator and the state price density only
i
though the rating of the firm. In other words, covt (Dt+y,1
, πt,y+1 ) is the same for all firms in rating
i
cohort R(i, t) at time t, as is covt (Dt,y , πt,y+1 ). While one may criticize the modeling approach
in this section for generating rating-cohort-level rather than firm-level credit risk premia, several
considerations speak in its favor:
• The focus of this study is on understanding the information revealed through CDS spreads,
and not on testing a specific credit risk model. I therefore impose as little structure on credit
risk premia as possible. While choosing a parametric specification for {πt,y+1 }t,y cannot be
avoided, given the state price density process I am interested in a statistical rather than
i
model-driven measure of covt (Dt+y,1
, πt,y+1 ).22 But since most firms stay alive throughout
i
i
the sample period, I cannot estimate covt (Dt+y,1
, πt,y+1 ) based on time-series data for Dt+y,1
and πt,y+1 alone. Instead, it becomes necessary to link default of firm i to the performance of a
cohort of firms for which reliable default statistics can be observed over time. The assumption
n
o
i
i
et Dt+y,1
e C(i,t) − D
e C(i,t) − Et D
e C(i,t) − D
e C(i,t) ,
E
= Et Dt+y,1
+ D
t,y
t,y
t,y+1
t,y+1
(18)
e C is the realized cumulative y-year default
where C(i, t) denotes firm i’s cohort at time t and D
t,y
rate for firms in cohort C at time t, allows me to establish such a link in a parsimonious
way.23 At this time, the smallest cohorts for which time-series data of realized default rates
are available to me are rating cohorts, hence assumption (13). For example, I could imagine
to refine the analysis presented here by defining cohorts based on firm-specific distances to
default or other leverage- or volatility-based variables that may explain a firm’s exposure to
systematic credit risk, and to obtain default data from proprietary sources such as Moody’s
Default Risk Service to compute cohort-specific default rates.24 Following steps similar to
22
i
Similar arguments apply to covt (Dt,y
, πt,y+1 ).
C(i,t)
i
e C(i,t) − D
e t,y
I opt for Equation (18) instead of simply assuming that covt (Dt+y,1
, πt,y+1 ) = covt (D
, πt,y+1 )
t,y+1
since (18) lends itself to extensions that take into account the comovement between LGD and the default indicator,
as described in Section 9.
24
While Feldhütter and Schaefer (2014) do not separately identify the expected loss and credit risk premium
23
17
those in Equations (14) and (15) would then yield (16) and (17) with R(i, t) replaced by
C(i, t).
i
• An alternative route I have explored is to specify covt (Dt+y,1
, πt,y+1 ) by directly modeling
i
a
a
e
e
the link between Dt+y,1 and Dt,y+1 − Dt,y , and by calibrating that link to the conditional
i
ea
ea
expectations or variances for Dt+y,1
and D
t,y+1 − Dt,y . Appendix B describes in detail the
drawbacks of such an approach.
• Measuring credit risk premia at the rating cohort level, even in the presence of firm-specific
default risk measures, is consistent with recent work by Hilscher and Wilson (2013), who compute rating-cohort-level “failure betas” using firm-specific PD estimates. Classic credit spread
puzzle papers that calibrate structural-model-implied PDs to their historical counterparts also
assume within-rating-cohort homogeneity of credit risk premia. (See, for example, Huang and
Huang (2012) and Chen, Collin-Dufresne, and Goldstein (2009).)
• Lastly, it is important to point out that while Equations (16) and (17) imply that time-t credit
risk premia are the same for all firms in a given time-t rating cohort, over time a firm’s credit
risk premium may change not only because the conditional comovement between aggregate
default rates and the state price density changes, but also because the firm’s credit quality
improves or worsens.
5.2
State price density process
The next step towards quantifying credit risk premia is to specify the state price density process.
In what follows, πt,y is modeled using the framework of Campbell and Cochrane (1999). I choose
the Campbell and Cochrane framework for a number of reasons. First, it is parsimonious. Second,
it is successful at capturing key features of historical equity returns, such as high equity risk premia
and strong time variation in Sharpe ratios. Third, Chen, Collin-Dufresne, and Goldstein (2009)
show that for certain specifications of the countercyclicality of default rates, the Campbell-Cochrane
pricing kernel can be consistent with the level of and variation in historical Baa-Aaa corporate bond
yield spreads.
I rely on the Chen, Collin-Dufresne, and Goldstein adaptation of the Campbell-Cochrane framereal as
work and specify the real relative state price density, πt,∆
real
πt,∆
= e−α∆ e−γ(st+∆ −st ) e−γ(ct+∆ −c1 ) ,
(19)
where α is a scalar, γ is the degree of risk-aversion, ct is log real per capita consumption and st is
components of corporate bond yield spreads, one of many appealing features of their work is that asset volatility—a
crucial parameter in any structural credit risk model—is modeled as a function of the firm’s leverage ratio and equity
volatility.
18
the log surplus consumption ratio. The dynamics of ct and st are given by
√
ct+∆ − ct = g∆ + σ ∆ zt,∆
√
st+∆ − st = κ(s − st )∆ + λ(st )σ ∆ zt,∆ ,
where zt,∆ are independent, normally distributed disturbances with mean √
zero and variance one,
q
1−2(st −s)
γ
− 1 if st ≤
independent of time-t information, S = σ κ , s = log S and λ(st ) =
S
2
s + 12 (1 − S ) and λ(st ) = 0 otherwise.
The parameters reported in Chen, Collin-Dufresne, and Goldstein (2009) are α=0.133, γ=2.45,
g=0.0189, σ=0.015 and κ=0.138. The real risk-free rate is constant at
r = α + γg −
γ 2σ2
2S
2
= 0.0094.
(20)
I obtain quarterly consumption data from 1947.I to 2010.II from the Bureau of Economic Analysis
and assume that the initial surplus consumption ratio is equal to its long-run average, that is,
S1947.I = S.
With regard to the computation of the covariance terms in Equations (16) and (17) it is important to point out that conditional on time-t information, the only sources of uncertainty in πt,y+1
are the random realizations of the disturbances zl,∆ , for l = t, t + ∆, . . . , t + y + 1 − ∆. To highlight
this fact, I express πt,y+1 as
y+1
πt,y+1 = e−ft
({zl,∆ }l )
.
(21)
Assuming a constant rate of inflation of i = 3%, the function fty (·) is given by
fty ({zl,∆ }l )
√
= (i + α + γg)y + γσ ∆
y/∆−1
X
zt+k∆,∆ + γgty ({zl,∆ }l ),
k=0
where gty ({zl,∆ }l ) = st+y − st =
Equations (16) and (17) as
Py/∆−1
k=0
i
covt (Dt+y,1
, πt,y+1 ) =
i
covt (Dt,y
, πt,y+1 )
=
(st+(k+1)∆ − st+k∆ ). Equation (21) allows me to restate
δ R(i,t),y+1
R(i,t)
wt
a
a
e t,y+1
e t,y
covt D
−D
, fty+1 ({zl,∆ }l ) ,
y−1 R(i,t),k+1
X
δ
R(i,t)
k=0
wt
19
ea
e a , f y+1 ({zl,∆ }l ) .
covt D
−
D
t,k+1
t,k t
(22)
(23)
5.3
Conditional covariation between aggregate default rates and the state price density
The final step towards quantifying credit risk premia is to estimate the conditional covariance
terms on the right-hand side of Equations (22) and (23). Figure 3 reveals that 1-year-ahead
ea = D
ea − D
e a , can be forecasted well using aggregate RMI PDs.
aggregate default rates, D
t,1
t,1
t,0
a , is the weighted average of firm-specific cumulative y-year
The y-year RMI U.S. PD index, RM It,y
a
RMI PDs across all public U.S. firms in the RMI database (currently about 3,800 firms). RM It,y
is available from 1991 onwards.
600
Aggregate default rate
Aggregate RMI PD (standardized)
500
Basis points
400
300
200
100
0
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
Figure 3: Aggregate default rates and RMI forecasts The figure shows the time series of 1-year-ahead
a
a
e t,1
aggregate default rates, D
, and the associated 1-year-ahead aggregate RMI PDs, RM It,1
(standardized). The
shaded areas indicate NBER recessions. The data cover the period from 1991 to 2010.
a tracks D
e a rather closely. Indeed, the forecasting R2 is 81%.
Figure 3 shows that RM It,1
t,1
While the forecasting power of aggregate RMI PDs for 1-year-ahead default rates is impressive,
it drastically decreases as the forecasting horizon increases: The forecasting R2 from regressing
a
a
ea
ea
D
t,y+1 − Dt,y on RM It,y+1 − RM It,y , for y = 1, . . . , 4, is 10% or less. Figure A.5 in the appendix
shows the time series of 5-year-ahead aggregate default rates and 5-year-ahead aggregate RMI PDs,
and confirms that there is only a moderate amount of comovement.
ea
ea
While it may be difficult to forecast future aggregate default rates D
t,y+1 − Dt,y for y > 0, I can
capture their comovements with the state price density process by exploiting the fact, visualized
in Figure 4, that aggregate default rates tend to be lower during economic expansions and higher
during recessions. This business cycle pattern of aggregate default rates, and the fact that the log
surplus consumption ratio is pro-cyclical, point to a positive contemporaneous link between aggregate default rates and negative changes in the log surplus consumption ratio, which is confirmed
20
in Figure 4.25
600
Aggregate default rate
Negative change in log surplus consumption ratio (standardized)
500
Basis points
400
300
200
100
0
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
Figure 4: Aggregate default rates through the business cycle The figure shows the time series of 1-yeara
e t,1
ahead aggregate default rates, D
, and the associated 1-year-ahead negative change in the log surplus consumption
ratio, −(st+1 − st ) (standardized). The shaded areas indicate NBER recessions. The data cover the period from 1991
to 2010.
Motivated by the findings in Figures 3 and 4, I model y-year ahead aggregate default rates as
a linear function of y-year-ahead aggregate RMI PDs and y-year-ahead changes in the log surplus
consumption ratio:
a
a
a
a
e t,y+1
e t,y
D
−D
= δ0D,y+1 + δ1D,y+1 RM It,y+1
− RM It,y
.
+ δ2D,y+1 {st+y+1 − st − Et (st+y+1 − st )} + D,y+1
t
(24)
I assume that D,y+1
are i.i.d. disturbances that are conditionally independent of future changes in
t
the log surplus consumption ratio and of the state price density process.26 I demean the changes
in the log surplus consumption ratio, so that all variation in time-t conditional expected future
27
ea
ea
aggregate default rates, Et (D
t,y+1 − Dt,y ), can be attributed to variation in aggregate RMI PDs.
OLS regression results are summarized in Table 5. As anticipated, the loadings on y-yearahead aggregate RMI PDs are positive and the loadings on y-year-ahead changes in the log surplus
25
At a first glance, Figures 3 and 4 may seem to suggest that aggregate RMI PDs forecast innovations to log
consumption. In a forecasting regression, however, the RMI PD coefficient is not statistically significant.
26
I have confirmed that expanding the vector of predictor variables in regression (24) to include additional, possibly
non-linear, functions of log consumption growth (that is, of the disturbances zl,∆ ) does not improve the goodness of
fit in any dramatic way.
27
The conditional expectation Et (st+y+1 − st ) is computed using Monte Carlo simulations of 100,000 sample paths
with antithetic sampling.
21
consumption ratio are negative. All loadings are statistically different from zero, with the exception
of δ1D,5 . The regression R2 is reassuringly high, ranging from 85% for y = 0 to 46% for y = 4.
Table 5: The link between aggregate default rates and the log surplus consumption ratio The table
a
a
a
e t,y+1
e t,y
reports the regression results for Equation (24). D
−D
is computed according to Equation (14), RM It,y
is
the aggregate cumulative y-year RMI PD, and st is the log surplus consumption ratio in the Chen, Collin-Dufresne,
and Goldstein (2009) adaptation of the Campbell and Cochrane (1999) framework. All variables are measured in
nominal terms. The data include annual observations from 1991 to 2010. T-statistics are reported in parentheses.
y
0
1
2
3
4
δ0D,y+1
0.0004
-0.0072
-0.0100
-0.0059
0.0041
(0.1404)
(-1.0579)
(-1.1009)
(-0.5995)
(0.5750)
δ1D,y+1
1.3599
2.3001
2.9828
2.5660
1.4576
δ2D,y+1
(7.8198)
(3.7380)
(2.9407)
(2.1205)
(1.4351)
-0.0132
-0.0486
-0.0487
-0.0420
-0.0307
(-2.1671)
(-4.6863)
(-4.3927)
(-3.7780)
(-3.1250)
R2
0.8521
0.6211
0.5853
0.5140
0.4645
Substituting Equation (24) into Equations (22) and (23), and using gty ({zl,∆ }l ) = st+y − st as
introduced earlier, yields
i
covt (Dt+y,1
, πt,y+1 ) =
i
covt (Dt,y
, πt,y+1 )
=
δ R(i,t),y+1
R(i,t)
wt
,
δ2D,y+1 cy+1,y+1
t
y−1 R(i,t),k+1
X
δ
k=0
R(i,t)
wt
,
δ2D,k+1 ck+1,y+1
t
(25)
(26)
where
ck+1,y+1
= covt (st+k+1 − st+k , πt,y+1 )
t
y+1
= cov gtk+1 ({zl,∆ }l ) − gtk ({zl,∆ }l ), e−ft ({zl,∆ }l ) .
An important feature of Equations (25) and (26) is that the covariance terms ck+1,y+1
can
t
be computed as unconditional covariances. The reason is that according to Equation (24), the
conditional covariance between k-year-ahead aggregate default rates and the state price density
is proportional to the conditional covariance between k-year-ahead changes in the log surplus
consumption ratio and the state price density. But given st , the only sources of uncertainty in
y+1
st+k+1 − st+k = gtk+1 ({zl,∆ }l ) − gtk ({zl,∆ }l ) and πt,y+1 = e−ft ({zl,∆ }l ) are the disturbances zl,∆ ,
which are independent of time-t information.
I compute the covariance terms ck+1,y+1
using Monte Carlo simulations.28 Their temporal
t
pattern is displayed in Figure 5. The plot highlights the fact that any variation in the covariance
terms is linked to variation in the log surplus consumption ratio. Indeed, the covariance terms can
28
The Monte Carlo simulations are based on 100,000 sample paths and use antithetic sampling.
22
be closely approximated by affine functions of st .
5
5
−st
−c1,1
(standardized)
t
−c5,5
(standardized)
t
4.5
4.5
4
4
3.5
3.5
3
3
2.5
2.5
2
Dec01
Dec03
Dec05
−st
Dec07
2
Dec09
Dec01
Dec03
Dec05
Dec07
Dec09
Figure 5: Time series of log surplus consumption ratio and cy+1,y+1
The figure shows the negative of the
t
quarterly time series of the log surplus consumption ratio and c1,1
(left panel) or c5,5
(right panel). The covariance
t
t
terms are standardized to have the same sample mean and standard deviation as the log surplus consumption ratio.
The data cover the period from January 2001 to June 2010.
5.4
Credit risk premium estimates
i
i ,π
Using Equations (25) and (26), I quantify covt (Dt+y,1
, πt,y+1 ) and covt (Dt,y
t,y+1 ). These
conditional covariance estimates are then plugged into Equations (11) and (12) to compute WtJ,i
and WtM,i , and eventually JtDti = WtJ,i /Ait and M tMti = WtM,i /Ait .
For the benchmark specification of EDF-based PDs and constant loss given default, summary
statistics for jump-to-default risk premia, JtDti , and survival-conditional mark-to-market credit
risk premia, M tMti , are reported in Table 6. I find jump-to-default risk premia to be minuscule:
The median value of JtDti is less than one basis point.29 Measured as a fraction of credit spreads,
median jump-to-default risk premia amount to only about 0.01, or 1%. Survival-conditional markto-market credit risk premia, on the other hand, are of moderate size, with a median value of 11
basis points, or 17% of credit spreads.
The median of the overall credit risk premium component, RPti = JtDti + M tMti , is 12 basis
points, and the median RP -to-C ratio is about 0.17. Credit risk premia are estimated to be higher
for HY firms (81 basis points, or 27% of the CDS spread) and lower for IG firms (7 basis points,
or 13% of the CDS spread). They are fairly flat throughout the early part of the sample, but start
increasing drastically in 2008, consistent with the fact that much of the time-series variation in
29
This is in line with the findings in Bai, Collin-Dufresne, Goldstein, and Helwege (2012).
23
Table 6: Median credit risk premium component and risk-premium-to-CDS ratio The table reports
the median jump-to-default component (columns marked “JtD”), survival-conditional mark-to-market credit risk
premium component (columns marked “MtM”) and overall credit risk premium component (columns marked “RP”)
of 5-year CDS spreads in basis points, and the associated median component-to-CDS ratios. The results are for
EDF-based PDs and benchmark loss given default. The data include 414,816 firm-day observations for 240 firms and
cover the period from January 2001 to June 2010. They are stratified by year and credit quality.
JtD
All firms
M tM
RP
All
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
0
0
0
0
0
0
0
0
1
1
1
11
6
7
8
7
7
7
9
16
26
24
12
6
7
8
8
7
8
10
17
27
25
All
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
0.01
0.00
0.00
0.00
0.01
0.01
0.01
0.01
0.01
0.01
0.02
0.17
0.04
0.06
0.12
0.16
0.17
0.20
0.22
0.15
0.28
0.35
0.17
0.04
0.06
0.12
0.16
0.17
0.21
0.23
0.15
0.29
0.37
JtD
0
0
0
0
0
0
0
0
1
1
1
IG firms
M tM
RP
In basis points
7
7
6
6
6
7
8
8
7
7
7
7
7
7
8
8
13
13
22
23
23
25
As a fraction of CDS spreads
0.00
0.13
0.13
0.00
0.04
0.04
0.00
0.05
0.05
0.00
0.10
0.11
0.00
0.13
0.13
0.00
0.13
0.14
0.01
0.17
0.17
0.01
0.19
0.20
0.00
0.11
0.11
0.01
0.19
0.20
0.01
0.23
0.24
JtD
HY firms
M tM
RP
3
1
1
1
1
1
1
1
3
11
15
78
28
36
39
36
36
37
46
89
249
312
81
28
37
40
37
37
38
48
92
260
328
0.01
0.00
0.00
0.01
0.01
0.01
0.01
0.01
0.01
0.02
0.03
0.26
0.07
0.10
0.16
0.23
0.24
0.28
0.28
0.21
0.40
0.62
0.27
0.07
0.10
0.17
0.23
0.26
0.30
0.29
0.22
0.43
0.67
credit risk premia stems from time-series variation of the log surplus consumption ratio over the
business cycle (Figure 5).
The credit risk premium estimates depend on the choice of the PD measure only though the
survival-conditional scaling factor Ait . The higher the likelihood of default, the lower Ait and,
everything else the same, the higher the credit risk premium. Since rating-based PDs tend to be
higher than their EDF- or RMI-based counterparts, the median RPti estimate is somewhat higher
for rating-based PDs than for EDFs or RMI PDs. Other than that, the results remain largely the
same when an alternative PD measure is used, as shown in Tables A.3 and A.4 in the appendix.
24
6.
The Residual Component of CDS Spreads
In standard CDS pricing models, residual spreads Resit are zero and CDS spreads are computed
as the sum of expected losses and credit risk premia:
Ctm,i = ExpLit + RPti ,
where Ctm,i denotes the standard-model-implied CDS spread, or model spread for short. Residual
spreads measure the deviation of actual spreads from their standard-model-implied counterparts:
Resit = Cti − Ctm,i .
Summary statistics for Resit are reported in Table 7. For EDF-based PDs and benchmark loss given
default, median residual spreads amount to 41 basis points, or 56% of the level of credit spreads.
The data are therefore consistent with a sizable residual component Resit . This result poses a credit
spread level puzzle in the sense that at the median—over time and across firms—the fraction of
credit spreads that is neither due to expected losses nor credit risk premia is positive and sizable.
Early in the sample, between 2001 and 2003, the puzzle is more pronounced for HY firms than
for IG firms, consistent with the lower ExpL-to-C ratios for HY than for IG firms reported in
Table 3. Between 2008 and 2010, however, that pattern is reversed: Model spreads for IG firms
underpredict actual spreads much more than model spreads for HY firms do, due to a sharp relative
increase in PDs and credit risk premia for HY firms (see Tables 3 and 6). For 2010, model spreads
even exceed actual spreads for the median HY firm and date.
The results for RMI PDs are, at least in the aggregate, fairly similar to those obtained for EDFs.
The main difference is that RMI-implied residual spreads for HY firms tend to be higher than their
EDF-implied counterparts, largely because RMI PDs for HY firms tend to be lower than EDFs
(see Table 2). As a result, for RMI PDs the credit spread level puzzle is almost as pronounced for
HY firms in 2008-10 than it is for IG firms.
Interestingly, I find that rating-based PDs imply a median residual component that at 17
basis points, or 33% of the level of credit spreads, is much lower than that obtained for EDFs
or RMI PDs. In other words, I show that the credit spread level puzzle tends to be much more
pronounced when expected losses and credit risk premia are calibrated to forward-looking firmby-firm default probabilities than when they are calibrated to historical rating-based PDs. This
insight is an important one for the credit spread puzzle literature, in which structural models
are often calibrated to historical rating-based default rates (see, for example, Huang and Huang
(2012), Cremers, Driessen, and Maenhout (2008) and Chen, Collin-Dufresne, and Goldstein (2009)).
Closer inspection of Table 7 reveals that the difference between EDF-/RMI-based model spreads
and rating-based model spreads is particularly stark between 2004 and 2007, when EDFs/RMI PDs
are significantly lower than their rating-based counterparts (Table 2) and credit risk premia are
fairly flat (Figure 5 and Table 6). During those years, rating-based model spreads tend to yield
25
Table 7: Median residual component and residual-spread-to-CDS ratio The table reports the median
residual component of 5-year CDS spreads in basis points, and the median residual-spread-to-CDS ratio. Results are
reported for EDF-based PDs (columns marked “EDF”), RMI PDs (columns marked “RMI”) and rating-based PDs
(columns marked “Rtg”), and for benchmark loss given default. The data include 414,816 firm-day observations for
240 firms and cover the period from January 2001 to June 2010. They are stratified by year and credit quality.
EDF
All firms
RMI Rtg
EDF
IG firms
RMI Rtg
All
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
41
68
70
35
32
35
29
35
110
33
16
47
81
79
46
39
36
22
23
114
76
32
17
84
74
24
8
7
0
3
73
48
17
In basis points
27
27
21
54
61
74
52
61
62
19
23
23
20
21
12
22
20
11
17
11
6
22
12
8
71
65
63
32
43
49
25
27
36
All
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
0.56
0.59
0.64
0.47
0.53
0.62
0.58
0.62
0.70
0.24
0.14
0.57
0.72
0.76
0.57
0.57
0.59
0.46
0.47
0.70
0.49
0.31
0.33
0.77
0.72
0.43
0.17
0.15
0.01
0.09
0.58
0.40
0.20
As a fraction of CDS spreads
0.59
0.56 0.49
0.55
0.68 0.80
0.62
0.74 0.76
0.42
0.47 0.54
0.53
0.54 0.33
0.65
0.59 0.33
0.62
0.44 0.22
0.66
0.42 0.27
0.76
0.71 0.70
0.37
0.49 0.58
0.35
0.37 0.49
EDF
HY firms
RMI
Rtg
119
273
320
193
105
109
89
122
230
36
-46
156
331
373
253
133
117
83
109
276
216
79
-24
218
222
38
-71
-65
-83
-62
119
40
-103
0.51
0.68
0.72
0.63
0.53
0.57
0.52
0.57
0.58
0.09
-0.14
0.59
0.82
0.82
0.73
0.61
0.59
0.48
0.52
0.69
0.48
0.21
-0.12
0.53
0.51
0.12
-0.35
-0.41
-0.59
-0.39
0.31
0.10
-0.37
a closer approximation—from below or above—of actual spreads than EDF/RMI-based model
spreads. For the remaining years, 2001-03 and 2008-10, the results are more mixed. Whereas
EDFs/RMI PDs tend to yield more accurate model spreads for IG firms (except for 2008), the
opposite is true for HY firms (except for 2010).
6.1
Connecting with the broader credit spread puzzle literature
While I use the term “credit spread level puzzle” to describe the observation that the median
fraction of credit spreads that is neither due to expected losses nor credit risk premia is positive and
sizable, several other definitions of this term can be found in the literature. The most commonly
used version defines the credit spread (level) puzzle as follows: “The difference between Baa- and
Aaa-rated corporate bond yield spreads is too high to be explained by standard structural credit
risk models.” (For references, see Footnote 5.) My use of the term is different in several aspects: (i)
It refers to the level of 5-year CDS spreads rather than the difference between long- or short-term
Baa and Aaa corporate bond yield spreads, (ii) it relies on PD measures that are widely used by
26
credit market participants and a pricing kernel that is successful in explaining the equity premium
puzzle rather than a specific structural model,30 and (iii) it is based on the median relative difference
between firm-by-firm model and actual spreads rather than the difference between index spreads
and spreads computed using average firm-specific parameters.
The latter distinction is particularly important as most credit-spread-puzzle papers use aggregate data (i.e., the average of firm-specific balance sheet or equity-market-based variables) as
model inputs. As long as the functional form between input parameters and model spreads is convex, this may lead to a downward bias in model spreads, as explained theoretically in David (2008)
and Zhang, Zhu, and Zhu (2009), and shown empirically in Feldhütter and Schaefer (2014). There
are only a few papers that I am aware of that analyze the puzzle on a firm-by-firm or bond-by-bond
basis: Eom, Helwege, and Huang (2004), Ericsson, Reneby, and Wang (2007) and Feldhütter and
Schaefer (2014).
Eom, Helwege, and Huang (2004) use 182 corporate bond price quotes during the period 1986 to
1997 and show that the Merton (1974) model consistently underestimates bond yield spreads, but
that most other structural models (i.e., Geske (1977), Longstaff and Schwartz (1995), Leland and
Toft (1996) and Collin-Dufresne and Goldstein (2001)) predict spreads that are too low for IG firms
and too high for HY firms. Ericsson, Reneby, and Wang (2007) use about 1,400 bond and CDS
price quotes from the period 1997-2003 and find that while many structural models systematically
underestimate bond yield spreads that is not the case for CDS spreads. Feldhütter and Schaefer
(2014) use over 500,000 bond transaction prices from TRACE for the period 2002-2012. For 4-year
bond yield spreads, the maturity closest to the one I analyze in CDS markets, they find that the
Merton model underpredicts IG spreads but overpredicts HY spreads.
As long as model spreads are calibrated to rating-based PDs, my findings are consistent with
theirs: Table 7 shows that rating-based model spreads tend to be lower than actual spreads for IG
firms, and tend to be higher than actual spreads for HY firms. Model spreads for HY firms are
particularly high relative to actual spreads between 2004 and 2007, and in 2010. Additional details
are provided in Table A.5 in the appendix, which stratifies the data by letter ratings.31
The insight I offer is that the severity of the credit spread level puzzle—that is, the bias of model
spreads relative to actual spreads—is highly sensitive to the choice of the PD measure. When model
spreads are calibrated to forward-looking firm-by-firm default probabilities such as EDFs or RMI
PDs rather than rating-based PDs, the data are consistent with positive residual spreads, both for
IG and HY firms.
Even though model spreads are consistently biased low for EDFs and RMI PDs, accuracy still
tends to be higher than in the existing literature: By comparing the results in Table A.5 to those for
30
According to Berndt, Douglas, Duffie, Ferguson, and Schranz (2005), 40 of the world’s 50 largest financial
institutions subscribe to Moody’s Analytics. EDFs are the most widely used name-specific source of conditional
default probability estimates of which I am aware, covering, as of 2005, over 26,000 publicly traded firms.
31
The table shows that rating-based model spreads tend to underpredict actual spreads most severely for Aaa- or
Aa-rated firms and overpredict actual spreads most severely for B-rated firms, and that they tend to be closest to
actual spreads for Ba-rated firms.
27
4-year bonds in Table 7 of Feldhütter and Schaefer (2014, FS), I find that my results exhibit smaller
percentage differences between median model and actual spreads, across all rating categories with
the exception of Baa-rated firms, for whom the percentage difference is slightly lower in FS. For
example, median actual and EDF-based model spreads for Aaa-Aa firms are 23 and 6 basis points,
compared to 24 and 1 basis points in FS. A model spread of 6 versus 1 basis point is noteworthy,
since credit spreads for the highest-credit-quality firms are notoriously difficult to explain. For Brated firms, median actual and EDF-based model spreads are 355 and 179 basis points, compared
to 463 and 856 basis points in FS.32
While this comparison underscores that consistent underpricing does not immediately translate
into low accuracy of model spreads, it by no means takes away from the findings in FS. On the
contrary, FS offer the important insight that if the Merton (1974) model is calibrated not to
historical default frequencies as is common practice in the prior literature, but to the leverage-based
asset volatility estimates in Schaefer and Strebulaev (2008), then even this simplest structural credit
risk model no longer consistently underestimates corporate bond yield spreads.
6.2
Benchmarking against Chen, Collin-Dufresne, and Goldstein (2009)
Since I rely on the same state price density specification as Chen, Collin-Dufresne, and Goldstein
(2009), or CCG for short, benchmarking my findings against theirs is of particular interest. CCG,
whose work is insightful and pathbreaking in many ways, find that the difference between modelimplied Baa and Aaa corporate bond yield spreads is in agreement with historical findings, as long
as the countercyclical nature of defaults is taken into account. My findings in Table A.5, on the
other hand, suggest that the difference between Baa and Aaa-Aa median CDS spreads is 43 basis
points for actual spreads and 33 basis points for rating-based model spreads.
Why is the discrepancy between actual and model Baa-Aaa/Aa spreads larger than in CCG?
While there are differences in the sample period (2001-2010 here versus 1970-2001 in CCG), measurement of PDs (firm by firm here and by rating cohort in CCG) and many aspects of the calibration exercise, I believe the most striking deviation of my approach from that in CCG is that I do
not use credit spread information to calibrate credit risk premia.33 Specifically, CCG regress 4-year
cumulative default rates for Baa-rated bonds on the composite Baa-Aaa spread, with the intent to
calibrate the countercyclicality of defaults to that regression coefficient as well as the unconditional
variation of the Baa-Aaa distribution.
To better understand the implications of these restrictions, consider the decomposition of actual
32
The higher median actual spreads in FS are likely due to the fact that they are based on corporate bond yield
spreads rather than CDS spreads, given that default swaps are generally more liquid than the underlying corporate
bonds.
33
Although some of the data presented by CCG extend much further into the past, other data used to calibrate
the model go back only to 1970.
28
(Ct ) and CCG model (Ctccg ) spreads:
Ct = ExpLt + RPt + Rest
Ctccg
ccg
= ExpLccg
t + RPt .
For the purpose of this exercise, I think of ExpLt , RPt and Rest as actual expected losses, credit
risk premia and residual spreads of a representative Baa firm in excess of those for a representative
ccg
Aaa firm, and ExpLccg
and RPtccg = 0 as their model counterparts. Loosely translated,
t , RPt
the restrictions that CCG propose are (i) cov(Ctccg , ExpLt )/var(Ctccg ) = cov(Ct , ExpLt )/var(Ct )
and (ii) var(Ctccg ) = var(Ct ). They immediately imply that the fitted model will be successful in
explaining the covariation between actual spreads and excess spreads:
cov(Ctccg , RPtccg ) = cov(Ct , RPt ) + cov(Ct , Rest ).
(27)
What CCG show—using the Cambpell-Cochrane pricing kernel—is that the countercyclicality
of defaults can indeed be specified in such a way that (i) and (ii) are satisfied simultaneously. More
importantly, CCG document that once the model is calibrated to (i) and (ii), it is successful in
explaining the average level of credit spreads,
E(Ctccg ) = E(Ct ).
(28)
Since CCG also calibrate their model to average historical default rates, E(ExpLccg
t ) = E(ExpLt ),
Equation (28) is equivalent to
E(RPtccg ) = E(RPt ) + E(Rest ).
(29)
While (27) is a direct implication of the restrictions (i) and (ii), Equation (28) does not follow
immediately, and hence offers interesting new insights.
Nevertheless, (i) and (ii) are self-imposed restrictions and Equations (27) through (29) may or
may not be an accurate description of the data. For example, if Baa-Aaa spreads were to contain
positive residual spreads, as suggested by the evidence in Table 4 of Longstaff, Mithal, and Neis
(2005), then Equation (29) would overestimate credit risk premia. And if Baa-Aaa spreads were
to contain residual spreads that covary positively with credit spreads, then Equation (27) would
overestimate the covariation between credit risk premia and credit spreads.
In contrast, I do not use CDS data to calibrate credit risk premia. Instead, estimates for RPt rely
primarily on the statistical comovement between realized default rates and the state price density.
By imposing fewer constraints on the data, I do find evidence of a sizable residual component, Rest .
29
7.
Model Spreads as a Function of the Level of Risk Aversion
Residual spreads may compensate investors for CDS market liquidity risk or liquidity risk
premia, or they may be due to clientele effects or measurement noise. But any misspecification of
the state price density process would also be absorbed into Resit . I explore the notion of potential
model misspecification by allowing equity and CDS markets to be populated by investors with
different levels of risk aversion. In particular, I quantify that level of risk aversion of CDS market
investors, γ c , at which the median residual-spread-to-CDS ratio is zero or, equivalently, the median
model-spread-to-CDS ratio is one.
The idea here is that as the level of risk aversion γ in the Campbell and Cochrane pricing kernel
increases, the countercyclicality of πt,y becomes more pronounced. Given the countercyclical nature
of defaults, an increase in γ is likely to result in higher Resit estimates. To facilitate different levels
of risk aversion while holding the dynamics of log consumption and the log surplus consumption
ratio the same, I use Campbell and Cochrane’s extension of their framework to time-varying real
interest rates:
rt = r − B(st − s),
where B is a non-positive scalar.
For B = 0, the model defaults to the one described in Section 5.2. But for B < 0, real rates
are low in bad times (low st ) and high in good times (high st ). The steady-state interest rates r is
computed as in Equation (20), after generalizing the expression for the steady-state gap between
consumption and habit levels to
s
S = σ
γc
.
κ − B/γ c
As γ c increases, B is lowered to keep the steady-state surplus-consumption ratio at its initial
level, and α is recomputed to ensure that steady-state interest rates remain unchanged, that is,
c2 2
α = r − γ c g + γ σ2 .
2S
Median model-spread-to-CDS ratios, as a function of γ c , are reported in Table 8. I find γ c = 3.8
for EDFs, γ c = 3.75 for RMI-based PDs and γ c = 3.3 for rating-based PDs. In that sense, model
spreads can be reconciled with the data only if CDS market investors are more risk averse than
equity market investors. But the implied discrepancy between risk aversion in equity markets
(γ e = 2.45) and credit markets is more pronounced when model spreads are calibrated to forwardlooking firm-by-firm PDs than when they are calibrated to historical rating-based PDs.
8.
Variance Decomposition of CDS Spreads
In addition to the decomposition of the level of CDS spreads, I am interested in the information
revealed through variation in credit spreads. Equation (7) implies that credit spread variation is
30
Table 8: Median model-spread-to-CDS ratio for various levels of risk aversion The table reports, for
various levels of risk aversion γ c , the median value for (ExpLit + RPti )/Cti . Results are reported for EDF-based
PDs (rows marked “EDF”), RMI PDs (rows marked “RMI”) and rating-based PDs (rows marked “Rtg”), and for
benchmark loss given default. The data include 414,816 firm-day observations for 240 firms and cover the period
from January 2001 to June 2010. They are stratified by credit quality.
γc =
2.00
2.45
3.00
3.25
EDF
RMI
Rtg
0.372
0.361
0.602
0.439
0.431
0.675
0.575
0.573
0.839
EDF
RMI
Rtg
0.358
0.391
0.458
0.413
0.443
0.515
EDF
RMI
Rtg
0.396
0.323
0.988
0.492
0.431
1.116
3.30
3.35
3.70
3.75
3.80
3.85
0.669
0.675
0.963
All firms
0.692
0.715
0.699
0.724
0.992 1.023
0.931
0.956
1.287
0.971
1.000
1.333
1.015
1.046
1.383
1.062
1.095
1.436
0.523
0.556
0.647
0.599
0.636
0.736
IG firms
0.616
0.634
0.655
0.675
0.757
0.778
0.804
0.857
0.963
0.837
0.890
0.998
0.871
0.927
1.034
0.908
0.966
1.073
0.684
0.601
1.419
0.821
0.737
1.620
HY firms
0.855
0.891
0.770
0.806
1.666
1.714
1.234
1.134
2.147
1.298
1.196
2.227
1.366
1.262
2.311
1.441
1.334
2.398
due to covariation of credit spreads with expected losses, credit risk premia or residual spreads:
var(Cti ) = cov(Cti , ExpLit ) + cov(Cti , RPti ) + cov(Cti , Resit ).
(30)
For each firm i, I estimate the fraction of the time-series variation in credit spreads that
corresponds to variation in the CDS component Y , Y ∈ {ExpL, RP, Res}, as the coefficient βYts,i
in the time-series regression
ts,i
+ βYts,i Cti + εts,i
Yti = βY,0
Y,t .
(31)
ts,i
ts,i
ts,i
= 1. The regression results are reported in Table 9.
+ βRes
+ βRP
Equation (30) implies βExpL
ts,i
coefficient of 0.24, implying that
Using benchmark PDs and LGD, the median firm has a βExpL
less than one quarter of the temporal credit spread variation reflects expected loss variation. The
ts,i
βExpL
estimate is somewhat higher for the median HY firm (0.26) than the median IG firm (0.22).
The temporal comovement between CDS spreads and credit risk premia is rather limited: The
ts,i
median βExpL
estimate is only 0.07. As a result, the median fraction of the time-series spread
variation that corresponds to model spread variation is only 0.33. This points to a pronounced
positive comovement between credit spreads and residual spreads. Indeed, the median fraction of
the temporal spread variation that corresponds to residual spread variation is 0.67. The positive
covariation between credit spreads and residual spreads gives rise to a credit spread volatility puzzle
in the sense that actual spread variation is too high to be explained by model spread variation.
The temporal volatility puzzle is somewhat more pronounced for IG firms than for HY firms.
The time-series variation of credit spreads is less aligned with that of expected losses, and
31
Table 9: Decomposing the time-series variation of CDS spreads I compute firm-by-firm estimates for βYts,i
in Equation (31) and report their median (columns marked “Med”) and interquartile range (columns marked “IQ
range”), across all firms and by median credit rating. Results are reported for EDF-based PDs (rows marked “EDF”),
RMI PDs (rows marked “RMI”) and rating-based PDs (rows marked “Rtg”), and for benchmark loss given default.
The data include 414,816 firm-day observations for 240 firms and cover the period from January 2001 to June 2010.
Med
All firms
IQ range
Med
IG firms
IQ range
Med
HY firms
IQ range
EDF
RMI
Rtg
0.24
0.07
0.07
[0.13, 0.38]
[0.04, 0.13]
[0.01, 0.22]
Expected losses
0.22 [0.10, 0.38]
0.09 [0.05, 0.13]
0.05 [0.01, 0.17]
0.26
0.06
0.13
[0.15, 0.40]
[0.03, 0.10]
[0.02, 0.30]
EDF
RMI
Rtg
0.07
0.06
0.07
[0.03, 0.16]
[0.03, 0.14]
[0.03, 0.15]
Credit
0.06
0.05
0.06
risk premia
[0.02, 0.10]
[0.02, 0.10]
[0.02, 0.10]
0.12
0.10
0.10
[0.04, 0.22]
[0.04, 0.19]
[0.04, 0.20]
EDF
RMI
Rtg
0.67
0.83
0.83
[0.46, 0.81]
[0.74, 0.91]
[0.64, 0.95]
Residual spreads
0.71 [0.54, 0.83]
0.84 [0.77, 0.91]
0.90 [0.74, 0.96]
0.60
0.82
0.76
[0.39, 0.77]
[0.70, 0.93]
[0.54, 0.90]
ts,i
hence model spreads, when EDFs are replaced by RMI PDs or rating-based PDs. The median βRes
estimate is 0.83 for both alternative PD measures. In that sense, the time-series volatility puzzle
is even more striking for RMI PDs and rating-based PDs than for EDFs.34 This is confirmed in
Figure 6, which shows actual and model spreads for one particular IG firm, Target, and one HY
firm, Goodyear. While EDF-based model spreads for both firms exhibit a substantial amount of
time-series variation and track CDS spreads reasonably well, RMI-based model spreads are fairly
flat across time. Rating-based model spreads are flat for the IG firm, but exhibit somewhat more
covariation with actual spreads for the HY firm, a pattern that is reflected in the broader sample,
as shown in Table 9.
For each day t, I estimate the fraction of the cross-sectional credit spread variation that corresponds to variation in Y as the coefficient βYcs,t in the cross-sectional regression
cs,t
Yti = βY,0
+ βYcs,t Cti + εcs,t
Y,i .
(32)
cs,t
cs,t
cs,t
Equation (30) implies βExpL
+ βRP
+ βRes
= 1.
cs,t
For benchmark PDs and LGD, the results reported in Table 10 reveal that the median βExpL
estimate is 0.27, meaning that on most days less than 30% of the cross-sectional credit spread
cs,t
variation signals expected loss variation. The βExpL
estimates vary dramatically over time, from a
low of 0.16 in 2005 to a high of 0.55 in 2010. They tend to be higher towards the end and shortly
after the 2001-02 and 2007-09 credit crunch, and lower for the years in between. The median
fraction of the cross-sectional credit spread variation that corresponds to model spread variation,
34
ts,i
As a robustness check, I re-estimate βRes
using monthly and quarterly data. The results remain nearly unchanged.
32
Target: EDF−based
Goodyear: EDF−based
300
2000
C
ExpL+RP
ExpL
250
C
ExpL+RP
ExpL
1750
Basis points
1500
200
1250
150
1000
750
100
500
50
0
250
Dec01
Dec03
Dec05
Dec07
0
Dec09
Dec01
Target: RMI−based
Dec03
Dec05
Dec07
Dec09
Goodyear: RMI−based
300
2000
C
ExpL+RP
ExpL
250
C
ExpL+RP
ExpL
1750
Basis points
1500
200
1250
150
1000
750
100
500
50
0
250
Dec01
Dec03
Dec05
Dec07
0
Dec09
Dec01
Target: Rating−based
Dec03
Dec05
Dec07
Dec09
Goodyear: Rating−based
300
2000
C
ExpL+RP
ExpL
250
C
ExpL+RP
ExpL
1750
Basis points
1500
200
1250
150
1000
750
100
500
50
0
250
Dec01
Dec03
Dec05
Dec07
0
Dec09
Dec01
Dec03
Dec05
Dec07
Dec09
Figure 6: CDS spreads, model spreads and expected losses The figure shows daily 5-year CDS spreads (C),
model spreads (ExpL + RP ) and expected losses (ExpL), for Target Corporation (left panel) and The Goodyear Tire
& Rubber Company (right panel). The top panel shows results for EDF-based PDs, the middle panel shows results
for RMI PDs, and the bottom panel shows results for rating-based PDs. Expected losses and credit risk premia are
based on benchmark loss given default.
cs,t
cs,t
βExpL
+ βRP
, is 0.44, meaning 56% of cross-sectional credit spread variation signals residual spread
variation. This is evidence that the credit spread volatility puzzle is present not just over time but
33
also across firms.
Table 10: Decomposing the cross-sectional variation of CDS spreads I compute day-by-day estimates for
βYcs,t in Equation (32) and report their median (columns marked “Med”) and interquartile range (columns marked
“IQ range”), across all days and by year. Results are reported for EDF-based PDs (rows marked “EDF”), RMI PDs
(rows marked “RMI”) and rating-based PDs (rows marked “Rtg”), and for benchmark loss given default. The data
include 414,816 firm-day observations for 240 firms and cover the period from January 2001 to June 2010.
Med
IQ range
2001
2001
2003
2005
2006
2007
2008
2009
2010
Expected losses
0.35
0.28
0.16
0.08
0.09
0.07
0.69
0.71
0.56
0.22
0.16
1.21
0.18
0.15
1.01
0.27
0.15
0.32
0.38
0.08
0.31
0.55
0.09
0.74
0.20
0.18
0.26
0.23
0.23
0.29
0.15
0.13
0.15
0.29
0.19
0.24
0.56
0.42
0.60
0.53
0.65
-0.47
0.56
0.62
-0.32
0.58
0.70
0.54
0.32
0.67
0.46
-0.11
0.45
-0.35
EDF
RMI
Rtg
0.27
0.10
0.58
[0.20, 0.35]
[0.08, 0.14]
[0.33, 0.79]
0.21
0.07
0.28
0.30
0.09
0.28
EDF
RMI
Rtg
0.16
0.14
0.19
[0.12, 0.21]
[0.10, 0.19]
[0.13, 0.27]
0.05
0.04
0.05
0.08
0.07
0.08
EDF
RMI
Rtg
0.56
0.74
0.21
[0.49, 0.65]
[0.65, 0.81]
[-0.16, 0.55]
0.75
0.88
0.67
0.62
0.84
0.64
2004
Credit risk premia
0.17
0.18
0.11
0.15
0.15
0.10
0.19
0.23
0.15
Residual spreads
0.48
0.54
0.74
0.78
0.76
0.83
0.11
0.06
0.29
The cross-sectional comovement of CDS spreads and expected losses is lower when EDFs are
replaced by RMI PDs. As a result, the median fraction of cross-sectional spread variation that
signals residual spread variation is higher for RMI PDs (0.74) than for EDFs (0.56). Interestingly,
for rating-based model spreads the cross-sectional volatility puzzle is much less pronounced than
for EDF-based model spreads. At times, the rating-based model spread variation even overshoots
the actual spread variation. While on the median day almost 80% of the cross-sectional spread
variation corresponds to rating-based model spread variation, only 44% corresponds to EDF-based
model-spread variation. Together, Tables 9 and 10 indicate that ratings do a better job in aligning
the cross-sectional ranking of actual and model spreads, whereas EDFs do a better job in aligning
the time-series ranking.
One potential explanation for the “ratings for cross-sectional ranking, EDFs for time-series
ranking” finding is that ratings supply a lot of information about relative credit quality across firms,
but are more stable through the business cycle than point-in-time absolute measures of default risk
such as EDFs. My findings suggest that combining EDFs and ratings for default prediction—for
example, by creating EDF-centered rating PDs—may yield an overall improved alignment between
actual and model spread variation.
9.
Robustness Checks
In this section, I consider alternative loss given default specifications and explicitly model the
covariation between loss given default and the default indicator.
34
9.1
Alternative loss given default specifications
Table 11 describes alternatives to the benchmark LGD specification. “Historical LGD” in any
given year is set equal to one minus the historical average senior unsecured recovery rate between
1982 and the end of the previous year, as reported in Moody’s annual “Corporate Default and
Recovery Rates” studies. “Markit LGD” relies on quotes of composite recovery rates, provided
by Markit on a daily basis for each firm, seniority and restructuring clause. My understanding is
that these quotes are based on estimates of expected future recovery rates provided to Markit by
market makers. Abstracting from recovery risk premia, these recovery rate quotes represent the
most detailed information on expected LGD available to me.
Table 11: Alternative loss given default specifications
LGD specification, conditional on time-t information
Benchmark
Historical
Markit
Rating-based
Regression-based
Lit+k∆,∆ = 0.63
Lit+k∆,∆ = Lt , where 1 − Lt is the average issuer-weighted recovery rate for senior
unsecured bonds between 1982 and the beginning of year t, as reported in Moody’s
“Corporate Default and Recovery Rates” studies
Lit+k∆,∆ = Lit , where 1 − Lit is the time-t recovery rate for firm i reported by Markit
IG/HY (i,t)
IG/HY (i,t)
1
for k = 0, .., 1−∆
Lit+k∆,∆ = Lt,1
for k = ∆
, Lit+k∆,∆ = Lt,2
, .., 2−∆
,
∆
∆
IG/HY (i,t)
and so on, where 1 − Lt,y
is the average issuer-weighted recovery rate for
senior unsecured bonds between 1982 and the beginning of year t for firms with
IG/HY status y years prior to default, as reported by Moody’s
e at+1,1 for k = 1 , .., 2−∆ , and so
e at,1 for k = 0, .., 1−∆ , Lit+k∆,∆ = Et L
Lit+k∆,∆ = Et L
∆
∆
∆
L,y+1
a
L,y+1
a
a
e t+y,1 = δ
e t,y+1 − D
e t,y
on, where Et L
+δ
E t (D
) per Equation (33)
0
At time t, the “rating-based LGD” specification sets the y-years-ahead LGD for investmentgrade firms equal to one minus the average senior unsecured recovery rate between 1982 and the
end of the previous year for firms with IG status y years prior to default, as reported in Moody’s
“Corporate Default and Recovery Rates” studies. For high-yield firms, historical average recovery
rates for firms with HY status y years prior to default are used.35
The “regression-based LGD” specification exploits the close contemporaneous link between
ea
realized aggregate default and LGD rates. I define L
t+y,1 as the aggregate loss given default for
senior unsecured bonds in year (t + y, t + y + 1]. It is measured as one minus the annual issuerweighted senior unsecured bond recovery rate based on post-default trading prices, as reported
in Moody’s Investors Service (2014). Figure A.6 in the appendix documents a tight link between
e a and D
e a . Using data from 1991 to 2010, the estimated model is
L
t,1
t,1
ea =
L
t,1
e a + L,1 ,
0.452 + 5.892 D
t,1
t
(15.376) (4.735)
35
Recovery rates by rating y years prior to default is available from 2004 onwards. Prior to that, rating-based LGD
is set equal to historical LGD.
35
with an R2 of 0.55.36 More generally, I perform OLS regressions
a
a
e at+y,1 = δ L,y+1 + δ L,y+1 (D
e t,y+1
e t,y
L
−D
) + L,y+1
.
t
0
(33)
The results are summarized in Table 12. For each y, I obtain an R2 greater than 40%.
a
a
e t,y+1
e t,y
Table 12: Regression-based LGD estimates This table reports the results from regressing Leat+y,1 on D
−D
per Equation (33), using annual data from 1991 to 2010.
y=0
constant
a
a
e t,y+1
e t,y
D
−D
2
R
0.452
5.892
0.555
(15.376)
(4.735)
y=1
0.442
6.155
0.581
(14.664)
(4.851)
y=2
0.447
6.257
0.550
(13.728)
(4.424)
y=3
0.443
6.818
0.547
(13.096)
(4.258)
y=4
0.456
6.762
0.433
(11.127)
(3.269)
e a for k = 0, .., 1−∆ , Li
ea
Conditional on time-t information, I set Lit+k∆,∆ = Et L
t,1
t+k∆,∆ = Et Lt+1,1
∆
L,y+1
1
, .., 2−∆
are i.i.d. disturbances that are
for k = ∆
∆ , and so on. I take the simple view that t
conditionally independent of future aggregate default rates and the state price density process, and
ea
use Equations (24) and (33) to compute Et L
t+y,1 as
a
a
e at+y,1 = δ L,y+1 + δ L,y+1 Et (D
e t,y+1
e t,y
Et L
−D
)
0
a
a
− RM It,y
.
= δ0L,y+1 + δ L,y+1 δ0D,y+1 + δ L,y+1 δ1D,y+1 RM It,y+1
The median expected loss given default is 0.63 for the benchmark and historical LGD specification, 0.60 when Markit quotes are used, 0.62 for 1-year-ahead LGDs (Lit,1 ) and 0.58 for 5-year-ahead
LGDs (Lit+4,1 ) based on ratings, and 0.50 for 1-year-ahead and 0.53 for 5-year-ahead regressionbased LGDs. Table A.6 in the appendix offers additional details on the four alternative LGD
specifications. It reveals that historical LGD for 2001—which is equal to one minus the aggregate
recovery rate for senior unsecured bonds between 1982 and 2000—is fairly low at 0.53. During
the 2001-2 credit crunch, however, a large number of firms defaulted and aggregate annual LGD
was at an all-time high (0.79 for 2001 and 0.70 for 2002), resulting in an increase in the historical
LGD estimates post-2001. Between 2002 and 2010, historical LGD ranges between 0.62 and 0.64.
Median Markit LGDs are close to 0.60 across sectors and rating categories, especially after 2002.
Compared to historical and Markit LGD, rating-based LGD exhibits somewhat more variation
over time and across firms. In particular, the expected loss given default tends to be higher for
HY firms than for IG firms. Comparing rating-based LGD for shorter versus longer horizons, I
find that for IG firms, 1-year-ahead LGD tends to be smaller than 5-years-ahead LGD, except for
2009 and 2010. For HY firms, short-term LGD tends to be higher than long-term LGD. Overall,
however, historical, Markit and rating-based LGD estimates are fairly stable across firms, time and
maturity. As a result, I expect only marginal changes in the CDS variance decomposition when
36
Further analysis of the relationship between aggregate recovery and default rates is provided, among others,
by Altman, Brady, Resti, and Sironi (2005) and Moody’s Investors Service (2008).
36
benchmark LGDs are replaced by one of these three alternative specifications. This is confirmed
in Table 13.
Table 13: Decomposition of CDS spreads for alternative LGD specifications This table reports the median
estimates for Yti in basis points, Yti /Cti , βYts,i and βYcs,t , for Y ∈ {ExpL, RP, Res}. Results are reported for EDF-based
PDs (columns marked “EDF”), RMI PDs (columns marked “RMI”) and rating-based PDs (columns marked “Rtg”),
and for alternative LGD specifications. The data include 414,816 firm-day observations for 240 firms and cover the
period from January 2001 to June 2010.
EDF
βYts,i
βYcs,t
Y
0.23
0.23
0.22
0.21
0.19
0.24
0.25
0.24
0.24
0.21
0.27
0.27
0.27
0.26
0.22
23
23
22
22
19
12
12
11
10
10
0.17
0.17
0.17
0.16
0.14
0.05
0.07
0.07
0.07
0.06
0.16
0.16
0.17
0.16
0.14
41
41
42
43
50
0.56
0.56
0.58
0.59
0.64
0.68
0.67
0.66
0.69
0.71
0.56
0.56
0.54
0.57
0.64
Y
Y
C
Benchmark
Historical
Markit
Rating-based
Regression-based
22
22
20
20
18
Benchmark
Historical
Markit
Rating-based
Regression-based
Benchmark
Historical
Markit
Rating-based
Regression-based
Y
C
RMI
βYts,i
βYcs,t
Rtg
βYts,i
βYcs,t
0.46
0.47
0.45
0.43
0.38
0.07
0.07
0.08
0.08
0.06
0.58
0.59
0.62
0.58
0.47
12
12
11
10
10
0.18
0.18
0.17
0.16
0.14
0.04
0.07
0.07
0.07
0.06
0.19
0.19
0.19
0.18
0.16
17
17
20
20
26
0.33
0.32
0.34
0.39
0.45
0.85
0.83
0.83
0.84
0.85
0.21
0.21
0.17
0.22
0.38
Y
Y
C
Expected losses
0.21 0.07 0.10
0.21 0.07 0.10
0.20 0.07 0.10
0.20 0.08 0.10
0.18 0.07 0.08
37
37
36
33
32
12
12
11
10
10
Credit risk premia
0.17 0.04 0.14
0.17 0.06 0.14
0.17 0.06 0.15
0.15 0.06 0.14
0.14 0.06 0.12
47
47
48
50
55
Residual spreads
0.57 0.85 0.74
0.57 0.83 0.74
0.58 0.83 0.70
0.60 0.84 0.74
0.65 0.85 0.78
Table A.6 reveals that except for 2002-03 and 2008-09, regressions-based Lit,1 estimates match
realized aggregate LGD at least as well if not better than any other LGD specification. This is
particularly noteworthy since between 2004 and 2007 and in 2010, regression-based LGD estimates
are significantly lower than other LGD estimates. It follows that for regression-based LGD, the
median model spread is somewhat smaller and the median residual spread is somewhat larger than
for other LGD, as documented in Table 13. Lower regression-based LGD estimates also dampen the
fraction of cross-sectional credit spread variation that reflects model spread variation, and increase
the fraction that signals residual spread variation. Overall, however, the results for the level and
variance decomposition of CDS spreads remain in the same ballpark, and hence are robust to a
wide range of LGD specifications.
9.2
Conditional covariance between loss given default and the default indicator
A common feature of the LGD specifications in Table 11 is that conditional on time-t information, there is no uncertainty about future LGD. I now extend the regression-based LGD specification
37
in Table 11 to allow for uncertainty:
e at+y,1 ,
Lit+k∆,∆ = L
for k =
y
y+1−∆
,...,
.
∆
∆
(34)
Both for regression-based LGD without uncertainty defined in Table 11 and regression-based
37 According to the exea
LGD with uncertainty defined in Equation (34), Et Lit+k∆,∆ = Et L
t+y,1 .
pected loss formula (8), this implies that expected losses for regression-based LGD with uncertainty
are equal to expected losses for regression-based LGD without uncertainty plus the scaled covariT
P∆
−1
i
δt,(k+1)∆ covt (Lit+k∆,∆ , Dt+k∆,∆
)/Ait . The scaled covariance term is approximately
ance term k=0
equal to
T
P∆
−1
k=0
i
δt,(k+1)∆ covt (Lit+k∆,∆ , Dt+k∆,∆
)
Ait
PT −1
≈
y=0
e i
ea
δt,y+1 covt (L
t+y,1 , Et Dt+y,1 )
Ait
,
(35)
et is defined in Section 5. The approximation is exact as long as default, should it occur in
where E
year (t + y, t + y + 1], occurs towards the end of the year.
Equations (24) and (33) yield
i
e at+y,1 , E
et Dt+y,1
covt (L
) =
δ R(i,t),y+1
R(i,t)
wt
2
) .
δ L,y+1 δ2D,y+1 vart (st+y+1 − st,y ) + var(D,y+1
t
R(i,t)
Descriptive statistics for these conditional covariances are reported in Table 14.38 δ R(i,t),y+1 /wt
ea
e i
and hence covt (L
t+y,1 , Et Dt+y,1 ), are generally higher for HY firms and lower for IG firms.
,
Table 14: Estimating the conditional covariance between LGD and the default indicator The table
i
e at+y,1 , E
et Dt+y,1
reports median estimates for covt (L
) in basis points. The data include 414,816 firm-day observations
for 240 firms and cover the period from January 2001 to June 2010. They are stratified by credit quality.
All firms
IG firms
HY firms
y=0
y=1
y=2
y=3
y=4
0.413
0.363
2.210
2.575
1.805
23.329
3.223
2.715
27.031
3.314
2.837
28.368
3.456
3.033
17.720
The level and variance decomposition results for regression-based LGD with uncertainty are
summarized in Table 15. As expected, extending the regression-based LGD specification to allow
for uncertainty in future LGD leads to an increase in expected losses due to the positive comovement
between future loss given default and default rates. For EDF-based PDs, median expected losses
are 24 basis points for regression-based LGD with uncertainty, compared to 18 basis points for
regression-based LGD without uncertainty. Measured as a fraction of CDS spreads, the associated
37
e at+y,1 is
I use the term “regression-based” in both instances, since “regression-based” refers to the fact that L
modeled via the regression equation in (33).
38
The conditional variance vart (st+y+1 − st ) is computed using Monte Carlo simulations of 100,000 sample paths
with antithetic sampling.
38
figures are 0.24 and 0.19. The median fraction of credit spread variation that corresponds to
expected loss variation is also higher when the covariance terms in (35) are taken into account,
cs,t
especially for the cross-sectional variation. The median βExpL
estimate is 0.25 for regression-based
LGD with uncertainty, compared to 0.22 for regression-based LGD without uncertainty. Overall,
however, the results remain in the same ballpark and the qualitative findings do not change.
Table 15: Accounting for covariance between LGD and the default indicator This table reports the median
estimates for Yti in basis points, Yti /Cti , βYts,i and βYcs,t , for Y ∈ {ExpL, RP, Res}. Results are reported for EDF-based
PDs (columns marked “EDF”), RMI PDs (columns marked “RMI”) and rating-based PDs (columns marked “Rtg”),
and for alternative LGD specifications including that in Equation (34) (rows marked “Regr-based w/ uncertainty”).
The data include 414,816 firm-day observations for 240 firms and cover the period from January 2001 to June 2010.
EDF
βYts,i
βYcs,t
Y
0.23
0.19
0.24
0.24
0.21
0.22
0.27
0.22
0.25
23
19
24
12
10
10
0.17
0.14
0.15
0.05
0.06
0.09
0.16
0.14
0.19
41
50
40
0.56
0.64
0.58
0.68
0.71
0.67
0.56
0.64
0.55
Y
Y
C
Benchmark
Regression-based
Regr-based w/ uncertainty
22
18
24
Benchmark
Regression-based
Regr-based w/ uncertainty
Benchmark
Regression-based
Regr-based w/ uncertainty
Y
C
RMI
βYts,i
βYcs,t
Rtg
βYts,i
βYcs,t
0.46
0.38
0.43
0.07
0.06
0.07
0.58
0.47
0.51
12
10
11
0.18
0.14
0.16
0.04
0.06
0.07
0.19
0.16
0.24
17
26
20
0.33
0.45
0.38
0.85
0.85
0.83
0.21
0.38
0.25
Y
Y
C
Expected losses
0.21 0.07 0.10
0.18 0.07 0.08
0.23 0.08 0.11
37
32
35
12
10
10
Credit risk premia
0.17 0.04 0.14
0.14 0.06 0.12
0.15 0.07 0.16
47
55
45
Residual spreads
0.57 0.85 0.74
0.65 0.85 0.78
0.58 0.84 0.71
To obtain credit risk premia for the LGD specification in Equation (34), I replace Equation (6)
with
Wti ≈
T
−1
X
T
−1
X
i
i
ea
et,y+1 Di
covt L
(
E
),
π
+
C
covt Dt,y
, πt,y+1 .
t,y+1
t+y,1
t+y,1
t
y=0
y=1
The second term is calculated according to Equation
(26), and Appendix C describes in detail how
a
e
et,y+1 Di
to compute the conditional covariances covt Lt+y,1 (E
t+y,1 ), πt,y+1 .
Table 15 reports the fraction of the level and the variation of CDS spreads due to credit risk
premia. I find that the regression-based LGD specification with uncertainty yields slightly higher
credit risk premia than the equivalent specification without uncertainty. Accounting for the covariance between LGD and the default indicator also leads to a somewhat larger fraction of credit
spread variation that corresponds to risk premium variation. Taken together, EDF-based expected
losses and credit risk premia, at the median, account for 42% of the level and 45% of the crosssectional variation of credit spreads when the LGD specification with uncertainty is used, compared
to 36% of the level and cross-sectional variation when LGD uncertainty is not permitted. While
the magnitude of these estimates increases somewhat when the covariance between Lit+k∆,∆ and
39
i
Dt+k∆,∆
is accounted for, Table 15 confirms that my findings are qualitatively robust to covariance
effects.
10.
Sources of Variation in Residual Spreads
To gain a better understanding of the sources of variation in residual spreads, I estimate panel
regressions of the form
Resit = b0 + Xti bX + zti ,
(36)
where Xti is a state vector of firm-specific and macro-economic variables. The predictor variables
are chosen by identifying potential sources of variation in credit spreads that cannot be explained
by variation in measured expected losses or credit risk premia. They are described below.
10.1
Potential mismeasurement of model spreads
Although PD measures such as EDFs and RMI PDs include the impact of equity volatility
through distance to default, they could suffer from functional misspecification and noise. Hence an
additional direct measure of expected future equity volatility—such as a firm’s implied volatility
(IVti )—may help control for variation in expected losses.
Furthermore, if volatility risk is priced, the survival-conditional mark-to-market credit risk
premium should be higher for more volatile firms, even after controlling for expected losses. Since
implied volatility includes a risk premium for volatility, it may capture variation in credit risk
premia that may have been ignored by the model in Section 5.
In addition to IVti , sector-fixed effects are included to account for potential cross-sectional
differences in senior unsecured recovery rates. What I have in mind are selection effects—such as
differences across sectors in subordination protection for senior bonds—that may cause the liability
stack to be structured differently in one sector than another.
10.2
Supply and demand of risk capital
One explanation for the temporal variation in CDS spreads, even after controlling for expected
losses and credit risk premia, may be based on variations over time in the supply of, and demand
for, risk bearing, potentially exacerbated by limited mobility of capital across different classes
of asset markets. Along the lines of the explanation suggested by Froot and O’Connell (1999)
for time variation in catastrophe insurance risk premia, and the arguments in Berndt, Douglas,
Duffie, Ferguson, and Schranz (2005), capital moves into and out of credit markets in response to
fluctuations in risk premia, but perhaps not instantaneously so. Generally, when there are large
losses or large increases in risk in a particular market segment, if capital does not move immediately
out of other asset markets and into that segment, then prices would tend to adjust so as to match
the demand for capital with the supply of capital that is available to the segment. Investors or asset
managers with available capital take time to be found by intermediaries, to be convinced (perhaps
40
being unfamiliar with the particular asset class) of the available risk premia, and to exit from the
markets in which they are currently invested. For the catastrophe risk insurance market, Froot and
O’Connell (1999) show that this process can take well over a year, in terms of the half-life of the
mean reversion of risk premia to long-run levels. Similar explanations, albeit with shorter half-lives,
have been offered by Gabaix, Krishnamurthy, and Vigneron (2007) for variation of prepayment risk
premia in the market for mortgage-backed securities, and by Greenwood (2005) for the price impact
of supply shocks in equity markets. Duffie and Strulovici (2012) present a theoretical model for the
movement of capital between asset markets.
In order to explore the role of limited capital mobility in determining residual spreads, I follow Berndt, Douglas, Duffie, Ferguson, and Schranz (2005) and control for stock-market volatility,
as measured by V IXt . V IXt is a popular measure of the implied volatility of S&P 500 index
options. As it goes up, a given level of capital available to bear risk represents less and less capital
per unit of risk to be borne. If replacement capital does not move into credit markets immediately,
the supply and demand for risk capital will match at a higher price per unit of risk. This effect
would be present even with perfect capital mobility, but the magnitude of the effect is increased
with partially segmented markets.
10.3
Corporate bond and CDS market liquidity effects
Bühler and Trapp (2009) present evidence that illiquidity in corporate bond markets has an
impact on CDS pricing. They argue that, everything else the same, CDS spreads increase as the
underlying corporate bonds become more illiquid and, as a result, expected LGD becomes larger.
I therefore include the average monthly U.S. corporate bond trading volume, V olt , as a predictor
variable.39
A number of recent studies, including, Tang and Yan (2008), Chen, Fabozzi, and Sverdlove
(2010), Bongaerts, de Jong, and Driessen (2011), Arakelyan and Serrano (2012), and Chen, Cheng,
and Wu (2013), also find significant CDS market liquidity effects. I use the aggregate CDS-bond
basis, which measures the difference between CDS spreads and cash-bond implied credit spreads,
for investment-grade firms, Basist , to capture relative CDS market liquidity.40 Both V olt and
Basist are available from 2005 onwards.
10.4
Market microstructure effects
I explore the notion that there are many investors who will consider buying an IG bond, but
that there are fewer investors competing to buy speculatively rated debt. What I have in mind are
not only portfolio holdings regulations and risk-based capital requirements for financial institutions
but also supply and demand effects associated with different types of investors.41 Unsophisticated
investors may not trust their information as much as professional investors in the junk bond market,
39
U.S. corporate bond trading volume data are available at sifma.org.
I thank Mark Mitchell for time-series data on the CDS-bond basis.
41
I thank Darrell Duffie for pointing me in this direction.
40
41
and they may not know as much about managing credit risk, even if they were to get fair pricing. As
a result, less sophisticated investors may prefer higher rated bonds of a given expected loss more
than lower rated bonds of the same expected loss. This would imply a permanent rating-based
investor clientele effect, in the sense that even after controlling for expected losses, credit spreads
would be higher for lower rated debt. To control for such a clientele effect, I include rating dummies
as predictor variables.
Ambrose, Cai, and Helwege (2012), Chen, Lookman, Schürhoff, and Seppi (2012) and Ellul,
Jotikasthira, and Lundblad (2012), among others, analyze price pressure effects in corporate bond
markets associated with the rating downgrades from IG to HY status. Chen, Lookman, Schürhoff,
and Seppi (2012) point to the price pressure associated with the rating downgrade of General
Motors in 2005. Ellul, Jotikasthira, and Lundblad (2012) document a large drop in bond prices
around the time of a downgrade, and show that a large fraction of the decline in prices is reversed
within 20 to 30 weeks, which may indicate that any permanent effects associated with information
were already available to sophisticated bond investors well before the downgrade. Ambrose, Cai,
and Helwege (2012), on the other hand, are reluctant to interpret the findings in Ellul, Jotikasthira,
and Lundblad (2012) as evidence of price pressure associated with “forced sales” of downgraded
bonds, holding the view that most of the price impact of a downgrade is based on the adverse credit
information associated with the downgrade.
To control for potential temporary price pressure effects, I include a dummy variable that
is turned on for six months following a downgrade of a firm from investment-grade to high-yield
status, 1DG (i, t). Any permanent rating-based clientele effect that I may find need not necessarily be
accompanied by a temporary price pressure effect. That is, if bond markets efficiently re-allocate
the corporate bonds that are sold on ratings downgrades by restricted investors to unrestricted
investors, there would be an immediate and unreversed increase in credit spreads, even if there
is no actual change in default risk, because of the sudden downward shift in the demand curve
for HY bonds. Similarly, any temporary price pressure effects do not have to be accompanied by
permanent clientele effects.
10.5
Estimation results
In summary, the base state vector Xti consists of IVti , V IXt , 1DG (i, t), V olt , Basist , and rating
and sector dummies. I also control for linear and non-linear functions of expected losses and credit
risk premia. The panel regression results are shown in the second panel of Table 16. They are based
on Driscoll and Kraay (1998) standard errors that are robust to heteroskedasticity, and temporal
and cross-sectional correlation.
For benchmark PDs and LGD, I find that 35% of the variation in residual spreads—over time
and across firms—is statistically explained by variation in the state vector Xti . For the full sample
period, January 2001 to June 2010, all predictor variables are statistically significant and have the
expected sign. All else the same, higher firm-specific implied volatility is associated with higher
residual spreads, as is higher market-wide volatility. The data are consistent both with a permanent
42
Table 16: Sources of variation in residual spreads This table summarizes the results for the panel regression
in (36), using Driscoll and Kraay (1998) standard errors. Residual spreads, expected losses, credit risk premia and
Basist are measured in basis points, IVti and V IXt are measured in percent, and V olt is measured in billions USD.
Results are reported for EDF-based PDs (columns marked “EDF”), RMI PDs (columns marked “RMI”) and ratingbased PDs (columns marked “Rtg”), and for benchmark loss given default. The data in the first three columns
cover the period from January 2001 to June 2010 and consist of 400,486 firm-day observations for 234 firms (396,174
observations for RMI PDs), whereas the data in the last three columns consist of 268,108 firm-day observations for
231 firms (264,828 observations for RMI PDs). T-statistics are reported in parentheses.
EDF
Constant
IVti − V IXt
V IXt
R2
1/2001-6/2010
RMI
Rtg
-87.10
(-13.08)
7.41
(22.11)
2.77
(10.39)
0.277
-177.01
(-32.82)
11.78
(40.86)
4.93
(21.78)
0.482
-236.38
(-30.63)
7.00
(15.81)
7.41
(23.70)
0.272
EDF
1/2005-6/2010
RMI
Rtg
-81.14
(-9.57)
6.72
(17.63)
2.65
(8.13)
0.235
-176.38
(-26.38)
11.12
(34.04)
4.88
(20.32)
0.455
-237.30
(-24.55)
6.88
(13.21)
6.95
(25.74)
0.269
Adding (ExpLit )k and (RPti )k , k = 1, 2, 3, rating and sector
dummies, 1DG (i, t), V olt , and Basist
Constant
-114.35
(-22.64)
8.72
(26.19)
4.98
(22.71)
-14.17
(-3.35)
-21.53
(-8.98)
52.68
(5.21)
110.32
(6.00)
343.57
(13.63)
45.95
(3.09)
-163.16
(-29.23)
8.94
(35.81)
5.91
(19.96)
-15.24
(-2.75)
-21.73
(-6.27)
103.44
(7.01)
183.30
(6.95)
459.38
(15.16)
15.31
(1.45)
-176.51
(-27.41)
10.99
(42.63)
5.80
(26.15)
5.69
(1.19)
-7.06
(-2.23)
-22.34
(-1.74)
-144.78
(-5.57)
-145.86
(-3.90)
67.50
(4.04)
R2
0.350
0.572
0.490
R2
0.517
0.692
R2
0.535
0.706
IVti − V IXt
V IXt
1Aaa−Aa (i, t)
1A (i, t)
1Ba (i, t)
1B (i, t)
1Caa−C (i, t)
1DG (i, t)
-6.65
(-0.28)
8.56
(32.04)
5.32
(10.05)
-6.83
(-1.05)
-22.71
(-5.45)
68.71
(3.86)
121.40
(4.17)
367.04
(11.10)
23.40
(1.69)
-9.89
(-6.67)
-0.10
(-1.22)
0.565
-17.52
(-0.78)
10.44
(38.34)
4.89
(10.16)
18.02
(3.42)
-5.45
(-1.62)
-65.38
(-4.47)
-211.36
(-7.21)
-249.86
(-5.99)
81.98
(3.56)
-10.60
(-7.61)
-0.08
(-1.02)
0.541
0.725
0.670
Adding monthly dummies
0.642
0.567
0.731
0.678
V olt
Basist
30.65
(1.43)
8.64
(23.18)
4.51
(10.10)
7.34
(1.70)
-14.06
(-4.89)
-4.30
(-0.37)
31.54
(1.58)
232.19
(8.45)
59.56
(3.02)
-10.67
(-8.49)
-0.04
(-0.55)
0.336
Adding firm dummies*
0.625
0.558
*When controlling for firm dummies, sector dummies are removed.
rating-based clientele effect and a temporary price pressure effect. The top panel in Table 16 reveals,
however, that most of the explanatory power for residual spreads—up to 28%—stems from firm43
specific and market-wide volatility. For later part of the sample period, January 2005 to June 2010
when V olt and Basist are available, a marginal increase in bond market liquidity is associated
with a significant decrease in residual spreads. The CDS-bond basis, however, is not statistically
significant.
Allowing for firm-fixed effects yields a dramatic increase in the percentage of residual spread
variation explained: From 35% to 52% for the full sample, and from 34% to 56% for the 20056/2010 subsample. But if I further include monthly dummy variables, the R2 remains nearly the
same, implying that the predictor variables in Xti are successful in capturing most of the common
time-series variation in residual spreads.
When EDFs are replaced by RMI PDs, variation in Xti explains 57% of the variation in residual
spreads. As for EDFs, a large fraction of the variance explained is due to IVti and V IXt . For RMI
PDs, however, the temporary price pressure effect appears to be much smaller than for EDFs, and
it is no longer statistically significant. For rating-based PDs, the state vector explains 49% of the
variation in residual spreads, out of which 27% can be captured by firm-specific and market-wide
volatility alone. Since rating dummies are directly linked to rating-based expected losses, they can
no longer be used to make statements about a possible rating-based clientele effect. Nevertheless,
I find that even after controlling for credit ratings, residual spreads for a given firms tend to be
higher during the six months following a downgrade from IG to HY status.
11.
Conclusion
In this paper, I analyze the information that is revealed through the prices that investors charge
for bearing corporate default risk. I quantify what fraction of these prices reflects compensation
for expected losses and what fraction reflects compensation for credit risk premia, both in terms of
levels and variation. To do so, I decompose 5-year CDS spreads into an expected loss component, a
credit risk premium component and a residual component. For EDF-based PDs and constant LGD,
the median fraction of CDS spreads due to expected losses is 0.23. Less than one quarter of the
temporal credit spread variation, and less than 30% of the cross-sectional credit spread variation,
corresponds to expected loss variation.
I decompose credit risk premia into jump-to-default risk premia and survival-conditional markto-market credit risk premia, and use the Chen, Collin-Dufresne, and Goldstein (2009) adaptation
of the Campbell and Cochrane (1999) pricing kernel to obtain a direct measure of both risk premia
components. I find that jump-to-default risk premia are minuscule, and that most of the credit
risk premia are due to mark-to-market risk premia. The median fraction of credit spreads due
to credit risk premia is 0.17. Taken together, expected losses and credit risk premia account for
44% of the level of credit spreads. In that sense, the data are consistent with a sizable residual
component. I also find pronounced positive comovement between credit spreads and the residual
component. The median percentage of the time-series credit spread variation that corresponds to
residual spread variation is 67%, and the median percentage of the cross-sectional credit spread
44
variation that corresponds to residual spread variation is 56%.
My findings give rise to a credit spread level and volatility puzzle in CDS markets. Importantly,
I show that the extent of the puzzle is rather sensitive to the choice of the PD measure. I document
that the level puzzle is more pronounced for forward-looking firm-by-firm PDs than for backwardlooking rating-based PDs, and that ratings do a better job in aligning the cross-sectional ranking
of actual and standard-model-implied spreads, whereas EDFs do a better job in aligning the timeseries ranking. This offers new insights for the broader credit spread puzzle literature, which so
far relies almost exclusively on rating-based PDs when calibrating structural models to physical
default probabilities.
I elaborate on the level puzzle, and the importance of the choice of the PD measure, by exploring
the notion of market segmentation. I show that model spreads can be reconciled with actual
spreads only if CDS market investors are more risk averse than equity market investors. But the
implied discrepancy between risk aversion in equity and credit markets is more pronounced when
model spreads are calibrated to forward-looking firm-by-firm PDs than when they are calibrated
to historical rating-based PDs.
My findings are qualitatively robust to a wide range of alternative loss given default specifications, and to covariance effects between LGD and the default indicator. I build a prediction model
for residual spreads and show that much of their variation—both over time and across firms—can
be statistically explained by firm-specific and market-wide equity-option-implied volatility. After
controlling for benchmark expected losses, I also find evidence of permanent rating-based clientele
and temporary price pressure effects in CDS markets.
45
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48
A.
Additional Figures and Tables
14
12
Percent
10
8
6
4
2
0
0
5
10
15
20
25
30
Composite depth
Figure A.1: Distribution of composite depth The figure shows the composite depth distribution of 5-year
Markit CDS quotes. The data include 414,816 firm-day observations for 240 firms and cover the period from January
2001 to June 2010.
700
Finance, Insurance, Real estate
Information
Manufacturing
Mining, Utilities, Construction
Trade, Transportation
IG firms
HY firms
900
600
800
700
500
Basis points
Basis points
600
500
400
300
400
300
200
200
100
100
0
Dec01
Dec03
Dec05
Dec07
Dec09
0
Dec01
Dec03
Dec05
Dec07
Dec09
Figure A.2: Median CDS spreads by credit quality and sector The left panel shows the daily median 5-year
CDS spreads for IG and for HY firms. The right panel displays the daily median 5-year CDS spreads by sector. In
the left (right) panel, only days for which CDS quotes are available for 30 (10) or more firms are shown. The data
include 414,816 firm-day observations for 240 firms and cover the period from January 2001 to June 2010.
49
Downgrades: Alphanum rating
70
60
60
50
50
Percent
Percent
Upgrades: Alphanum rating
70
40
30
40
30
20
20
10
10
0
2001
2003
2005
2007
0
2009
70
60
60
50
50
40
30
20
10
2003
2005
2007
2007
2009
30
10
2001
2005
40
20
0
2003
Downgrades: Refined rating
70
Percent
Percent
Upgrades: Refined rating
2001
0
2009
2001
2003
2005
2007
2009
Figure A.3: Rating upgrades and downgrades The figure shows the average annual number of rating upgrades
(left panel) and rating downgrades (right panel) per firm. The top panel is based on Moody’s alphanumeric rating
whereas the bottom panel is based on the refined rating. The data include 240 firms and cover the period from
January 2001 to June 2010.
35
Aaa−Aa
A
Baa
Ba
B
Caa
30
Percent
25
20
15
10
5
0
1990
1992
1994 1996 1998 2000 2002 2004 2006
Year at the beginning of which cohort was formed
2008
2010
Figure A.4: Rating cohort weights The data are available from Moody’s annual “Corporate Default and Recovery
Rate” studies and cover the period from 1991 to 2010.
50
600
5−year−ahead aggregate default rate
5−year−ahead aggregate RMI PD (standardized)
500
Basis points
400
300
200
100
0
−100
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
Figure A.5: 5-year-ahead aggregate default rates and RMI forecasts The figure shows the time series of 5a
a
a
a
e t,5
e t,4
year-ahead aggregate default rates, D
−D
, and the associated 5-year-ahead aggregate RMI PDs, RM It,5
−RM It,4
(standardized). The shaded areas indicate NBER recessions. The data cover the period from 1991 to 2010.
600
Aggregate default rate
Aggregate LGD (standardized)
500
Basis points
400
300
200
100
0
−100
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
Figure A.6: Aggregate default rates and LGD The figure shows the time series of 1-year-ahead aggregate
a
e t,1
e at,1 (standardized). The shaded areas indicate
default rates, D
, and of 1-year-ahead aggregate loss given default, L
NBER recessions. The data cover the period from 1991 to 2010.
51
Table A.1: Ex-ante versus ex-post rating-based default rates The table reports ex-ante and ex-post rating
based 1- and 5-year annualized default rates in basis points, for cohorts of Ba and B rated firms. The ex-ante default
rates for year t are based on the average cumulative issuer-weighted global default rates by letter rating from 1983
to year t − 1, for t = 2006-10, and from 1970 to year t − 1, for t = 2001-05. These data are extracted from Moody’s
annual “Corporate Default and Recovery Rates” studies. The ex-post default rates for year t are computed using the
cumulative issuer-weighted global default rates for the rating cohort formed at the beginning of year t (see Moody’s
Investors Service (2014)).
Cohorts
formed
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
Ba
1yr
B
5yr
1yr
Ba
5yr
Ex-ante default rates
127 235 616
634
127 240 666
715
128 238 651
751
126 234 621
742
122 224 581
701
131 249 569
681
127 236 526
617
117 224 466
560
115 219 433
546
121 240 455
591
1yr
B
5yr
1yr
5yr
Ex-post default rates
151 135 995
733
152 148 467
404
105
68 219
217
45
86
88
186
0 206 105
442
22 225 121
461
0 193
0
510
122 194 214
488
233 115 753
301
0 55*
51
128*
*Based on 4-year cumulative default rates.
Table A.2: Specifying the link between rating cohort and aggregate default rates The left panel reports
the R2 s for the regression in (15), and the right panel reports the R2 s when the dependent variable is replaced by
R
R
R
e t,y+1
e t,y
e t,y
D
−D
. The values for D
are observed as the realized cumulative y-year issuer-weighted default rates by
a
a
e t,y+1
e t,y
annual letter rating cohort, and D
−D
is computed according to Equation (14). The data include annual
observations from 1991 to 2010 and are obtained from Moody’s annual “Corporate Default and Recovery Rates”
studies.
Aaa-Aa
A
Baa
Ba
B
Caa or lower
y=0
eR
eR
wtR D
t,y+1 − Dt,y
y=1 y=2 y=3
y=4
0.003
0.227
0.550
0.260
0.629
0.764
0.001
0.375
0.594
0.608
0.880
0.695
0.036
0.162
0.781
0.618
0.949
0.715
0.000
0.344
0.803
0.496
0.952
0.807
0.017
0.276
0.750
0.702
0.934
0.752
52
y=0
eR
eR
D
t,y+1 − Dt,y
y=1 y=2 y=3
y=4
0.004
0.209
0.596
0.421
0.426
0.538
0.001
0.390
0.584
0.733
0.683
0.674
0.021
0.170
0.686
0.763
0.851
0.739
0.000
0.337
0.784
0.630
0.883
0.714
0.006
0.271
0.724
0.758
0.911
0.579
Table A.3: Median credit risk premium component and risk-premium-to-CDS ratio: RMI PDs The
table reports the median jump-to-default component (columns marked “JtD”), survival-conditional mark-to-market
credit risk premium component (columns marked “MtM”) and overall credit risk premium component (columns
marked “RP”) of 5-year CDS spreads in basis points, and the associated median component-to-CDS ratios. The
results are for RMI PDs and benchmark loss given default. The data include 414,816 firm-day observations for 240
firms and cover the period from January 2001 to June 2010. They are stratified by year and credit quality.
JtD
All firms
M tM
RP
All
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
0
0
0
0
0
0
0
0
1
1
1
11
6
7
8
7
7
7
9
16
26
24
12
6
7
8
8
7
8
10
17
27
25
All
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
0.01
0.00
0.00
0.00
0.01
0.01
0.01
0.01
0.01
0.01
0.02
0.16
0.04
0.06
0.12
0.16
0.17
0.20
0.22
0.14
0.25
0.33
0.17
0.04
0.06
0.12
0.16
0.17
0.21
0.23
0.15
0.27
0.35
JtD
0
0
0
0
0
0
0
0
1
1
1
IG firms
M tM
RP
In basis points
7
7
6
6
6
7
8
8
7
7
7
7
7
7
8
8
13
13
21
22
23
25
As a fraction of CDS spreads
0.00
0.13
0.13
0.00
0.04
0.04
0.00
0.05
0.05
0.00
0.10
0.11
0.00
0.13
0.13
0.00
0.13
0.14
0.01
0.17
0.17
0.01
0.19
0.20
0.00
0.11
0.11
0.01
0.19
0.20
0.01
0.23
0.24
53
JtD
HY firms
M tM
RP
3
1
1
1
1
1
1
1
3
9
15
77
27
35
38
36
35
36
46
80
210
301
80
28
36
39
37
36
37
47
83
220
316
0.01
0.00
0.00
0.01
0.01
0.01
0.01
0.01
0.01
0.02
0.03
0.25
0.07
0.09
0.15
0.22
0.24
0.28
0.28
0.20
0.35
0.58
0.26
0.07
0.10
0.15
0.23
0.25
0.30
0.29
0.21
0.37
0.62
Table A.4: Median credit risk premium component and risk-premium-to-CDS ratio: Rating-based
PDs The table reports the median jump-to-default component (columns marked “JtD”), survival-conditional markto-market credit risk premium component (columns marked “MtM”) and overall credit risk premium component
(columns marked “RP”) of 5-year CDS spreads in basis points, and the associated median component-to-CDS ratios.
The results are for rating-based PDs and benchmark loss given default. The data include 414,816 firm-day observations
for 240 firms and cover the period from January 2001 to June 2010. They are stratified by year and credit quality.
JtD
All firms
M tM
RP
All
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
0
0
0
0
0
0
0
0
1
1
1
11
6
7
8
7
7
7
9
16
26
24
12
6
7
8
8
7
8
10
16
27
25
All
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
0.01
0.00
0.00
0.00
0.01
0.01
0.01
0.01
0.01
0.01
0.02
0.17
0.04
0.06
0.12
0.16
0.17
0.21
0.23
0.15
0.27
0.35
0.18
0.04
0.06
0.12
0.17
0.18
0.22
0.24
0.15
0.28
0.37
JtD
0
0
0
0
0
0
0
0
1
1
1
IG firms
M tM
RP
In basis points
7
8
6
6
6
7
8
8
7
7
7
7
7
7
8
8
13
13
21
22
23
25
As a fraction of CDS spreads
0.00
0.13
0.13
0.00
0.04
0.04
0.00
0.05
0.05
0.00
0.10
0.11
0.00
0.13
0.13
0.00
0.14
0.14
0.01
0.17
0.17
0.01
0.19
0.20
0.00
0.11
0.11
0.01
0.19
0.20
0.01
0.23
0.24
54
JtD
HY firms
M tM
RP
3
1
1
1
1
1
1
2
3
10
16
86
28
37
41
38
37
39
48
83
251
335
90
29
38
42
39
39
41
50
86
262
352
0.01
0.00
0.00
0.01
0.01
0.01
0.01
0.01
0.01
0.02
0.03
0.28
0.07
0.10
0.17
0.25
0.28
0.33
0.32
0.22
0.40
0.66
0.29
0.07
0.10
0.17
0.25
0.29
0.34
0.33
0.23
0.42
0.70
Table A.5: Actual versus model spreads The table reports the median 5-year CDS spreads in basis points and
the median associated model spreads. Results are reported for EDF-based PDs (columns marked “EDF”), RMI PDs
(columns marked “RMI”) and rating-based PDs (columns marked “Rtg”), and for benchmark loss given default. The
data include 414,816 firm-day observations for 240 firms and cover the period from January 2001 to June 2010. They
are stratified by year and rating cohort.
Actual
CDS
Model spreads
EDF RMI Rtg
Aaa-Aa
6
13
All
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
23
3
36
26
18
14
10
13
20
17
6
2
1
1
13
20
15
7
7
9
4
3
3
2
2
2
All
2002
2003
2004
2005
2006
2007
2008
2009
2010
200
391
290
173
148
125
156
293
269
205
Ba
76
101
85
67
54
52
52
97
252
225
73
62
68
60
56
67
65
90
160
176
155
154
157
152
145
139
143
158
207
227
Actual
CDS
Model spreads
EDF RMI Rtg
Actual
CDS
35
68
64
33
26
24
20
22
65
66
58
A
15
34
28
23
13
9
7
7
14
40
37
19
26
21
24
15
11
13
15
20
38
40
12
10
12
11
13
11
11
11
13
20
23
66
142
114
69
53
47
40
42
112
106
90
355
B
179
154
525
834
487
321
256
221
272
568
611
433
308
176
123
109
105
258
637
510
166
137
113
116
122
189
361
396
563
527
486
441
398
432
629
682
783
565
816
505
467
1,518
1,457
777
55
Model spreads
EDF RMI
Rtg
Baa
26
55
44
40
24
17
14
14
25
64
57
29
38
30
35
23
19
20
23
31
54
57
36
30
34
36
35
34
34
33
38
50
50
Caa
538
331
1,086
559
460
366
264
221
620
1,103
926
259
271
241
200
232
391
475
495
1,164
1,179
1,096
1,020
872
912
1,350
1,094
Table A.6: Descriptive statistics for alternative LGD specifications The table reports median values for
Lit+y,1 , for alternative LGD specifications and different strata of the data. The data include 414,816 firm-day observations for 240 firms and cover the period from January 2001 to June 2010. The bottom panel reports one minus
realized issuer-weighted senior unsecured corporate bond recovery rates, measured by post-default trading prices, as
reported in Moody’s Investors Service (2014).
2001
2002
2003
2004
0.53
0.63
0.63
0.64
Finance, Insurance, RE / IG
Finance, Insurance, RE / HY
Information / IG
Information / HY
Manufacturing / IG
Manufacturing / HY
Mining, Utilities, Constr / IG
Mining, Utilities, Constr / HY
Trade, Transportation / IG
Trade, Transportation / HY
Others / IG
Others / HY
0.55
–
0.56
–
0.57
–
0.56
–
0.58
0.63
0.56
0.57
0.59
–
0.59
0.68
0.59
0.58
0.58
0.60
0.59
0.61
0.59
0.60
0.59
0.60
0.60
0.63
0.60
0.60
0.60
0.60
0.60
0.62
0.60
0.61
0.62
0.60
0.62
0.63
0.61
0.61
0.60
0.62
0.60
0.62
0.61
0.62
y
y
y
y
0.53
0.53
0.53
0.53
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.54
0.65
0.59
0.61
y=0
y=4
0.59
0.52
0.57
0.51
y=0
0.79
0.70
=0
=0
=4
=4
/
/
/
/
IG firms
HY firms
IG firms
HY firms
2007
2008
2009
2010
Historical LGD
0.64
0.64
0.62
0.63
0.64
0.63
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.61
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.63
0.60
0.65
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.62
0.60
0.60
Rating-based LGD
0.54
0.51
0.51
0.62
0.65
0.64
0.57
0.56
0.54
0.61
0.58
0.58
0.52
0.62
0.55
0.56
0.58
0.64
0.57
0.59
0.60
0.64
0.56
0.60
0.50
0.53
Regression-based LGD
0.48
0.49
0.48
0.48
0.54
0.53
0.54
0.54
0.55
0.52
0.59
0.50
0.53
0.57
0.58
0.48
Realized LGD
0.45
0.45
0.67
0.63
0.49
56
2005
Markit
0.60
0.60
0.61
0.61
0.61
0.61
0.61
0.60
0.60
0.60
0.60
0.60
2006
LGD
0.60
0.60
0.60
0.61
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.46
B.
i
ea
ea
The Pitfalls of Linking Dt+y,1
Directly to D
t,y+1 − Dt,y
Consider the model
i,y
i
ea
ea
Dt+y,1
= η0i,y + η i,y (D
t,y+1 − Dt,y ) + ut ,
(B.1)
i,y
2
where ui,y
t is assumed to have a constant conditional variance σu = vart (ut ), and to be condiea
ea
tionally independent of future aggregate default rates, D
t,y+1 − Dt,y , and the state price density
process. Equation (B.1) has the drawback that it links a binomial variable on the left-hand side to
an affine function of multinomial, or nearly continuous, variables on the right-hand side, but it is
appealing due to its simplicity. Given Equation (B.1), the parameter of interest is η i,y , as
i
ea
ea
covt (Dt+y,1
, πt,y+1 ) = η i,y covt (D
t,y+1 − Dt,y , πt,y+1 ).
Equation (B.1) implies
a
a
i
e t,y+1
e t,y
−D
) + Et ui,y
Et Dt+y,1
= η0i,y + η i,y Et (D
t .
(B.2)
i
ea
ea
If one were to estimate η i,y by regressing Et Dt+y,1
on Et (D
t,y+1 − Dt,y ), as might seem appealing
at first glance, one would ignore the time variation in Et ui,y
t . For example, consider the scenario
i,y
i,y
i,y
i,y
a
a
i
e
e
where η = 0 and ut = Et (D
t,y+1 − Dt,y ) + (vt − Et vt ) in Equation (B.1). Then Et Dt+y,1 =
ea
e a ), and regressing Et Di
ea
e a ) would result in an estimate
−D
on Et (D
−D
η i,y + Et (D
0
t,y+1
t,y
t+y,1
t,y+1
t,y
of η i,y = 1!
Equation (B.1) also implies
i
a
a
e t,y+1
e t,y
vart (Dt+y,1
) = (η i,y )2 vart (D
−D
) + σu2 .
(B.3)
i
ea
Given Equation (B.3), one may be tempted to estimate η i,y by regressing vart (Dt+y,1
) on vart (D
t,y+1
a ). I explore this approach and perform these regressions after computing var (D i
e t,y
−D
t
t+y,1 ) as
i
i
a
a
e
e
Et (Dt+y,1 )(1 − Et (Dt+y,1 )), and vart (D
t,y+1 − Dt,y ) using the specification of aggregate default
rates in Equation (24). In addition to obtaining estimates only for |η i,y | and not η i,y , I find that the
regressions yield negative (η i,y )2 estimates for some of the firms. And when the (η i,y )2 estimates
are positive, they are often unreasonably large. The reason is that because of the binomial nature
i
i
i
of the default indicator, its conditional covariance vart (Dt+y,1
) = Et (Dt+y,1
)(1 − Et (Dt+y,1
)) is on
i
the order of Et (Dt+y,1 ), whereas the conditional covariance of realized aggregate default rates is
roughly on the order of aggregate default rates squared. This reinforces the notion that modeling
a binomial variable as an affine function of nearly continuous variables as in Equation (B.1) may
not be a viable approach.
i
ea
ea
In summary, while linking Dt+y,1
directly to D
t,y+1 − Dt,y in a simple way that lends itself to
i
computing covt (Dt+y,1
, πt,y+1 ) may seem appealing, there are clear drawbacks to calibrating such
i
ea
ea
a link to the conditional expectations or variances for Dt+y,1
and D
t,y+1 − Dt,y .
57
C.
Credit Risk Premia Computations for LGD Specification With Uncertainty
To compute credit risk premia for the LGD specification in Equation (34), I replace Equation (6)
with
Wti ≈
T
−1
X
T
−1
X
i
i
e at+y,1 (E
et,y+1 Dt+y,1
), πt,y+1 + Cti
covt Dt,y
, πt,y+1
covt L
y=1
y=0
=
T
−1
X
covt Zti,y+1 , πt,y+1 + Cti
y=1
T
−1
X
i
covt Dt,y
, πt,y+1 .
y=1
i,y+1
i ,π
The terms covt Dt,y
is given by
t,y+1 are computed as in Equation (26), and Zt
Zti,y+1
=
δ
δ0L,y+1
!
R(i,t),y+1
R(i,t)
wt
+
ai,y+1
δ L,y+1
t
2
δ R(i,t),y+1 e a
a
a
a
e t,y+1
e t,y
e t,y
D
−D
+ δ L,y+1
D
−
D
,
t,y+1
R(i,t)
wt
where
i
ai,y+1
= Et Dt+y,1
+
t
δ0R,y+1
R(i,t)
wt
e R(i,t) − D
e R(i,t) .
− Et D
t,y
t,y+1
D,y+1
ea
ea
covt (st+y+1 − st , πt,y+1 ), and that
Note that covt D
t,y+1 − Dt,y , πt,y+1 = δ2
covt
a
a
e t,y+1
e t,y
D
−D
2
, πt,y+1
= 2aD,y+1
δ2D,y+1 covt (st+y+1 − st , πt,y+1 )
t
+ (δ2D,y+1 )2 covt (st+y+1 − st )2 , πt,y+1 ,
a
a
− RM It,y
− δ2D,y+1 Et (st+y+1 − st ). As before, the
where aD,y+1
= δ0D,y+1 + δ1D,y+1 RM It,y+1
t
conditional covariance terms covt (st+y+1 − st , πt,y+1 ) are computed using Monte Carlo simulations
of 100,000 sample paths with antithetic sampling. The same applies to covt (st+y+1 − st )2 , πt,y+1 .
58