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Decomposing the Returns on European Debt1
Antje Berndt2 and Iulian Obreja3
Current version: March 13, 2008
Abstract
Common variation in the prices of European corporate debt may not always
be associated with a rational response to an increase in the relative importanceof a macroeconomic risk factor. Building on Campbell’s ICAPM framework,
we show that risk premia of assets with nonlognormal return distributions represent compensation not only for exposure to macroeconomic factors but also
for unexpected revisions to these assets’ return distributions, such as sudden
increases in the likelihood of extreme events. If such revisions happen acrossassets almost simultaneously, perhaps as a systemic response to a large credit
event, they can induce covariation in risk premia unrelated to the time variation of the priced macroeconomic factors. Our study presents evidence from theEuropean debt markets which supports this theory. The asset pricing tests also
document patterns consistent with the “flight to quality” effect for European
corporate bonds.

JEL Classifications: G12, G13, G15
Keywords: European credit and bond markets, decomposing returns, risk factors
1

We are grateful to Jan Ericsson, Rick Green, Philipp Hartmann, Burton Hollifield, Andrew
Karolyi, and Frank Packer for extended discussions.
2
Antje Berndt is with the Tepper School of Business at Carnegie Mellon University.
3
Iulian Obreja is with the Leeds School of Business at the University of Colorado at Boulder.
Iulian Obreja acknowledges financial support from the Lamfalussy Fellowship Program sponsored by
the European Central Bank. Views expressed are the authors’ alone and do not represent the viewsof the ECB or the Eurosystem. Corresponding author: Iulian Obreja (iulian.obreja@colorado.edu).

1

1.

Introduction

This paper investigates the sources of common time-series variation in the prices ofEuropean corporate debt. Standard asset pricing models state that to the extent
that CDS spreads represent compensation for exposure to systematic risk factors,common variation in default swap spreads arises mainly because the price of risk of asystematic factor increases relative to the price of risk of other systematic factors. In
other words, as long as systematic risk factors are the only source of common variationin credit spreads, equity and credit markets should be well integrated. This, however,does not seem to be the case at all times. One episode of weak integration between
equity and debt markets dates back to 2002, when the European high-yield market
saw one quarter of its bonds default. On a par basis, nearly 90% of that year’s
defaults came from the telecommunications and cable sectors. Over the first quarterof 2002, European credit spreads computed as the difference between the redemption
yield on the iBoxx EURO corporate bond index and the 10-year Euro-vs.-Euribor rateincreased by almost 50%, whereas the Morgan Stanley’s MSCI EURO stock market
index remained flat for the same time period. About two years later, during thethree-month period from June until August 2004, European credit spreads rose by
more than 60% while European stocks climbed rather steadily, rising by more than
10%. Although one could argue that certain macroeconomic factors are the source
for the common variation in credit spreads, it is at best doubtful that these would
suddenly become important for the credit market but not for the equity market.
In this study, we propose an alternative explanation for the common variationin the returns of corporate bonds which is not necessarily related to systematic risk
factors. We argue that for assets with nonlognormal return distributions, such as defaultable bonds, common variation in risk premia unrelated to systematic factors canarise when investors make systemic revisions to the distribution of extreme events
that affect these assets’ payoffs. For instance, the dramatic increase in European
credit spreads during the first few months of 2002 could have resulted, at least partially, from investors realizing almost simultaneously that the payoff structure of their
credit-contingent claims was more downward skewed than previously anticipated. As
a result, investors in European credit markets may have repriced their credit-sensitive
securities based on a revision of the cash flow distribution rather than on a revisionof the discount rates.
Our argument is rooted in a theoretical result derived in the context of Campbell’s(1993) intertemporal capital asset pricing model (ICAPM). Within this framework,2

we show that the risk premia of defaultable bonds represent compensation not only
for exposure to systematic risk, but also for unexpected revisions to these bonds’ nonlognormal return distributions. Such unexpected revisions can arise in the aftermathof an adverse credit-market event, such as a corporate default, when investors havethe opportunity to gather more information about the distribution of extreme events
that may lead to corporate default. For instance, an unexpected corporate default
could present investors with the opportunity to learn that certain extreme events are
more likely than previously anticipated and that the payoff distribution of the defaultable bonds they hold is more downward skewed than they previously thought.4
If such revisions in the distribution of default events happen nearly simultaneously
across the entire universe of firms, the returns of all defaultable bonds will move inthe same direction. More importantly, this common movement in defaultable bondreturns across firms cannot be attributed to macroeconomic risk.
The advantage of building on Campbell’s ICAPM framework is that it allows us
to identify likely sources of macroeconomic risk that are priced by security markets.
In our context, these sources turn out to be the market factor for Europe and thelong–short spread on European benchmark government bonds. At the same time, it
is important to point out that Campbell’s framework cannot be applied directly todefaultable bonds. The original framework was developed for securities with lognormal returns and defaultable bonds are quite unlikely to fall into this category. Indeed,
Berndt (2007) provides empirical evidence that instantaneous returns on zero-coupon
defaultable bonds with zero recovery are more likely to follow conditionally lognormal
dynamics as opposed to conditionally normal dynamics.
The theoretical contribution of our paper is the decomposition of the common
variation in the returns of defaultable bonds into a portion that is due to exposure
to macroeconomic risk, proxied by the market and the long–short government term
spread, plus a portion that stems from systemic revisions in the distribution of thedefault event across all firms.
For the remainder of our study, we investigate whether these theoretical implications have support in European corporate bond markets. Our empirical approachconsists of three steps: First, we identify a latent factor which has the potential of
capturing the comovement in defaultable bond returns that is due to systemic revi4

The notion of investors revising the payoff distribution for their holdings in corporate debt issimilar in spirit to the credit contagion model by Schönbucher (2003), who introduces the notion offrailty models to the credit-risk literature, and by Collin-Dufresne et al. (2003), who model default
contagion via the updating of investors’ beliefs.

3

sions of the distributions of the default event across firms. Second, we test whether
this latent factor induces common variation in bond returns above and beyond thetwo macroeconomic risk factors suggested by the theory. Third, we test whether thislatent factor does actually capture the systemic revision of the distributions of thedefault event across firms.
We now describe each of these steps in detail. Regarding the first step, to theextent that investors do revise the distributions of the default events across firms in
the aftermath of adverse credit-market events, we should see price comovement not
only in defaultable bonds, but also in other credit-contingent securities such as credit
default swaps (CDS), over-the-counter securities that provide default insurance on
debt. Based on this idea, we use CDS data to identify a latent factor which has
the potential to capture the comovement in CDS spreads induced by the systemic
revision of the distributions of the default event across firms. For this procedure we
only use the CDS spreads of the European non-financial firms with the most liquid
CDS market from January 2003 to October 2006. To extract the latent factor we firstuse the pricing information in the CDS market quotes to construct excess returns on
zero-coupon zero-recovery defaultable bonds. Then, guided by our model, we extract
our fact as the common component of the residuals of the orthogonal projectionsof these excess returns on the market and the long-short term spread on European
benchmark government bonds. We call this latent factor the credit market factor,
or short CMF. Crucial for this step is the extraction of the pricing information fromthe available CDS spreads. This is carried out by estimating a time-series model forrisk-neutral default probabilities using credit default swap data.
This step in our empirical approach reveals that CMF captures on average between
30% and 54% of the risk-adjusted excess returns of the zero-coupon zero-recovery
defaultable bonds constructed from the CDS data. This suggests that a potentially
important fraction of the CDS spread captures compensation for unexpected revisions
to the distribution of the default event.
The second step of our empirical analysis, we first test whether the CMF has
any impact on corporate bonds. As mentioned, a revision of the default event’s
distribution should impact the pricing of all credit-contingent assets. This implies thatwhenever we observe comovements in default swap rates across the CDS market, even
if they cannot be accounted for by changes in macroeconomic risk factors, we shouldobserve a similar pattern not just in CDS-inferred prices of zero-coupon zero-recovery
bonds, but the corporate bond market as a whole. Given the limited availability offirm-level pricing data for European credit and corporate bond markets, this is not4

a tautology, despite the fact that owners of corporate debt use CDS contracts as anatural hedge against default risk. With 55 European firms serving as the reference
entities for the CDS data in our sample, the price data for these corporate bondissuers is bound to be variable and it is unlikely that the hedging effect alone willaccount for the comovements in the prices of these two credit-contingent asset classes.
In that sense, it is important to verify that a wide range of European corporatebond returns responds contemporaneously to the CMF that we extract from CDS
rates for the limited set of liquidly traded reference entities in the European credit
market. In an effort to cover the whole cross section of European corporate bondreturns, we consider a large variety of corporate bond portfolios sorted on one ofthree characteristics: credit rating, time-to-maturity and sector. The corporate bondportfolios used in this study are preconstructed by either Merrill Lynch or Lehman
Brothers and span the entire universe of European corporate bonds.
If investors revise a default event’s distribution, the returns on risky bonds suchas those with low credit ratings or long times to maturity should reflect the impactof these revisions to a larger extent than returns on relatively safe investments incorporate debt. To illustrate this point, let us consider a simple stylized scenario
where an investor holds two zero-coupon zero-recovery defaultable bonds A and B
with the same time to maturity of one year. Assume that as of today, bond A pays
one unit of account in states s1 and s2 and zero in state s3 . Bond B, on the other
hand, pays one unit of account for sure. Suppose that one week from today, investors
realize that state s1 is also a default state in which neither bond will pay anything.
Suppose further that each of these three states is equally likely and that discountrates remain the same throughout the year. If r denotes the weekly discount rate,
the price of bond A prior to the revision is P0A = 32 e−50r , while the price of bond B isgiven by P0B = e−50r . After revising the distribution of the default event, these prices
change to P1A = 31 e−49r and P1B = 32 e−49r , respectively. The net holding returns onPAthese bonds can be computed as r1A = log P1A = r log 12 and r1B = r log 23 . If there were0

no revisions, clearly the net return on both bonds should be r. A revision, however,is bad news for both bonds, leading to negative net returns. Thus, consistent with
our intuition, the revision induces the net returns of these two bonds to move in thesame direction. More importantly, the riskier bond A records a larger loss, r − r1A ,than bond B since r1A < r1B .
Our asset pricing test results strongly support the two hypotheses: (i ) Corporatebond portfolios respond contemporaneously to innovations in the CMF. (ii ) This
response is commensurate with the riskiness of the portfolio, as measured a priori by5

a set of risk characteristics. Without exception, all corporate bond portfolios load
positively on the CMF, and their loadings increase with the riskiness of the portfolio.
The last step of our empirical analysis answers the question of whether the CMF
does capture the systemic revision of the default event’s distributions across firms.
To the extent that the returns on equity portfolios are more likely to be lognormally
distributed, we test whether equity portfolios react to innovations in the CMF in any
way. If our story is correct, we should not see any response in the equity portfolios.
Indeed, the data indicate that equity portfolios respond only weakly, and sometimes
even in the wrong direction, to innovations in the CMF. In summary, our results
support the theory that the CMF captures a price behavior that originates frominvestors’ systemic revisions of the distribution of the default event, across firms.
In addition to the aforementioned results, we also document another interesting
pattern in the returns on defaultable debt. Most of the European corporate bondportfolios load negatively on the excess returns on the market. These loadings become
more negative as the maturity of the assets in the portfolio increases and less negative,
sometimes even positive, as the rating of the assets decreases. In the asset pricing
literature this behavior is referred to as the flight to quality effect. As the economy
goes through a recession period, investors’ appetite for risk decreases and they invest
in safer assets with longer maturities. Similarly, as the economy goes through an
expansion period, investors’ appetite for risk increases and they invest in riskier highyield bonds.
This study contributes to the growing literature concerned with the measurementof the default risk premia that includes Elton et al. (2001), Collin-Dufresne et al.
(2001), Blanco et al. (2004), Longstaff et al. (2004), Driessen (2005), Amato and Remolona (2005), Berndt et al. (2005), Saita (2006) and Berndt et al. (2007). With the
exception of Denzler et al. (2005), these studies all focus solely on the U.S. corporatebond market. In addition to the fact that we study European capital markets, thecontribution of our work to the existing literature stems from extending the ICAPM
to accommodate nonlognormal returns of defaultable securities. We compare themodel’s results to the data, construct a CMF and find strong empirical support forthe model’s theoretical implication: the CMF captures the price behavior due to investors’ systemic revisions to the distribution of default events in the aftermath ofadverse credit-market news.
The remainder of the paper is structured as follows. Section 2 describes in detailour source for European CDS data and pricing data on European corporate bond andequity portfolios used in this study. Section 3 describes how to compute CDS-inferred6prices on zero-coupon zero-recovery defaultable bonds of various maturities and estimates a time-series model of risk-neutral default probabilities using the information
embedded in the CDS spreads. Section 4 presents the theoretical determinants ofdefault risk premia and constructs an expected-returns beta representation for the
return on defaultable assets. In Section 5, we identify the common variation in thenonlognormal component of returns on zero-coupon zero-recovery defaultable bonds,
and Section 6 implements a two-step methodology to disentangle the different sourcesof common variation in excess returns of zero-coupon zero-recovery bonds and constructs our CMF. In Section 7, we investigate the nature of the CMF in more detailand show that it captures compensation for unexpected revisions to the nonlognormal component of the return distribution of credit-sensitive assets. Finally, Section 8
implements a number of robustness checks, summarizes the results of our paper andconcludes.

2.

Data

This section discusses our data sources for European CDS rates, systematic factors
and test assets.

2.1.

Credit Default Swaps

Credit default swaps are single-name over-the-counter credit derivatives that provide default insurance. The payoff to the buyer of protection covers losses up to thenotional value in the event of a default by the reference entity. Default events are triggered by bankruptcy, failure to pay, or, for some CDS contracts, a debt-restructuring
event. The buyer of protection pays a quarterly premium, quoted as an annualized
percentage of the notional value, and in return receives the payoff from the sellerof protection should a credit event occur. Fueled by participation from commercial
banks, insurance companies, and hedge funds, the CDS market has been doubling insize each year for the past decade, reaching $12.43 trillion in notional amount outstanding by mid-2005.5 In this paper, we use CDS spreads instead of corporate bondyield spreads as our primitive source for prices of default risk because default-swap
5See, for example, the International Swaps and Derivatives Association mid-2005 market survey.
The CDS market is still undergoing rapid growth. The notional amount of default swaps grew byalmost 48% during the first six months of 2005 to $12.43 trillion from $8.42 trillion. This represents
a year-on-year growth rate of 128% from $5.44 trillion at mid-year 2004.

7

spreads are less confounded by illiquidity, taxes and various market microstructure
effects that are known to have a marked effect on corporate bond yield spreads.6
In particular, we use default-swap spreads for five-year CDS contracts for Eurodenominated senior unsecured debt. The data is provided by Credit Market Analysis (CMA) Thomson through Datastream. It contains daily CDS bid/ask quotes
contributed by active market participants including banks, hedge funds and active
managers. CMA assures full transparency for its clients by providing a qualifier (Veracity Score) for each data point of any time-series of CDS prices. The Veracity Score
indicates the liquidity or, if applicable, the extent to which a value has been modelderived. We focus exclusively on firms with very liquid five-year CDS markets forthe sample period between January 2003 and November 2006. The CDS contractsof these firms typically make up the iTraxx CDS Europe index of 150 most liquid
nonfinancial five-year CDS contracts. To optimally mitigate the tradeoff between the
microstructure effects of high-frequency quotes and the statistical power of our tests,
we focus on weekly CDS quotes. Most of the quotes have a Veracity Score of 3 or
better. This indicates that the quote is associated with an actual trade or that thequote is an indication provided by a market participant. We do not consider quotes
with a Veracity Score higher than 3.5. The final sample of default swap rates used inthis study consists of 55 firms from 11 European countries and 16 different industries,
based on Moody’s industry classification. Detailed summary statistics are provided
in Table I. A typical firm in our sample has 150 valid weekly CDS observations out of
196 maximum possible weekly quotes. The minimum number of weekly observations
is 95. Figure 1 plots the distribution of the credit quality of the firms in our sample,
showing a concentration of medium credit quality.
The fact that our sample has only 55 firms is an important caveat of this paper.
The typical major concern with small samples such as ours is whether the sample isrepresentative enough to support unbiased results. Despite its small size, we believe
our sample is quite diverse because the firms in our sample are distributed across 16
different industries. In addition, since the goal of this paper is to extract information
about the compensation rewarding investors for bearing risk, we believe this information can be extracted more precisely7 from the quotes on the CDS contracts of thosefirms with very liquid five-year CDS markets. To this extent, we are confident that
6

Recent papers that analyze the contribution of noncredit factors to bond yields include Zhou
(2005), Longstaff et al. (2004) and Ericsson and Renault (2001).
7

In order to extract this information, we use the approach in Berndt et al. (2005) which requiresa relatively long time series of prices (or quotes, in our case).

8the results in the paper are not biased by the size of our sample.

2.2.

Interest Rates, Systematic Factors and Test Assets

Throughout our empirical analysis, we rely on information about the Euro term
structure of riskless bonds. This data is obtained from the Datastream Euro zero
curves constructed relative to Euribor.8 All the excess returns and the zero-cost
portfolios are computed relative to the one-month zero yield.
For the purposes of Sections 6 and 7, we need to compute zero-cost portfolios thatare long the market portfolio and short the one-month zero yield or long the 30-year
zero yield and short the one-month zero yield. For the latter zero-cost portfolio, we
use the data in the Euro zero curves with the corresponding maturities. For theformer zero-cost portfolio, we construct two types of market portfolios: one thatincorporates the entire universe of European stocks and one the incorporates only
the stocks from a specific country. To maintain consistency with previous studies oncapital markets integration, whenever possible we use portfolios constructed from thedata disseminated in the electronic version of the MSCI. For those countries where
MSCI data is not available, we use the local portfolios constructed by FTSE. All theseportfolios are available through Datastream.9
Finally, for the purposes of Section 7, we need to compute realized returns on a
range of test assets in excess of the one-month zero yield. We consider the following
test assets: the Merrill Lynch nonfinancial corporate bond portfolios sorted on rating
and time to maturity (two separate sorts), the Merrill Lynch AAA-, AA-, A- andBBB-rated corporate bond portfolios sorted on maturity, and the Lehman Brothers
Euro-aggregate industrial corporate bond portfolios sorted on rating, maturity andsector. The time-series data for all these portfolios comes from Datastream.10
8The mnemonics for the yield of a zero-coupon Euro bond with time to maturity of n years andm months is EMnYm. For instance the mnemonic corresponding to the maturity of one year andfour months is EM01Y04.
9

The mnemonic for the MSCI European market portfolio is MSEURIL. The mnemonics for thecountry-specific market portfolios are WIDNMKE (Denmark), MSFINDL (Finland), MSFRNCL(France), MSGERML (Germany), MSGDEEL (Greece), MSITALL (Italy), MSNETHL (Netherlands), WINWAYE (Norway), MSSPANL (Spain), WISWDNE (Sweden) and FTSE10E (UK).
10

These portfolios have respectively the following mnemonics: MLNF3AE, MLNF1AE,
MLNF3BE, MLENFAE, MLENFCE, MLENFDE, MLENFGE, MLEC3AE, MLEC3EE, MLEC3GE,
MLEC3KE, MLEC2CE, MLEC2GE, MLEC2JE, MLEC1CE, MLEC1GE, MLEC1JE, MLEC1KE,
MLEC8CE, MLEC8GE, MLEC8JE, LHAI3AE, LHAI2AE, LHAI1AE, LHAIBAE, LHEHYBA,
LHAC1YE, LHAC3YE, LHAC5YE, LHAC7YE, LHAC10E, LHEAEDE, LHEBANK, LHEBMAT, LHECAPG, LHECHEM, LHECOMM, LHACCYE, LHACNCE, LHEDMAN, LHAFBVE,

9

3.

Prices of Defaultable Zero-Coupon Bonds

This section describes how to compute CDS-inferred prices on zero-coupon zerorecovery defaultable bonds of various maturities. Let us take as given a probability
space (Ω, F , P ) and information filtration {Ft : t ≥ 0}. In the absence of arbitrage and market frictions, there exists a stochastic discount factor, denoted by M(see, for instance, Duffie, 2001). Moreover, under mild technical conditions, Harrison
and Kreps (1979) and Delbaen and Schachermayer (1999) show that there exists a“risk-neutral” probability measure Q associated with M. Note that in our setting,markets are not necessarily complete, so the stochastic discount factor and the associated risk-neutral measure may not be unique. This pricing approach neverthelessallows us to express
h the pricei at time t of a security paying Z at time T > t, asRT

Et [Mt,T Z] = EtQ e− t rs ds Z , where r is the short-term interest rate and EtQ de-

notes the expectation operator with respect to Q, conditional on the information

available up to and including time t. In particular, the time-t market value of a defaultable zero-coupon bond that pays one unit of account in the event that a currently
surviving firm does not default before time T and zero otherwise is given by

P (t, T − t) = Et Mt,T 1{τ >T }



= EtQ

i
h RT
− t rs ds
1{τ >T } ,e(1)where τ denotes the default time of the firm. To compute the prices in Equation (1),we rely on the reduced-form arbitrage-free approach to pricing defaultable bonds
where the risk-neutral distribution of the default time τ is fully described by a riskneutral default intensity process λQ . If we assume the doubly stochastic property
under Q, Equation (1) reduces toh RTiQ
P (t, T − t) = EtQ e− t rs +λs ds ,(2)

subject to sufficient conditions given in Duffie (2001).
Motivated by Berndt et al. (2005), we suppose that the risk-neutral default intensity for firm i, λQ,i, statisfies



p
p
Q,i
Q
Q,i
Q
Q dB c + σ Q
d log λQ,i=κρ
1 − ρQ dBti ,
θ
−
log
λ
dt
+
σ
t
t
t

(3)

where Btc and Bti are independent standard Brownian motions with regard to thephysical measure P , and κQ , θQ,i , σ Q and ρQ are scalars to be estimated. The riskLHALODE, LHAREFE, LHATLPE, LHATBCE, LHAWRSE, and LHAMNCE.

10

neutral distribution of λQ is specified by assuming that



p
p
Q,i
Q
Q,i
Q
Q dB Q,c + σ Q
d log λQ,i
=
κ̃
θ̃
−
log
λ
dt
+
σ
ρ
1 − ρQ dBtQ,i ,tttwhere κ̃Q and θ̃Q,i are scalars to be estimated. Btc and Bti are independent
standard
Brownian motions with regard to Q. Given a set of parameters {θ̃Q,i }, κ̃Q , σ Q , wecan compute model-implied values for λQ,i using data on five-year CDS rates andan assumed risk-neutral loss given default of LQ = 0.6. For details, we refer thereader to Section 5.1 in Berndt et al. (2005). To improve the interpretability andthe reliability of our parameter estimates, we impose the overidentifying restrictionthat θQ,i equals the sample mean of log λQ,iand that the ratio of θ̃Q,i to the sampletmean of log(CDS it /LQ ) is constant within a given country. Using country-by-country
maximum likelihood estimation (MLE), we obtain estimates for the parameters thatgovern the processes for λQ . The estimated values of these parameters are presented
in Table II.

4.

An Asset Pricing Model for Assets with Nonlognormal Returns

In this section we use Campbell’s (1993) discrete ICAPM to identify likely sourcesof macroeconomic risk and to understand the impact of these sources of risk on theprices of defaultable bonds. Suppose the economy is populated with identical agentswith nonexpected-utility preferences:θ
n1−γ
 1−γ  1θ o 1−γ
Ut = (1 − β)Ct θ + β Et Ut+1
,

(4)

1−γ
where γ is the coefficient of relative risk aversion, θ = σ σ−1and σ is the elasticity ofintertemporal substitution.11 As Epstein and Zin (1989, 1991) show, the first-ordercondition of the representative agent in this economy can be stated as(11

1 = Et  β



Ct+1
Ct

− σ1 )θ 

1
m
Rt+1

For more details on the parameters see Campbell (1993).

11

1−θ



i 
Rt+1
,

(5)

m
where C is the aggregate consumption, Rt+1
is the return on the market portfolio andi
Rt+1
is the return on a security i.

Campbell (1993) shows that under the assumption that asset returns and consumption growth are jointly conditionally homoscedastic and lognormally distributed,
the aggregate budget constraint can be exploited to substitute out consumption andto simplify the Euler equation to1
fi
Et rt+1− rt+1= − Vii + γVim + (γ − 1)Vih ,
(6)
2


i
i
i
m
where r ∗ (∗ = i,f ) denotes log returns, Vii = cov
r
,
r
,
V
=
cov
r
,
r
t
im
t
t+1
t+1
t+1
t+1

P
i
j m
and Vih = covt rt+1
, (Et+1 − Et ) ∞ρr
t+1+j . The second argument of the lastj=1
covariate captures the news about the future returns on the market. ρ is the steady-state ratio of invested wealth to total wealth.12b
Furthermore, if rt+1
denotes the return on a riskless consol bond that pays one

unit of account every period, Campbell (1993) shows that the above equation can befurther simplified to1fi
Et rt+1− rt+1= − Vii + γVim + (1 − γ)Vib ,
(7)
2

i
b
where Vib = covt rt+1
, rt+1.
b ,r mcovt (rt+1
b,m
b,⊥
t+1 )
b
m
b
Let rt+1 = rt+1
−βt rt+1
with βtb,m =. Substituting rt+1
in the above
Vmm
equation yields
h
i
1
f
i
Et rt+1− rt+1= − Vii + γ + βtb,m (1 − γ) Vim + (1 − γ)Vib⊥ ,
(8)
2


b,⊥
b,⊥
i
where Vib⊥ = covt rt+1
, rt+1. If we further assume that rt+1
and the consumption
growth are both jointly conditionally homoscedastic and lognormally distributed, web,⊥mcan apply the above relation to both rt+1
and rt+1. Using the unconditional versionsof these relations we obtain Ertm,e 1
γ + β̄ b,m (1 − γ) =
−
Vmm
2
b,⊥,e
Ert
1
1−γ =
−
,
2
Vbb⊥

(9)

where E denotes the unconditional expectation operator, β̄ b,m = Eβtb,m , rtm,e = rtm −rtf
12

See Campbell (1993) for the exact definition.

12and rtb,⊥,e = rtb,⊥ − rtf . Substituting these formulas back into Equation (8) and taking
expectations yields the following expected-returns beta representation.
1Erti,e + Vii = βim
2





1
1 ⊥
m,e
b,⊥,e
⊥
Ert + Vmm + βib Ert+ Vbb ,
2
2

(10)

where βim = Vim /Vmm and βib⊥ = Vib⊥ /Vbb⊥ .13
The expected-returns beta representation in Equation (10) suggests that the time
variation in returns is due mainly to two factors: time variation in the returns on themarket portfolio in excess of the riskless short rate and time variation in the returns ofa portfolio that longs a riskless console bond and shorts the riskless short rate. Close
relatives of this latter portfolio have been previously used in the financial economic
literature. One of the best known is the spread between long- and short-term treasury
bonds, or TERM for short. For the exact definition, see Fama and French (1993).
The representation in Equation (10) applies to any returns that are both jointly homoscedastic and conditionally lognormally distributed with the consumption growth
and the market return. However, returns on certain assets are less likely to satisfy thelatter condition. For instance, Berndt et al. (2005) document that the instantaneous
excess returns on defaultable zero-coupon bonds are more likely to be lognormally
distributed rather than normally distributed (recall that the instantaneous returns
are natural logs of the gross returns). Thus, the above pricing equation might notwork as well for this type of returns. Under certain conditions, the expected-returns
beta representation model in Equation (10) can be slightly generalized to accommodate instantaneous excess returns that are not necessarily conditionally normally
distributed. We describe this modified model below.
Suppose the returns on a defaultable bond rtD can be decomposed into two components. The first component, rtD,c, is jointly homoscedastic and lognormally distributedwith the consumption growth and the market return. The second component, rtD,n ,is orthogonal on the information contained on both the consumption growth and themarket.14 This latter component is going to capture the impact of the departure
13

Notice that βim and βib are in fact the conditional betas, which happen to be constant under
the homoscedasticity assumption. Thus, they can be different from the unconditional betas.
Here is one way to implement suchLet µ = ErtD and kr =
 D a decomposition.
Dccov(rt+1 , rt )/ cov(rt , rt ). Define νt+1 = rt+1 − µ − kr (rt − µ) . Let νt+1
denote the linear projection of νt+1 onto the space generated by the consumption growth and the market return. Let⊥cνt+1= νt+1 − νt+1
denote the orthogonal residual. Since both the consumption growth and thecmarket return are conditionally normally distributed, νt+1
will be also conditionally normally disc⊥tributed. In addition, since νt+1 has zero mean, both νt+1 and νt+1
can be normalized to have zero
D,c
D,c
D,c
c
mean. Define rt recursively as follows: rt+1 − µ = kr (rt − µ) + νt+1
, with r0D,c = r0D . Also,
14

13

from the conditional lognormality assumption on prices. Under these assumptions it
can easily be shown that the expected-returns beta representation in Equation (10)becomes
1 cc
ErtD,c,e + VDD
= βDm
2



1
Ertm,e + Vmm
2



c⊥
+ βDb



1 ⊥
b,⊥,e
Ert
+ Vbb + Ezt ,
2

(11)





D,c
D,c m
c
c
c⊥
where rtD,c,e = rtD,c − rtf , VDD= vart rt+1
, βDm
= covt rt+1
, rt+1 /Vmm , βDm
=


D,n
D,c b,⊥
covt rt+1, rt+1 /Vbb⊥ , and zt = − log Et ert+1 . Making use of the fact that r D,n isorthogonal on the information contained in the market returns and the long–shorttreasury portfolio,15 we can rewrite the above equation as
1 c
= βDm
ErtD,e + VDD
2



1
Ertm,e + Vmm
2



⊥
+ βDb



1 ⊥
b,⊥,e
Ert
+ Vbb + E∆zt ,
2

(12)

D,n

D,n
where ∆zt = Et rt+1
− log Et ert+1 . The returns model associated with this represen-tation can be summarized as⊥ b,⊥,e
rtD,e = α + βDm rtm,e + βDbrt+ ∆ztD + D
t .

(13)

where Dt is an iid normally distributed error term with mean zero.
Thus, as do conditionally lognormal returns, the returns of defaultable bonds
vary over time in response to changes in excess market returns and the returns on thelong–short treasury portfolio. However, unlike conditionally lognormal returns, thereturns of defaultable bonds are also moved by changes in the shape of the conditional
distribution of r D , via ∆zt . This latter source of time variation could host both acommon component as well as an undiversifiable firm-specific component. More importantly, these two components affect the level of expected returns directly rather
than through covariances. This follows from the fact that rtD,n is orthogonal on thestochastic discount factor M. An important consequence of this result is the factthat to the extent that ∆zt hosts a common component, this component cannot be
related to macroeconomic risk (i.e., the stochastic discount factor M). Of course,
this argument is viable as long as our model of returns is sufficiently well specified.
D,n⊥define rtD,n recursively as follows: rt+1
= kr (rtD,n − 0) + νt+1
, with r0D,n = 0. Then rtD = rtD,c + rtD,nD,cD,nand rt and rt
satisfy the desired properties.
15

Campbell (1993) shows that the informational content of this portfolio overlaps with that of themarket returns and the consumption growth.

14Admittedly Campbell and Vuolteenaho (2004) show that a close version of the modelin Equation (10)—the “bad beta, good beta” model—does a very good job in capturing important time-series properties of U.S. stock returns—including size and value“anomalies”—and, as such, could be a well-specified model for them. However, there
is still the possibility that the same model is not well specified for European stock
returns.16 If our model is misspecified, then the common component in ∆zt could becorrelated with an omitted macroeconomic factor. We take up this empirical matterin a subsequent section. Before we do so, however, we first need to identify the common component in the firm-specific time-varying terms ∆zt . The next section deals
precisely with this issue.

5.

Identifying the Common Variation in the Nonlognormal Component of Returns

Our methodology for identifying the common component of the terms ∆zti , where iis an index for firms, is reminiscent of the fixed time effects in panel regressions. It
consists of two steps: In the first step, we identify the residuals from the firm-specificregressions of the bond excess returns on the market and term-spread excess returns.
That isrti,e = αi + βim rtm,e + βib⊥ rtb,⊥,e + it ,(14)

where it are iid normally distributed errors with zero means. In the second step, foreach time stamp s, we average these residuals across all firms represented at time s:
N
1 X i
ˆ
fs =
α̂ + ˆis .
N i=1

(15)

The time series fˆs captures the common variation of the residuals ˆis . The extent
to which fˆs captures only the common variation in these residuals depends on thenumber of bonds in the sample as well as the degree of correlation across residuals.
For instance, if is = ρi Zs + ξsi , where Zs and ξsi are iid normal variables and if N is
16

Several studies have addressed the issue of market integration between the U.S. and European
stock markets. Some of the best known studies include: Karolyi and Stulz (2003), Bekaert andHarvey (1995), Bekaert et al. (2005), Harvey (1995), Ferson and Harvey (1993), Griffin (2002) andothers.

15

sufficiently large, fˆs becomes
N
1 X i
ˆ
fs =
ρ Zs
N i=1

(16)

and therefore it captures only the common variation in the residuals.
While there are other ways to construct measures of common variation for our
residuals—we construct and test a few alternatives in a later section—we choose thisparticular methodology for its simplicity and ease of interpretation. Intuitively, fs
picks up the fixed time effects of the residuals, which are the output of the firm-specificregressions of bond excess returns on the market and term-spread excess returns. We
will refer to fs as the CMF.
To implement this methodology, we need to compute the bond returns on the lefthand side of Equation (14). In fact, the pricing restriction in Equation (10), while
exemplified so far with bond returns, holds true for any kind of returns, as long as
long as they are holding returns.
Given the superior liquidity of the CDS market over the corporate bond market,
for instance, our intent is to use the quote data on CDS spreads to compute the lefthand side returns. Holding returns on CDS contracts are neither readily available noreasy to compute given that we only have data on the CDS premium (spread) set forthat the outset of the contract. One way to circumvent this problem is to use the pricing
information of the CDS spread quotes to construct holding returns on zero-coupon
bonds of various maturities and then use these holding returns in excess of the riskfree rate as our primary dependent variable in the first step of our methodology. The
drawback of this approach is that returns computed in this manner will reflect not
only the informational content of the CDS spread quotes, but also the assumptions
used in order to extract this information. This critique applies, however, wheneverreturns are inferred rather than readily available. For robustness, we also consider analternative approach which approximates holding returns with the difference between
two consecutive quotes.17 The results under this specification are presented in a later
section.
The next step is to compute returns on defaultable zero-coupon bonds. Relying
on the notation from Section 3, the holding return between time t and time t + 1 for
17

Since the quotes at time t and t − 1 are essentially quotes on different issues (same type of CDS
contract on the same entity but originated at different times), this measure of holding return will bea very coarse approximation of actual holding returns. Hence results under this specification should
be interpreted with this caveat in mind.

16

a defaultable zero-coupon zero-recovery bond with maturity T > t + 1 is given by
rt+1 = log P (t + 1, T − t − 1) − log P (t, T − t),(17)where P (t, T − t) denotes the time-t market value of the bond. As a special case, theholding return for the period [T − 1, T ] is computed asrT= − log P (T − 1, T ).

(18)

It is important to notice that these returns cannot be computed directly because we
do not have data on defaultable zero-coupon corporate bonds. They will be inferredfrom CDS rates as explained in Section 3.
We start with the formula in Equation (2). Suppose the risk-neutral default
Q,cintensity λQ+ λQ,n such that λQ,c and rs are correlated Gaussian processes (int = λparticular, they are joint homoscedastic and conditionally normally distributed) and
λQ,n is orthogonal on the information contained in the consumption growth rates andthe market returns.18 Then,hihi h R T Q,n i
R
R− tT λQ− tT λQ,cs dssP (t, T − t) = Et Mt,T e= E Mt,T e
Et e− t λs
h R T f Q,c i h R T Q,n i= EtQ e− t rs +λs Et e− t λs .

(19)

Since rt and λQ,care correlated Gaussian processes, it is easy to established thatth R T f Q,c i
log EtQ e− t rs +λs = A(T − t) − B(T − t)rtf − C(T − t)λQ,c
t ,

(20)

where A(T − t), B(T − t) and C(T − t) depend on T − t only.19 Thus, log P (t, T − t)can be rewritten as
h R T Q,n i− t λslog P (t, T − t) = A(T − t) − B(T − t)rtf − C(T − t)λQ,c
+
log
E
.
t e
t

(21)

Combining, we obtain the following expression for rt+1 .
hif
rt+1 = B(T − t)rtf − B(T − t − 1)rt+1
h
i
Q,c
n
+ C(T − t)λQ,c
−
C(T
−
t
−
1)λ
t
t+1 + rt+1 ,
h

RT

Q,n

− t+1 λsn = A(T − t) − A(T − t − 1) + log E
where rt+1
t+1 e
18

See Footnote 14 for a way to construct such a decomposition19

See the appendix for the derivation of these coefficients.

17

i

(22)

h R T Q,n i− log Et e− t λs
capturesthe nonlognormal component of the returns. For a firm i, the expected-returns beta
representation in Equation (11) can be restated in the following form.
i
h
i
h
f
i,Q,c
i
+ E C i (T − t)λi,Q,c
E B i (T − t)rtf − B i (T − t − 1)rt+1
−
C
(T
−
t
−
1)λ
t
t+1




1
1
1
f
m,e
c,⊥
b,⊥,e
c
i
− Ert+1+ Viic = βim+ Vmm + βib+ Vbb⊥ + Ezt+1
(23)
Ert+1
Ert+1
2
2
2

or equivalently
h
i
h
i
f
i,Q
f
i
E B i (T − t)rtf − B i (T − t − 1)rt+1+ E C i (T − t)λi,Q
−
C
(T
−
t
−
1)λ
t
t+1 − Ert+1



 

1
1
1 ⊥
1 n
m,e
b,⊥,e
⊥
i
+ Vii = βim Ert+1 + Vmm + βib Ert+1 + Vbb + E∆zt+1 + Vii , (24)
2
2
2
2

h
i
i,n
i,Q,n
i
i
i
i
where zt+1
= − log Et ert+1 , ∆zt+1= zt+1− C i (T − t)λi,Q,n
−
C
(T
−
t
−
1)λ
t
t+1

and Viin is the variance of the nonlognormal return component r i,n . The return modelin Equation (13) becomesfi,Qi
B i (T − t)rtf − B i (T − t − 1)rt+1+ C i (T − t)λi,Qt − C (T − t − 1)λt+1
fm,e⊥ b,⊥,ei− rt+1= αi + βim rt+1
+ βib
rt+1 + ∆zt+1+ it+1 (25)and thus the excess returns to be used in the first step of our methodology, Equation (14), can now be computed as a function of the default intensities:
i,e
f
i,Q
f
i
rt+1
= B i (T − t)rtf − B i (T − t − 1)rt+1+ C i (T − t)λi,Qt − C (T − t − 1)λt+1 − rt+1 . (26)Given the estimated time series for the risk-neutral default intensities from Section 3, we can now compute returns of defaultable zero-coupon bonds. In what follows,
we will formally test the expected-returns beta representation derived in Section 4.

6.

Time Variation in Defaultable Debt Returns

This section deals with the implementation of the two-step methodology for identifying the sources of common variation in zero-coupon zero-recovery bond excess
returns of various maturities.
In order to implement the first step, we need to proxy for the two zero-cost portfolios capturing the systematic risk, namely the market portfolio and the portfoliothat longs a riskless console bond paying one unit of account every period and shorts
the short interest rate. For this exercise, we set the holding period to a week.
18

The reference entities behind the CDS contracts in our data set are from various
European countries. Most of these countries are also part of the European Monetary
Union20 but there are few countries that are not (Denmark, Norway, Sweden andthe UK). Since capital markets throughout Europe are more or less integrated,21
we proxy for the market portfolio with both a portfolio tracking the largest stocks
throughout Europe as well as local portfolios tracking the largest, most liquid stocks
within a specific country. To maintain consistency with the previous studies on capital
markets integration, whenever possible we use portfolios constructed from the data
disseminated in the electronic version of the MSCI. For those countries where MSCI
data is not available, we use the local portfolios constructed by FTSE. For more
information on these portfolios, see Section 2.2. Since the CDS spreads in our data
set are reported relative to the Euro term structure, it is important that the returns onthese portfolios are extracted from prices reported in Euros. We denote with rtEMKT ,ethe weekly returns on the European market portfolio in excess of the riskless short
rate and with rtCMKT ,e the weekly returns on the local market portfolio in excess of theriskless short rate. The riskless short rate corresponds to the yield of the one-monthzero-coupon Euro bond. For more information on the Euro term structure curves,see Section 2.2.
We proxy for the other source of systematic risk—captured by the portfolio whichlongs a riskless console bond paying one unit of account every week and shorts theshort interest rate—with the term-spread portfolio, which longs the 30-year zerocoupon riskless Euro bond and shorts the one-month Euro bond. We denote theweekly returns of this portfolio with rtTERM .
The first step in extracting the common variation in the returns inferred from the
CDS spreads is to extract the residuals from the following firm-specific regressions.
i,e
EMKT ,e
CMKT ,e
i
i
i
TERM
rt+1
= αi + βEMKT
rt+1+ βCMKT
rt+1+ βTERM
rt+1+ it+1(27)

The excess returns on the left-hand side are excess returns on zero-coupon zerorecovery corporate bonds and they depend on the risk-neutral default intensity andmaturity. The default intensities are extracted from the CDS data as indicated in theprevious section. Since these excess returns are inferred from CDS spreads data, we
20

See Table I for more details

21

There is quite a bit of literature on this topic. Some of the most well known studies include
Fama and French (1998), Griffin (2002), Ferson and Harvey (1993), Bekaert and Harvey (1995), andKarolyi and Stulz (2003).

19

have no a priori preference for a specific maturity. To limit the potential impact of
the choice of maturity on our results, we perform the same empirical exercise for six
maturities: one week, one year, two years, three years, four years and five years. In
addition, as mentioned in Section 4, the excess returns on the left-hand side are notlognormally distributed, in general, and thus the residuals will not be lognormally
distributed either. This observation is important because the typical t-statistic might
not be very informative in the context of these regressions.
Table III presents the averages of the estimated coefficients of these regressions
along with their average standard errors. We present the results across the six maturities to assess the impact of the choice of maturity.
For all choices of maturity, the results show that the common risk factors considered explain relatively little from the time variation of the LHS excess returns—in all
instances the adjusted R2 is around 15%. Also notable is the size of the pricing errors
and their standard errors. Except for the one-week maturity, the absolute value ofthe pricing error increases with the maturity as well, and it ranges from 6 bps toabout 21 bps. These errors are consistently different from 0, with t-statistics of atleast 4. These are relatively sizable pricing errors (for weekly returns), suggestingthat the return model based only on the market and the TERM factors could be
misspecified. This is not surprising given that the independent variables are close tobeing lognormally distributed while the dependent variable is not. More importantly,
from the perspective of the model derived in Section 4, these results are reassuring.
We now proceed with implementing the second step of our methodology as described in Section 5. Using the residuals extracted from the first step and Equation (15), we construct estimates for the CMF at every point in time.
The next to last column of Table III reports the average of the fraction of thepricing error explained by the CMF. We notice that the CMF captures as much as
53.84%, on average, of the time variation of the pricing errors of the risk-adjusted
excess returns of zero-coupon zero-recovery bonds with a maturity of five years. This
fraction decreases to 29.83% as we decrease the maturity of the bonds to one year.
In the special case, when the maturity is one week, the CMF captures 35.28% of thepricing error. The last column of Table III also reports the increase in the R2 of theregression of the CDS-inferred excess bond returns on the two macroeconomic factor
and the CMF. We notice that by accounting for the CMF, the R2 of the regression
increases by at least 4 times.
While the change in the average magnitude from one week to one year mightseem a bit puzzling, we should keep in mind that the returns of the zero-coupon
20

zero-recovery bonds with time to maturity of one week are special because the payoffof these bonds is 1. This is unique to the time to maturity of one week, because theholding period throughout this study is fixed at one week as well.
According to the model developed in Section 4, the CMF captures the common
variation of the nonlognormal component of CDS-inferred returns. The same model
tells us that this factor impacts expected returns level directly rather than throughcovariances. If our model is correct, this result suggests that the CMF picks up
common variation in the nonlognormal component of CDS-inferred returns that does
not originate from exposure to the common risk factors endogenous to the model.
Such a scenario is not completely implausible and here is why.
The spreads of a new CDS contract or the value of a corporate bond at some point
in time depend on the investors’ assessment of the distribution of the default eventat that time. This dependency is particularly strong for zero-coupon zero-recovery
bonds given that the uncertainty in the payoff structure of these assets reduces only
to the uncertainty about the occurrence of the default event. In this context, theoccurrence of a new credit event gives investors an opportunity to learn more about(update) the distribution of firm-specific extreme events leading to default. In particular, if the observed credit event reveals that certain extreme events are more likely
than previously thought, investors would react rationally by updating the distributionof the default event across the entire universe of credit-contingent assets—including
corporate bonds and CDS contracts. This systemic reaction could lead to commonality in the price (or return) behavior of these assets inducing common variation. It
is important to notice that such commonality in price behavior might not be related
to changes in the underlying macroeconomic risk factors. That is because the updateon the investor side is essentially a revision of the distribution of payoffs and thisrevision might not lead to changes in discount rates.
To illustrate this argument, consider the period surrounding the U.S. corporate
scandals in 2001–2002. The default of Enron and Worldcom in 2001 uncovered thatcertain “value-enhancing” accounting practices were more commonly employed acrossfirms then previously thought. The downward correction in firm value demandedby rational investors in the aftermath of these credit events led mechanically to an
upward correction in the likelihood of the default event (because firms were now closer
to their default boundary). These revisions led to a drop in the corporate bond prices
and, at the same time, induced bondholders to scramble for insurance—driving the
CDS spreads up.
Thus, one can argue that the CMF is capturing the systemic revision of the default
21

event’s distribution across firms, following the occurrence of a seemingly unrelated
corporate default. The next section investigates whether this theory is supported by
the data.

7.

Supporting Evidence

The CMF is extracted from the CDS data of the firms with the most liquid CDS
market during our sample period. As documented in the previous section, this factor
is able to capture a large fraction of the time variation in the returns of zero-coupon
zero-recovery corporate bonds of the firms in our sample. The asset pricing model
developed in Section 4 suggests that the common variation in these returns reflects
exposure to macroeconomic risk factors only to the extent that this common variation
originates from exposure to two sources of systematic risk: the market and the term
spread. By construction, the CMF captures common variation beyond whatever canbe explained by these sources of macroeconomic risk. One possible explanation suggested by our model is that the CMF captures the systemic response of the investors
who act upon observed corporate defaults by revising their assessment of the default
event’s distribution. In this section, we investigate whether this theory has support
in the data.
The revision of the distribution of the default event should impact the pricing of
all credit-contingent assets. This means that, to the extent that we observe comovement in spreads across the CDS market, we should observe a similar phenomenon in
the corporate bond market as well. This is not necessarily a tautology, despite thefact that CDS contracts are a natural hedge for corporate bonds. When the reference
entities behind corporate bonds and CDS contracts are different firms with different
characteristics, the hedging effect is not likely to be the source of the comovement in
the prices of these two classes of credit-contingent assets. To ensure that the reference
entities behind the corporate bonds used in our tests are different enough from thefirms in our CDS sample, we consider a variety of corporate bond portfolios sorted on
three different characteristics: rating, time to maturity and sector. These portfolios
are preconstructed by either Merrill Lynch or Lehman Brothers and they focus on either the entire universe of European corporate bonds or on the nonfinancial/industrial
sectors.22 .
Riskier bonds—such as those with lower ratings or longer time to maturity—are
22

For more information on these portfolios see Section 2.2.

22

more likely to display skewness and fat tails in their return distributions. Our theory
implies that if investors revise the default event’s distribution in the aftermath of acorporate default, the returns of the riskier bonds should reflect the impact of theserevisions to a larger extent.
To illustrate this point consider the following stylized example. Suppose an investor holds two zero-coupon bonds with the same maturity of one year. As of rightnow, Bond A pays 1 in states s1 and s2 and 0 in state s3 . Bond B pays 1 for sure.
Suppose that one week later the investor realizes that s1 could lead to default andso both bonds will now pay 0 in that state. Suppose further that each of these threestates is equally likely and that discount rates remain the same throughout the year.
Let r denote the weekly discount rate. Prior to the revision, the price of bond A is
P0A = 32 e−50r , while the price of bond B is P0B = e−50r . After the revision, these prices
change to P1A = 31 e−49r and P1B = 32 e−49r , respectively. The net holding returns onPAthese bonds can be computed as r1A = log P1A = r log 12 and r1B = r log 23 , respectively.
0

If there were no revision, the net return on both bonds should be r. However, therevision is bad news for both bonds and their prices drop, leading to negative net
returns. Thus, consistent with our theory, the revision induces the net returns ofthese two bonds to move in the same direction. More importantly, the riskier bond,bond A, records a larger loss: r − r1A > r − r1B
We now present evidence supporting the dual hypothesis that systemic revisions
to a default event’s distribution—as captured by the CMF—lead to comovement incorporate bond returns and that these revisions have more impact on the riskier
bonds.
We run time-series regressions,
i,e
EMKT ,e
i
i
TERM
i
rt+1
= αi + βEMKT
rt+1+ βTERM
rt+1+ βCMFCMF t+1 + it+1 ,(28)

where r i,e is the excess return on the portfolio i used as the test asset.
Tables IV–X report the estimated coefficients for the corporate bond portfolios
we use as test assets. The CMF used in these tests is extracted from defaultablezero-coupon bonds maturing in five years. The results for the other five choices oftime to maturity are illustrated in Figures 2–5.
The results reported in these tables overwhelmingly support our dual hypothesis.
All portfolios load positively on the CMF and these loadings trend in the direction
suggested by the characteristic used in constructing these portfolios. For instance,
the Merrill Lynch portfolios sorted on credit quality (rating) load heavier on the CMF
23

when the rating is lower. Similarly, the Merrill Lynch portfolios within a rating class,sorted further on maturity, load heavier on the CMF when the time to maturity
is larger. The same pattern can be observed for the Merrill Lynch portfolios of
nonfinancials sorted on maturity. These patterns are further supported by all the
Lehman Brothers portfolios sorted on either rating or time to maturity. Noticeable
here is the loading of the high-yield portfolio which is almost 3 times higher thanthe loading of the BAA-rated corporate bonds and more than 80 times higher thanthe loading of the AAA-rated corporate bonds. These results transgress the choicefor time to maturity when constructing the CMF. Figures 2–6 show that most of thepatterns continue to hold when the time to maturity varies from one week to five
years.
To increase the power of our test, we can run the previous time-series regressions
as pooled time-series regressions. Specifically, for each group of portfolios—Lehman
Brothers sorted on rating, maturity or sector and Merrill Lynch sorted on rating,maturity or both—we run the following pooled regression.
i,e
EMKT ,eTERM
rt+1= α + βEMKT rt+1
+ βTERM rt+1
+ βCMF CMF t+1 + it+1 ,(29)

where i is an index for corporate bond portfolios in a given group.
Table XI presents the results. In all instances the loading on the CMF is always
positive and significant at a 5% level (after correcting for lags using the Newey–Westprocedure). Once again, this provides support for the importance of the CMF in
explaining the time-variation of corporate bond returns.
While we expect the CMF to have an impact on corporate bond portfolios—thereturns of these portfolios are more likely to be nonlognormally distributed—we also
expect that the CMF will have no impact on equity portfolios—the returns of theseportfolios are more likely to be lognormally distributed. To test if this is the case, we
run pooled time-series regressions similar to the ones in Equation (29) for countryspecific equity portfolios sorted on sector. These portfolios are based on the price-levelsector indexes for each country, available from Datastream. The results are presented
in Table XII. For all countries considered, our results support the hypothesis that the
CMF has no impact on the returns of the equity portfolios. This result reassures us
once again that the CMF is likely to capture a price behavior specific to assets withnonlognormal returns only.
Tables IV–X also reveal another interesting pattern. Most of the corporate bondportfolios load negatively on the market. These loadings become more negative as the24maturity of the assets in the portfolios increases and less negative (and even positive)
as the rating of the assets deteriorates. This fact seems to confirm the flight toquality effect. As the economy goes through an expansion, investors’ appetite for riskincreases and they are more likely to invest in riskier assets such as high yield (lower
rating) corporate bonds. As the economy goes through a recession, investors’ appetite
for risk turns sour and they prefer to invest in safer assets with longer maturity—suchas high-rating long-term corporate bonds.

8.

Discussion and Conclusion

To ensure the robustness of our findings, we investigate two alternative ways of extracting the CMF. We then investigate whether the results of the previous section
remain valid.
The first alternative computes the returns of the zero-coupon zero-recovery defaultable bonds used on the left-hand side of the regression in Equation (14) simply
as the difference between two consecutive five-year CDS market rates. This approachhas the advantage of being nonparametric (and model free). However, it only provides
a very coarse approximation of the actual returns.
The second alternative proposes a different way to extract the CMF. Essentially,
the extracted value for the CMF at time t equals the loading on the time dummy attime t, δt in the following pooled regression.
rti,e = αi + βim rtm,e + βib⊥ rtb,⊥,e +X

δt + it ,

(30)

t

where δt = 1 if the time stamp is t, and 0 otherwise. In both cases, our unreported
results support qualitatively and, sometimes, quantitatively the results reported in
the previous section.
In summary, the CDS market is one of the largest and most liquid markets andcomparable in many respects with the equity market. Yet, there are times when these
two markets seem to move very differently on an aggregate level. The question is,Why? If common variation in these markets arises exclusively as a consequence of
exposure to the same macroeconomic risk, these markets should move in sync. Yet
that does not seem to happen all the time. In this paper, we try to understand thesources of common time-series variation in the premiums of the CDS contracts.
We use a slightly modified version of the Campbell’s ICAPM to characterize therisk premia of the assets with and without lognormal returns. According to the model,
25the common variation in the returns of assets with lognormal returns can only arise
from exposure to two macroeconomic risk factors: the market and the term spread—the spread between the long and the short ends of the term structure of interest rates.
However, common variation in the returns of the assets with nonlognormal returns
can also arise if investors systemically revise the distribution of the default event in
the aftermath of a corporate default.
Using European CDS, corporate bond and equity data, we provide evidence insupport of this theory. To the extent that investors learn from corporate defaults andupdate the distribution of the default event, the impact of these revisions should beparticularly high for defaultable zero-coupon zero-recovery corporate bonds. Returns
on these type of bonds are not readily available, but they can be inferred from CDS
data, which is typically available. We construct such returns and identify a common
component in these returns that captures the systemic updating on the part of theinvestors, as suggested by the theory. We call this component the CMF.
Our tests concentrate around corporate bond and equity portfolios. To overcome
the potential hedging bias, we consider a large variety of bond portfolios sorted on
rating, time to maturity and sector. We find that corporate bond portfolios respond toinnovations in the two macroeconomic risk factors, but they also respond positively toinnovations in the CMF. All our equity portfolios—presorted on sector and country—
show little or no response to innovations in the CMF. This is consistent with ourtheory since equity returns are more likely to be lognormally distributed and should
only respond to innovations in the macroeconomic risk factors.
The model and the evidence provided in this paper seem to suggest that thesources of common variation for a particular market do not necessarily have to be
associated with macroeconomic risk factors. This point has been made previously by
Daniel and Titman (1997) for equity markets. We expand the focus of this point tothe credit and corporate bond markets.

A.

Derivation of the Coefficients for Equation (20)The coefficients A(T − t), B(T − t) and C(T − t) can be derived in a recursive fashion
as it is typically done in the affine term-structure literature. Suppose rtf and λQ,ctfollow jointly Gaussian dynamics of the following form.
fr
rt+1= kr r̄ f + (1 − kr )rtf + σr ξt+1
Q,c
λ
r
λQ,c
+ (1 − kλ )λQ,c+ σλ ξt+1+ σr,λ σr ξt+1
t
t+1 = kλ λ̄

26Then, for any t < T , we haveA(T − t) = A(T − t − 1) − [B(T − t − 1) + 1] kr r̄ f − [C(T − t − 1) + 1] kλ λ̄Q,c1+ [(B(T − t − 1) + 1) + σr,λ (C(T − t − 1) + 1)]2 σr2
2
1+ [C(T − t − 1) + 1]2 σλ22B(T − t) = [B(T − t − 1) + 1] (1 − kr )C(T − t) = [C(T − t − 1) + 1] (1 − kλ ),with the initial conditions A(0) = B(0) = C(0) = 0. Notice that under the decomposition suggested in Footnote (14), kλ can be computed as follows:
h
i
h
i
Q
Q
Q,c
Q,c
cov λt , λt+1
cov λt , λt+1
h
i
h i .
=
1 − kλ =Q,cvar λtvar λQt

27

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30

Tables
Table I: Distribution of firms across industries and countries
Industry Name
Advertising
Aerospace/Defense
Airlines
Automotives
ChemicalsEntertainment
Food/Soft Drinks
Hotels
Machinery
Media
Paper
Printing/Publishing
Retail Grocery Chains
Steel
Telecom
Utilities
Total

No. of Firms
1
2
4
6
5
1
2
1
1
2
3
3
6
1
13
4
55

Country
Denmark
FinlandFranceGermany
Greece
Italy
Netherlands
NorwaySpain
SwedenUK

No. of Firms
1
5
13
10
1
2
4
1
2
5
11

55

Firms are grouped into industries according to the Moody’s industry classification.

31

Table II: Estimation of the risk-neutral default intensities

Denmark
FinlandFranceGermany
Greece
Italy
Netherlands
NorwaySpain
Sweden
UK

κQ

σQ

ρQ

mean(θQ )κ̃Q

mean(κ̃Q θ̃Q )no firms

0.154
0.290
1.363
1.764
4.533
1.086
3.339
2.640
2.512
0.673
1.143

0.444
0.410
1.455
1.274
1.043
1.872
0.938
0.257
0.510
1.098
0.849

–
0.309
0.683
0.591
–
0.767
0.902
–
0.287
0.220
0.357

3.624
3.870
2.959
2.962
3.182
3.257
2.848
2.758
3.070
3.246
3.530

−0.123
0.003
0.158
0.224
0.186
0.306
−0.206−0.159
0.015
0.289
0.182

−0.220
0.242
0.402
0.722
0.768
0.611
−0.854−0.214
0.325
1.401
0.903

1
5
13
10
1
2
4
1
2
5
11

Summary statistics for the country-by-country parameter estimates describing the
dynamics of the risk-neutral default intensities in Equation (4).

32

Table III: Estimates for the return model in Equation (27)α0.0000(3.3160)

βEMKT−0.0005(−0.0101)

βCMKT0.0001(0.0077)

βTERM
0.1607(2.6410)

P erc(C)35.28

P erc(S)404.04−0.0006(−4.3896)

0.0045(0.7568)

0.0028(0.1908)

1.1120(3.7908)

29.83

405.40−0.0010(−4.8849)

0.0064(0.6559)

0.0031(0.1730)

1.9146(4.1127)39.16

460.73−0.0014(−5.1307)0.0073(0.5875)

0.0030(0.1595)

2.6169(4.2915)45.85

497.29−0.0018(−5.2826)

0.0080(0.5376)

0.0027(0.1498)

3.2480(4.4050)50.50

521.64−0.0021(−5.3858)

0.0086(0.4992)

0.0024(0.1426)

3.8203(4.4827)53.84

538.83This table reports the results of the panel regression of the excess returns of defaultable zero-coupon bonds on the excess market returns (EMKT ), the excess local
market return (CMKT ), the spread between long and short Euro bonds (TERM )and the dummies controlling for specific weeks between January 2003 and October
2006, 197 weeks. The left-hand side excess returns correspond to defaultable bonds
with the following times to expiration: one week, one year, two years, three years,
four years and five years. The first line in the table corresponds to the estimates of
the returns model where the left-hand side returns correspond to corporate bonds
with the shortest time to maturity. The CMF at time t corresponds to the slope
coefficient of the dummy controlling for time t. The P erc(C) column reports thefraction of the pricing error, obtained from the first step, explained by the CMF.
The P erc(S) column reports the increase in adjusted R2 when the CMF is added tothe independent variables in the regression of step 1. The t-statistics are reported inparentheses. The reported values for the estimates are averages across firms of thecorresponding firm-specific estimates.

33

Table IV: The Merrill Lynch nonfinancial corporate bond portfolios sorted on rating
α0.0005(0.5557)

βEMKT−0.0595(−4.3825)

βTERM
0.3449(0.1998)

βCMF
0.2627(0.9530)

adj R20.1995E [R]0.00060.0003(0.2899)−0.0449(−3.2363)

1.3751(0.6576)

0.3485(1.1975)

0.09990.00070.0003(0.2616)−0.0182(−1.1118)

2.0377(1.0694)0.4504(1.6468)

0.04580.0009This table reports the results of the time-series regressions of the excess realized
returns of three Merrill Lynch nonfinancial corporate bond portfolios sorted on rating(AAA, A and BBB), on the excess market returns (EMKT ), the spread between longand short Euro bonds (TERM ) and the CMF between January 2003 and October
2006, 197 weeks. The CMF is extracted from returns on defaultable zero-coupon
bonds maturing in five years according to the model in Equation (27). The first line
corresponds to the higher-rating portfolio. The Newey–West t-statistics (adjustingfor 3 lags) are reported in parentheses.

34

Table V: AAA and AA Merrill Lynch corporate bond portfolios sorted on maturity

0.0001(0.1321)

βEMKT
βTERM
βCMF
adj R2AAA Portfolios Sorted on Maturity−0.02280.69240.18270.1864(−4.9212)(0.9058)(1.7263)0.0003(0.3379)−0.0561(−4.6869)

0.8716(0.4915)0.3009(1.1967)

0.18820.00050.0007(0.4863)−0.0759(−4.3608)

0.6578(0.2709)

0.3428(0.9914)0.17540.00070.0012(0.5301)−0.1068(−4.0024)−0.1108(−0.0278)

0.2672(0.4943)

0.14130.00100.0002(0.2564)

AA Portfolios Sorted on Maturity−0.0405
0.97820.26990.1756(−4.6479)(0.7125)(1.4026)0.00050.0006(0.4663)−0.0735(−4.2552)

0.9079(0.3741)0.3674(1.0345)0.16890.00070.0011(0.6194)−0.0852(−4.0652)

0.4482(0.1486)

0.3962(0.9308)

0.14720.0009αE [R]0.0005This table reports the results of the time-series regressions of the excess realized
returns of four AAA-rated Merrill Lynch corporate bond portfolios sorted on maturity(1–3 years, 3–5 years, 5–7 years and 10+ years) and three AA-rated Merrill Lynch
corporate bond portfolios sorted on maturity (1–5 years, 5–7 years, and 7–10 years)
on the excess market returns (EMKT ), the spread between long and short Euro bonds(TERM ) and the CMF between January 2003 and October 2006, 197 weeks. The
CMF is extracted from returns on defaultable zero-coupon bonds maturing in five
years according to the model in Equation (27). The first line in each of the twopanels corresponds to the lower-maturity portfolio. The t–statistics are reported inparentheses.

35

Table VI: A and BBB Merrill Lynch corporate bond portfolios sorted on maturity

0.0002(0.2502)

βEMKT
βTERM
βCMF
R2
A Portfolios Sorted on Maturity−0.0333
1.1517
0.27640.1416(−3.9213)(0.8837)(1.4436)0.0007(0.5141)−0.0638(−3.7003)

1.1890(0.5009)

0.4313(1.1918)

0.13980.00080.0010(0.5703)−0.0683(−3.1622)

1.0885(0.3642)0.4733(1.1045)

0.10680.00100.0015(0.6192)−0.0607(−2.2942)

1.6356(0.4022)0.7687(1.3974)

0.05770.00120.0001(0.1794)

BBB Portfolios Sorted on Maturity−0.0107
1.77700.3616
0.0546(−1.1048)(1.4401)(1.9255)

0.00080.0004(0.2502)−0.0341(−1.5129)

2.4134(1.0277)

0.5214(1.3538)

0.05680.00090.0008(0.4681)−0.0236(−0.9869)

2.5044(0.8136)0.6602(1.5680)

0.03590.0012αE [R]0.0006This table reports the results of the time-series regressions of the excess realized
returns of four A-rated Merrill Lynch corporate bond portfolios sorted on maturity(1–5 years, 5–7 years, 7–10 years and 10+ years) and three BBB-rated Merrill Lynch
corporate bond portfolios sorted on maturity (1–5 years, 5–7 years, and 7–10 years),on the excess market returns (EMKT ), the spread between long and short Eurobonds (TERM ) and the CMF between January 2003 and October 2006, 197 weeks.
The CMF is extracted from returns on defaultable zero-coupon bonds maturing infive years according to the model in Equation (27). The first line in each of thetwo panels corresponds to the lower-maturity portfolio. The Newey–West t-statistics(adjusted for 3 lags) are reported in parentheses.

36

Table VII: The Merrill Lynch nonfinancial corporate bond portfolios sorted on maturity
α0.0000(0.01947)

βEMKT−0.0145(−2.1089)

βTERM
1.2130(1.5237)

βCMF
0.2207(1.8600)

adj R20.0909E [R]0.00060.0002(0.1957)−0.0328(−2.8268)

1.8151(1.0831)0.3889(1.6571)

0.09940.00070.0004(0.3192)−0.0493(−2.5540)

1.7211(0.7296)0.4422(1.2075)

0.09270.00080.0014(0.5520)−0.0597(−1.6539)

2.1434(0.5077)0.7049(1.2781)0.04740.0015This table reports the results of the time-series regressions of the excess realized
returns of four Merrill Lynch nonfinancial corporate bond portfolios sorted on maturity (1–3 years, 3–5 years, 5–7 years and 10+ years) on the excess market returns(EMKT ), the spread between long and short Euro bonds (TERM ) and the CMF
between January 2003 and October 2006, 197 weeks. The CMF is extracted fromreturns on defaultable zero-coupon bonds maturing in five years according to themodel in Equation (27). The first line corresponds to the lower-maturity portfolio.
The Newey–West t-statistics (adjusted for 3 lags) are reported in parentheses.

37

Table VIII: The Lehman Brothers Euro-aggregate industrial corporate bond portfolios
sorted on ratingα−0.0007(−0.7199)

βEMKT−0.0550(−3.8647)

βTERM
0.7849(0.4435)

βCMF
0.1591(0.5781)R2
0.1627E [R]−0.0003−0.0002(−0.1897)−0.0605(−4.2683)

0.1914(0.0889)

0.2726(0.8825)0.1377−0.0003−0.0003(−0.2315)−0.0496(−3.0092)

0.4817(0.2214)0.2871(1.0274)

0.0914−0.0002−0.0004(−0.3551)−0.0370(−2.7172)

1.3619(0.6662)0.4976(1.6431)0.0708−0.00030.0015(0.5592)

0.0554( 1.9091)

3.1904(0.9239)

1.3851(1.7404)

0.06180.0011This table reports the results of the time-series regressions of the excess realized
returns of five Lehman Brothers Euro-aggregate industrial corporate bond portfolios
sorted on rating (AAA, AA, A, BAA and High Yield) on the excess market returns(EMKT ), the spread between long and short Euro bonds (TERM ) and the CMF
between January 2003 and October 2006, 197 weeks. The CMF is extracted fromreturns on defaultable zero-coupon bonds maturing in five years according to themodel in Equation (27). The first line corresponds to the higher-rating portfolio.
The t-statistics are reported in parentheses.

38

Table IX: The Lehman Brothers Euro-aggregate corporate bond portfolios sorted on
maturityα−0.0006(−1.1988)

βEMKT−0.0188(−3.4722)

βTERM
1.0872(1.1677)

βCMF
0.2645(2.1299)

adj R20.1186E [R]−0.0001−0.0003(−0.3513)−0.0385(−3.4332)

0.8861(0.5141)0.3908(1.6555)

0.1066−0.0002−0.0008(−0.5271)−0.0693(−3.6313)

2.1092(0.8244)

0.3910(1.0410)0.1417−0.00020.0000(0.0072)−0.0763(−3.2838)

0.0900(0.0279)

0.4878(1.1081)0.1089−0.0005

0.0012(0.3786)−0.1013(−2.9269)

2.6832(0.5325)

1.1192(1.5013)0.07940.0007This table reports the results of the time-series regressions of the excess realized
returns of five Lehman Brothers Euro-aggregate corporate bond portfolios sorted on
maturity (1–3 years, 3–5 years, 5–7 years, 7–10 years and 10+ years) on the excess
market returns (EMKT ), the spread between long and short Euro bonds (TERM ) andthe CMF between January 2003 and October 2006, 197 weeks. The CMF is extracted
from returns on defaultable zero-coupon bonds maturing in five years according to themodel in Equation (27). The first line corresponds to the lower-maturity portfolio.
The Newey–West t-statistics (adjusted for 3 lags) are reported in parentheses.

39

Table X: The Lehman Brothers Euro-aggregate corporate bond portfolios sorted on
sector
α
βEMKT
βTERM
βCMF
R2
E [R]
Cross-Sectional Averages of the Estimates−0.0004−0.0416
1.19840.43010.0703−0.00020.0004(−0.3498)(0.2960)

Cross-Sectional Standard Deviations of the Estimates
0.01590.82030.2036
0.03370.0001

Cross-Sectional Averages of the t-Statistics(−3.3661)(0.4835)(1.5052)
Cross-Sectional Standard Deviations of the t-Statistics(1.1992)(0.2949)(0.5092)

This table reports the results of the time-series regressions of the excess realized
returns of 16 Lehman Brothers Euro-aggregate corporate bond portfolios sorted on
sector (Aero/Defense, Banking, Building Materials, Capital Goods, Chemicals, Communications, Consumer Noncyclical, Consumer Cyclical, Diversified Manufacturing,Food and Beverages, Lodging, Refining, Telephone, Tobacco, Wireless and Media
Noncable), on the excess market returns (EMKT ), the spread between long andshort Euro bonds (TERM ) and the CMF between January 2003 and October 2006,
197 weeks. The CMF is extracted from returns on defaultable zero-coupon bonds
maturing in five years according to the model in Equation (27). Each panel reports
an average statistic across portfolios.

40

Table XI: Pooled regressions for corporate bond portfolios

0.0004(0.6173)

βTERM
βCMF
adj R2Merrill Lynch sorted on rating−0.0409
1.2526
0.35390.0963(−4.4799)(1.1253)(2.1702)0.0006(1.5949)

Merrill Lynch sorted on rating/maturity−0.0539
1.15750.40140.0917(−9.2750)(1.6852)(4.0677)0.00030.0005(0.6617)

Merrill Lynch sorted on maturity−0.0391
1.72310.4392
0.0535(−3.2384)(1.3380)(2.3995)

0.0004−0.0000(−0.0502)

Lehman Brothers sorted on rating−0.02931.20210.52030.0329(−3.0176)(1.0575)(2.7349)−0.0005−0.0001(−0.1112)

Lehman Brothers sorted on maturity−0.0609
1.37110.5306
0.0767(−5.3811)(1.0059)(2.7049)−0.0005−0.0004(−1.1990)

Lehman Brothers sorted on sectors−0.0416
1.19840.43010.0706(−9.5535)(2.0372)(4.7138)−0.0007

α

βEMKT

E [R]
0.0003This table reports the results of the pooled regressions of the excess realized returns ofcorporate bond portfolios on the excess market returns (EMKT ), the spread between
long and short Euro bonds (TERM ) and the CMF between January 2003 and October
2006, 197 weeks. The CMF is extracted from returns on defaultable zero-coupon
bonds maturing in five years according to the model in Equation (27). The first line
corresponds to the lower-maturity portfolio. The Newey–West t-statistics (adjusted
for 3 lags) are reported in parentheses.

41

Table XII: Pooled regressions for equity portfolios
α

βEMKT

−0.0008(−0.8230)

0.2686(11.908)

βTERM−0.5073(−2.3018)

0.1643 0.0031
n = 750.0026(3.9095)

France0.0945−3.60290.6705(1.9018) (−3.0799) (12.803)−0.0862(−0.5036)

0.3437 0.0022
n = 1130.0018(2.4525)

0.0799−1.3920(1.6718) (−1.0053)

Germany
0.6187(16.535)

0.0080(0.0403)

0.3352 0.0023
n = 1120.0023(2.6127)

Netherlands
0.3318−2.42200.3678(9.1078) (−1.5138) (10.7255)

0.2469(1.0287)

0.2984 0.0019
n = 760.0017(2.2894)

Sweden0.0631−2.64210.6434(2.3808) (−1.8971) (32.703)−0.3013(−1.7799)

0.4751 0.0028
n = 69UK
0.6924(30.818)0.2251(1.2429)0.2987 0.0018
n = 750.0021−0.0095−1.9132(2.7902) (−0.5649) (−1.4125)

βCMF

adj R2βCMKTFinland5.49260.2005(2.9977) (14.839)

E [R]This table reports the results of the pooled regressions of the excess realized returnsof country-specific equity portfolios sorted on sectors on the excess market returns(EMKT ), the spread between long and short Euro bonds (TERM ), the countryspecific excess market return (CMKT ) and the CMF between January 2003 andOctober 2006, 197 weeks. The CMF is extracted from returns on defaultable zerocoupon bonds maturing in five years according to the model in Equation (27). The
first line corresponds to the lower-maturity portfolio. The Newey–West t-statistics(adjusted for 3 lags) are reported in parentheses.

42Figures25

numer of firms

20

15

10

5

0

AA

A

BAA

BA

B

Figure 1: Distribution of firms by median credit rating during the sample period.

43

8
1Y
2Y
3Y
4Y
5Y

7

6

5

4

3

2

1

0

−1

AAA

AA

A

BAA

HY

Figure 2: The estimates of the slope coefficient on the CMF extracted from returns on
defaultable zero-coupon bonds with maturity varying from one to five years. These
slopes are estimated from the time-series regressions of the excess realized returns offive Lehman Brothers Euro-aggregate industrial corporate bond portfolios sorted on
rating (AAA, AA, A, BAA and High Yield) on the excess market returns (EMKT ),the Euro term spread (TERM ) and the CMF between January 2003 and October
2006, 197 weeks.

44

6
1Y
2Y
3Y
4Y
5Y

5

4

3

2

1

0

1−3 Years

3−5 Years

5−7 Years

7−10 Years

10+ Years

Figure 3: The estimates of the slope coefficient on the CMF extracted from returns on
defaultable zero-coupon bonds with maturity varying from one to five years. These
slope coefficients are estimated from the time-series regressions of the excess realized
returns of five Lehman Brothers Euro-aggregate corporate bond portfolios sorted on
maturity (1–3 years, 3–5 years, 5–7 years, 7–10 years and 10+ years) on the excess
market returns (EMKT ), the Euro term spread (TERM ) and the CMF between
January 2003 and October 2006, 197 weeks.

45

1.5
1Y
2Y
3Y
4Y
5Y

1

0.5

0

AAA

A

BBB

Figure 4: The estimates of the slope coefficient on the CMF extracted from returns on
defaultable zero-coupon bonds with maturity varying from one to five years. These
slopes are estimated from the time-series regressions of the excess realized returns of
three Merrill Lynch nonfinancial corporate bond portfolios sorted on rating (AAA,A and BBB) on the excess market returns (EMKT ), the Euro term spread (TERM )and the CMF between January 2003 and October 2006, 197 weeks.

46

6
1Y
2Y
3Y
4Y
5Y

5

4

3

2

1

0

1−3 Years

3−5 Years

5−7 Years

7−10 Years

10+ Years

Figure 5: The estimates of the slope coefficient on the CMF extracted from returns on
defaultable zero-coupon bonds with maturity varying from one to five years. These
slope coefficients are estimated from the time-series regressions of the excess realized
returns of four Merrill Lynch nonfinancial corporate bond portfolios sorted on maturity (1–3 years, 3–5 years, 5–7 years, and 10+ years) on the excess market returns(EMKT ), the Euro term spread (TERM ) and the CMF between January 2003 andOctober 2006, 197 weeks.

47

2
1Y
2Y
3Y
4Y
5Y

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

A/1−5

A/5−7

A/7−10

A/10+

BBB/1−5

BBB/5−7

BBB/7−10

Figure 6: The estimates of the slope coefficient on the CMF extracted from returns on
defaultable zero-coupon bonds with maturity varying from one to five years. These
slope coefficients are estimated from the time-series regressions of the excess realized
returns of four A-rated Merrill Lynch corporate bond portfolios sorted on maturity(1–5 years, 5–7 years, 7–10 years and 10+ years) and three BBB-rated Merrill Lynch
corporate bond portfolios sorted on maturity (1–5 years, 5–7 years, and 7–10 years)
on the excess market returns (EMKT ), the Euro term spread TERM and the CMF
between January 2003 and October 2006, 197 weeks.

48