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Working Paper 8604
THE DEFAULT PREMIU:V A N D CORPORATE BOND EXPERIENCE
By Jerome S. Fons
Jerome S. Fons is a n economist at t h e Federal
Reserve Bank of Cleveland. The author would like t o
thank Jim Balazsy for excellent research assistance,
Bill Gavin and Ross S t a r t for helpful comments, and
t h e participants of t h e Federal Reserve System's
Financial Analysis Committee Meeting at t h e Board
of Governors in November 1985.
Working Papers of t h e Federal Reserve Bank of
Cleveland a r e preliminary materials, circulated t o
stimulate discussion and critical comment. The views
s t a t e d herein a r e t h e author's and not necessarily
those of t h e Federal Reserve Bank of Cleveland or
of t h e Board of Governors of t h e Federal Reserve
System.
June 1986
Federal Reserve Bank of Cleveland
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The Default Premium and Corporate Bond Experience
I. Introduction
The emergence of organized markets for low-rated corporate (or junk)
bonds has provided financial researchers with an opportunity t o address a
fundamental question: a r e holders of default- prone debt compensated (actuarially) f o r t h e risk of default? Past research on this market has focused on t h e
default experience of corporate debt. A quite different a r e a of research involves modeling t h e spreads between t h e returns of bonds of different credit
quality. Few (if any) research e f f o r t s have combined these approaches by
using past default experience t o explain differential r a t e s of return on lowrated bonds.
In this study we develop a risk-neutral model of t h e expected probability
of default for low-grade bonds as a function of t h e additional required r a t e
of return on these instruments over default- free bonds. Within this framework, securities a r e priced as functions of t h e first moment of t h e return distribution. The techniques a r e used t o express this pricing relationship in terms
of t h e yields t o maturity of risky bonds as well as their holding period returns. W e then compare t h e default r a t e s implied in corporate bond yields t o
a series based on r e c e n t corporate bond default experience. We also discuss
why implied default r a t e s cannot b e obtained from measured holding period
returns. Finally, attention is paid t o macroeconomic indicators of expected
default rates.
In a n early paper on t h e subject of default risk premia, Fisher (1959) suggested t h a t t h e risk premium required on a corporate bond (holding maturity
constant) depends on t h e likelihood t h a t t h e issuing firm will default (defined
here as a failure t o pay any coupon or principal payments when due) and on
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t h e "marketability u of t h e bond. In addition, modern approaches acknowledge
t h e influence on required returns t h a t result from call provisions, t h e t a x effect f o r deep discount bonds (due t o t h e d i f f e r e n t t a x treatment of o r d ~ n a r y
Income vs. capital gains), and sinking fund payments (which reduce t h e avera g e maturlty of a firm's debt). Isolating t h e influence of default likelihood on
interest- rate differentials involves controlling f o r these o t h e r effects.
As Jones, Mason and Rosenfeld (1984) pointed out, studies t h a t a t t e m p t
t o explain corporate bond prices c a n b e identified 3s being e i t h e r "macro N in
nature, in t h a t relative bond prices a r e modeled as being "functions of t h e
supply and demand of various assets, and/or t h e position of t h e economy in
t h e business cycle," o r as being "microw in t h a t relative prices a r e modeled .as
a function of firm specific characteristics. The approach taken in this paper
i s macro; t h a t is, t h a t a g g r e g a t e returns on a sample of bonds a r e used t o
infer a v e r a g e default probabilities for t h e population of bonds with similar
characteristics.
In order to test hypotheses concerning t h e models derived in this paper,
w e assume t h a t bond market participants a r e in complete agreement as to t h e
probability of d e f a u l t f o r a particular issue. W e f u r t h e r assume t h a t all bonds
a r e perceived, and t h e r e f o r e priced, as having t h e same likelihood of default
as o t h e r s in t h e same rating category. Ideally, t h e assigned rating gives, in a
single measure, t h e rating agency's estimate of t h e issue's probability of default.
Studies t h a t h a v e a t t e m p t e d to measure t h e importance of rating
changes on bond price movements tend to differ in t h e i r conclusions. Hettenhouse and Satoris (1976) as well as Weinstein (1977) conclude t h a t market
participants incorporate new information before a rerating. On t h e o t h e r
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hand, Eder~ngton, Yawitz, and Roberts (1984) flnd t h a t t h e market responds
t o rating changes in addition to publicly available information. They also conclude t h a t ratings by Moody's and Standard & Poor's a r e equally reliable indic a t o r s of an issue's creditworthiness.
In order to isolate expected default probabilities, we will restrict our
attention to two broad classes of ratings: investment grade and speculative
grade. Investment grade corporate bonds carry a rating of Baa3 or higher
(from Moody's) and/or BBB- or higher (from Standard & Poor's), while speculative grade bonds consist of issues with ratings below these, a s well a s corporate bonds t h a t a r e nonrated.
Section I1 contains a discussion of t h e construction of t h e default r a t e
series used in this paper. In section 111 w e present a model of t h e pricing ofdefault- prone bonds in terms of their required y ~ e l d st o maturity and compare
derived implied default r a t e s with a c t u a l default experience. Section IV rep e a t s this exercise for holding period returns and discusses t h e complications
of using holding period returns. Section V investigates t h e relationship between changes in expected corporate default r a t e s and certain macroeconomic
measures. In section VI w e present a summary of t h e paper and some closing
remarks.
11. Measuring Corporate Defaults
Economists have tracked t h e performance of corporate d e b t beginning
with a study by W.B. Hickman (1958). His and almost every subsequent study
defines t h e r a t e of default as t h e value of issues defaulting during t h e period
examined divided by t h e value of bonds outstanding during some part of t h e
p e r ~ o d(usually t h e beginning). Altman and Nammacher (1985) (A&N hereafter)
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a r g u e that, since almost all defaults over t h e last few years have occurred in
t h e low-rated sector, t h e appropriate measure of t h e corporate default r a t e
includes only t h e value of low-rated bonds in t h e denominator.
In A&N, and in this study, only publicly held, straight (non-convertible)
corporate debt with a speculative (or no) rating is included in t h e denominator. Convertible bond defaults w e r e included in t h e numerator for our measurements, however, because of t h e likelihood t h a t market participants do not
differentiate between losses in this s e c t o r and t h e straight bond sector. In
t h e six-year period from t h e beginning of 1980 through 1985, a total of
$3.586 billion of corporate d e b t defaulted, roughly $1.021 billion of which
consisted of convertible issues. Eliminating t h e s e defaults would substantially
reduce our measured default rates. The a c t u a l default r a t e series presented
below i s therefore biased upwards.
r
A complication arises in t h e construction of a measure of t h e default
r a t e f o r bonds of a given rating: by t h e time a n issue defaults, i t h a s usually
descended in rating until i t h a s reached t h e rating D (for Default). We, therefore, limit our analysis t o t h e performance of all low-rated corporate debt.
Figure 1 presents a monthly time series plot of annualized default r a t e s for
January 1980 through December 1985. This was constructed by dividing t h e
par value of bonds defaulting at e a c h month by t h e par value of outstanding
low- rated bonds at e a c h date. T h e par value of defaulting issues was obtained
from A&N (up to December 1984) a n d Standard & Poor's Bond Guide (through
December 1985). Observations o n t h e par value of speculative grade bonds
outstanding were taken at t h e end of e a c h year from Standard & Poor's Bond
Guide. Estimates of outstanding bonds, by month, w e r e obtained by interpolating annual measures. The mean of t h i s default series is 1.883 percent, with a
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standard deviation of 3.297 percent. The series reaches a minimum of zero a t
several points, including part of 1981, and attains a maximum of 19.504 perc e n t in April 1982.
For some of t h e analysis, a smoothed default r a t e series is employed,
partly because of t h e volatile nature of actual default rates. A smoothed version was constructed by summing at e a c h d a t e defaults occurring over t h e
past 12 months and dividing by outstanding low-rated bonds six months
earlier. This is essentially a 12-month moving average of t h e monthly default
r a t e series. From January 1980 through December 1985, t h e average value of
our smoothed default r a t e series is 1.796 percent, roughly corresponding t o
A&N1s estimate of 1.507 percent (obtained from year-end observations for
January 1978 through December 1984). Our smoothed default r a t e series h a s a
standard deviation of 1.273 percent. The maximum value of 4.756 percent
occurs in November 1982, while t h e minimum of 0.045 percent occurs in October 1981. A summary of these series, and all subsequent d a t a series, c a n be
found in table I.
W e note t h a t t h e constructed series a r e based on the assumption t h a t
defaults result in a t o t a l loss t o bondholders. In fact, AhN find t h a t defaulting bonds continue to t r a d e at 41 p e r c e n t of par within one month following
t h e default. Therefore, t h e a c t u a l ttloss rate t 1 i s somewhat lower than our default r a t e estimates.
111. Default R a t e s and Yields t o Maturity
Our theoretical model is based on t h e pioneering work of Bierman and
Hass (1975), with subsequent extensions by Yawitz (1977). The proposed model
is in t h e same spirit as t h a t used by Yawitz, Maloney,and Ederington (1983)
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to model yield spreads in the municipal bond market on the basis of differing
default probabilities and tax effects.
1
Most asset-pricing models are based upon the first and second moments
of the return distribution. With risk-neutral preferences, on the other hand,
agents consider only the first moments of the distributions of return: the security's expected return completely determines i t s market price. This framework facilitates the construction of -a certainty-equivalence pricing relationship.
Assume that a promised coupon (or principal) payment will be rendered
at the end of a given period with a perceived probability
P. A payment pro-
.A
t
mised t periods from now i s expected to be received with probability P
default occurs (and applies only to payment streams for which there havebeen no previous defaults) with probability (1-PI. In the event of a default, a
fraction of the promised coupon and principal payments is received, denoted
here by p.
I f capital markets are frictionless, and information is costless, arbitrage
will force the market price of a certainty-equivalent (default-risk-adjusted)
payment stream, discounted at the riskless rate of interest to be equal t o a
risky stream, discounted at the appropriate risky rate of interest. Algebraically:
where i is the riskless rate of interest, r is the risky rate of interest, C is
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t h e promised risky coupon rate, and N is t h e number of years to maturity.
Using a geometric sum formula to express (1) without summations, we
have:
The yields to maturity, i and r, a r e for bonds t h a t a r e identical in all
respects except for t h e likelihood of default. Further simplifications of t h e
above expression a r e possible, if one approximates t h e finite-maturity coupon
bond with a perpetuity, and if one assumes t h a t default results in a total loss
2
t o debt holders.
A more general specification of equation ( I ) would involve time-sub-
scripts for t h e variable P, so t h a t payment r a t e s would b e allowed t o vary
over calendar time (hence t h e term structure). The product of t h e PIS from
t h e initial d a t e t o t h e relevant payment d a t e s would replace P t in t h e first
term in t h e numerators of t h e l e f t hand side of (1). The product of t h e P's
from t h e initial d a t e t o t h e d a t e preceeding t h e payment d a t e would b e multiplied by 1 minus t h e expected payment r a t e in t h e relevant period for t h e
second term. Of course, t h e r e is no way to identify t h e values of t h e separ-
ate expected probabilities of payments. In addition, a closed-form solution
like t h a t of equation (2) could not b e found. The use of a single, constant
measure of P c a n b e interpreted as a n "average" likelihood of payment, summarizing expectations of f u t u r e payment rates.
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A. Yield t o Maturity Data.
An index of yields to maturity for low-rated bonds was obtained from
Salomon Brothers 1 Corporate Bond Research department. The index used begins at t h e end of 1979 and is constructed from a sample of 176 bonds (as of
September 19851, weighted by t h e outstanding principal amount of each issue
(to control for e a c h issue's relative influence on market rates) t h a t meet t h e
following criteria: 1) more than $25 million in principal outstanding (assuring
adequate marketability), 2) ratings below Baa31BBB-, o r not rated but of
lower than Baa31BBB- quality, 3) a coupon of 10 percent o r more, and 4)
longer than 10 years in maturity.) In addition, we w e r e able t o obtain t h e
weighted coupon r a t e s and weighted maturity d a t e for t h e sample at e a c h point in time. Defaulting bonds a r e removed from t h e sample, as a r e issues
t h a t are upgraded t o investment- grade status.
Complications arise in t h e analysis because of several uncontrolled factors. First, nearly all corporate bonds contain call provisions. In a sample of
702 currently outstanding, publicly held, low- rated (or nonrated) issues, all
but 32 had call provisions, and 97 w e r e being called as of January 1986. In
practice, many (high-coupon) low- rated bonds t r a d e on a yield-to-call basis.
Of course, high- grade corporate bonds also c a r r y call provisions. The f a c t
t h a t t h e low-rated sample consists of high-coupon issues, increases t h e likelihood t h a t they would b e called if interest r a t e s fall significantly (or if t h e
f irmls financial condition warrants a n up-grading). This and o t h e r f a c t o r s
imply t h a t t h e r e is no comparable high-grade index t h a t will exactly match
e a c h of t h e characteristics (apart from default risk) of t h e low- rated sample.
As a compromise, w e chose to use Salomon Brothers' New Medium Term
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Industrials index f o r Aaa rated bonds, found in their Analytical Record of
Yields and Yield Spreads. This series is based on estimates (by Salomon's Syndicate Department) of t h e required yields on issues coming t o market t h a t a r e
rated Aaa and will mature in 10 years. These estimates were made a t t h e
beginning of month t + l and were aligned with t h e low-rated index observations t h a t were taken on t h e last day of month t.
It has been observed t h a t new issues a r e priced at yields slightly higher
than t'seasoned" issues, d u e t o their relative lack of liquidity. The Aaa/AAA
rated yields were chosen t o represent t h e default- risk- free r a t e s largely because of t h e lack of defaults by bonds originally issued with this rating in t h e
past 15 years. Using t h e yields on long-term U.S.
Treasury issues a s t h e
default- free yield would complicate t h e analysis, because these securities lack-.
call provisions, and because their returns a r e subject t o different t a x t r e a t ment. In addition, t h e sheer volume of transactions involving Treasury bonds
introduces t h e possibility t h a t yield differentials r e f l e c t a marketability
factor.
4
Finally, cross- sectional variations in t h e measured returns of a sample of
representative bonds can b e attributed t o firm-specific idiosyncrasies. It i s
assumed t h a t t h e a v e r a g e measured returns will vary systematically a s t h e
result of a f a c t o r t h a t is tied t o t h e default experience of corporate bonds.
The use of a weighted average of bond returns causes t h e influence of t h e
idiosyncratic variations t o cancel one another. Hickman (1958, p.66) discusses
t h e difficulties of using (weighted) average returns as measures of t h e return
on a pooled investment portfolio. He concludes t h a t under most conditions,
t h e e r r o r will b e negligible.
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8. Comparison of Actual and Implied Default R a t e s
A numeric solution program was employed t o solve (2) for t h e expected
"payment rate", P, given supplied values for r,
I,
p, C,
and N, at t h e end of
each month t. This is a measure of t h e cross-sectional average of ~ m p l i e d
expected payment rates, based on t h e yields of a cross-section of low-rated
bonds. A problem emerges, however, because of t h e aggregation procedures
used.
L e t us assume t h a t P i s a n implicit function of r (with i, C, p, and N held
fixed). Since (2) cannot b e solved explicitly f o r P, a computer simulation was
employed to graph t h e implicit function with restrictions on t h e values of t h e
o t h e r variables and a n assumption about t h e relationship between C and r.
Figure 2 i s a graph of t h e simulation. Note t h a t when t h e payment r a t e P i s '
equal t o 1, t h e risky r a t e takes on t h e supplied value of t h e riskless r a t e (10
percent here). The relationship between P and r is shown t o b e convex in t h e
relevant range. Jensenls inequality, therefore, suggests t h a t t h e cross-sectional average of P will b e g r e a t e r than, o r equal to, t h e measured payment rate.
This implies t h a t our e s t i m a t e of (I-PI, t h e implied expected default rate, is
biased downwards.
A plot of (I-PI, t h e expected default r a t e implied by our model of yield
differentials, is presented in figure 3 along with a plot of t h e moving average
default series. The f a c t t h a t t h e implied expected default r a t e series appears
t o track, and even lead, ltactualll default r a t e s so well i s surprising, given
t h a t t h e implied r a t e represents a n average of expected f u t u r e default rates.
This behavior indicates a d e g r e e of myopia on t h e p a r t of market participants. The spread between implied and a c t u a l (smoothed) default r a t e s is also
surprisingly large and persistent over this period.
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Acknowledging t h e statistical complications introduced by t h e construction of these variables, one may gain additional insight by using regression
techniques. Cochrane- Orcutt adjusted regressions of (1-P) on constants and
the "raw tt default r a t e series, ADR, a s well a s t h e smoothed default r a t e series, SADR, a r e presented below in table 2. These regressions indicate t h a t
t h e r e is some connection between measured implied default r a t e s (based on
risk-neutral preferences) and t h e two actual default r a t e series. The adjusted
R-squares of 10.2 p e r c e n t and 11.1 percent, respectively, indicate t h e perc e n t a g e variation in t h e implied default r a t e series t h a t is "explained" by t h e
two measures of a c t u a l default rates. The large t- statistics for t h e constant
terms cause us to r e j e c t t h e null hypothesis t h a t t h e market's (risk-neutral)
estimate of default r a t e s equals actual default r a t e experience. In fact,'.
evidence suggests t h a t market prices imply default r a t e s t h a t exceed a c t u a l
default r a t e s by roughly 5 percentage points.
IV. Default Experience, Holding Period Yields, and Ex-post Performance
In this section, w e apply t h e default-risk- neutral framework t o t h e
pricing of risky d e b t in terms of t h e expected holding period yields on
default-prone and default- free bonds. A bond's holding period return embodies
changes in t h e market price as well as coupon earnings (pro-rated for t h e
holding period). Define Bt t o b e t h e default- free bond's market price at t h e
end of period t, and C t t o b e t h e promised coupon payment earned in period
t. Now l e t t h e holding period return f o r a default-risk-free bond b e defined
by Ht, such that:
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The corresponding gross return to t h e holder of a default- prone bond with
price B l i and coupon C l i in period i is represented by:
Note t h a t H and h a r e period-specific returns, in turn, convertible t o annual
t
t
rates.
Now l e t mt b e t h e percieved probability t h a t a n issuer will not default
over period t, conditional upon a default not having previously occurred. If
t h e period under consideration is a single month, then (mt)12 i s t h e expected
likelihood t h a t t h e firm will not default over a given year.
L e t us further assume t h a t in t h e e v e n t of a default, t h e holder of t h e
risky bond will r e c e i v e with certainty a fraction p of t h e beginning period
price B*t-l.
The investor's expected (net of default) r e t u r n o n t h e risky bond,
E(ht), i s therefore given by:
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For t h e certain case in which m equals 1, E(ht) will equal h
t
t'
whereas in t h e
c a s e of certain default, E(h ) will equal (p-l), resulting in a loss to t h e
t
bondholder. In t h e absence of market imperfections, equilibrium in t h e riskneutral setting requires t h a t t h e expected net-of-default return on t h e
default-prone and the default- free bond will be equal. Setting t h e right-hand
side of ( 5 ) equal to Ht and using equation (4), we have:
Subtracting both sides of (6) from ht and rearranging, gives:
where (I-mt) is t h e period- specific expected default r a t e embodied in t h e
holding period yields of t h e default-prone and default- free securities, given
a n assumed recovery rate, pO5 Note t h a t (7) represents a risk-neutral, e x a n t e
relationship between expected holding period returns and expected default
rates.
A bond's realized holding period return, however, i s a n e x post measure
of performance. Conversely, measured yields-to-maturity a r e based on expect e d performance and embody e x a n t e expected default rates. Bond holding
period returns may deviate from expected returns, limiting our ability to
measure implied default r a t e s from t h e difference between holding period
yields of risky and risk-f r e e bonds.
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Consider a short-run increase in t h e expectation of corporate defaults.
C e t e r i s paribus, this would have t h e effect of reducing t h e prices of outstanding low-rated bonds, thereby reducing t h e measured holding period return
h
. IJnder
t
most conditions, this would lower t h e "implied" default r a t e (1-mt). 6
Indeed, below we show t h a t relatively short-run price movements (resulting
from new default information) c a n cause t h e r~ght-hand side of (7) t o take on
negative values, thereby violating t h e definition of a probability. Therefore,
(7) cannot b e used t o obtain implied default rates. What o n e obtains from
applying this formula t o e x post returns is a differential "performance rate"
for low-rated bonds.
A. Holding Period Data.
A proxy for ht was constructed monthly by Blume and Keim (1984) based
on t h e price movements and coupon payments of t h e bonds used in Salomon
Brother's Low-Rated (or High Yield) Bond Index (discussed above). The
"merged1' series s t a r t s at t h e end of January 1980 and covers through J u n e
1984. It has a mean of 1.14 p e r c e n t (for a n equivalent annual average return
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of 14.57 percent) and a standard deviation of 4.09 percent.
As a measure of t h e holding period returns on default- free bonds, Ht, w e
used Salomon Brother's High G r a d e Index f o r t o t a l rate-of-return
found in
their Analytical Record of Yields and Yield Spreads (up to December 1985).
The index was formed by calculating t h e t o t a l returns of roughly 900 issues
with weights based on issue size. The weights a r e revised monthly, and bond
issues a r e included and deleted as ratings a r e updated. The a v e r a g e weighted
maturity of t h e issues at t h e end of 1985 was 22.1 years. This series i s also
used as a benchmark return in Blume and Keim (1984).
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8. Comparison of Actual and Implied Default Rates.
In order to minimize extraneous influences on holding period yields, a
holding period of one year was selected, in addition to the one-month holding
period. The Blume-Keim series was converted t o an annual return series by
accumulating monthly returns over t h e past year a t each month. That is, the
measured annual holding period return at each d a t e is based on t h e returns to
bond holders who sold a security purchased one year earlier (and collected
coupon payments for t h e period). With these measures of return, (7) implies
t h a t annual performance r a t e s a r e estimated.
In figure 4, we present a plot of t h e performance r a t e implied by equation (7), obtained from annual holding period measures, along with t h e historic
moving average default rate, SADRt. Confirming our intuition, negative performance r a t e s exist when actual d e f a u l t experience is highest. The performance rate, (I-mt), reaches a minimum value of -0.1009 in November 1982,
t h e month following t h e maximum value reached by t h e smoothed actual
default r a t e series. It i s clear t h a t periods corresponding t o negative performance r a t e s a r e those in which one- year holders of low-rated bonds realized
significant losses. In general, t h e performance r a t e series descends as actual
default r a t e s rise, and vice-versa.
In t a b l e 2, w e present t h e regressions of t h e measured performance r a t e s
(expressed in annual terms and based on one- and 12-month holding period
yields) on t h e two actual default r a t e series. The low R-squares indicate t h a t
relatively l i t t l e of t h e variation in performance r a t e s is explained by a c t u a l
default rates. The negative coefficients on actual default r a t e s and t h e significant t- statistic on t h e smoothed default r a t e series supports t h e observa15
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tion of a negative correlation between a c t u a l default r a t e s and t h e performance of low-rated bonds. The significant (and positive) t- statistics on t h e
constant terms of t h e regressions, using a performance r a t e series formed
from 12-month holding period yields suggest t h a t , on average, holders of lowr a t e d bonds realized significant holding period gains relative t o their highgrade counterparts.
V. Default Expectations and Macroeconomic Measures
In this section, a n a t t e m p t is made t o allow for t h e influence of o t h e r
macroeconomic variables, in addition t o a c t u a l corporate default rates, on
implied default and performance rates. P a s t studies of differential quality
spreads have used a n assortment of macroeconomic indicators. J a f f e e (1975)
examines f a c t o r s t h a t influence t h e risk spread of corporate yields in a cyclic a l fashion. He finds t h a t t h e most significant variable in explaining t h e risk
spread i s a measure constructed by Fair (19711, based on d a t a collected by
t h e University of Michigan Survey Research Center, which acts as a proxy f o r
consumer sentiment. This f a c t o r was also used by Cook and Hendershott
(19781, in addition t o others, to explain t h e spread between high-grade corpor a t e and Treasury securities. Rather than t a k e this approach, implied default
and performance r a t e s a r e t e s t e d f o r correlation with new default information and surprises in macroeconomic measures.
I t i s well known t h a t in periods of (unanticipated) rising prices, firms
with fixed nominal contractual obligations tend to benefit. Conversely, (unanticipated) reductions in prices may c a u s e hardship to some firms. Since expect e d inflation will already b e incorporated into t h e contracts, it is t h e unan16
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ticipated part of inflation t h a t will a f f e c t t h e probability of default. Therefore, a natural macroeconomic proxy is t h e deviation of t h e percentage
change in t h e consumer price level from expectations. Other indicators of
macroeconomic activity a r e the Board of Governors of t h e Federal Reserve's
industrial production index and t h e Labor Department's unemployment r a t e
estimate.
Two characteristics of our sample period tend t o limit t h e effectiveness
of this exercise, however. The first is t h e relatively short sample period
available to us. The size of t h e market f o r low-rated bonds approached significance only towards t h e end of t h e 1970s. The identification of long-run relationships is, thus, seriously hampered. Secondly, in t h e sample period of this
study, t h e overall inflation r a t e was, on average, falling, a f t e r a long period
of accelerating inflation. The e f f e c t s of this regime switch on t h e reported
results is indeterminate, introducing t h e possibility t h a t t h e behavior of
market participants over a longer period may well differ from t h e behavior
exhibited here.
To test f o r a relationship between unanticipated inflation r a t e s and our
estimates of implied default and performance rates, w e constructed a n unanticipated inflation series by subtracting one-month-ahead forecasts of t h e perc e n t a g e change in t h e CPI (obtained from Money Market Services) from
a c t u a l monthly percentage changes. Similar series were constructed f o r
measures of t h e unemployment r a t e and t h e percentage change in industrial
production (a monthly proxy for GNP). O n e would expect that, if a g e n t s
incorporate new information about t h e economy (in addition t o firm-specific
factors) into their expectations of default rates, these proxies will b e related
to changes in expected default rates.
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For vat-ious sample periods, we regressed t h e first differences of implied
default rates, obtained from differential yields t o maturity (I-P), on a constant, a c t u a l d e f a u l t r a t e s (in levels a s well as first differences), unanticipated inflation, unanticipated industrial production, and unanticipated unemployment. The macroeconomic surprises were lagged o n e month, a s the tirning
of t h e actual series normally lags t h e reported period by a f e w weeks. The
results, found in t a b l e 3, indicate t h a t of t h e t h r e e macroeconomic indicators,
surprises in reported measures of industrial production have t h e highest correlation with implied expected default rates, although t h e level of actual
d e f a u l t r a t e s contributes slightly more. When t h e f i r s t differences of a c t u a l
default r a t e s a r e used, t h e surprise in inflation appears t o have t h e highest
(negative) correlation with expected default rates. However, no variable
e n t e r s significantly in e i t h e r regression at t h e 95 percent confidence level.
The low adjusted R-squares also leads us t o conclude t h a t current macroeconomic surprises a r e poor indicators of expected default rates.
The same regressions, adjusted for serial correlation of t h e error terms,
were run using t h e implied performance r a t e (based on one-month holding period yields, converted t o annual rates) in place of expected default rates. Also
found in t a b l e 3, t h e s e results suggest that, though insignificant at t h e 95
percent confidence level, surprises in inflation a r e most closely related (positively) t o performance rates. I t must b e t h e case t h a t firm-specific f a c t o r s
dominate t h e formation of default expectations t o t h e point t h a t surprises in
macroeconomic measures a r e poor predictors of overall quality spreads.
VI. Summary & Conclusions
This paper represents t h e f i r s t e f f o r t t o tie together the. differential
18
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returns required by holders of low-rated corporate bonds and t h e actual default experience of these issues. A model of t h e behavior of low-rated bond
pricing was developed in a risk-neutral setting. W e applied t h e model t o the
observed returns of a sample of bonds and compared t h e default r a t e s implied
in these returns to t h e default experience of low-rated debt. W e conclude
t h a t t h e default r a t e s implied in corporate bond returns exceed those experienced in recent years. In this sense, holders of well-diversified portfolios of
low-rated corporate bonds a r e rewarded for bearing default risks. It was also
shown t h a t measured holding period returns cannot be used t o e x t r a c t implied
default rates.
Finally, w e examined t h e relationship between a s e t of macroeconomic
variables and expected measures of default and performance rates. W e conclude t h a t expected corporate default r a t e s a r e not related t o any of t h e
macroeconomic variables at t h e 5 percent critical level, although expected
default r a t e s were most strongly related t o surprises in inflation measures and
actual default rates. Surprises in output proxies appear t o have less of a relationship t o expected default rates.
Further study in this a r e a will require t h e accumulation of b e t t e r (and
more detailed) measures of corporate bond returns. The construction of a
standardized d a t a base, modeled a f t e r t h e C e n t e r for Research on Security
Prices (or CRISP) tapes, would most benefit f u t u r e endeavors in this field. In
addition, a longer sample period would increase our understanding of both t h e
pricing of default risk and t h e relationship between expected default r a t e s
and macroeconomic activity.
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Footnotes
1. Yawitz, Maloney, and Ederington (1983) do not compare their estimates of
default r a t e s in t h e municipal market with a c t u a l rates.
2. If o n e approximates t h e finite- maturity coupon bond with a perpetuity--
multiply t h e left-hand side of (2) by ( ~ + i ) - ~ / ( l + i )and
' ~ t h e right hand side by
( ~ + r ) - ~ / ( l + r and
) - ~ l e t N approach infinity--then (2) becomes:
If we assume t h a t default results in a t o t a l loss t o holders
(pol, then this
becomes:
3. For t h e 176 issues in Salomon Brother's Low-Rated Index, as of September
1985, 23 w e r e r a t e d BB (by Standard & Poor's), 26 w e r e r a t e d B+, 34 w e r e
r a t e d B, 45 w e r e r a t e d B-, and 48 w e r e r a t e d CCC. AdtN find t h a t t h e
highest default- risk group (in terms of rating at issuance) w e r e bonds r a t e d
single 0. This index, therefore, represents t h e average returns of t h e riskiest
corporate bonds.
4. Coupon payments received from Treasury securities a r e currently exempt
from state and local income taxes. Ibbotson and Sinquefieldls (1982) measured
default premium, constructed by subtracting t h e ex-post holding returns o n
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Treasury bonds from AAA/Aaa rated corporates, may mostly reflect this tax
differential.
5. As in footnote 2, the assumption that default results in a total loss to
bondholders (that is, p=0) gives:
6. The partial derivative of (1-mt) with respect to ht is:
{Ht + (l-p)IAht + (1-p#
2
,
and will be positive when Ht>(p-1). The smallest value reached by the onemonth holding period return on the high-grade series from January 1980
through June 1984 is -0.0799. The 12-month holding period minimum for this
rate is -0.1296. Based on Altmants estimates, (p-1) equals -0.59, implying that
this condition will be met under most circumstances.
7. See Blume and Keim (1984) for a description of this series.
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Table 1 Summary of Measured and Constructed Series
Mean
Standard deviation
Minimum
(date)
12/79 through 12/85
ADRt
0.01857
(I-month)
ADR .
0.0 1776
(12-month)
1/80 through 6/84
12/80 through 6/84
(1-m)
0.08858
(fr. 12-month HPY)
t Expressed as an annual rate.
0.00000
(23 points)
Maximum
(date)
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Table 2 Implied Default and Performance Rates on Actual and Smoothed
Default Rates
Dependent variable
Const.
ADR
SADR
R~
DW
(I-m)
(fr. I-month HPY)
(1-m)
(fr. 12-month HPY)
(I-m)
(fr. 12 month HPY)
Note: All regressions were run using the Cochrane-Orcutt procedure for
first-order serial correlation. The reported Durbin-Watson statistics are less
powerful when the serial adjustment technique is used. The t-statistics are
reported in parentheses.
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Table 3 Implied Default and Performance R a t e s on Actual Default R a t e s and
Macroeconomic Surprises
(Dependent Variable First-Differenced)
Dependent
variable
Const.
ADR
Infl.
1nd.Prod.
Unemp.
(Dependent Variable and ADR First-Differenced)
*-.
'K
DW
.
t Run using t h e Cochrane-Orcutt procedure for first-order serial correlation.
The t- statistics a r e reported in parentheses.
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Figure 1: Actual Default R a t e Series (in Annual Rates).
Source: Altman and Nammacher (1985), and Standard & Poor's Bond Guide.
25
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Figure 2: Plot of r against P (with p=.41, i=.l,
C=.913rt,and N=14).
t: The partial adjustment for coupon rates is based on the fact that corporate
bonds were trading, on average, at 91.3 percent of par value.
Source: See text.
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Figure 3: Time S e r i e s of (1-P) and Smoothed Actual Default Rates.
-
SADR
a - 0 - 0 - - -
I
I
1
-
/- - -
I
I
I
I
--J
~
~
Source: See text.
-1
I
I
I
~
~
~
~
/\
\
I
I '
\"
/'' \,*I
4
'
t
b
1I
I
I
I
',
/
\
,\
I
J
/
I'
I\
~
\
I
I
I
I
I
I
I
I
-
-
I
I
~
~
~
~
I
~
~
~
~
~
~
~
~
~
I
I
I
I
~
I
I
I
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F ~ g u r e4: Time Series Plot of (I-m), Constructed from Annual Holding Period
Yields and Smoothed Actual Default Rates.
Legend
-
--------
-
I
---
I - - - - -
1
1
1
1
1
1
1
1
'@------e-
1
1
Source: See text.
1
1
1
1
1
1
1
1
1
t
l
l
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
I
I
I
I
I
.
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References
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High Yield Corporate Debt," Morgan Stanley & Co., Incorporated, March
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Weinstein, Mark. "The Effect of a Rating Change Announcement on Bond
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~ l f f e r e nDefault
t
~ecoveries,"Journal of Financial and Quantitative
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Risk, and Yield Spreads," NBER Working Paper, No. 1215, October 1983.
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