The Cross-Section of Expected
Corporate Bond Returns: Betas or
Characteristics?
William R. Gebhardt
Søren Hvidkjær
Bhaskaran Swaminathan
Journal of Financial Economics, 2005
Introduction

This paper investigates the ability of betas and
characteristics in explaining the cross-section of
corporate bond returns.


In equities: Fama-French (1992) vs. Fama-French
(1993) vs. Daniel-Titman (1997) vs. Davis-FamaFrench (2000)
Two identifiable sources of risk: Default risk and
term risk. Betas are then loadings on these
factors, while characteristics are bond ratings and
duration.

Focus on systematic vs. idiosyncratic risk, rather
than risk vs. mispricing.
2
Key Findings


Betas, rather than characteristics, appear to be
priced in the corporate bond market

Default betas vs. ratings: Default betas matter,
weaker evidence of ratings being priced

Term betas vs. duration: Term betas matter, but
evidence is weaker
Strong (but less surprising) evidence that YTM
predicts the cross-section of returns
3
Outline


Data: LBFI Database and our sample
The explanatory variables



Bond market factors: term and default betas
Bond characteristics: ratings and duration
The two test methods:

Portfolio tests
• Univariate sorts on betas and characteristics
• Conditional sorts on betas and characteristics
– And on Yield-to-maturity


Fama-MacBeth regressions
Conclusion and practical implications
4
Data



Lehman Bros. Fixed Income Database.
Month-end bid prices, ratings, yields, etc. in
the period 1973-1997 for a large cross-section
of U.S. corporate bonds.
Trader quotes and matrix prices.


No transactions prices, but input to traded indices
Some missing values.
5
Our Sample

To be included in our sample, each bond must






have no more than 4 missing values
be non-convertible
be coupon-bearing
have at least 3 years to maturity
be investment grade
This leaves around 3000 bonds per year.
6
Bond Market Factors

Following Fama-French (1993), we define:


TERM: The difference between the monthly returns
on a portfolio of long-term U.S. government bonds,
and one month T-bills.
DEF: The difference between the monthly returns
on a value-weighted portfolio of all investmentgrade corporate bonds with at least 10 years to
maturity, and a portfolio of long-term government
bonds.
• Results qualitatively identical with alternative DEF
factor based on BBB bonds only (not presented)
7
Factors: Summary Statistics
Mean
Std
Min
Max
DEF
0,04%
1,20%
-5,37%
5,48%
TERM
0,20%
3,18%
-9,37%
13,95%
Percentage monthly returns.
Table 1a
8
The Factor Model:
Portfolio-level regressions
rp  rf     d DEF  t TERM   u
AAA
AA
A
BBB

-0,01
d
0,82
t
0,92
(-0,60)
(28,67)
(72,45)
-0,02
0,87
0,90
(-1,46)
(44,50)
(109,17)
0,00
0,96
0,88
(-0,26)
(38,69)
(109,51)
0,04
1,08
0,88
(0,97)
(15,69)
(46,56)
R2
0,98
0,99
0,99
0,93
Gibbons, Ross, Shanken Ftest:
F-statistic
0,781
P-value
0,538
Table 1b
9
The Factor Model: ex ante betas as
measures of systematic risk
ri  rf     d DEF   t TERM   u

Regression is estimated for each bond each
month, using prior 60 months return. Beta
estimates are then used to form portfolios, and
in Fama-MacBeth regressions.
10
Bond characteristics
as measures of total risk

Default risk measure:



Corporate bond ratings from Moody’s/Standard &
Poor: AAA, AA, A, BBB
S&P rating used if available. Otherwise, Moody’s
Term risk measure:

Modified duration: price sensitivity of the bond to a
parallel shift in the yield curve
11
Univariate Sorts: Ratings
Table 2a
AAA
AA
A
BBB Diff t(diff)
N
210
787
1226
674
d
0,72
0,81
0,98
1,26
t
0,96
0,95
0,94
0,95
Maturity
Duration
Rating
18,79
7,29
2
16,42 15,23 14,6
6,79 6,43 6,15
4,2
6,93 9,89
Return
0,26
0,27
Duration adjusted return
-0,02 -0,01
0,27
0
Yield
10,16 10,8
9,75
9,92
0,33 0,07 (1,31)
0,04 0,06 (1,51)
12
Univariate Sorts: Duration
N
d
Low
579
0,73
t
Maturity
0,70
5,23
Duration
Rating
3,50
6,46
5,33
7,01
6,43
6,73
7,54
6,41
8,73
5,07
Return
0,24
0,29
0,30
0,31
0,28
Dur/Rating Adj. Ret.
0,00
0,01
0,00
0,01
-0,01 -0,01 (-0,67)
Yield
9,97
Table 2b
580
1,00
580
1,05
580
1,07
High
579
0,93
diff t(diff)
0,87 0,97 1,03 1,08
10,39 14,60 18,53 24,83
0,04 (0,37)
10,18 10,21 10,30 10,13
13
Univariate Sorts: Default beta
N
d
Low
579
0,40
t
Maturity
0,77
14,04
Duration
Rating
6,00
5,39
6,65
5,41
6,83
5,87
6,85
6,59
6,77
7,95
Return
0,21
0,25
0,28
0,31
0,35
0,13 (2,54)
Dur/Rating Adj. Ret.
-0,02
-0,02
0,00
0,01
0,05
0,07 (3,37)
Yield
10,05
10,00 10,10 10,21 10,48
Table 2c
580
0,76
580
0,95
580
1,15
High
579
1,65
diff t(diff)
0,93 0,97 1,00 1,09
16,69 16,81 16,50 16,53
14
Univariate Sorts: Term beta
N
d
Low
579
0,60
t
Maturity
0,60
9,46
Duration
Rating
4,65
6,43
5,76
6,30
6,67
6,12
7,45
5,92
7,94
6,16
Return
0,22
0,24
0,28
0,30
0,33
0,11 (1.00)
Dur/Rating Adj. Ret.
-0,03
-0,02
0,00
0,01
0,04
0,07 (1.66)
Yield
10,26
10,04 10,09 10,13 10,20
Table 2d
580
0,85
580
0,99
580
1,03
High
579
1,24
diff t(diff)
0,85 0,97 1,06 1,21
13,09 16,46 19,09 20,26
15
Returns to default beta portfolios
within rating/duration portfolios
Panel A: Pre-ranking Default Beta
Char. portfolio
bd
Rating
Duration
1
2
3
Diff
high
short
0,219 0,199
0,220
0,002
high
med.
0,266 0,272
0,303
0,038
high
long
0,242 0,280
0,315
0,073
t(diff)
(0,05)
(1,18)
(2,05)
med.
short
0,213
0,214
0,256
0,043
(1,19)
med.
med.
0,263
0,297
0,350
0,087
(2,18)
med.
long
0,236
0,289
0,316
0,080
(2,37)
low
low
low
short
med.
long
0,280
0,284
0,201
0,314
0,308
0,242
0,378
0,371
0,305
0,098
0,088
0,104
(1,97)
(1,81)
(1,56)
245
0,268
0,313
0,068
(2,73)
Average
High default beta portfolios exhibit high returns. Effect (diff)
monotonically increasing in rating, duration.
Table 3a
16
Ex ante default betas predict ex post betas
Rating
Duration
1
2
3
bd
high
high
short
med.
0,565
0,883
0,648
0,973
0,776
1,026
high
long
0,886
0,953
0,974
med.
short
0,662
0,796
0,860
med.
med.
1,003
0,999
1,089
med.
low
low
long
short
med.
0,961
0,787
1,187
1,027
1,016
1,218
1,066
1,112
1,204
low
long
1,116
1,160
1,233
0,894
0,977
1,038
Average
High pre-ranking default beta portfolios have high post-ranking
default betas
Table 4b
17
Long-short default portfolio regressions:
the Daniel-Titman test
Panel B: Regression results for high default beta minus low default beta portfolio
Rating Duration
high
short
high
med.
high
long
med.
short
med.
med.
med.
long

-0,049
0,007
0,067
-0,004
0,068
0,068
d
0,211
0,143
0,088
0,198
0,086
0,105
t
0,162
0,094
0,012
0,150
0,061
0,033
t(t)
14,80
6,64
0,77
14,83
3,90
2,19
Adj. R2
-2,15
0,28
1,84
-0,19
2,26
1,99
t(d )
8,99
3,97
1,47
6,95
1,96
1,88
t()
0,65
0,30
0,01
0,62
0,13
0,04
low
low
short
med.
0,054
0,076
0,326
0,017
0,121
0,042
1,23
1,64
4,60
0,27
4,05
1,81
0,23
0,03
low
long
0,067
0,116
0,125
1,15
1,48
2,45
0,13
0,039
0,143
0,089
2,06
5,86
10,00
0,44
Average
Table 4d
18
Summary: default betas predict returns

So far:




Strong relationship between default betas and the
cross-section of future stock returns in univariate
portfolio setting
Same results within portfolios sorted on ratings and
duration.
Default betas survive the Daniel-Titman test
How about term betas?
19
Returns to term beta portfolios
within rating/duration portfolios
Char. portfolio
Rating
Duration
high
short
1
0,197
bd
2
0,207
3
0,235
Diff
0,038
t(diff)
(0,62)
high
high
med.
med.
med.
long
short
med.
0,252
0,236
0,215
0,270
0,275
0,275
0,217
0,291
0,307
0,307
0,259
0,352
0,055
0,071
0,045
0,082
(1,06)
(1,21)
(0,70)
(1,59)
med.
low
low
low
long
short
med.
long
0,230
0,282
0,271
0,176
0,296
0,322
0,336
0,268
0,315
0,387
0,355
0,316
0,085
0,105
0,085
0,140
(1,42)
(1,23)
(1,25)
(1,62)
0,237
0,276
0,315
0,078
(1,51)
Average
Relationship between term betas and returns is weaker, but monotonic.
Table 3b
20
Ex ante term betas predict ex post betas
Charac. Portfolio
Rating
Duration
Preranking Term Beta Portfolio
1
2
bt
0,452
0,609
0,806
0,934
0,955
1,075
0,489
0,663
0,824
0,938
0,909
1,071
3
high
high
high
med.
med.
med.
short
med.
long
short
med.
long
0,743
0,979
1,128
0,782
1,008
1,098
low
short
0,519
0,697
0,839
low
low
med.
long
0,862
0,916
0,993
1,084
1,050
1,136
High pre-ranking term beta portfolios have high post-ranking term
betas
Table 5c
21
Long-short term portfolio regressions:
the Daniel-Titman test
Panel B: Regression results for high term beta minus low term beta portfolio
2
Rating Duration
a
bd
bt
t(a)
t(b d )
t(b t ) Adj. R
high
short
-0,047
0,242
0,291
-1,98
7,78
24,94
0,86
high
med.
0,007
0,076
0,172
0,17
1,47
7,76
0,47
high
long
0,030
-0,108
0,174
0,73
-2,04
8,56
0,55
med.
short
-0,044
0,296
0,293
-1,48
8,58
21,48
0,79
med.
med.
0,030
0,083
0,184
0,86
1,64
9,93
0,55
med.
long
0,036
-0,014
0,189
0,83
-0,24
8,51
0,51
low
short
0,003
0,476
0,319
0,05
4,15
9,07
0,49
low
med.
0,034
0,035
0,188
0,62
0,47
7,64
0,36
low
long
0,078
0,101
0,221
1,15
0,94
3,43
0,28
0,014
0,132
0,226
0,59
3,79
18,31
0,79
Average
Table 5d
22
Summary: default betas predict returns, term
betas less clear

So far:




Strong relationship between default betas and the
cross-section of future stock returns in univariate
portfolio setting. Weak for term betas
Same default beta results within portfolios sorted
on ratings and duration. Weak, but monotonic
results for term betas
Default and terms betas survive the Daniel-Titman
test
How about characteristics: ratings and
duration?

Reversing the sorting order
23
Returns to ratings portfolios within default
and term beta portfolios
Factor portfolio
d
t
1
1
Diff
0,071
t(diff)
(2,38)
1
2
0,227
0,244
0,249
0,022
(0,57)
1
3
0,309
0,264
0,314
0,055
(1,23)
2
1
0,202
0,233
0,328
0,126
(2,69)
2
2
0,253
0,360
0,279
0,026
(0,76)
2
3
0,295
0,302
0,319
0,023
(0,61)
3
3
3
1
2
3
0,254
0,307
0,341
0,354
0,335
0,930
0,259
0,359
0,770
0,005
0,052
0,036
(0,08)
(1,03)
(0,64)
0,259
0,274
0,304
0,045
(1,49)
Average
Table 6a
Panel A: Ratings
Rating portfolio
high
2
low
0,183
0,241
0,255
24
Returns to duration portfolios within
default/term beta portfolios
Factor portfolio
d
t
1
1
Diff
-0,003
t(diff)
(-0,03)
1
2
0,258
0,450
0,232
-0,026
(-0,41)
1
3
0,317
0,258
0,262
-0,056
(-0,98)
2
1
0,246
0,284
0,258
-0,013
(0,19)
2
2
0,277
0,272
0,289
-0,020
(0,43)
2
3
0,332
0,295
0,319
-0,043
(0,94)
3
3
3
1
2
3
0,271
0,346
0,385
0,339
0,326
0,350
0,244
0,313
0,330
-0,027
-0,032
-0,055
(-0,36)
(-0,57)
(-0,88)
0,293
0,288
0,263
-0,030
(-0,62)
Average
Table 6b
Panel B: Duration
Duration portfolio
short
2
long
0,208
0,219
0,206
25
Yield-to-maturity

In the time-series, yield variables have been shown to
be excellent predictors of aggregate bond returns.



Fama-Bliss (1987), Campbell-Shiller (1991), Campbell
(1995), and Cochrane-Piazzesi (2002)
The YTM is likely to be a catch-all proxy for
information about default and term risk, call provisions
and other bond covenants not captured by the default
beta, differences in liquidity, mispricing, and any
omitted sources of risk beside the default and term
risk.
As a result, one would expect the yield-to-maturity to
be a significant predictor of average bond returns.
26
Returns to YTM portfolios
within default/term beta portfolios
Factor portfolio
bd
bt
1
1
1
2
1
3
Yield-to-Maturity Portofolios
1
2
3
0,082
0,200
0,325
0,143
0,235
0,315
0,200
0,267
0,353
Diff
0,243
0,172
0,153
t(diff)
(5,60)
(3,29)
(3,23)
2
2
2
1
2
3
0,121
0,181
0,233
0,236
0,270
0,288
0,341
0,332
0,361
0,220
0,152
0,128
(4,45)
(3,96)
(3,30)
3
3
3
1
2
3
0,091
0,207
0,243
0,298
0,302
0,330
0,385
0,437
0,458
0,294
0,230
0,215
(4,14)
(4,58)
(4,16)
0,167
0,270
0,368
0,201
(5,83)
Average
YTM strongly predicts returns in the cross-section
Table 6c
27
Risk-adjusted returns to long-short YTM portfolios
within default/term beta portfolios
Panel A: Regression results for portfolios that are long high yield and short low yield
2
t(a)
bd
bt
a
bd
bt
t(bd ) t(bt) Adj. R
1
1
1
2
0,245
0,139
0,076
0,233
-0,018
0,094
5,46
2,86
1,16
3,45
-0,67
3,55
0,03
0,11
1
3
0,148
0,138
-0,004
3,18
2,71
-0,20
0,04
2
1
0,205
0,205
0,028
4,26
2,81
1,10
0,06
2
2
0,136
0,157
0,035
3,71
3,35
2,08
0,06
2
3
3
1
0,128
0,270
0,046
0,254
-0,006
0,055
3,38
3,94
1,03
2,92
-0,30
1,68
0,00
0,04
3
2
0,209
0,191
0,049
4,33
2,75
2,28
0,05
3
3
0,200
0,200
0,030
3,83
2,78
1,14
0,05
0,187
0,167
0,029
5,69
3,66
1,90
0,08
Average
Table 7a
28
Sharpe ratios from yield and default beta
investing: Is it risk?
Portfolio Annualized
sorting
Ex Ante
variable
Sharpe
Ratio Sh
q (H0:
Sh=0.6)
NonNon- Z-statistic p-value
centrality central F
(H0:
parameter p-value Sh=0.6)
l
Yield
1,37
3,75
6,81
0,02
2,13
0,02
Default
beta
0,79
1,26
6,81
0,73
0,60
0,27
Sharpe ratios from portfolios in tables 4 and 7 (cond. sorts).
Table 8
29
Fama-MacBeth regression (1):
excess returns
The standard Fama-MacBeth regression for month 
with K factor loadings and M characteristics:
K
M
k 1
m 1
ri  rf  a   Fk  ki    m Cmi  ui
where ri-rf is the excess return on the bond, ki is the
factor loading/beta for risk factor Fk and m is the slope
coefficient for characteristic Cmi.


Regressions with individual bonds are potentially more
powerful than portfolios sorts.
The FM estimate is then the average of the slope estimates
across time.
30
Fama-MacBeth regression (2):
risk-adjusted returns
Returns are risk-adjusted before running the FM regression:
r  ri  rf  ˆdi DEF  ˆtiTerm
*
i
Where ri* is the risk-adjusted return, and beta hats are the
estimated default and term betas. Using these risk-adjusted
returns we estimate the following FM regression:
r  a   1Rating i   2 Durationi  u
*
i
*
i
31
Fama-MacBeth regression (3)-(4):
excess/risk-adj. and purged returns
A final test relies on the fact that the coefficients from the
Fama-MacBeth regression are returns on portfolios. To
purge these coefficients of possible influences from factor
realizations, we regress the time-series of slope coefficients
on the default and term factors:
ˆ j   j  dj DEF  tjTERM   u j
The intercept from this regression, j, is the purged estimator
which can be used to test whether regressor j is related to
average returns.
32
FM regressions with characteristics
Panel A: Characteristics
Dep. var.
Raw, Excess
Purged, Excess
Raw, Risk-adjusted
Purged, Risk-adjusted
Table 9a
Rating
0,011
(1,72)
0,007
(1,30)
0,010
(0,49)
0,012
Duration
0,020
(0,72)
-0,020
(-1,75)
-0,006
(-0,41)
0,019
Avg. R2
0,16
0,06
33
FM regressions with characteristics and
betas
Panel B: Betas and characteristics
Dep. var.
Rating
Duration
bd
bt
Avg. R2
Raw, Excess
0,007
(1,25)
0,004
(0,17)
0,235
(2,95)
0,046
(0,28)
0,20
Purged, Excess
0,005
(0,98)
-0,024
(-1,75)
0,229
(2,73)
-0,109
(-1,02)
Table 9b
34
FM regressions with characteristics, betas
and yields
Panel C: Betas, characteristics and Yields
Dep. var.
Raw, Excess
Rating Duration
bd
bt
Yield
-0,015
(-2,56)
-0,023
(-0,92)
0,161
(1,91)
0,232
(1,65)
0,160
(5,46)
Purged, Excess -0,015
(-2,51)
-0,054
(-4,21)
0,170
(2,15)
0,095
(0,97)
0,15
(5,30)
Table 9c
Avg. R 2
0,24
35
Conclusions

Betas, rather than characteristics, appear to be
priced in the corporate bond market

Default betas vs. ratings: Default betas matter,
weak evidence of ratings being priced

Term betas vs. duration: Term betas matter, but
evidence is weaker

Strong (but less surprising) evidence that YTM
predicts the cross-section of returns

Overall, the evidence indicates that systematic
risk factors are priced – arguably in contrast to the
36
equity market.
Practical implications

A parsimonious empirical model containing two
systematic risk variables, default beta and
term beta, and the yield-to-maturity could be
used to compute the expected bond
return/cost of debt.
 The conventional approach is to rely on bond
ratings and maturity or duration as a proxy of a
bond’s default and term risk. Our results show
that default and term betas are more
important.
37
Future research/extension

TRACE (and other) data sets allows extension
of sample by 16 years using high-quality data
 Extension to non-investment grade bonds?
Possibly larger default beta effect.
38