DBJ Discussion Paper Series, No.1214

Corporate Bond Premia

Yoshio Nozawa(Development Bank of Japan)

March

2013

Discussion Papers are a series of preliminary materials in their draft form. No quotations,reproductions or circulations should be made without the written consent of the authors inorder to protect the tentative characters of these papers. Any opinions, findings,
conclusions or recommendations expressed in these papers are those of the authors and do
not reflect the views of the Institute.

白紙

Corporate Bond Premia
Yoshio Nozaway
March 13, 2013

Abstract
I identify level and slope factors in corporate bond returns. I show that these two factors
can explain 93% of the variation in average excess returns on corporate bonds. To show thisresult, I describe expected excess returns and risks as functions of characteristics of corporate
bonds such as bond spreads and use a parametric characteristic-based asset pricing test. Thisapproach allows one to test the model’s ability to explain the variation in average excess returns
associated with multiple characteristics. The two factor model does well for all characteristics
except equity momentum.

I thank my advisors John Cochrane, Lars Hansen, Ralph Koijen and Pietro Veronesi for their continuous guidance.
I am grateful for comments and suggestions from Eugene Fama, Zhiguo He, Bryan Kelly, Tobias Moskowitz, Diogo
Palhares, Shri Santosh, Brian Weller and participants in the Finance Workshop and Economics Dynamics working
group at the University of Chicago and Development Bank of Japan. Research support from the Fama-Miller Center
for Research in Finance is gratefully acknowledged.
yDevelopment Bank of Japan, University of Chicago Booth School of Business and Department of Economics.
Email: ynozawa@chicagobooth.edu

11Introduction

I study empirically the behavior of the corporate bond premia and corporate bond spreads using
the historical data of corporate bond prices since 1973 in the US. The corporate bond premiumis de…ned as the expected return on corporate bonds in excess of synthetic treasury bonds with
an identical cash ‡ow schedule.

The corporate bond spread is de…ned as the relative price of acorporate bond to its corresponding treasury bond, scaled by the time to maturity. I construct a
factor pricing model based on the bond spread to understand the variation of the corporate bondpremia observed in the data.
My asset pricing model consists of the …rst two principal components of the excess returns on
corporate bond portfolios sorted on bond spreads. The …rst principal component is a level factor,
on which all corporate bonds have the same loading. The second principal component is a slope
factor, which a¤ects the bonds with high bond spread positively and the bonds with low bondspread negatively. I show that this simple two factor model can account for the majority of thevariation in bond premia when bonds are sorted into portfolios based on 14 characteristics such as
bond spread, coupon yields and momentum.
The two factor model has a problem in pricing the corporate bond portfolios sorted on theissuers’ past equity returns.

That is, the equity momentum in the bond market identi…ed byGebhardt, Hvidkjaer and Swaminathan (2005b) causes a problem for the two factor model.
I also show that these two factors explain more than 70% of the variation in realized excess
returns on portfolios sorted on other characteristics. This …nding suggests that there is a strong
factor structure in corporate bond excess returns and shows that the empirical result of this article
can be interpreted as an application of the Arbitrage Pricing Theory of Ross (1976).
Next, I show a novel approach to conduct an asset pricing test which complements the classic
test result using portfolio sorts.
As pointed out by Cochrane (2011), the asset pricing test based on portfolio sorts amounts tomatching the expected excess return function (as a function of securities’characteristics) to the riskfunction produced by the model. The classic test based on portfolio sorts implicitly assumes thatboth expected excess returns and risk are functions of characteristics. I parametrically characterize
the expected excess return and risk functions of characteristics, using individual security data. Thisapproach yields a number of bene…ts for the test.
With the parametric characteristic-based asset pricing test, I can test the model using multiple
characteristics at once.

With the classic test based on portfolio sorts, using multiple character-istics is challenging due to the limited data amount. The parametric characteristic-based test isparticularly useful in analyzing my sample, where the sample size varies signi…cantly over time.
With a few number of observations in the early 1970s, it is impossible to conduct multidimensionalsorts using my corporate bond data. With the parametric characteristic-based test, using four or2

…ve characteristics is fairly easy and can be done without losing precision in the estimated bondpremia.
To implement the test, I use a three-step procedure based on pooled OLS regressions using
individual bond-level observations. In the …rst and second steps, I estimate the expected excess
return function and the risk function parametrically using a large panel of individual security
returns. That is, I forecast returns and comovements with the characteristics of the bonds, suchas bond spread and time to maturity. I deliberately remove the ‘name’ of the security from theregressor of the forecasting regression.

By conditioning on characteristics and not on the nameof the security, I can estimate the expected excess returns and expected comovements precisely.
This result shows that the corporate bond premia and the risks associated with corporate bonds
are more stable functions of the bond characteristics than the security name.
In the third step, I project the estimated expected excess returns onto the estimated risk byminimizing the sum of squared pricing errors. These pricing errors are small if the expected excess
return function is close to a linear combination of the risk functions. That is, if the estimated bondpremia is matched well by a linear combination of the estimated risks produced by the model, themodel is successful in pricing these test assets.
Using the parametric characteristic-based asset pricing test, I show that the two factor model
explains 93% of the variation in estimated bond premia even when the four characteristics thatbest predict excess returns are used as instruments.
From a methodological perspective, the parametric characteristic-based test is attractive for anumber of reasons. In addition to the capability of handling multiple characteristics in the test,
using individual securities gives rise to a large variation in estimated bond premia and risks. A
large variation in estimated risks leads to more e¢ cient estimates of the model parameters thanclassic portfolio sorts. Furthermore, the parametric characteristic-based test o¤ers the ‡exibilityof testing both time-varying bond premia as well as cross-sectional variation in bond premia.
Corporate bond premia can arise due to an exposure to systematic risk, transaction costs or
mispricing in bonds with some characteristics. The transaction cost to trade corporate bonds canbe large.

Since corporate bonds are traded in the Over-the-counter (OTC) market, an investor

may have to search for a trading counterparty. As she anticipates costly search when selling thebond in the future, the price today may re‡ect a discount1 . My empirical result does not identify
which one of these three sources is the main reason for the bond premia. The fact that 93% of thevariation in bond premia can be explained by the exposure to the systematic risk only shows thatrisk-based explanation is a plausible explanation of corporate bond premia.
The rest of the article proceeds as follows: In section 2, I discuss the related literature.
1

In

Acharya and Pedersen (2005) show that the existence of liquidity can lead to liquidity risk, which may then show
up as a systematic risk. My factor model may be a proxy for liquidity risk and I do not separate liquidity risk fromcredit risk. Rather, with the two factor model I separate the risk premia (the sum of credit and liquidity risk premia)
and the liquidity (transaction cost) itself.

3

section 3, I explain the data to be used for the empirical analysis. I conduct the asset pricing testof the two factor model based on classic portfolio sorts in Section 4. I introduce the parametriccharacteristic-based approach in Section 5 and show the test result.

The last section provides

concluding comments.

2

Literature Review

There is a strand of literature which identi…es and explains the risk premia priced in the corporatebond market. Gebhardt, Hvidkajaer and Swaminathan (2005), the paper closest to mine, use a
two factor model of corporate bonds consisting of default (the long-term aggregate corporate bondreturns minus the long-term treasury bond returns) and term (the long-term treasury bond returns
minus T-bill rates) factors. They …nd that the two-factor model prices the cross-sectional variationin expected excess returns of corporate bonds associated with duration and credit ratings. However,
the model fails to price the variation associated with yield to maturity.
The key di¤erence in this article is that I adopt two factors consisting of the …rst two principal
components of the bond excess returns. I …nd that the second principal component, a slope factor,
is particularly important in explaining the variation in expected excess returns on corporate bonds.
With my two factor model, I can price the variation in corporate bond premia associated not onlywith the three characteristics tested in Gebhardt, Hvidkajaer and Swaminathan (2005), but also
many others.
Another strand of literature identi…es the risk premia priced in corporate bonds based on models.
By constructing either a reduced form model or a structural model which is consistent with theobserved credit spread and default probability, one can study the source and the magnitude of risk
premia. Papers based on a reduced form model includes Driessen (2005), while papers that use astructural model include Leland (1994), Chen, Collin-Dufresne and Goldstein (2009) and Bharmra,
Kuehn and Strebulaev (2010), among others. The focus of these papers are limited to explaining
the variation in risk premia associated with time to maturity and credit ratings. In this article, I
explore a wider range of characteristics.
This article proposes a novel approach to conduct asset pricing tests based on characteristics. The idea of explicitly expressing moments of the returns on securities using characteristics ispioneered by Rosenberg (1974), who models the beta of stocks that depend on the stocks’characteristics. The use of characteristics of securities in an asset pricing test is proposed by Cochrane(2011) and partially implemented by Gao (2009). Cochrane (2011) points out that the traditional
asset pricing test based on portfolio sorts can be thought of as a characteristics-based test by"An implicit assumption underlies everything we do: Expected returns, variances, and covariances are stable functions of characteristics such as size and book-to-market ratio, and not security
names. This assumption is why we use portfolios in the …rst place."4

I formalize Cochrane (2011)’s idea and show that the parametric characteristic-based test canbe conducted using an instrumental variable regression. Gao (2009) estimates non-parametricallythe expected excess returns and covariance as a function of stocks’characteristics. However, Gao(2009) applies a classic asset pricing test of Fama and MacBeth (1973), and tests if his measure ofcovariance is priced in the cross-section of equities. In this article, I present a coherent frameworkof both the measurement of moments, and the asset pricing test based on characteristics.
More recently, Connor, Hagmann and Linton (2012) propose an e¢ cient non-parametric asset
pricing test for their characteristic-based factor models.

My approach is parametric and allows

risks to depend on multiple characteristics at once. Therefore my work is complimentary to Conner,
Hagmann and Linton (2012).
The parametric characteristics-based test presented in this paper allows both the expectedexcess returns and risk measures to vary over time.
varying expected excess returns di¤erently.

The existing research approaches the time-Ferson and Harvey (1991) attempt to explain thetime-variation of expected excess returns for 10 size-sorted portfolios and industry portfolios using
standard 5-year rolling betas.

The characteristics-based test I present in this article allows one

to measure the time-varying risk more precisely without limiting the focus to portfolios sorted ina particular way.

Other articles regarding asset pricing tests with time-varying expected excess

returns include Nagel and Singleton (2011), and Adrian, Crump and Moench (2011).
The parametric characteristic-based test uses individual securities for the test, not the portfolios.
Ang, Liu and Schwartz (2010) show that using individual stocks instead of portfolios helps reduce
the standard errors for the estimated factor weights of the stochastic discount factor.

Gagliardini,

Ossola and Scaillet (2011) use individual panel data to test the three factor model of Fama andFrench (1993). Avramov and Chordia (2006) also propose an asset pricing test using individual
securities, extending Fama and MacBeth (1973).

All of these studies use the noisy estimates ofbetas based on time-series regressions for each security.

In contrast, I condition the measure ofrisk on characteristics and reduce noises in the estimates signi…cantly.

3

Data

I obtain the monthly price observation of senior unsecured corporate bonds from the following
four data sources. First, from 1973 to 1997, I use the Lehman Brothers Fixed Income Databasewhich provides month-end bid prices. Since Lehman Brothers used these prices to construct the
Lehman Brothers bond index while simultaneously trading it, the traders at Lehman Brothers had
an incentive to provide correct quotes. Thus although the prices in the Lehman Brothers Fixed
Income Database are quote-based, they are considered to be reliable.
In the Lehman Brothers Fixed Income Database, some observations are dealers’ quotes whileothers are matrix prices.

Matrix prices are set using algorithms based on the quoted prices of5other bonds with similar characteristics. Since matrix prices are less reliable than actual dealer
quotes (Warga and Welch (1993)), I remove them from the sample and treat the data points as
missing from the sample. The exception is December 1984, where all the prices are recorded as
matrix priced, and thus I retain these matrix prices only for this month.
Second, from 1994 to 2011, I use the Mergent FISD/NAIC Database. This database consistsof actual transaction prices reported by insurance companies.
Third, from 2002 to 2011, I use TRACE data which provides actual transaction prices. TRACE
covers more than 99 percent of the OTC activities in the US corporate bond markets after 2005.
The data from Mergent FISD/NAIC and TRACE are transaction-based data and therefore theobservations are not exactly at the end of months. Thus, I use only the observations that are in
the last …ve days of each month. If there are multiple observations in the last …ve days, I use thelatest one and treat it as a month-end observation.
Lastly, I use the DataStream database which provides month-end price quotes from 1990 to2011.
Since there are some overlaps among the four databases, I prioritize in the following order: theLehman Brothers Fixed Income Database, TRACE, Mergent FISD/NAIC and DataStream. As
Jostova, Nikolova, Philipov and Stahel (2012) …nd, the number of the overlaps are not large relative
to the total size of the dataset, with the largest overlaps between TRACE and Mergent FISD being
3.3% of the non-overlapping observations. To check the data consistency, I compare the di¤erence
across databases in Appendix A using the overlapping observations.
The Lehman Brother’s Fixed Income Database and Mergent FISD2 provide the other characteristics speci…c to the issue of bonds, such as the maturity dates, credit ratings, coupon rates andoptionalities of the bonds. I remove bonds with ‡oating rates and with any option features otherthan callable bonds. Until the late 1980s, there are very few bonds that are non-callable. Thus,
removing callable bonds would reduce the length of the sample period signi…cantly and it is for thisreason that I include callable bonds in my sample. As the callable bond price re‡ects the discountdue to the call option value, the yield on these bonds are not exactly comparable to the yield on
non-callable bonds. Crabbe (1991) estimates that call options contribute nine basis points to thebond spread on average for investment grade bonds. Moreover, Crabbe and Helwege (1994) showthat speculative grade bonds typically have lower call option values than investment grade bonds.
Therefore, the e¤ect of call options does not seem large enough to a¤ect my results signi…cantly3 .
I apply several …lters to remove the observations that are likely to be subject to erroneousrecording. First, I remove the pricing observations that are higher than matching treasury bond2
Mergent FISD provides relatively limited price information but provides bond characteristic information for mostof the bonds since 1994.
3
As a recent example, Gilchrist and Zakrajsek (2010) also use the Lehman Brothers Fixed Income Databaseincluding callable bonds in their analysis.

6prices (170,784 observations). Some corporate bond price observations seem to be unreasonably
high with a number even exceeding 1,000,000 per 100. By comparing the bond’s price with theprice of the synthetic treasury with the same repayment schedule (explained below), I determine
whether the corporate bond price is unrealistically high or not. On the other hand, several bonds
have extremely low price observations, including zeros. Zero prices are unrealistic as the bond is asecurity with limited liability. Furthermore, unless the …rm is in default, a price very close to zero
may not be reasonable, either. Thus, I drop the price observations below one cent per dollar, except
for bonds that are in default (4,794 observations)4 . Some bonds have price observations after thematurity date or before the issue date. I drop these observations as well (3,202 observations).
In addition, I remove the return (de…ned below) observations that show a large bounce back.
Speci…cally, I compute the product of the adjacent return observations and remove both observations if the product is less than

0:04. That is, if the same bond jumps up more than 20% in one

month and comes down more than 20% in the following month, I assume that the price observationin the middle is recorded with errors. This …lter removes 2,394 observations. Finally, I see somebonds whose prices do not change for extended period (observed frequently in DataStream). As
these constant price observations are likely to be subject to lack of liquidity or data entry errors5 , I
remove returns that are exactly zero for more than three consecutive months. This …lter removes
162,827 observations.
For the information regarding defaults, I use Moody’s Default Risk Service which provides thehistorical record of bond defaults from 1970. I compute the returns only up to the month in which
the default occurs.

The same source also provides the secondary market value of the defaulted

bond one month after the incidence. If the price observation in the month when a bond defaults
is missing, I add Moody’s secondary market price to my dataset. I also add a price observation of
100 if the price at maturity is missing and the bond does not default nor is called.
After the …ltering, I have an unbalanced panel of 1,454,363 bond-month price observations with
39,120 bonds over 468 months.
With the …ltered price, I compute the return on corporate bond i by
Ri;t+1 =(Pi;t+1 + AIi;t+1 + Couponi;t+1 )Pi;t + AIi;t(Pi;t + AIi;t )where Pi;t is the price for corporate bond i at time t, AIi;t is the accrued interest for bond i at timet and Couponi;t is the coupon payment for bond i at time t.
To compute some of the characteristics of bonds, I use accounting information from Compustat.
The short-term debt ratio is computed by dividing the amount of short-term debt by the amount4
One might be concerned by the fact that the bond price falls before default and removing low prices then mightremove low returns before default, which is informative. The concern does not invalidate this …lter as only 0.09% of
the price observations upon default is less than one cent.
5Chen, Lesmond and Wei (2007) …nd that zero return observations are highly related to illiquidity.

7

of total debt outstanding.

The tangibility ratio is computed by dividing the value of property,

plant and equipment by the value of total assets. I use CRSP for monthly stock price observations
and return volatilities.
To compute the corporate bond premia and the corporate bond spread, I need to construct
prices of the synthetic treasury bonds that match corporate bonds. To this end, I use the Federal
Reserve’s constant maturity yields data. First, I interpolate the treasury yield curve using cubic
splines and construct zero coupon curves for treasuries by bootstrapping. At each month and foreach corporate bond in the dataset, I construct the future cash ‡ow schedule for the coupon andprincipal payments.

Then I multiply each cash ‡ow with the zero coupon treasury bond pricewith the corresponding time to maturity6 . I add all the cash ‡ows to obtain the synthetic treasury
bond price which matches the corporate bond. I do this process for all corporate bonds at eachmonth to obtain the panel data of matching treasury bond prices. I compute the returns on thefsynthetic treasury Ri;t+1
using the same de…nition as the corporate bonds.

To compare my result with the previous literature, I form a term factor termt based on thedi¤erence in returns between long-term treasury bonds and one month t-bills.

I also form adefault factor deft which is the di¤erence in returns between long-term corporate bonds and longterm treasury bonds.

I obtain the data for these variables from Ibbotson’s Stocks, Bonds, Billsand In‡ation Yearbook.
Finally, I merge all four databases using the CUSIP identi…ers at the …rm and at the issue level.
In what follows I de…ne the key variables that I use for the analysis.

The bond premium isde…ned by the expected returns on the corporate bond in excess of the matching treasury bond.
he
E Ri;t+1= E Ri;t+1

f
Ri;t+1iThroughout this article, I use the word ‘excess returns’ on a corporate bond to mean Ri;t+1
f
Ri;t+1. By focusing on the returns in excess of the matching treasury bond returns, I eliminate

the mechanical e¤ect of shocks to the treasury yield curve on the corporate bond returns. Thus,
this de…nition of excess returns allows me to study the risk-return trade o¤ that is unique in thecorporate bond market.
The bond spread is de…ned by the relative price scaled by time to maturity

si;t =

1
i;t

log

i;t

f
Pi;t

Pi;t

6
I match the maturity of the zero coupon treasury prices to the cash ‡ow exactly by linearly interpolating continuous compounding forward rates.

8

Lastly, the default loss is de…ned bylossi;t = log

Pi;t
f
Pi;t

where t is the time of default.

4

Cross-section of Corporate Bond Premia

4.1

Portfolios Sorted by Bond Spread

To understand the information contained in the variation of the individual bond spreads, I startwith a classic portfolio analysis of the cross-section of corporate bonds sorted on bond spreads.
At the end of July every year, I form ten portfolios based on the average bond spreads between
January and June.

I record the monthly value-weighted average returns for each portfolio from

August to the following July. The one month gap between the time to form portfolios and the timeof the information used to sort bonds is necessary to remove high frequency market microstructurenoise which may induce spurious return forecastability. I also take six-month averages of the bondspread to average out the potential measurement errors in bond prices.
A preliminary analysis shows that there is a distinct jump in the average excess returns between
the ninth and tenth portfolios. To see the information in the last decile better, I split the tenthportfolio into three sub-portfolios based on bond spreads when rebalancing portfolios.
The top panel of Table 1 shows the average monthly portfolio returns in excess of matchingtreasuries.

The average excess returns on corporate bonds rise monotonically from -0.01% permonth7 to 0.64% per month as they move from the lowest decile to the last decile.

The di¤erencein average excess returns between the highest and the lowest decile is 0.64% with t-statistics 3.2.
The excess returns on these portfolios are serially correlated, as shown in the lag k autocorrelation coe¢ cients ARk . The signs of the AR1 coe¢ cients suggest that bonds with a high price (lowbond spread) tend to be followed by a bad return while bonds with a low price (high bond spread)also tend to be followed by a bad return. Except for the seventh and eighth deciles and the last
subportfolio (10c), the AR1 coe¢ cients are statistically signi…cant at the 5% level. On the other
hand, the AR2 and AR3 coe¢ cients (not reported) are all statistically insigni…cant.
There are two potential reasons why the autocorrelations are present in the data. First, it ispossible that high frequency measurement errors cause spurious mean reversion in prices. Second,
individual bond returns should be negatively autocorrelated if the variation in the bond spread is
associated with the news about expected excess returns, as opposed to the news about expected
7

The negative average excess returns might be due to i) the interpolation in treasury yield curve, ii) Omissionof callablility of corporate bonds and iii) recording errors in the data. Since the negative average excess returns is
economically and statistically small, I will not impose further …lters just to remove it.

9Table 1: Portfolios Sorted On Bond Spreads (Percentage Per Month)
Sample Statistics:
1(low)
2
3
4
5
6
7
8
9
10
10a
10b
10c 10-1e
E(R ) -0.01 0.04
0.04 0.05 0.04 0.05 0.17 0.18 0.21 0.64 0.42 0.68 0.91 0.64ese[E(R )] (0.08) (0.08) (0.07) (0.07) (0.08) (0.08) (0.09) (0.10) (0.13) (0.23) (0.19) (0.25) (0.31) (0.20)
AR1 (Re ) -0.12 -0.14 -0.17 -0.14 -0.19 -0.14-0.06 0.05 0.14 0.26 0.19 0.29
0.07e
AR2 (R ) -0.04 -0.06 -0.05 -0.05 -0.05 -0.04 -0.06 -0.05 -0.03 0.01 -0.03 0.04 -0.03
E[s]0.03 0.04
0.05 0.06 0.07 0.09 0.11 0.14 0.20 0.30 0.28 0.33 0.37
E[loss]0.00 0.00
0.00 0.00 0.03 0.02 0.01 0.01 0.03 0.20 0.09 0.19 0.41
E[ ] 14.94 14.44 13.48 12.67 12.01 11.60 11.23 10.20 8.94 9.58 8.31 7.63 11.86
E[M V ]
7.91 7.05
6.70 6.53 6.23 5.79 5.36 5.28 4.82 4.36 1.51 1.43 1.42
Issuer Information:
1(low)
2
M Veq 22.71 14.06
B=Meq
0.29 0.36
momeq
0.32 0.40

3
10.70
0.40
0.42

4
7.63
0.38
0.38

5
6.32
0.42
0.39

6
4.56
0.51
0.41

7
3.80
0.52
0.44

8
2.56
0.56
0.51

9
1.22
0.60
0.42

10
1.16
0.52
0.43

10a
0.93
0.57
0.46

10b
0.66
0.59
0.47

10c 10-1
1.74 -20.97
0.46 0.17
0.34 0.02eTime-series Regression: Ri;t
= i + 1;i Levelt + 2;i Slopet + "i;t1(low)
2
3
4
5
6
7
8
9
10
10a
10b
10c 10-1-0.03 0.01
0.00 0.01 0.01 -0.01 0.03 0.01 -0.08 0.03 -0.08 0.05 0.17 0.05
se(a) (0.04) (0.02) (0.02) (0.03) (0.02) (0.04) (0.02) (0.03) (0.06) (0.03) (0.06) (0.10) (0.13) (0.03)
0.27 0.30
0.30 0.29 0.31 0.33 0.33 0.31 0.31 0.39 0.34 0.41 0.45 0.12
1
se( 1 ) (0.02) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.02) (0.02) (0.04) (0.01)-0.24 -0.25 -0.22 -0.19 -0.23 -0.17 -0.05 0.04 0.26 0.80 0.62 0.79 0.99 1.04
2
se( 2 ) (0.03) (0.02) (0.01) (0.01) (0.03) (0.02) (0.02) (0.02) (0.05) (0.02) (0.09) (0.07) (0.12) (0.03)
R2
0.79 0.89
0.94 0.91 0.89 0.84 0.88 0.85 0.77 0.96 0.67 0.80 0.63

GMM 2 test:
10.82 [0.545]
At the end of July, the bonds are sorted into portfolios according to their average bond spreads between
January and June. The returns are recorded from August to next July, when portfolio rebalancing occurs.
The time period of the data is from 1973 to 2011. Level and Slope correspond the …rst two principal
components of excess returns. ARk ( ) is the univariate autocorrelation coe¢ cient with lag k and showsthat the coe¢ cient is statistically signi…cant at 5% level. s denotes the value-weighted average of the bondspreads on the portfolios. loss is the value-weighted average default loss rate.
is time to maturity and M V
is the total value of the portfolio in billion dollars. M Veq is the (bond) value-weighted market value of theissuers’equities in million dollars. B=Meq is the value-weighted average of the issuers’book-to-market ratio.
momeq is the value-weighted average of the issuers’average equity returns from t 12 to t 2. Standard
errors (in parenthesis) are adjusted for the serial correlations up to 12 lags following Newey and West (1987).
The GMM 2 test shows the test of the hypothesis that the intercepts of the regressions are jointly zero.
The p-value is shown in the bracket.

10default. If the bond price falls due to the news about discount rates, the return on the bond going
forward must be higher on average.
The second source of serial correlation in returns is analyzed using a term structure model inNozawa (2012). The lesson from this exercise is that the individual bond returns show negative
autocorrelations when the bond is held until maturity. Even so, the serial correlation is not too
persistent and accounting for 12 to 24 lags is enough to conduct statistical inference. As I analyze
portfolios of bonds where each individual bond can leave or join the portfolio in the middle of its
life, the serial correlation of portfolio returns must be attenuated relative to the model-implied
serial correlations in Nozawa (2012). Thus, in the following analysis, I account for 12 lags using
the weights of Newey and West (1987) when computing standard errors.
The average ex-post8 bond spread (per month) E [s] and the default loss rate E [loss] for the
portfolios rise almost monotonically from the lowest decile to the highest, with a sharp rise fromthe ninth decile to the tenth decile. The bonds in the …rst four deciles never default in this sample.
The loss rate in the ninth decide is only 3 basis points per month while it rises to 20 basis points
for the highest decile. The fact that the …rst four deciles have positive spreads and yet have nodefaults at all does not imply that there is an arbitrage opportunity. An investor who buys a bondin the …rst four deciles may still su¤er from the news about the credit event, as such news can send
the bond into lower deciles with lower prices. A default may occur after the bond migrates to thelower deciles, but the lower returns due to the news is recorded in the original low spread portfolio.
The bonds in the lowest spread decile have longer time to maturity on average (high E [ ]) andlarger total market value (high E [M V ]). Also, an analysis on the equities on the issuers of thesebonds shows an interesting pattern. The issuers of the high spread bonds have small equity market
values. On the other hand, there is no signi…cant variation in the issuers’book-to-market ratio9
B=Meq and the past average equity returns momeq . Though B=Meq rises from the …rst decile tothe tenth decile, the variation is not economically large.

This result seems contrary to the ideathat the book-to-market ratio is the proxy for …nancial distress.

4.2

Two Factor Model

To examine if the variation in average excess returns across the bond spread sorted portfolios canbe explained by a risk exposure, I form a simple factor model consisting of the …rst two principal
components of excess returns on ten portfolios sorted on bond spread.
The third panel of Table 1 shows the loadings of the two principal components on the ten
portfolios.

The …rst principal component is a level factor (Levelt ) which has nearly the sameloadings on all ten portfolios, as shown in1.

Levelt explains 73% of the common variation in

8

The ex-post bond spread is the spread observed after the portfolio formation, which can be di¤erent from thebond spread used to form the portfolios.
9
Though the high spread portfolios tend to have slightly higher book-to-market ratio, the di¤erence is economicallysmall. The value …rms in Fama and French (1996) have the book-to-market ratio greater than one.

11

Table 2: Summary Statistics of Two Principal Components (Percentage Per Month)
Descriptive Statistics of Principal Components:
Factor Correlations:E[ ][ ] skew kurt E[ ]= [ ]RM RFt SM Bt HM Lt termt deft dV IXt
Levelt 0.49 5.71 0.04 1.24
0.09
0.26
0.11
-0.03 0.03 0.33
-0.43(se) (0.27)(0.11) (0.23)
Slopet 0.52 2.61 0.52 3.87
0.20
0.29
0.28
0.02 -0.08 0.35
-0.23(se) (0.12)(0.11) (0.23)
Monthly returns from 1973 to 2011. Levelt and Slopet are the …rst and the second principal componentsof the excess returns on ten bond spread sorted portfolios. skew is skewness and kurt is excess kurtosis.
RM RFt , SM Bt and HM Lt are the three factors of equity returns of Fama and French (1993). termt isa return on long-term treasury bonds minus t-bills. deft is the return on long-term corporate bond index
minus long-term treasury bonds. dV IXt is the change in VIX index. The correlation between dV IXt is
computed using the data from 1990 to 2011.

realized excess returns. The pattern in

2 shows that the second principal component is a slope

factor (Slopet ) whose loadings monotonically go up from the lowest decile to the highest decile.
Slopet captures another 15% of the common variation. The fact that Levelt and Slopet explain
nearly 90% of the common variation in excess returns shows that there is a strong factor structure
in excess returns on the ten portfolios.
The summary statistics of Levelt and Slopet are shown in the left panel of Table 2. Levelt has
statistically insigni…cant average excess returns. The average excess returns scaled by the standard
deviation is low at 0.09. Slopet , on the other hand, has the high ratio of average excess returns
to standard deviation (0.20). The high risk price of Slopet raises the possibility that Slopet may
proxy for a shock to an important state variable that is of concern to the average investor.
To study the nature of the risk in the two factors, the top panel of Figure 1 shows the cumulative
log returns on Levelt and Slopet . The gray area in the background shows the recessions identi…ed by
NBER. Slopet is highly cyclical and tends to fall in the …rst half of recessions before subsequently
recovering. For the last three recessions, the fall in Slopet actually precedes each recession, whichsuggests that Slopet somewhat forecasts these recessions. Furthermore, Table 2 shows that Slopet
is fat-tailed with excess kurtosis of 3.87. The cyclicality and fat-tailness of Slopet show that thehigh ratio of average excess returns to the standard deviation for Slopet is not necessarily a proofof market ine¢ ciency where a good investment opportunity is left on the table.

In fact, Slopet

seems like a risky investment strategy.
The right panel of Table 2 shows the correlation of Levelt and Slopet with the …ve bond andequity pricing factors of Fama and French (1993). The correlations with the three equity factors
are in general low. The highest correlation is only 0.29 between Slopet and the excess returns onequity market portfolio (RM RFt ): Though issuers of high spread bonds have small equities, thecorrelation between Slopet and SM Bt is only 0.28. This low correlation between Slopet and SM Bt
suggests that the equity and the bond markets are not fully integrated at monthly frequency.

12

(log) Cumulative Returns

Cum ulative Factor Returns
2.5
2

Level
Slope

1.5
1
0.5
0
1980

1990

2000

2010

Default Loss (% per m onth)1
M arket-wide Default Loss

0.5

0

1980

1990

2000

2010

Figure 1: Cumulative Factor Returns and Aggregate Default Loss Rate (valueweight average of default loss of individual bonds)

13

The correlations between the two factors and the change in the VIX index dV IXt are negative.
This is expected as an increase in volatility tends to increase an option value. Since a long position
in defaultable bonds are equivalent as a short position in put options on …rm’s asset value and along position in a treasury bond, an increase in the VIX should a¤ect the defaultable bond returns
negatively. However, the correlation between dV IXt and Levelt is only -0.43 and I cannot replace
one of my factors with dV IXt to keep the pricing capability of the two factor model.
I test if the two factor model consisting of Levelt and Slopet can explain the cross-sectionalvariation in average excess returns. The third panel in Table 1 shows the estimated coe¢ cients forthe following time-series regression
e
Ri;t
=

i+

1;i Levelt +2;i Slopet + "i;t(1)

and the corresponding standard errors. If the variation in bond premia is explained by the twofactor model, thene
E Ri;t=1;i E [Levelt ] +2;i E [Slopet ]must hold.
I estimate (1) for all deciles (which I use to form my factors) and the three subportfolios of thelast decile.
In Table 1,
the spread factor
decile).

1

is around 0.3 for all portfolios but the extreme deciles, and the loading on
2 rises monotonically from -0.24 (the lowest decile) to 0.99 (10c of the highest

The high R-squared in each regression shows that the two factors explain most of thetime-series variation in excess returns, which we expect from the fact that my factors are principal
components. The pricing errors, , are small in general, ranging from -0.06% to 0.17% per monthand statistically insigni…cant. The GMM

2 test which tests if the intercepts are jointly zero fails

to reject the two-factor model at the 10% level.

(This test would be the same as the joint testof Gibbons, Ross and Shanken (1989) if there is no lags in computing standard errors. Since thereturns show autocorrelation, I account for it for robust statistical inference.)
The fact that the betas with respect to Slopet match the cross-section of bond premia suggeststhat corporate bond premia are risk premia.

That is, investors may demand the premia on

corporate bonds due to their comovements with some systematic risks.
The corporate bonds with a high bond spread earn high average excess returns for three potential
reasons. First, the average investor may dislike the bond that is likely to default and therefore
demand a premium. Second, the high spread bonds are illiquid and costly to trade.

Thus, thebond premia for high spread bonds re‡ect the expected transaction cost which incurs at the timeof sale in the future. Third, the high spread bonds comove positively with the average investor’s
marginal utility of consumption and are exposed to the systematic risk proxied by Slopet .
14

The fact that the Slopet betas match the pattern of average excess returns of corporate bonds
can be consistent with any three of these explanations above.

A challenge for the irrational

preference or transaction-based explanations is that they also need to be consistent with the factor
structure of excess returns. A bond with high transaction costs should earn a high bond premium,but why does it have to comove with other bonds with high transaction costs? Either way, in thisarticle, I do not o¤er any direct evidence in support of one explanation over the other.

4.3

Other Characteristics

To further examine the performance of the two-factor model consisting of the level and slope
factors, I sort the bonds into deciles according to the fourteen characteristics listed in Table 3. I
use previous literature in …nding the characteristics associated with bond premia. I use four pricerelated variables: current yield cy, market value of a bond size, the past one year average excess
returns (excluding the last month) mom and the yield to maturity on the synthetic treasury that
corresponds to the corporate bond i, try. The bond speci…c characteristics used in the analysis
are time to maturity , duration dur and the time after the issuance of the bond age. The issuer’s
fundamental variables include the (risk neutral) distance-to-default DD, the ratio of short-term
debt to the total debt sdtd, the ratio of the …xed assets over the total assets tangible and issuer’s
credit ratings rating.

Since credit ratings are expressed with symbols, I transform them into

numerical variables such that AAA is 1 and C is 21. Finally, I use equity anomaly variables suchas equity size M Veq , equity book-to-market ratio B=Meq and equity momentum momeq .
The (risk neutral) distance-to-default for …rm i at time t is de…ned byA

DDi;t =

ln Di;t
+ rt
i;t

i;t

1 2
2 i;t;Ai;t;Awhere Ai;t is the market value of the …rm, Di;t is the default boundary, rt is the risk-free rate,is the payout rate andi;t;A is the conditional volatility of the returns on the …rm asset.

i;t

The

computational detail is shown in Appendix B. Roughly speaking, the market value of the …rm isidenti…ed such that the call option on the …rms’value calculated using the Black-Scholes formula isequal to the observed stock price in the data. If Merton’s (1973) model is correct, the cumulative
density of standard normal distribution evaluated at

DDi;t should be equal to the risk-neutral

probability of default.
The motivation for using the characteristics listed in Table 3 is clear for some variables suchas equity anomaly variables but not for others. I use sdtd because the maturity structure of the…rm is associated with the rollover risk suggested by He and Xiong (2012).

I test if tangible isassociated with expected excess returns since it is used as a proxy for the tangibility of assets which
serve as collateral for borrowing under the incomplete contract theory.

15

Though the corporateTable 3: List of Characteristics for Corporate Bonds
Name
Price Related:s Bond spreadcy Current yieldsize Sizemom Bond momentum
try Treasury yield

De…nitionf=Pi;t1= log Pi;tCoupon rate (per face value) divided by Pi;tAmount outstanding multiplied by Pi;t
Average bond return between t 12 and t 2
Yield to maturity of the matching treasury bond at time t

Issue-speci…c:Time to maturity
dur Duration
age AgeTime to maturity of the bond in years
Macaulay duration of the bondTime passed since the issuance of the bond in years

Issuer’s fundamentals:
DD Distance to default
Sdtd Sdtdtangible Tangibility
rating Ratings

Risk-neutral distance-to-default of Merton’s model
Short-term debt (book value) / Total debt (book value)
Plant, Property and Equipment (book value) / Total asset (book value)
Credit ratings by Moody’s or S&P

Equity Anomalies:
M Veq Equity size
Market value of equity
B=Meq Equity book-to-market ratio Book value of equity divided by market value of equity
momeq Equity momentum
Average equity return between t 12 and t 2

bonds I analyze are uncollateralized bonds, higher debt capacity of the issuer might still a¤ect its
bond premia.
The use of age as a forecaster of the returns is motivated by Bao, Pan and Wang (2010) whoshow that the bond’s age is related to their measure of liquidity10 . The use of try is motivatedby the …ndings of Gebhardt, Hvidkjaer and Swaminathan (2005) that the yield to maturity ofcorporate bonds is a strong predictor of returns.
The expected excess returns on the ten portfolios sorted on characteristics as well as the regression estimates of equation (1) for each portfolio are shown in Table 4. For brevity, I report thetest result only for the …rst and last deciles.
Table 4 shows that the two factor model explains most of the variation in average excess returns
for portfolios sorted on all characteristics but momeq . Among the characteristics I test, cy, size,
DD, sdtd, M Veq and B=Meq give rise to statistically signi…cant variations in estimated bond premia.
The root mean squared alpha RM SE

and the mean squared alpha M AE

are economicallysmall for these characteristics, as the pricing errors are mostly less than 5 basis points per month.
Also, these alphas are small relative to the variation in estimated bond premia (as shown in thedi¤erence between high and low portfolios), which shows that the model performs well in pricing
these portfolios.
10

Driessen (2005) also uses age as a proxy for liquidity.

16

Table 4: Asset Pricing Test with Variety of Characteristics (Percentage Per
Month)
1 (low)
Characteristics E[Re ]Price-relatedcy
0.07 0.02
size
0.33 0.17
mom
0.23 -0.04
try
0.14 0.06

10 (high)
E[Re ]

10

1

2

pv[ 2 ] avg R2

(se)

RM SE

M AE

0.41 -0.01
0.02 -0.08
0.11 -0.03
0.04 -0.05

0.34 (0.16)-0.31 (0.11)-0.12 (0.10)-0.10 (0.09)

0.03
0.07
0.02
0.05

0.02
0.05
0.02
0.05

10.6
13.7
5.7
12.8

[0.39][0.19][0.84][0.23]

0.81
0.76
0.75
0.71

0.08 0.01
0.09 0.02
0.07 -0.01

0.01 -0.07-0.04 -0.08
0.08 0.00-0.07 (0.09)-0.13 (0.08)
0.00 (0.05)

0.04
0.04
0.03

0.03
0.03
0.03

24.3 [0.01]
20.3 [0.03]
16.1 [0.10]

0.78
0.77
0.78

Issuer’s fundamentals
DD
0.43 0.03
sdtd
0.11 0.00
tangible
0.05 -0.08
rating-0.01 -0.01

0.03 0.01
0.01 -0.06
0.06 -0.03
0.33 -0.06-0.40 (0.13)-0.10 (0.05)
0.01 (0.06)
0.34 (0.17)

0.03
0.02
0.05
0.03

0.02
0.02
0.04
0.02

5.7
4.4
14.1
5.0

[0.84][0.93][0.17][0.89]

0.75
0.73
0.72
0.76

Equity Anomalies
M Veq
0.31 0.01
B=Meq
0.05 -0.03
momeq
0.03 -0.26-0.03 -0.07
0.24 0.03
0.21 0.09-0.34 (0.12)
0.19 (0.08)
0.18 (0.12)

0.04
0.04
0.09

0.03
0.03
0.06

22.2 [0.01]
10.3 [0.42]
44.6 [0.00]

0.76
0.76
0.77

Issue-speci…cdurage

All Bond/Issuer Characteristics (High and Low only)
0.05
0.04 34.21 [0.06]
All Characteristics (High and Low only)
0.07
0.05 69.81 [0.00]eAsset pricing test using time series regressions: Ri;t
= i + 1;i Levelt + 2;i Slopet +"i;t . Monthly returns from1973 to 2011. The bonds are sorted into 10 portfolios every year according to characteristics except for mom
and momeq . For mom and momeq , the portfolios are sorted every month. Low is the …rst decide
qandPHigh is
10
1
2
the tenth decile and H-L shows the di¤erence between the tenth and …rst deciles. RM SE
i=1 i
10
P
10
1
2
and M AEis the GMM test statistics of the null that the intercepts are jointly zero for
i=1 j i j.
102
all ten deciles and[pv] is the corresponding p-value. avg R2 is the R-squared of the time-series regression
averaged over 10 portfolios. The characteristics used are de…ned in Table 3. The standard errors are inparenthesis and computed using Newey and West (1987) 12 lags.

17

The GMM

2 test which tests if the intercepts

i are jointly zero for all of the ten portfolios

mostly fails to reject the model except for , dur, M Veq and momeq . For , dur and M Veq , thepricing errors are economically small as RM SE is 4 basis points and M AE is 3 basis points.
Therefore, the statistical rejection of the model seems to come from the fact that these alphas arevery well measured. In such cases, the two factor model can still be used as a reasonable descriptionof the bond premia.
On the other hand, the two factor model fails to explain the variation in estimated bond premia
associated with momeq . In this case, the model works in the ‘wrong’direction in the sense thatthe magnitude of alphas is greater than the magnitude of estimated bond premia.

For the …rst

decile (low portfolio), the estimated bond premium is only 3 basis points while the magnitude of
the alpha is as large as -26 basis points. This large alpha leads to relatively large RM SE (9 basis
points) and the statistical rejection of the model.
The last two rows of Table 4 show the joint test if the intercepts are zero for multiple characteristics. The second last row shows the test if intercepts are jointly zero for the …rst and the lastdeciles of the …rst eleven characteristics (from cy to rating). In other words, I am testing if the 22
portfolios in the high end and the low end of the 11 univariate sorts have jointly signi…cant alphas.
RM SE is small at 5 basis point, which is good given that I am using only extreme deciles and thus
the variation in estimated bond premia is large. The model cannot be rejected at the conventional
5% level. This result suggests that the two factor model can price the portfolios associated withmultiple characteristics at once, when portfolios are formed using univariate sorts.
On the other hand, when I test if the intercepts are jointly zero for the …rst and the last decilesof all of the 14 characteristics (i.e. 28 portfolios in total), the model is rejected at the 5% level.
Including the equity momentum sorted portfolios causes a problem for the two factor model again.
Another interesting point in Table 4 is that the factor structure of corporate bond excess returns.
avg R2 in Table 4 shows the R2 of time-series regressions averaged across the ten portfolios for eachcharacteristic.

avg R2 is in general high and ranges from 0.7 to 0.8.

The high avg R2 implies

that the …rst two principal components constructed from the portfolios sorted on bond spread also
explain the common variation in excess returns on other portfolios. The common factor structure
in corporate bond returns seems to exist no matter what characteristics we use to sort bonds.
The high avg R2 in realized return is comforting as I can motivate my two factor model by the
Arbitrage Pricing Theory of Ross (1976). On the other hand, the high avg R2 raises a question why
I use the two principal components of the bond spread-sorted portfolios rather than other principal
components. Maybe the …rst two principal components of the portfolios sorted on current yields
will do an equally good job. Instead of showing the empirical result using 14 other potential sets
of factors against hundreds of potential test assets, I support my choice by claiming that the bondspread is the cleanest price variable associated with bond premia.
For an individual bond, the bond spread must forecast its excess return or its default loss. For18

a portfolio, the bond spread must forecast the portfolio excess returns or the credit shocks to theportfolio which cause individual bonds in the portfolio to migrate to another portfolio.

Bondspreads are the variables that should have information about the distribution of future excess
returns on corporate bonds. Therefore, it is the most natural variable to sort corporate bonds andform factors.

By sorting securities using bond spreads, I can alleviate the concern that my two

factors earn premia and thus work as a pricing kernel only in my sample period. Unless one …rmly
believes that the greater bond spread always forecasts negative credit events (such as defaults), it
must forecast excess returns and therefore Slopet should earn the premia.

4.4

Comparison with Gebhardt, Hvidkjaer and Swaminathan (2005)

To clarify the contribution of this paper to the literature, I also test the two factor model ofGebhardt, Hvidkjaer and Swaminathan (2005) which is based on termt and deft of Fama andFrench (1993), using the same test assets as in Table 1 and Table 4.
available upon request and I summarize my …ndings here.

The detailed result is

When I test if the two factor model

consisting of termt and deft can price the ten portfolios sorted on the bond spread, I …nd that themodel has no power to explain the variation in estimated bond premia.
The reason for the poor performance of termt and deft is simple: these two factors have nearlyzero average returns in my sample and the resulting alphas are basically the same as the estimatedbond premia. For example, deft , which takes a long position in long-term corporate bonds andtakes a short position in a long-term (about 20 years) treasury bond, has the average excess returnsof -1 basis point per month in my sample. This value is lower than 2 basis points reported in Fama
and French (1993) who use the sample period from 1963 to 1990.

Gebhardt, Hvidkajaer andSwaminathan (2005) construct deft in a slightly di¤erent manner but following the same principle(long-term corporate bond returns based on their sample minus long-term treasury bond returns)
and …nd that deft earns the average excess returns of 4 basis points over the sample period from1973 to 1996. Since my sample includes the …nancial crisis of 2008, it makes sense that deft haseven lower average excess returns in my test.

Due to the near zero average excess returns, thetwo factor model consisting of termt and deft have nearly no explanatory power of the test assets
analyzed in this article.

5

Parametric Characteristic-based Asset Pricing Test

5.1

Idea

In the previous section, I test the two factor model using traditional portfolio sorts based on one
characteristic in each test.

This is not satisfactory as investors can take advantage of multiple

characteristics at once in forming their investment strategy. A simple way to address this concern
19

is multidimensional sorts.

Just as Fama and French (1993) sort stocks based both on size andbook-to-market ratio, I can do two-way sorts of corporate bonds. However, a higher dimensionalsort is impractical particularly when there is a lot of information in the tail of the distributionof characteristics.

In Table 1, splitting the last decile into three subportfolios uncovers a large

variation in bond premia.

If an economist does the same for multiple characteristics, she willquickly run out of data points to form portfolios. This problem is even worse for corporate bonds
since the size of my sample is small in the 1970s. For example, I only have 19 bonds in January
1973. The small sample size of the corporate bond data makes it necessary for a researcher to be
more resourceful in using the data.
In this section, I characterize bond premia as a smooth function of characteristics.

I alsocharacterize risk as a function of characteristics, and test if these two functions match. I call thisapproach the parametric characteristic-based approach.
To get an idea of the di¤erence between portfolio sorts and the approach adopted here, I maketwo plots which describe these tests in Figures 2 and 3. Figure 2 is a graphical reproduction of
the result of Table 1.

In the classic test, we express the bond premia and betas as a functionof portfolio rankings which re‡ect the variation in bond spreads. As in the …gure, the estimatedbetas for Slopet go up as we move from left to right, so do the average excess returns. Since theincreasing pattern in the average excess returns is matched by the increasing patterns in the betas,
the model prices these portfolios successfully.
The top panel of Figure 3 shows the corresponding result of the parametric characteristic-based
test. Before going into the detail, I will explain the intuition of the test. In this case, I express
the bond premia and a measure of the risk, expected comovements (that will be de…ned clearly
below), as a smooth function of bond spreads.

In this example, I express the bond premia andexpected comovements as functions of a bond spread, a squared bond spread and square-root of abond spread.
Figure 3 also shows that the model seems successful in pricing the variation in bond premia
associated with bond spreads.

The expected excess return function seems to be matched wellby the model-implied counterpart, which is a linear combination of the two expected comovement
functions E [V ]. In particular, the expected comovements with the slope factor Slope E [V ] seems
parallel to the expected excess return function in the data, which suggests that the slope factor is
the key to the success of the model.
Using a smooth function reduces the number of parameters to be estimated, which makes it
easy to accommodate multiple characteristics in an asset pricing test. Of course, the bene…t comeswith some costs. If the assumed functional form is incorrect, then the test becomes misspeci…ed.
However, in this case, there seems to be little nonlinearity in the expected excess returns. To see
this, I re-express the point estimates of bond premia from portfolio sorts as a function of bondspreads (not a function of ranking) in the top panel of Figure 3.

20

The dotted line with circles

Asset Pricing Test Based on Portfolio Sort
2

1

1

0.5

Beta

Expected Returns (%)

E[Rx]Model E[Rx]Level beta
Slope Beta

Level beta

00Slope beta

-1

1

2

3

4

5

6
Portfolio number

7

8

9

10a

10b

-0.5
10c

Figure 2: Asset Pricing Test Based on Portfolio Sorts
shows the estimated bond premia based on portfolio sorts11 .

This non-parametric estimates ofbond premia do not produce a complex functional form of bond premia.

Rather, the estimatedbond premia seem to go up smoothly from left to right.
When we look at the pattern in estimated bond premia in Table 1, we might conclude thatthe bond premia jump up in the last decile and it is hard to replicate such a jump with a smoothfunction. When we express the bond premia as a function of underlying characteristics, the jump
goes away and it seems feasible to …t a smooth function.
The bottom panel of Figure 3 shows the histogram of the bond spread.

As we can see, thedistribution is highly skewed and the most of the observations are below 0.2% per month.

The

white and gray colors in the panel correspond to the cuto¤ values (averaged over time) for the ten
portfolios analyzed in Table 1.

For example, the white area at the left end of the …gure showsthese bonds will be mostly in portfolio one, while the gray area at the center to the right shows thatthese bonds will be mostly in the last decile. This means that when sorting securities into bins,
we are throwing a lot of variation in characteristics away by averaging all the extreme observationsin the last decile.
Of course, the point of …tting a functional form is to use multiple characteristics at once, notjust bond spreads as shown in Figures 2 and 3. Based on the intuition developed by comparing
Figures 2 and 3, I formally de…ne the parametric characteristic-based approach in the next section.
11
Strictly speaking, the data points in Figure 3 di¤ers from the portfolio sorts in Table 1 as the …gure shows theresult of the regression (3) using dummy variables. Since the weights assigned to each security in regression (3) isdi¤erent from portfolio sorts, there is a slight gap in the estimated bond premia between Table 1 and Figure 3.

21

Expected Returns (%)

Level E[V|z]

1

20

0.5
10
0
Slope E[V|z]-0.5

0Data E[R|z]
M odel E[R|z]
0

0.2

0.4
0.6
Bond Spread (% per m onth)
Histgram of Bond Spread

0.8

0.2

0.3
0.4
0.5
0.6
0.7
Bond Spread (% per m onth)

0.8

Expected Comovements

Characteristic-based Asset Pricing T est
1.5

1

10000

5000

0

0.1

0.9

1

Figure 3: Parametric Characteristic-based Test using s, s2 and s0:5 as instruments(Top) and the Distribution of Bond Spread (Bottom): E[Rjz] is the expected excess
return function and E[Vjz] is the expected comovement function. The dotted linewith circles are expected excess returns using 12 dummy variables corresponding
to portfolios.

22

5.2

Three-step Procedure

I describe the parametric characteristic-based asset pricing test using a simple three-step estimation
procedure. I test the model which speci…es K factors ft = (f1;t ; : : : ; fK;t ) driving the stochastic
discount factor. Let zi;t be the L dimensional vector of instruments for bond i observed at time t.
Traditionally, these instruments are macro variables such as lagged market returns or consumption
growth rates (for example, see Ferson and Harvey (1991)).

In the characteristic-based test, zi;tincludes smooth functions of characteristics ci;t of each asset i at time t. The characteristics drive
the individual securities’bond premia and expected comovements with factors.
Let Vi;t+1e
Ri;t+1
ft+1 . The asset pricing model states thate
E Ri;t+1
jci;t = E [Vi;t+1 jci;t ]where ci;t is a vector of characteristics and(2)

is a K dimensional vector of factor risk weights. Thatis, if the model is correct the expected excess return function is equal to a linear combination of
the expected comovement functions E [Vi;t+1 jci;t ].
I test (2) using a simple three-step estimation procedure. First, I pick an instrument vector
zi;t which is a function of characteristics ci;t . I estimate the expected excess return function by thepooled OLS regressioneRi;t+1= zi;t br + "ri;t+1

i = 1; : : : ; N; t = 1; : : : ; T(3)where N is the number of bonds and T is the number of months. Recall that i denotes individualbond i as opposed to a portfolio.

e
I forecast individual security excess returns Ri;t+1
with thefunction of its characteristics zi;t .
Second, I estimate the expected comovement function by the pooled OLS regression
Vi;t+1 = zi;t bv + "vi;t+1

i = 1; : : : ; N; t = 1; : : : ; T

(4)

i
h
^ [Vi;t+1 jci;t ] = zi;t^bv , where^ Re jci;t = zi;t^br and E(3) and (4) yield the estimated functions E
i;t+1^br and ^bv are the OLS estimates of br and bv respectively.
Third, I estimate the factor risk weightby regressing estimated expected excess returns on

estimated expected comovements. That is, I run the pooled OLS regression
zi;t^br = zi;t^bv +i = 1; : : : ; N; t = 1; : : : ; T

i;t

(5)

Let ^ be the estimate of . The sample counterpart of equation (2) implies that^br = ^bv ^

23

(6)

That is, the parameters that characterize the expected excess return function must be completely
matched by a linear combination of the parameters of the expected comovement functions. Sincezi;t is L dimensional, we need to match the L dimensional vector br with a combination of K vectorsthat are the columns of bv (bv is a L

K matrix).

This procedure is closely related to the classic asset pricing test based on portfolio sorts. In
the characteristic-based test, the expected excess returns are estimated using the average excess
returns on the dynamic investment strategy based on zi;t . Each entry of zi;t gives us the weighton security i to execute the strategy.
In the classic test, the portfolios are formed by sorting securities into bins.

Suppose thatzi;t are dummy variables which take a value of one if the observation is for a particular portfolio.
Then running regression (3) is equivalent to taking sample average of realized portfolio returns andrunning (4) is essentially the same as taking average of realized comovements at the portfolio level.
The third regression (5) is a cross-sectional regression using portfolio average excess returns andportfolio average comovements. Therefore, the asset pricing test based on classic portfolio sorts
is a special case of the characteristic-based approach. The key di¤erence between classic portfolio
sorts and the parametric approach lies in the choice of instruments.

The instruments I use aresmooth functions of characteristics, not dummy variables associated with characteristics.
The average excess returns on the portfolios formed by sorting securities into bins are thenon-parametric estimates of the expected excess return function. In the parametric characteristicbased approach, I essentially form portfolios by running regressions and estimate the expectedexcess return function parametrically.
The dimension of zi;t corresponds to the number of test assets in the classic test. Thus for thecharacteristic-based test to have power, one needs to have L > K.

Fama and French (1996) test

the three factor model using 25 size and book-to-market sorted portfolios and thus have 25 assets toprice. In the characteristic-based approach, if expected excess returns are linear functions of sizeand book-to-market ratio, zi;t will be a two-dimensional vector, and we have two assets to price.
To see if we statistically reject the null hypothesis (6), we can apply the standard

2 test resultof the GMM estimator. That is, compute000
ET zi;thi;t+1 cov ET zi;thi;t+1

where ET [ ] is a sample average operator and hi;t+1

10
ET zi;thi;t+1eRi;t+1

2
L K(7)

Vi;t+1 . This test asks if the alphas

for L strategies formed based on zi;t are jointly statistically signi…cant or not.

5.3

Bene…ts of Parametric Characteristic-based Test

There are a number of bene…ts of using the parametric characteristic-based asset pricing test insteadof the classic test based on portfolio sorts. Rather than going into detail, I list the bene…ts here
24

and provide supporting evidence in the empirical analysis that follows.
First, with the parametric characteristic-based approach, one can conduct an asset pricing
test with multiple characteristics. By sorting securities into portfolios along some characteristics,
variations in other characteristics will be lost.

Practically speaking, we can only sort securities

using up to two or three characteristics even for equities that have a larger sample size. Thus,
sorting securities into portfolios makes it impossible to test the performance of the proposed asset
pricing model along more than three characteristics at once.

Due to this limitation researchers

often test the model along one or two characteristics at a time, neglecting the interaction between
multiple characteristics. This is the main reason why I adopt this approach in analyzing corporate
bonds as there are a few securities in the earlier part of my sample.
Second, for the asset pricing test to be powerful, greater variation in expected comovements isdesirable. By sorting securities into bins along particular characteristics, some of the variation in
expected comovements is lost, which results in (statistically) less e¢ cient estimates of the factor
weight parameter0.

This is why Ang, Liu and Schwartz (2010) propose to use individual securities

to conduct an asset pricing test.
Third, when sorting securities into portfolios, we essentially impose time …xed e¤ects in estimating expected excess return functions and expected comovement function. That is, a researcher
focuses on cross-sectional variation in expected excess returns in evaluating the asset pricing model.
In a classic asset pricing test based on sorts, the test assets are the portfolios sorted on thepast characteristics. These portfolios are created based on the strategies of buying the bonds withrelatively high characteristics at each point in time. If the characteristics of all bonds increase atsome time, this trading strategy does not do anything in capturing the variation in expected excess
returns observed then relative to the past. Thus in the standard asset pricing test using portfolios,
economists ask if the asset pricing model can price the cross-sectional variation in expected excess
returns, but not the time-series variation.
With the characteristic-based test with no time …xed e¤ects, the model can be asked to pricevariation of bond premia both over time and in a cross section. Therefore, using all the individual
bonds in a characteristics-based test helps us test an implication of the model other than crosssection of expected excess returns.

Since the goal of this paper is to explain the cross-sectionalvariation in bond premia, I will not take advantage of this ‡exibility in this article and include time
…xed e¤ects in the empirical analysis that follows.
In the classic test based on sorts, a key motivation to form portfolios is to reduce the noisein the estimated betas.

In the parametric characteristic-based asset pricing test, the expectedexcess return function (3) and the expected comovement function (4) can be estimated with all thepanel observations of individual bonds. The use of the entire panel of individual bonds makes theestimated expected excess returns and comovements precise given the speci…cation of the functional
form. As a result, the concern about noise in the estimates does not invalidate the use of individual25

bonds as test assets. This point will be quantitatively supported in the next section.

5.3.1

Standard Errors of Expected Comovements

One reason why researchers started to use portfolio sorts was to reduce the noise in estimatedbetas. The parametric characteristic-based approach addresses the issue by projecting the realized
comovements onto characteristics. To show the reliability of the estimated expected comovements, I
use a bond spread, a squared bond spread and square-root of a bond spread for zi;t as an illustrating
example. I run a regression (4) and I computethe corresponding standard errors of the estimated
ii r
h h
[k]
[k]
0 where V [k]^ VE= zi;t cov ^bv zi;t
expected comovementsi;t+1 jzi;ti;t+1 is the k-th element in[k]Vi;t+1 and ^bv is the k-th column of the matrix ^bv . Since the standard error depends on zi;t , I pick
the percentiles of the distribution of bond spreads.
Table 5 shows the estimates of the standard errors and the estimated expected comovements forthe percentiles of the bond spread. As we can see, the standard errors of the expected comovements

are small relative to the variation in expected comovements. For Slopet+1 , the expected comovements vary from -0.7 to 11.9 as the spread increases from the 1st percentile to the 99th percentile,
while their standard errors are 0.37 and 3.09, respectively.

The di¤erence between the 5th and95th percentiles are highly statistically signi…cant. Since the precision of the estimates depends on
the noise in ^bv , this conclusion does not depend on the particular choice of the percentiles. Overall,these …gures show that the estimated expected comovements are well measured despite the factthat I use individual securities.
Sorting securities into portfolios is an e¤ective way to reduce the standard errors of the estimatedexpected comovements. However, portfolio sorts are not the only way to improve precision of theestimates. What is truly needed is a projection of comovements onto a set of characteristics that
forecast comovements well. Portfolio sorts can be thought of as a non-parametric regression, while
in contrast the parametric characteristic-based approach uses a parametric regression.

The keyis the choice of instruments. The fact that the betas of individual securities are typically poorly
measured suggests that the dummy variables for the name of securities are not good instruments
to project on.

The bond spread seems to be a good instrument in this respect and we can doprojections either parametrically or non-parametrically.

5.4

Estimating Expected Excess Return Function

To describe the variation in expected excess returns on individual bonds, I run regression (3)using instruments zi;t that are linear functions of characteristics associated with expected excess
returns. To see whether the assumption of linearity is satisfactory, in Appendix C, I show a seriesof analysis about the functional form using non-parametric regression with Gaussian kernel for eachcharacteristic I use. Based on the non-parametric regression result, I transform size and time to26

Table 5: Standard Errors of Estimated Expected Comovements

si;t
1%
5%
50%
95%
99%

Levelt+1^^ i;t+1 jzi;t ]]E[Vi;t+1 jzi;t ][E[V3.3(0.74)
4.6(1.01)
9.1(1.79)
20.2(3.54)
26.3(6.68)

Slopet+1^^ i;t+1 jzi;t ]]E[Vi;t+1 jzi;t ]
[E[V
-0.7
(0.37)-0.8(0.51)-1.0(0.94)
3.2(1.94)
11.9(3.09)

95% 5%
15.6(3.01)
4.0(1.56)^^^^E [Vi;t+1 ] = zi;trbv , where bv is the OLS estimate. The values in parenthesis are standard errors of E [Vi;t+1 ]computed by[k]
0 where z
zi;t cov ^bv zi;ti;t are functions of the percentile values of the bond spread. The

F-statistics for ^bv for Levelt+1 is 16.7 with the p-value of 0.00. The F-statistics for ^bv for Slopet+1 is 11.0
with the p-value of 0.00.

maturity with log to make the relationship with returns closer to be linear. I also demeaned market
value-related variables such as size by the cross-sectional average in each month to maintain thestationarity.

After such transformations, when I analyze each characteristic separately, a linear

function seems to …t the data reasonably well. I also scale each characteristic so that it has a unit
standard deviation.
Table 6 shows the estimation result. I run both univariate and multivariate regressions using
the characteristics which give rise to statistically signi…cant variations in estimated bond premia
when I use them to sort corporate bonds into deciles. Speci…cally, I use s; cy, log size, DD, sdtd,log M Veq and B=Meq .

Furthermore, I include momeq in this test as my two factor model has atrouble pricing the portfolios sorted on momeq .
The top panel of Table 6 shows the result of the univariate regressions. Since I am using only
the characteristics that are associated with estimated bond premia based on portfolio sorts, theslope coe¢ cient br should be statistically signi…cant. The result in the top panel of Table 6 showsthat cy, log size and B=Meq do not have statistically signi…cant slope coe¢ cients possibly due tothe linearity assumption.
The bottom panel of Table 6 shows the result of the multivariate regression.

When all thecharacteristics are used at the same time, only …ve characteristics have economically signi…cantcoe¢ cients: s, cy, log size, log M Veq and momeq . Thus, in my main empirical result using theparametric characteristic-based approach, I use these …ve characteristics as instruments.

27

Table 6: Multivariate Return Forecasting Regression
Characteristicss
Univariate Regression:br
0.51t(br ) (4.18)
q
P
2
1
0.26
i;t (zi;t br )NTcylog sizeDD

sdtd

log M Veq

B=Meq

momeq

0.02
(0.61)-0.04(-1.41)-0.06(-3.73)-0.03(-2.25)-0.06(-2.91)0.01(0.79)

0.06(2.69)

0.01

0.03

0.05

0.03

0.06

0.01

0.05

Multivariate Regression:
br
0.52
0.06
-0.14
-0.02
-0.01
0.05
-0.01
0.07
t(br ) (4.32) (0.92)(-2.57) (-1.07) (-0.57)(1.85) (-0.89)(3.51)qP
2
1
0.35
F-statistics:
10.0 [0.000]i;t (zi;t br )NTeMonthly observations from 1973 to 2011. I estimate br by regressions Ri;t+1 = zi;t br + "ri;t+1 . si;t is thebond spread for bond i at time t. To put time …xed e¤ects, I demeaned the vector zi;t by subtracting thePNt
1time-speci…c mean zt
i=1 zi;t . Each observation is weighted by the square root of relative value ofNt
the bond at time t. The numbers in parenthesis are t-statistics, where standard errors are computed taking
into account 12 lags of serial correlations (with Newey and West (1987)’s weighing).

5.5
5.5.1

Practical Issues in Parametric Characteristic-based Test
Attenuating Extreme Observations

In the characteristic-based test using individual bonds, the estimated coe¢ cients br and bv may
be sensitive to extreme observations. This is true especially when I include higher-order terms ofcharacteristics (such as squared bond spread) in zi;t . Small di¤erence in br and bv

can translate

into a large pricing error for a large value of the characteristic.
In the classic asset pricing test using portfolio sorts, a security with an extreme value of characteristics is averaged out with other securities in the extreme portfolio.

This is especially thecase for the size and book-to-market sorted 25 portfolios of Fama and French (1993) where thethreshold of characteristics are set using NYSE stocks only. As a result, the stocks with extreme
characteristics (that are typically small and more likely to be NASDAQ stocks) will end up in
one or two portfolios that have many securities to average across.

This practice is justi…ed fornoise-reduction reasons.
To downweight the extreme observations in the characteristic-based asset pricing test, I valueweight the observations.

Since the characteristic-based test involves panel data and the market

value of bonds grows over time, a naive use of the market value at time t as a weight will downweighte
old observations too much. Thus I multiply Ri;t+1; Vi;t+1 ; zi;t with weight wi;t de…ned by

wi;t

s

M Vi;tj2Jt M Vj;tP

28

where Jt is the set of the bonds that are in sample at time t.
5.5.2

Computing Standard Errors

The three step estimation ofdescribed in (5) boils down to a one step minimum distance estimah
i
h
i 1
0
e
0 z
tion of the moment conditions E zi;t.
Ri;t+1Vi;t+1= 0, with a weighting matrix E zi;ti;tThus it is straightforward to compute standard errors using the GMM framework. The panel data
I use has a large number of securities that is highly correlated with each other. To account for thecorrelation across securities, I …rst compute the standard errors clustered by time and then add
several time lags. The exact number of lags added is speci…ed in each test result.
As the cross section is large relative to the time-series, one may be concerned about the reliability of the estimated standard errors.

To address this concern I also compute the standard

errors imposing some structural assumptions on shock vectors. I apply the structural assumptions
suggested by Ang, Liu and Schwartz (2010) and compare the performance of the computed standard errors in Appendix D. Appendix D shows that these alternative methodologies of computing
standard errors seem to produce similar results. Thus in the following empirical analysis, I show
the standard errors clustered by time (with several lags added).

5.6

Empirical Result

Below I implement the parametric characteristics-based test of the two-factor asset pricing modelusing various set of characteristics.

5.6.1

All Characteristics but Equity Momentum

First, I use all the characteristics except for momeq in the set of instruments. That is, I use s, cy,
log size and log M Veq as my test characteristics. I apply the three step procedure and report theresult in Table 7.
The …rst step regression (in the top panel) shows estimated bond premia as a linear function of
the characteristics that I use. The corporate bond with a high spread, a high current yield, a low
market value and a high issuer’s market value of equity
earns higher excess returns going forward.
r
2
P
The resulting variation in estimated bond premia T1N i;t zi;t^br is 38 basis points per month.

In comparison, the same statistic for the variation in E (Re ) of the ten portfolios in Table 1 is 23
basis points. Thus, in the parametric characteristic-based approach, we have economically sizablevariations in estimated bond premia which the model needs to match.
The second step regression (in the middle panel) shows the estimates of the expected comovement function. As I have two factors, there are two regressions. The point estimates for bv show
the way in which the estimated risk varies along with the characteristics. The comovements with29

Levelt has a large positive loading on cy. Thus, Levelt helps explain the variation in estimatedbond premia associated with cy. The comovements with Slopet , on the other hand, has a large
positive loading on s and thus helps explain the variation in bond premia along s. Both factors
have the loadings on log size and log M Veq that have the same sign as the estimated br in the …rst
step. That is, both factors help explain the variation in bond premia associated with log size andlog M Veq .
The third step regression is reported in the third panel of Table 7.
and 0.10 for Levelt and Slopet , respectively. The estimated

The estimatedis 0.01

for Levelt is insigni…cant while it ishighly signi…cant for Slopet . This is consistent with the fact that Levelt does not earn statisticallysigni…cant average excess returns in my sample while Slopet earns signi…cant average excess returns(as shown in Table 2).
If the model performs well, (6) should hold. To see this, I report ^bv ^ in the third panel. As
we can see, ^bv ^ are very close to ^br , which implies that the model does well in pricing these test
assets.

In other words, the model can produce the expected comovement functions that closely

match the expected excess return function.
The third panel of Table 7 shows another diagnosis of the model.

RM SE and M AE are

0.11% and 0.09% per month, respectively. They are larger than the case of univariate sorts and it
is reasonable given that I use all the four characteristics jointly here. The R-squared is as high as
0.93, meaning 93% of the variation in estimated bond premia is matched by the two factor model.
The pricing errors are jointly statistically insigni…cant, as the GMM

2 test fails to reject the modelat the 10% level.
Multiplying

with E [f 0 f ] yields the estimated bond premium for each factor.

The bondpremia for Levelt and Slopet are 0.37% and 0.71% per month, respectively. Levelt is assigned asmaller premium than its average excess returns (0.49%) and Slopet is assigned larger premium
than its average excess returns (0.52%).

The gap between the average excess returns and theestimated bond premium using the parametric characteristic-based approach raises the concernthat I may be mispricing the factors to price the test assets.
To alleviate this concern, I also test if the two factors are priced correctly or not, using thepoint estimates ^ . This is a calibration-veri…cation exercise of Hansen (2008) which accounts forthe fact that ^ are estimated using the test assets. The test (not in the table) fails to reject thenull that alphas for the two factors are jointly zero and thus supports the model’s performance.
In the parametric characteristic-based approach, it is important to have large values of Fstatistics for the second step regression. As the three step procedure is an application of a standard
instrumental variable regression, the usual caution about weak instruments applies. Staiger andStock (1997) point out that including instruments that weakly forecast endogenous variables (Vi;tin my case) causes a problem in the small sample behavior of the estimates. To detect weak instruments, one can compute the statistics of Cragg and Donald (1993), a relative of F-statistics which30

accounts for the interactions between multiple endogenous variables. In my speci…c setting, theinteraction between Vi;t for Levelt and Slopet does not matter as these two factors are uncorrelated
by construction. Thus, the statistics of Cragg and Donald (1993) reduce to the minimum of thetwo F-statistics computed in the second step.
In Table 7, I have the minimum F-statistics of 15.2. This value is large relative to the cuto¤values of a weak instrument test reported in Stock and Yogo (2002). Thus, the set of instrumentss, cy, log size and log M Veq jointly forecasts comovements well enough.
In conclusion, the two factor model matches well the variation in estimated bond premia associated with s, cy, log size and log M Veq jointly.
5.6.2

All Characteristics but Equity Momentum with Interaction

The advantage of the parametric characteristic-based approach over the joint test using univariate
sorts (as in Table 4) is that one can allow for interactions among characteristics.

If one uses

univariate sorts of equities based on size and book-to-market ratio and tests if the intercepts arejointly zero for all portfolios, the ‘small growth stocks’ e¤ect (Fama and French (1993)) will bemissed.
With the parametric characteristic-based approach, one can easily account for interactions by
including the interaction terms in the set of instruments and conducting the test in the samemanner.
As an example, I include the four characteristics tested in the previous section (s, cy, log sizeand log M Veq ) and the interaction terms with log time to maturity ~. Though log time to maturity
itself does not lead to a signi…cant variation in bond premia, it is interesting to ask if the e¤ect of
the characteristics becomes greater for long-term bonds than for short-term bonds.
Table 8 shows the test result. Now I have eight instruments as I add s, cy, log size and log M Veq
multiplied by ~ to the previous set of instruments. s ~ and log size ~ give rise to a statisticallysigni…cant variation in estimated bond premia. In the …rst step, the e¤ect of the bond spread ismore pronounced for long-term bonds (br for s ~ is positive) and the e¤ect of size is attenuatedfor long-term bonds (br for log size ~ is positive).
The eight characteristics jointly signi…cantly forecast comovements as well, as we can see fromthe F-statistics in the second step.
The result for the third step regression shows that despite the greater number of instruments,
the two factor model has no problem in pricing these test assets. RM SE and M AE are small
at 0.06% and 0.04% per month, respectively. The model cannot be rejected at the 10% level.

31

Table 7: Asset Pricing Test with Four Characteristicse= zi;t br + "ri;t
1st step: Ri;t+1^brt(^br )scy

log sizelog M Veq

0.53
(4.32)

0.06(0.95)-0.16(-2.38)0.04(1.63)

2nd step: Vi;t+1 = zi;t bv + "vi;tscy

log sizelog M Veq

^bv;Levelt(^bv;Level )1.05(0.54)

4.56(4.98)-1.05(-1.25)0.29(0.92)^bv;Slopet(^bv;Slope )5.41(6.58)-0.39(-0.74)-0.62(-1.45)0.17(1.35)

3rd step: zi;t^br = zi;t^bv + i;tscy log size
^bv ^
0.55
0.01
-0.07

log M Veq
0.02

q

1
NT

q

1
NT

P
(z i;t^br )20.38

P(z i;t^bv )2

F-stats
6.7[0.000]F-stats

2.99

15.8[0.000]2.80

15.2[0.000]

2
Level
Slope
RM SE
M AE
R2
0.01
0.10
0.11
0.09
0.93
2.04
1.10
4.70
[0.360]Premium
0.37
0.71
eMonthly observations from 1973 to 2011. I estimate br , bv and by three regressions Ri;t+1= zi;t br + "ri;t+1 ,v
Vi;t+1 = zi;t bv + "i;t+1 and zi;t^br = zi;t^bv + i;t . To put time …xed e¤ects, I demeaned the vector zi;tPNt
1by subtracting the time-speci…c mean zt
Each observation is multiplied by the square
i=1 zi;t .
Ntroot of relative value of the bond at time t. The numbers in parenthesis are t-statistics, where standard
errors are computed taking into account 1,12 and 12 lags of serial correlations (with Newey and West0(1987)’s weighing) for , br and bv , respectively. Premium is computed by E ft+1
ft+1
and expressed in2
percentage permonth.
ischi-squared
statistics
ofHansen(1982).
The
numbers
inbracketare p-values.
q
P 2
P
P
1
1
2
2
R is R-squared de…ned by 1 P(z bit )2 .
RM SEi;t i;t and M AE
i;t j i;t j.
NTNT^t( ^ )

it r

32

Table 8: Asset Pricing Test with Four Characteristics and Interactionse
1st step: Ri;t+1= zi;t br + "ri;tscy log size log M Veq^br 0.28
0.02
-0.33
^
t(br ) (2.39) (0.30) (-3.38)

0.18(2.59)s ~cy ~ log size ~ log M Veq ~0.25 -0.01(3.28) (-0.65)

0.07(2.40)-0.04(-1.78)

2nd step: Vi;t+1 = zi;t bv + "vi;tscy log size log M Veqs ~cy ~ log size ~ log M Veq ~^bv;Level 3.42 -2.41-1.00
t(^bv;Level ) (2.93) (-3.22) (-0.79)

1.38(1.95)

2.06
1.80(2.28) (7.59)

0.45(1.25)-0.22(-0.78)^bv;Slope 2.11
0.59
-1.19
t(^bv;Slope ) (2.77) (1.70) (-1.98)

0.59(1.59)

2.24 -0.47(6.57) (-4.00)

0.23(1.30)-0.11(-0.91)

3rd step: zi;t^br = zi;t^bv + i;tscy log size log M Veq^bv ^ 0.25
0.02
-0.13
0.08

s ~
0.25

q

1
NT

q

1
NT

P
(zi;t^br )2 F-stats
0.37

P(zi;t^bv )2 F-stats
5.16

11.6
[0.000]2.80

9.7[0.000]cy ~ log size ~ log M Veq ~
-0.02
0.03
-0.01

2
Level Slope
RM SE M AE
R2
^ 0.02
0.09
0.06
0.04
0.97
10.52
^
t( ) 0.42
4.64[0.105]Premium 0.56
0.67eMonthly observations from 1973 to 2011. I estimate br , bv and by three regressions Ri;t+1= zi;t br + "ri;t+1 ,Vi;t+1 = zi;t bv + "vi;t+1 and zi;t^br = zi;t^bv + i;t . ~ is log time to maturity. To put time …xed e¤ects,
PNt
1
Each observation isI demeaned the vector zi;t by subtracting the time-speci…c mean zt
i=1 zi;t .
Ntmultiplied by the square root of relative value of the bond at time t. The numbers in parenthesis are tstatistics, where standard errors are computed taking into account 1,12 and 12 lags of serial correlations (with0Newey and West (1987)’s weighing) for , br and bv , respectively. Premium is computed by E ft+1
ft+1
2
and expressed in percentage per month.
is chi-squared statistics of Hansen (1982). The numbers in
q
P
P
1
1
2
2
bracket are p-values. RM SE
i;t i;t and M AE
i;t j i;t j. R is R-squared de…ned by
NT
NT

1

P 2
it
P
.
(zit br )233

4.6[0.000]5.6.3

Including Equity Momentum

Finally, I include momeq in the set of instruments and test if the model can match the variation inestimated bond premia associated with the …ve characteristics. The test result is shown in Table
9.
The …rst step regression shows that ^br for momeq is positive and statistically signi…cant inpresence of other characteristics. That is, the bonds issued by past equity winners tend to earnhigh bond returns going forward.
The second step regression shows that the model works in the opposite direction. Both Levelt
and Slopet have negative loadings ^bv on momeq . That is, from the model’s perspective, the bonds
issued by past winners should be less risky. As a result, the model implied expected excess return
function has a wrong sign for momeq . Table 9 shows ^bv ^ for momeq is -0.03 while it is 0.07 for ^br .
As I include momeq in the set of instruments, RM SE and M AE become large at 0.18% and0.15% per month, respectively.

The alphas are jointly statistically signi…cant and the GMM

2test rejects the model at the 1% level. In conclusion, the two factor model cannot price these test
assets as the pricing errors are both economically and statistically signi…cant.

6

Conclusion

I show that the simple two factor model consisting of the …rst two principal components of the 10
portfolios sorted on bond spreads prices much of the variation in corporate bond premia. Whenthe asset pricing tests are done with univariate sorts, the model performs well regardless of the
choices of sorting variables except equity momentum.
I introduce the parametric characteristic-based asset pricing test, and show its ‡exibility ofincorporating multiple characteristics into the test. It can describe expected excess returns andcovariances of returns with factors as a function of characteristics and check if these functions
are equal. By taking a stand on the functional form of the expected comovement function, I canestimate risk for individual securities precisely without sorting securities into bins.
The two factor model consisting of Levelt and Slopet cannot be rejected using the four characteristics that best forecast excess returns as instruments, providing a further evidence for themodel. The model can price the variation in bond premia associated with the characteristics otherthan the bond spread used to form the factors. However, the model is rejected when it is tested
using the …ve instruments including equity momentum.

34

Table 9: Asset Pricing Test Including Equity Momentume= zi;t br + "ri;t
1st step: Ri;t+1^brt(^br )scy

log sizelog M Veq

momeq

0.54
(4.53)

0.06(0.91)-0.17(-2.57)0.05(1.88)

0.07(3.47)

2nd step: Vi;t+1 = zi;t bv + "vi;tscy

log sizelog M Veq

momeq

^bv;Levelt(^bv;Level )0.96(0.50)

4.55(4.94)-0.97(-1.16)0.27(0.83)-0.38(-1.38)^bv;Slopet(^bv;Slope )5.37(6.62)-0.40(-0.76)-0.57(-1.33)0.15(1.29)-0.23(-1.78)

3rd step: zi;t^br = zi;t^bv + i;t
s
cy
^bv ^
0.52
0.00

log size-0.06

log M Veq
0.02

momeq
-0.03

q

1
NT

q

1
NT

P
(zi;t^br )2P

F-stats

0.40

12.6[0.000](zi;t^bv )2

F-stats

2.99

12.8[0.000]2.79

12.2[0.000]

2
Level
Slope
RM SE
M AE
R2
0.01
0.10
0.18
0.15 0.80
19.54
0.93
4.45
[0.000]Premium
0.32
0.68
eMonthly observations from 1973 to 2011. I estimate br , bv and by three regressions Ri;t+1= zi;t br + "ri;t+1 ,v
Vi;t+1 = zi;t bv + "i;t+1 and zi;t^br = zi;t^bv + i;t . To put time …xed e¤ects, I demeaned the vector zi;tPNt
1by subtracting the time-speci…c mean zt
Each observation is multiplied by the square
i=1 zi;t .
Ntroot of relative value of the bond at time t. The numbers in parenthesis are t-statistics, where standard
errors are computed taking into account 1,12 and 12 lags of serial correlations (with Newey and West0(1987)’s weighing) for , br and bv , respectively. Premium is computed by E ft+1
ft+1
and expressed in2
percentage permonth.
ischi-squared
statistics
ofHansen(1982).
The
numbers
inbracketare p-values.
q
P 2
P
P
1
1
2
2
it
P
.
RM SE
i;t i;t and M AE
i;t j i;t j. R is R-squared de…ned by 1NTNT(z b )2^t( ^ )

it r35

References[1] Adrian, Tobias, Richard K. Crump and Emanuel Moench, 2011, E¢ cient, Regression-Based
Estimation of Dynamic Asset Pricing Models, Working Paper[2] Ang, Andrew, Jun Liu and Krista Schwarz, 2010, Using Stocks or Portfolios in Tests of Factor
Models, Working Paper[3] Avramov, Doron and Tarun Chordia, 2006, Asset Pricing Models and Financial Market Anomalies, Review of Financial Studies 19, 3, 1001-1040[4] Bao, Jack, Jun Pan and Jiang Wang, 2011, The Illiquidity of Corporate Bonds, Journal of
Finance 66, 911-946[5] Bhamra, Harjoat S., Lars-Alexander Kuehn and Ilya A. Strebulaev 2010, The Levered Equity
Risk Premium and Credit Spreads: A Uni…ed Framework, Review of Financial Studies 23, 2,
645-703[6] Chen, Long, David A Lesmond and Jason Wei, 2007, Corporate Yield Spread and Bond
Liquidity, Journal of Finance 62, 1, 119-149[7] Chen, Long, Pierre Collin-Dufresne and Robert S. Goldstein, 2009, On the Relation Between
the Credit Spread and the Equity Premium Puzzle, Review of Financial Studies 22 (9), 33673409[8] Cochrane, John H., 1996, A Cross-Sectional Test of an Investment-Based Asset Pricing Model,
Journal of Political Economy 104, 572-621[9] Cochrane, John H., 2011, Discount Rates, Journal of Finance 66, 1047-1108[10] Collin-Dufresne, Pierre, Robert S. Goldstein and J. Spencer Martin, 2001, The Determinantsof Credit Spread Changes, Journal of Finance 56, 6, 2177-2207[11] Connor, Gregory, Matthias Hagmann and Oliver Linton, 2012, E¢ cient Semiparametric Estimation of The Fama–French Model and Extensions, Econometrica 80, 2, 713-754[12] Crabbe, Leland E., 1991, Callable corporate bonds: A vanishing breed, FEDS working paper#155, Board of Governors of the Federal Reserve System[13] Crabbe, Leland E. and Jean Helwege, 1994, Alternative Tests of Agency Theories of Callable
Corporate Bonds, Financial Management 23, 4, 3-20
[14] Cragg, John G. and Stephen G. Donald, 1993, Testing Identi…ability and Speci…cation in
Instrumental Variable Models, Econometric Theory 9, 2, 222-240

36

[15] Driessen, Joost, 2005, Is Default Event Risk Priced in Corporate Bonds?, Review of Financial
Studies 18 (1), 165-195[16] Fama, Eugene F. and Kenneth R. French, 1993, Common Risk Factors in the Returns on
Stocks and Bonds, Journal of Financial Economics 33, 23-49[17] Fama, Eugene F. and James D. MacBeth, 1973, Risk, Return and Equilibrium: Empirical
Tests, Journal of Political Economy 81, 3, 607-636[18] Ferson, Wayne E. and Campbell R. Harvey, 1991, The Variation of Economic Risk Premiums,
Journal of Political Economy 99, 2, 385-415[19] Gagliardini, Patrick, Elisa Ossola and Olivier Scaillet, 2011, Time-varying risk premium inlarge cross-sectional equity datasets, Working Paper[20] Gao, Pengqin, 2010, Firm Characteristics, Covariances and Cross-Sectional Expected Returns,
Working Paper[21] Gebhardt, William R., Soeren Hvidkjaer and Bhaskaran Swaminathan, 2005, The cross-sectionof expected corporate bond returns: Betas or characteristics?, Journal of Financial Economics
75, 85-114[22] Gebhardt, William R., Soeren Hvidkjaer and Bhaskaran Swaminathan, 2005b, Stock and bondmarket interaction: Does momentum spill over?, Journal of Financial Economics 75, 651-690[23] Gilchrist, Simon, and Egon Zakrajsek, 2012, Credit Spreads and Business Cycle Fluctuations,
American Economic Review 102, 4, 1692-1720[24] Hansen, Lars P., 2008, Generalized Method of Moments Estimation, New Palgrave Dictionaryof Economics, 2nd edition[25] Rosenberg, Barr, 1974, Extra-Market Components of Covariance Among Security Prices, Journal of Financial and Quantitative Analysis 9, 263-274[26] He, Zhiguo and Wei Xiong, 2012, Rollover Risk and Credit Risk, Journal of Finance 67,
391-429[27] Jostova, Gergana, Stanislava Nikolova, Alexander Philipov and Christof W. Stahel, 2010,
Momentum in Corporate Bond Returns, Working Paper[28] Merton, Robert C., 1973, Theory of Rational Option Pricing, Bell Journal of Economics andManagement Science 4, 1, 141-183[29] Nagel, Stefan and Kenneth J. Singleton, 2011, Estimation and Evaluation of Conditional Asset
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Instruments, Econometrica 65, 3, 557-586[35] Stock, James H. and Motohiro Yogo, 2002, Testing For Weak Instruments In Linear IV Regression, Working Paper[36] Warga, Arthur and Ivo Welch, 1993, Bondholder Losses in Leveraged Buyouts, Review of
Financial Studies 6, 959-982

AComparing Overlapping Data Sources

Table 10 compares the mean and percentile di¤erences between the observations used in my analysis
and the overlapping observations in alternative databases.

The distribution of the di¤erence inprices is skewed so that the mean is signi…cantly greater than the median.

Though the mediandi¤erence is small, some extreme observations exist (shown in 90 percentile values). This is because
the database in comparison has a unreasonable price probably due to recording errors. Using thedata in comparison instead of the one I use in the main analysis will not change my main result,
as when I apply the …lters described in Section 3, these unreasonable prices will be completely
eliminated.
The …ndings of Jostova, Nikolova, Phikipov and Stahel (2010) also support my conclusion.
They use similar data as mine and prioritize the data source in the same way as I did. They showthat whether using only the data with the highest priority and using the average across overlapping
observations do not a¤ect the resulting returns.

B

Estimation of Distance to Default

To create a characteristic which measures the probability of default, I will use Merton’s model. The
value of the assets of a …rm At follows a geometric Brownian motion.
38

Table 10: Comparing the Overlapping Observations Across Databases
RefDatLehman
Lehman
Lehman
Trace
Trace
Trace

CompareMergent
DataStream

Mean
1.807
1.674
1.895

10ptle
0.125
0.113
0.125

50ptle
0.935
0.853
1.000

90ptle
4.470
4.191
4.583

n
12,9295,504
7,425

Mergent
DataStream

1.322
1.356
1.293

0.010
0.000
0.062

0.521
0.477
0.613

3.304
3.375
3.250

107,770
55,861
51,909Mergent DataStream 1.646
0.047
0.552
3.678
23,815The mean and percentiles of price di¤erence (per 100 dollars) are reported. RefDat is the reference data
used in the analysis and Compare is the data presented here for comparison. 10ptle, 50ptle and 90ptle are10, 50 and 90 percentile values respectively. n is the number of observations.

dAt= dt +
At

A dWt

(8)

Let Dt be the book value of the debt of the …rm at time t. If the value of the …rm’s asset isless than the book value of debt at the maturity date, then it cannot repay the debt and defaults.
When in default, the bond holders take over the …rm immediately and the equity holders receive
zero. If the assets exceed the debt, then the equity holder receives the di¤erence between At and
Dt . This way, the market value of equity St can be considered as the price of a call option. The
equity value is given by Black-Scholes formula for a call option

St = At (d1 )Dt (d2 )

(9)

where
d1 =

log(At =Dt ) + r + 12

2
A

A

d2 =

log(At =Dt ) + r

1 2
2 A

A

and r is a risk-free rate andis a cumulative density function of a standard normal distribution.

We cannot directly observe the market value of the asset At and its volatility
can observe the market value of equity St and its volatilityS.

A . Instead, we

By applying Ito’s lemma to theequity and imposing no-arbitrage condition, we have the risk neutral dynamics of equity St :dSt = rSt dt +@St
At A dWt@At39

By matching coe¢ cient, we obtain@St AtA@At StAt= (d1 )
A
St

S =

(10)

Equations (9) and (10) give a system of two equations with two unknowns: At and

A.

Since

they are non-linear equations, I solve them numerically using KNITRO solver.

C

Non-parametric Estimates of the expected excess return function

To investigate an appropriate functional form, I provide graphical investigations based on nonparametric kernel regression of the formeRi;t+1= f (zi;t ) + "i;t+1

where the function f (zi;t ) is estimated non-parametrically using Gaussian kernel. Speci…cally, thekernel regression estimator at z0 is given by
zi z0
1 PN
e
Ri;t+1i=1 K
Nhhft (z0 ) =
zi z0
1 PN
i=1 K
Nh
h

N
X

e
wi0;h Ri;t+1

i=1

where K ( ) is the normal probability density function and h is the bandwidth of the kernel chosen
to minimize the cross-validation CV (h),CV (h)f^ i (zi ) =
and

N
X

e
Ri;t+1f^ i (zi )

2

(zi )

i=1
PN
e
j6=i wji;h Rj;t+1
PN
j6=i wji;h( ) is an indicator function which is one if zi is between 5 percentile and 95 percentile of the

distribution and zero otherwise. The idea of cross-validation procedure is that one has to strike abalance between minimizing errors and improving e¢ ciency of an estimate. Putting a large weighteon Ri;t+1
to estimate f (zi ) will reduce an error but will not use as much information from theadjacent observations Re , which results in ine¢ ciency. I penalize ine¢ ciency by using f^ i (zi ),j;t+1

or a leave one-out estimates in computing CV (h).

Minimizing CV (h) amounts to choose an

optimal balance between error minimization and e¢ ciency of the estimates. I estimate the function40Non-P arametric E s tim ate of Expec ted Returns
7
6

kernellinear43e

E(R ) % per month

5

2
1
0
-1
-5

0

5
10
Charac teris tic s

15

20

Figure 4: Bond Spreadft (z) for each month and take average of them. The standard errors of the estimated expectedexcess returns are computed using the time variation of ft (z) accounting for 12 lags. Figures 4 to
17 show the estimated expected excess return function with error bars for each characteristic. I
also plot the …tted line of linear regressions on zt . The result is robust to the choice of the kernel
function and Epanechnikov (quadratic) kernel yields a similar result.
From the …gures, we can see that for most of the characteristics, a linear approximation provides
a decent …t. Though statistical tests for the …t of parametric estimates to the kernel regression
estimates are available, I do not pursue them here as I run a multivariate regression (3) instead ofa univariate regression in the asset pricing test. Thus, the estimated kernel regression provides alimited guidance about how to choose the functional form, requiring a researcher to try multiple
functional form for robustness of the test result.

D

Comparing the performance of the standard errors

To compute standard errors, I need to computeSnwhere "i;t 2 "ri;t ; "vi;t ; "ri;tgiven by0
E zi;t"i;t+1 "0i;t+1 zi;to"vi;t 0 . If I cluster the standard errors by time, the estimates of S is
T

1 X 0
S^ =
Zt
NT
t=1

41

t Zt

Non-Parametric Estimate of Expected Returns
8kernellinear

7

5
4
3

e

E(R ) % per month

6

2
1
0
-1

0

0.05

0.1

0.15
0.2
Characteristics

0.25

0.3

0.35

Figure 5: Current Yield

Non-Parametric Estimate of Expected Returns
0.25
kernellinear

0.2

0.1
0.05
0

e

E(R ) % per month

0.15

-0.05
-0.1
-0.15
-0.2

2

4

6

8

10
Characteristics

12

14

16

Figure 6: Distance to Default

42

18

Non-Parametric Es timate of E xpec ted Returns
3
kernellinear2.5e

E(R ) % per month

2

1.5

1

0.5

0

-0.5
-6

-4

-2

0

2
4
Charac teris ti c s

6

8

10

Figure 7: Momentum (-12 to -2 month returns)

Non-Parametric Es timate of Expected Returns
0.7
kernellinear

0.6

0.4
0.3

e

E(R ) % per month

0.5

0.2
0.1
0
-0.1
-3

-2

-1

0
Characteristics

1

Figure 8: Log Size

43

2

3

Non-Parametric Estimate of Expected Returns
0.2

0.15

E(R ) % per month

0.1

0.05

e

0

-0.05

-0.1
kernellinear
-0.15
-0.5

0

0.5

1

1.5
2
Characteristics

2.5

3

3.5

Figure 9: Log Time to Maturity

Non-Parametric Estimate of Expected Returns
0.15
kernellineare

E(R ) % per month

0.1

0.05

0

-0.05

-0.1
-0.2

0

0.2

0.4
0.6
Characteristics

0.8

1

1.2

Figure 10: Short-term Debt/Total Debt Ratio

44Non-Parametric E s timate of Expec ted Returns
0.15
kernellineare

E(R ) % per month

0.1

0.05

0

-0.05

-0.1
-0.2

0

0.2

0.4
0.6
Charac teris tic s

0.8

1

1.2

Figure 11: Tangibility (Property, Plant and Equipment/Total Asset)

Non-Parametric Es timate of Expec ted Returns
0.35
0.3

kernellinear

0.2
0.15
0.1

e

E(R ) % per month

0.25

0.05
0
-0.05
-0.1

0

5

10
15
Charac teris tic s

20

25

Figure 12: Credit Ratings (AAA as 1, C as 21)45

Non-Parametric E s timate of Expec ted Returns
0.25
kernellinear0.2

e

E(R ) % per month

0.15

0.1

0.05

0

-0.05

-0.1
-5

0

5

10

15
20
Charac teris tic s

25

30

35

Figure 13: Age

Non-Parametric Estimate of Expected Returns
1.5
kernellineare

E(R ) % per month

1

0.5

0

-0.5

-1
6.2

6.3

6.4

6.5

6.6
6.7
Characteristics

6.8

6.9

7

Figure 14: Yield to Maturity on Matching Treasury Bonds

46

Non-Parametric E s timate of Expec ted Returns
2kernellineare

E(R ) % per month

1.5

1

0.5

0

-0.5

3

4

5

6

7
8
Charac teris tic s

9

10

11

3.5

4

Figure 15: Equity Size

Non-Parametric E s timate of Expec ted Returns
0.6

0.5

kernellineare

E(R ) % per month

0.4

0.3

0.2

0.1

0

-0.1

0

0.5

1

1.5

2
2.5
Charac teris tic s

3

Figure 16: Equity Book-to-Market Ratio

47

Non-Parametric Es timate of Expec ted Returns
0.2

0.15

kernellineare

E(R ) % per month

0.1

0.05

0

-0.05

-0.1

-0.15
-1

-0.5

0
0.5
Charac teris tic s

1

1.5

Figure 17: Equity Momentum (-12 to -2 Equity Returns)
where Zt is Nt

by

L data matrix of instruments andt is the sample covariance matrix of error

term "i;t+1 . Since the number of bonds at time t, Nt , is large relative to length
h of time (Nt >>i T ),
t may not be reliable estimates of true conditional covariance matrix E"i;t+1 "0i;t+1 jzi;t .

Toovercome this issue, I impose a structure on the data-generating process of the errors to obtainreliable estimates. Speci…cally, as suggested in Ang, Liu and Schwartz (2010), I compute standard
errors using two methods. First, I impose one-factor structure in error terms. Second, I assume
the deciles de…ned by distance-to-default drive comovements among error terms12 .
In the …rst methodology of computing standard errors, I assume that the error terms have theone-factor structure"i;t+1 =where ut+1 and vi;t+1 is iid. Let

i ut+1 + vi;t+1

2 denote the variance of u
t+1 anduwhose diagonal element is the variance of vi;t+1 .
t =

where

t is a Nt

by

1 vector which stacks

v denote a diagonal matrix

Then

0 2
t t u+

vi for all i that exist at time t.

Empirically, the shock ut+1 can be obtained as average errors
Nu~t+1 =t
1 X"i;t+1Nt

i=112Instead of using the characteristic-based deciles, Ang, Liu and Schwartz (2010) use ten industries to categolize
securities.

48

and the sensitivity parameter

is obtained by regressing "i;t+1 onto ut+1 . That is, regress"i;t+1 =+~t+1 + vi;t+1
iu

and obtain the OLS estimates ^i and v^i;t+1 .
In the second methodology, I assume that the bond i in decile D has the shock following"i;t+1 =where

i is 1

byi uD;t+1 +i;t+1

10 indicator vector whose D-th entry is 1 and 0 otherwise. uD;t+1 is 10

vector of decile-wide shocks with covariance matrix

u and

by

1

i;t+1 is an idiosyncratic shock that is

independent across securities and time. Then we have
t =

where

t is Nt

by

0
u t+

t

10 matrix constructed by stacking

diagonal entry is the variance ofi;t+1 .

i andis a diagonal matrix whose

In the article, I use distance-to-default as a characteristics

that de…nes the deciles. At each time t, I classify bonds that exist at time t into deciles dependingon their value of distance-to-defaults.

The decile-wide shock uD;t is constructed by the sample

averageu~D;t =

1 X
"i;t
Nt;D
i2D

where Ni;D is the number of securities in decile D at time t.
covariance matrix of u
~D;t . The idiosyncratic shock~i;t = "i;tu is estimated using the samplei;t is constructed by

~D;t
iu

and the sample variance of ~i;t is used to estimate the diagonal entries of.

Collin-Dufresne, Goldstein and Martin (2001) show that the residuals of the bond price change
after accounting for fundamentals have a one factor structure. Thus, the one-factor error model
has some empirical support. On the other hand, the distance-to-default decile model is logicallymore consistent with the objective of this article in the sense that all the moments of the model
should be estimated as a function of characteristics, not the name of the individual bond.
I show the result of the characteristic-based asset pricing test of the two factor model using threedi¤erent measures of standard errors. Here I use the set of instruments that best forecast returns,
as in Table 7. Since it is time consuming to account for time lags in the methodologies of Ang,
Liu and Schwartz (2010), I compute standard errors with no lags for all the three methodologies.
Table 11 shows the estimated standard errors andassumptions.

2statistics using the three alternative

I show the result with the same instruments as Table 7 as an example but other49

Table 11: Comparing Alternative Models of Standard Errors
Point Estimates
br bv;Level bv;Slope bv
s 0.53
1.05
5.41 0.55
cy 0.06
4.56-0.39 0.01
log size -0.16-1.05-0.62 -0.07
log M Veq 0.04
0.29
0.17 0.02

Cluster by time
t-statistics
br bv;Level bv;Slope(3.89) (0.73) (7.17)(0.80) (5.98) (-1.04)(-1.94) (-1.35) (-1.60)(1.35) (1.10) (1.32)

One factor error modelt-statistics
br bv;Level bv;Slope(4.78) (0.81) (6.61)(0.87) (8.08) (-1.04)(-2.41) (-1.78) (-1.87)(1.81) (1.42) (1.34)

DD-decile error modelt-statistics
br bv;Level bv;Slope(3.38) (0.84) (8.64)(1.55) (15.44) (-2.38)(-3.14) (-2.78) (-3.08)(1.54) (1.42) (1.58)

Levelt Slopet
0.01
0.10

Levelt Slopet
Levelt Slopet
Levelt Slopet(1.07) (4.65)(1.02) (6.32)(2.27) (3.55)
2[pv]
1.81 [0.404]
2.12 [0.346]
4.02 [0.134]
Monthly observations from 1973 to 2011. Each observation is multiplied by the square root of relative value ofthe bond at time t. The numbers in parenthesis are t-statistics, where standard errors are computed taking
into account 0, 0 and 0 lags of serial correlations for , br and bv , respectively. 2 is chi-squared statistics
for the test of the overidentifying restrictions and the numbers in bracket are corresponding p-values.

choices of instruments yield qualitatively the same result. Comparing

2 statistic and t-statistics

of , the statistical inference coincides with all the three measures of standard errors except for
t-statistic of ^ for Levelt using the distance to default decile model. That is, with any measure ofstandard errors, the model cannot be rejected at 10% level, ^ for Levelt is statistically insigni…cantand ^ for Slopet+1 is statistically signi…cant. The only di¤erence is that ^ for Levelt using
the distance to default decile model becomes marginally signi…cant.

Overall, the distance todefault decile model seems to produce standard errors that are too small.

This may be due tothe assumption that the sensitivity of the individual error terms to the industry level shock is
restricted to be one. If there is signi…cant heterogeneity in an industry such that the assumption
is unrealistic, the resulting standard errors might be wrong. Nonetheless, it is comforting that thetwo factor model is still far from being statistically rejected.
On the other hand, the one factor model produces very similar result to the clustered standard
errors. Overall, with alternative measures of standard errors, I did not …nd signi…cant evidence
against clustered standard errors.

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