Journal of Empirical Finance 16 (2009) 306–317
Contents lists available at ScienceDirect
Journal of Empirical Finance
j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / j e m p f i n
On the explanatory power of firm-specific variables in cross-sections
of expected returns ☆
Chu Zhang ⁎
Department of Finance, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
a r t i c l e
i n f o
Article history:
Received 4 September 2007
Received in revised form 2 September 2008
Accepted 7 October 2008
Available online 15 October 2008
JEL classification:
G12
Keywords:
Factor-mimicking portfolios
Firm-specific variables
Principal component factors
a b s t r a c t
This paper pertains to the controversy surrounding the explanatory power of certain firmspecific variables such as size and the book-to-market ratio in cross-sections of average stock
returns. To investigate whether these firm-specific variables capture the sensitivity of returns to
unobserved systematic risk, two sets of principal component factors are used. The first set is
constructed from individual stock returns and the second set is from size- and book-to-marketsorted portfolio returns. The evidence from the first set of factors shows that size and the bookto-market ratio have little to do with factor betas. The evidence from the second set of factors
shows that the forces underlying size and the book-to-market ratio are indeed systematic risks,
although they explain very little return variation at the firm level, and that the betas of size- and
book-to-market-sorted portfolio returns with respect to the corresponding systematic factors
do explain the size and book-to-market effects.
© 2008 Elsevier B.V. All rights reserved.
The finding in Fama and French (1992) that two firm-specific variables, size and the book-to-market equity ratio, can explain
average returns across firms sparked a serious debate among researchers. This finding is controversial because it potentially
invalidates the standard asset pricing theory that expected returns across assets are linear in the betas (i.e., standardized
covariances) of the returns with respect to a few marketwide factors. The Fama-French finding certainly contradicts the wellknown Capital Asset Pricing Model (CAPM), which includes the market portfolio as the only factor. The question is then whether
these two firm-specific variables are proxies for other unidentified systematic factors. To that end, Fama and French (1993, 1996)
construct two factors, one the return on small stocks minus the return on large stocks (SMB) and the other the return on high bookto-market stocks minus the return on low book-to-market stocks (HML). Together with the excess return on the market portfolio
(MKT), the three Fama-French factors pass Gibbons, Ross and Shanken (1989) multivariate test and explain most asset pricing
anomalies. Therefore, Fama and French (1996) interpret the predictability of the two firm-specific variables as being consistent
with the standard beta pricing theory. The three-factor model has since caught on in the literature of empirical asset pricing and it
is treated by many researchers as the empirical representation of the Intertemporal Capital Asset Pricing Model (ICAPM) of Merton
(1973) and of the Arbitrage Pricing Theory (APT) of Ross (1976).
The Fama-French interpretation of the explanatory power of the size and book-to-market effects, however, is not agreed by all
researchers. Lakonishok, Shleifer and Vishny (1994), for example, contend that the explanatory power of a firm's characteristics
comes from the irrational behavior of the investors who overreact to the past performance of the firms and drive the stock prices
away from their fundamental values. When the prices eventually return to their fair values, the reversals are picked up in
regression equations on firm characteristics related to the previously distorted prices. Their empirical evidence is that high book-
☆ I would like to thank Raymond Kan, Robert Savickas, Yexiao Xu, Yong Wang and Guofu Zhou for their helpful comments on an earlier version. The financial
support from the Direct Allocation Grant DAG03/04.BM36 of HKUST is gratefully acknowledged. All remaining errors are mine.
⁎ Corresponding author. Tel.: +852358 7684; fax: +852 2358 1749.
E-mail address: [email protected].
0927-5398/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.jempfin.2008.10.001
C. Zhang / Journal of Empirical Finance 16 (2009) 306–317
307
to-market portfolios are not necessarily riskier than are low book-to-market portfolios. Another challenge to the rationality of the
size and book-to-market effects comes from Daniel and Titman (1997) who use a clever two-way independent sorting of portfolios
by the book-to-market ratio and the beta with respect to HML. They show that average returns vary with the book-to-market ratio
holding the HML beta constant and that average returns do not change with the HML beta holding the book-to-market ratio
constant. The Daniel–Titman result casts doubt on the claim that size and the book-to-market ratio are proxies for the betas of
unidentified factors.
A lingering issue is that, although the Fama-French factors fail the Daniel–Titman test, other mimicking portfolios can be
constructed to explain the size and book-to-market effects.1 The issue then becomes moot because it is impossible to consider all
the ways that mimicking portfolios are constructed.
This paper addresses these issues. Are the size and book-to-market effects proxies for betas with respect to unobserved
systematic factors and how systematic are these factors in terms of the variation they explain in returns? Can factors be
constructed in a way that avoids the difficulty faced by the Fama-French factors? To address these questions, we construct two sets
of principal component factors. The first set of principal component factors is constructed from individual stock returns and the
second set is constructed from one-hundred portfolios sorted by size and the book-to-market ratio. The reason for constructing
two sets of principal component factors is the following. If the forces underlying the explanatory power of size and the book-tomarket ratio are the only systematic factors, or at least the dominant ones, they will then undoubtedly be present in the principal
component factors constructed from individual stock returns. Existing evidence by Brennan et al. (1998) has shown otherwise.
Suppose that the forces underlying the size and book-to-market effects are not dominant factors, but still systematic. It is possible
that these factors may be swamped by other more important systematic factors (in the sense of accounting for return variations) in
the principal component factors extracted from individual stock returns. If portfolios are sorted by size and the book-to-market
ratio, factors unrelated to size and the book-to-market ratio are averaged out and factors related to size and the book-to-market
ratio stand out, if they are indeed systematic. These factors can then be extracted by principal component analysis.
Empirical results confirm this intuition. The betas of individual stock returns with respect to the first ten principal component
factors constructed from individual stock returns have little cross-sectional correlation with size and the book-to-market ratio, so
size and the book-to-market cannot be proxies for the betas with respect to these factors. These principal component factors also
do not explain the size and book-to-market effects in an extended multivariate test. On the other hand, the principal component
factors constructed from the one-hundred portfolios sorted by size and the book-to-market ratio have the following features. First,
the factor structure is more obvious. The first three principal components explain more than 85% of the total return variation in the
one-hundred portfolios, much more than do those extracted from individual stock returns. Second, the factor loadings on the onehundred portfolio returns have clear patterns that are similar to the patterns of Fama-French factors. The loadings of the first factor
are relatively flat, reminiscent of the equally weighted market portfolio. The loadings of the second factor are positive for large
stock portfolios and negative for small stock portfolios, resembling the negative SMB, The loadings of the third factor are positive
for high book-to-market portfolios and negative for low book-to-market portfolios, resembling HML. Despite the resemblance, the
principal component factors extracted from the size- and book-to-market-sorted portfolios avoid the difficulty faced by the FamaFrench factors. They pass the two-way sorting test introduced by Daniel and Titman (1997) and they pass an extended multivariate
test to explain the size and book-to-market effects.
The rest of this paper is organized as follows. Section 1 lays out the basic structure of the beta pricing models. Section 2 presents
the empirical results on the principal component factors extracted from individual stock returns. Section 3 presents the empirical
results on the principal component factors extracted from one-hundred portfolios sorted by the size and book-to-market. Section 4
summarizes the results and discusses the implications.
1. Beta pricing models and econometric tests
Suppose that rt is the vector of returns on N assets in excess of the riskfree rate over the period t − 1 to t, following a stationary
process with mean ν and variance matrix Σ. The beta pricing theory states that the expected excess returns are linear in the betas
of the return with respect to some marketwide factors, ft, that can be traced by factor-mimicking portfolios. The asset pricing
model assumes that the returns are generated from
rt = α + Bft + et ;
Eðet jIt−1 ; ft Þ = 0N ;
ð1Þ
where It − 1 represents the information set at t − 1. Without loss of generality, we can assume that α′B = 0, i.e., the alpha is orthogonal
to the beta matrix. If ft is the excess returns on K factor-mimicking portfolios, the beta pricing theory asserts, in its exact form, that
α = 0N. Therefore,
Ert = BEft :
ð2Þ
The well-known CAPM is an example of a one-factor model in which ft is the excess return on the market portfolio. In the face of
the empirical difficulty for the CAPM, many researchers now resort to multifactor models that do not depend on the criterion of
mean-variance efficiency. Prominent examples of multifactor models include the ICAPM and APT in which the number and the
identity of the factors are unspecified. The essence of the beta pricing theory is that a large cross-section (N) of expected returns is
1
A discussion on this possibility is presented in the conclusion section of Daniel and Titman (1997).
308
C. Zhang / Journal of Empirical Finance 16 (2009) 306–317
explained approximately by their betas with respect to a small number (K) of systematic factors. In conditional versions of these
models, the beta matrix can be time-varying.
The ICAPM and APT do not specify what the systematic factors are. In empirical research, there are three major approaches to
constructing systematic factors, other than the return on a market portfolio. The first approach, pioneered by Chen, Roll and Ross
(1986), uses macroeconomic variables as systematic factors. The advantage of this approach is its straightforward interpretation of
the systematic factors, if they are found to be priced. The second approach, adopted by many researchers and exemplified by Fama
and French (1993), is to form factor-mimicking portfolios based on the explanatory power of some firm-specific variables and take
the returns on these portfolios as the systematic factors. The advantage of the second approach is also its easy interpretation of the
constructed factors and its suitability in addressing issues related to the explanatory power of the firm-specific variables. The third
approach uses purely statistical methods to extract systematic factors, relying on the factor structure of the return-generating
process, that is assumed in most asset pricing models. The advantage of the third approach is that extracted factors are guaranteed
to be systematic in the sense of accounting for return variations. The disadvantage is the potential lack of economic interpretation
of the extracted factors.
A widely used methodology in testing whether a given set of systematic factors is priced, i.e., demanding nonzero factor
premiums, is the so-called two-pass cross-sectional regression test in which the betas of the given factors are estimated in time
series and the returns are then regressed on the estimated betas to obtain the factor premiums. For the case in which the
systematic factors, ft, are excess returns on portfolios, a popular methodology to test the beta pricing model is the multivariate test
developed by Gibbons, Ross and Shanken (1989). It tests the null hypothesis, α = 0N, against the general alternative, α ≠ 0N, with a
statistic that is χ2N distributed asymptotically under the null, or FK,N − K distributed in finite sample under the additional normality
assumption.
Suppose that Xt = (x1t,…,xMt) is an N × M matrix of predetermined firm-specific variables and that Xt ∈ It − 1 with the possibility
that x1t ≡ 1. The firm-specific variables are said to explain the cross-section of expected returns if, in the empirically motivated
panel data regression model,
rt = Xt θ + ηt ;
E ηt jXt = 0;
ð3Þ
the vector of the slope coefficients is significantly different from the zero vector. Theoretically, there are two extreme cases in
which firm-specific variables may explain the cross-section of expected returns. The first is that the beta pricing theory is correct
and the firm-specific variables are perfect proxies for the betas of unidentified systematic factors. In this case, the columns of Xt
belong to the space spanned by columns of (possibly time-varying) B for some unspecified ft. The other extreme is α ≠ 0 and Xt lies
in the orthogonal complement of B. In this case, Xt can explain the cross-section of returns because it is cross-sectionally correlated
with α. More generally, Xt can be decomposed into two parts, one in the space spanned by B and the other in its orthogonal
complement, correlated with α. To argue that size and the book-to-market ratio are proxies for betas of systematic factors, Fama
and French (1993) construct the two factor-mimicking portfolios, SMB and HML, and show that the hypothesis that the α with
respect to MKT, SMB and HML equals zero cannot be strongly rejected.
In this paper, an extension to the multivariate test will be used that suits the purpose of examining the consistency between the
explanatory power of firm-specific variables and the beta pricing model. The model takes the form
rt = Xt θ + Bft + et ;
E½et jXt ; ft = 0N ; Var½et jXt ; ft = ∑;
ð4Þ
with the null hypothesis being θ = 0M and the alternative hypothesis being θ ≠ 0M. If the first column of Xt is the vector of ones, and
θ = (θ1,θ′2)′ where θ2 is the vector of slope coefficients, then the null hypothesis and the alternative hypothesis can be revised to
θ2 = 0M − 1 and θ ≠ 0M − 1, respectively. The difference between the extended multivariate test and the original one is that, in the
extended test, the alternative hypothesis is specifically tied to the given firm-specific variables, while in the original test, the
alternative is general and unspecified. The extended test is more powerful than the original one if the mispricing, α, is indeed
related to the given firm-specific variables, but it is less powerful if the mispricing is unrelated to the given firm-specific variables. If
the question is about a given set of firm-specific variables, then the extended multivariate test is more appropriate. Despite the fact
that both time-series parameters, B, and cross-sectional parameters, θ, are involved, the model is linear in B and θ, so that they can
be easily estimated given Σ, while Σ can be estimated in iterations as usual. A hypothesis regarding θ can be tested with a test
statistic that is asymptotically χ2 distributed under the assumptions that the error terms, εt, are conditionally heteroskedastic and
autocorrelated.
2. Principal component factors extracted from individual stock returns
2.1. Features of the principal component factors
The original APT model in Ross (1976) and Huberman (1982) assumes a strict factor structure for the return-generating process
in which firm-specific error terms are uncorrelated with each other. Factor analysis is required to extract factors in such models.
Realizing that it is unnecessary to make such a strict assumption, Chamberlain and Rothschild (1983) develop the APT implication
under an approximate factor model for the return-generating process in which firm-specific error terms may be weakly correlated
with each other, paving the way for principal component analysis to be used to extract factors. Based on the asymptotic APT of
Chamberlain and Rothschild (1983), Connor and Korajczyk (CK 1986, 1988) design a scheme of extracting factors from individual
C. Zhang / Journal of Empirical Finance 16 (2009) 306–317
309
Table 1
Principal component factors of individual returns
A. Proportional variances (%)
i
1
1966–1970
33.2
1971–1975
35.8
1976–1980
26.8
1981–1985
17.8
1986–1990
17.1
1991–1995
16.4
1996–2000
16.5
2001–2005
20.2
i
15
1966–1970
1.5
1971–1975
1.3
1976–1980
1.5
1981–1985
1.8
1986–1990
1.8
1991–1995
1.7
1996–2000
1.8
2001–2005
1.8
B. Cross-sectional regression test of
k
1
1966–1970
0.298
1971–1975
0.721
1976–1980
0.002
1981–1985
0.702
1986–1990
0.581
1991–1995
0.246
1996–2000
0.272
2001–2005
0.393
2
4.0
4.9
3.4
4.2
3.5
6.7
6.2
5.0
20
1.3
1.1
1.4
1.6
1.6
1.5
1.4
1.5
3
3.1
3.6
3.0
3.1
2.8
3.0
4.4
3.8
25
1.1
1.0
1.2
1.4
1.4
1.4
1.3
1.3
factor premiums
2
3
0.581
0.780
0.475
0.666
0.001
0.002
0.001
0.002
0.382
0.377
0.017
0.041
0.331
0.414
0.137
0.036
4
2.3
2.7
2.6
2.9
2.6
2.9
4.0
3.1
30
1.0
0.9
1.1
1.2
1.3
1.2
1.1
1.1
5
2.3
2.4
2.2
2.7
2.6
2.6
3.6
2.9
35
0.9
0.8
1.0
1.1
1.2
1.1
1.0
1.0
6
2.2
2.2
2.1
2.6
2.5
2.5
3.3
2.6
40
0.8
0.7
0.9
1.0
1.1
1.0
0.9
0.9
7
2.0
2.0
2.1
2.4
2.3
2.3
2.7
2.5
45
0.7
0.6
0.8
0.9
1.0
0.9
0.8
0.8
8
1.9
2.0
2.1
2.3
2.2
2.3
2.4
2.4
50
0.5
0.5
0.7
0.8
0.9
0.8
0.7
0.7
9
1.8
1.8
2.0
2.2
2.1
2.1
2.2
2.2
55
0.4
0.4
0.6
0.7
0.8
0.7
0.6
0.6
10
1.7
1.7
1.9
2.1
2.0
2.1
2.1
2.2
60
0.3
0.3
0.5
0.6
0.6
0.5
0.4
0.4
4
0.873
0.801
0.003
0.002
0.542
0.059
0.484
0.053
5
0.941
0.823
0.004
0.003
0.437
0.096
0.466
0.083
6
0.467
0.902
0.008
0.007
0.436
0.117
0.584
0.110
7
0.214
0.943
0.009
0.001
0.553
0.178
0.665
0.168
8
0.296
0.957
0.009
0.002
0.516
0.117
0.708
0.228
9
0.386
0.975
0.016
0.002
0.385
0.170
0.515
0.294
10
0.423
0.986
0.025
0.001
0.368
0.232
0.593
0.313
Panel A of this table reports proportional eigenvalues δi / ∑60
j = 1 δj of the second-moment matrix of excess returns. Panel B reports the p-value of the cross-sectional
regression test that the factor premiums of the first k principal component factors are all zero for k = 1,…,10. The calculations are done for each of eight sixty-month
subperiods during 1966–2005.
stock returns where the number of assets is much greater than the number of time-series observations. In this section, we follow
the CK approach to construct the principal component factors from individual stock returns.
Since the CK approach requires a constant number of stocks, it is usually applied to, say, five-year subperiods in the literature.
From the monthly returns of individual NYSE/AMEX/NASDAQ stocks with non-missing data in eight five-year subperiods during
1966–2005, we construct the principal component factors of the returns using the CK methodology. The number of stocks ranges
from 579 in 1966–1970 to 3130 in 1996–2005. Panel A of Table 1 reports the relative eigenvalues of the monthly return variance
matrix, δi / ∑60
j = 1 δj for i = 1,…,10, and, to save space, for i from 15 to 60 in steps of 5. Each number represents the proportion of the
total variation a factor explains in all the stocks that have non-missing returns during the subperiod. Panel B reports the p-values of
the two-pass cross-sectional regression test that the factor premiums associated with the first k principal component factors are
jointly zero, for k = 1,…,10.
The results in Panel A of Table 1 indicate that the first principal component explains 16.4% to 35.8% of the total variation in the
individual stock returns. Each of the remaining factors explains much less, but the proportion of explained variation declines
slowly. The p-values in Panel B show that, except for the 1976–1980 and 1981–1985 subperiods, the factor premiums are not
significantly different from zero. The results in Panel B are exploratory only. It is well known that the two-pass cross-sectional
regression test is not n-consistent and it does not work well when the number of stocks is much greater than the number of timeseries observations.2
We then examine the relations between extracted principal component factors and the firm-specific variables at the level of
individual firms. If size and the book-to-market ratio are proxies for the betas with respect to the major factors, then the crosssectional correlations of size and the book-to-market ratio with the betas of the first few extracted factors should be high. Panel A (B)
of Table 2 reports pairwise correlations between the size (book-to-market ratio) and one factor. The cross-sectional correlations are
calculated for each month and are averaged across months within each of the five-year periods.3 The correlations are low except for
the first factor or for certain periods. The numbers reported in Table 2 are pairwise correlations between the size (or book-to-market
ratio) and one factor. We also calculate multiple correlations between the size (or book-to-market ratio) and a set of factors (not
reported here). These multiple correlations are also low. The low correlations cast doubt on the notion that size and the book-to-
2
The factors extracted using the CK approach are unique only up to a nonsingular transformation, as are the factor betas and factor premiums. Testing the
significance of the premium for each individual factor is meaningless.
3
Alternatively, size and the book-to-market can be averaged across months within each of the five-year periods. One cross-sectional correlation can then be
calculated between the average size (or the book-to-market ratio) and the factor beta of the period. Theoretically, the two ways of calculation are different, but
the actual results turn out to be similar, so the results of the alternative calculation are not reported.
310
C. Zhang / Journal of Empirical Finance 16 (2009) 306–317
Table 2
Features of the loadings of principal component factors from individual returns
Factors
3
4
5
6
7
8
9
A. Average correlations
1966–1970
1971–1975
1976–1980
1981–1985
1986–1990
1991–1995
1996–2000
2001–2005
between the betas and SZ
0.51
0.56
0.47
0.45
0.33
0.18
0.13
0.16
0.05
0.33
0.18
0.02
0.24
0.28
0.18
0.29
1
0.25
0.36
0.29
0.43
0.05
0.01
0.22
0.02
0.09
0.04
0.22
0.19
0.19
0.21
0.01
0.03
0.04
0.24
0.02
0.11
0.03
0.00
0.19
0.17
0.11
0.04
0.17
0.04
0.22
0.01
0.17
0.01
0.06
0.08
0.26
0.09
0.07
0.02
0.05
0.05
0.02
0.11
0.12
0.17
0.07
0.12
0.04
0.09
0.00
0.05
0.12
0.00
0.15
0.28
0.09
0.03
0.05
0.06
0.01
0.06
0.06
0.05
0.07
0.12
B. Average correlations
1966–1970
1971–1975
1976–1980
1981–1985
1986–1990
1991–1995
1996–2000
2001–2005
between the betas and BM
0.10
0.14
0.15
0.35
0.10
0.15
0.28
0.01
0.21
0.14
0.03
0.23
0.24
0.02
0.09
0.03
0.28
0.14
0.20
0.05
0.07
0.08
0.06
0.15
0.09
0.08
0.14
0.10
0.03
0.07
0.09
0.04
0.10
0.10
0.03
0.02
0.04
0.09
0.04
0.08
0.12
0.06
0.04
0.16
0.10
0.05
0.03
0.04
0.04
0.04
0.03
0.08
0.02
0.01
0.00
0.04
0.10
0.08
0.13
0.07
0.00
0.05
0.01
0.01
0.09
0.20
0.07
0.07
0.00
0.00
0.10
0.00
0.04
0.01
0.06
0.05
0.06
0.12
0.06
0.06
1.7
0.3
0.7
1.3
0.9
1.4
3.4
8.9
0.3
0.4
0.9
0.2
0.9
0.3
7.1
1.2
0.8
0.2
0.1
1.0
0.5
1.5
0.6
1.5
0.1
0.6
0.7
1.2
0.0
2.2
0.2
0.1
0.0
0.3
0.3
1.6
0.2
0.3
0.4
0.7
0.1
0.9
0.8
0.6
0.1
1.0
1.2
0.2
0.1
1.0
1.5
0.3
0.1
0.9
2.2
0.0
1.0
1.3
0.3
0.3
0.2
0.7
0.7
0.1
0.02
0.03
0.05
0.01
0.05
0.01
0.26
0.06
0.06
0.01
0.01
0.07
0.03
0.08
0.02
0.07
0.01
0.04
0.05
0.08
0.00
0.11
0.01
0.00
0.00
0.02
0.02
0.11
0.01
0.02
0.02
0.04
0.01
0.06
0.05
0.05
0.01
0.05
0.05
0.01
0.01
0.07
0.10
0.02
0.00
0.05
0.10
0.00
0.09
0.10
0.02
0.02
0.02
0.04
0.03
0.01
C. Average beta
1966–1970
1971–1975
1976–1980
1981–1985
1986–1990
1991–1995
1996–2000
2001–2005
43.3
56.2
47.5
37.0
43.4
19.1
33.3
42.8
2
5.4
4.6
1.0
0.2
1.5
23.9
19.0
1.3
D. Average beta divided by standard deviation of beta
1966–1970
1.90
0.30
0.11
1971–1975
2.26
0.21
0.02
1976–1980
2.12
0.06
0.04
1981–1985
1.91
0.01
0.08
1986–1990
2.38
0.07
0.05
1991–1995
0.38
1.01
0.06
1996–2000
0.74
0.66
0.11
2001–2005
1.20
0.05
0.39
10
Panel A of this table presents the average cross-sectional correlations between the betas with respect to the first ten principal component factors extracted from
individual stock returns and the size of the individual firms. Panel B of the table presents the average cross-sectional correlations between the betas with respect to
the first ten principal component factors extracted from individual stock returns and the book-to-market ratio of the individual firms. Panels C and D present the
average beta and the average beta divided by the standard deviation of the beta, respectively. The calculation is done on eight 60-month periods during 1966–2005.
market ratio are proxies for the betas with respect to the factors extracted from individual stock returns. Panels C and D of Table 2
report the average beta and the average beta divided by the standard deviation of the beta, respectively, for each principal
component factor from 1 to 10. With few exceptions, the mean beta of the first factor is different from zero, while other factors have
their mean betas near zero. This pattern and the results in Panels A and B have implications that are discussed later.
2.2. Extended multivariate test
We now conduct a formal test of the consistency between the size and book-to-market effects and the beta pricing model using
the first ten principal component factors extracted from individual stock returns. The testing assets to be used here are the twentyfive portfolios sorted by size and the book-to-market ratio which are used by Fama and French (1993, 1996) and many other
empirical studies. The portfolios' returns and corresponding sizes and book-to-market ratios are obtained from French's website.
The time-series averages of the returns and the time-series standard deviations of the returns are reported in Table 3. For each
month, the cross-sectional average of the portfolios' sizes and book-to-market ratios are subtracted from the corresponding
variables, so that both size and the book-to-market ratio to be used in the extended multivariate test are in excess of their crosssectional means. The demeaned sizes and book-to-market ratios have more stable time-series properties, needed for econometric
evaluation, than do the original sizes and book-to-market ratios.
The twenty-five portfolios sorted by size and the book-to-market ratio are widely used in the literature. Their properties are well
understood. In Panel A of Table 3, we see that except for the five lowest book-to-market portfolios, there is a decreasing pattern
between the average return and size. Holding size constant, there is an increasing pattern between the average return and the book-
C. Zhang / Journal of Empirical Finance 16 (2009) 306–317
311
Table 3
Descriptive statistics of twenty-five portfolios sorted by size and book-to-market
A. Time-series means of portfolio returns
Low
Small
0.385
2
0.313
3
0.392
4
0.503
Big
0.392
2
0.893
0.657
0.703
0.525
0.554
3
1.047
0.930
0.711
0.742
0.563
4
1.196
0.943
0.866
0.844
0.615
High
1.451
0.981
1.035
0.839
0.650
B. Time-series standard deviation of portfolio returns
Low
Small
8.778
2
7.930
3
7.282
4
6.343
Big
5.380
2
7.281
6.331
5.698
5.401
4.844
3
6.432
5.577
5.124
5.019
4.662
4
5.970
5.297
4.924
4.858
4.405
High
6.346
5.960
5.583
5.549
4.819
C. Time-series mean of the relative size of portfolios
Low
Small
−2.402
2
−0.898
3
−0.053
4
0.867
Big
2.884
2
−2.365
−0.893
−0.038
0.859
2.591
3
−2.394
−0.880
−0.038
0.870
2.554
4
−2.469
−0.884
−0.038
0.878
2.408
High
−2.737
−0.908
−0.041
0.881
2.247
D. Time-series mean of the relative book-to-market ratio of portfolios
Low
2
Small
−0.567
−0.265
2
0.567
−0.293
3
−0.569
−0.300
4
−0.566
−0.290
Big
−0.578
−0.305
3
−0.062
−0.080
−0.084
−0.081
−0.082
4
0.183
0.158
0.158
0.179
0.159
High
0.906
0.802
0.753
0.745
0.649
This table presents the time-series mean and standard deviation of returns on twenty-five portfolios sorted by size and the book-to-market ratio. The returns are
monthly, measured in percentages. The sizes and book-to-market ratios of these portfolios are relative to their cross-sectional mean. The sample period is 1966.01–
2005.12.
to-market ratio. From Panel B, we see that the standard deviation of the portfolio return is decreasing in size. There is no clear
pattern for the standard deviation with respect to the book-to-market ratio. This fact is used by Lakonishok et al. (1994) to argue that
the book-to-market is not a valid measure of risk. Panels C and D report the demeaned sizes and book-to-market ratios for the
twenty-five portfolios. These two firm-specific variables are used in the extended multivariate test below. Since portfolios are sorted
by them, there is a clear increasing pattern in the numbers in Panel C across size and a clear increasing pattern in the numbers in
Panel D across the book-to-market ratio, but, overall, there are no clear patterns between size and the book-to-market ratio.
To implement the extended multivariate test, we need to determine the set of systematic factors. As we can see, the first factor
accounts for 16.4%–35.8% of the total variation in various periods. It is well known in the APT literature that the first principal
component factor has a strong correlation with the equally weighted market returns. The contribution of the remaining factors is
much smaller. A rule of thumb in empirical studies to determine the number of factors is to choose a K such that the first K eigenvalues account for a large proportion of the total variance while additional eigenvalues do not add much. Practically, however, it is
difficult to determine the cut-off point. There is no consensus in the literature on what should be the number of factors for the US
stock returns. Connor and Korajczyk suggest any number from one to six. Brennan et al. (1998), for example, choose K = 5. Others
use various numbers and check the robustness. In the following study, we consider three different choices: K = 3,5 and 10. We then
conduct an extended multivariate test on the twenty-five size/book-to-market ratio portfolios using no factors and using three sets
of principal component factors. The test pits firm-specific variables against factor betas in explaining average returns in cross
sections. Table 4 reports the results.
The model without factors is just the cross-sectional regression (3) at the portfolio level and the result shows that the returns
are negatively related to the size and positively related to the book-to-market ratio. The slope coefficients are significantly different
from zero. In the next three models with three, five, and ten principal component factors, the size and book-to-market effects
remain. The slope coefficients actually become slightly more significant because, as the number of factors increases, the variance of
the error term in the regression model decreases, and the standard error of the estimator becomes smaller. The results of the
extended multivariate tests indicate that the size and book-to-market effects cannot be explained by the betas of the first ten
principal component factors.
Brennan et al. (1998) examine similar issues as those in this paper and their results should be discussed here. Using a variant of
the two-pass test, they contrast the firm-specific variables with two sets of factors, one the first five principal component factors
extracted from individual stock returns and the other the three Fama-French factors. They note that the principal component
factors and the Fama-French factors are quite different. In the model with principal component factors, they obtain basically the
same result as those obtained in this paper: the size and book-to-market effects are strong in the presence of the principal
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C. Zhang / Journal of Empirical Finance 16 (2009) 306–317
Table 4
Extended multivariate tests of the size and book-to-market effects with factors extracted from individual returns
Model
θ1N
θSZ
θBM
No factor
1.3057
(7.7925)
1.3704
(8.3467)
1.3783
(8.5447)
1.3734
(8.5260)
− 0.1254
(− 4.3040)
− 0.1302
(− 4.5930)
− 0.1328
(− 4.6957)
− 0.1349
(− 4.7803)
0.1516
(2.6964)
0.1590
(2.9610)
0.1549
(2.9016)
0.1655
(3.1325)
Three factors
Five factors
Ten factors
This table reports the result of the extended multivariate test of the size and book-to-market effects in the model without factors
rt = θ1N 1N + θSZ SZt + θBM BMt + et
and in the model with principal component factors
K
rt = θ1N 1N + θSZ SZt + θBM BMt + ∑ bk hkt + et
k=1
where rt is the return on the twenty-five size- and book-to-market-sorted portfolios, SZt and BMt are their sizes and book-to-market ratios, hkts are the principal
component factors extracted from individual stock returns. The number of factors, K, is three, five, and ten. The numbers in parentheses are t-ratios. The sample
period is 1966.01–2005.12.
component factors. In the model with Fama-French factors, the size and book-to-market effects are much attenuated. They merely
report the difference without concluding whether the explanatory power of size and the book-to-market ratio is or is not
consistent with the beta pricing theory. The main conclusion of their paper is that there are many other firm-specific variables,
such as trading volume, dividend yield, stock price level, and past returns, not considered by Fama and French (1992) that can
explain returns and the Fama-French factors do not explain the explanatory power of these additional firm-specific variables.
In this paper, the apparent inconsistency between the size and book-to-market effects and the beta pricing model with the
principal component factors extracted from individual stock returns is the puzzle to be solved. There are two scenarios with which
the results in Table 4 are consistent. The first scenario is that the size and book-to-market effects are indeed inconsistent with the
beta pricing theory in general. They are not proxies for the betas of unidentified factors at all. The second scenario is that the size
and book-to-market effects are consistent with the beta pricing theory, but the corresponding factors (or their linear combinations)
are too small to be included in the first ten Connor–Korajczyk factors extracted from the individual stock returns. To find out which
scenario is more likely the case, we proceed to analyze principal component factors extracted from portfolios that are sorted by size
and the book-to-market ratio.
3. Principal component factors extracted from size and book-to-market sorted portfolios
3.1. Features of the principal component factors
Since the exact number of factors in the individual stock returns is difficult to determine, it is possible that the actual number of
factors is greater than ten and the size and book-to-market effects can be explained by the remaining factors. Adding more factors
to the extended multivariate test turns out to be an ineffective way of determining this. Instead, we choose to extract principal
component factors from size- and book-to-market-sorted portfolios. When portfolios are sorted by certain firm-specific variables,
patterns in the stock returns related to the given firm-specific variables are strengthened, while patterns unrelated to the given
firm-specific variables are canceled out. By working on portfolios sorted by size and the book-to-market ratio and on principal
component factors extracted from the portfolios, we can identify the systematic factors associated with size and the book-tomarket ratio more easily. We can then examine the consistency between the size and book-to-market effects and beta pricing
theory more rigorously in the extended multivariate tests.
We use the one-hundred portfolios sorted by size and the book-to-market ratio, also obtained from French's website. Monthly
time series from January 1966 to December 2005 are used for the analysis. Out of the one-hundred portfolios, four of them contain
no stocks in certain years. These four portfolios are excluded from the analysis. But, for convenience, we continue to call the
remaining portfolios the one-hundred portfolios. From the variance matrix of the one-hundred portfolios, we can construct the
principal components of the returns. Their features are reported in Table 5.
Panel A of Table 5 lists the proportional variances and cumulative proportional variances of the first ten principal components.
The first principal component factor accounts for 75.70% of the total variation in the one hundred portfolios and the first three
principal component factors account for 85.72% of the total variation. The remaining factors are not very important in the sense
that they explain a very small amount of return variation. The factor structure is more obvious for portfolios than for individual
stocks. The pattern of the variances of the principal components indicate that there are common systematic factors in the onehundred portfolios sorted by size and the book-to-market ratio.
Panels B, C and D list the portfolio weights for the first, second and third principal component factors. These weights are of great
interest. The first principal component factor has portfolio weights relatively evenly across all one-hundred portfolios with small
stocks and low book-to-market stocks receiving relatively higher weights. The first factor, therefore, resembles a weighted market
C. Zhang / Journal of Empirical Finance 16 (2009) 306–317
313
Table 5
Principal component analysis of 100 size/book-to-market-sorted portfolios
A. Variances of the first ten principal components (%)
i
1
2
Prop. var.
75.70
6.46
Cumu. prop. var.
75.70
82.15
3
4
3.57
85.72
1.07
86.79
5
0.74
87.54
6
0.51
88.05
3
1.34
1.34
1.27
1.22
1.13
1.06
0.99
0.97
0.85
0.81
4
1.28
1.22
1.20
1.11
1.07
1.00
0.97
0.97
0.86
0.76
5
1.19
1.15
1.09
1.08
1.01
0.95
0.95
0.84
0.80
0.75
C. Weights of the second principal component portfolio (%)
Low
2
3
Small
−17.96
− 14.17
−12.48
2
−13.03
−9.39
− 6.00
3
−9.08
−5.82
− 3.73
4
−8.20
−1.58
− 0.80
5
−7.03
−0.59
1.00
6
−5.22
0.27
2.42
7
−3.22
2.38
5.68
8
−1.97
3.67
4.56
9
0.05
4.14
6.29
Big
3.30
5.30
6.33
4
−10.92
−4.65
−0.39
−0.98
2.13
3.94
5.63
6.30
7.74
7.72
D. Weights of the third principal component portfolio (%)
Low
2
3
Small
12.09
9.94
23.56
2
− 35.89
−18.05
− 0.11
3
−47.48
−24.00
−15.19
4
−60.77
−31.31
− 3.24
5
−68.51
− 25.87
−12.11
6
−72.52
− 34.07
−16.77
7
− 74.99
−33.77
−10.75
8
−82.92
−33.15
−25.06
9
−75.37
−41.23
−20.41
Big
−68.29
− 34.52
−30.45
4
32.15
6.57
8.91
−0.99
−4.92
− 11.22
−10.56
−9.54
−15.67
−20.66
B. Weights of the first principal component portfolio (%)
Low
2
Small
1.52
1.38
2
1.64
1.46
3
1.58
1.39
4
1.48
1.26
5
1.46
1.20
6
1.36
1.18
7
1.18
1.10
8
1.19
1.00
9
1.08
0.90
Big
0.81
0.80
7
8
9
0.48
88.53
0.45
88.98
0.39
89.37
10
0.34
89.71
6
1.14
1.08
1.08
0.98
0.96
0.86
0.89
0.92
0.77
0.70
7
1.07
1.08
0.99
0.99
0.91
0.86
0.90
0.83
0.80
0.64
8
1.06
1.05
0.99
0.98
0.92
0.85
0.86
0.80
0.69
–
9
1.06
1.09
1.03
1.04
0.91
0.94
0.93
0.84
0.72
–
High
1.10
1.17
1.14
1.21
1.16
1.05
–
1.02
0.75
–
5
− 9.13
− 2.25
− 0.02
1.86
3.73
4.07
5.09
6.66
8.41
7.81
6
−8.01
−1.76
0.56
1.44
3.55
5.54
6.52
7.62
7.63
7.42
7
− 7.06
− 1.95
1.28
3.39
4.51
5.99
5.03
7.77
7.80
8.67
8
− 6.98
− 0.89
1.79
1.79
4.65
5.39
5.36
7.08
7.91
–
9
−6.81
−1.17
0.90
2.89
5.03
4.72
6.12
6.92
8.09
–
High
−8.35
−1.74
2.27
1.10
2.02
3.60
5
38.26
11.31
9.65
9.26
2.93
4.41
− 1.97
− 3.48
− 7.58
−16.38
6
38.58
26.02
21.58
13.45
10.82
7.64
1.37
−1.06
−3.06
−16.31
7
42.51
25.36
26.30
16.41
17.37
13.26
8.07
5.44
1.47
2.10
8
46.42
26.01
29.86
19.12
23.76
19.90
13.81
15.81
9.42
–
9
45.94
36.65
34.30
27.36
29.46
27.04
26.66
13.55
11.19
–
High
61.69
52.79
47.33
40.36
27.12
43.25
–
30.89
13.71
–
6.85
7.67
–
Panel A reports the proportional variances and cumulative proportional variances of the first 10 principal components extracted from the one-hundred size- and
book-to-market-sorted portfolio returns. Panels B, C and D report the weights of the first three principal component factors.
portfolio where the weights are proportional to the product of the numbers in Panel B and the number of stocks in each of the onehundred portfolios. The second principal component factor has long positions in the big stocks and short positions in the small
stocks. Thus, the second principal component factor resembles the negative of the second Fama-French factor, SMB. It is not exactly
the same as it is tilted toward having more short positions on low book-to-market stocks. The third principal component factor
tends to have long positions in high book-to-market firms and short positions in low book-to-market firms, so it resembles the
third Fama-French factor, HML. The exception is for the smallest size decile portfolios that all receive positive weights. Roughly
speaking, the three principal component factors resemble the Fama-French factors, (MKT, -SMB, HML). There are several
differences between the two. The first is that the principal components are constructed to explain maximum variance among all
uncorrelated unit length linear combinations, while the Fama-French factors are designed to capture expected returns without
taking the variances into consideration. The second is that MKT is value-weighted, while the first principal component here is not.
The third is that MKT, SMB and HML are zero-cost portfolios while the principal component factors are not. The fourth difference is
that, while the Fama-French factors are correlated, the principal components are uncorrelated with each other by design. The tilt
seen from their weights reflects the necessary rotation to achieve zero correlation.
The phenomenon that exactly three systematic factors show up in the one-hundred portfolios and that the second and third
factors are related to size and the book-to-market ratio have something to do with the fact that the portfolios themselves are sorted
by size and the book-to-market ratio and other patterns in the return are smoothed out. For the results in Table 5 to hold, three
conditions are crucial. The first condition is that much systematic risk at the level of individual stocks, except for the systematic risk
similar to the equally weighted average return, has betas uncorrelated with size and the book-to-market ratio at the firm level. The
second condition is that these betas average to zero. These two conditions guarantee that that systematic factors whose betas are
unrelated to the market beta, size and the book-to-market ratio drop out in the size- and book-to-market-sorted portfolios. The
314
C. Zhang / Journal of Empirical Finance 16 (2009) 306–317
Table 6
Goodness-of-fit for various multivariate regressions
A. rt = a + BftK + εt
1966–1970
1971–1975
1976–1980
1981–1985
1986–1990
1991–1995
1996–2000
2001–2005
B. rt = a + Bxt + εt
f t3
f t5
f10
t
xt = g t
xt = ht
0.404
0.445
0.316
0.248
0.236
0.250
0.270
0.284
0.450
0.496
0.366
0.305
0.288
0.307
0.347
0.344
0.546
0.594
0.471
0.422
0.399
0.423
0.476
0.464
0.391
0.420
0.298
0.222
0.215
0.171
0.220
0.258
0.126
0.113
0.098
0.070
0.088
0.097
0.143
0.097
C. gt = a + Bf tK + εt
1966–1970
1971–1975
1976–1980
1981–1985
1986–1990
1991–1995
1996–2000
2001–2005
D. ht = a + Bf tK + εt
f t3
f t5
f 10
t
f t3
f t5
f10
t
0.625
0.697
0.465
0.284
0.555
0.342
0.382
0.487
0.719
0.803
0.641
0.293
0.661
0.509
0.455
0.658
0.764
0.882
0.796
0.418
0.727
0.633
0.590
0.703
0.614
0.689
0.451
0.267
0.534
0.336
0.375
0.477
0.710
0.798
0.632
0.277
0.646
0.505
0.448
0.652
0.757
0.879
0.791
0.404
0.715
0.629
0.585
0.697
f t3
f t5
f 10
t
f t3
f t5
f 10
t
0.777
0.671
0.583
0.595
0.514
0.413
0.526
0.679
0.499
0.447
0.396
0.363
0.339
0.282
0.386
0.467
0.259
0.258
0.238
0.202
0.213
0.191
0.220
0.243
0.467
0.344
0.323
0.300
0.273
0.157
0.322
0.376
0.308
0.249
0.234
0.190
0.188
0.128
0.233
0.285
0.162
0.158
0.156
0.113
0.127
0.108
0.138
0.152
E. f tK= a + Bgt + εt
1966–1970
1971–1975
1976–1980
1981–1985
1986–1990
1991–1995
1996–2000
2001–2005
F. f Kt= a + Bht + εt
This table reports the goodness-of-fit for the multivariate regressions of the form yt = a + Bxt + εt, defined as 1 − vε / vy where vy is the sum of the variances of the
components in yt and vε is the sum of the variances of the components in εt. The variables are defined as follows.
rt: individual stock returns.
ftK: first K principal component factors extracted from individual returns.
gt: the first three principal component factors extracted from the one-hundred size- and book-to-market-sorted portfolios.
ht: the second and third principal component factors extracted from the one-hundred size- and book-to-market-sorted portfolios.
results in Table 2 indicate that this is indeed the case. The third condition, of course, is that there are systematic factors, not
necessarily corresponding to small numbers of the principal component factors extracted from the individual stock returns, whose
betas are correlated with size and the book-to-market ratio.4
Before we move to examine the consistency between the explanatory power of the firm-specific variables and the beta pricing
model with factors extracted from size- and book-to-market-sorted portfolios, we compare the two sets of principal component
factors. To reinforce the notion that the forces underlying the size and book-to-market effects are systematic, but they do not
account for the most important variations in individual returns, we calculate a goodness-of-fit measure for various multivariate
regressions of the form yt = a + Bxt + εt. The goodness-of-fit measure is defined as 1 − vε / vy where vy is the sum of the variances of the
components in yt and vε is the sum of the variances of the components in εt, consistent with principal component analysis.
The calculated goodness-of-fit measures are reported in Table 6.
Panel A reports the goodness-of-fit measures for individual stock returns regressed on the principal components factors
extracted from individual stock returns. These numbers are similar to those in Panel B of Table 1.5 Panel B reports the goodness-offit measures for individual stock returns regressed on the principal components factors extracted from the one-hundred size- and
book-to-market-sorted portfolio returns. The goodness-of-fit measures for the first three principal components factors extracted
from portfolios are smaller, but not too much smaller, than those for the first three principal components factors extracted from
individual returns. The main contribution, however, comes from the first principal component factor extracted from portfolios,
which also resemble equally weighted market portfolios. For the second and third principal component factors extracted from
4
It should be noted that the extracted factors from portfolios sorted by certain firm-specific variables may not represent the most important factors in terms of
their contribution to explaining the total variation in the individual returns. They may not be relevant for patterns in the expected returns associated with other
firm-specific variables either.
5
The CK method is based on second moment matrices while the goodness-of-fit measures here are based on variance matrices, causing some slight
differences.
C. Zhang / Journal of Empirical Finance 16 (2009) 306–317
315
Table 7
Daniel–Titman test on 100 portfolios sorted by SZ and BM
Average number of portfolios
β
Low
A. All portfolios
Low
BM
High
Average excess returns on portfolios
High
Low
β
High
26.3
5.6
0.1
5.0
19.2
7.8
1.0
7.2
23.8
0.50
0.50
0.06
0.63
0.75
0.72
1.10
0.97
1.03
8.8
2.2
0.0
2.4
4.9
2.7
0.0
2.9
8.1
0.65
0.70
–
1.13
0.77
0.98
–
1.18
0.95
C. Medium stock portfolios
Low
10.1
BM
0.9
High
0.0
1.1
8.0
0.9
0.0
1.0
10.0
0.86
0.56
–
0.61
0.82
0.54
–
0.70
0.92
D. Large stock portfolios
Low
BM
High
1.2
8.2
0.6
0.0
0.8
10.3
0.87
0.66
–
0.92
0.83
0.44
–
0.41
0.91
B. Small stock portfolios
Low
BM
High
9.9
1.1
0.0
This table reports the test of Daniel and Titman (1997) using the returns on portfolios independently sorted by book-to-market (BM) and the beta (β) with respect
to the third factor extracted from the returns on the one-hundred portfolios sorted by size and the book-to-market ratio. The analysis is conducted for all portfolios
(Panel A) and each of three size groups (Panel B for small stock portfolios, Panel C for medium stock portfolios, and Panel D for large stock portfolios).
portfolios, the explanatory power is substantially lower. This is the main evidence that the factors that track size and book-tomarket effects in returns are not major factors in individual stock returns. Panels C and D report the goodness-of-fit for the
regressions of the principal component factors extracted from the portfolio returns on the principal component factors extracted
from the individual stock returns. As we see, these goodness-of-fit numbers tend to be large. This means that part of the forces
underlying size and the book-to-market ratio is contained in the first ten principal component factors extracted from individual
stock returns, but they are scattered and are not strongly associated with any of the factors. Panels E and F report the goodness-offit for the regressions of the principal component factors extracted from individual stock returns on principal component factors
extracted from the portfolio returns. Two patterns show up. The explanatory power of the second and third principal component
factors extracted from the portfolio returns is much smaller than that of the first three principal component factors extracted from
the portfolio returns. As the number of factors extracted from individual stock returns on the left-hand side increases from three to
ten, the explanatory power of the factors extracted from the portfolios declines quickly. These patterns reflect the fact that much of
non-market factors in individual stock returns is canceled out in the size- and book-to-market-sorted portfolios and, as a result, the
factors extracted from portfolios have low explanatory power for the factors extracted from individual stock returns, except for the
common market factor.
3.2. Firm-specific variables vs. factor betas
Since the weights of the principal component factors have patterns related to the firm-specific variables, we can conduct some
tests comparing the role of firm-specific variables and factor betas. We pay more attention to the book-to-market ratio which is at
the center of the debates in the recent literature.
We first conduct an informal test, introduced by Daniel and Titman (1997), using portfolios independently sorted by the bookto-market ratio and the beta with respect to the third principal component from the one-hundred portfolios. The procedure is as
follows. For each year from 1966 to 2005, the one-hundred portfolios are sorted by their book-to-market ratio and are divided into
three equal groups. The one-hundred portfolios are also sorted by their beta with respect to the third principal component factor
and divided into three groups.6 These two sets of portfolios are sorted independently. For each of the nine intersections of the two
groups, monthly portfolio returns are averaged within the year and within the intersection. Also calculated are the number of
portfolios in each intersection for a given year. These annual numbers are then averaged over the forty years in the sample period.
The results are in Panel A of Table 7. We also conduct the same analysis for three equally divided size groups and the results are in
Panels B, C, and D.
Before we discuss the results, we note the differences between the analysis conducted here and that in Daniel and Titman
(1997). The first difference is that the beta they use is with respect to the Fama-French factor, HML, while the beta used here is with
respect to the third principal component factor extracted from the one-hundred portfolios. Their HML beta is estimated in rolling
regressions using return data in the previous three years, while the beta used here is the unconditional beta. The second difference
6
We divide them into three beta groups only, rather than five as Daniel and Titman do, because we do this for portfolios rather than for individual stocks.
Finely divided groups may end up with too few portfolios in many groups.
316
C. Zhang / Journal of Empirical Finance 16 (2009) 306–317
Table 8
Extended multivariate tests of the size and book-to-market effects with factors extracted from size- and book-to-market-sorted portfolio returns
Model
θ1N
θSZ
θBM
First factor
− 0.0026
(− 0.3090)
0.8123
(6.2751)
0.7482
(5.1590)
0.0023
(0.4712)
0.0023
(0.3567)
0.1791
(2.4731)
0.0003
(0.0555)
−0.0417
(−1.5445)
−0.1168
(−6.7447)
0.0124
(0.8248)
−0.0149
(−2.5109)
0.0048
(0.3308)
−0.0200
(−2.6527)
−0.0024
(−0.8701)
0.1459
(2.6207)
0.0643
(1.3340)
−0.0197
(−0.6292)
0.1180
(2.5040)
0.0280
(0.9575)
−0.0311
(−1.3363)
0.0164
(1.2164)
Second factor
Third factor
First & second factors
First & third factors
Second & third factors:
First three factors
This table reports the result of the extended multivariate test of the size and book-to-market effects in the model
rt = θ1N 1N + θSZ SZt + θBM BMt + ∑ bk hkt + et
k
where rt is the return on the twenty-five size- and book-to-market-sorted portfolios, SZt and BMt are their sizes and book-to-market ratios, respectively. hkts are
principal component factors extracted from the one-hundred size- and book-to-market-sorted portfolios. The numbers in parentheses are t-ratios. The sample
period is 1966.01–2005.12.
is that they sort individual stocks according to their beta with respect to HML, while we sort portfolios according to their beta with
respect to the third principal component factor. Even for the same factor, these two approaches may yield different groups.
The results reported in Table 7 are interesting and can be contrasted with those in Daniel and Titman (1997). First, from the left
panels, we see that the portfolios tend to fall in intersections on the main diagonal lines. There are few portfolios that have low
book-to-market ratios and high betas. Likewise, there are few portfolios that have high book-to-market ratios and low betas. The
sorting by the book-to-market ratio and the sorting by the beta with respect to the third principal component from the onehundred portfolios are in agreement. This feature is anticipated if the book-to-market ratio is a proxy for the beta with respect to a
systematic factor. In Daniel and Titman (1997) where the factor is HML and the sorting is done for individual stocks, this feature is
absent. The presence of this feature here bodes well for the results on average returns in the right panels. In Panel A on the right,
the average return no longer increases with the book-to-market ratio, but it increases with the beta. In Panels B, C and D, the
pattern is less clear, probably because there are fewer portfolios in each intersection of the groups. At least, the pattern that the
average return increases with the book-to-market ratio instead of the beta, observed by Daniel and Titman (1997) for HML, does
not emerge. The results in this table show that with a right factor and at a more aggregated level, the book-to-market effect is
consistent with the beta pricing theory.
We then conduct a formal test of the consistency between the size and book-to-market effects and the beta pricing model with
the principal component factors from the one-hundred portfolios. Table 8 contains the results of the extended multivariate test on
the twenty-five size- and book-to-market-sorted portfolios with various combinations of the principal component factors
extracted from the one-hundred size- and book-to-market-sorted portfolios.
From the table, we see that for models with first and/or second factors, the size and book-to-market effects remain strong, but
for the model with three factors, or the mode with just the third factor, the slope coefficients of the firm-specific variables become
much smaller and their t-ratios indicate that they are no longer significantly different from zero. The size and book-to-market
effects can indeed be explained by the betas of the three principal component factors extracted from the one-hundred size- and
book-to-market-sorted portfolios.
4. Conclusion
This paper deals with the difficulty that the Fama-French factors face in reconciling the size and book-to-market effects with the
beta pricing theory. We construct principal component factors from both individual stock returns and returns on portfolios sorted
by size and the book-to-market ratio. As has been shown in the previous literature, principal component factors extracted from
individual stock returns fail to explain the size and book-to-market effects. We also show that the individual stocks' betas with
respect to these factors are poorly correlated with the size and book-to-market ratio at the firm level. The principal component
factors extracted from the one-hundred size- and book-to-market-sorted portfolios tell a different story. The one-hundred sizeand book-to-market-sorted portfolios have a more obvious factor structure. The factor loadings are clearly related to size and the
book-to-market ratio, in a way resembling those of the Fama-French factors. The empirical performance of these factors is better
than that of the Fama-French factors. They pass the Daniel–Titman test using two-way independent sort by the book-to-market
and the beta of the third factor. They also pass the formal extended multivariate test.
C. Zhang / Journal of Empirical Finance 16 (2009) 306–317
317
The different results for the two sets of principal component factors rise from the following scenario. The forces underlying the
size and book-to-market effects are systematic, but they do not account for much of the return variation at the firm level and they
do not manifest themselves in the principal component factors extracted from individual stocks returns. In portfolios sorted by size
and the book-to-market ratio, most variation in returns unrelated to size and the book-to-market ratio is averaged out and
systematic factors underlying the size and book-to-market effects show up.
The main conclusion of this paper is that the size and book-to-market effects are consistent with the beta pricing theory,
although the systematic factors underlying these effects are not the most important ones in the sense of accounting for return
variation. Beside the main conclusion, the results from this paper have some implications for empirical asset pricing studies. The
twenty-five size- and book-to-market-sorted portfolios are often used in empirical studies as testing assets and the Fama-French
factors have been used as a yardstick in the empirical asset pricing literature against which the performance of other proposed
systematic factors is judged. In view of the results presented in this paper, these practices are unwarranted if the issues under study
are unrelated to the size and book-to-market effects.
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