The Size and Book-to-Market Effects Revisited by M. LAMBERT a and G. HÜBNER b Abstract. Fama and French (F&F) factors do not represent pure estimates of the size and book- to-market effects. We argue that the independent sorting procedure underlying the formation of the F&F mimicking portfolios distorts the rankings of US stocks along the size and book-to-market dimensions, causing spurious correlations between the premiums. Replacing independent rankings by conditional ones improves the properties of the individual risk premiums. As a major improvement, the technique delivers less specification errors when pricing passive investment indices. Keywords: Fama-French, Carhart, Size, Book-to-market, Momentum, Mimicking Portfolios JEL Classification: G11, G12 a Post-doctoral researcher, School of Business and Economics, Maastricht University, The Netherlands, and Solvay Brussels School of Economics and Management, Free University of Brussels, Belgium. Mailing address: School of Business and Economics, Maastricht University, Tongersestraat 53, Room B 1.05A, 6211 LM Maastricht, The Netherlands. Tel.: (+31) 43 388 39 49. Email: [email protected]. b Corresponding author. The Deloitte Professor of Financial Management, HEC Management School, University of Liège, Belgium; Associate Professor, School of Business and Economics, Maastricht University, the Netherlands; Chief Scientific Officer, Gambit Financial Solutions, Belgium. Mailing address: HEC-University of Liège, Rue Louvrex 14 – Bldg N1, B-4000 Liège, Belgium. Tel: (+32) 4 2327428. Fax: (+32) 4 2327240. E- mail: [email protected].
Transcript
The Size and Book-to-Market Effects Revisited
by
M. LAMBERTa and G. HÜBNERb
Abstract. Fama and French (F&F) factors do not represent pure estimates of the size and book-
to-market effects. We argue that the independent sorting procedure underlying the formation of the
F&F mimicking portfolios distorts the rankings of US stocks along the size and book-to-market
dimensions, causing spurious correlations between the premiums. Replacing independent rankings
by conditional ones improves the properties of the individual risk premiums. As a major
improvement, the technique delivers less specification errors when pricing passive investment
a Post-doctoral researcher, School of Business and Economics, Maastricht University, The Netherlands, and Solvay Brussels School of Economics and Management, Free University of Brussels, Belgium. Mailing address: School of Business and Economics, Maastricht University, Tongersestraat 53, Room B 1.05A, 6211 LM Maastricht, The Netherlands. Tel.: (+31) 43 388 39 49. Email: [email protected]. b Corresponding author. The Deloitte Professor of Financial Management, HEC Management School, University of Liège, Belgium; Associate Professor, School of Business and Economics, Maastricht University, the Netherlands; Chief Scientific Officer, Gambit Financial Solutions, Belgium. Mailing address: HEC-University of Liège, Rue Louvrex 14 – Bldg N1, B-4000 Liège, Belgium. Tel: (+32) 4 2327428. Fax: (+32) 4 2327240. E-mail: [email protected].
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The Size and Book-to-Market Effects Revisited
1. Introduction
Anomalies in the US stock markets have been documented since the early 1980’s. Banz (1981)
uncovers a small size effect: firms with low market capitalization tend to outperform large cap stocks.
The research conducted by Fama and French (1992, 1998) also reveals that value stocks (i.e. stocks
with high book equity value with regard to their market value) outperform growth stocks (i.e. stocks
with low book-to-market ratio) over various sample periods. Finally, Carhart (1997) points out a
significant momentum effect in the US stock market by showing that significant gains can be realized
from long positions in persistent winner stocks and short positions in loser stocks.
Although these effects have long been related to both risk and mispricing, the seminal work of
Fama and French (1993) relates the first two market anomalies to proxy for, respectively, liquidity risk
and for market distress. The Fama and French (1993) 3-factor model and its 4-factor extension given
by Carhart (1997) have become a standard in performance evaluation. Using a set of data from CRSP
(The Center for Research in Security Prices), Fama and French consider two ways of scaling US
stocks, i.e. an annual two-way sort on market equity and an annual three-way sort on book-to-market
according to NYSE breakpoints (quantiles). They then construct six value-weighted (two-dimensional)
portfolios at the intersections of the annual rankings (performed each June of year y according to the
fundamentals displayed in December of year y-1). The size factor or SMB factor (“Small minus Big”)
measures the return differential between the average small cap and the average big cap portfolios,
while the book-to-market factor or HML factor (“High minus Low”) measures the return differential
between the average value and the average growth portfolios. French make these two factor series
available online1. Carhart (1997) completes the Fama and French three-factor model by computing,
along a similar method, a momentum (i.e. a 1-year prior-return) or UMD (“Up minus Down”) factor
that reflects the return differential between the highest and the lowest prior-return portfolios. On his
online data library, French computes a similar momentum premium by replacing the book-to-market
with the momentum risk dimension. The set of 2x3 size/momentum-sorted portfolios is rebalanced on
a monthly basis.
The challenge in the Fama and French (F&F) heuristics is to constitute mimicking or hedge
portfolios that are able to capture the marginal returns associated with a unit exposure to each attribute.
Although the factor construction method developed by Fama and French (1993) has become a
standard in constructing size, the value-growth and the momentum risk premiums, recent works have
suggested that the premiums obtained with the F&F technique could be misevaluated. For instance, the
study of Cremers et al. (2010) shows that the Fama and French 3-factor model displays significant
levels of specification errors when pricing passive indexes like the Russell 3000 or the MSCI value
Index or even size- and book-to-market-sorted portfolios. According to these authors, the value
premium is overestimated in the F&F framework as the latter methodology puts the same weight on
the small and big size portfolios while the value effect is in fact more important in small caps than in
big caps. Besides, following Huij and Verbeek (2009), the F&F mimicking portfolios could suffer
from an overestimation of the value premium and an underestimation of the momentum factors.
According to Brooks et al. (2008), the size premium even captures some part of the value premium
when defined using book-to-market.
In this paper, we investigate the Fama and French methodology and argue that the F&F premiums
are contaminated by cross-effects that are not adequately taken into account when performing an
independent sorting procedure. Besides, as in Cremers et al. (2010), we show that the F&F method
creates disproportion between the portfolios constituting the premiums: disproportionate weight is
placed on small value stocks. In our view, the independent ranking procedure does not optimally
diversify the other sources of risk than the one to be priced and does not sufficiently take into account
the correlations across risk dimensions. Our research gives indeed empirical and theoretical evidence
that the independent sorting procedure – because of the correlation between the rankings and the
disproportionate weights between portfolios – cause spurious estimates of the returns related to size
and value-growth effects. Therefore, we propose to replace the independent sorting procedure by a
sequential sorting procedure where the sort on the risk dimension to be priced is made conditional on
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the sorts over the two control risk dimensions. To concur with this objective of improving the
construction method of risk premiums, we consider a finer size classification by performing a triple
sort on market capitalization. The three sources of empirical risks (size, book-to-market, and
momentum) are treated within a cubic framework. Two breakpoints are used for all fundamentals and
are based on the whole equity market. Hence, 27 portfolios, instead of 6 in the original F&F
methodology, are formed per cube. Our common objective is to produce pure estimates of the returns
associated with each risk exposure.
Moreover, as our purpose is to define general guidelines that could be valid for any market
fundamentals and for any markets, we review some of the methodological choices made by Fama and
French that, according to us, are specific to the US stock market. For instance, while Fama and
French’s study covers the AMEX, NASDAQ and NYSE markets, only data from the NYSE are used
to form the breakpoints. The cubic method however considers the whole sample when defining the
breakpoints.
We bring all these modifications to the Fama and French methodology and propose an improved
set of size, book-to-market, and momentum premiums. The new procedure is tested against the F&F
one on a sample of monthly data downloaded from Thomson Financial Datastream Inc2, and on a
recent period of time: the actual sample for the risk premiums range from May 1980 to April 2007, i.e.
a total of 324 monthly observations. Our specification tests show that the cubic factors deliver less
specification errors than the F&F premiums when estimated on a set of 2x3 F&F portfolios (sorted on
size and book-to-market) and even insignificant specification errors for most passive indexes contrary
to F&F estimates. We also point out that the cubic factors proportionally better explain and provide
less specification errors than the F&F premiums do in explaining a set of 11,377 stock returns.
Concluding, we argue that if one has to choose one or the other specification, all evidence indicates
that the cubic construction should be preferred.
The rest of the paper is organized as follows. Section 2 addresses the problems related to the
independent sorting procedure performed in the F&F methodology. Section 3 presents the alternative
2 The use of alternative databases for the same market does not influence our results.
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methodology for constructing mimicking portfolios on size, book-to-market, and momentum. Section
4 carries out an analysis of the properties of the cubic and the F&F samples of empirical risk factors.
Section 5 performs comparative tests about the specification power of each pair of premiums. Section
6 concludes.
2. Preliminary evidence: problems related to the independent sorting procedure
2.1. Theoretical framework
The F&F approach works on a 2x3 approach while the cubic method, as it name indicates, works
on a 3x3x3 dimension. Suppose that the cubic and the F&F methods were working on similar
dimensions, i.e. along the 2x3 original approach. In this case, both the original Fama and French
methodology and the alternative method proposed in this paper would define their mimicking
portfolios as the following:
iiii
iiiii
iiii
iiiii
iiii
iiiii
iiii
iiiii
MVBTMMV
MVRBTMMV
MVBTMMV
MVRBTMMV
MVBTMMV
MVRBTMMV
MVBTMMV
MVRBTMMVSMB
)3&2.(Pr
)3&2.(Pr
)3&1.(Pr
)3&1.(Pr
)1&2.(Pr
)1&2.(Pr
)1&1.(Pr
)1&1.(Pr
(1)
and
iiii
iiiii
iiii
iiiii
iiii
iiiii
iiii
iiiii
MVBTMMV
MVRBTMMV
MVBTMMV
MVRBTMMV
MVBTMMV
MVRBTMMV
MVBTMMV
MVRBTMMVHML
)1&2.(Pr
)1&2.(Pr
)3&2.(Pr
)3&2.(Pr
)1&1.(Pr
)1&1.(Pr
)3&1.(Pr
)3&1.(Pr
(2)
where i = all stocks in the US markets (NYSE, AMEX, NASDAQ), MV = market value, BTM = book-
to-market and Pr.(.) = 1 or 0, MV {1=small, 2=large}, BTM {1=low, 2=medium, 3= high}.
In order to group together US stocks with small/large market capitalization and with low/high book-
to-market ratios, Fama and French perform two independent rankings on market capitalization and on
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book-to-market. Mathematically, the independent sorting procedure underlying the F&F framework
considers the following equality in order to classify stocks:
).(Pr).(Pr)&.(Pr bBTMaMVbBTMaMV iiii (3)
where )&.(Pr bBTMaMV ii is the probability of a stock i being included in group a {small,
large} sorted on market value and also in group b {low, medium, high} sorted on book-to-market.
The authors thus consider independent probabilities when classifying the US stocks into portfolios.
Indeed, the stock book-to-market ratio (resp. market capitalization) does not intervene in the stock
ranking according to size (resp. book-to-market). For instance, a stock enters the small/value portfolio
if and only if the probability of the market capitalization to be small is equal to 1 and the probability of
the same stock to be among value stocks is 1. In case one of these probabilities is zero (independently
of the other one), the stock will not be considered in the portfolio.
Such a classification is only valid if and only if the probabilities for a stock of having, for example,
low capitalization and low book-to-market are independent, i.e. if there is not important correlation
between the risk fundamentals. However, market capitalization and book-to-market are correlated. The
study of Fama and French (1993) points out that “using independent size and book-to-market sorts of
NYSE stocks to form portfolio means that the highest book-to-market/market equity quintile is tilted
toward the smallest stocks” (Fama and French, 1993, pp. 12). Besides, the use of NYSE breakpoints in
the F&F approach involves an over-representation of small stocks in the portfolios. The effects these
facts have on the formation of the independent portfolios are illustrated at Figure 1.
< Insert Figure 1 >
Figure 1 emphasizes two consequences of the independent sorting on the validity of the premiums.
First, the figure shows disproportionate weights among the shaded portfolios. The return spread
between the small value portfolio and the small growth portfolio captures the returns related to the
book-to-market effect after controlling for the small size effects. The disproportion between these
portfolios induces that the size effect could not be equivalently diversified in both portfolios and
therefore could not be eliminated by difference. A similar effect is also observed within the book-to-
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market spread in large cap stocks. The same reasoning could be applied to the estimation of the size
effects after controlling for book-to-market (using a vertical reading of Figure 1). The second
consequence, as already noted in Cremers et al. (2010), is that the F&F methodology places
disproportionate weight on small value stocks. This appears clearly in Figure 1 as well.
Not only this would produce correlation between the premiums but also the overweight of some
portfolios could cause spuriousness in the definition of the premiums.
The sequential sorting procedure proposed in this paper rather considers conditional probabilities as
follows:
)|.(Pr).(Pr)&.(Pr aMVbBTMaMVbBTMaMV iiiii
)|.(Pr).(Pr bBTMaMVbBTM iii (4)
where ).(Pr bBTM i is the probability of a stock i being included in group b {low, medium, high}
sorted on book-to-market, ).(Pr aMVi is the probability of a stock i being included in group a
{small, large} sorted on market value, )|.(Pr bBTMaMV ii is the probability of a stock i
already included in group b being included in group a, and finally )|.(Pr aMVbBTM ii is the
probability of a stock i already included in group a being included in group b.
Contrary to independent probabilities, such a definition of the joint probability takes into account
the correlation between the two risk dimensions. Indeed, the market capitalization of a stock will be
taken into account when ranking stocks according to their book-to-market ratios and vice versa. The
ranking on book-to-market (resp. size) will indeed be carried out conditionally on the market
capitalization (book-to-market) of the stock. We therefore argue that a sequential sorting procedure
could solve both of these problems, as illustrated in Figure 2.
< Insert Figure 2 >
By dividing each size (resp. book-to-market) sample into three equivalent groups of book-to-market
(resp. market capitalization), the conditional approach places the same weight on each portfolio. Note
that the F&F approach is working on NYSE breakpoints, while the sequential approach computes their
8
quantiles based on all US stocks. Therefore, in presence of correlation between the market
fundamentals, the use of conditional probabilities would better rank stocks into each specific portfolio.
As a consequence, it would also provide a better estimate of the return related to each dimension.
So far, we have worked as if the cubic and the F&F premiums were considering the same
breakpoints for breaking stocks into portfolios. The fact that the cubic method uses a 3x3x3 sorting –
instead of the original 2x3 F&F approach – results in a finer classification of stocks into size
portfolios. This also contributes to equilibrate the weights between the sorted portfolios and
contributes to a better distinction between small and large cap stocks.
2.2. Empirical evidence
We discuss here one intuitive illustration showing concrete pitfalls related to the use of an
independent sorting procedure as embedded in the original F&F premiums.
Table 1 displays the results of an “acid test” performed on a set of F&F portfolios made available
on French’s website. These portfolios are based on a 2x3 sort of stocks into size and book-to-market.
For instance, the Low/Mid portfolio stands for a portfolio made of stocks with low market
capitalization and medium levels of book-to-market. For each of these portfolios, the table considers
the original 4-factor F&F and Carhart model (model M.1) and evaluates the changes in the regression
coefficients when successively eliminating one risk factor (models M.2 to M.4), and then a second one
(models M.5 to M.7). SMBff, HMLff, and UMDff, stand respectively for the F&F estimates of the size,
book-to-market and momentum premiums. All changes superior to 80% with regard to the 4-factor
model are reported in bold.
< Insert Table 1 >
Table 1 indicates that the exposures to the SMBff factor displayed by the High/High
( ffSMB 541.18%) and High/Mid (
ffSMB 87.39%) portfolios are highly sensitive to the inclusion
of the HMLff factor in the regression-based analysis. Out of the analysis of the Low/Low and Low/Mid
portfolios, the loadings on the HMLff factor also appear to be unstable when other risk factors are not
9
included, particularly the SMBff factor ( ffHML 124.62% for the L/L portfolio, 96.47% for the L/M
portfolio when SMBff is not included in the regression). Finally, the UMDff factor is sensitive to the
inclusion of the two other empirical factors in the regression for 5 out of 6 portfolios. Although the
sort is not performed on momentum, the UMDff factor of F&F is significant in the 4-factor Carhart
model for all 2x3 portfolios, but the significance of the coefficient vanishes for small cap portfolios in
the absence of the size premium.
From this table, it appears that the exposures to the F&F empirical factors become unstable when
all three premiums are not considered together in one single regression-based analysis. Despite the fact
that the portfolios chosen in our example are supposed to reflect the size and value dimensions, the
F&F and Carhart 4-factor model does not deliver “pure” estimates of the exposures attached to each
type of risk, i.e. whose loadings are robust to a change in specification.
3. The modified Fama and French technique: the cubic method
In this section, we replace the independent rankings by a sequential sorting procedure. We
advocate that such a technique would lead to the purification of the risk factors by ensuring the
homogeneity of each constructed portfolio on all three fundamental risk dimensions (book-to-market,
momentum and size). Working on a 3x3x3 dimension, we call this the “cubic” method, by analogy to
the creation of a cube built with 27 identical cubic components.
3.1 The principle
The cubic approach differs from the F&F methodology on various points. First, we consider a
comprehensive framework that analyzes together the three empirical dimensions of risks: size, book-
to-market, and momentum. Each form of risk is equally considered. Besides, we propose a consistent
and systematic sorting of all listed stocks, while F&F perform a heuristic split according to NYSE
stocks only. Second, a monthly rebalancing of the portfolios captures more realistically the returns
associated with some time-varying dimensions of risk like liquidity issues or market distress. Third
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and finally, our sequential sort avoids spurious cross-effects in risk factors due to any correlation
between the rankings underlying the construction of the benchmarks.
The following subsections go into the details of the construction of this cube.
3.1.1. Sequential sorting procedure
Our objective is to detect whether, when it is made conditional on two of the three risk
dimensions, there is still enough variation related to the third risk criterion. Therefore, we substitute
the F&F “independent sort” with a “sequential or conditional sort”, i.e. a multi-stage sorting
procedure. Namely, we perform successively three sorts. The first two sorts are operated on “control
risk” dimensions, while we end with the risk dimension to be priced.
The sequential sorting produces 27 portfolios capturing the return related to a low, medium, or a
high level on the risk factor, conditional on the levels registered on the two control risk dimensions.
Taking the simple average of the differences between the portfolios scoring high and low on the risk
dimension to be priced, but scoring at the same levels for the two control risk dimensions, we obtain
the return variation related to the risk under consideration.
Figure 3 illustrates this procedure.
< Insert Figure 3 >
In the sequential sort, we end up with the risk dimension to be priced. Therefore, there are only
two possible ways to create the risk premiums, depending on the ordering of the first two sorts. We
choose the one that maximizes the number of stocks into the smallest final portfolio.3 First, the size-
sorted portfolios are formed by performing a 3-stage sequential sorting procedure on, successively,
book-to-market, momentum and market capitalization. Second, the book-to-market-sorted portfolios
are formed by performing a 3-stage sequential sorting procedure on, successively, market
capitalization, momentum, and book-to-market. Finally, the momentum-sorted portfolios are formed
3 We assume that the larger the portfolio, the better the accuracy of the risk premiums. Our conjecture is
confirmed by the empirical results. The control check tests are available upon request.
11
by performing a 3-stage sequential sorting procedure on, successively, book-to-market, market
capitalization, and momentum.
Contrary to an independent sorting, the sequential sorting places the same weight on all 27
portfolios.
3.1.2. Three-way sort
We consider the cross-section of US stock returns and model this risk space as a cube. We split the
sample according to three levels of size, BTM, and momentum4. Two breakpoints (1/3rd and 2/3rd
percentiles) are used for all fundamentals. Thus, not 6 but 27 portfolios are formed. The breakpoints
are based on all US markets, not only on NYSE stocks. The finer size classification also contributes to
equilibrate the proportion between the small/value, small/growth, large/value and large/growth
portfolios. It also provides a better distinction between small and large cap stocks.
3.1.3. Monthly rebalancing
To comply with a monthly rebalancing strategy, we assume that market participants refer to the
last quarterly reporting to form their expectations about each stock. Therefore, we use a linear
interpolation to transpose annual debt and asset values into quarterly data, as this is the usual
publishing frequency on the US markets:
)(12 1,,1, yiyiyiik DDkDD (5)
)(
12 1,,1, yiyiyiik AAkAA (6)
for k = 3,6,9,12, i.e. kth month of year y. Second, we ignore unrealistic values5 of BTM for the US
markets, i.e. higher than 12.5, in line with the empirical study of Mahajan and Tartaroglu (2008).
4 Like in Jegadeesh and Titman (2001) and Carhart (1997), the one-year momentum anomaly for month t is defined as the trailing eleven-month returns lagged one month (t-11 to t-1). Stocks that do not have a price at the end of month t-12 are not considered for that period.
5 We allow a variation of up to one standard deviation around the US average BTM.
12
3.2. The setup
The sample used in this paper is formed of all NYSE, AMEX, and NASDAQ stocks collected
from Thomson Financial Datastream for which the following information is available6: company
annual total debt7, the company annual total asset8, the official monthly closing price adjusted for
subsequent capital actions and the monthly market value. Monthly returns and market values9 are then
recorded for observations whose stock return does not exceed 100% and whose market values are
strictly positive. This is to avoid outliers that could result from errors in the data collection process.
We then define the book value of equity as the net accounting value of the company assets, i.e. the
value of the assets net of all debt obligations.
From a total of 25,463 dead and 7,094 live stocks available as of August 2008, we retain 6,579
dead and 4,798 live stocks with all criteria respected for the period ranging from February 1973 to
June 2008. The usable sample for the risk premiums ranges from May 1980 to April 2007 due to some
missing accounting data. The analysis covers 324 monthly observations. The market risk premium
inferred from this space corresponds to the value-weighted return on all US stocks minus the one-
month T-Bill rate.
We illustrate our methodology with the HML factor construction. We start by breaking up the
NYSE, AMEX, and NASDAQ stocks into three groups according to the market capitalization
criterion. We then successively scale each of the three size-portfolios into three classes according to
their 2-12 prior return. Each of these 9 portfolios is in turn split in three new portfolios according to
their book-to-market fundamentals. We end up with 27 value-weighted portfolios. The rebalancing is
performed on a monthly basis. For each month t, each stock is ranked on the selected risk dimensions.
It integrates one side, then one row, then one cell of the cube and thus enters one and only one
6 As for the risk space of F&F, temporary data non-availability excludes the stock from the analysis at that
time. 7 The company total debt at year y (D) concerns all interest bearing and capitalized lease obligations (long and
short term debt) at the end of the year. These variables have been collected on Compustat. 8 The company total asset at year y (A) is the sum of current and long term assets owned by the company for
that year. These variables have been collected on Compustat. 9 We designate by market value at month t, the quoted share price multiplied by the number of ordinary shares
of common stock outstanding at that moment. As in Fama and French (1993), negative or zero book values that result from particular cases of persistently negative earnings are excluded from the analysis.
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portfolio. The stock specific value-weighted return in the month following the ranking is then related
to the reward of the risks incurred in this portfolio.
To create a risk factor, we only consider, among the 27 portfolios inferred from the cubic risk
space, the 18 that score at a high or a low level on the risk dimension. 9 portfolios are then constituted
from the difference between high and low scored portfolios, which display the same ranking on the
size and momentum dimensions (used as control variables). Finally, the HML cubic risk factor is
computed as the arithmetic average of these 9 portfolios.
3.3. The acid test revisited
This subsection revisits the preliminary evidence presented in the Section 2.2 and contrasts it with
a similar acid test on the exposures obtained with the cubic premiums. We argue that as we rely on
conditional probability for classifying stocks into portfolios, our technique provides a more controlled
estimate of the return related to each portfolio.
Table 2 reproduces the analysis presented at Table 1 using a cubic 4-factor Carhart model. SMBc,
HMLc, and UMDc, stand respectively for the cubic estimates of the size, book-to-market and
momentum premiums.
< Insert Table 2 >
Contrary to the SMBff factor, the cubic estimate of the size premium resists much better to the
inclusion of the other empirical risk factors. In Table 1, the total average relative difference in the
SMBff coefficient resulting from removing one or two factors ranges from 6.99% to 384.6%, with a
mean change of 88.2%. The corresponding range in Table 2 is 1.08% to 34.45%, with a mean change
of 14.14%. The exposure to HMLc factor remains also fairly insensitive to the inclusion of the SMBc
factor in the regression for all portfolios but the Low/Mid one. The UMDc factor is finally largely
independent from the other two empirical risk factors: the UMDc portfolio is sensitive to the inclusion
of the SMBc factor only for one out of six portfolios.
Such results suggest that the SMBc, HMLc, and UMDc factors could be used separately in
regression-based analyses. Their exposures do not substantially vary when the other factors are
14
introduced or eliminated from the regression, unlike the SMBff, HMLff, and UMDff factors. The
maximum relative change in Table 1 is 594.92% from model M.1 to M.2-M.7, while it is only
272.49% in Table 2. The average instability is 52.97% when removing the first factor in the F&F
analysis while it is only 23.83% in the cubic framework. When removing the second factor, the
difference is even more important: the average instability evaluated from the absolute percentage of
change when two factors are jointly removed rises to 87.30% in the F&F analysis (Table 1), whereas it
is only 44.56 % in Table 2 with the cubic premiums. Besides, it has to be noted that the significant
changes in the cubic loadings only occur in cases where the loadings in the four-factor empirical
model are not significant from zero. These loadings even stay insignificant when one or two factors
are removed. Contrarily, the significant changes in the F&F factors are affecting the results as it
concerns explanatory variables that are significant in explaining the portfolio returns. This result is
confirmed by the very low values taken by the average absolute differences in exposures in the cubic
analysis compared to the F&F one.
Relying on a systematic and sequential sorting technique has two main advantages. First, our
factor construction method enables us to maximize the dispersion in the related source of risk while
keeping minimal dispersion in correlated sources of risk. Thus, it better captures the return spread
exclusively related with the source of risk to be priced. We expect our method to produce more
consistent estimates of the returns attached to any risk exposures than the ones produced by F&F.
Second, the cubic technique leads to risk premiums with a much lower level of correlations (see Table
6). By not conditioning the use of a risk premium to the inclusions of all other factors in the model, we
circumvent a strong limitation of the original Fama and French factors. Their risk exposures are indeed
highly sensitive to the inclusion of the other risk factors in the regression because of the levels of
cross-correlations. By contrast, the cubic risk factors ought not to be necessarily used jointly in a
regression-based model.
15
4. Properties of the cubic and the original Fama and French factors
The previous sections only present preliminary evidence over the stability of the cubic risk
premiums compared to F&F factors. We now turn to a deeper and more systematic analysis of the
properties of the competing sets of factors.
Table 3 below displays a full table of descriptive statistics for both sets of premiums. The table
covers the period May 1980-April 2007.
< Insert Table 3 >
All the F&F premiums display positive average returns over the period but only the HMLff and
UMDff premiums are significant over the period (at the usual significance levels). The momentum
strategy has the strongest returns, with an average value that is more than five times higher than the
one displayed by the size premium, and almost the double of the one displayed by the HMLff strategy.
The momentum premium is also more volatile. Regarding the cubic version of the premiums, not all
premiums present a positive average return. The HMLc premium displays a very small, insignificant
negative average return over the total period. To confirm our results, we downloaded data about the
S&P 500/Citigroup Growth and Value Indexes over the same period. Our results match empirical data
as the S&P 500/Citigroup Growth Index is slightly outperforming the value of the S&P 500/Citigroup
Value Index over this period.10 Note also that the importance of the SMBc premium becomes similar to
the one of the momentum strategy in the cubic framework. They produce approximately the same
(significant) positive average return over the period. The UMDc premium presents characteristics very
similar to the corresponding F&F premium.
In order to analyze the impact of our modifications on the original F&F method and the
differences in descriptive statistics between the F&F and the cubic risk premiums reported in Table 3,
we examine the 9 return spreads that result from each of our 3-stage sequential sorting procedures and
the return spreads that result from the F&F construction. This analysis helps us to understand the
differences resulting from applying the alternative methodologies.
10 The indexes display an average return of 0.9403% and 0.8784% for respectively the Growth and the Value
Index over the period May 1980-April 2007.
16
Table 4 reports descriptive statistics for the 3 sets of 9 return spreads related respectively to the
SMBc, the HMLc, and the UMDc factors. For each panel, the ordering sequence ends up with the
dimension to be priced, as explained in the methodological section. We closely examine the
correlations of these three sets of 9 portfolios with the SMBc, the HMLc, and the UMDc factors11.
< Insert Table 4 >
Panel A of Table 4 shows that each of the 9 return spreads related to the SMBc factor values the
premium related to the size effect equivalently. All portfolios offer comparable levels of mean returns
and volatility. The coefficient of variation of the series of average returns across portfolios is quite low
(i.e. CV=0.26/0.88 or 0.30). Besides, the portfolios seem to display strong correlation with the SMBc
factor but weak correlations (inferior to 30%) with the HMLc and UMDc factors.
Panel B shows that the 9 differences are correlated on average at 54.77% with the HMLc factor, but
display only weak correlation (less than 30%) with other types of risk. The book-to-market risk
premium is the highest in portfolios formed of stocks of low (resp. medium) market capitalizations and
presenting low (resp. medium) levels of prior returns. The table shows very large variations within the
series of mean returns across the different book-to-market spreads. This preliminary descriptive
analysis suggests the absence of a book-to-market effect in our sample. Indeed, as there appears to be
no stable BTM effect in the cubic framework, this might indicate that the sort on BTM captures only
noisy returns that could not be related to a source of risk priced on the market. Several papers pointed
out the possibility of a mispricing for explaining the positive return spread between value and growth
stock, even considering the latter riskier than the former one. Besides, the research conducted by
Mohanram (2004) and Michou (2007) shows that the distinction between growth and value stocks
could help distinguishing winner from loser stocks. Besides, the book-to market effect has also been
presented as being the strongest in low capitalized stocks (Griffin and Lemmon, 2002) or even as
being explained partly by the size effect (Brooks et al., 2008). Therefore, we expect that after having
controlled for the size and the momentum effects, the book-to-market effect would be seriously
mitigated if not vanishing.
11 Note that all correlations are significantly different from 1 at the usual significance levels.
17
Panel C shows that the momentum effect decreases with market capitalization. The momentum
spreads tend to be the highest in stocks presenting small or medium levels of market capitalization.
Table 5 repeats the same analysis on the 2 spread portfolios forming respectively the HMLff and
UMDff factors and the 3 spread portfolios forming the SMBff factor.
< Insert Table 5 >
The size spreads forming the SMBff factor displayed in Table 5 are all strongly correlated with the
SMBff factor but, contrary to the return spreads forming the SMBc factor, they also display substantial
correlations with the HMLff factor (superior to 30% for all 3 portfolios). Besides, while our
specification delivers portfolios that are quite homogeneous regarding the return spreads related to
size, here the low book-to-market-sorted portfolios display a very different average size spread
compared to the ones of the two other portfolios. The coefficient of variation even increases from 0.30
to 2.14 (i.e. CV=0.30/0.14). Such evidence suggests that the F&F empirical size factor is contaminated
by a book-to-market effect, as also indicated by the values taken by the cross-correlations between the
size return spreads and the HMLff factor. It even results in a negative size return spread in the low
book-to-market portfolio (where the reward associated with the book-to-market effect is in fact
negative). Besides, as already mentioned, our size factor is formed on the basis of the return
differential between portfolios of extremely small caps and portfolios of big stocks. By considering all
the NYSE, NASDAQ, and AMEX stocks, our breakpoints are tilted towards small caps compared to
the F&F premium. This could also explain the larger average spread observed for this premium.
Similarly, the two book-to-market spreads forming the HMLff factor display strong correlation with
the HMLff factor, but still present moderate levels of correlation with the SMBff factor. The
characteristics of the book-to-market return spread portfolios confirm evidence that the book-to-
market effect is the highest in low size portfolio. Finally, the momentum constructed according to our
cubic specification displays half the level of volatility compared to the F&F UMDff factor. There exists
substantial variation in returns related to the F&F momentum risk between small and big
capitalizations. The returns are more stable across the 9 cubic difference portfolios, showing that the
18
size effect has been eliminated. The coefficient of variation of the series of cross-sectional mean
returns is evaluated at 0.80 while it stands as a moderate 0.55 when considering the cubic sorts.
Concluding, the cubic construction method induces a large correlation of the post-formation spread
portfolios with the related factor but at the same time isolates the effects of the other two sources of
risk. The F&F factors do not seem to purely price the returns attached to respectively the size and
book-to-market effects but appear to be contaminated with correlated sources of risks. The book-to-
market premium, which is insignificant in the cubic framework, might be responsible for this
contamination effect. As in Cremers et al. (2010), we argue that the value-growth effect could be
overvalued. Our analysis even suggests that the book-to-market effect does not capture any kind of
systematic risk priced on the stock market.
Table 6 displays the correlation matrix of these two sets of premiums.
< Insert Table 6 >
The bottom-left corner displays the cross-correlations between the two sets of premiums. The
SMBc and HMLc factors are correlated at respectively 67.16 % and 68.25% with their F&F
counterparts. These levels indicate that, although the original and the modified size and value
premiums are intended to price the same risk, approximately one third of their variation provides
different information. The analyses conducted in Tables 4 and 5 have highlighted the potential reasons
for this difference. The momentum premium displays a higher correlation with the UMDff factor.
Contrary to the SMBff and HMLff factors, the French’s momentum premium does not exactly follow the
Fama and French (1993) methodology. The premium is rebalanced monthly rather than annually. It
differs from our momentum premium only with regard to the breakpoints used for the rankings and the
sequential sorting. The bottom-right corner presents the intra-correlations among the F&F premiums.
The SMBff and HMLff factors are highly negatively correlated over the period (-40.83%). The UMDff
premium also displays a negative correlation with the HMLff factor, but a positive correlation with the
SMBff factor. Such evidence contrasts with the top-left corner that presents the intra-correlations
among the cubic premiums. The signs are consistent with the ones displayed by the F&F premiums
but the levels of the correlations are considerably lower, which is consistent with our objective of
19
designing uncorrelated premiums. The intra-correlations among the F&F premiums are all statistically
significant, whereas the correlations among our alternative factors are only significant (but at an
inferior level) between the SMBff and HMLff factors12.
5. Specification tests
Two types of specification tests are conducted in this section. To begin with, we perform a basic
efficiency test – similar to the one that has been performed in Cremers et al. (2010) – on the empirical
asset pricing model to evaluate whether the F&F and the cubic specifications are able to price passive
indexes and passive investment portfolios without specification errors. Then, we carry out a direct and
rigorous comparison of the competing models. The procedure features a test of non-nested models on
individual stocks. The outcome of this test delivers the proportions of stock return series for which
there is a statistical dominance of one specification over the other.
5.1. Factor efficiency test
We evaluate the specification errors displayed by a cubic or an original 4-factor Carhart analysis
on the set of 2x3 F&F portfolios and on a set of passive benchmark indices.
5.1.1. The 2x3 set of F&F portfolios
These portfolios are constructed on the basis of a two-way sort into size and a three-way sort into
book-to-market. The time-series are downloadable on the French’s website.
We consider the following multivariate linear regression and test the values taken by the alphas:
ptmomtmomHMLtHMLSMBtSMBmtmppt RRRRR for p=1,…,N (7)
Instead of testing N univariate t-statistics based on each equation, we use the Gibbons et al. (1989)
test on the joint significance of the estimated values for p across all N equations:
0:0 pH for p=1,…,N (8)
12 Note that all correlations are statistically different from 1 at the usual significance levels.
20
Following Gibbons et al. (1989), under the null hypothesis (H0) that p is equal to 0 for all N
portfolios, the statistics ppFF RRT 1'1' )]ˆ1/([ follows a central F distribution with degrees
of freedom N and (T-N-L), where FR is a vector of sample means for the L factors ( FtR~ ), is the
sample variance-covariance matrix for FtR~ , p is a vector of the least squares estimates of the p
across the N equations.
We apply the Gibbons et al. methodology on the set of size and book-to-market-sorted portfolios
using returns from May 1980 to April 2007. We consider the case where L = 4 (the market index, the
SMB, the HML, and the UMD factor) and N = 6 for the 6 independent portfolios. The F statistic to test
hypothesis (4) when using the set of F&F premiums is 0.0597, so we cannot reject efficiency of the
F&F model at the usual levels of significance. When using the cubic premiums, the F statistics is even
reduced to 0.0000272. Thus, both sets of premiums seem to efficiently explain stock returns, with a
slight advantage to the cubic approach. In other words, the different changes performed on the original
F&F methodology does not seem to affect the efficiency of the factors.
5.1.2. Passive benchmark indices
We follow the study of Cremers et al. (2010) and apply a four-factor Carhart model to a set of
passive indexes. We evaluate both the F&F original factors and the cubic versions developed in this
paper. We consider the following passive benchmarks (All, Growth and Value): Russell 1000, Russell
2000, Russell 3000, S&P500, S&P MidCap, S&P SmallCap. The data were downloaded from
Thomson Financial Datastream over the period April 1997-April 2007, which constitutes a common
sample for all benchmarks.
< Insert Table 7 >
Table 7 shows that the original 4-factor model of F&F and Carhart produces significant levels of
specification errors for almost all passive benchmark indices. This result is fully consistent with what
Cremers et al. (2010) demonstrate in their study conducted over the period 1980-2005. The
modifications brought to the Fama and French methodology enables us to deliver a new set of risk
21
premiums which seem to be more able to price passive benchmark indices. Indeed, alphas of the 4-
factor Carhart model are mostly insignificant across all the regressions.
5.2. Non-nested models on individual stocks
This sub-section attempts to identify the potential superiority of one set of empirical premiums
(either the F&F ones or our updated version of the premiums) over the other one. We follow the
literature on model specification tests against non-nested alternatives (MacKinnon, 1983; Davidson
and MacKinnon, 1981, 1984). Such tests have already been used in the financial and the
macroeconomics literature13.
We consider the following two models:
1. M1 or the F&F model:
ittitiiit XR ',1 (9)
2. M2 or the cubic model :
ittitiiit ZR ',2 (10)
where Ri stands for the excess return on asset i, for the market premium, X’ for the F&F premiums,
and Z’ for the cubic risk premiums.
Two tests are jointly conducted. First, the model to be tested is M1, and the alternative model M2.
To test the model specification, we set up a composite model within which both models are nested.
The composite model (M3) writes:
ittiitiitiiti ZXR ')1( '1,
'1,,3, (11)
Under the null hypothesis 01, i , M3 reduces to M1; if 01, i , M1 is rejected. Tests are
conducted on the value of 1,i . Davidson and MacKinnon (1981, 1984) prove that under H0, can be
13 Bernanke et al. (1986) and Elyasiani and Nasseh (1994), among others, use non-nested models to compare
some model specifications about investment and U.S. money demand, respectively. Elyasiani and Nasseh (2000) differentiate between the performance of the CAPM and of the consumption CAPM through non-nested econometric procedures. Al-Muraikhi and Moosa (2008) test the impact of the actions of traders who act on the basis of fundamental or of technical analysis on financial prices based on non-nested models.
22
replaced by its OLS estimate from M2 so that 1,i and i (and i,3 , i ) are estimated jointly. This
procedure is called the “J-test”. We define iii )1( 1,* so that M3 can be rewritten as follows:
ittiititiiti ZXR 'ˆ '1,
'*,3, (12)
To test M2, we reverse the roles of the two models. We construct model M4:
ittitiitiiti ZXR ''ˆ '*'2,,4, (13)
We replace i by its estimate along M1 ( i ) and estimate i* (and i,4 , i ) jointly with 2,i . If
02, i , M4 reduces to M2; if 02, i , M2 is rejected. Tests are conducted on the value of 2,i .
We evaluate the goodness-of-fit of the two alternative asset pricing models on 11,377 individual
stocks. The following hypotheses are jointly tested on all the individual test assets:
Hypothesis I: ;0:0: 1,11,0 ii HagainstH
Hypothesis II: 0:'0:' 2,12,0 ii HagainstH
Each j follows a normal distribution with mean j and volatility j. Therefore, under the null
hypothesis, the statistics j
j
follows a Student distribution with 315 degrees of freedom – the
number of observation in each time-series (i.e. 324) minus the number of factors in each regression
(i.e. 9: the constant, the market portfolio, the 2 sets of 3 empirical premiums, and the estimate).
Among the four possible scenarios, we consider the following two cases:
)',( 10 HH , M1 is not rejected but M2 is;
),'( 10 HH , M2 is not rejected but M1 is14.
Table 8 presents the results of the tests over the significance of 1 and 2 across the assets, for
different confidence levels. We perform the following tests about the value of 1 and 2 :
14 Note that the rejection of H0 does not tell anything about the validity of H0’.
23
andtHtH stati
istat
i
i 1
11
1
10 ,
stati
istat
i
i tHtH 2
21
2
20 ','
(14)
< Insert Table 8 >
The table displays, for different levels of significance, the frequency of non-rejections of the F&F
model i.e. H0 (resp. of the cubic model, H’0) while rejecting the cubic premiums i.e. H’1 (resp. of the
F&F premiums, H1). It also reports the frequency of assets for which M1 and M2 are both rejected or
not. The first quarter of the table reflects the performance of the cubic model (Not reject M2 & Reject
M1), while the second quarter identifies the frequency of dominance of the original F&F model. We
report evidence that the cubic version is less frequently rejected and the F&F premiums more often
rejected for individual stocks than the opposite. The gap is largest at the 10% significance level, where
the test leads to the non-rejection of the cubic premiums 6.54% more often than with F&F premiums.
Overall, the non-nested econometric analysis shows that in most cases the F&F and the cubic
models are both not rejected when compared to the augmented model. This result does not imply that
any of the models provides a good fit, as this is not the scope of such test when performed on a
database of individual stocks. For a limited subset of stocks however (up to ca. one third), we can
discriminate between these models. Our cubic premiums seem to outperform the F&F specification.
The extent of this superiority is economically quite important, as the adoption of F&F factors instead
of the cubic ones would be (statistically) a wrong choice for almost 4,000 individual stocks.
6. Conclusions
This paper proposes an alternative way to construct the empirical risk factors of Fama and French
(1993). The original Fama and French (F&F) method performs a 2x3 sort of US stocks on market
capitalization and on book-to-market and forms six two-dimensional portfolios at the intersections of
the two independent rankings. The premiums are defined as the spread between the average low- and
high-scoring portfolios. Our main argument motivating the modifications brought to the original F&F
method is that the independent sorting procedure underlying the formation of the six F&F two-
24
dimensional portfolios distorts the way stocks are ranked into portfolios by placing disproportionate
weights between the portfolios.
Our paper aims at addressing some of the drawbacks identified in this heuristic approach to
construct risk factors. The main innovations of our premiums reside in a monthly rebalancing of the
portfolios (underlying the construction of the risk premiums) in order to capture the time-varying
dimensions of risk, in a finer size classification and in a conditional sorting of stocks into portfolios.
We consider three risk dimensions. The conditional sorting procedure answers the question whether
there is still return variation related to the third risk criterion after having controlled for two other risk
dimensions. It consists in performing a sequential sort in three stages. The first two sorts are
performed on control risks, while we end by the risk dimension to be priced.
Compared to the Fama and French method, our factor construction method better captures the
return spread associated with the source of risk to be priced. It is able to maximize the dispersion in
the related source of risk while keeping minimal dispersion in correlated sources of risk. The
conditional sorting and the finer size classification contribute to better equilibrate the weights placed
on the small/large value/growth portfolios. The great improvement of the new method lies in the
reduction of the specification errors when pricing passive benchmark investment portfolios. Besides,
without losing in significance power, in factor efficiency, the cubic technique is neater and leads to
risk premiums that may not necessarily be used jointly in a regression-based model, unlike the original
Fama and French factors whose risk exposures are highly sensitive to the inclusion of the other Fama
and French risk factors in the regression.
More generally, as in Cremers et al. (2010) and in Huij and Verbeek (2009), we argue that the
book-to-market premium of F&F is overvalued. We moreover argue that a sequential sorting
procedure could be more appropriate to take into consideration the contamination effects between the
premiums. We show that the premiums constructed along this way deliver more consistent risk
properties while reaching at least the same specification level as the F&F premiums.
At the same time, our paper tackles an important gap in the literature: how to best construct
fundamental risk factors. It has become standard practice to use the Fama and French method to
25
construct multiple size and book-to-market portfolio sorts and to use them in the cross-sectional asset-
pricing literature to evaluate models (Daniel and Titman, 1997; Ahn et al., 2009; Lewellen et al.,
2009). But there are, to our knowledge, only very few articles that use such multiple portfolio sorts for
pricing fundamental risk premiums. The usefulness of such an analysis is obvious for at least two
reasons. First, a method that could be systematically applied enables us to apply the method to other
exchange markets or to price other risk fundamentals. Second, by insulating as much as possible the
effects of other sources of risk when evaluating one risk factor, each of them can be used
independently of the others. This property is very useful for stepwise factor selection procedures, for
instance in style analysis or hedge fund models.
Acknowledgments
This paper has benefitted from comments by Antonio Cosma, Dan Galai, Aline Muller, Pierre
Armand Michel, Patrick Navatte, Christian Wolff, Jeroen Derwall as well as the participants to the
European Financial Management Association 2010 (Aarhus, Denmark), the French Finance
Association 2010 (St-Malo, France), the 1st World Finance Conference (Viana do Castelo, Portugal),
the Maastricht-Liège seminar day (Liège, September 2010) and the Luxembourg School of Finance
seminar participants. The present project is supported by the National Research Fund, Luxembourg
and cofunded under the Marie Curie Actions of the European Commission. Georges Hübner
acknowledges financial support of Deloitte Luxembourg. All remaining errors are ours.
a We perform a 4-factor Carhart analysis (M.1) on the F&F 2x3 portfolios sorted on size and book-to-market. In Models M.2 to M.4, we estimate the exposures to SMBff, HMLff or UMDff when one of these factors is removed from the regression. In Models M.5 to M.7, we estimate the exposures to SMBff, HMLff or UMDff when two factors are removed from the regression. We report in parentheses the percentage of change of the estimated coefficient. The average relative – resp. absolute – change reports, for each factor, the average percentage of change – resp. the average difference – (in absolute value) when one other factor is removed. The total average relative – absolute – change reports, for each factor, the average percentage of change – difference – (in absolute value) when either one or two other factors are removed from the regression. Percentages over 80% are reported in bold. T-tests are performed over the estimated coefficients: *,** ,and *** stand for significant at 10%, 5%, and 1%, respectively
Table 2
Independence test of the cubic empirical risk factorsb
b We perform a 4-factor Carhart analysis (M.1) on the F&F 2x3 portfolios sorted on size and book-to-market. In Models M.2 to M.4, we estimate the exposures to SMBc, HMLc or UMDc when one of these factors is removed from the regression. In Models M.5 to M.7, we estimate the exposure to SMBc, HMLc or UMDc when two factors are removed from the regression. We report in parentheses the % of change of the estimated coefficient. The average relative – resp. absolute – change reports, for each factor, the average percentage of change – resp. the average difference – (in absolute value) when one other factor is removed. The total average relative – absolute – change reports, for each factor, the average percentage of change – difference – (in absolute value) when either one or two other factors are removed from the regression. Percentages over 80% are reported in bold. T-tests are performed over the estimated coefficients: *,** , and *** stand for significant at 10%,5%, and 1%, respectively.
Table 3
Descriptive statistics over the empirical risk premiums (May 1980-April 2007)c
# Obs. 324 324 324 324 324 324 c Table 3 displays descriptive statistics for size (SMB), book-to-market (HML), and momentum (UMD) premiums over the period ranging from May 1980 to April 2007. T-tests of the significance of the different time-series are conducted. The values of the t-stats have been corrected for the presence of autocorrelation in the time-series. Panel A presents the statistics for the empirical risk premiums of F&F, while Panel B presents the statistics for the updated F&F premiums built along our cubic methodology. *,** ,and *** stand for significant at 10%, 5%, and 1% respectively.
Table 4
Descriptive statistics over the return spread portfolios forming each cubic risk factord
ic spreadUMD , 71.86 79.82 64.07 70.05 78.44 62.13 59.75 30.14 69.62 65.10 (14.77) d Table 4 displays descriptive statistics for the 9 return spreads forming the SMBc, HMLc, and UMDc factors. The correlations (in %) of each spread portfolio with the SMBc, HMLc, and UMDc factors are reported. The last column reports the average and the standard deviation of the statistics for the different portfolios. The size (resp. book-to-market, resp. momentum) spread portfolios are formed by performing a 3-stage sequential sorting procedure on, successively, book-to-market, momentum and market capitalization (resp. market capitalization, momentum, and finally book-to-market; resp. book-to-market, market capitalization, and momentum). Each spread portfolio is defined from a difference between two portfolios defined by 3 letters describing the 3-stage sequential sorting procedure. L stands for a low scoring portfolio, M for a medium scoring portfolio, and H for a high scoring portfolio. S.D. = Standard Deviation. The row corresponding to the dimension sought after by the spread portfolios is grayed.
Table 5
Descriptive statistics over the return spread portfolios forming each F&F risk factore
e Table 5 displays descriptive statistics for the return spreads forming the SMBff (Panel A), HMLff (Panel B), and UMDff (Panel C). The correlation (in %) of each spread portfolio with the SMBff, HMLff, and UMDff factors are reported. The last line in each panel reports the average and the standard deviation of the statistics for the different portfolios. The size spread portfolios are formed from the return spreads between small and big caps for 3 levels of book-to-market. The book-to-market (resp. momentum) spread portfolios are formed from the return spreads between high and low levels of book-to-market (resp. momentum) for two levels of market capitalization. Each spread portfolio is defined from a difference between two portfolios formed at the intersection of a two-way sort of stocks on size and a three-way sort on book-to-market or on momentum. S.D. = Standard Deviation. J-B = Jarque-Bera. The column corresponding to the dimension sought after by the spread portfolios is grayed.
Table 6
Correlation matrix of the empirical risk premiums (May 1980-April 2007)f
SMBc HMLc UMDc SMBff HMLff UMDff
SMBc 1
HMLc -15.50*** 1
UMDc 2.56 -3.19 1
SMBff 67.16*** -34.46*** 3.24 1
HMLff -18.87*** 68.25*** 2.35 -40.83*** 1
UMDff 9.61* -12.03** 82.63*** 10.66* -12.85** 1
f Table 6 reports the paired correlations (in %) among the cubic and among the F&F empirical risk premiums, as well as across these two sets of factors. Tests over the significance of the pair-wise correlations are performed: *,** , and *** stand for significant at 10%, 5%, and 1% respectively.
Table 7
Specification errors of passive investment indexesg
g Table 7 performs a 4-factor Carhart analysis on a set of passive indexes: Russell 1000/2000/3000 and the S&P 500/MidCap/SmallCap: only the levels of specification errors (alphas) and their significance are displayed. For each index, the results for the composite, the growth and the value index are presented.
Panel A Panel B 4-Factor Carhart Model :F&F specification 4-Factor Carhart Model: Cubic Specification All Growth Value All Growth Value Russell 1000 -0.0010*** 0.0002 -0.0021** 0.0000 -0.0015 0.0024 Russell 2000 -0.0043*** -0.0043*** -0.0040*** -0.0040 -0.0065* -0.0002 Russell 3000 -0.0013*** -0.0002 -0.0023** -0.0003 -0.0020 0.0021 S&P 500 -0.0031*** -0.0021* -0.0047*** -0.0021** -0.0037** -0.0008 S&P Mid Cap -0.0033* -0.0034* -0.0035** -0.0009 -0.0039 0.0021 S&P Small Cap -0.0060*** -0.0062*** -0.0065*** -0.0044 -0.0070** -0.0025
Table 8
Tests over the value of 1 and 2 in the Nested Models M3 and M4 for individual assetsh
H’0 and H1: Not reject M2 & Reject M1
H0 and H’1: Not reject M1 & Reject M2 H1 and H’1: Reject M1 & M2 H0 and H’0: Not reject M1 & M2 signif Test # % Test # % Test # % Test # %
10%
65.1' t and
65.1t 3786 34.15
65.1t and
65.1' t 3061 27.61 65.1t and
65.1' t 1890 17.05 65.1t and
65.1' t 2350 21.20
5%
97.1' t and
97.1t 3431 30.95
97.1t and 97.1' t 2884 26.01
97.1t and
97.1' t 997 9.00 97.1t and
97.1' t 3775 34.05
1%
59.2' t and
59.2t 2275 20.52
59.2t and
59.2' t 2017 18.19 59.2t and
59.2' t 265 2.39 59.2t and
59.2' t 6530 58.90
h Table 8 estimates the models M3 (testing the F&F model) and M4 (testing the cubic model) for the 11,377 individual assets (only 11,087 were available for the analysis). It jointly tests the significance of θ1 and θ2 using Equation (14). The table reports, for different levels of significance, the number of assets (and the frequency) for which both models are “accepted”, rejected, or accepted while the other one rejected.
Fig. 1. The formation of Fama-French mimicking portfolios. Fama and French sort stocks independently according to their market value (MV) and their book-to-market ratio (BTM) and form group of stocks at the intersection of the two rankings. Figure 1 is based on the result displayed at Table 1 of Fama and French (1993, pp.11). The shaded areas illustrate the formation of the following four portfolios: small/value (top right corner), small/growth (top left corner), large/value (bottom right corner) and large/growth (bottom left corner). Because of the use of the NYSE breakpoints for defining the mimicking portfolios, the portfolios within the lowest size quantiles have the most stocks. Therefore, the small/value and small/growth portfolios include proportionally more stocks than respectively the large/value and the large/growth portfolios. Moreover, Fama and French show that the number of stocks within each portfolio decreases from lower- to higher-BTM portfolios, except for small cap stocks. Finally, as the highest BTM-stocks are to be found within the smallest cap stocks, the small/value portfolio is expected to display more stocks than both the small/growth and large/value portfolios.
Fig. 2. The formation of mimicking portfolios using conditional rankings. The left-hand figure illustrates the formation of the book-to-market return spreads. The US stocks are first ranked according to their market capitalization. The light grey area represents the sample made of small cap stocks, while the white area corresponds to the sample of large cap stocks. Within each size sample, the stocks are then sorted according three levels of book-to-market (BTM). The difference between the portfolio of low and high BTM (shaded areas with plain hatching for small caps and dotted hatching for large caps) form the BTM return spreads. The right-hand figure illustrates the formation of the size mimicking portfolios. The dark grey area corresponds to the sample of US stocks with high BTM, the light grey area to the sample of US stocks with low BTM and the white area comprises stocks with medium BTM. Each area is further divided into two subsets according to the stock’s market capitalization. The spread between the dotted and the plain hatching area corresponds to the size return spread for each level of BTM.
Fig. 3. Sequential three-stage sorting procedure. This figure illustrates the sequential three-stage sorting procedure. The stocks are first sorted into three portfolios according to one control risk dimension. Within each portfolio, the stocks are sorted into three portfolios according to another control risk dimension. Finally, the stocks within the nine portfolios are sorted into three portfolios according to the risk dimension to be priced. Out of the 27 portfolios, we take the 9 return spreads on the risk dimension to be priced and compute the simple average of these 9 portfolios.
