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How to Construct Fundamental
Risk Factors?
January 2011
Georges Hübner
Affiliate Professor of Finance, EDHEC Business School
Marie Lambert
Researcher, School of Business and Economics, Maastricht University
Abstract
This paper proposes an alternative way to construct the Fama and French (1993) empirical risk
factors. Without losing in significance power, in beta consistency or in factor efficiency compared
to the Fama and French factors, our technique insulates the effects of other sources of risk as
much as possible when evaluating one risk factor. Consequently, the approach 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 factors in the regression. This property is very useful for stepwise factor selection
procedures. Besides, the methodology developed in this paper is easily extendable to price risk
fundamentals other than the empirical size, book-to-market and momentum effects and to other
markets (even to small exchange markets as the technique requires less in terms of data histories).
Concluding, this paper creates a theoretical framework for pricing the returns attached to a unit
exposure to any particular source of risk.
Keywords: Fama and French Factors, Momentum, Hedge/Mimicking Portfolios, Market Risk
Jel codes: G11, G12
This paper has benefited from comments by Antonio Cosma, Dan Galai, Martin Gruber, Thorsten
Lehnert, Pierre Armand Michel, Patrick Navatte, Christian Wolff, as well as Luxembourg School of
Finance seminar participants, the 2010 French Finance Association Conference (St Malo, France),
the 2010 EFMA Conference (Aarhus, Denmark), the 2010 annual joint Maastricht-Liège seminar
and the 1st World Finance Conference 2010 (Viana do Castelo, Portugal) attendees. This paper is a
substantially revised version of chapter 3 of the first author’s Ph.D dissertation at the Universities
of Liège and Luxembourg. The present project is supported by the National Research Fund,
Luxembourg and co-funded under the Marie Curie Actions of the European Commission. Georges
Hübner acknowledges financial support of Deloitte Luxembourg. All remaining errors are ours.
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2
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Copyright © 2011 EDHEC
Introduction
Fundamental factor models use market fundamentals or firm characteristics to construct factor
betas. Among this class of models, the empirical three-factor model of Fama and French (1993)
and the four-factor Carhart (1997) model capture the size, book-to-market, and momentum
effects. They have largely been used in the literature for explaining stock or mutual fund returns.
With the growth of the alternative investments universe since the mid-nineties, scholars have
also tried to estimate the returns attached to non-linear co-movements of asset returns with
market index returns. Their idea is to use the levels of skewness (3rd moment) and excess kurtosis
(4th moment) of stock, mutual fund, or hedge fund return distributions as factor betas.1
The challenge in such models is to constitute mimicking or hedge portfolios that are able to capture
the marginal returns associated with a unit exposure to each attribute. The factor construction
method developed by Fama and French (1993) has become a standard in constructing fundamental
risk factors. Using a set of data from CRSP (The Center for Research in Security Prices), Fama and
French consider two ways of scaling US stocks, — 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.2 Carhart (1997) completes the Fama and French threefactor 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 replaces the book-to-market
risk dimension by the momentum risk dimension and provides an estimation of the momentum
factor. The set of 2x3 size/momentum-sorted portfolios is rebalanced on a monthly basis.
The Fama and French (henceforth F&F) methodology specifically sticks to the US stock market,
and cannot be extended as such to other contexts for two reasons. First, their factor construction
method relies only on a limited equity market segment (the NYSE stocks) to define the sorting
breakpoints in order not to be tilted towards the numerous small stocks of the NASDAQ and AMEX
exchanges. The F&F methodology is also rather heuristic regarding the way risk fundamentals are
priced. The authors suggest performing a double sorting for size and a triple sorting for booktomarket on the judgmental basis that size is less informative than book-to-market.
The recent works by Cremers et al. (2008) and Huij and Verbeek (2009) have both shown that the
F&F method tends to misevaluate some of the premiums. According to Cremers et al. (2008), 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. As a corrective action, Cremers et al. (2008) consider small and big caps
separately when pricing the value factor. Following Huij and Verbeek (2009), an overestimation
of the value premium and an underestimation of the momentum factor should be related to
the ignorance of transaction costs when using stock returns for constructing the mimicking
portfolios.
In this paper, we address and confirm the issues raised in the paper of Cremers et al. (2008) and
Huij and Verbeek (2009) concerning the overvaluation of the HML premium. We argue that the
F&F premiums are contaminated by cross-effects that are not well taken into account when
performing an independent sorting procedure (causing correlation between the rankings). In our
view, the independent ranking procedure does not optimally diversify the other sources of risk
than the one to be priced or does it sufficiently take into account the correlations across risk
dimensions.
3
1 - See a.o. the analysis of Moreno and Rodriguez (2009) for mutual funds, and the models of Kat and Miffre (2006) and Agarwal et al. (2009) for hedge funds.
2 - French has made these series available for download on his website http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
If our view is true, the F&F methodology is not appropriate for pricing market fundamentals that
would be highly correlated as it would lead to a seriously unbalanced set of portfolios due to the
high correlation levels between the rankings. Therefore, we propose to replace the independent
sorting procedure by a sequential sorting procedure (the sort on the risk dimension to be priced
is made conditional on the sorts over the two control risk dimensions) in order to produce pure
estimates of the returns associated with each risk exposure. Moreover, as the 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 and to the analysis of the empirical sources of risk. We
particularly rely on a cubic representation of the risk exposures: we consider three sources of
risk and measure them using a three-way sort, forming a cube. 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.
Relying on a systematic and sequential sorting technique has two main advantages. First, our
factor construction method enables us to maximise 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.
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 necessarily be used
jointly in a regression-based model.
Besides, since our premiums do not rely on market-specific guidelines, we anticipate them to be
more easily applicable to other markets and other risk fundamentals.
Based on the same empirical risk factors example as F&F, we test our guidelines against those of
F&F. We conduct our analysis on a sample of monthly data downloaded from Thomson Financial
Datastream Inc3, and on a recent time period: the actual sample for the risk premiums range from
May 1980 to April 2007, i.e. a total of 324 monthly observations.
An analysis of the statistical properties of both samples of risk premiums shows that the cubic
and the original F&F factors are highly correlated (at about 70%). They aim at capturing the same
type of risk, but at the same time display very different descriptive statistics. In the F&F original
analysis, the size factor is considered to be less important than the momentum and the bookto-market effects. On the contrary, only the size and the momentum factors (and thus not the
book-to-market effect) generate a significant risk premium in the cubic framework. Through a
very simple example, we show that the F&F method does not deliver pure estimates of the return
attached to each type of risk. The exposures to the F&F empirical factors become erratic when
all three premiums are not considered together in one single regression-based analysis. We find
evidence that the cubic construction method better isolates the cross-effects between premiums
and, as a consequence, delivers purer estimates for the factors.
Beyond the study of each method’s own pricing abilities, we devote the second part of the study
to test the dominance of one method over the other. It is first shown that the cubic premiums
more efficiently price the returns of the F&F set of 2x3 portfolios sorted on size and book-tomarket than the F&F factors do. Nevertheless, the impact of correlations between the F&F risk
factors does not contaminate the ordering of their coefficient from one portfolio to another
when performing the time series analysis as shown from the analysis of a set of 5x5 portfolios
4
3 - The use of alternative databases for the same market does not influence our results.
sorted on size and book-to-market. We also point out that the cubic factors proportionally better
explain and provide less specification error than the F&F premiums do in explaining a set of
11,377 stock returns. In conclusion, we argue that if one has to choose one specification or the
other, all evidence indicates that the cubic construction should be preferred.
The rest of the paper is organised as follows. Through one simple example, the first section
addresses the problems related with the F&F methodology. Section 2 presents the alternative
cubic methodology proposed in the paper. Section 3 carries out the analysis of the properties of
the cubic and the F&F samples of empirical risk factors. Section 4 performs comparative tests
about the specification power of each pair of premiums. Section 5 concludes.
1. Preliminary Evidence: An Acid Test on Factor Exposures
We discuss one intuitive illustration showing concrete pitfalls with the use of the F&F premiums
as they stand.
Table 1 displays the results of an acid test 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
capitalisation 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.
Table 1: Independence test of the F&F empirical risk factors
Table 1 indicates that the exposures to the SMBff factor displayed by the High/High (541.18%)
and High/Mid (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 included,
5
particularly the SMBff factor (Δ HMLff =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 erratic 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 method does not deliver pure estimates of the returns attached to each type
of risk.
2. The Cubic Method
Although it does not suggest any method to improve the F&F shortcomings, the previous example
clearly outlines the need for a method that delivers stable and consistent fundamental risk
premiums. In this section, we propose an alternative approach that leads to the purification of the
risk factors by ensuring the homogeneity of each constructed portfolio on all three fundamental
risk dimensions. We call this the “cubic” method, by analogy to the creation of a cube built with
27 identical cubic components.
2.1. The principles
The cubic approach differs from the F&F methodology on various points. First, we consider a
comprehensive approach that jointly analyses the three empirical dimensions of risks: size, booktomarket 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 is more realistic in capturing the
returns associated with some time-varying dimensions of risk like higher-moment exposures. Third
and lastly, our sequential sort avoids spurious significance 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.
i. 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 momentum.4 Two breakpoints (1/3th and
2/3th 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.
ii. 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:
6
(1)
(2)
4 - We borrow Jegadeesh and Titman’s (2001) and Carhart’s (1997) definition of momentum. 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. Our momentum strategy benefits from the outperformance of winners and from the
underperformance of losers by combining long positions in winners and short positions in losers.
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).
iii. 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.
To be precise, we perform three sorts successively. 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 1 illustrates this procedure.
Figure 1. Sequential three-stage sorting procedure. This figure illustrates the sequential 3-stage sorting procedure. The stocks are first
sorted into 3 portfolios according to one control risk dimension. Within each portfolio. the stocks are sorted into 3 portfolios according
to another control risk dimension. Finally. the stocks within the 9 portfolios are sorted into 3 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.
5 - We allow a variation of up to one standard deviation around the US average BTM.
7
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.6 First,
the size-sorted portfolios are formed by successively performing a 3-stage sequential sorting
procedure on book-to-market, momentum and market capitalisation. Second, the book-tomarket-sorted portfolios are formed by successively performing a 3-stage sequential sorting
procedure on market capitalisation, momentum and book-to-market. Finally, the momentumsorted portfolios are formed by successively performing a 3-stage sequential sorting procedure
on book-to-market, market capitalisation and momentum.
2.2 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 available:7 company annual
total debt,8 the company annual total asset,9 the official monthly closing price adjusted for
subsequent capital actions and the monthly market value. Monthly returns and market values10
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 capitalisation
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 made on a monthly basis. For each month t, every 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 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.
2.3. The acid test revisited
This subsection revisits the preliminary evidence presented in the previous section and contrasts
it with a similar acid test on the exposures obtained with the cubic premiums.
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.
8
6 - 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.
7 - As for the risk space of F&F, temporary data non-availability just excludes the stock from the analysis at that time.
8 - The company total debt at year y (D) concerns all interest bearing and capitalised lease obligations (long and short term debt) at the end of the year. These variables have been collected on Compustat.
9 - 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.
10 - 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-French (1993), negative or zero
book values that result from particular cases of persistently negative earnings are excluded from the analysis.
Table 2: Independence test of the cubic empirical risk factors
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 change reports, for each factor, the average percentage of change (in absolute
value) when one other factor is removed. The total average relative change reports, for each factor, the average percentage of change (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.
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 (last
column) 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 of 14.14%. The exposure to HMLc factor appears as well to remain fairly insensitive
to the inclusion of the SMBc factor in the regression for all portfolios but the Low/Mid one. The
UMDc factor is also 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 6 portfolios.
9
Such results suggest that the SMBc, HMLc, and UMDc factors could be used separately in regressionbased analyses. Their exposures do not substantially vary when the other factors are 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 2 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.
3. Properties of the Cubic and the Original F&F Factors
The previous sections only present preliminary evidence over the stability of the F&F factors
compared to the cubic risk premiums. 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.
Table 3: Descriptive statistics over the empirical risk premiums (May 1980-April 2007)
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.
All the F&F premiums display positive average returns over the period, but only the HMLff and
UMDff premiums are significantly positive 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 that 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. The
importance of the SMBc premium becomes similar to the one of the momentum strategy. They
report approximately the same (significant) positive average return over the period. The UMDc
premium presents characteristics very similar to the corresponding F&F premium.
10
In order to analyse 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.
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 3 sets of 9 portfolios with the SMBc, the HMLc, and the UMDc factors.11
Table 4: Descriptive statistics over the return spread portfolios forming each cubic risk factor
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 capitalisation,
momentum, and finally book-to-market; resp. book-to-market, market capitalisation, 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.
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
11
11 - Note that all correlations are significantly different from 1 at the usual significance levels.
HMLc factor, but display only weak correlation (less than 30%) with other types of risk. The bookto-market risk premium is the highest in portfolios formed of stocks of low (resp. medium) market
capitalisations 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. As
there appears to be no stable BTM effect in the cubic framework, this might indicate that the sort
on BTM captures noisy returns that could not be related to a source of risk priced on the market.
The BTM effect might thus be overvalued in the F&F framework. Panel C shows that the effect
of momentum is decreasing with market capitalisation. The momentum spreads tend to be the
highest in stocks presenting small or medium levels of market capitalisation.
Table 5 repeats the same analysis on the 2 spread portfolios respectively forming the F&F HMLff
and UMDff factors and the 3 spread portfolios forming the F&F SMBff factor.
Table 5: Descriptive statistics over the return spread portfolios forming each F&F risk factor
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-tomarket (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 capitalisation. 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-tomarket 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.
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 which are quite homogeneous regarding the return spreads
related to size, here the low book-to-market-sorted portfolios display an average size spread very
different 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). 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.
Only the two return spread portfolios underlying the UMDff factor appear to be almost only
correlated with the UMDff factor, even though their mean values are again very different. 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.
12
Finally, 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 the size and
book-to-market effects respectively, but appear to be contaminated with correlated sources of
risks. Our analysis even suggests that the book-to-market effect does not capture any kind of
systematic risk priced on the stock market.
The comparison of Table 4 against Table 5 explains much of the differences in the risk properties
between the cubic and the original F&F SMB, HML and UMD factors displayed in Table 3.
Evidence shown in Table 5 reflects that the F&F empirical size factor is contaminated by a bookto-market effect, as 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). The book-to-market effect does not systematically affect average returns across the
9 cubic spread portfolios constructed on size (Table 4, Panel A). Consequently, our specification
gives a premium whose average return is superior to the F&F equivalent. Besides, as already
mentioned, our size factor is formed from 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.
Second, as to the HML factor, the decomposition of the cubic premium into 9 portfolios shows
that the negative sign of the premium comes mostly from the return spread in stocks displaying
big capitalisation and high levels of prior returns. When pricing the HML factor, the cubic
specification associates highly negative return to this risk in a high size/high momentum-portfolio.
This particular portfolio concentrates high performing companies that demonstrate persistence in
performance over several periods. Since the HML factor somehow rewards a contrarian strategy,
it is thus not surprising to find a negative reward for the book-to-market effect in this portfolio.
Besides, by sorting stocks according to the NYSE only, the F&F method tends to over-represent
small stocks by comparison with big stocks as both the size and BTM quantiles are higher in the
F&F framework than it would be based on a median sort. The HMLff factor overstates the returns
attached to the book-to-market effect in the small size portfolio due to its contamination through
the size effect. In turn, by taking a simple average of these two portfolios, the F&F premium tends
to overestimate the return spread attached to the book-to-market effect.
Finally, the momentum constructed according to our cubic specification displays half the level of
volatility compared to the F&F UMDff factor. Substantial variation exists in returns related to the
F&F momentum risk between small and big capitalisations. The returns are more stable across the
9 cubic difference portfolios, showing that the size effect has been eliminated.
Correlation analysis
Table 6 displays the correlation matrix of these two sets of premiums.
Table 6: Correlations matrix of the empirical risk premiums (May 1980-April 2007)
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.
13
The bottom-left corner displays the cross-correlations between the two sets of premiums. The SMBc
and HMLc factors are correlated at 67.16% and 68.25% respectively 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 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
factors.12
4. Specification Tests
The three types of specification tests proposed in this section obey a progressive logic. To begin
with, we perform a basic efficiency test on the asset pricing model to ensure that both the F&F and
the cubic specifications are worth being examined further. The second test is a relevance check on
the individual betas of each specification. We study whether the risk factor exposures of portfolios
that are supposed to borrow certain risk characteristics faithfully reflect these characteristics. The
third test allows us to 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.
4.1. Factor efficiency test
We evaluate the specification error displayed by a cubic or an original 4-factor Carhart analysis
on the set of 2x3 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 French’s
website.
We consider the following multivariate linear regression and test the values taken by the alphas:
(3)
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 αp across all N equations:
(4)
H0 : αp = 0 for p=1,…,N Following Gibbons et al. (1989), under the null hypothesis (H0) that αp is equal to 0 for all N
follows a central F distribution with degrees of
portfolios, the statistics
is a vector of sample means for the L factors
,
is the
freedom N and (T-N-L), where
,
is a vector of the least squares estimates of the
sample variance-covariance matrix for
αp across the N equations.
14
12 - Note that all correlations are statistically different from 1 at the usual significance levels.
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
statistic 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.
We investigate the properties of the alternative methods further below.
4.2 Test on the magnitudes of betas
We perform a cubic and an original 4-factor Carhart analysis and test the value of the loadings
for different portfolios sorted on size and book-to-market. We expect the loadings on the size
premium to be highly significant and high in magnitude for portfolios made of stocks of small
capitalisation. Besides, we expect the exposures to the HML factor to be higher in portfolios with
a high book-to-market level. In order to have more variability in the exposures, we consider the
F&F set of 5x5 portfolios sorted into size and book-to-market.
Table 7 displays the values of the loadings on the market, size, book-to-market and momentum
premiums for the 2 models.
Table 7: Cubic and F&F empirical 4-factor model: Magnitude of the betas for a set of 5x5 portfolios sorted on size and BTM
15
We estimate a cubic and a F&F 4-factor Carhart model on a set of 5x5 F&F portfolios sorted on size and book-to-market. Table 7 reports
the estimates of the beta loadings for the different portfolios. T-tests over the significance of the beta coefficients are reported: *,** ,and ***
stand for significant at 10%, 5%, and 1%, respectively. Average betas and their standard deviations are also reported for the 5 size- and the
5 BTM- portfolios. For instance, the beta of the portfolio of Size 1 (resp. BTM 1) corresponds to the average beta of the 5 book-to-market(resp. size-) portfolios with a size 1 (resp. BTM 1).
The left part of the table displays the results for the cubic premiums. The portfolios display
consistent loading on the size premium: presenting positive and significant exposure in small
portfolios, medium level of exposure in mid portfolios and finally negative loadings for portfolios
made of big stocks. While the premium is statistically significant for almost all portfolios, the
factor seems to matter only for the first two sets of size-sorted portfolios, the exposures being
considerably reduced – as expected – for medium size-sorted portfolios. In the F&F framework
(right half of the table), the size premium, while consistently significant across all portfolios, is
still highly priced in medium size-sorted portfolios of levels 3 or 4. The F&F size premium captures
more than the risks embedded in small size stocks.
For both sets of premiums, the HML premiums behave consistently with the book-to-market level
of the related portfolio. The exposures to the HML premium increase within each size portfolio
with the level of book-to-market. As expected, the book-to-market effect is less important in the
cubic framework than in the F&F analysis.
Consistently with the sort underlying the construction of the 5x5 portfolios, the exposure to
momentum is close to 0 for most of the portfolios.
The table also analyses the stability of the exposures to the SMB, HML and UMD factors within
each of the size-portfolios for different levels of book-to-market and within each book-to-market
portfolios for different levels of market capitalisation. The estimates of the beta loadings on the
HML factor are less volatile in the cubic framework than in the F&F one. The volatility of the SMB
loadings are however comparable in the cubic and the original F&F framework.
4.3. 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 financial macroeconomic
literature.13
16
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.
We consider the following two models:
1. M1 or the F&F model:
2. M2 or the cubic model :
(5)
(6)
where Ri stands for the excess return on asset i, μ 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:
(7)
Under the null hypothesis θi1 = 0, M3 reduces to M1; if θi1 ≠ 0, M1 is rejected. Tests are conducted
on the value of θi1. Davidson and MacKinnon (1981, 1984) prove that under H0, can be replaced
by its OLS estimate from M2 so that θi1 and δi (and α3,i, βi) are estimated jointly. This procedure is
called the “J-test”. We define δi* = (1 - θi1) δi so that M3 can be rewritten as follows:
(8)
To test M2, we reverse the roles of the two models. We construct model M4:
(9)
We replace δi by its estimate along M1 ( ) and estimate γ*i (and α4,i , βi) jointly with θi,2. If θi,2,
M4 reduces to M2; if θi,2 ≠ 0 , M2 is rejected. Tests are conducted on the value of θi,2.
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: H0 : θi,1 = 0 against H1 : θi,1 # 0;
Hypothesis II: H'0 : θi,2 = 0 against H'1 : θi,2 # 0
Each θj follows a normal distribution with mean θj and volatility σj. Therefore, under the null
follows a Student distribution with 315 degrees of freedom – the
hypothesis, the statistics
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 θ estimate).
Among the four possible scenarios, we consider the following two cases:
• (H0, H'1), M1 is not rejected but M2 is;
• (H'0, H1 ), M2 is not rejected but M1 is.14
Table 8 presents the results of the tests over the significance of θ1 and θ2 across the assets, for
different confidence levels. We perform the following tests about the value of θ1 and θ2 :
14 - Note that the rejection of H0 does not reveal anything about the validity of H0’.
(10)
17
Table 8: Tests over the value of θ1 and θ2 in the Nested Models M3 and M4 for individual assets
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 (10). 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.
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 are 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, which does
not imply that any of the models provides a good fit (as it is not the scope of the test), could
be expected from a database of individual stocks. For a limited subset of stocks (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 (statistically) be a wrong choice for almost 4,000
individual stocks.
5. Concluding Remarks
This paper proposes an alternative way to construct the empirical risk factors of Fama and French
(1993). The original F&F method performs a 2x3 sort of US stocks on market capitalisation 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 paper aims to address the drawbacks of this heuristic approach to constructing risk factors,
and to tackle an important gap in literature: how to best construct fundamental risk factors. It
has become standard practice to use the Fama and French method (F&F) to construct multiple
size and BTM portfolios 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 the effects of other sources of
risk as much as possible 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.
18
With regard to these objectives, we raise three main issues in applying the F&F methodology.
First, the annual rebalancing is consumptive in long time-series which sometimes simply do not exist
for small exchange markets. Besides, it would also not easily capture the time-varying dimension of
risk such as higher-moment exposures. Second, the independent sorting procedure underlying the
formation of the 6 F&F two-dimensional portfolios causes moderate, but economically significant
levels of correlation between premiums. Finally, the breakpoints as defined along the NYSE stocks
produce an over-representation of small caps in portfolios.
The main innovations of our premiums reside in a monthly rebalancing of the portfolios underlying
the construction of the risk premiums, 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 F&F method, our factor construction method better captures the return spread
associated with the source of risk to be priced as it is able to maximise the dispersion in the
related source of risk while keeping minimal dispersion in correlated sources of risk. Put another
way, without losing in significance power, in beta consistency or 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 F&F factors whose risk exposures are highly sensitive
to the inclusion of the other F&F risk factors in the regression.
In conclusion, as in Cremers et al. (2008) and in Huij and Verbeek (2009), we argue that the
book-to- market premium of F&F is overvalued. Moreover, we argue that a sequential sorting
procedure could be more appropriate to take the contamination effects between the premiums
into consideration. We show that the premiums constructed in this way deliver more consistent
risk properties while having at least the same specification as the F&F premiums.
As shown is the very last analysis, there is still some risk to be captured in both sets of premiums.
As there is evidence that empirical risk premiums could be proxies for higher-order risks (BaroneAdesi et al., 2004; Chung et al., 2006, Hung, 2007; Nguyen and Puri, 2009), future research must
consider the incremental value of higher-order moment-related factors for benchmark models.
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