Portfolio Overlapping Bias in Tests of the Fama–French Three‐Factor Model

DOIhttp://doi.org/10.1111/eufm.12064
Date01 June 2016
Published date01 June 2016
Portfolio Overlapping Bias in Tests of
the FamaFrench Three-Factor Model
Kathrin Tauscher and Martin Wallmeier
Department of Finance and Accounting, University of Fribourg/Switzerland, Bd. de P
erolles 90,
CH-1700, Fribourg, Switzerland
E-mails: kathrin.tauscher@gmail.com; martin.wallmeier@unifr.ch
Abstract
In the standard approach of the three-factor model of Fama and French (1993),
both the test portfolios and the SMB and HML factor portfolios are formed on the
basis of size and the book-to-market ratio. Thus, a potential overlapping bias in
time-series regressions arises. Based on a resampling method and a split-sample
approach, we provide an in-depth analysis of the effect of overlapping for a broad
sample of European stocks. We nd that the overlapping bias is non-negligible,
contrary to what seems to be the general opinion.
Keywords: Asset pricing, three-factor model, portfolio overlapping, size effect,
value premium
JEL classification: G12, G14
1 Introduction
The FamaFrench three-factor model has long been established as one of the most
widely accepted asset pricing models.
1
It is based on two foundations: (i) the nding of
Fama and French (1992) (hereinafter FF92) that the two variables size and book-to-
market ratio (B/M) explain the cross-sectional variation in average stock returns, and (ii)
the nding of Fama and French (1993) (hereinafter FF93) that mimicking factors for
returns related to size and B/M explain a signicant part of the variation of returns over
time. The mimicking factors introduced by Fama and French are known as Small Minus
Big (SMB) and High Minus Low (HML). Using SMB,HML and a market proxy as
We would like to thank the Editor (John Doukas) and two anonymous referees for their
helpful suggestions. We are also grateful to Paulo Maio, Steffen Mahringer (discussants),
and participants at the 2013 FMA European Conference (Luxembourg), the SGF 2013
Conference (Z
urich), and the AFFI 2013 Conference (Lyon) for their comments. We also
thank Thomson Reuters Switzerland for providing data.
1
See, e.g., Bauer et al. (2010), p. 171: Our test assets are 25 portfolios formed on size and
B/M, which have become standard in asset pricing tests after the failure of CAPM.
European Financial Management, Vol. 22, No. 3, 2016, 367393
doi: 10.1111/eufm.12064
©2015 John Wiley & Sons, Ltd.
explanatory variables, FF93 run time-series regressions for 25 portfolios sorted on size
and B/M. The intercepts are all close to zero, which indicates that the three factors seem
to do a good job explaining the cross-section of average stock returns.
2
A peculiar characteristic of the time-series regression setup in FF93 is that the sorting
variables are the same for both the dependent and independent variables. FF93 remark:
In the time-series regressions for stocks, the dependent returns and the two
explanatory returns SMB and HML are portfolios formed on size and book-to-market
equity. Many readers worry that the apparent explanatory power of SMB and HML is
spurious, induced by the regression setup.
3
However, the authors argue:
We think this is unlikely, given that the dependent returns are based on much ner size
and BE/ME sorts (25 portfolios) than the SMB and HML returns.
4
In fact, an independent test of FF93 supports this view. The idea of the test is to use two
disjoint groups of stocks to measure the independent and dependent variables separately,
thus excluding any overlap. Further support for the three-factor model comes from
results based on different sorting variables for the test portfolios (see FF93, p. 47 ff; Fama
and French, 1996).
Largely, the results of FF92 for the US with size and B/M as the determinants of
expected returns have been conrmed for other markets, including Asia Pacic, Japan
and European countries.
5
There is less international evidence, however, on the validity
of the risk-based interpretation of FF93. In many countries, the time-series regressions
are more difcult to replicate because a substantially smaller number of stocks are
available. For this reason, Ziegler et al. (2007) and Schrimpf et al. (2007) for example,
use a (4 4)-classication of size and B/M for German stocks in contrast to the (5 5)-
classication used by FF93. Thus, the portfolios are closer to the (2 3) building blocks
of SMB and HML, leading to a higher similarity between the independent and dependent
variables. The independent test of FF93 is also not available if the number of stocks is too
small to split the stocks into two disjoint groups. Thus, the impact of portfolio overlaps
might be more important in these markets than in the US market.
We are not aware of any direct evidence on the impact of portfolio overlaps in
applications of the three-factor model. Some authors note that results remain largely the
same when the independent FF93 test with disjoint groups is applied.
6
They tend to
explain any remaining differences based on the smaller number of stocks in the test
portfolios after the sample is split into two groups.
7
Overall, it seems to be generally
accepted that overlapping does not signicantly inuence the empirical results in typical
tests of the three-factor model. We aim to provide an in-depth analysis of the impact of
2
Fama and French (1993), p. 5.
3
Fama and French (1993), p. 46 f. Similarly in Fama and French (1996), p. 76: It may not be
surprising, however, that portfolios like SMB and HML that are formed on size and BE/ME
can explain the returns on other portfolios formed on size and BE/ME (albeit with a ner
grid).
4
Fama and French (1993), p. 47. The authors denote book-to-market equity by BE/ME
instead of B/M in this paper.
5
See, e.g., Fama and French (1998), Grifn (2002) and Fama and French (2012).
6
See Fama and French (1993), p. 46 f., and Guidi and Davies (2000), p. 10 f.
7
See Guidi and Davies (2000), p. 11.
©2015 John Wiley & Sons, Ltd.
368 Kathrin Tauscher and Martin Wallmeier

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