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

• Author: Kathrin Tauscher,Martin Wallmeier
• DOI: http://doi.org/10.1111/eufm.12064
• Date: 01 June 2016
• Published date: 01 June 2016

Portfolio Overlapping Bias in Tests of

the Fama–French 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 find 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 Fama–French 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 finding 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 finding of Fama and French (1993) (hereinafter FF93) that mimicking factors for

returns related to size and B/M explain a significant 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, 367–393

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 finer 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 confirmed for other markets, including Asia Pacific, 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 difficult 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)-classification of size and B/M for German stocks in contrast to the (5 5)-

classification 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 significantly influence 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 finer

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), Griffin (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