Median momentum

DOIhttp://doi.org/10.1111/eufm.12204
Date01 September 2019
AuthorPin‐Huang Chou,Tsung‐Yu Chen
Published date01 September 2019
DOI: 10.1111/eufm.12204
ORIGINAL ARTICLE
Median momentum
Tsung-Yu Chen1Pin-Huang Chou2
1Research Center of Finance, Shanghai
Business School, Shanghai, China
Email: tychen67@gmail.com
2Department of Finance, National
Central University, Taoyuan City,
Taiwan
Email: choup@cc.ncu.edu.tw
Abstract
The median is a better measure of a sample’s central ten-
dency in the presence of extreme observations. We propose
an alternative momentum strategy formed by buying (short-
ing) stocks with high (low) average median returns over a
formation period of 3--12 months. The median momentum
strategy outperforms the traditional price momentum strat-
egy for all holding periods from 1 month to 5 years, with no
long-term reversal. This same return pattern is observed for
all G7 countries. Further analysis indicates that median mo-
mentum profitability is an underreaction-only phenomenon
and shows behavioral patterns related to short-sale restric-
tions and investor sentiment.
KEYWORDS
median, momentum, regret, cognitive dissonance, short-sale restrictions,
investor sentiment
JEL CLASSIFICATIONS
G12, G14
1INTRODUCTION
Jegadeesh and Titman (1993) have discovered that winning (losing) stocks over the past six months
to one year tend to continue to be winners (losers) over the next six months to one year, giving rise
to the well-known momentum anomaly, which is followed by another anomalous pattern of long-term
reversals initially documented by De Bondt and Thaler (1985, 1987).1After 25 years, the momentum
anomaly remains one of the most robust and persistent anomalies in finance (e.g. Schwert, 2003; Fama
We thank John Doukas (the editor) and an anonymous referee for very helpful comments and suggestions that greatly improved
the paper. We also thank Chuan-Yang Hwang, Kuan-Cheng Ko, Kuo-Chiang John Wei, Jin-Huei Yeh, Hsin-Yi Yu, Guofu Zhou,
and seminar participants at the 24th Annual Conference on Pacific Basin Finance, Economics, Accounting, and Management
(Hsinchu, Taiwan) for helpful comments. Chou acknowledges financial support from the Ministry of Science and Technology,
Taiwan (grant no. 104-2410-H-008-010-MY3). All errors are our own.
1The coexistence of short-term (or intermediate-term) momentum and long-term reversals in stock returns has been one of the
most robust puzzles in financial research that cannot be fully explained by neoclassical risk-based theories. Famous models,
such as those of Daniel, Hirshleifer, and Subrahmanyam (1998), Barberis, Shleifer, and Vishny (1998), and Hong and Stein
Eur Financ Manag. 2018;1--39 wileyonlinelibrary.com/journal/eufm © 2018 John Wiley & Sons, Ltd. 1
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2CHEN AND CHOU
& French, 2015). While there is a large volume of literature on momentum,2the performance measure
for identifying winners and losers remains the same: average or cumulative returns over the formation
period. A stylized fact about stock returns, at both the individual and portfolio levels, is that they
are skewed and leptokurtic (e.g. Albuquerque, 2012). Average or cumulative past returns thus could
be problematic in measuring the true performance of stock returns, because they are sensitive to the
presence of extreme observations, or outliers.
We argue that the presence of extreme observations overstates the performance of some winners
and losers, thus contributing to subsequent return reversals. In contrast, the median could be a better
alternative, because it is less affected by extreme observations.3We use the median as the performance
measure to define winners and losers and construct the ‘‘median’’ momentum in the same way as Je-
gadeesh and Titman (1993). Unlike traditional momentum strategies that rank stocks based on monthly
returns over the formation period, our strategies are formed based on the monthly median, defined as
the median of daily returns over a month. Specifically, for each stock, we first calculate the monthly
median return based on the daily returns over that month. The monthly medians are then averaged to
provide a median measure of performance over a formation period of, say, 𝐽months, where 𝐽ranges
from 3 to 12. Based on this median performance measure, we sort all stocks into decile portfolios and
buy those stocks in the top decile portfolio (referred to as the median winners) and sell those in the
bottom decile (referred to as the median losers). The median momentum is the return on the long--short
portfolio, which is held for 3--12 months. We expect the median momentum to be more persistent in
future performance than the traditional price momentum.
Using Center for Research in Security Prices (CRSP) data from July 1963 to December 2015, we
find strong supportive evidence that the median momentum strategy earns significant profits and out-
performs the original price momentum strategy over a one-year holding period, regardless of the choice
of formation months. The results are robust when the returns are adjusted for risk using either Fama and
French’s (2015) five-factor model or the skewness-augmented capital asset pricing model (CAPM) of
Kraus and Litzenberger (1976). Median momentum profits are stronger outside of January and are even
more persistent when we control for confounding effects such as the size effect and bid--ask bounce.
Specifically, when both strategies are considered simultaneously in a Fama--MacBeth regression that
controls for other confounding factors, the median momentum yields an average monthly net return
of 0.548%, whereas the net return is 0.083% for the price momentum for a one-year holding period.
Outside of January, the average monthly profits for the median momentum and the price momentum
are 0.758% and 0.339%, respectively.
Another striking finding associated with the median momentum is that it does not exhibit the unde-
sirable long-term reversal pattern of the price momentum, suggesting that it is an underreaction-only
phenomenon.4To see the economic rationale behind our results, we consider early theoretical expla-
nations for the reversal of the price momentum. Lakonishok, Shleifer, and Vishny (1994) and Hong
and Stein (1999), among others, suggest that investors may overreact to good or bad news and gener-
ate large outliers in the returns, leading to a reversal in returns later on. Based on the sorting of past
(1999), all seek to explain the coexistence of short-term momentum and long-term reversals documented in US stock markets.
George and Hwang (2004, 2007) and Conrad and Yavuz (2017), however, show that they are separate phenomena.
2The latest number of Google citations of Jegadeesh and Titman (1993) in 2018 exceeds 10,000.
3A major advantage of the median over the mean is that it is not ‘‘skewed’’ so much when the sample contains observations
with extremely large or small values. In descriptive statistics, the median is often used to compare against the mean to evaluate
the asymmetry of the sample distribution.
4Our empirical results show that median momentum profitability lasts for up to 5 years. However, because long-term profitabil-
ity beyond a one-year holding period can be explained by Fama and French’s five-factor model, our analysis focuses only on
the short-term holding period of 1 year.
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CHEN AND CHOU 3
average returns, the price momentum is likely to pick many ‘‘overreaction’’ stocks, which causes its
subsequent reversals (e.g. Chen, Chou, & Hsieh, 2017). Recent evidence also indicates that the price
momentum suffers severe losses, because loser stocks experience severe rebounds following extreme
market downturns (e.g. Barroso & Santa-Clara, 2015; Daniel & Moskowitz, 2016; Han, Zhou, & Zhu,
2017).
The median, by contrast, is much less affected by outliers and, hence, overreaction stocks play a
lesser role in the median momentum. Indeed, our empirical results indicate that median-sorted decile
portfolios exhibit no clear patterns in terms of return skewness, whereas portfolios sorted by past re-
turns exhibit an upward pattern in skewness. Perhaps most striking is that median-sorted portfolios
show a hump-shaped pattern in kurtosis, whereas portfolios sorted by past returns have an opposite
pattern, a U-shape. The results confirm our conjecture that the median as a performance measure is
less affected by extreme observations. In fact, our empirical evidence indicates that, during periods of
momentum crashes, the median momentum experiences better performance than the price momentum.
During the price momentum’s 10 worst months, an average loss of −26.361% is incurred. By contrast,
the average loss for the median momentum’s 10 worst months is −21.435%.
Based on a sample for the other G7 countries for the period 1970--2015, we find strong supportive
out-of-sample evidence for median momentum, suggesting that median momentum profitability is
prevalent and not the result of data mining. For all six countries, including Japan, whose markets
are well known to exhibit no price momentum (e.g. Chou, Wei, & Chung, 2007; Chui, Titman, &
Wei, 2010), the median momentum earns significant short-term profits and experiences no long-term
reversals.
What drives the persistent profitability of the median momentum beyond common risk factors?
While investors can overreact to news in some firms, they can underreact in others. It seems that the
latter explains the continuation of the median momentum profitability, which is consistent with Anto-
niou, Doukas, and Subrahmanyam (2013) and Stambaugh, Yu, and Yuan (2012). Based on insights
from Antoniou et al. (2013), we argue that noise traders are reluctant to sell (buy) losers (winners)
during periods of optimism to avoid regret and cognitive dissonance and, hence, the performance per-
sistence of losers (winners) is stronger following optimism (pessimism) than following pessimism
(optimism). Persistence is stronger for losers than for winners, because rational arbitrageurs face short-
sale restrictions when they aim to eliminate the mispricing embedded in losers. Altogether, this argu-
ment implies that the median momentum is stronger following periods of optimism and the pattern is
stronger among stocks subject to tighter short-sale restrictions. Based on the sentiment index of Baker
and Wurgler (2006) and the consumer confidence index and using institutional ownership and short
interest as proxies for short-sale restrictions, we find strong evidence supporting these implications.
The remainder of this paper proceeds as follows. Section 2 describes the data and presents prelim-
inary results concerning the performance of median-sorted portfolios. Section 3 presents the perfor-
mance of the median momentum based on a Fama--MacBeth style regression, and Section 4 provides
further tests from both rational and behavioral perspectives. Section 5 reports the results of robust-
ness checks. Section 6 presents out-of-sample evidence based on the other G7 countries. Section 7
concludes.
2DATA AND METHODOLOGY
Our sample consists of the ordinary common equities of all firms (with share codes 10 and 11) listed
on the New York Stock Exchange (NYSE), American Stock Exchange (AMEX), and Nasdaq for the
sample period from July 1963 to December 2015. We obtain market data, including daily returns,
monthly returns, share prices, and market equities, from the CRSP database. The accounting data are
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