Exploring Momentum Premium---the Relation Between
Ranking Horizon and Rebalancing Frequency
Master Thesis ZINING LUO (S2711621)
University of Groningen Faculty of Economics and Business
MSc Finance z.luo.3@student.rug.nl
Abstract: This study tests momentum premium in US stock market with different ranking
and holding horizon. When evaluating the momentum profitability, we argue the necessity of looking at the persistent momentum returns over the holding time. Academics and investors should be cautious when frequently rebalance momentum portfolios. Our results present the relation between ranking and holding period and its impact on momentum premiums. Overall, the result verifies the representativeness of the conventional 6-month/ 6-month momentum strategy.
Key words: Investment Strategies, Momentum, Stock selection,Portfolio management
Table of Contents
1. Introduction ... 1
2. Literature Review ... 3
2.1 Profitability of Momentum Strategies ... 3
2.2 Explanation Theory of Momentum Anomaly... 5
2.3 Predictive Power of Past Performance ... 6
2.4 Holding Horizon and Momentum Reversal ... 7
2.5 Market States and Momentum Crashes ... 9
3. Data and Methodology ... 10
3.1 Data Description ... 10
3.2 Momentum Portfolio Formation ... 10
3.3 Methodology ... 11
3.4 Portfolio Turnover ... 12
4. Empirical Analyses ... 13
4.1 Horizon of Ranking Period ... 13
4.2 Horizon of Holding Period ... 16
4.3 Momentum Persistence and Market State ... 18
4.4 Transaction Costs for Various Momentum Strategies ... 20
4.5 Momentum Strategies and Size Factor ... 22
4.6 Double-Sorted Portfolio ... 23
5. Robustness Test ... 25
6. Conclusion ... 27
1 1. Introduction
In financial world, momentum describes the continuation of an investment’s relative strength. Asness, Frazzini, Israel, and Moskowiz (2014, pp 75) define that “momentum is the phenomenon that securities which have performed well relative to peers (winners) on average continue to outperform, and securities that have performed relatively poor (losers) tend to continue to underperform.”.
This relative strength is first introduced by Levy (1967) that stocks with higher ratio of current price compared to the moving average price of preceding 26 weeks can generate significant higher return over the subsequent 26 weeks than stocks with lower ratios. Later, Jegadeesh and Titman (JT, 1993) introduce the concept of momentum to define this relative strength. Different from Levy’s work, JT simplify the ranking criteria by selecting stocks based on their average returns over a past period and extend the holding strategy by taking both long position on stocks with strong return performance and short position on stocks with bad performance.
According to JT, a momentum strategy (J, K) consists of a ranking period (J), over which winner stocks and loser stocks are selected, and a holding period (K), during which the selected winners are bought long and losers are sold short. In their study, JT test the
profitability of 16 momentum strategies by combining different J (=3, 6, 9, 12) and K (=3, 6, 9, 12). The afterward momentum studies have followed this convention, generally using 6- or 12- month symmetric ranking and holding horizon.
The pervasive consistency of using the similar horizon of ranking and holding period in the current momentum studies draws our attention. To the best of our knowledge, researchers have neglected to mention the reasoning behind the choice of six months or 12 months. If there exists an optimal momentum strategy other than (6,6) or (12,12), studies related to momentum returns may be limited by implementing an inferior strategy. It is therefore necessary to explore the profitability of unconventional momentum strategies.
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Many researches (Jegadeesh, 1990; DeBondt and Thaler, 1985) find out that short-term (less than one month) and long-horizon (three to five years) past returns are negatively related to future price movements, while that intermediate-horizon (two months to two years) past returns are positively related to average future returns. Novy-Marx (2011) compares returns of momentum strategies with recent past performance and with intermediate horizon past performance. By fixing the holding period as 1-month, Novy-Marx (2011) draws the conclusion that stock’s past performance measured from 12 to 7 months prior to portfolio holding date has more predictive power on momentum returns than that of recent past 6 to 1 months. However, when we extend the holding horizon, momentum portfolios formulated in his way not only encounter price reversal sooner than portfolios that are based on recent past performance but also result in extensive trading costs.
The aforementioned finding emphasizes the necessity of looking at the persistency of momentum premiums over the holding period. To consider about this question, we
interpreted it as two sub-questions: a) how much transaction costs investors can expect to save because of less infrequent trading; b) how long the selected winners (losers) can potentially continue their upward (downward) trend. Momentum reversal is the main hurdle when holding momentum portfolio for long time as it is almost impossible to incorporate the periodicity into stock expected return models (Figelman, 2007). This trade-off between the risk of price reversal in momentum premium and the cost of frequent trading is discussed in more detail in the later sections.
Based on the results, we summarize two types of optimal momentum strategies, that is, 1) strategies that built upon evaluating stocks’ performance from intermediate past returns and holding for 3 to 5 months; 2) strategies that rank stock returns during recent past 3 to 7 months and have holding horizon of around 10 months long. However, it should be notice that the optimal holding horizon may change across market conditions, which implies that investors should adjust their rebalancing frequency flexibly.
Moreover, we construct a double-sort momentum strategy that sort stock returns of the recent 5 months as well as the returns over the period from 11 to 6 months prior to the
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This study contributes to the current literature in two dimensions. Firstly, for the academic interest, we show that symmetric ranking and holding horizon is generally not optimal for testing momentum premiums. We also address that momentum studies using a short holding horizon, such as 1 month, should be cautious because they may leave out the capacity of a momentum strategy to continually create superior returns. For the practical interest, we present the relation between cost and benefit of portfolio rebalancing frequency and throw light on enhancing momentum premium by adjusting ranking and holding horizon.
It is necessary to distinguish trend momentum with the relative momentum that we discuss in this paper. Trend following strategy has existed for a long time. It employs that stocks have increased in value over time will continue their upward trend. In contrast, relative momentum is that a stock has outperformed other stocks over time. In practice, traders also describe a stock’s upward (downward) trending as its momentum indicator and some academies call this trending as time-series momentum or absolute momentum (Moskowitz, Ooi & Pedersen, 2011). So it should be noted that we refer the word “momentum” throughout the context as stocks’ relative strength.
The rest of the paper is organized as follows. Section 2 shows relevant literatures. Section 3 briefly describes the data and methodology. Section 4 discusses the empirical results. Section 5 gives a robustness check on the result consistency. Section 6 summarizes the findings and provides suggestions for further researches.
2. Literature Review
This section discusses the relevant momentum studies and explains how different horizon of ranking period and holding period may impact momentum premiums.
2.1 Profitability of Momentum Strategies
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When JT (1993) introduce this investment method, they define a momentum strategy, noted as [ J/ K], to consist of a ranking horizon of J months and a holding horizon of K months. At the beginning of each month, stocks are ranked in ascending order according to their returns over the ranking horizon. Subsequently, they construct the momentum portfolio with long position in stocks that ranked in the bottom decile and short position in stocks in the top decile. Skipping one week as the formulation period, the momentum portfolio enters the holding period and is closed out K months later.
JT test 16 momentum strategies with {J, K} = {3, 6, 9, 12} and chooses [ 6/ 6] to conduct further analysis. The afterward momentum researches follow the convention and generally sort stocks based on their returns over the past six months or the past one year. Table 1 gives a partial overview of the tested momentum strategies shown on previous momentum literatures.
Table 1 Momentum Strategy and the Return on Previous Studies
This table briefly presents the momentum strategies that are tested on previous momentum literatures focusing on US stock market. The first column is a brief reference on that literature. The second column displays the sample period on that research. The third column and the fourth column record the momentum strategy and the average monthly return, respectively. J stands for the horizon of ranking period and K means the horizon of holding period. 1 in the mid of the bracket means there is one month/week gap between the ranking and holding and 0 otherwise.
Literature Data Period Momentum
strategy [J,1/0,K]
Momentum Monthly Return
Gezcy and Samonov (2013) 1801-1926 [10,1,1] 0.28%
1927-2012 [10,1,1] 0.40%
Jegadeesh and Titman (1993) 1965-1989 [6,0,6] 0.95%
Fama and French (1996) 1963-1993 [11,1,1] 1.31%
Moskowitz and Grinblatt (1999) 1963-1964 [6,0,6] 0.73% Griffin, Ji, and Martin (2003) 1926-2000 [6,1,6] 0.59%
Asem (2008) 1927-2005 [5,1,6] 1.21%
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On the other side of the story, there are also studies questioning the arbitrage premium of momentum strategies since the implementation requires frequent trading and generates considerable trading costs. The leading academic papers are Lesmond, Schill, and Zhou (2003) and Korajczyk and Sadka (2004). For example, Lesmond et al (2003) present that the stocks that contribute most of the momentum profitability are generally small cap and less liquid firms associated with high bid-ask spread. They aggregate the daily or intra-daily data to estimate the transaction costs and claim that the result completely destroys the momentum premiums.
However, until today, the cost theory has failed to challenge the status of momentum profitability. Israel, and Moskowitz (2012) suggest that momentum strategies can reduce trading costs considerably in real world and generates strong net returns. Asness et al (2014) also address that the estimated costs of Lesmond et al (2003) are indeed for the average investor but is in practice ten times larger than the costs of an institutional investor.
2.2 Explanation Theory of Momentum Anomaly
To defend for the efficient market hypothesis (EMH) or to understand more about momentum premiums, numerous studies have been conducted to explain the momentum phenomenon.
The first type of scholars tries to challenge the existence of momentum as a market anomaly. They either claim that the empirical finding on momentum is due to data snooping or try to prove that momentum is actually a compensation for risks. Chordia and Shivakumar (2002) propose that momentum profits can be explained by lagged macroeconomic variables and maintain that momentum effect disappears once returns are adjusted for their expected returns based on these variables. Avramov, Chordia, Jostova, and Philipov (2007) show that winner and loser portfolios are comprised mainly of firms with highest credit risk. Sagi and Seasholes (2007) also suggest that high revenue volatility and low cost of goods sold can deliver strong momentum profits.
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Comparably, the behavioural models are more pervasive. This theory mainly explains momentum returns via two dimensions: 1) the market friction regarding to information diffusion; and 2) investor biases towards public information.
Chan (2003) compares stock returns following public news and returns with no identifiable news and finds out that momentum only exists in the former stocks, especially stocks with bad news. Hong and Stein (1997) and Hong (2000) also show that information, especially negative information, diffuses gradually across investors, leading to the post-event abnormal momentum returns.
Besides, even more researches attribute momentum premiums to investor sentiment and perception bias (Daniel, Hirshleifer and Subramanyam, 1998; Barberis, Shleifer, Vishny, 1998; Grinblatt and Han, 2005; Frazzini, 2006). The perception bias generally describes that investors tend to underreact to news in the short run and overreact to price movements in the long run. This theory is informative for understanding the periodicity of momentum
strategies. For holding a momentum portfolio, the underreaction bias in the short term may result in profits in intermediate horizon, while the overreaction in the long run unavoidably lead to negative returns. Therefore, momentum investors should close their positions before encountering return reversals.
2.3 Predictive Power of Past Performance
The publication of momentum theory challenges the EMH, even the weak-form
efficiency that all past prices of a stock shall be incorporated in the stock’s current price. The predictive power of stock past returns has been intensively studied and proved (DeBondt and Thaler, 1985; Jegadeesh, 1990; Jegadeesh et al., 1993; Conrad and Kaul, 1998). Researches show that different horizons of past performance may deliver opposite trading signals. For instance, short-horizon (less than one month) and long-horizon (3 to 5 years) past returns are negatively related to average future returns and intermediate-horizon (3 to 12 months) past returns are positively related to average future returns. The first conclusion gives rise to the contrarian strategy, which buys loser and sells winners, while the second suggests the momentum strategy in our paper, which trades the exact contrary direction.
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They also emphasize the importance of stocks continually outperforming, that is, stocks’ ability to exhibit positive returns in at least eight months of the ranking period.
In a related vein, Bianchi, Drew, and Fan (2015) sort commodity futures based on their past performance over two periods rather than one period. Specifically, they select
commodities that are winners (losers) both during past one year and also on the year before. Result presents that these commodities exhibit even stronger upside (downside) potential than commodities that are purely winners (losers) over the past two years or over the past one year. This finding also supports the role of the consistency in outperformance on discussing the predictive power of stocks’ past returns.
In discussing the predictive power of past one-year price movements, Brush (2001) compares the performance of the most commonly used momentum strategies and convinces the 1-month reversal effect and thus the necessity of skipping one month between ranking and holding period. He also suggests that by adjusting for the month-to-month volatility effect, excluding extreme price changes in the past performance, or omitting returns in January, investors expect to construct a more reliable momentum model than that based on pure price changes.
In another direction, Novy-Marx (2011) questions the conventional momentum strategy about that ranking period ends one month before holding period. He states that momentum is mainly driven by stock’s intermediate past performance, that is, the performance over the 12 months to 7 months prior to portfolio holding. By using multiple tests, Novy-Marx proposes that conventional momentum strategy is suboptimal because intermediate past performance has stronger ability for predicting momentum return than recent past performance. However, as we will examine later, the momentum return measurement over one-month holding horizon is problematic.
2.4 Holding Horizon and Momentum Reversal
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Some explanations are given. JT (1993) conjecture that it is the time varying exposure to common factors in the momentum portfolio that causes the sign reverse. It is also possible that weak companies in the loser portfolio either went to bankruptcy or survived to be strong companies, leading to a loss for momentum investors. The aforementioned overreaction theory also anticipates the long-term reversal effect, or mean reversion, in stock prices.
Whatever the reason may be, the consensus is that momentum effect decays over long-term horizon. Timing the momentum reversal would be both difficult and impractical. To extend the holding horizon, previous literatures provide methods on improving screening criterion to select winner and loser stocks with potential longevity of momentum premiums.
Lee and Swaminathan (2000) find out that past trading volume can help to predict the persistence of momentum returns. They show that all else being equal, winner (loser) stocks with low trading volume in the look-back period can generate higher return and longer return continuation than those with high trading volume. They also propose a conceptual framework called momentum life cycle (MLC) to describe the interaction between price momentum, reversals, and trading volume.
Relating to information diffusion speed, Verardo (2009) shows that the heterogeneity of beliefs about a firm’s fundamental value can benefit return continuation. Consistently, Chen, Chou, Hsieh (2015) state that stocks with greater information asymmetry experience stronger and longer-term momentum premiums.
Chen and Yu (2011) introduce a momentum reversal strategy (MRS) with a new performance measurement that compares a stock’s geometric average rate of return (GARR) over the ranking horizon with the past 12-month GARR of the winner (loser) portfolio. They report that winner/ loser stocks sorted according to this performance measure are less
possible of running into reversals.
Blitz et al (2011) develop the strategy of “residual momentum” that aims to ease the time varying exposure to Fama and French factors. In their strategy, they standardize the residual returns by adjusting the risk exposure and sort winners/losers based on the standardized returns.
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return when holding for 18 months. The result is particularly striking when compared with the pure JT momentum strategy using the same data set, which generates not statistically significant 1% annual return when holding for the same horizon.
In sum, abundant researches have drawn attention on the momentum reversal effect, expecting to explore more return continuation premiums. In our study, we do not claim an optimal holding period as it is always a trade-off between the cost of frequent trading and the risk of running into price reversal. The focuses in this study are to discuss this trade-off and to explore the link between ranking horizon and holding horizon. It is necessary to stress that with more complex signals or ranking criteria, investors are hopeful to reduce the trading costs as well as to bear less risk exposure on momentum reversal. We refer readers to those papers to determine their target holding time.
2.5 Market States and Momentum Crashes
In spite of its attractive profitability, momentum returns appear to be extreme volatile and negatively skewed. Researches show that momentum strategy infrequently incurs grave losses. In the back-testing section of JT (1993), they show that momentum strategy
experience enormous losses in the 1930-1932 period. The close relation between momentum profitability and market states draws many researches attention.
Daniel and Moskowitz (2013) present that two worst time for momentum strategy, July/August of 1932 and March/April of 2009, with monthly return of -69.54% and -42.81%, respectively. Daniel et al. refer this as momentum crash and show that the crash usually occurs after severe market downturns, when the short side of the momentum portfolio, or the past losers rebound promptly compared to winners. Consistently, Cooper, Gutierrez, Hameed (2004) reveal that momentum portfolio is profitable after UP market states while incurs loss after DOWN market states.
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Predicting the upcoming market states is not realistic and well beyond the scope of this paper, but investors are suggested to pay attention to the market states and adjust their investing philosophy across time.
3. Data and Methodology
3.1 Data Description
Data for the one-month T-bill and the three factors are obtained from Kenneth’s French website. Stock sample file consists of all firms listed on the NYSE and AMEX extracted from the Centre for Research in Security Prices (CRSP) monthly stock file during the period from July 1993 to December 2015. This period has seen several market booms and regressions that are interesting to study for. Meanwhile, the study also eliminates American Depository Receipts (ADRs), real estate investment trusts (REITs), close-end funds, primes and foreign companies, that is, only stocks with share code of 10 or 11 in CRSP are selected. We exclude all stocks with share prices less than $1 as of the portfolio formation date just to alleviate the impact from small and illiquid stocks.
There are 2372 stocks at the first month and 1137 stocks by the end of the sample file. Concerning the survivor bias, we corporate the delisting return information into the stock return profile in case that the selected stocks are delisted during the holding period.
3.2 Momentum Portfolio Formation
The portfolio formation method follows the conventional momentum strategy designed by JT (1993). At the beginning of each month t, all eligible stocks are ranked in a descending order according to their J-month lagged cumulative returns, that is, the cumulative return from the first day of month (t-J) to the last day of month (t-1). The stocks in the top decile and bottom decile are assigned to winner portfolio and loser portfolio, respectively. In order to be selected, eligible stocks must have complete consecutive J-month lagged returns. All the portfolios are formulated in an equal-weighted convention. The total momentum portfolio that buys winners and sells losers is then hold for the next K month starting from the first day of month (t+1). In this way, we skip one month between the ranking period and the holding period, mitigating the impact of price pressure and bid-ask bounce1. Additionally, to make
profitability of different strategies comparable at any time point, the first portfolio is
1 Chordia and Shivakumar (2002) propose the importance of skipping one month for an implementable strategy
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formulated at the beginning of July 1994 regardless of the choice of J. The last portfolio is constructed at (K-1) month prior to December 2015.
Similar with JT (1993) and other momentum studies, this paper also constructs
overlapping portfolios. To be specific, the momentum return for any month is the average of returns from the momentum strategies implemented in the current month as well as in the previous K-1 months. For example, for K=3, the monthly return for month t includes the returns of momentum portfolios constructed at month t, (t-1) and (t-2). Therefore, in each month, we revise 1/K of the stocks in the entire portfolio and carry over the rest to the next month.
Lastly, Antonacci (2014)2 introduces a new concept of dual momentum, which
addresses that assets to be selected into the long side momentum portfolio should not only have positive relative momentum but also positive absolute momentum. We take this into account and rank stocks according to their past excess returns compared to the return on one-month T-bill. Stocks that are ranked on the top decile but underperform than the risk free rate will not be selected. In that case, we go long for the 1-month T-bill instead.
3.3 Methodology
Our objective in this study is to compare the impact on momentum profitability of implementing strategies with different combination of J and K. We base our analyses on the sample data and apply various momentum strategies by changing the length of J or K. We range both J and K from one to 12 in this study as it is mentioned by many academies that they are representative for formulating profitable momentum strategy (Moskowitz el al, 1999; Fama et al, 1996). In addition, Novy-Marx3 (2011) proposes that momentum strategies based
on recent past performance (i.e. from month t-5 to t-1, where t is the formulation month) are less profitable than those based on intermediate horizon past performnce, say performance from t-11 to t-6. We re-examine this issue in this paper by adding the intermediate past performance defined by Novy-Marx into testing procedure.
2 This is mainly for the consideration that the large price deduction during 2008 financial crisis. However, our
results are robust based on pure return.
3 Novy-Marx (2010) use a different time horizon expression. In his context, he defines recent past as 6) to
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For either analysis of ranking horizon or analysis of holding horizon, we keep the other part to be constant. For example, we firstly restrict the holding period to be one month and examine the impact on returns of momentum strategies with different ranking horizon.
3.4 Portfolio Turnover
Ignoring trading costs are suitable to show the relation between risk and returns while not implementable in practice. As noted in Section 2.1, several studies question the
profitability of momentum strategies since the execution requires too frequent trading and faces large transaction costs. Korajczyk et al (2004), Lesmond et al (2003) claim that tradings in momentum strategies are dominated by small, off-NYSE stocks and propose several ways to estimate the actual transaction costs. However, their conclusions are challenged by
Frazzini et al (2012) and Asness et al (2014).
There is no conclusion regarding to the debate. In this study, we recognize that gross returns do overstate the profitability of momentum strategies and that frequent trading is the major concern of consuming the returns. Take one-month and six-month holding horizon for example. The former holding strategy certainly requires more frequent rebalancing. However, investors may not need to change all the stocks in the portfolio because some stocks could consistently be the winner/ loser at the time when portfolio is rebalanced. In that case, one-month holding strategy has more potential to see these consistent winners/ losers and thus lower the level of trading activities.
Therefore, in this study we address this issue by using monthly portfolio turnover as the proxy for transaction costs. It is worthy to distinguish portfolio turnover in our context from the turnover describing stocks’ trading volume. Portfolio turnover measures the percentage of how many new stocks are bought in terms of the total stocks in the portfolio. Based on
historical data, we calculate the average portfolio turnover (PT) for each momentum strategy and then scale it into a monthly basis.
Monthly portfolio turnover = PT × 1 𝐾⁄ , (1)
where K is the length of holding horizon and PT is obtained from historical data.
The estimated transaction cost, C, is calculated as
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where cost per switch measures the commission fees and multiple it by two to calculate the roundtrip trading costs. We use 1% as estimate for cost per switch here, which is also used by Balvers et al (2005). It is worthy to mention that the 1% estimation is rather conservative compared to the cost in real life. The monthly portfolio turnover is doubled to reflect the trading activities in both winner portfolio and loser portfolio. As suggested by the above equations, a large portfolio turnover ratio and a high trading frequency imply expensive trading costs.
This study does not specifically address bid-ask bounce issue for two reasons. Firstly, stocks in our sample file are relatively liquid and we also skip one month between the ranking period and holding period, both relaxing the price pressure. Additionally, we overestimate the cost per switch to compensate the lack of bid-ask spread. Secondly, as suggested by Frazzini et al (2012), for large institutions in practice, transaction costs of implementing momentum strategies can be minimized substantially, making it unreliable to measure the costs using the theoretical models.
4. Empirical Analyses
In this section, we discuss about the empirical results relating to the impacts on momentum profitability of choosing different duration of ranking period (J) and holding period (K).
4.1 Horizon of Ranking Period
Table 2 summarizes the overall performance for momentum portfolios using various J strategies. The momentum portfolios in this section are constructed using monthly
rebalancing frequency, that is, K is set to be 1. The returns, volatility and Sharpe ratios for the winner portfolio, loser portfolio and the total momentum portfolio (winner minus loser) are shown on the table separately. The t-statistics shown in parentheses test the significance level of momentum premiums.
Several finding can be seen from table 2. Firstly, different from the finding of Hong, Lim, and Stein (2000) that a momentum strategy’s profitability is driven by losers’
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Table 2 Portfolio Performance for Different J-Month Momentum Strategies
This table presents average performance for winner portfolio, loser portfolio, and the total momentum portfolios based on J-month past returns from July 1994 through November 2015 At the beginning of each month t, all eligible NYSE/AMEX listed stocks are ranked in descending order based on their
previous J months’ returns. J ranges from 1 to 12 in addition to an intermediate past ranking period (J=11-6) that measures return from month t-11 until month t-6. The top 10 decile stocks and the bottom 10 decile stocks are equally weighted assigned to Winner portfolio with long position and Loser portfolio with short position, respectively. The return on the momentum portfolio is equal to the return on Winner portfolio minus the return on Loser portfolio earned on the subsequent month t+1. All results are reported in percentage. The numbers shown in parentheses are t-statistics. *, ** and *** indicate the underlying result is significant a level of 10%, 5% or 1%, respectively.
J=1 J=2 J=3 J=4 J=5 J=6 J=7 J=8 J=9 J=10 J=11 J=12 J=11-6
Panel A: Average Returns for Portfolios
Winners 0.92** 1.20*** 1.28*** 1.32*** 1.41*** 1.53*** 1.50*** 1.56*** 1.45*** 1.49*** 1.44*** 1.30*** 1.23*** (2.48) (3.30) (3.49) (3.65) (3.89) (4.16) (4.07) (4.27) (3.91) (4.03) (3.80) (3.44) (3.27) Losers 0.75 0.75 0.69 0.65 0.58 0.63 0.46 0.46 0.40 0.40 0.32 0.29 0.33 (1.52) (1.48) (1.38) (1.31) (1.13) (1.23) (0.88) (0.86) (0.75) (0.73) (0.61) (0.56) (0.80) Momentum 0.17 0.45 0.60* 0.67* 0.83** 0.90** 1.04** 1.10** 1.05** 1.09** 1.12** 1.01** 0.89*** (0.60) (1.39) (1.74) (1.89) (2.10) (2.21) (2.57) (2.61) (2.46) (2.54) (2.72) (2.53) (3.25) Panel B: Standard Deviation of Portfolio Returns
Winners 5.93 5.81 5.88 5.79 5.81 5.88 5.89 5.83 5.93 5.92 6.05 6.06 5.99
Losers 7.83 8.09 7.99 7.92 8.27 8.28 8.28 8.56 8.44 8.66 8.37 8.31 6.65
Momentum 4.64 5.19 5.48 5.68 6.33 6.48 6.50 6.73 6.86 6.90 6.57 6.39 4.39
Panel C: Sharpe Ratios for Portfolios
Winners 11.9 17.0 18.2 19.1 20.7 22.4 21.9 23.0 20.9 21.6 20.2 18.0 16.9
Losers 6.8 6.6 6.0 5.5 4.5 5.1 3.0 2.9 2.2 2.1 1.3 0.9 1.8
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out that the long side of the Kenneth French’s UMD portfolio contributes to most of the excess returns during the period from 1926 through 2013. Additionally, the major historical losses in momentum portfolio are mainly due to the extreme positive returns from its short side (Daniel et al, 2013). If it is not for the purpose of constructing self-finance momentum portfolio, recent decades have seen little benefit of shorting loser portfolios.
Secondly, consistent with the conclusion of Moskowitz et al (1999) that short-term historical returns (less than 3 months) have poor reliability for predicting return continuation. Momentum portfolios with ranking horizon equal to one or two month(s) are not able to earn significant profits and have much lower Sharpe ratios than other portfolios. Besides, Table 2 also shows that for winner portfolio and the total momentum portfolio, expected returns seem to increase monotonically when extending the horizon of ranking period. And for loser portfolio, the effect is opposite. To put it more concretely, the result suggests that with holding horizon of one month, formulating momentum portfolio based on relative long horizon of ranking period, i.e. past six to twelve months, is averagely more profitable. This finding is presented graphically in Fig. 1.
The second conclusion is consistent with Novy-Marx (2011), who presents that intermediate past performance, that is measured from 11 to 6 months prior, shows more predictive power than return data measured on recent past months. Meanwhile, another important argument in his paper is that stocks that were intermediate horizon winners (losers) but are losers (winners) based on recent data show more upward (downward) price trending compares to recent winners (losers). For convenience, we name the former type of stocks as inconsistent outperformers. As can be seen in Fig. 1, momentum portfolios using these stocks are still inferior compared to momentum portfolios holding outperformers for a longer
horizon.
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Figure 1 Cumulative Momentum Return (K=1).
This figure illustrates the cumulative returns (in percentage) for momentum portfolios formulated with different length of evaluation period. J represents the J months prior to formulation month t. J=11-6 means that the returns are measured from month t-11 until month t-6. All portfolios are held on month t+1 and rebalanced monthly.
4.2 Horizon of Holding Period
Cooper et al. (2004, page1347) summarize that ‘when there is momentum, there is ultimately long-run reversal’. As reversal effect has long been verified, holding a momentum portfolio for long-horizon is certainly not recommended. We now extend the holding horizon to 20 months and look into the profitability of momentum portfolios after its formulation. The result is shown on Fig. 2. For brevity, we only report the return for momentum portfolio with ranking horizon of J={4, 7, 10, 11-6}.
The Fig. 2 a) and b) are more close to the momentum returns in a general market state. They show that, for strategies with 4- and 7- month evaluation duration, the cumulative momentum returns averagely peak around six months to ten months after its formulation, while for portfolios with longer-term outperformance, they are expected to experience reversal in a shorter run. To put it another way, despite the fact that intermediate
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outperformance is able to predict most of the momentum effect in the first month, it fails to tell the persistency of the momentum returns. The advantage of using recent past returns to formulate momentum portfolios also outstands as shown in Fig. 2 d). Momentum portfolio that sorts stocks based on their 4-month lagged returns manages to deliver positive returns even when other momentum portfolios fail.
a) 1994-2014 b) 1994-2000
c) 2001-2008 d) 2009-2014
Figure 2 Cumulative Momentum Premiums
This figure illustrates the cumulative returns for 4 momentum portfolios with NYSE/AMEX stocks. J represents the J months prior to formulation month t. J=11-6 means that the returns are measured from month t-11 until month t-6. The x-axis presents the time period after month t. The y-axis presents the returns in percentage.
This section addresses the necessity of looking at longer holding horizon. For short holding horizon, intermediate past performance indeed predicts momentum returns well. However, the reliability is a term effect. On the contrary, ranking stocks based on short-term past performance delivers persistent and stable returns over the long holding period.
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4.3 Momentum Persistence and Market State
The Fig.2 also presents the relation between holding horizon and market state. As illustrated in Section 2.5, the probability of momentum portfolio running into reversal is closely related to the market state. In consistency with Cooper et al (2004), Fig. 2 c) displays the good time of momentum portfolios during expanding and post-expanding market from 2001 to 2008. The reversal is nearly invisible at this time since the market is running irrational exuberance before October, 2007. Once the financial crisis breaks out, almost all investors promptly stand in the short side of the market, trying to sell the overpriced equities. Although both winner stocks and loser stocks take a hit during this stage, the latter candidate jump deeper than the former, leaving investors decent momentum premiums. This is
comparable to the post-crisis market state, as presented in Fig. 2 d), where there is a short rebound in the depressed market. The rebound effect is strongest on March to May 2009, when the past losers show exceptional performance after they were deep underestimated by the market. In this stage, the mean-reversion effect stands out and the momentum premium reaches to its historical trough.
The above finding gives further support to the behaviour theory in explaining
momentum. To see this more clearly, we conduct time-series regressions based on the Fama-French (1993) 3-factor model to obtain the risk-adjusted returns for momentum strategies. To be specific, we run the following time-series regression using the returns on K month after holding the momentum portfolio:
(𝑟𝑖 − 𝑟𝑓) = 𝛼𝑖 + 𝑏𝑖 ∙ 𝑅𝑀𝑅𝐹 + 𝑠𝑖∙ 𝑆𝑀𝐵 + ℎ𝑖 ∙ 𝐻𝑀𝐿 + 𝜀𝑖, (3)
where ri is the return on the Kth month after holding the momentum portfolio; i ranges from 1 to 20; rf is the risk-free rate; RMRF, SMB, and HML are Fama-French 3 factors; bi, si, hi are the corresponding sensitivity coefficients; and αi is the alpha or risk-adjusted return of that month.
We report the alphas in Fig. 3. The result is consistent with previous conjecture.
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Comparably, when the market is on post-crisis state, the momentum profits not only decrease in magnitude but also face the price reversal closely at hand.
a) 1994-2014 b) 1994-2000
c) 2001-2008 d) 2009-2014
Figure 3 Risk-adjusted Momentum Premiums
This figure illustrates the 4 momentum portfolios returns adjusted with Fama-French (1993) three-factor model. J represents the J months prior to formulation month t. J=11-6 means that the returns are measured from month t-11 until month t-6. The x-axis presents the time period after month t. The y-axis presents the adjusted returns in percentage. J represents the duration of ranking period.
To sum up, the optimal holding horizon is closely related the ranking strategy and also the market state. When we restrict to hold a momentum portfolio for a short time, i.e. one month, the stock intermediate past performance indeed predicts more momentum returns on that period. However, this momentum persistence significantly underperforms than those ranking strategies with less duration. The question then becomes that investors should chase for a short-term high momentum premium or a persistent momentum performance. We look into this problem from the perspective of transaction costs in the next subsection.
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Moreover, the impact of market state on the momentum persistence should also be addressed. Timing the market is nonetheless a tricky question, but some general rules are worthy to study for. Momentum portfolios can be profitable in a bad market state when the state is post-exuberance but can run into crash in a good market state when the state is
rebound from a depression. Especially, strategy based on past 4-month stock returns is able to be averagely profitable on the post-depression market. If holding momentum portfolio cannot be avoided in its bad time, ranking stocks based on their recent performance can give rise to more flexible momentum premium.
4.4 Transaction Costs for Various Momentum Strategies
To assess the profitability of momentum strategies, we further investigate the portfolio turnover and transaction costs issue.
The choice of ranking period and holding period has direct influence on momentum portfolio turnover. For holding period, this relation is obvious. Holding for one month means that the momentum portfolio has to rebalance every month, while holding for 12 month means that trading frequency is once a year. However, the turnover rates for these two strategies are possible to be less than one and 1/12 due to the roll-over of consistent winner/loser stocks.
The results are reported in Table 3. As expected, for each K strategy, the corresponding portfolio turnover is less than 1/K and when extending the holding horizon, the monthly portfolio turnover jumps dramatically, especially from one-month holding to five-month holding.
Strikingly however, when extending the ranking horizon, investors are also expected to buy and sell less. One possible reason is that some stocks have remarkable performance in a short-term and may trigger an accelerating overreaction among investors. For these stocks, they are less possible to consistently stay in the momentum portfolio. Thus, recent past performance, for J= {1:3}, gives more volatile trading signal than longer-horizon of ranking period and leads to higher turnover. Nonetheless the effect eliminates when portfolio
rebalancing is less frequent.
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less than 8 basis points. This delivers informative intuition for investors that instead of running the risk of price reversal, rebalancing the portfolio semi-annually can enjoy less risk and comparable transaction costs.
Table 3 Portfolio Turnover and Transaction Costs for Various Momentum Strategies
This table presents the monthly portfolio turnover (Panel A) and the corresponding estimated transaction costs (Panel B) for 156 momentum portfolios with NYSE/AMEX stocks from July 1994 through November 2015. At the beginning of each month, all eligible stocks are ranked in descending order based on their previous J-month returns. J=11-6 means that returns are measured from month t-11 until month t-6. The momentum portfolio is then held from month t+1 till t+K with a long position in the top decile stocks and a short position in the bottom decile stocks.
Panel A: Portfolio Turnover for Various Strategies
K=1 K=2 K=3 K=4 K=5 K=6 K=7 K=8 K=9 K=10 K=11 K=12 J=1 84.0% 41.6% 27.6% 20.8% 16.8% 13.9% 12.0% 10.6% 9.4% 8.5% 7.8% 7.1% J=2 60.5% 41.6% 27.4% 20.7% 16.6% 13.8% 11.9% 10.4% 9.3% 8.5% 7.7% 7.0% J=3 49.4% 34.4% 27.5% 20.5% 16.4% 13.7% 11.8% 10.4% 9.3% 8.4% 7.6% 7.0% J=4 43.4% 30.2% 24.2% 20.5% 16.4% 13.7% 11.8% 10.3% 9.2% 8.3% 7.6% 7.0% J=5 39.0% 27.2% 21.8% 18.6% 16.4% 13.6% 11.7% 10.3% 9.2% 8.3% 7.6% 7.0% J=6 35.5% 24.9% 20.0% 17.1% 15.1% 13.6% 11.6% 10.2% 9.1% 8.3% 7.5% 7.0% J=7 33.1% 23.2% 18.7% 16.0% 14.1% 12.7% 11.6% 10.2% 9.1% 8.2% 7.5% 7.0% J=8 30.8% 21.7% 17.5% 15.0% 13.2% 11.9% 11.0% 10.2% 9.1% 8.2% 7.5% 6.9% J=9 29.0% 20.5% 16.6% 14.2% 12.6% 11.4% 10.4% 9.7% 9.1% 8.2% 7.5% 6.9% J=10 27.7% 19.5% 15.8% 13.6% 12.0% 10.9% 10.0% 9.3% 8.7% 8.2% 7.5% 6.9% J=11 26.3% 18.6% 15.0% 13.0% 11.6% 10.5% 9.7% 9.0% 8.4% 7.9% 7.5% 6.9% J=12 25.1% 17.8% 14.5% 12.5% 11.1% 10.1% 9.3% 8.7% 8.1% 7.7% 7.2% 6.9% 11-6 35.6% 24.9% 20.0% 17.1% 15.1% 13.6% 11.7% 10.2% 9.2% 8.3% 7.6% 7.0%
Panel B: Estimated Transaction Costs for Momentum Strategies
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4.5 Momentum Strategies and Size Factor
Either risk-based or behaviour-based explanation predicts that momentum is more profitable among small size stocks than big stocks. In this section, we compare the momentum premium for big size stocks and small size stocks. By achieving this, in the beginning of month t, we split the active stocks into two subgroups. Stocks with market capitalization greater (smaller) than the median market cap of the stock universe are assigned to the group “Large” (“Small”). We then implement momentum strategies within the two group separately. The results are shown on Table 4. For brevity, we only practice the test with nine types of momentum strategies.
However, the result does not favour the outstanding momentum effect among small cap stocks. Although we find 0.16% and 0.18% return for small cap stocks, this return is far below the associated transaction costs when the rebalancing frequency is once per month. Based on the same sample file, Table 4 suggests that various momentum strategies are not able to generate sizable momentum premium in either of the size group, especially when we compare this result with table 2. Theories that momentum are stronger in small stocks may have points to certain degree, but recent two decades have seen the necessity of building momentum portfolio based on the total stock universe.
It is important to summarize the main findings at this point. To formulate profitable momentum portfolio, the choice of the duration of ranking period and holding period are closely related to each other and the profitability is also affected by market state as well as the resulting trading costs. Our empirical findings show that when the holding period is short, i.e. 1 or 2 months, momentum portfolios formulated based on an intermediate time, say 10 to 12 months, have better performance. When holding period is relatively long, i.e. 5 to 10 months, recent past performance (3 months to 7 months) generates more persistent
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performance can contribute to a more flexible momentum strategy to mitigate large loss due to the loser rebound.
Table 4 Momentum Premiums for Large Cap and Small Cap Stocks
This table presents the average percentage monthly profits for momentum portfolio with
NYSE/SMEX small cap and large cap stocks from 1994 to 2015. In the beginning of each month t, stocks that are larger (smaller) than the median market cap of the all stocks are assigned to “Large” (“SMALL”) group. [J, K] momentum strategy is then implemented within either group of stocks. J represents the horizon of ranking period, where the past performance is measured. Based on the performance, momentum portfolios are formulated with long position in the top decile stocks and short position in the bottom decile stocks. The momentum portfolio is then held for K month starting from month t+1. “W” stands for winner portfolio and “L” stands for loser portfolio. The momentum return is equal to the return on Winner portfolio minus the return on Loser portfolio. The t-statistics are shown in parentheses to show whether the underlying result is significantly difference from zero or not. Bond font represents that the underlying return is significantly different from zero at least 0.1 level. SMALL LARGE W L W-L W L W-L [4,1] 0.38 0.21 0.16 0.33 0.48 -0.14 (3.27) (1.89) (2.09) (2.66) (3.95) (-1.56) [7,1] 0.41 0.23 0.18 0.34 0.37 -0.03 (3.53) (1.95) (2.33) (2.91) (2.96) (-0.38) [11-6,1] 0.46 0.33 0.13 0.43 0.30 0.13 (3.78) (2.86) (1.49) (3.45) (2.38) (1.63) [4,7] 0.34 0.39 -0.06 0.35 0.39 -0.05 (7.16) (8.18) (-1.82) (6.53) (7.55) (-1.40) [7,7] 0.32 0.37 -0.06 0.36 0.38 -0.03 (6.54) (7.65) (-1.81) (7.13) (7.40) (-1.10) [11-6,7] 0.35 0.37 -0.03 0.35 0.35 -0.01 (6.83) (7.40) (-0.93) (6.63) (7.33) (-0.39) [4,12] 0.34 0.39 -0.06 0.36 0.38 -0.03 (8.69) (10.25) (-2.11) (8.56) (9.29) (-1.09) [7,12] 0.32 0.38 -0.07 0.36 0.38 -0.02 (8.23) (10.19) (-2.76) (9.14) (9.25) (-0.89) [11-6,12] 0.35 0.38 -0.04 0.35 0.37 -0.03 (8.51) (9.25) (-1.50) (8.05) (9.40) (-1.20) 4.6 Double-Sorted Portfolio
Based on above, we observe that although the intermediate past performance
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portfolio to improve the persistency in momentum premiums. To be concrete, we will select stocks that are winners/losers based on both recent past performance and intermediate past performance. If strategies based on recent past performance are able to explore more return continuation in the momentum portfolios, stocks that are able to stay outperforming on two periods shall be able to produce more persistent momentum premium than those “old” winners/losers, which are the target stocks in Novy-Marx (2011). This consistency is also proposed by Grinblatt et al (2003), who disclose that the consistency in stock past
performance has as important predictive power as the magnitude of past performance. In the beginning of each month t, we assign the stocks that presents in the top (bottom) decile based on the performance measure over 11 months to 6 prior to month t and the performance measured over the last 5 months. We test the double-sort strategy with the other four
momentum strategies and set the rebalancing frequency as annually and semi-annually. The results are presented in Table 5.
The result suggests that when the portfolio is held for 12 months, the double-sort momentum strategy, that based on both recent past performance and intermediate past performance, is able to produce return of 5.73% annually, which is more profitable than strategies that just based on intermediate past performance, 4.75% and 0.61% for J=10 and J=11-6, respectively. Additionally, when rebalancing momentum portfolios once half a year, the double-sort strategy is even more profitable than strategies that selected just based on recent performance, generating 6.65% in half year.
Moreover, the Fig. 4 graphically shows the trending of the cumulative returns for the five strategies along the holding time. The double-sort strategy not only captures the
predictive power of the intermediate past performance for the momentum premium in a short time but also improve its ability to select stocks with more return continuation.
Table 5 Comparison of Momentum Strategies Performance
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statistics are shown in parentheses to show whether the underlying result is significantly difference from zero or not.
K=12 J=4 J=7 Double-Sort J=10 J=11-6 Mean 6.34% 5.57% 5.73% 4.75% 0.61% (5.49) (4.74) (2.21) (3.96) (0.73) Standard error 18.07% 18.37% 40.58% 18.78% 13.05% Sharpe ratio 0.34 0.29 0.14 0.24 0.03 K=6 J=4 J=7 Double-Sort J=10 J=11-6 Mean 4.41% 5.52% 6.65% 5.31% 2.01% (5.26) (6.14) (3.80) (5.66) (3.01) Standard error 13.27% 14.24% 27.70% 14.85% 10.58% Sharpe ratio 0.31 0.37 0.23 0.34 0.17 a) K=12 b) K=6
Figure 4 Cumulative momentum profits for various momentum strategies.
This figure illustrates the cumulative returns for 5 momentum portfolios with NYSE/AMEX stocks with holding horizon of one year and six months, respectively. J represents the J months prior to formulation month t. J=11-6 means that the returns are measured from month t-11 until month t-6. The double-sorted strategy selects stocks that are winners/losers both in recent past five months and in the period from month t-11 to month t-6. The x-axis presents the time period after month t. The y-axis presents the returns in percentage.
5. Robustness Test
In the previous context, we see the ex-post performance of different momentum portfolios formulated with different evaluation horizon. We summarize two types of
strategies for constructing profitable momentum portfolios, that is, A) intermediate portfolio
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holding (K= {6:10}) based on recent past performance (J= {3:6}); B) short holding (K= {1:5}) based on intermediate past performance (J= {7:12}). In this section, we conduct an out-of-sample test to verify the robustness of our conjecture. Specifically, we replicate the test procedure by using the same source of data file but during the period from July 1970 to December 1992. In this way, if we can observe the same pattern with before, we can have our conclusions verified. The results can be seen on Fig. 5 and Table 6.
Fig. 5 compares momentum strategies profitability for different rebalancing frequency. Fig. 5 a) suggests that the intermediate past performance, measured from 11 to 6 months before the portfolio formation date, has strongest predictive power for momentum premiums among other period, which is in line with the conclusion in Novy-Marx (2011). However, Fig. 5 b) implies that the persistence of this momentum premium is also weakest. Momentum strategies formulated with this way of intermediate past performance encounter price reversal twice faster than other strategies.
Figure 5 Robustness test
The figure displays the average returns for four momentum portfolios with NYSE/AMEX stocks from 1970 to 1992. Portfolios are formulated based on different horizon of past performance. J represents the length of the ranking period. “11-6” means the performance is measured from 11 to 6 months prior to portfolio formation month. a) shows the cumulative returns of momentum portfolios across time. b) shows the average cumulative return on that month.
Table 6 further gives numerical illustration on the profitability of these momentum strategies. Panel A implies that the return on momentum strategies based on intermediate past performance decreases, from 1.42% to 0.40%, as we extending the holding horizon, while the portfolio returns based on recent past performance increases, from 0.68% to 0.92% when
-50.00 0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.00 400.00 J=4 J=7 J=10 J=11-6 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 J=4 J=7 J=10 J=11-6
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there is less rebalancing. All momentum portfolios encounter price reversal within one year, making rebalancing once a year a suboptimal strategy considering that the cost reduction is less than 20 basis points. In sum, the robustness test convinces our previous findings.
Table 6 Robustness Test for Target Momentum Strategies
This table presents the average monthly profits for momentum portfolios based on J-month past returns from 1970 through 1992. At the beginning of each month, all eligible NYSE/AMEX listed stocks are ranked in descending order based on their previous J months’ returns. “11-6” means the performance is measured from 11 to 6 months prior to portfolio formation. The top 10 decile stocks and the bottom 10 decile stocks are equally weighted assigned to Winner portfolio with long position and Loser portfolio with short position, respectively. The formulated momentum portfolio is then held for K month starting from month t+1. The numbers shown in parentheses are t-statistics. *, ** and *** indicate the underlying result is significant a level of 10%, 5% or 1%, respectively.
Panel A Portfolio Return Panel B Transaction Costs
J=4 J=7 J=10 J=11-6 J=4 J=7 J=10 J=11-6 K=1 0.42% 0.68%** 0.94%*** 1.42%*** 1.80% 1.38% 1.16% 1.49% (1.53) (2.11) (2.94) (6.65) K=3 0.60%*** 0.76%*** 1.18%*** 1.13%*** 0.99% 0.77% 0.65% 0.82% (3.61) (4.00) (6.55) (8.57) K=5 0.62%*** 0.90%*** 1.06%*** 0.78%*** 0.67% 0.58% 0.49% 0.62% (4.97) (6.59) (8.24) (8.22) K=7 0.66%*** 0.92%*** 0.87%*** 0.40%*** 0.48% 0.48% 0.41% 0.48% (6.62) (8.18) (8.60) (5.32) K=10 0.75%*** 0.73%*** 0.63%*** 0.09% 0.34% 0.34% 0.33% 0.34% (9.15) (8.48) (7.93) (1.39) K=12 0.58%*** 0.53%*** 0.45%*** 0.00% 0.28% 0.28% 0.28% 0.28% (8.18) (7.28) (6.83) -(0.03)
6. Conclusion
Since JT (1993) publish the strategy of constructing momentum portfolios, momentum studies almost follow their tradition. But no one address the difference between the
profitability of various momentum strategies. This thesis examines the relation between momentum returns and the choice of ranking horizon and holding horizon.
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intermediate past performance are more profitable, whereas when the holding period is longer, i.e. six months or longer, portfolios formulated based on stocks recent past
performance can produce more persistent and greater momentum premiums. We also conduct an out-of-sample test with the same data file during the period from 1970 to 1992. The result is robust across time.
For investors, it is important to know that either type of strategy has its own cost and benefit. When the holding horizon is relatively short, investors have less potential to
encounter price reversal compared to long holding horizon, while on the other hand, they are facing large trading costs. Long holding horizon has the contrary effect. Therefore, it is of investors own interest to determine the rebalancing frequency and the corresponding ranking horizon. Nonetheless, we address that in the second part of this thesis that with more complex selecting signal and technical analysis, investors are able to downsize the reversal probability.
For academics, this thesis provides justice for the most commonly used momentum strategy among academic studies. Our result shows that the [6,6] strategy is durable due to its neutral position to gain momentum returns but that studies use one-month holding horizon should be cautious on drawing conclusions. Because results present that monthly rebalancing not only gives rise to large trading costs but also omits the portfolios’ ability to continuingly gain momentum premiums. In this vein, we re-examine the theory proposed by Novy-Marx (2011). We agree with Novy-Marx on the predictive power of intermediate past performance. However, we raise our point that the abnormal return in a one-month holding horizon cannot fully stand for the momentum premium. The persistence in the return continuation over longer holding horizon is of same research meaning.
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