The post-cost profitability of a momentum strategy and the size of
winner and loser portfolios;
An analysis of the Dutch stock market
Master Thesis MSc Finance, By: H.C.J.M. Castelijns, Thesis supervisor: W.P. Forbes Abstract
This thesis studies the profitability of a momentum strategy on the Dutch stock market for different portfolio sizes. For all different sizes the strategy yields a loss after accounting for transaction costs. When excluding low-priced stocks and small market capitalization stocks the transaction costs are reduced for all portfolio sizes but do not realize statistically significant profits. This thesis shows that transaction costs are larger for strategies with smaller sized portfolios, suggesting that prior research overstates the profitability of a momentum strategy. Opposed to prior research covering different stock markets, this thesis finds that for the Dutch stock market the majority of the momentum returns are generated by the winner portfolio instead of the short position in the loser portfolio. Furthermore, this study illustrates the tradeoff between the higher returns of small portfolio sizes and the benefits of diversification reaped by larger portfolios.
JEL classification: G10, G11, G14
Keywords: Momentum strategy, portfolio size, trading cost, portfolio turnover, bid-ask spread, Sharpe ratio, Sortino ratio, market efficiency
Author: H.C.J.M. Castelijns Mail: hcastelijns@gmail.com Phone: +31623800380
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1 Introduction
In their search for riches people attempt to forecast the stock market. The industry of technical analysis employs historical data on stock prices in an attempt to recognize patterns on which investment strategies can be based. Though different strategies as a result of technical analysis are widespread amongst practitioners, academic researchers debate the feasibility of these strategies. The efficient market hypothesis in its weak form states that it is impossible to have a strategy based upon technical analysis of past stock prices that grants an investor a profit after considering transaction costs.
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Up until recently the academic literature has focused on strategies employing a very large number of different stocks, neglecting the impact of the choice for various amounts of different stocks comprising the momentum portfolio. Apart from transaction costs another less attractive aspect to the seemingly simple strategy is that monitoring large stock universes and investing in many different stocks demands relatively large amounts of attention and capital. Whereas investors’ attention is limited (Barber and Odean, 2008) as well as investors’ capital, this reduces attractiveness of a momentum strategy. Secondly, varying the number of stocks in the momentum portfolio impacts the size of momentum returns, as illustrated by Siganos (2007). In an attempt to bring insight to the size of momentum returns for various portfolio sizes Siganos (2007) finds that for the UK stock market a portfolio of 40 different stocks maximizes momentum returns. With momentum returns sharply increasing as additional stocks are added to the momentum portfolio for portfolio sizes under 40 and momentum returns gradually decreasing as one adds additional stocks beyond 40. Siganos (2007) states that since a momentum strategy is prone to large transaction costs it is in an investor’s best interest to employ the strategy in such a way that returns are maximized. However, maximizing return is not always the sole goal of an investor, as investors are also interested in the tradeoff between return and risk.
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Hofstede index1 (2001). This is of importance as Chui, Titman and Wei (2010) suggest a country’s individualism impacts the magnitude of the momentum profits observed in that countries domestic market. Thirdly, by showing the tradeoff between the higher returns of small portfolio sizes and the benefits of diversification reaped by larger portfolios, by showing the rate of return over standard deviation and the rate of return over downside deviation. Fourth, by extending the research of Siganos (2007) by not only varying the amount of stocks employed in a momentum strategy but also the length of the ranking and holding period. Showing that the result that momentum returns diminish as the number of stocks in the momentum portfolio becomes larger is robust for different applications of the momentum strategy. Besides adding to the existing academic literature this thesis is of practical interest to investors. First of all, this thesis provides evidence that a momentum strategy on the Dutch stock market is unprofitable and suggests that prior research for the UK stock market overestimates the profitability of a momentum strategy. Second, this thesis shows different ways to applicate the momentum strategy as it comes to the holding and evaluation periods, the preferences in time span for these periods can differ amongst investors depending on their goals. Third, this thesis discusses ratio between the size of momentum returns and the fluctuations in these returns for investors with different risk preferences.
The rest of this paper is structured in the following way; the next section discusses the present literature on the momentum strategy. Section 3 describes the data and methodology applied by this research. The fourth section presents and discusses the empirical results. Section 5 concludes with the most important findings, discusses the limitations of this research and brings suggestions for further research.
2 Literature Review
This section explains how the momentum effect seems to conflict with the efficient market hypothesis and illustrates the findings of previous literature on the topic of a momentum strategy.
5 Efficient market hypothesis vs. Technical analysis
In his paper on the efficiency of modern day stock markets Fama (1970) characterizes three forms of market efficiency. First of all, the weak form stating that a stock’s future price movements are independent of the past price path. Secondly, the semi-strong form that states that all publicly available information is incorporated in a stock’s price. Third, the strong form of market efficiency, for which both public and private information is incorporated into stock prices. The hypotheses are formulated in a hierarchical way such that if the strong form holds, the semi-strong form also holds and if the semi-strong form holds, the weak form will hold too. Under all three forms of Fama’s efficient market hypothesis, future stock prices are independent of a stock’s past price path. This leads to the expression that stock prices follow a random walk, displaying no “memory”. A trading strategy based on past stock prices would therefore not be possible. However, practitioners of technical analysis contradict this belief as many professionals attempt to analyze stocks based on past performance and trade stocks and advice clients based upon this analysis. In a research investigating the behavior of US mutual funds between the mid-seventies and the mid-eighties, Grinblatt and Titman (1989, 1991) state that 70% of the mutual funds showed a tendency to buy recent winners. In addition, in a survey conducted amongst chief foreign exchange dealers in London, Taylor and Allen (1992) found that more than 90 percent of the executives opposed the efficient market hypothesis and viewed technical analysis of past prices as a tool to help predict the future.
Momentum effect
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stocks that are winners (losers) over the past three to twelve months keep performing (underperforming) over the subsequent three to twelve months. Within a time window of twelve months the stocks seem to have a certain momentum that causes them to keep moving in the same direction, similar to objects that are in motion and only come to rest when an external force stops their movement. Jegadeesh and Titman (1993) construct a zero sum investment portfolio, comprised out of a long position in the top performing decile, financed by a short position in the decile of worst performing stocks. The stocks are ranked based upon their performance over the previous three to twelve (J) months and held for the subsequent three to twelve (K) months, with one-month skip between J and K to filter out the effects of bid-ask spread price pressure and lagged reaction effects as are found by Jegadeesh (1990) and Lehmann (1990). As the research of Jegadeesh and Titman (1993) points out, this strategy leads to a significant positive return of 1.10% per month. In a replicating study for European markets Rouwenhorst (1998) observes the same effects as Jegadeesh and Titman (1993).
In contrast to a momentum effect in a period between three to twelve months De Bondt and Thaler (1985) find that stocks that have been losing money over the previous three to five years tend to outperform past winners over the subsequent period of three to five years. Thereby showing that just as trees, stock prices do not grow to the sky. Not only in the long run the contrarian effect is observed, but also in the very short run Jegadeesh (1990) and Lehmann (1990) both find that in a period of within a month reversals in stock prices are observed.
Economic significance of momentum returns
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rendered unprofitable, in contrast to Korajezyk and Sadka (2004) who state that a momentum strategy remains profitable even when one accounts for larger trading costs. Agyei-Ampomah (2007) states that transaction costs destroy the profitability of a momentum strategy for strategies with a holding period shorter than one year.
Explanations to the momentum anomaly
As post cost momentum returns contradict the usual belief that stock markets are efficient, many researchers try to explain the presence of momentum returns. To date research has attempted numerous possible explanations to the momentum anomaly. Conrad and Kaul (1998) suggest cross-sectional dispersion in expected returns as a possible source of momentum returns. Claiming that a stock’s return contains a component related to the stock’s expected return and that stocks with a high return in one period also have a high expected return for the next period. Nonetheless, Jegadeesh and Titman (2001) argue that the by Conrad and Kaul (1998) proposed effect is not driving the positive momentum returns observed in their research. Another explanation that might lead to the expectation that momentum returns are caused by a higher risk is the possible existence of serial correlation in factor returns. But, research rejects the explanations that pose higher risk as a cause of momentum returns. Jegadeesh and Titman (1993) corrected for risk using a CAPM benchmark, Fama and French (1996), Grundy and Martin (2001) and Jegadeesh and Titman (2001) correct for risk using the Fama and French three-factor model.
Behavioral explanations
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due to naïve extrapolation of high returns caused by the representative heuristic Barberis et al. (1998). Research in favor of under reaction by investors is conducted by Grinblatt & Hann (2005) arguing that under reaction is caused by investors’ unwillingness to capitalize on their losses, also known as part of the disposition effect. Mental accounting and prospect theory cause investors to hold onto their losing stocks, creating a spread between fundamental value and market price, this under reaction advances an explanation for negative momentum in losing stocks. George & Hwang (2004) claim that investors underreact to news on earnings since they anchor prices based on the highest price level attained in the past year, they thus wait to retrieve what their stock holding is “worth” if its price falls. Daniel, Hirshleifer and Subrahmanyam (1998) claim that short term momentum is driven by self-attribution bias; investors address more value to news coming out that confirms the beliefs they had when they traded, than news that opposes their beliefs. This strengthens the confidence of investors on their point of view and thereby leads to short term autocorrelation in returns.
Portfolio size
The groundbreaking research of Jegadeesh and Titman (1993) has attracted a vast amount of research investigating the momentum effect and the puzzle it forms as it contradicts the efficient market hypothesis. However, little focus is on the impact of the number of stocks comprising the winner and loser portfolio on the magnitude of momentum returns. Whereas most studies seem to arbitrarily choose a fraction of the number of stocks available for the number of stocks comprising the momentum portfolio. Such as the top 10% past performing stocks and the worst performing 10% by Jegadeesh and Titman (1993, 2001), Rouwenhorst (1998), Lui et. al (1999) or the top and bottom 30% as researched by Hong et. al (2000) and Doukas and McKnight (2005). In addition, Doukas and McKnight (2005) also investigate a momentum strategy selecting the top and bottom 20%, for the smaller portfolio size they observe larger momentum returns; respectively 8.76% per annum and 10.68% per annum.
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other research does not explicitly search for the optimal portfolio size, published results do imply that the momentum effect is stronger in smaller size portfolios. Furthermore, Siganos (2007) argues that the momentum returns observed by him remain profitable even after an assumed level of trading costs is factored into the calculation of trading profit, hereby neglecting the impact of portfolio size on the size of transaction costs.
Diversification
Do not put all your eggs in one basket, is a very common known phrase and one that is also applicable to investing. As Noble prize winner Markowitz (1952) shows in his modern portfolio theory, adding stocks to a portfolio which returns are negatively correlated to those of the stocks which are already in the portfolio decreases idiosyncratic risk in the portfolio. As rational investors attain value to expected return and dislike risk, the tradeoff between risk and return is of importance to these investors.
10 Added value
This research attempts to add to the existing literature by investigating the profitability of a momentum strategy for various portfolio sizes after transaction costs by estimating transaction costs explicitly. To my best knowledge there is no previous research that does this as previous literature investigating transaction costs of a momentum strategy employs large fractions of the total stock universe instead of various numbers of portfolio sizes. Furthermore, this study not only focuses on the optimal number of stocks to be held in the winner and loser portfolio in order to maximize returns but also to maximize the ratio of return over units of risk. Thirdly, this research investigates the impact of portfolio size on momentum returns for various different holding and formation periods as previous research only investigated the most common J6K6 strategy. Investigating strategies with different holding and formation periods is of interest since the size of returns can differ and since some investors might prefer to liquidate portfolios more often.
3 Data and Methodology
The purpose of this section is to introduce the data and to elaborate the research methods and equations used to measure the momentum returns and transaction costs.
Data
This research investigates a momentum strategy using monthly data on all stocks traded on the Dutch stock market between March 2000 and March 2016. The data are extracted from the Datastream database and include surviving and non-surviving stocks. During the sample period in total 420 different stocks are traded, on 73 stocks there is no information available, the maximum number of stocks available at one point in time is 245 and the minimum 102, on average 150 different stocks are available in any given month.
Methodology
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period J has ended, “skipping” one month, to filter out a the effects of bid-ask price pressure and lagged reaction effects as are found in Jegadeesh (1990) and Lehmann (1990). The returns over the evaluation period J are calculated using the total return index extracted from Datastream as shown in equation 1, the returns over holding period K are calculated in a similar way as denoted by equation 2. Datastream’s total return index displays a stock’s total value assuming that all dividends are reinvested to purchase additional stocks.
𝑅𝑖,𝑡 = 𝑅𝐼𝑖,𝑡𝑅𝐼−𝑅𝐼𝑖,(𝑡−𝐽)
𝑖,(𝑡−𝐽) (1)
𝑅𝑖,(𝑡+1) =𝑅𝐼𝑖,(𝑡+1+𝐾)−𝑅𝐼𝑖,(𝑡+1)
𝑅𝐼𝑖,(𝑡+1) (2)
Where Ri,t denotes the return of stock i over evaluation period J, Ri,(t+1) is the return of stock i over holding period K, which as previously stated starts one month after the evaluation period which ends at time t. RIi,t is the total return index of stock i at time t, RIi,(t-J) the return index of stock i at the start of the evaluation period J, RIi,(t+1+K) is the total return index of stock i at the end of the holding period K and RIi,(t+1) is the total return index of stock i at the beginning of evaluation period K.
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Where Rp is the return of respectively the winner or loser portfolio, which is equal to the average return of the stocks included in the portfolio. The annual pre-cost profit of the WML portfolio is calculated as the annualized pre-cost return as shown in equation 4. Where RWML is the pre-cost return on the long position in the winner portfolio Rw and the short position in the loser portfolio RL annualized by multiplying with twelve over K, the length of the holding period.
𝑅𝑝(𝐽, 𝐾, 𝑁) = 1
𝑁 . ∑𝑁𝑖=1𝑅𝑖,(𝑡+1) (3)
𝑅𝑊𝑀𝐿(𝐽, 𝐾, 𝑁) = (𝑅𝑊− 𝑅𝐿) .12
𝐾 (4)
To test the statistical significance of the observed momentum returns t-values in this paper are calculated using Newey and West (1987) heteroskedastic and autocorrelation consistent (HAC) standard errors with the truncation lag set equal to the length of the holding period, as proposed by Agyei-Ampomah (2007), since the constructed portfolios are overlapping in time period and therefore suspected of autocorrelation.
Performance measures
As investors are not solely interested in the expected return on their investments but also in the risk associated with this returns, this paper addresses the optimum number of stocks to be held in a momentum strategy not only as the number of stocks that maximize momentum returns but also as the number of stocks that maximize returns relative to risk. To measure this thesis makes use of two risk adjusted performance measures; the Sharpe ratio and the Sortino ratio. The Sharpe ratio is one of the most widely used performance measures in modern literature. The measure assumes that investors care about return and risk. The Sharpe ratio captures how much return a portfolio earns in excess of the risk free rate for each unit of risk.
𝑆 = 𝑅𝑝− 𝑅𝑓
𝜎𝑝 (5)
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French European three factor model, extracted from the data library on the website of Kenneth R. French2.
In a way the Sharpe ratio can be viewed as if the investor has to pay, or take certain amounts of risk, in order to earn a higher return. For higher Sharpe ratios the amount of risk an investor is paying for an amount of return is less, which is preferred by investors.
Sortino and van der Meer (1991) propose to distinguish risk from volatility by measuring the returns below a minimal acceptable rate of return (MAR) as true risk, and all the upside deviations as a positive aspect of volatility. Here investors are not concerned if a stock exceeds in mean performance over the period measured. Equation (6) shows how downside deviation is defined with δ as downside deviation and Rmar as minimal acceptable rate of return.
𝛿 = √1
𝑇(𝑅𝑡− 𝑅𝑚𝑎𝑟)2 ∀ 𝑅𝑡 < 𝑅𝑚𝑎𝑟 (6)
As an estimate of the minimal acceptable return this paper will use the risk free rate and to illustrate investors that are only worried about large substantial losses -3%, -5% and -10% per annum.
Trading costs
As stated previously a momentum strategy asks for frequent trading since the portfolio needs to be rebalanced after each holding period K. Naturally this frequent trading comes at a cost. Therefore, this study also examines the impact of trading costs on the optimal portfolio size for a momentum strategy. Previous literature uses different methods to estimate trading costs associated with a momentum strategy. Jegadeesh and Titman (1993) assume trading costs of 0.5% per annum, based on average trading costs for the US. But the trading cost to a momentum strategy tend to be larger than the by Jegadeesh and Titman (1993) assumed costs, due to fact that momentum portfolios are often biased towards stocks with larger trading costs and require frequent rebalancing (Lesmond et al., 2004; Agyei-Ampomah, 2007; Korajzyk and Sadka, 2004). Moreover, since constructing portfolios consisting of only the few most extreme stocks is more likely to be prone to these effects, this research estimates trading costs by observing the bid and
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ask price spreads of the associated stocks more directly by following the quoted spread estimate approach of Lesmond et. al (2004). This method measures the round trip cost of a transaction as the average spread of a stock – calculated as a stock’s average spread over a period of 12 months starting 18 months before the actual trade - plus two commissions. Equation 7 shows this estimation equation for the quoted spread.
𝑄𝑢𝑜𝑡𝑒𝑑 𝑆𝑝𝑟𝑒𝑎𝑑 (𝑖, 𝑡) = 1 12 ∑ 2 ((𝐴𝑠𝑘(𝑖, 𝑡 + 𝑇) − 𝐵𝑖𝑑(𝑖, 𝑡 + 𝑇)) (𝐴𝑠𝑘(𝑖, 𝑡 + 𝑇) + 𝐵𝑖𝑑(𝑖, 𝑡 + 𝑇)) 𝑡=−6 𝑡=−12 (7)
Secondly, to estimate the trading costs of a momentum strategy one has to account for the cost per trade and the frequency of trading. The frequency of trading is calculated by the turnover rate; the number of stocks in the portfolio at month t that were not in the portfolio at month t-K divided by the total number of stocks in the portfolio, as shown in equation 8.
𝑇𝑢𝑟𝑛𝑜𝑣𝑒𝑟 𝑟𝑎𝑡𝑒𝑡 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑒𝑤 𝑠𝑡𝑜𝑐𝑘𝑠𝑡
𝑛 (8)
Thus to estimate the total trading costs the turnover rate is multiplied with the round trip cost as shown in equation 9.
𝑡𝑟𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 = 𝑡𝑢𝑟𝑛𝑜𝑣𝑒𝑟 𝑟𝑎𝑡𝑒 ∗ (𝑞𝑢𝑜𝑡𝑒𝑑 𝑠𝑝𝑟𝑒𝑎𝑑 + 2 𝑐𝑜𝑚𝑚𝑖𝑠𝑠𝑖𝑜𝑛𝑠) (9)
As a commission rate this paper uses an estimate of 0,18% as reported by the Dutch bank ABN Amro3.
In line with Agyei-Ampomah (2007) this study investigates the trading costs when employing a momentum strategy ranking all available stocks and a restricted sample. In the restricted sample the penny stocks and 30% smallest market capitalization are left out, as these stocks tend to be less liquid and have larger transaction costs, Agyei-Ampomah (2007) reports lower transaction costs for the restricted sample. Liu et al. (1999), Hong et al. (2000) and Doukas and McKnight (2005) find that market capitalization affects the profitability of momentum strategies and as size proxies for risk according to Fama and French (1993).
3 Source:
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4 Results
This section discusses the empirical results. Section 4.1 discusses the results for the entire sample. Section 4.2 limits the sample by excluding small cap stocks and penny stocks since these increase trading costs in line with the research of Agyei-Ampomah (2007).
4.1 Momentum returns versus portfolio size in the unrestricted sample
The annualized returns for a momentum strategy employing different portfolio sizes are shown in graph 1. The momentum returns of graph 1 are realized by employing a momentum portfolio using the methods of Jegadeesh and Titman (1993) with an evaluation period J and holding period K of both six months each. As is shown, the largest momentum returns are realized with a momentum portfolio of 14 stocks (7 winners and 7 losers). The results displayed in graph 1 are similar to those observed in the research of Siganos (2007). At first momentum returns sharply increase as the number of stocks in the portfolio increases, suggesting the benefits of diversification dominate initially. After the first extreme winners and losers the momentum returns tend to decline as the size of the portfolio increases.
Pre-cost momentum returns for different portfolio sizes
Graph 1: This graph shows the annualized momentum returns before transaction costs for
different number of stocks in the momentum portfolio. The graph shows a momentum strategy with evaluation period J and holding period K both equal to six months.
0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00% 1 5 9 13 17 21 25 29 33 37 41 45 49 Ann u al ized Ret u rn (%) Size
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Graph 2 shows the momentum returns for different portfolio sizes for the winner and the loser portfolio separately. These results differ from earlier findings of Jegadeesh and Titman (1993), Agyei-Ampomah (2007), Rouwenhorst (1998) who find - using large fractions of available stocks such as deciles and triciles - that most of the return of a momentum strategy is realized by the short position in the loser portfolio. Whereas graph 2 shows that for the Dutch stock market during the sample period this is the case only for small portfolios, using larger portfolio sizes the momentum returns are mainly driven by the winner portfolio. Furthermore, short positions in the loser portfolio consisting out of more than 16 shares show negative returns. This result is of importance to investors, since in the real world investors face restrictions to short selling. Not only do they have to pay larger costs for shorting or are they asked to put down extra collateral for their short position, in some cases authorities prohibit short selling in certain funds as they view it as damaging to the market and immoral to speculate on for instance bankruptcy of companies is distress (Diether, Lee and Werner, 2009; Grünewald, Wagner, and Weber, 2010).
Pre-cost returns of the Winner and Loser portfolio for different portfolio sizes
Graph 2: The annualized momentum returns before transaction costs for different number of stocks in the momentum portfolio. The graph shows a momentum strategy with evaluation period
J and holding period K both equal to six months. The blue line depicts the return of the winner
portfolio (W) and the red line the return of the loser portfolio (L).
The Sharpe and Sortino ratios for different portfolio sizes using an evaluation period J and holding period K of both six months are shown in graph 3. Displaying the ratio between
-20.00% -15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00% 1 5 9 13 17 21 25 29 33 37 41 45 49 Ann u al ized Ret u rn (%) Size
(number of stocks in the Winner and Loser portfolio)
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return and risk for each portfolio size. Both the Sharpe Ratio and the Sortino Ratio are maximized with a momentum portfolio consisting out of 28 stocks. The relationship between the number of stocks and the Sharpe ratio seems similar to the relation between portfolio size and return. A notable difference though is that to maximize the Sharpe ratio the portfolio contains twice as many stocks as the portfolio that maximizes the momentum returns. However the Sortino ratio shows a different pattern. Just as the momentum returns and the Sharpe ratio the Sortino ratio seems to increase sharply at first as additional stocks are added to the portfolio, but a clear decline at a certain point remains absent. These results show that both the risk neutral investor - who wants to optimize - the Sharpe ratio and the more risk averse investor – who would be more interested in preventing a loss and would therefore monitor the Sortino ratio – initially benefit from diversification over a more extreme portfolio with higher momentum returns. However, for the more risk averse investor the benefit of diversification stays present more clearly as portfolio size increases while for the risk neutral investor the tradeoff between more extreme momentum returns and the benefit of diversification seems to move towards the less diversified portfolios.
Risk versus pre-cost return for various portfolio sizes
Graph 3: The Sharpe and Sortino ratio for different sizes of the momentum portfolio, applying a
ranking period J of six months and a holding period K of 6 months. The blue line depicts the Sharpe ratio (return in excess of the risk free rate divided by standard deviation of the returns), the red line depicts the Sortino ratio (return in excess of a minimum acceptable rate divided by the downside deviation below this rate). The minimum acceptable rate in this figure is set to the risk free rate.
0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 1 5 9 13 17 21 25 29 33 37 41 45 49 Ann u al ized Ret u rn (%) Size
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Above results are shown in more detail in table 1. Table 1 also includes the associated t-statistic of the momentum return, noticeable is that none of the returns are significant, this due to the large fluctuations in returns.
Table 1: Momentum returns versus portfolio size.
The annualized returns for various portfolio sizes for the entire Dutch stock universe between March 2000 and March 2016. The portfolios are formed by selecting the top performing stocks for the winner portfolio and the worst performing stocks for the loser portfolio based on their previous J=6 months return and then held – with one-month skip in between – for a period of K=6 months. The momentum portfolio (WML) is constituted out of a long position in the winner portfolio and a short position in the loser portfolio. The t-statistic is calculated using Newey West hac standard errors. The Sortino ratio is based on a minimum acceptable rate equal to the risk free rate.
Winner portfolio Loser portfolio
WML
portfolio size Return (%) Return (%)
19 0 1000 2000 3000 4000 5000 1 5 9 13 17 21 25 29 33 37 41 45 49 Av er age M ar k et Capi tal izat io n p er st ock (m il lion € ) Size
(number of stocks in the Winner and Loser portfolio)
avg MC W avg MC L avg MC W-L
Portfolio characteristics
The smaller the number of different stocks in the momentum portfolio, the more the portfolio tends to be weighted towards small cap stocks and penny stocks. The rationale behind this is that small cap stocks tend to fluctuate more in prices than larger stocks with larger market capitalization and that minor price changes in penny stocks show already large returns when expressed as a percentage of price. This in line with the findings of previous literature as Agyei-Ampomah (2007), Jegadeesh and Titman (2001), Rouwenhorst (1998) and Lesmond et al. (2004) who all find that momentum portfolios are weighted towards less liquid stocks. As shown in graph 5 the average market capitalization of the momentum portfolio increases as extra stocks are added to the momentum portfolio.
Average market capitalization per share for different portfolio sizes
Graph 5: Plotted is the average market capitalization of the stocks in the Winner (W), Loser (L) and
momentum portfolio (WML) for various sizes of the number of stocks in the Winner and Loser portfolio based on a momentum strategy with evaluation period J=6 and holding period K=6.
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Average percentage of penny stocks per portfolio for different portfolio sizes
Graph 6: This graph represents the percentage of stocks in the portfolio that are penny stocks (defined as
shares with a share price below €1,-) when applying a J6K6 momentum strategy. The blue line depicts the winner portfolio (W), the red line the loser portfolio (L) and the green line the momentum portfolio (WML) consisting out of a long position in W and a short position in L.
Alternative strategies for J and K
The most commonly researched holding and evaluation period to research a momentum strategy is with J=6 and K=6. However, researchers find differences in return for different holding and evaluation period strategies. Therefore, this research also displays the effects of different portfolio sizes for strategies with different J and K. Graph 7 displays the relationship between momentum returns and number of stocks in the winner and loser portfolio for various strategies of J and K. This is in order to assess if the relationship between portfolio size and corresponding momentum return is consistent across the different strategies.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 Av er age amo u n t o f p enn y st ock s (%) Size
(number of stocks in the winner and loser portfolio)
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Pre-cost momentum returns per portfolio size for various strategies of J and K
Graph 7: This graph shows the annualized momentum returns before transaction costs for different
number of stocks in the momentum portfolio. The graph shows momentum strategies with various different evaluation period J and holding period K.
The most important observation is that for all different proposed strategies positive momentum returns are observed when applying respectively small portfolios of four and more stocks. Secondly, all strategies show an increase or a capricious jumping up and down pattern when adding the first few stocks to the portfolio and after a certain point a graduate decrease in return by diversifying the extremity of the first few stocks. For the strategies J3K9, J9K3, J9K9 the results show – similar to the J6K6 strategy discussed above – a sharp increase of momentum returns after adding the first extra stocks to the portfolio and more or less gradually decreasing returns as more stocks are added. A notable difference however is that the maximum is already observed employing only the four most extreme stocks in both the winner and loser portfolio. The strategy J3K3 shows a capricious pattern when adding the first extra stocks to the portfolio, similar to the J6K6 strategy the returns gradually decline when one diversifies beyond the first fourteen stocks. The strategy J6K9 shows a small increase when adding the first two extra stocks and a gradual decrease of returns when adding more. The strategies J6K3 and J3K9 both show a fluctuating growth in momentum return up until the first respectively sixteen and fourteen stocks and a small graduate decrease when adding extra stocks.
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 1 5 9 13 17 21 25 29 33 37 41 45 49 Ann u al ized moment u m ret u rns (%) Size
(number of stocks in the winner and loser portfolio)
22 Trading costs
As stated previously, researchers differ in the point of view whether momentum returns are significant after considering trading costs. Jegadeesh and Titman (1993) argues that a momentum strategy is profitable after transaction costs, assuming transaction costs are 0.5%. However, Lesmond et al. (2004) claim that the costs of a momentum strategy are higher than the assumed 0.5% and leave a momentum strategy unprofitable. Siganos (2007) claims that in the UK a profitable momentum strategy can be observed, assuming transaction costs similar to those reported by Lesmond et al. (2004). By finding the portfolio size that optimizes momentum returns and incorporating a fixed level of transaction costs into the calculation of profit, hereby neglecting the impact of portfolio size on the transaction costs and assuming that costs in the UK are equal to those in the US. This section discusses the observed trading costs for the unrestricted sample on the Dutch stock market following the previously elaborated methods of Lesmond et al (2004). For this analysis a smaller time period of the total sample is observed since the first 18 months of the sample are needed to estimate the transaction costs.
Turnover rate for various portfolio sizes
Graph 8: The turnover rate, expressed as the percentage of stocks that need to be rebalanced at the end of
holding period K for various sizes of the momentum portfolio applying a J6K6 strategy. The blue line depicts the turnover rate of the winner portfolio (W), the red line the turnover rate of the loser portfolio (L) and the green line the turnover rate of the momentum portfolio (WML), consisting of a long position in W and a matching short position in L.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 Tu rno ver r at e (%) Size
(number of stocks in the winner and loser portfolio)
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A principal driver of the costs of a momentum strategy is the frequent rebalancing. The turnover rate shown in graph 8 shows which percentage of the portfolio needs to be rebalanced at the end of every period with length K. From this chart it can be observed that the turnover ratio declines as the number of stocks in the portfolio increases, leading to lower transaction costs for more diversified portfolios. This result is as one might logically expect since the chance that the top performing stock of a previous period is also the top performing stock in the next period must be lower than the chance that this stock is somewhere within the top ten of best performing stocks. The results for the portfolio sizes between 20 stocks and 40 stocks are similar to those found by Lesmond et al. (2004) and Agyei-Ampomah (2007). The turnovers found are within the range of between 20 and 40 stocks are between 84% and 78% whereas Lesmond et al. (2004) finds a portfolio turnover of 81% and Agyei-Ampomah (2007) of 80.5%.
The other component used to estimate the costs to rebalancing is the roundtrip cost. Graph 8 shows the roundtrip cost for the various portfolio sizes when implementing a J6K6 strategy for the restricted sample. Graph 9 shows that the roundtrip costs for the most extreme sizes are substantially larger than for the more moderate and large size portfolios.
Roundtrip costs per portfolio size for the momentum portfolio
Graph 9: Plotted is the average roundtrip cost for different sizes of the momentum portfolio employing a J6K6 strategy. The roundtrip costs consist out of the quoted spread plus two commissions, the
commissions are based on the tariff posted on the website of the Dutch bank ABN Amro.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 R ou nd trip c osts (%) Size
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The post-cost returns are shown in graph 10, note that these differ from the earlier reported pre-cost returns minus the pre-costs since the transaction pre-costs are not calculated over the entire sample period but exclude the first 18 months of the sample period, since this period is needed to estimate the quoted spread. As shown the large drop in roundtrip cost as well as the graduate decline in the turnover rate cause trading costs to decline as the portfolio size increases. These results are consistent with the theory of Barber and Odean (2000) who state that frequent trading comes at large costs and is mostly destroying the wealth of investors. As shown for all portfolios negative post-cost returns are realized during the sample period. As shown in table 3 the loss realized is statistically significant.
Post-cost momentum returns for various portfolio sizes
Graph 10: Plotted are the pre-cost momentum returns over the sample period august 2001 until march
2016 the transaction costs and the post-cost momentum returns for a J6K6 momentum strategy.
This leads to the important conclusion that during the researched sample period exploiting a momentum strategy, when considering all available stocks on the Dutch market, is unprofitable after taking transaction costs in consideration. This result differs from Siganos (2007) who argues that in the UK many opportunities to execute a profitable momentum strategy were then present, foregoing that transaction costs are larger for smaller sizes of the momentum portfolio. These results are consistent with the findings of Agyei-Ampomah (2007) and Lesmond et al. (2004) as it provides evidence that transaction costs are more substantial than assumed by Jegadeesh and Titman (1993), leaving the J6K6 momentum strategy unprofitable.
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 1 3 5 7 9 1113151719212325272931333537394143454749 Annu al iz ed R etu rn (%) Size
(number of stocks in the winner and loser portfolio)
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Table 2: Post-cost momentum returns versus portfolio size.
The annualized returns for various portfolio sizes for the entire Dutch stock universe between August 2000 and March 2016. The portfolios are formed by selecting the top performing stocks for the winner portfolio and the worst performing stocks for the loser portfolio based on their previous J=6 months return and then held – with one month skip in between – for a period of K=6 months. The momentum portfolio (WML) is constituted out of a long position in the winner portfolio and a short position in the loser portfolio. The transaction costs are estimate using the quoted spread and turnover ratio. The t-statistic is calculated using Newey West hac standard errors. The *, **, *** show significance at the 1%, 5% and 10% level respectively.
pre-cost cost post-cost
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4.2 Restricted sample
This section discusses the momentum returns for various portfolio sizes when excluding the smallest 30% in market capitalization and stocks with a stock price below €1. The annualized returns for a momentum strategy employing different portfolio sizes are shown in graph 11. The momentum returns of graph 11 are realized by employing a momentum portfolio using the methods of Jegadeesh and Titman (1993) with an evaluation period J and holding period K of six months each.
Pre-cost momentum returns for different portfolio sizes of the restricted sample
Graph 11: This graph shows the annualized momentum returns before transaction costs for different
number of stocks in the momentum portfolio, excluding penny stocks and small cap stocks from the sample. The graph shows a momentum strategy with evaluation period J and holding period K both equal to six months.
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returns for all portfolio sizes, whereas this study observes the larger returns only for the more extreme sizes. This result is in line with Agyei-Amponah (2007) who finds for a moderate portfolio size that the pre-cost momentum returns are smaller for a sample that excludes small caps. A notable difference with the unrestricted sample is that the momentum returns for the restricted sample statistically significant differ from zero, whereas the momentum returns of the unrestricted sample do not. This is mainly due to due to lower fluctuations in the observed returns leading to smaller Newey West hac standard errors.
Pre-cost returns of the Winner and Loser portfolio for different portfolio sizes (restricted sample)
Graph 12: The annualized momentum returns before transaction costs for different number of stocks in the momentum portfolio, excluding small caps and penny stocks from the sample. The graph shows a momentum strategy with evaluation period J and holding period K both equal to six months. The blue line depicts the return of the winner portfolio (W) and the red line the return of the loser portfolio (L).
Similar to the findings for the unrestricted sample, for the less extreme portfolio sizes the momentum returns are mainly realized by the winner portfolio, whereas the short position in the loser portfolio delivers a loss for portfolio sizes larger than thirteen. Again this differs from Agyei-Ampomah (2007), Rouwenhorst (2007) who find that most of the return of a momentum strategy is realized by the short position.
28 Table 2: Pre-cost momentum returns versus portfolio size. (restricted sample)
The annualized returns for various portfolio sizes for the Dutch stock universe excluded penny stocks and small cap stocks between August 2000 and March 2016. The portfolios are formed by selecting the top performing stocks for the winner portfolio and the worst performing stocks for the loser portfolio based on their previous J=6 months return and then held – with one month skip in between – for a period of K=6 months. The momentum portfolio (WML) is made up of a long position in the winner portfolio and a short position in the loser portfolio. The t-statistic is calculated using Newey West hac standard errors. The *, **, *** show significance at the 1%, 5% and 10% level of significance.
Winner portfolio Loser portfolio WML portfolio
size Return (%) Return (%)
29 Trading costs
This displays the observed trading costs for the restricted sample on the Dutch stock market following the previously elaborated methods of Lesmond et al (2004).
Turnover rate for various portfolio sizes (restricted sample)
Graph 13: The turnover rate, expressed as the percentage of stocks that need to be rebalanced at the end
of holding period K for various sizes of the momentum portfolio applying a J6K6 strategy, restricting small caps and penny stocks from the sample. The blue line depicts the turnover rate of the winner portfolio (W), the red line the turnover rate of the loser portfolio (L) and the green line the turnover rate of the momentum portfolio (WML), consisting of a long position in W and a short position in L.
The turnover rate shown in graph 13 shows what percentage of the portfolio needs to be rebalanced at the end of every period with length K. From this chart it can be observed that the turnover ratio declines as the number of stocks in the portfolio increases, leading to lower transaction costs for more diversified portfolios. These results are similar to the results of the unrestricted sample
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the restricted sample. Graph 14 displays that the roundtrip costs for the most extreme sizes are substantially larger than for the more moderate and large size portfolios.
Roundtrip costs per portfolio size for the momentum portfolio (restricted sample)
Graph 14: Plots the average roundtrip cost for different sizes of the momentum portfolio employing a
J6K6 strategy. The roundtrip costs consist out of the quoted spread plus two commissions, the
commissions are based on the tariff posted on the website of the Dutch bank ABN Amro.
The post-cost returns are shown in graph 15. As shown only for the portfolio sizes smaller than a total of 18 stocks positive post-cost returns are realized during the sample period. As shown in table 3 none of the reported profits are statistically significant.
Post-cost momentum returns for various portfolio sizes (restricted sample)
Graph 15: Plotted are the pre-cost momentum returns over the sample period august 2001 until march
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Table 2: Post-cost momentum returns versus portfolio size (restricted sample)
The annualized returns for various portfolio sizes for the Dutch stock universe excluding small caps and penny stocks between August 2000 and March 2016. The portfolios are formed by selecting the top performing stocks for the winner portfolio and the worst performing stocks for the loser portfolio based on their previous J=6 months return and then held – with one month skip in between – for a period of K=6 months. The momentum portfolio (WML) is made up of a long position in the winner portfolio and a short position in the loser portfolio. The transaction costs are estimate using the quoted spread and turnover ratio. The t-statistic is calculated using Newey West hac standard errors. The *, **, *** show significance at the 1%, 5% and 10% level.
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5 Conclusion
This study investigates the impact of portfolio size on pre- and post-cost momentum returns on the Dutch stock market between 2000 and 2016. The momentum strategy invests in the past top performing shares and finances this with a short position in the past worst performing shares. This research finds that when using a ranking period J and a holding period K both equal to six months momentum returns are maximized to 13% per annum by a portfolio of 14 different stocks. Note that this return is not statistically different from zero. Different from the findings of Jegadeesh and Titman (1993) most of the returns are generated by the winner portfolio instead of the short position in the loser portfolio. For investors who not only seek out to maximize return, but who are interested in return versus risk it is advisable to diversify more than the point to where returns are maximized. The Sharpe ratio of the strategy climbs further up until the point of 28 different stocks in the portfolio. As investors are mostly concerned about fluctuations of their returns below a certain level and are only happy to see returns fluctuate upwards above this level the Sortino ratio measures the tradeoff between return and deviations below a certain rate. Setting this minimum acceptable rate equal to the risk free rate the benefits of diversification seem to remain present instead of being deterred as by the Sharpe ratio.
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significant profit after transaction costs. For portfolio sizes of 14 different stocks or more the strategy still generates a loss. This result is similar to Agyei-Ampomah (2007) in the way that both studies observe smaller transaction costs when excluding small market capitalization stocks and find that for the J6K6 strategy the strategy is not profitable after transaction costs. A difference is that Agyei-Ampomah (2007) observes smaller pre-cost momentum returns for the restricted sample, whereas this research finds that for the small portfolio sizes pre-cost momentum returns increase and for various portfolio sizes the returns are statically significant when leaving out small market cap shares. The transaction costs leave only the smaller sizes, portfolios up to 12 different shares, profitable but this profit is not statistically significant. An investor choosing to employ a momentum strategy excluding the small cap stocks with a very small portfolio size will probably face months with large losses. When the minimum acceptable rate is set equal to the risk-free rate the Sortino ratio is optimized when choosing the smallest possible portfolio, however when an investor is mainly looking to prevent large losses and the minimum acceptable rate is set to 5% per annum or -10% per annum the benefits of diversification dominate over the larger return.
Overall this research shows that, when excluding small market capitalization stocks, significant momentum returns can be realized. and that picking only the two most extreme stocks maximizes these returns. Furthermore, this thesis finds that for the Dutch stock market the majority of this return is earned by the long position in the winner portfolio. Yet, after accounting for transaction costs the strategy does not remain significantly profitable. As the size of transaction costs depends on the size of the portfolio this thesis suggests prior research may have been reporting illusory profits.
Limitations and suggestions for further research
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