Size matters: the influence of portfolio size on
momentum returns
Melvin R. de Vries1 Master thesis Finance University of Groningen
January 2015 Supervisor: dr. J.O. Mierau
Abstract
This thesis studies the relationship between the size of the momentum portfolio and the magnitude of returns by constructing unconventional portfolio sizes using US stock date between 1993 and 2013. My main conclusion is that small portfolio sizes already show strong momentum returns. Another conclusion is that to maximize momentum returns the amount of shares in the portfolio has to be disproportionately increased. Lastly, for the smallest portfolio sizes positive post-cost returns only emerge when excluding the smallest companies. Overall, my research shows that profitable momentum strategy execution is possible with small portfolio sizes.
Keywords: momentum strategy, market efficiency, portfolio size
JEL classification: G10, G11, G14
1 University of Groningen, Faculty of Economics and Business, MSc Finance
Student number: 1875361
2
1. Introduction
An investment strategy as simple as buying previous winners and selling previous losers which is both statistically as well as economically significant according to many researchers; the momentum strategy. The momentum phenomenon has been one of the most extensively researched anomalies of the efficient market hypothesis. Although statistical significance has been shown, researchers have not yet reached consensus about the economic significance.
Levy (1967) was the first to state that a strength strategy of buying stocks with prices substantially above their 27 week average yielded significant abnormal positive returns. In 1993 Jegadeesh and Titman built on the idea of Levy, which implied stock returns continuation for a medium time period. They rank stocks based on their previous six-month returns. Subsequently, they form one portfolio employing stocks with ranking in the top decile (winner portfolio) and one portfolio containing stocks ranked in the bottom decile (loser portfolio). They buy the winner portfolio and short the loser portfolio for the next six months. This strategy yielded significant above market returns. Under the assumption of efficient markets, all information should be incorporated in current stocks price. Thus, the historical price path should have no influence on future stock returns. The momentum strategy is a clear violation of the efficient market hypothesis.
In attempts to rationalize the momentum anomaly, researches have described these abnormal returns as a result of data mining (Black, 1993; MacKinlay, 1995), compensation for bearing higher risk (Conrad and Kaul, 1998), short selling constraints (Agyei-Ampomah, 2007) and even statistical “chance results” (Fama, 1998). However, extensive studies have shown the momentum effect for a wide range of stock markets and confirmed that the effect is more than just a “statistical anomaly” (Rouwenhorst, 1998; Griffin, Ji and Martin, 2003; Doukas and McKnight, 2005; Gupta, Locke and Scrimgeour, 2013).
Whatever the explanation of momentum returns may be, researchers have not conclusively shown the economic significance of the momentum strategy. The momentum strategy requires frequent trading to effectively work and is therefore very costly. Besides, momentum portfolios consist of several hundreds of shares, which often is out of reach for private investors. Several variations of the momentum strategy have been suggested to enhance momentum returns. Academics have researched different transaction costs estimations (Lesmond, Schill and Zhou, 2004), restricting sample to large caps (Chan, Jegadeesh and Lakonishok, 1999), excluding share prices below $5 (Jegadeesh and Titman, 2001), neglecting short positions (Grinblatt and Moskowitz, 2004) and focusing on single blue chip stocks (Rey and Schmid, 2007).
3
employing the top 40 and bottom 40 ranked shares substantially increases momentum returns. Why would the portfolio size influence momentum returns? Firstly, basic finance theory teaches us that altering portfolio size influences the level of diversification of a portfolio. Secondly, altering portfolio size includes or excludes certain stocks and might alter the characteristics of
the stocks included in the portfolio.
Suppose a world without transaction costs in which we have two extreme sizes (A and B) of the momentum portfolios. Size A consists of a winner portfolio containing 20 shares and a loser portfolio containing 20 shares. For size B, the winner and loser portfolio both contain 200 shares. Size A and B obviously differ in level of diversification but how can a difference in size influence the portfolio characteristics? Note that the momentum strategy first ranks stocks based on their previous six-month returns. This means that the top and bottom ranked stocks had the most extreme, either positive or negative, returns over these six months. The extreme returns of these stocks imply that their volatility is very high. After the ranking, the strategy creates momentum portfolios. The size of the portfolio determines how many stocks of the top (winners) and bottom (losers) of the ranking are included in the momentum portfolio. A portfolio containing only the top 20 and bottom 20 ranked stocks (size A) is clearly more focussed on volatile stocks than a portfolio employing the 200 top and bottom stocks (size B). Generally speaking, small and illiquid stocks are more volatile than their larger and more liquid counterparts. This suggests that size A is more weighted towards small and illiquid stocks than portfolio size B. What does this imply? Firstly, the average market cap and liquidity of size A should be lower than size B. Secondly, the average volatility of size A should be higher than size B due to more weight on more volatile stocks. Besides that, increased diversification benefits decrease the volatility of portfolio B even further. Most important, Hong, Lim and Stein (2000) show that momentum returns are mostly driven by stocks with low liquidity and small size. It follows that the average return of portfolio A should be higher than the average return of portfolio B. The above example implies a relationship between portfolio size and returns. However, one has to include transaction costs to execute the strategy. How will limiting the portfolio size affect the post-cost momentum returns?
4
These observations result in an important question: can researchers and private investors take better advantage of the momentum strategy? In the words of Gupta, Locke and Scrimgeour (2013, p. 223) “The literature to date has not achieved consensus on appropriate rules for
determining the optimal portfolio size for calculation momentum returns. The lack of precision may affect momentum profit as portfolios may be formed with only extreme return stocks or a large number of stocks.”
The contribution of this research is twofold. Firstly, I contribute to the literature focussed on optimizing the momentum strategies by investigating the relationship between the portfolio size and the magnitude of momentum returns. This not only leads to maximization of returns, but also to the discussion of the reasoning behind this relationship. Secondly, there is a practical contribution: maximizing returns is always of interest to investors. Besides, private investors sometimes face practical limitations when employing the momentum strategy. They often have limited financial resources inhibiting them from creating the large sized momentum portfolios. Also, limited time might restrict their abilities to rebalance the large portfolios every month. Profitable implementation of the momentum strategy with a smaller portfolio size could therefore be very interesting for private investors.
The structure of the paper is as follows. The next section describes relevant literature regarding momentum and the relationship with portfolio size. In addition, Section 3 describes the data and methodology. Furthermore, Section 4 discusses the empirical results. Section 5 concludes and provides options for further research.
2. Literature review
The purpose of this section is twofold; it discusses relevant literature on the momentum anomaly, momentum strategy and the drivers of momentum returns, and it explains how these drivers can influence the momentum returns for different portfolio sizes.
2.1 Momentum
5
and exploits the fact that the movement to the new value continues in the near future.
In 1993, Jegadeesh and Titman introduced the momentum strategy. They find that buying stocks that performed well in the past tend to perform well in the future. The opposite is true for stocks that performed badly. They rank stocks based on their performance during a formation period with a J-month length. The length of J ranges in their research between 3 and 12 months. Subsequently, they construct a winner portfolio with stocks ranked in the top ten percent. Similarly a loser portfolio is constructed with stocks in the bottom ten percent. A long position is taken in the winner portfolio and a short position in the loser portfolio. These positions are held for another K-months, where K ranges from 3 to 12 months. Jegadeesh and Titman report that between 1965-1989 trading strategies that bought past winners and sold past losers yielded an average return of 1.10% per month. After this, numerous researchers have extensively tested and confirmed the existence of the momentum effect throughout the world. Table 1 presents an overview of several studies regarding the momentum effect.
Author(s) Market Period Portfolio size
Monthly return
Jegadeesh and Titman (1993) US 1965-1989 Decile 1.10%
Chan, Jegadeesh and Lakonishok (1996) US 1977-1993 Decile 1.42%
Rouwenhorst (1998) EU 1980-1995 Decile 1.28%
Lui, Strong and Xu (1999) UK 1977-1996 Decile 1.42%
Hong, Lim and Stein (2000) US 1980-1996 Triciles 0.53%
Jegadeesh and Titman (2001) US 1965-1998 Decile 1.23%
Chen and Hong (2002) US 1928-1999 Decile 0.64%
Griffin, Ji and Spencer (2003) WO 1928-2000 Quintile 0.49%
George and Hwang (2004) US 1963-2001 Triciles 0.48%
Doukas and McKnight (2005) EU 1988-2001 Quintiles 0.89%
Triciles 0.73%
Chelley-Steeley and Siganos (2005) UK 1975-2001 Decile 1.53%
Quintiles 1.18%
Triciles 0.96%
Siganos (2007) UK 1975-2001 40 stocks 2.09%
Rey and Schmid (2007) SW 1994-2005 1 stock 2.79%
Gupta, Locke and Scrimgeour (2013) WO 1973-2007 Decile 0.90%
Bandarchuk and Hilscher (2013) US 1964-2008 1/50 portfolio 2.43%
1964-2008 1/25 portfolio 2.20%
1964-2008 Decile 1.61%
1964-2008 Quintile 1.11%
This table presents literature in the field of momentum strategies using a six month ranking period and a six month holding period. Authors, markets, sample periods, portfolio size and average monthly returns are shown. US, EU, SW, UK and WO are the abbreviations for United States, Europe, Switzerland, United Kingdom and world.
6
Firstly, Table 1 reveals positive momentum returns independent of research country or time period. Secondly, it also shows that researchers tend to use different sizes for the momentum portfolio. Most important, Siganos (2007) noticed a pattern when evaluating the returns of the different portfolio sizes. He finds that smaller portfolio sizes seem to have higher returns. He uses an UK dataset and finds an optimal portfolio sizes employing only the 40 top and bottom performing shares. Barndarchuk and Hilscher (2013) find that a portfolio size of 1/50 of the universe performs substantially better than the basic decile portfolios. Doukas and McKnight (2005) and Chelley-Steeley and Siganos (2008) show that changing the portfolio size from triciles to quintiles increases momentum returns.
Several papers thus suggest a relationship between momentum portfolio size and the strength of momentum returns. In the introduction I already introduced two basic consequences of altering portfolio size. Firstly, adjusting the size will influence the level of diversification. Secondly, altering portfolio size changes the stocks included in the portfolio. In the following subsections I will discuss why momentum portfolios are not exempt from the influences of these two basic theories.
2.2 Diversification
In their ground-breaking paper Jegadeesh and Titman (1993) opted for using deciles to define their momentum portfolios on all the stocks in CRSP data file (containing several thousands of stocks). In more recent studies, researchers even used as much as 1/3 of their stock universe size for the winner and loser portfolio. Hong et al. (2000) justify their choice for triciles to define their portfolios because for their research they needed to split their stock universe into 120 portfolios. This means that they reach a point “where some of the individual portfolios are quite
undiversified, thereby creating larger standard errors in our test statistics”(Hong et al. 2000, p.
276). They mention that unlike Jegadeesh and Titman (1993) they are not so much interested in establishing the existence of momentum per se, but in comparing momentum effects across subsamples of stocks. Although the purpose of both their studies is different than mine, an important observation can be made from their choice of portfolio size. The goal of their research is not to optimize returns for investors, but to investigate the dynamics of the momentum effect. In the context of diversification, which I will explain in the next paragraph, this might actually negatively influence returns.
7
However, for private investors it is often not realistic to employ hundreds of stocks. These large portfolios will also significantly increase the transaction costs of the strategy. Besides, at a certain portfolio size diversification will diversify momentum returns and eventually even erase them. Jegadeesh and Titman were not so much interested in maximizing momentum returns but rather establishing the existence of this anomaly. However, for an investor maximizing returns is of the utmost importance. Any possible “over-diversification” is therefore unwanted.
Statman (1987) finds that a well-diversified portfolio should hold a minimal of 30 to 40 stocks, which is far less than currently used for momentum portfolios. In the context of diversification there is no need to employ the hundreds of shares as Jegadeesh and Titman (1993) do in their momentum portfolios. Reverting back to the example portfolios A and B. It is obvious that A has a lower level of diversification than B. However, is it really critical, in terms of increased returns and diversification benefits, to employ the large amount of shares in portfolio B? Considering the above, one might wonder if smaller portfolio sizes exist that have similar levels of diversification and momentum returns as the standard decile portfolios.
Interestingly, Siganos (2007) finds a significant optimum portfolio size including 40 winners and losers (i.e., portfolio size 40). In between portfolio size 10 and 40 he finds continuously increasing returns. Also the volatility decreases from 4.9% at portfolio size 10 to 3.5% at portfolio size 40. That means that up until the 40th stock, including more stocks increases portfolio returns while decreasing risk. This seems a contradiction with diversification. It implies that not only diversification influences the relationship between size and momentum risk and return, but that momentum returns differ between certain stocks which are included or excluded in the portfolios for various sizes.
2.3 Stock characteristics
From the above literature can be derived that not only diversification plays an important role in the relationship between portfolio size and momentum returns. There is also a second important driver. Several studies confirm that certain stock characteristics increase the strength of momentum returns (Hong et al., 2000; Daniel et al., 2001; Zhang, 2006). Siganos’ (2007) research suggests that at a certain portfolio size we reach an optimal balance of these characteristics and diversification benefits to maximize momentum returns. It is therefore highly relevant to identify these stock characteristics and their relationships with momentum returns. Although researchers have not reached consensus regarding the sources of momentum, most turn to behavioral models for an explanation.
8
on their fundamentals enhances psychological biases. This implies that overconfidence should be stronger when a firm is harder to value. Daniel et al. (1998 and 2001) confirm this and find that investors are more overconfident when a firm is harder to value and therefore conclude that return predictability should be stronger in firms with greater uncertainty. Additionally, Zhang (2006, p. 109) concludes that “misvaluation effects of almost any mistaken-beliefs model should be strongest among firms about which there is high uncertainty and poor information”. Zhang confirms his theory and indeed finds that investors tend to underreact more to new information when there is more ambiguity with respect to its implications for firm value.
Hong and Stein (1999) propose an alternative explanation. They argue that momentum comes from gradual information flows and thereby confirm the theory of underreaction. However, they do not attribute this to overconfidence of the investor. They divide the universe in “newswatchers” and “momentum traders”. The first act solely on signals regarding future returns and the latter only react on past price changes. They find that momentum is a consequence of the gradual diffusion of private information, combined with the failure of newswatchers to extract this information from prices. Hong et al. (2000) find more momentum in stocks where information diffuses more slowly.
Although both Hong and Stein (1999) and Zhang (2006) take a different angle to explain momentum returns, their conclusion is the same; higher information uncertainty increases momentum. This conclusion is critical for my research. This implies that portfolios containing shares with a higher level of information uncertainty have stronger momentum returns compared to portfolios with lower information uncertainty. In the next subsection I will explain how limiting the portfolio size increases the weight on information uncertain stocks. However, one first needs to know how information uncertainty is measurable in stocks. Information uncertainty is an abstract term and for an unbiased measurement one has to use proxies. In the literature liquidity and market capitalization are most common as proxies for information uncertainty.
2.4 Proxies for information uncertainty
Firstly, Zhang (2006) proposes market cap as a proxy for information uncertainty. Information is scarce for smaller companies and the cost to obtain information is higher. Thus, one can expect that information uncertainty is higher for smaller firms.
9
as a whole. From that point they decline monotonically to the point where they are essentially zero for the largest stocks. Hong et al. (2000) argue that although smaller firms have slower information diffusion, they probably also have more limited investor participation (i.e., thinner market making capacity), which may lead to more pronounced supply-shock reversals and thereby inhibiting underreaction.
Secondly, liquidity is also a popular proxy for information diffusion and a strong determent of momentum profits. For instance, Agyei-Ampomah (2007) and Lesmond et al. (2004) find that illiquid stocks tend to generate most of the momentum returns. Although small stocks are often less liquid and vice versa, I will treat these separately for the simple reason that they are measured in a different way. Size is measured in terms of market capitalization and bid ask spread percentage is used to determine liquidity.
From the above literature can be derived that stocks with lower market cap and liquidity (i.e., “high information uncertainty”) show stronger momentum. Most important now, how do these two proxies relate with the size of the momentum portfolios? Generally speaking, smaller companies are not as financially stable, lack the resources to easily weather economic downturns, and are more likely to be less diversified. All of these reasons lead to increased volatility. Besides that, lower liquidity may increase price movement shocks in share prices. Note that the momentum strategy ranks stocks based on their past returns. Stocks with higher volatility will have an increased chance of either very high or very low returns. Thus, they are more likely to have a high or a low ranking. The size of the portfolio determines how many stocks of the top and bottom of the ranking are included in the momentum portfolio. A portfolio containing only the top 20 and bottom 20 ranked stocks is more focussed on volatile stocks than a portfolio employing the 200 top and bottom stocks. Since, we have established that higher volatility often goes paired with a low market cap and low liquidity, I can assume that portfolio A has more weight on small and illiquid stocks than B. It follows that the average return of portfolio A is higher than the average return of portfolio B.
One remark regarding the previous conclusion. Although Hong et al. (2000) acknowledge the negative relationship between firm size and returns, they argue that this might not apply to the smallest shares. This implies that if the companies in portfolio size A are too small, supply-shock reversals might negatively influence momentum returns. In the robustness I will test for this theory of Hong et al. (2000). However, the sizes of portfolio A and B are merely an illustration to show that portfolio size matters. Thus, for now I conclude that the relationship between market cap and returns evolves monotonically to continue the example.
2.5 Profitability of the momentum strategy
10
that is the heart of the problem. Transaction costs can increase rapidly and researchers have not reached consensus on the correct size of the transaction costs. Jegadeesh and Titman (1993) and Lui, Strong and Xu (1999) impose a fixed one way transaction cost of 0.5%. Their estimate is based on average trading costs and ignores stock specific characteristics. However, differences in stock characteristics are an important determinant of bid ask spreads and thereby transaction costs. Therefore, several studies have indicated that this estimation underestimates the real transaction costs (Lesmond, et al., 2004; Korajczyk and Sadka, 2004; Agyei-Ampomah, 2007). Lesmond et al. (2004) indicate that momentum strategies require frequent trading in disproportionately high cost stocks, which prevents a profitable execution of the momentum strategy. These high cost securities are identified in their research as very small and illiquid. This implies, when reverting back to example portfolio A and B, that we can expect substantially higher trading costs per share for portfolio A compared to portfolio B. Still, one has to pay commission costs for every transaction. Since portfolio B comprises of more shares, commission costs will also be substantially higher.
2.6 Research objective
In summary, there is a basis in the literature for the theorized differences in momentum returns between example portfolios A and B. The literature predicts that momentum returns are stronger in relatively small and illiquid stocks. These stocks also have a higher volatility on average. This implies increased chances of a very high or low ranking. Following this logic, portfolio A will be more weighted towards small and illiquid stocks. This implies that the average return of portfolio A is higher compared to portfolio B. Additionally, the combination of more weight on volatile stocks and less diversification benefits will also lead to a higher volatility of portfolio A. Including transaction costs may change the situation. A larger portfolio size results in extra commission costs. However, the extra weight on smaller stocks in portfolio A might disproportionately increase the transaction costs per share.
The key objective of this research is to establish the relationship between the size of the momentum portfolio and the magnitude of returns. The focus is on limiting the conventional portfolio size to show if more investors are able to pursue momentum returns.
3. Data and Methodology
In this section I will elaborate on the data and methods used in this research. I will also show and explain the equations used.
3.2 Data
11
Exchange (AMEX), and the NASDAQ stock market (NASDAQ). The time period examined in this research ranges from 1993 to 2013. The sample period focuses on the post 1993 period. Bid ask spreads for all companies are required, which for small NASDAQ firms were only included from 1993. Considering that at least 13 months of returns are needed, the first return of the momentum strategy is available on the 31st of January 1994. Throughout this research I exclude ADRs, REITS, closed-end funds, and primes and scores—stocks that do not have a CRSP share type code of 10 or 11. This restricts my data set to common shares and ensures that this thesis is comparable with the most prominent papers in momentum research. The number of firms analysed in any given period ranges from 3,313 to 6,117 with an average of 4,663. All monthly returns are calculated using month-end closing prices and all dividends are assumed reinvested in the same stock. For calculations of bid or ask prices, which are necessary in several definitions (e.g. quoted spreads and liquidity), a monthly average is calculated instead of taking the last day of the month. I do this to cope with turn-of-the-month behaviour in spreads as reported by Lesmond et al. (2004). The sample of stocks extracted from CRSP includes surviving and non-surviving stocks to cope with survivorship bias. To be included, a firm must have no missing values in the CRSP database.
3.3 Methodology
The formation of the portfolios is similar to the strategy developed by Jegadeesh and Titman (1993). At time t stocks are ranked from best to worst on their previous cumulative J months´ returns, where J is the ranking period. The ranking of stocks is essential in the momentum strategy because it will determine which stocks will be included in the winner and loser portfolio. In month t+1, I form winner-loser (W-L) portfolios, where a long position is taken in the winner portfolio (top ranked stocks) and a short position is taken in the loser portfolio (bottom ranked stocks) for the following K-month horizon in which K is the holding period. Jegadeesh and Titman (1990) propose a month skip between the ranking date at month t and the start of the holding period at t+1 to mitigate effects of bid ask spread, price pressure, and lagged reaction. I restrict the J and K to the basic “6x6” strategy. This strategy sets the formation period
J to six months and the holding period K to six months, which is the most common form of the
momentum strategy. To illustrate, for each month t the portfolio that will be held during the holding period, months t+1 to t+6, is determined by the performance of stocks during the ranking period, month’s t-5 to t. However, as robustness check I will test with ranking periods
J-3, J-12 and holdings periods K-3 and K-12. At the end of the holding period the positions will
12
return per portfolio size over the total sample period. This way my research can be compared to other papers. To increase the power of the test, I will construct overlapping portfolios.
Jegadeesh and Titman create winner portfolios containing the top 10% ranked stocks and loser portfolios containing the bottom 10% ranked stocks at time t. Here, I will make one crucial alteration to their momentum strategy. At time t I construct winner (loser) portfolios containing the best (worst) 1,2,3,4,…,199,200 ranked shares over the previous six months. 200 shares is the maximum because this study focusses on limiting the conventional portfolio size and 200 shares is similar to the winner and loser portfolio of Jegadeesh and Titman (2001). Besides, similar to Siganos (2007), I do not find any new results when increasing this portfolio size. This alteration to the basic strategy will provide me with 200 different W-L portfolios of ascending sizes at time t, which allows me to assess how returns evolve over the ascending portfolio sizes. As mentioned before, the ranking of stocks at time t is crucial in determining which stocks are included at what portfolio size. For instance, a W-L portfolio size 1002 consists of the top ranked 100 shares (winner portfolio) and the bottom 100 shares (loser portfolio). All the t-values in this paper are calculated using Newey and West (1987) heteroskedastic and autocorrelation consistent standard errors with the truncation lag set to the holding period.
I also analyse the characteristics of the portfolios in the form of average market cap and liquidity. Liquidity is calculated, similar to Agyei-Ampomah (2007), as a one month average bid ask spread percentage. Both liquidity and market cap are extracted, consistent with Lesmond et al. (2004), the month before the holding period to avoid endogeneity issues. Market cap is extracted the last day of the month and liquidity is the average of the same month. The reason I calculate the monthly average of liquidity instead of extracting it at month-end is to avoid any turn-of-the-month effects in bid ask spread behaviour as reported by Lesmond et al. (2004).
A common issue in assessing the profitability of the momentum strategy is the inclusion of the transaction costs. Lesmond et al. (2004) indicate that especially small and illiquid stocks are associated with disproportionally high transaction costs. Since limiting portfolio size will increase the weight on these small and illiquid stocks, it is sensible to include a transaction cost estimation that accounts for the differences in costs. This thesis follows the method used in Lesmond et al. (2004) and Agyei-Ampomah (2007) which produces estimates of “spread plus commission” costs by examining quoted market bid-ask spread data and prevailing commission schedules. The advantage of this method is that it directly examines cost data. However, due to the fact that trades frequently occur off the quoted prices and commissions are often not fixed the estimation is not perfect. The quoted spread for stock i at month t is estimated by calculating
2 There is a distinctive difference in how I address the size of a winner and loser portfolio or a W-L portfolio. I refer
13
the one month average of the ask price (ask) minus the bid price (bid) over the bid ask midpoint. This can be expressed as:
Quoted spread(𝑖, 𝑡) = ∑ 1(𝐴𝑠𝑘(𝑖, 𝑡 + 𝜏) − 𝐵𝑖𝑑(𝑖, 𝑡 + 𝜏)) 2 (𝐴𝑠𝑘(𝑖, 𝑡 + 𝜏) − 𝐵𝑖𝑑(𝑖, 𝑡 + 𝜏))
(1)
Opposed to Lesmond et al. (2004), I calculate the one month average of bid and ask the month before the holding period instead of a 12 month average. The reason for this is that, contrary to Lesmond et al. (2004), transaction costs are not the main subject of this research. Completely following Lesmond’s definition would lead to exclusion of a significant amount of companies that have missing data in the estimation period of the transaction costs. The commission rate of 0.67% is similar to Agyei-Ampomah (2007). The total transaction cost formula can be expressed as:
Roundtrip cost = Quoted spread + (2 ∗ commissions) (2)
4. Empirical Results
The primary results to answer the research question of this paper are shown in Section 4.1 – does portfolio size influence momentum returns? Section 4.2 and 4.3 provide deepening of these results. Firstly, I break down the W-L returns in winner and loser returns. Secondly, I show how information uncertainty interacts with portfolio size. From Section 4.4 onwards I test robustness of the results for various alterations to the momentum strategy.
4.1 Unrestricted returns
Table 2 presents the average returns per month for various portfolio sizes in the total sample period 1994-2013 which is graphically displayed in Figure 1. The results are, as mentioned, based on Jegadeesh and Titman’s 6x6 momentum strategy. The returns of the winners, losers and W-L portfolios are shown. The t-statistics are included to evaluate the significance of the results. Also the standard deviations are shown for every portfolio size to indicate the risk of the portfolios. The two most important results are that (1) strong and significant momentum returns can already found employing only 40 winners and losers; (2) to realize the maximum momentum return the amount of shares in the portfolio has to be disproportionately increased.
14 Winner Loser W-L
Size Ret (%) MC Liq (%) Ret (%) MC Liq (%) Ret (%) t-statistic SD (%)
1 -3.24 260 2.67 -3.42 81 7.72 0.19 0.13 16.91 2 -2.74 327 2.32 -2.62 72 7.33 -0.12 -0.11 12.36 3 -2.65 418 2.22 -2.36 70 6.77 -0.30 -0.29 10.83 4 -2.18 442 2.16 -2.55 79 6.40 0.37 0.41 9.58 5 -2.18 471 2.18 -2.42 75 6.48 0.24 0.77 8.29 6 -2.09 476 2.19 -2.44 80 6.32 0.34 0.46 7.47 7 -2.00 518 2.19 -2.35 79 6.33 0.35 0.53 6.85 8 -1.94 544 2.19 -2.33 77 6.26 0.39 0.60 6.66 9 -1.83 534 2.19 -2.33 77 6.33 0.50 0.76 6.39 10 -1.75 536 2.22 -2.18 80 6.27 0.42 0.66 6.11 20 -1.36 578 2.12 -2.10 92 5.87 0.74 1.27 4.89 30 -1.07 659 2.07 -2.10 101 5.78 1.04 1.83*** 4.62 40 -0.94 705 2.09 -2.09 108 5.68 1.16 2.22** 4.28 50 -0.84 753 2.10 -2.02 119 5.58 1.18 2.33** 4.15 60 -0.77 777 2.09 -1.95 125 5.48 1.18 2.31** 4.08 70 -0.72 802 2.08 -1.99 128 5.36 1.28 2.55** 3.95 80 -0.67 827 2.08 -1.96 135 5.32 1.29 2.60* 3.86 90 -0.60 867 2.07 -1.99 144 5.24 1.39 2.86* 3.80 100 -0.55 892 2.05 -1.95 148 5.19 1.40 2.87* 3.77 110 -0.52 929 2.03 -1.94 158 5.12 1.42 2.91* 3.76 120 -0.46 953 2.03 -1.95 172 5.07 1.49 3.09* 3.70 130 -0.41 975 2.03 -1.94 179 5.01 1.53 3.18* 3.67 140 -0.38 991 2.02 -1.92 184 4.96 1.54 3.24* 3.62 150 -0.33 1032 2.01 -1.91 196 4.92 1.59 3.35* 3.58 160 -0.31 1057 2.01 -1.90 220 4.89 1.59 3.36* 3.57 170 -0.28 1077 2.00 -1.87 236 4.85 1.59 3.39* 3.54 180 -0.26 1095 2.00 -1.86 245 4.81 1.59 3.46* 3.49 190 -0.25 1120 2.00 -1.84 260 4.78 1.59 3.47* 3.47 200 -0.22 1155 2.00 -1.82 265 4.74 1.59 3.49* 3.44
Each month stocks are classified into either a winner or loser portfolios based on their past performance in the ranking period. The ranking period is set to six months. A long position is taken in the winner portfolio (top ranked stocks) and a short position in the loser portfolio (bottom ranked stocks), which creates the winner-loser (W-L) portfolio. The stocks are equally weighted in the portfolio. Subsequently the portfolio is held for six months. The overlapping returns are calculated and averaged per month for the various portfolios sizes. The t-statistics show whether the returns of the W-L portfolios are significantly different from zero. Ret described the average monthly portfolio return. Size describes the number of shares in each winner and loser portfolio. Market cap (MC) describes the average market cap (in $ millions). Liquidity (Liq) describes the average percentage bid ask spread of the portfolio. SD shows the standard deviation of the portfolios. *,** and *** show significance at 1%, 5% or 10%.
15
portfolio size 150 there is minor deviation in returns and very little to gain when including more shares in the portfolio. The peak at portfolio size 156 seems arbitrary and could have been anywhere between size 150 and 200.
Secondly, Table 2 and Figure 1 show that returns increase the strongest over the first 40 portfolio sizes. At portfolio size 40, the average monthly return is 1.16% (t-statistic 2.22). This even is slightly higher than the 1.10% found by Jegadeesh and Titman (1993). At this portfolio size 72.05% of the aforementioned maximum momentum returns have already been realized. This implies that adding 116 shares to both the winner and loser portfolio (i.e., increasing to portfolio size 156) only gains an additional 28%. In perspective, one has to quadruple the amount of shares to gain the last quarter of maximum returns. An annualized return of 20.98% at portfolio size 40 indicates strong significant momentum returns at smaller portfolio sizes. Furthermore, at this portfolio size diversification decreased the standard deviation from 16.91% to 4.28% of the portfolio. At portfolio size 200, standard deviation decreases slightly more to 3.44%, but substantially less compared to the risk reduction over the first 40 portfolio sizes. Although portfolio size 30 shows interesting characteristics with 64.56% of the returns already realized and a standard deviation of 4.62%, the returns are only significant at 10%. Therefore I will focus on the portfolio size 40, which is significant at 5%. The results reveal that investors can already employ the momentum strategy with relatively small portfolios.
Lastly, as expected, returns are not significant until portfolio size 30. The results reveal very low returns between portfolio size 1 and 10, which can be understood following a theory
-4.0% -3.5% -3.0% -2.5% -2.0% -1.5% -1.0% -0.5% 0.0% 0.5% 1.0% 1.5% 2.0% 0 10 20 30 40 50 60 70 80 90 10 0 11 0 12 0 13 0 14 0 15 0 16 0 17 0 18 0 19 0 20 0 R e tu rn s Size W-L Winner Loser Winner, loser and winner-loser (W-L) returns for various portfolio sizes
Figure 1: Returns various portfolio sizes
16
sketched by Hong et al. (2000). Although smaller firms have lower information diffusion leading to higher momentum returns, they also have limited investor participation (i.e. thinner market making capacity), which may cause more pronounced supply-shock-induced reversals and thereby inhibiting underreaction.
4.2 Breakdown of returns
Siganos (2007) finds maximum returns at portfolio size 40, which is a substantially smaller size than the peak in my results at portfolio size 156. To understand the difference in portfolio sizes for maximum returns, I break down the W-L returns into winner and loser returns.
Interestingly, Table 2 shows that all the profits come from the loser portfolio. The minimum of the loser portfolio returns is at portfolio size 1 with an average monthly return of -3.42% and returns increase with every increase in portfolio size. Since the momentum strategy shorts the loser portfolio, negative returns have a positive contribution to the W-L portfolio returns. These results are similar to Siganos (2010). However, the results are more extreme than Jegadeesh and Titman (1993,2001) and Hong et al. (2000). They find that only the majority of the returns are generated by the loser portfolio. The trend of the loser portfolio is consistent with the theory that smaller portfolio sizes have higher momentum returns than larger portfolio sizes.
The maximum return (0.22%) of the winner portfolio is at the largest portfolio size. A minimum of -3.24% is, similar to the loser portfolio, found at portfolio size 1. However, note that a long position is taken in the winner portfolio and a short position in the loser portfolio. The winner portfolio’s profitability actually increases when increasing the size of the portfolio, whereas the loser portfolio’s profitability decreases. The winner portfolio gains over 3.02 percentage point between portfolio size 1 and 200, whereas the loser portfolio gains only 1.60 percentage point. The returns of the winner portfolio increase faster than the returns of the loser portfolio. This explains the peak of the W-L portfolio at a relatively large size. I find, which is addressed more thoroughly later on, that when excluding the financial crisis positive winner portfolios at larger portfolio sizes can be found. However, these are barely above zero and have a minimal contribution to the W-L portfolio.
4.3 Portfolio characteristics
The previous results show maximum momentum returns at portfolio size 156, but more than 2/3 of these returns are already achieved at portfolio size 40. This implies that one does not need large portfolios to generate strong momentum portfolios. Furthermore, the results show opposite behaviour in terms of profitability of the winner and loser portfolio to alteration of the portfolio size. The theory predicts that the level of information uncertainty influences the height of momentum returns. In this section I evaluate if this theory can explain the return patterns the W-L, winner and loser portfolio show over the various portfolio sizes.
17 0 200 400 600 800 1000 1200 1400 0 10 20 30 40 50 60 70 80 90 10 0 11 0 12 0 13 0 14 0 15 0 16 0 17 0 18 0 19 0 20 0 M ar ke t cap itali zation Size Winner Loser W-L
Average market capitalization
Figure 2: Average market cap over various portfolio sizes
Plotted is the average market cap in for the winner, loser and the W-L portfolios for the various portfolio sizes over the full sample. Average market cap is expressed in millions of dollars. Portfolios are created based on the performance of stocks during a six-month ranking period and subsequently hold for six months.
0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 0 10 20 30 40 50 60 70 80 90 10 0 11 0 12 0 13 0 14 0 15 0 16 0 17 0 18 0 19 0 20 0 Li q u id ity Size Winner Loser W-L Average liquidity
Figure 3: Average liquidity over various portfolio sizes
18
average more weighed to “information uncertain” stocks, given that the average market cap and liquidity of the W-L portfolios increase with every increase in portfolio size. The proxies for information uncertainty, liquidity and market cap, show an almost linear relation with every increase in portfolio size. The W-L portfolio has an average market cap and liquidity percentage of 171 million and 5.19% at portfolio size 1 and 710 million and 3.37% at portfolio size 200. For every increase in portfolio size the correlation between average returns and liquidity of the portfolio is -95.67%. For market cap and average returns the correlation is 91.77%. This implies that, inconsistent with Zhang (2006), the results show rising returns with decreasing information uncertainty. Substantial differences in how the winner and loser portfolio react to information uncertainty are the cause of this.
The results of the loser portfolios are consistent with the theory of information uncertainty. When discarding the fact that for the first 10 portfolio sizes the average market cap and liquidity bounce slightly up and down, which might be a result of randomness, the loser portfolio returns increase with every increase in information uncertainty. The market cap and liquidity of the loser portfolio increase with 229% and 39% and the returns increase with 47% between portfolio size 1 and 200. For every increase in portfolio size the correlation between average liquidity and average returns is an astounding -91.68%, whereas for market cap this is 69.93%. The winner portfolios show exactly the same pattern of increasing returns over every increase in portfolio size. The market cap and liquidity of the winner portfolio increase 343% and 25%, and the returns increase with 93%. The correlation between average liquidity of the portfolio and average returns is -92.00% and for market cap this is 93.99%. Although the correlation coefficients of the winner and loser have the same sign, their meaning is completely different because of one important reason, i.e., the strategy takes a long position in the winner portfolio and a short position in the loser portfolio. This means that every increase in liquidity percentage (implies lower liquidity) increases the profitability of the loser portfolio, but decreases the profitability of the winner portfolio. In other words, increased information uncertainty enhances underreaction for the loser portfolio, but inhibits underreaction for the winner portfolio.
19
4.4 Time period break-up
Several researchers state that momentum returns are affected by the time-period investigated. For instance, Jegadeesh and Titman (2011) show that momentum returns have diminished over the last decade. Daniel and Moskowitz (2013) argue that the momentum strategy significantly underperforms during crisis periods. They find that their loser portfolio gains an astounding 126% in the period March to May of 2009 alone. Considering these findings it is sensible to test whether my results remain robust over different sub periods. Figure 4 shows the average monthly returns of the W-L portfolios over the complete sample period for portfolio sizes of 5, 40 and 200. These sizes are chosen to evaluate how different levels of diversification influence momentum returns over the complete sample period.
The first thing that emerges from Figure 4 is, as reported by Daniel and Moskowitz (2013), the momentum strategy is highly volatile during crisis periods 2000-2002 and 2008-2009. This is for the greater part caused by the loser portfolio gaining strong positive returns in an upward moving market. During the bursting of the dot-com bubble and the financial crisis the momentum strategy experiences extreme negative returns with maxima of 28.89% and -31.03% for the size 5 W-L portfolio, -8.45% and -21.73% for the size 40 W-L portfolio and respectively -5.60% and -19.74% for the size 200 W-L portfolio. The different portfolio sizes also strongly differ in correlation with the S&P 500. For the portfolios 5, 40 and 200 the correlation with the S&P 500 is -7.66%, -9.42% and -12.76%. Consistent with Rey and Schmidt (2007) and Siganos (2010), momentum returns are negatively correlated with market returns (S&P 500). This indicates that momentum returns are mainly driven by the loser portfolio in bear markets. -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% 19 94 19 94 19 95 19 96 19 97 19 98 19 99 19 99 20 00 20 01 20 02 20 03 20 04 20 04 20 05 20 06 20 07 20 08 20 09 20 09 20 10 20 11 20 12 20 13 R e tu rn s Year 5 W-L 40 W-L 200 W-L W-L returns over time for portfolio sizes 5, 40 and 200
Figure 4: Returns portfolio sizes 5, 40 and 200
20
Figure 5 shows the W-L returns for various portfolio sizes over different sub periods. The sample period is broken up in two sets of 7 years and one of 6 years. The last period of 2008-2013 is chosen to fully include the financial crisis in the returns and the period 1994-2007; 2010-2013 is chosen to exclude the crisis. Figure 5 shows that the portfolios, except for the 2008-2013 period, perform consistent with earlier results. At portfolio size 40 the 1994-2000 period realized 86.52% of its peak with a return of 2.08% (t-statistic 4.45) compared to 67.73% and 1.25% (t-statistic 1.72) for the 2001-2007 period. The peak in returns is 2.41% (t-statistic 8.55) at size 156 for the 1994-2000 period and 1.84% (t-statistic 2.40) at size 184 for the 2001-2007 period. Again, returns stabilize from portfolio size 150 in any of the sub periods with minor deviations in returns when increasing the portfolio size.
As expected, during the crisis period 2008-2013 the momentum strategy shows substantial underperformance. Figure 5 shows that in the 2008-2013 period the returns for portfolio sizes 1 to 40 are very volatile with positive returns at size 1 and substantial negative returns between size 2 and 40. Although not shown in Figure 5, during the top of the financial crisis in 2008 and 2009 the W-L portfolio shows negative monthly returns for any size. The returns during the financial crisis are obviously influencing the overall results. The 1994-2007;2010-2013 period is therefore chosen to exclude the financial crisis from the total sample. The 1994-2007;2010-2013 period strongly outperforms the full sample period. The peak of 2.00% (t-statistic 5.87) at portfolio size 156 is substantially higher than the 1.60% found before. Also, at size 40 already 82.93% of this maximum return has been realized. Interestingly, at size
-1.5% -1.0% -0.5% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 1 10 20 30 40 50 60 70 80 90 10 0 11 0 12 0 13 0 14 0 15 0 16 0 17 0 18 0 19 0 20 0 Re tu rn s Size 1994-2013 1994-2000 2001-2007 2008-2013 1994-2007;2010-2013 W-L returns for different sub periods
Figure 5: Returns for different sub periods
21
20 already 68.10% of the maximum returns have been reached. The results show that the financial crisis, especially for smaller portfolios, negatively influenced the overall results.
The previous results show that altering the time period of the sample, especially when excluding the financial crisis, influences momentum returns. However, it does not change our previous conclusions. Over different sub periods we still observe that the most significant part of the maximum returns is already realized at portfolio size 40. Secondly, a portfolio size employing the 150 top and bottom shares is sufficient to realize maximum returns.
4.5 Alternative J and K periods
Although the six-month holding and six-month ranking period is the most extensively researched form of the momentum strategy, alternative periods are not uncommon. Also, Jegadeesh and Titman (1993) and Agyei-Ampomah (2007) find strong variation in momentum returns with alternative ranking and holding periods. I therefore test the robustness of my results with multiple holding and ranking periods.
Figure 6 present an overview of the returns of various portfolio sizes for multiple holding and ranking periods. I first keep the holding period constant at six months and vary between ranking periods of 3 and 12 months. Subsequently, I keep the ranking period constant at six months and vary between holding periods of 3 and 12 months.
Again we observe that the most substantial amount of returns is already realized at portfolio size 40. At W-L portfolio size 40 the percentage of the maximum significant returns achieved for the J6K3, J3K6, J12K6 and J6K12 portfolios are 72.77%, 59.24%, 58.56%, and 9.74% respectively. For the basic J6K6 portfolios this was 72.05%. Altering the holding period decreases the momentum profits realized at smaller portfolio sizes. However, almost 60% of the peak returns for the J3K12 and J12K6 portfolios at portfolio size 40 is still substantial. The J6K12 portfolio clearly is the outlier in Figure 6. The J6K12 portfolio shows negative returns up until the size 30 and strong increase in returns between portfolio size 30 and 100. However, substantial underperformance of the J6K12 portfolio is consistent with the results of Jegadeesh and Titman (1993), who found monthly returns of 0.90% for the J6K12 portfolio compared to 1.10% for the J6K6 portfolio. Thus, following the momentum strategy with limited portfolio size would not be beneficial when extending the holding period to 12 months.
22
If I break the W-L portfolios down, appendix table A1 shows that for any holding or ranking period the returns of the winner and loser portfolio show similar trends as in earlier findings. The winner portfolio substantially underperforms the loser portfolio. The loser portfolio’s profitability decreases with increasing portfolio sizes and the winner portfolios show the opposite pattern. Most noteworthy, for the J6K3 portfolios I observe positive returns for the winner portfolio at size 130, which keeps on rising to a maximum monthly return of 0.24% at size 190. It seems that we actually capture momentum returns for the winner portfolio with this holding and ranking period.
The results reveal that altering the holding period or reducing the ranking period does not materially change my previous results. We continue to observe the same return pattern. Firstly, we observe very low returns for the smallest portfolios. Secondly, a strong increase of returns between portfolio size 10 and 40. Thirdly, returns grow slowly towards a peak around portfolio size 150. From that point returns stabilize and show only minor deviation when adding more shares to the portfolios. However, extending the ranking period to 12 months has a strong negative influence, especially for smaller portfolio sizes.
4.6 Cuts on size and liquidity
My previous results show that strong momentum returns can already be realized at small portfolio sizes. However, these smaller portfolios show some extreme characteristics in the form of low market cap and illiquidity. In this section I will test the robustness of my results when
-2.0% -1.5% -1.0% -0.5% 0.0% 0.5% 1.0% 1.5% 2.0% 0 10 20 30 40 50 60 70 80 90 10 0 11 0 12 0 13 0 14 0 15 0 16 0 17 0 18 0 19 0 20 0 R e tu rn s Size J6K3 J6K6 J3K6 J12K6 J6K12 Figure 6: Returns various ranking and holding periods
Plotted are the monthly average returns for the W-L portfolios for various portfolio sizes for five different combinations of holding and ranking periods. J is the ranking period length in months and K is the holding period length in months
23
excluding either the smallest or most illiquid stocks. By doing so, I want to ensure that my results are not primarily driven by small and illiquid stocks.
Hong et al. (2000) reported that the smallest firms actually have negative momentum returns and I observe the same phenomenon in my results. Figure 7 shows the returns over various portfolios when adjusting the minimal market cap of stocks eligible for inclusion. The size thresholds are $20 million, $200 million and $1bn. $20 million is chosen to exclude the low return stocks as reported by Hong et al. (2000). $200 million and $1bn are chosen to test how returns evolve over the sizes. The results reveal, especially at smaller portfolio sizes, improved returns. At portfolio size 40, the size thresholds $20 million, $200 million and $1bn have realized 82.08%, 89.26% and 90.77% of their maximum returns. At this size the average return for the $20 million threshold is 1.52% (t-statistic 2.61), which is substantially higher than the 1.16% of the unrestricted momentum strategy. For all three thresholds stabilization commences in smaller portfolio sizes. For the threshold $20 million returns peak at size 148 with 1.85%, but were already 1.77% at size 80. The other two thresholds show even strong decreases in returns over the larger portfolio sizes. The presence of market cap thresholds increases returns for small portfolio sizes. Furthermore, these market cap thresholds limit the portfolio size needed for maximum returns.
Figure 8 shows the results over various portfolio sizes when excluding stocks based on liquidity decile breakpoints. These breakpoints were calculated by averaging the monthly decile
-1.5% -1.0% -0.5% 0.0% 0.5% 1.0% 1.5% 2.0% 0 10 20 30 40 50 60 70 80 90 10 0 11 0 12 0 13 0 14 0 15 0 16 0 17 0 18 0 19 0 20 0 R e tu rn s Size MC >0 MC >20m MC >200m MC >1000m
Plotted are the average monthly returns of the W-L portfolios for various portfolio sizes. Stocks are only eligible for inclusion in the momentum portfolio when their market capitalization (MC) at the end of the ranking period exceeded the following thresholds: (1) zero (2) $20 million (3) $200 million (4) $1bn
Returns W-L portfolios with cuts on market cap sizes
24
breakpoints over the total sample. For instance, the 10% liquidity decile excludes stocks when their liquidity at the end of the holding period is higher than the respective average breakpoint. Thus, the 10% decile excludes the most illiquid stocks. The results are, although less significant, similar to the previous results. The exclusion of the top 10%, 30% and 50% illiquid stocks result in 71.61%, 76.05% and 88.29% of the peak realized at portfolio size 40. Again we observe that the most substantial amount of the maximum returns is generated up until portfolio size 40. In absolute returns, only the exclusion of the top 10% illiquid stocks slightly increases the returns compared to the unrestricted sample. The 30% and 50% breakpoints show underperformance over the larger portfolio sizes compared to the unrestricted sample.
These results show that to the extent that momentum is driven by illiquid and small shares, this does not apply to the most extreme forms of both. Excluding stocks with extreme characteristics actually improve returns in smaller portfolio sizes. This may be a result of, as suggested by Hong et al. (2000), supply-shock reversals inhibiting underreaction in the smallest stocks. For larger liquidity and size thresholds returns even decrease in the larger portfolio sizes. Thus, when pursuing momentum returns in smaller portfolios, excluding the smallest stocks strongly increases momentum returns. However, for larger sized portfolios high market cap and liquidity thresholds decrease returns.
-1.0% -0.5% 0.0% 0.5% 1.0% 1.5% 2.0% 0 10 20 30 40 50 60 70 80 90 10 0 11 0 12 0 13 0 14 0 15 0 16 0 17 0 18 0 19 0 20 0 R e tu rn s Size Liq <0 Liq <10% Liq <30% Liq <50% Returns W-L portfolios with cuts on liquidity
Figure 8: Returns with sensitivity to liquidity
Plotted are the average monthly returns of the W-L portfolios for various portfolio sizeswhen
25
4.7 Alternative ranking methodology
Given that my economic story is all about finding ways for investors and researchers to take better advantage of momentum returns, it seems sensible to deviate from the standard structure of ranking. Several researchers suggest minor alterations to the basic momentum strategy to increase the strength of the strategy. Considering my results, an alteration proposed by Siganos (2007) is particularly interesting. He reports that removing the top and bottom 5% of the ranking actually improves returns.
Siganos (2007) argues that removing extreme returns actually enhances the momentum returns for a portfolio employing 40 winners and losers. As shown in Table 2 and Figure 1, I find continuously increasing returns between portfolio size 1 and 156. The momentum strategy includes stocks to the momentum portfolio based on ranking. The smallest portfolio size includes only the highest and lowest ranked stock and every increase in size means inclusion of stocks with less extreme ranking. Thus the increasing returns between size 1 and 156 can only mean that the lower ranked stocks increase the average return of the portfolio up until size 156. Therefore, considering Siganos (2007) and my results, it seems logical to create portfolios that actually exclude these extreme ranked stocks with relatively low momentum returns.
Figure 9 shows a moving average of the returns over the ranking of stocks for the W-L portfolio sizes 40, 60 and 80. For instance, a portfolio with maximum included rank 70 with portfolio size 40 includes the best and worst stocks ranked at the end of the ranking period
1.2% 1.4% 1.6% 1.8% 2.0% 2.2% 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 R e tu rn s
Maximum included rank
Size 40 Size 60 Size 80
Returns W-L portfolios for sizes 40, 60 and 80 with alternative inclusion method
Figure 9: Returns with sensitivity to ranking
26
between rank 30 and 70. As shown earlier, portfolio size 40 is the least amount to create a well-diversified and statistically significant portfolio. Most noteworthy, Figure 9 reveals that the strongest momentum returns can be found excluding the most high-ranked stocks. The returns of the size 40 portfolio show an upward trend towards a peak, containing stocks ranked between the 108th and 148th place, with a return of 2.11% (t-statistic 4.55). Hereafter, the returns show a gradual decline. All portfolios with size 40 show substantially improved returns compared to the unrestricted results for the same size. Similar results are found with portfolio sizes 60 and 80. In summary, stocks with extreme ranking have relatively low momentum. Excluding these extreme stocks increases, especially for the smaller portfolio sizes, momentum returns.
4.8 Transaction costs
A common issue in assessing the profitability of the momentum strategy is the inclusion of the transaction costs. Several researchers indicated that transaction costs completely destroy all momentum profits. Especially small and illiquid stocks are associated with disproportionally high transaction costs. Since limiting portfolio size will increase the weight on these small and illiquid stocks, it is sensible to test the economic significance of my results.
Figure 10 shows the results when including transaction costs based on a quoted spread measure plus commission. The most important takeaway is that for any size the W-L portfolios show negative monthly returns with a minimum of -2.26% at portfolio size 2 and a maximum of -0.05% at size 200. Transaction costs destroy, consistent with Lesmond et al. (2004) and Agyei-Ampomah (2007), any profits of the momentum portfolios. Compared to our earlier results without transaction costs, one can observe the strongest return reduction is in the smaller portfolio sizes. For instance, the average return reduction of the first ten portfolio sizes is 2.02%, while the average of all portfolio sizes is 1.75%. Although the difference of 0.27% seems insignificant, one has to note that this is the average monthly reduction. The reduction over the six-month holding period is therefore six times higher. The transaction estimation in this research uses a fixed commission cost and a variable quoted spread. The difference in transaction costs of the portfolio sizes can therefore be attributed to difference in quoted spreads.
27 -4.5% -4.0% -3.5% -3.0% -2.5% -2.0% -1.5% -1.0% -0.5% 0.0% 0 10 20 30 40 50 60 70 80 90 10 0 11 0 12 0 13 0 14 0 15 0 16 0 17 0 18 0 19 0 20 0 N e t R e tu rn s Size W-L W L 0.0% 0.5% 1.0% 1.5% 2.0% 0 10 20 30 40 50 60 70 80 90 10 0 11 0 12 0 13 0 14 0 15 0 16 0 17 0 18 0 19 0 20 0 Q u o te d S p re ad in % Size W L Net winner, loser and W-L returns for various portfolio sizes
Figure 10: Returns including transaction costs
Plotted are the monthly average net returns for the winner, loser and W-L portfolios for various portfolio sizes over the full sample. Portfolios are created based on the performance of stocks during a six-month ranking period and subsequently hold for six months. Transaction costs are estimated on basis of a quoted spread and commission costs
Figure 11: Quoted spreads
Plotted are the average monthly quoted spreads for the winner and loser portfolios for various portfolio sizes over the full sample. The quoted spreads per stock are divided by the number of months in the holding period to compare with the monthly returns.
28
considering their average market cap and liquidity. When we revert back to Figure 2 and 3, the market cap of the smallest sized winner portfolio is $250 million and its liquidity percentage is 2.6%, which increases to $1.2bn and 2.0% at portfolio size 200. However, the loser portfolio has an average market cap of $81 million and a liquidity percentage of 7.72% at size 1, which increases to $265 million and 4.74% at a portfolio size 200. Low market cap and especially low liquidity are the basic cause for high bid ask spreads and both are substantially lower for the loser portfolio for every size. Hence, the higher quoted spreads in the loser portfolios.
With the previous results in mind, it seems sensible to evaluate how excluding these high cost stocks will affect our net results. In Figure B1 and B2 from Appendix B, I included the alterations made in Subsection 4.6 with transaction costs. These alterations excluded stocks with extreme characteristics that are, as my results show, associated with high transaction costs. The results show that these alterations make the momentum strategy profitable, even at the more extreme sizes. However, the returns are never significant. Thus Figure B1 and B2 are solely to illustrate that profitable returns are possible with extreme sizes.
Although including transaction costs seem to deteriorate all the momentum profits, one must bear in mind that the debate concerning transaction cost estimation of the momentum strategy is still ongoing. Adjusting the transaction cost estimate to the one-way 0.5% cost used by Jegadeesh and Titman (1993) would substantially improve the results. Also, Lesmond et al. (2004) show that the quoted spread estimate is significantly higher than for instance a direct effective spread estimation. Lastly, excluding the smallest and illiquid stocks substantially improves post-cost returns especially for smaller portfolio sizes.
5. Conclusion
This thesis examines a relationship which is mostly neglected by the literature so far, i.e., the influence of momentum portfolio size on the magnitude of returns. Most of the research regarding the momentum strategy is focused on understanding why the strategy works, but overlooks the fact that the success of an investment strategy greatly depends on whether it is implementable by a wide range of investors. The conventional momentum strategy requires trading in large portfolios, which may prevent many private investors from pursuing momentum returns.
29
employing only a limited amount of shares. Another conclusion is that the marginal extra return gained by increasing the portfolio size quickly diminishes after the smallest portfolio sizes.
Contrary to many studies, I find that all the momentum profits for any portfolio size come from the loser portfolio, whereas the winner portfolio has negative returns for any size. Besides that, the winner and loser portfolio show distinctly different return patterns over the various portfolio sizes. The winner portfolio’s profitability rises with every increase in portfolio size whereas the loser portfolio shows exactly the opposite. Every increase in winner or loser portfolio size increases the average market cap and liquidity, which are proxies for information uncertainty. Thus, the winner portfolio benefits from less information uncertainty whereas the loser portfolio benefits from more information uncertainty. I conclude therefore that information uncertainty enhances underreaction for the loser portfolio, but inhibits underreaction for the winner portfolio.
My results are robust for different time periods and various holding and ranking periods. The momentum strategy, and especially the smallest portfolio sizes, perform very badly during crises and returns improve significantly when excluding the financial crisis. I also find that extending the holding period will strongly decrease momentum profits for smaller portfolio sizes.
When testing with alternative cuts on ranking, size and liquidity very interesting results are produced. Excluding either the smallest companies or stocks with extreme ranking substantially increases returns. Especially the smaller portfolio sizes seem to benefit from these alternative methodologies. Thus, to the extent that information uncertainty enhances momentum returns, this does not apply for stocks with extreme characteristics. However, in this research this is only part of the robustness test and therefore not extensively investigated. Researching this phenomenon more thoroughly might yield some interesting results and shed some light on the general implementation of this variation.
Transaction costs prevent profitable strategy execution for any size. My research shows that decreasing the portfolio size substantially increases the average transaction costs per share. However I want to make three remarks regarding these results. Firstly, the quoted spread estimate used in this research is relatively high compared to for instance a direct effective spread estimate. Secondly, I assume a 100% turnover of stocks in the portfolio, which in reality is not true. Thus, not closing these positions would decrease transaction costs. Lastly, I show that profitable execution is possible when adjusting the stocks eligible for inclusion. I find that excluding the smallest and illiquid stocks substantially improves post-cost returns especially for smaller portfolio sizes.
30
results may not be applicable to other smaller stock universes. For instance, portfolio size 40, which is considered small in this research, is relatively large when employed on the Dutch stock universe. Thus for further research, one may want to extend my work to other stock universes and use other ways to define portfolio sizes.
