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The effect of idiosyncratic volatility on
momentum returns
Thesis MSc Finance
Abstract
I investigate the relationship between the returns of momentum strategies and idiosyncratic volatility. I find excess returns of 2,14% per month of momentum portfolios with high idiosyncratic volatility over momentum portfolios with low idiosyncratic volatility. This difference between the average retuns on high and low idiosyncratic volatility momentum strategies is primarily caused by the underperformance of stock with low past-returns and high idiosyncratic volatility. This finding is in line with the results of Ang et al. (2006) and Arena et al. (2008). My results support the view of Schleifer and Vishny (1997) and Pontiff (2006) who conclude that idiosyncratic volatility is an important limit to arbitrage. This finding can help to explain the persistence of momentum returns in the financial literature. Because the returns on momentum strategies seem difficult to arbitrage away. The results also support the behavioural explanations of the existence of the momentum anomaly from
Barberis et al. (1998) and Daniel at al. (1998).
Keywords: price momentum, idiosyncratic volatility, limits to arbitrage, asset pricing, 3-factor asset pricing model
JEL Classifications: G12, G14
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1. Introduction
Ever since the development of Modern Portfolio Theory by Harry Markowitz (1952) and the development of the Capital Asset Pricing Model (CAPM) many financial research is carried out to describe the cross-section of expected returns. Eugene Fama and Kenneth French (1993) greatly improved on the CAPM by adding a size and value factor to the CAPMs single market factor and calculating alphas by applying the 3-factor model has become standard practice in the asset pricing literature (Blitz et al., 2016). More recently, Fama and French (2015) extended their 3-factor model to a 5-factor model adding a factor for profitability or RMW (robust minus weak) and investment or CMA (conservative minus aggressive). This new model aims to explain some prominent and pervasive patterns in the cross-section of average stock returns that their 3-factor model could not explain. Their 5-factor model is likely to become the new benchmark in asset pricing in the coming years (Blitz et al., 2016). However Blitz et al. (2016) address some concerns about this new 5-factor model.
One concern is the absence of a momentum factor (WML, winners minus losers), which was first found by Jegadeesh and Titman (1993) and added as a factor to the Fama and French 3-factor model by Carhart (1997). Jegadeesh and Titman (1993) find that firms with high returns over the past three to twelve months continue to outperform firms with low past returns over the same period. De Bondt and Thaler (1985, 1987) find return reversals over longer horizons. Stock with low three- to five-year past returns earn higher average returns than stock that performed better over this period. The momentum effect has been one of the most studied asset pricing anomalies in the financial literature. Although the evidence for the momentum anomaly is widely accepted, researchers still did not find consensus on what drives this cross sectional pattern in stock returns.
This paper wants to contribute to the current literature by exploring the relationship between the momentum anomaly and idiosyncratic volatility (Ivol). I will do this by following a relatively recent string of literature started by Arena et al. (2008) and McLean (2010) and more recently by Cheema and Nartea (2017). This thesis contributes to the literature by finding evidence on the relationship between momentum returns and idiosyncratic volatility (IVol) in a more recent and smaller sample from the Euronext Amsterdam Stock Exchange.
The main research question of this thesis is “Do stocks with higher idiosyncratic volatility
earn higher momentum returns?”
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models, markets are inefficient and new information is only slowly incorporated into security prices. So investors underreact to new firm-specific information. This leads to momentum in security returns over periods from three to twelve months. When more firm-specific
information of a firm is released, it is also likely that this firm has higher idiosyncratic volatility. Therefore, it is likely that firms with higher idiosyncratic volatility experience higher momentum returns.
Schleifer and Vishny (1997) and Pontiff (2006) conclude that idiosyncratic volatility is the primary holding cost for arbitrageurs. Therefore, it is an important limit to arbitrage. When firms with higher idiosyncratic volatility earn higher momentum returns, it could be that a momentum anomaly for these kind of firms is more difficult to arbitrage away. It could explain why the momentum effect is still observed in recent data more than twenty years after its discovery by Jegadeesh and Titman (1993).
I follow the methodology of creating quintile portfolios based on past returns. This
methodology is widely used in the momentum literature. Secondly, I use the methodolgy used by Arena at al. (2008) and Cheema and Nartea (2017) to create double-sorted portfolios based on past returns and idiosyncratic volatility.
This thesis will proceed as follows. In section 2, I review the existing literature on momentum returns, the cross-sectional relation between idiosyncratic volatility and expected returns and the relation between momentum returns and Ivol. Section 3 will explain the data and
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2. Literature review
2.1. Momentum
The literature on momentum is very extensive. Jegadeesh and Titman (1993) were the first to examine this momentum pattern is stock returns in the United States. Contrary to research by de Bondt and Thaler (1985, 1987) who find long-term return reversals in three to five year periods, they find evidence of continuation in expected returns in periods ranging from from three to twelve months. The methodology followed by Jegadeesh and Titman (1993) is straightforward. At the end of each month, stock are ranked and put into ten decile portfolios based on their past J-month return (J is 3, 6, 9 or 12). The bottom decile portfolio is the loser portfolio and the top decile portfolio is the winner portfolio. All stock have a return history of at least 12-months. Portfolios are equally weighted and held for k-months (k is 3, 6, 9 or 12) and are not rebalanced during this time.
Using the methodology of Jegadeesh and Titman (1993), Rouwenhorst (1998) also finds that an internationally diversified zero-cost momentum portfolio that invests in past medium-term winners and short sells past medium-term losers earns approximately 1 percent per month. These momentum returns are present in all twelve European markets in his sample. After controlling for country-specific risk factors and for both size and country-specific risks, the momentum effect is still present in all twelve individual countries. Rouwenhorst (1998) finds for all combinations of past J-month returns and k-month holding periods a winner-loser spread of around 1 percent. An interesting finding is that regardless of the interval used for portfolio formation, average returns tend to fall for longer holding periods.
Focusing on the 6-month ranked returns (J = 6) and 6-month holding period (K = 6) portfolio, Rouwenhorst (1998) finds that the standard deviation across the past return decile portfolios is U-shaped, with the loser and winner portfolios having a higher volatility than portfolios in the middle deciles. Momentum decile portfolios created by Blitz and van Vliet (2007), show a similar U-shaped pattern in volatility where winner and loser portfolios have higher standard deviations than the middle decile portfolios and volatilty is skewed towards the loser
portfolios, so loser portfolios are more volatile than winner portfolios.
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past losers have a smaller size than past winners. Suggesting the momentum premium cannot be due to systematic market risk or a size premium. Using subsamples stratified on market beta and firm size and using market risk-adjusted returns doesnt seem to have much effect on the momentum premium. These results further confirm that the momentum risk premium must be due to firm-specific risk. Consistent with my hypothesis, because the momentum effect must be due to firm-specific or idiosyncratic risk, it is plausible that stocks with higher idiosyncratic volatility earn higher momentum returns.
Rouwenhorst (1998) also concludes, after examining the market beta of the decile portfolios that the excess returns of zero-cost momentum portfolios is unlikely to be explained by market risk. He also finds the average firm size for both the winner and loser portfolio to be smaller than the average firm size in the whole sample. Consistent with Jegadeesh and Titman (1993) losers seem to be smaller than winners and have a stronger positive loading on the size factor. However, size-neutral portfolios have a negative size loading, suggesting that losers behave more like small stock irrespective of their size (Rouwenhorst, 1998). He raises the question whether the momentum effect is only limited to smaller stocks. However, using size-neutral zero-cost momentum portfolios, Rouwenhorst (1998) finds a momentum effect within and among all size decile portfolios. Interestingly the momentum effect seems larger for smaller firms.
Further analysis by Jegadeesh and Titman (1993) whereby the zero-cost momentum portfolio returns of each month (1 until 36) after portfolio formation are examined show that the zero-cost momentum portfolio earns positive excess returns from month 2 through 12 where the cumulative excess returns reach a maximum of 9,5% at the end of 12 months after portfolio formation. The zero-cost momentum portfolio returns in years 2 and 3 (month 12 through 36) are insignificantly negative. Rouwenhorst (1998) finds that looking at event time, zero-cost momentum portfolio returns are significantly positve in the first 11 months after portfolio formation, after which they turn negative. The results of Jegadeesh and Titman (1993) for the U.S. market and the results of Rouwenhorst (1998) for twelve international European markets are strikingly similar.
Chan et al. (1996) try to find the sources of predictability of future stock returns based on past returns. They make a distinction between earnings momentum and price momentum. Earnings momentum is based on earnings surprises and may benefit from market underreaction related to specific events like earnings announcements. While price momentum is based on past returns and may be caused by a broader set of information. They find zero-cost price
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lowest en highest decile portfolios. Although the price momentum effect seems stronger and longer lived than the earnings momentum effect. They conclude that prior returns as well as all three earnings suprise variables have marginal predictive power for the postformation drifts in returns and each momentum strategy thus draws upon the market`s underreaction to different pieces of information (Chang et al., 1996).
Asness et al. (2013) find return premia for momemtum strategies in eight different asset classes across four markets. These are the United States, the United Kingdom, continental Europe and Japan. Although the momemtum premium in Japan for individual equities they find to be statistically insignificant. Consistent with Rouwenhorst (1998) they find that momentum strategies in one market have a positive correlation with momentum strategies in the other markets.
Two papers who propose a model using a behavioural explanation to explain the momentum pattern in stock returns are from Barberis et al. (1998) and Daniel et al. (1998). Daniel et al. (1998) present a model of underreaction and overreaction based on investor overconfidence and biased self-attribtion. The model of Barberis et al. (1998) focuses on underreation and overreaction caused by the representativeness heuristic and conservatism bias. Both models are based on the efficient market hypothesis, which states that all relevant information should be incorporated and reflected in security prices. When stock experience positive excess returns after good news, it means the market has underreacted to this news. Stock prices are then slowly corrected for this news in the subsequent period, leading to higher returns following good news. Similarly, there is underreaction by the market if there is an announcement of bad news and this leads to lower or negative excess returns in the
subsequent period. Overreaction occurs over the long run following a period with subsequent good news or bad news. After a series of good (bad) news announcements, investors might get overly optimistic (pessimistic). This will eventually result in securities being overvalued (undervalued) and a subsequent correction and reversion in prices. According to these models, underreaction explains the momentum effect over horizons from three to twelve months, while overreaction explains the reversal of security returns over periods from three to five years.
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Hong and Stein (1999) further assume that private information only diffuses gradually across the population of newswatchers.
Blitz et al. (2011) and Blitz and Vidojevic (2017) claim that the momentum risk factor has significant time-varying risk exposure to systematic factors like the market beta and the SMB size factor. For example, a momentum strategy tends to select winner stock with high market beta in an increasing bull market and consequently this momentum portfolio will perform poorly during market reversals. They create a new momentum risk factor, which they call residual or idiosyncratic momentum. Residual momentum exhibits smaller time-varying factor exposures; therefore, it reduces the volatility of the momentum strategy and doubles the Sharpe ratio compared to traditional total return momentum. Blitz et al. (2011) and Gutierrez and Prinsky (2007) also find that while total return momentum profits revert at horizons beyond one year, residual momentum strategies continue to generate positive returns. This thesis wants to contribute to the previous literature of the momentum effect by focusing on the Dutch equity market and the relationship between momentum returns and idiosyncratic
volatility. Therefore, I will focus on the total return momentum factor instead of a residual momentum factor.
2.2. Momentum and idiosyncratic volatility
The Capital Asset Pricing Model (CAPM) assumes investors are only rewarded for bearing systematic risk when markets are complete, frictionless and in equilibrium. Investors don’t get rewarded for bearing idiosyncratic risk as this can be diversified away. However, the
Intertemporal Capital Asset Pricing Model (ICAPM) of Merton (1987) shows that markets are incomplete and investor’s dont hold perfectly diversified portfolios. Therefore, idiosyncratic risk should be rewarded. Although the cross-sectional relationship between expected returns and idiosyncratic risk should be either flat or positive according to these models, it has received some mixed evidence in the financial literature.
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that stock with low past IVol outperform stocks with high past IVol and this low-volatility effect cannot be explained by exposure to the market, size or value factor.
Ang et al. (2006) also investigate if the low-volatility effect can be explained by momentum returns. They create portfolios double-sorted on IVol and momentum. First stock are sorted based on past-returns. Then within each past-return quintile portfolio, they sort stock based on idiosyncratic volatility. In this way, they create quintile portfolios sorted by IVol that control for past returns. Controlling for past one, six and twelve month returns still produces
statistically significant negative FF-3 risk adjusted average returns. Therefore, momentum returns cannot explain the low-volatility anomaly. They also find that the low-volatility effect is stronger among loser stock.
Ang et al. (2006) find that momentum returns are asymmetric across the IVol sorts. Among stock with low idiosyncratic volatility, the returns from past losers towards past winners are symmetrical, but for stock with higher idiosyncratic volatilities, the momentum returns become highly skewed towards loser stocks. This result implies that momentum returns can be improved by short selling past losers with high past idiosyncratic volatility.
In there follow-up study Ang et al. (2009) find further evidence on the negative relationship between idiosyncratic volatility and expected returns. They find evidence for this in 23 developed markets. Using various cross-sectional regressions, they find a statistically significant negative relationship between expected returns and IVol for each individual G7 country as well as for various different poolings on developed countries. They conclude that the effect is strongest in the U.S. market, but it is still statistically significant in poolings excluding the U.S. market. Their results appear to be robust after using different formation periods and when using value-weighted portfolios instead of equal-weights. They also create quintile portfolios of stocks sorted on their IVol. They find statistically significant FF-3 risk-adjusted spreads between the highest and lowest IVol portfolios in the samples of Europe, G7 countries and all 23 countries. Spreads based on raw returns are only statistically significant when U.S. stocks are included.
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Many other explanations have been proposed to solve the puzzling negative relationship between idiosyncratic volatility and subsequent expected returns found by Ang et al. (2006, 2009). Hou and Loh (2016) provide a simple unified framework to evaluate a large number of candidate explanations to the puzzle. They conclude that many existing explanations explain less than 10% of the idiosyncratic volatility puzzle. When taken together, all existing
explanations still leave a sizeable portion of this puzzle unexplained.
This thesis wants to contribute to the literature by examing the effect of idiosyncratic
volatility on momentum returns. Arena et al. (2008) investigate the relationship between price momentum and idiosyncratic volatility. They find that high IVol stock have greater
momentum returns than low IVol stock. High IVol stock also experience greater and larger reversals. Similar to previous results they find a U-shaped pattern of volatility among the ten past return decile portfolios. Where the loser portfolio has IVol of 31,59% and the winner portfolio has IVol of 14,18%, the median portfolio only has an IVol of only 9,24%. They also find smaller market capitilization for the winner and loser portfolios. The winner and loser portfolio also experience higher turnover and have higher market betas. Financial distress risk (measured by Altmans Z-score) also increases from past winner to past loser portfolios.
They divide momentum returns of ten past-return decile portfolios in three categories of IVol (low, medium and high). They find an increase in winner minus loser spreads from low IVol to high Ivol portfolios. The difference between the high IVol and low IVol portfolio is 0,88% per month or 10,56% per year and this result is statistically significant. Consistent with the results of Ang et al. (2006) they find their result to be primarily driven by underperformance of high IVol losers. They find that the effect stays statistically significant after controlling for size, share price, turnover, beta, price delay and financial distress risk.
When looking at long-horizon momentum returns, it seems that high IVol stocks experience quicker and larger reversals after 12 months. The spread between high IVol and low IVol momentum returns is positive and statistically significant in the first year after portfolio formation, but becomes negative in the following four years. High IVol stocks also experience larger momentum return reversals in the four years after portfolio formation.
The results of Arena et al. (2008) support the behavioural models presented by Barberis et al. (1998) and Daniel et al. (1999). As I have discussed, the models present a theoretical
explanation for the existence of the momentum anomaly. They focus on the underreaction to firm-specific information as a result of behavioural biases and heuristics. If momentum
returns can be explained by the underreaction to firm-specific information, it naturally follows that firms that are more sensitive to firm-specific information should experience higher
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likely to have higher idiosyncratic volatility. Thus, a firm with more firm-specific information will have ceteris paribus higher idiosyncratic volatility and experience more price momentum (Arena et al., 2008). Therefore, the positive relationship between momentum returns and idiosyncratic volatility provides some evidence for the behavioural models of Barberis et al. (1998) and Daniel et al. (1998).
Arena et al. (2008) also conclude their results show that IVol is an important factor in limiting the successful arbitrage of the momentum anomaly and provides an explanation why the momentum anomaly is still present in data that is more recent. The model of Shleifer and Vishny (1997) identifies (idiosyncratic) volatility as the primary arbitrage holding cost and thus an important limit to arbitrage. As arbitrageurs are poorly diversified, they are exposed to excess firm-specific risk. Therefore, they tend to avoid stocks with high Ivol. If arbitrageurs truly avoid stocks with higher Ivol, it is likely that the momentum anomaly is more difficult to be arbitraged away for stock with higher Ivol and we would expect these stocks to exhibit higher momentum profits.
McLean (2010) further investigates whether Ivol can explain the persistence of both the long-term reversal effect and the short-long-term momentum effect. Arbitrageurs will trade on a
mispricing only to the point that the marginal benefit of a position is equal to its cost and idiosyncratic risk is seen as the primary arbitrage holding cost according to Schleifer and Vishny (1997) and Pontiff (2006). McLean (2010) uses this argumentation to find evidence for a relation between momentum returns and idiosyncratic volatility. Contrary to Arena et al. (2008), he finds no significant relation between momentum returns en idiosyncratic volatility. McLean (2010) shows that the results of Arena et al. (2008) are derived from a biased sample excluding small size and low-priced stocks. Their sample selection criteria eliminates most of the high Ivol stocks in their sample. Therefore I dont use any restrictions on my sample.
Cheema and Nartea (2017) extend the literature on momentum returns and idiosyncratic volatility by focusing on the Chinese stock market. They also do not find a significant positive relation between momentum returns and idiosyncratic volatility consistent with McLean (2010). Their results provide important out-of-sample evidence from the Worlds largest emerging market, considering the mixed evidence from Arena et al. (2008) and McLean (2010) on U.S. data. This thesis also contributes to the current literature by providing out-of-sample evidence for a positive relation between momentum returns and idiosyncratic
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3. Data and Methodology
In this section, I will give information about the used data and I will explain the methodologies used for this research.
3.1. Data and survivorship bias
Data on monthly stock prices are collected from Thomson Reuters Datastream. I restrict the investment universe to stock that are traded on the Euronext Amsterdam Stock Exchange. The investment universe consists of all constituents of the AEX All Share index from the period September 2002 until March 2017. The total sample includes 266 companies. To control for survivorship bias both surving and non-surving stock are included in the sample. On
Datastream this can be accomplished by executing requests of the constituents of the AEX All Share index for every month within the sample period. I request the MNOMIC code of the AEX All Share index, followed by the month and year. For example, the MNOMIC code LNLALSHR0607 gives all constituents of the AEX All Share index for June 2007. Combining all monthly constituents and removing duplicates results in a total of 266 companies. The total number of stock available each month ranges from 181 to 125. Stock that are delisted before or during the portfolio formation period are removed from the investment universe at that point in time.
3.2. Momentum portfolios
To find evidence for momentun returns on the Euronext Amsterdam Stock Exchange I use the methodology used by Jegadeesh and Titman (1993, 2001) and Rouwenhorst (1998). I sort stock based on their past J-month returns and create quintile portfolios based on these returns. I hold these quintile portfolios for K-months. I use portfolios with a 6-month formation period and a 6-month holding period. I use overlapping portfolios and portfolios are not rebalanced during the 6-month holding period. At the beginning of month t, stock are ranked in ascending order based on their average return over the past 6 months and placed in five equally weighted quintile portfolios. The top quintile portfolio (P5) is called the winner portfolio and the
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Because of a few extreme results regular past 6-month stock returns are calculated instead of log returns. The stock returns are calculated as follows:
𝑅𝑠𝑖,𝑡 = 𝑃𝑠𝑖,𝑡−1𝑃− 𝑃𝑠𝑖,𝑡−7
𝑠𝑖,𝑡−7 (1)
Where 𝑅𝑠𝑖,𝑡 is the return of stock i at the beginning of month t, 𝑃𝑠𝑖,𝑡−1 is the price of stock i at the beginning of month t-1 and 𝑃𝑠𝑖,𝑡−7 is the price of stock i at the beginning of month t-7.
The return of a quintile portfolio is calculated as the geometric mean of the average semi-annual returns of the stock that are included in the quintile portfolio.
3.3. Hypothesis
Momentum portfolio returns are calculated by creating zero-sum portfolios. A zero-sum portfolio is created by buying the winner portfolio and short selling the loser portfolio. Returns on these momentum portfolios must be statistically significantly different from zero. This leads to the following null and research hypothesis:
Null hypothesis: Returns on zero-sum momentum portfolios are equal to zero.
Research hypothesis: Returns on zero-sum momentum portfolios are statistically significantly different from zero.
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3.4. Idiosyncratic volatility
Following the example of Cheema and Nartea (2017) I use a simple market model to estimate the idiosyncratic volatility of individual stocks. IVol is measured from the residuals estimated using the following regression:
𝑟𝑖,𝑡 = 𝛼𝑖 + 𝛽𝑖 𝑟𝑚𝑡+ 𝜀𝑖,𝑡 (2)
Where 𝑟𝑖,𝑡 is the monthly return on stock i at the beginning of month t, 𝑟𝑚𝑡 is the monthly return on the AEX All Share total return index at the beginning of month t and 𝜀𝑖,𝑡 is the error
term. The above regression equation is estimated for each stock on the portfolio formation date using data over the previous twelve months. Then idiosyncratic volatility is calculated as the standard deviation of 𝜀𝑖,𝑡:
𝐼𝑉𝑜𝑙 = √𝑣𝑎𝑟(𝜀𝑖,𝑡) (3)
3.5. Momentum returns and idiosyncratic volatility
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3.6. Risk-adjusted returns
The methodology described above measures raw momentum returns. To test if these
momentum returns are not caused by any other known asset pricing anomaly, the momentum returns need to regressed on relevant asset pricing factors. The most widely used factors are the market, size and value factors from the Fama and French three-factor model (1993). As a robustness test, I will regress the momentum returns for each and between each IVol portfolio on the market, size and value factor. This will show if the effect of IVol on momentun returns can be explained by any of these factors. This will result in the following time-series
regression using monthly returns:
𝑟𝑖,𝑡 = 𝛼𝑖+ 𝑏𝑖 (𝑟𝑚,𝑡 − 𝑟𝑓,𝑡) + 𝑠𝑖 𝑆𝑀𝐵𝑡+ ℎ𝑖 𝐻𝑀𝐿𝑡+ 𝜀𝑖,𝑡 (4)
Where 𝑟𝑖,𝑡 is the monthly return for zero-sum momentum portfolio i at time; 𝑟𝑓,𝑡 is the
risk-free rate at time t. To estimate the risk-risk-free rate I use 10-year Dutch government bond indices. I take the geometric mean of the monthly returns on the total return index of three different bond indices. 𝑟𝑚,𝑡 is the monthly return on the price index of the AEX All Share index. 𝑆𝑀𝐵𝑡 and 𝐻𝑀𝐿𝑡 are the monthly returns on the Fama-French size and value factors respectively and
𝜀𝑖,𝑡 is the error term. Returns on the Fama-French size and value factors are collected from
Kenneth French’s website1. Unfortunately, data specified on Dutch equities is not available.
Therefore, I use data on the European three factors. This dataset includes data on the Netherlands.
Both heteroscedasticity-consistent (HC) standard errors of Huber (1967) and White (1980) and heteroscedasticity-autocorrelation-consistent standard errors of Newey and West (1987) are used to control for both heteroscedasticity and autocorrelation. It is also assumed that the regression residuals follow a normal distribution. To test for normality in the residuals I perform a Jarque-Bera test. The results are presented in appendix B. None of the regression residuals seems to follow a normal distribution.
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4. Main results
In this section, I will present my main results. The first part describes the characteristics of the past-return quintile portfolios and the zero-cost momentum portfolios. The second part sets outs the returns on the single-sorted momentum portfolios. I also divide the sample into subsamples based on different periods before, following and at the time of the financial crisis of 2007-2008 and the European debt crisis that started in 2009. The third part presents the results of the Ivol double-sorted momentum portfolios. Lastly, the last section presents the results of the Fama-French risk-adjusted momentum returns.
4.1. Portfolio characteristics
First, I will discuss the characteristics of the past-return quintile portfolios and the zero-cost momentum portfolios. Descriptive statistics are presented in table 1. We can see that the mean and median semi-annual returns of the zero-sum momentum portfolios are monotonically increasing from the low to the high Ivol portfolio. The mean (median) semi-annual returns for the low, medium and high Ivol-sorted portfolios are 3,4% (4,2%); 8,4% (7,4%) and 14,7% (13,9%) respectively. These results are already suggesting a positive relationship between momentum returns and idiosyncratic volatility.
For the winner portfolios P5 the mean and median semi-annual average returns are increasing from the low to the medium Ivol portfolio from 5,2% to 6,9% for the mean and from 6,4% to 8,9% for the median. Then in the high Ivol portfolio the mean and median returns drop to 3,5% and 4,3% respectively. For the loser portfolios P1, the mean and median semi-annual average returns are monotonically decreasing in the Ivolsorted portfolios. From 1,9% to -1,5% and -11% for the mean values and 3,8% to -0,1% and -12% for the median values. In a next section, I will draw some conclusions about this large drop of the mean and median semi-annual average returns on the high Ivol loser portfolio. On both the single-sorted
portfolio and the double-sorted Ivol portfolios semi-annual average returns are increasing over the past-return quintiles.
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I also test if the returns on the portfolios follow a normal distribution. I use the Jarque-Bera (JB) test statistic to see if the portfolio returns indeed follow a normal distribution. Under the null hypothesis of the JB-statistic, the sample follows a normal distribution. However, as shown in table 1, we have very low p-values for the JB test-statistic in most of the portfolios. Therefore, the null hypothsis has to be rejected. Which means most portfolios are not
normally distributed. Only for the portfolios with high Ivol and high past-returns, the JB null hypothesis is not rejected. So only high Ivol catgory past-return portfolios P3, P4 and P5 seem to be normally distributed.
4.2. Momentum returns
Table 2 presents the average semi-annual returns on the quintile portfolios sorted on past 6-month returns for different sample periods. The last row presents the returns of the zero-sum momentum portfolio. This is the return on a portfolio, which buys the stock in the winner portfolio and short sells the stock in the loser portfolio. The first column of table 1 presents the results on the past-return quintile and momentum portfolios over the full sample period from September 2002 until March 2017. The average return on the zero-sum momentum portfolio is 11,8%. This result is highly statistically significant with a t-statistic of 8,478. The returns on the quintile portfolios are monotonically increasing from the lowest to the highest past-return quintile. The table shows that the return on the momentum portfolio is asymmetric in the loser quintile. Loser stock lose on average -8,6% over the following 6 months after portfolio formation, while winner stock gain on average 3,2% over the following 6 months. The average returns on the winner portfolio and loser portfolio are statistically significant at the 1% and 5% respectively.
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Table 2: Zero-sum momentum portfolio returns for different periods. The table presents semi-annual returns. Portfolios are equally weighted. All stock are from the Euronext Amsterdam Stock Exchange. The sample contains constituents from the AEX All Share index. The first column contains average portfolio returns over the full sample period from September 2002 until March 2017. The last three columns contain average portfolio returns for different subsamples of 5-year periods. Portfolio P1 to P5 are the past-return quintile portfolios sorted on their past 6-month returns. Where P1 contains stock with lowest past 6-month returns and P5 contains stock with the highest past 6-month returns. All portfolios have a holding period of 6 months. Portfolio P5-P1 is a zero-cost momentum portfolio. It has a long position in stock from portfolio P5 and a short position in stock from portfolio P1. T-statistics are reported in parentheses. The critical values for the t-statistics are calculated with a Bonferroni-correction and using the appropriate number of degrees of freedom. *, ** and *** denote statistical significance of the test-statistic at the 10%, 5% or 1% level respectively.
The third column presents the returns of the past-return quintile portfolios during the bear market caused by the financial crisis in the 2007-2008 period and shortly followed by the Eurozone crisis in 2009. Not suprisingly the average returns on all quintile past-return portfolios are negative. While losers lost on average -18,6% semi-annually, the winners lost only -6% on average semi-annually. After a Bonferroni-correction, only the returns on the loser portfolio are statistically significant. In these extreme market conditions, winner stock outperformed loser stock by 12,6% on average. This result is highly statistically significant with a t-statistic of 6,545.
The last column shows the semi-annual average returns of the past-return quintile portfolios during the most recent period starting January 2012 and ending March 2017. Although the Eurocrisis continued after 2012, markets where mostly recovering from the losses of the financial crisis. In this period winner stock outperformed loser stock by 13,7% on average. These momentum returns are also highly statistically significant with a t-statistic of 6,828. For the most recent subperiod, the returns are also increasing from the lowest past-return quintile portfolio to the highest past-return quintile portfolio. While the worst performing stock lose on average -8% the following 6 months, the best performing stock gain on average 5,7% over the following 6 months. Only the returns on the three highest past-return quintile portfolios are statistically significant.
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4.3. Momentum returns and idiosyncratic volatility
This section outlines the relationship between momentum returns and idiosyncratic volatility. Firstly, I sort stock based on their idiosyncratic volatility. Ivol is estimated from the market model regression of equation 2. It is estimated as the standard deviation of the residual from equation 2 using monthly data over the twelve months prior to portfolio formation following equation 3. Next, I sort stock in ascending order from low Ivol to high Ivol. Then I divide the stock into equal-weighted tercile portfolios based on Ivol. For all three Ivol portfolios I sort stock based on their past 6-month returns and create equal-weighted past-return quintile portfolios. The results of the Ivol portfolios are presented in table 3 below.
Table 3: Zero-sum momentum portfolio returns sorted by idiosyncratic volatility. The table presents semi-annual returns. Portfolios are equally weighted. All stock are from the Euronext Amsterdam Stock Exchange. The sample contains constituents from the AEX All Share index from September 2002 until March 2017. Portfolio P1 to P5 are the past-return quintile portfolios sorted on their past 6-month returns. Where P1 contains stock with lowest past 6-month returns and P5 contains stock with the highest past 6-month returns. All portfolios have a holding period of 6 months. Portfolio P5-P1 is a zero-cost momentum portfolio. It has a long position in stock from portfolio P5 and a short position in stock from portfolio P1. In the low, medium and high Ivol columns, stock from the full sample are sorted based on their idiosyncratic volatility as measured by the standard deviations of the residuals from the market model presented in equation 2. The standard deviations of the residuals are estimated over the twelve months prior to portfolio formation. The low, medium and high Ivol portfolios include the terciles of stock with respectively the lowest, mid-range or highest idiosyncratic volatility. T-statistics are reported in parentheses. The critical values for the t-statistics are calculated with a Bonferroni-correction and using the appropriate number of degrees of freedom. *, ** and *** denote statistical significance of the test-statistic at the 10%, 5% or 1% level respectively.
Looking at the first column, it can be seen that the low Ivol zero-sum momentum portfolio has an average semi-annual return of 4% on average. This momentum return is statistically
significant at the 5% level with a t-statistic of 3,386. The average returns of the past-return quintile portfolios are all positive and increasing from the loser portfolio to the winner portfolio. The returns on the three highest past-return portfolios are statistically significant at the 1% level. Low Ivol losers earn on average only 0,4% semi-annually, while low Ivol winners earn 4,4% semi-annually on average.
LOW IV MEDIUM IV HIGH IV IV HIGH - IV LOW
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In the second column average semi-annual momentum returns on the medium Ivol portfolio are 8,7%. This result is highly statistically significant at the 1% level with a t-statistic of 8,075. Medium Ivol losers on average lose -3,5% over the following 6 months after portfolio formation, while the medium Ivol winners gain 5,3% on average over the same period. Only the returns on the two highest past-return quintile portfolios are statistically significant.
The high IVol zero-sum momentum portfolio earns the most on average as seen from the third column. Momentum returns are on average 17,6% semi-annually. The momentum portfolio return of high Ivol stock is also highly statistically significant at the 1% level with a t-statistic of 7,493. The high Ivol momentum return is predominantly realized by short selling the loser stock. The loser quintile portfolio has an average semi-annual return of -16,7%, while the winner quintile portfolio only earns a semi-annual return of 0,9% on average. Only the return on the loser portfolio is statistically significant.
The zero-sum momentum returns increase monotonically from the low Ivol to the high Ivol tercile portfolio going from 4% to 17,6%. The most important result is shown in the last column of table 3. I subtract the momentum portfolio returns of the highest Ivol portfolio from the returns of the lowest Ivol portfolio. The difference in momentum returns of the highest and lowest Ivol portfolio is 13,6% on average semi-annually and this result is highly statistically significant at the 1% level with a t-statistic of 5,942. As I mentioned this result is for a large part realized by the bad performance of high Ivol losers. Arena et al. (2008) find similar results where the difference between the high Ivol and low Ivol portfolio momentum returns is positive and statistically significant. They also find that this result is primarily driven by an underperfomance of high Ivol losers. Ang et al. (2006) also identify this pattern.
4.4. Risk-adjusted returns
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The first regression shows the results for the zero-sum momentum returns of the full sample. The coefficients on the size and value factors are negative, but only the coefficient on the value factor is statistically significant at the 5% level. So part of the momentum returns are explained by exposure to value risk. The coefficient on market risk is 0,099 and positive. However, it is not statistically significant.
The regression alpha is positive and highly statistically significant at the 1% level. This means that the single-sorted momentum returns cannot be fully explained by the market, size and value risk factors.
Table 4: Fama-French three-factor regressions on the zero-sum momentum portfolio returns. The dependent variable is the monthly return on the P5-P1 zero-sum momentum portfolios. Independent variables are the market risk premium and the size (SMB) and value (HML) risk premia introduced by Fama and French (1993). The market risk premium is the return on the market minus the return on the risk-free rate. The market return is calculated as the monthly return on the AEX All Share price index. The risk-free rate is calculated as the monthly return on 10-year Dutch government bond indices. The geometric mean of three different bond indices is used to estimate the risk-free rate. The monthly returns on the SMB and HML risk factors are taken from from Kenneth French’s website2. The results in panel A are estimated using heteroscedasticity-consistent (HC) Huber
(1967) and White (1980) standard errors. Results in panel B are estimated using heteroscedasticity-autocorrelation-consistent (HAC) Newey-West (1987) standard errors.
2http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/index.html, December 2017.
Variable Coefficient Std. Error t-Stat p-Value Adjusted R-squared FULL SAMPLE Alpha 0,015 0,002 6,350 0,000 0,025
mkt-rf 0,099 0,062 1,601 0,111 SMB -0,036 0,136 -0,261 0,794 HML -0,162 0,082 -1,983 0,049
LOW IVOL Alpha 0,004 0,002 2,606 0,010 0,001 mkt-rf 0,044 0,042 1,068 0,287
SMB -0,022 0,108 -0,206 0,837 HML -0,099 0,143 -0,688 0,492
MED IVOL Alpha 0,012 0,002 7,843 0,000 -0,010 mkt-rf 0,026 0,033 0,792 0,430
SMB 0,011 0,086 0,127 0,899 HML -0,045 0,073 -0,619 0,537
HIGH IVOL Alpha 0,016 0,006 2,766 0,006 0,021 mkt-rf 0,280 0,217 1,289 0,199
SMB -0,211 0,387 -0,545 0,586 HML -0,344 0,155 -2,223 0,028
HIGH - LOW Alpha 0,015 0,003 4,822 0,000 0,026 mkt-rf 0,076 0,073 1,045 0,298
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Next, I examine the regression results of the low, medium and high Ivol portfolios. All Ivol momentum returns have positive exposure to market risk, although these results are not statisctically significant. The coefficient of the market risk premium on the high Ivol tercile portfolio is 0,28 and this is higher than for the low and medium Ivol terciles. This suggests that stock with high idiosyncratic risk are also more exposed to market risk. The medium Ivol tercile portfolio has the lowest exposure to market risk.
All momentum portfolio returns have a negative exposure to the HML value factor.
Suggesting that the stock on the Euronext Amsterdam Stock Exchange are mostly glamour or growth stock. Exposure to the value factor is only statistically significant for the high Ivol momentum returns. So part of the momentum returns on the high Ivol portfolio are explained by exposure to value risk. However because this exposure is negative, momentum returns on high Ivol portfolios could be even higher without this exposure.
The momentum returns of the low and high Ivol tercile portfolios have a negative exposure to the SMB size factor, while the momentum return of the medium Ivol portfolio has a positive exposure to this small firm risk factor. I cannot draw any conclusions on this, as the
coefficients are far from statistically significant.
Variable Coefficient Std. Error t-Stat p-Value Adjusted R-squared FULL SAMPLE Alpha 0,015 0,004 3,998 0,000 0,025
mkt-rf 0,099 0,075 1,328 0,186 SMB -0,036 0,105 -0,338 0,736 HML -0,162 0,079 -2,045 0,042
LOW IVOL Alpha 0,004 0,003 1,562 0,120 0,001 mkt-rf 0,044 0,044 0,999 0,319
SMB -0,022 0,125 -0,178 0,859 HML -0,099 0,124 -0,796 0,427
MED IVOL Alpha 0,012 0,002 4,995 0,000 -0,010 mkt-rf 0,026 0,033 0,787 0,433
SMB 0,011 0,081 0,134 0,893 HML -0,045 0,072 -0,625 0,533
HIGH IVOL Alpha 0,016 0,008 1,989 0,048 0,021 mkt-rf 0,280 0,228 1,228 0,221
SMB -0,211 0,344 -0,614 0,540 HML -0,344 0,145 -2,379 0,019
HIGH - LOW Alpha 0,015 0,005 3,066 0,003 0,026 mkt-rf 0,076 0,083 0,918 0,360
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Most importantly, the alphas of all Ivol tercile momentum portfolios are positive and
statistically significant when using heteroscedasticity-consistent standard errors. After using Newey-West (1987) heteroscedasticity and autocorrelation consistent standard errors, only the alpha on the low Ivol regression becomes statistically insignificant. Therefore, within all Ivol categories neither market, size or value risk can fully explain the returns on the momentum portfolios.
Lastly, I want to test if the positive relationship between momentum returns and idiosyncratic volatility is still present after adjusting for these risk factors. Therefore, I take the difference between the zero-sum momentum returns of the high Ivol portfolio minus the zero-sum momentum returns of the low Ivol portfolio and regress it on the three Fama-French factors. The difference between Ivol tercile portfolios cannot be explained by exposure to market risk or size risk. Both the coefficient on the market risk factor and SMB factor is positive, but t-statistics are very low. However, the coefficient on the value factor is statistically significant at the 5% level. The coefficient is -0,333. This means that about a third of the positive relationship between mometun returns and idiosyncratic volatility is explained by a value factor. The coefficient is negative and therefore the positive relationship between momentum returns and idiosyncratic volatility could actually be higher without this exposure to value risk. It seems that either portfolios with low or high idiosyncratic volatility contain relatively more growth stocks. Value stock are perceived to be more risky and their idiosyncratic volatility is high. These value stocks have a low market value relative to their debt levels. This makes them more exposed to financial distress and therefore they face more
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5. Conclusions and limitations
I find a difference in momentum returns between high and low Ivol portfolios of 13,6% on average semi-annually. This compares to a difference in returns of 29% per year or 2,14% per month. The difference is statistically significant at the 1% level. The positive relationship is still significant at the 1% level after adjusting the momentum returns for marker risk, size risk and value risk. Although it seems the result is partially explained by a higher concentration of growth stock in the low Ivol portfolios. The results contradict the research by McLean (2010) and Cheema and Nartea (2017). Both papers don’t find a statistically significant positive relationship between momentum returns and idiosyncratic volatility.
My results support the research by Arena et. Al (2008). The momentum returns on their low, middle and high Ivol tercile portfolios are positive and monotonically increasing from the lowest to the highest Ivol portfolios. The difference between their momentum returns of the highest minus the lowest Ivol portfolios is 0,88% per month or 10,56% per year. This result is statistically significant.
Ang et al. (2006) find a negative relationship between expected returns and idiosyncratic volatility. The difference in risk-adjusted returns between low Ivol stock and high Ivol stock is -1,31% (Ang et al., 2006). They control their results for momentum effects. First stocks are sorted based on past-returns. Then within each past-return quintile portfolio, they sort stock based on idiosyncratic volatility. Although I take a different approach, looking at the data they provide, their momentum portfolio returns also seem to increase with idiosyncratic volatility. Zero-sum momentum portfolios sorted by past 12-month returns result in an average monthly risk-adjusted return of 0,86% in the lowest Ivol quintile portfolio and an average monthly risk-adjusted return of 2,63% in the highest Ivol quintile portfolio. This results in a difference in momentum returns between high Ivol and Low Ivol portfolios of 1,77% per month. My results are remarkably similar with a difference of 2,14% per month without adjusting for market, size and value risk. However regressing the difference in momentum returns between my high Ivol and low Ivol portfolios on the FF-3 factor model still produces a statistically significant alpha of 1,50%.
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returns on stocks with high idiosyncratic volatility. Hence, one way to improve the returns to a momentum strategy is to short past losers with high idiosyncratic volatility”.
The findings in this thesis provide some evidence for the behavioral explanations of the momentum effect by Daniel et al. (1998) and Barberis et al. (1998). The findings could also be seen as evidence in favor of idiosyncratic volatility as a primary arbitrage holding cost and therefore as an important limit to arbitrage according to the models of Schleifer and Vishny (1997) and Pontiff (2006).
This research focused on momentum returns based on portfolios with a 6-month formation period and a 6-month holding period. While these portfolios are the most used in the momentum literature, it is common practice to test for momentum returns by creating
momentum portfolios with different combinations of J-month formation en K-month holding periods. Wherby formation en holding periods of 3, 6, 9 and 12 months are most common. Due to limited time, I was not able to test for results on momentum porfolios with different formation and holding periods.
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Appendix A
Critical values for the test-statistics with Bonferroni-correction
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