How can momentum crashes be dampened?

Transcription

1 M.Sc. Finance Thesis Dimitrios Orfanakos January 28, 2014

2 M.Sc. Finance Thesis Tilburg University Tilburg School of Economics and Management Department of Finance Name: Dimitrios Orfanakos ANR: Supervisor: Dr. L.T.M. Baele Second Reader: Dr. J.C. Rodriguez Date: 28/01/2014

3 Acknowledgements I would like to express my appreciation and gratitude to my supervisor Dr. Lieven Baele for his guidance and valuable advices. Furthermore, I would like to thank my fellow students Stergios Axiotis, Cosmin-Ionut Mazilu, Erik Pedersen, Daan Rottger and their thesis supervisor Dr. Alberto Manconi, who kindly provided me data about international stock markets. In addition, I would like to thank my parents and my brother for all their support during my studies. Finally, I would like to thank all my friends and Marialena for this wonderful academic year. 2

4 Abstract Momentum strategy has been proved very lucrative through the years and across international market equities. Its high Sharpe ratio and abnormal returns made momentum very popular across investors. However, momentum has also experienced really severe crashes through years. These crashes are present in all equity markets around the world. Nonetheless, there is evidence that momentum crashes can be forecasted by the realized volatility of the strategy itself. Two different functions of realized volatility are being used in order to manage the exposure in momentum strategy based on volatility. Both lead to significant improvements in momentum Sharpe ratio and higher moments and nearly eliminate its crashes. In addition, the results from all 23 equity markets suggest that realized volatility forecasts not only momentum losses but also positive momentum returns. 3

5 Table of Contents 1. Introduction Literature Review and Hypothesis Development In general about momentum Momentum in International Stock Markets Momentum Risk and Momentum Crashes Hypothesis development Data and portfolio formation Portfolio formation Momentum data in US stock market International momentum data Empirical Methodology Ex-ante calculation of volatility Risk-managed momentum Timing Momentum Implementable momentum portfolio Statistical Tests Empirical Results Momentum portfolio performance in US equities Risk Managed Momentum in US equities Timing the momentum in US equities Momentum portfolio performance in International Equities Risk Managed Momentum in International Equities Timing the momentum in International Equities Conclusion References Appendices

6 1. Introduction Momentum is the tendency of assets to maintain their recent past performance in the short term future. Momentum strategy tries to exploit the aforementioned phenomenon. For that reason, it consists of long positions on assets with strong performance in the recent past and short positions on assets that experienced poor past performance. Momentum is well documented in US equities but it is not a local or asset-specific anomaly. Momentum is documented in several stock markets as well as in many other asset classes except equities. This research is focused on equity momentum, which is proved very profitable. More precisely, momentum has experienced statistically strong abnormal returns and its Sharpe ratio is by far the highest among those of the market, the value and the size factors in US stock market. As a consequence, momentum looks like an ideal investment in a mean variance framework. On the other hand, momentum has also experienced very severe crashes through time. During these crashes that usually last some months, momentum can realize extensive losses and it takes decades to recover. Note that if an investor had invested his wealth in momentum strategy in US equities just before the biggest crash of momentum in 1932, he would reach again the value of his initial investment in 1963 without controlling for inflation. In this research, I investigate the characteristics of momentum in US equities during a 86-year period from 1/1927 to 12/2012. Furthermore, this thesis studies momentum characteristics in international markets for a more restricted sample of 17 years. In US equities, momentum returns are significantly negative skewed, because of the crashes that momentum experienced through time. More precisely, the biggest crashes for momentum take place after recessions and at the point that the market recovers from extensive losses. At that time, market volatility as well as momentum volatility itself is very high. The two worst periods for momentum returns follow the two most severe international crises; the great depression of and the recent financial crisis of In international stock markets, momentum returns show similar patterns after the crisis of 2008, indicating that momentum returns are correlated across markets. The extensive downward risk of momentum makes the strategy unattractive for a reasonably risk averse investor. Many researchers have tried to reduce the crash risk of 5

7 momentum following mostly two different ways. The first way is to hedge momentum exposure on systematic risk or on all the three Fama-French factors. According to this theory, momentum is long on low beta stocks and short on high beta stocks, when the market has experienced large losses. Intuitively, this is straightforward, since in a bear market high beta stocks experience the largest losses. On the other hand, low beta stocks experience the highest returns during bear markets. As a result, momentum beta is negative and big in magnitude after a bear market. Thus, momentum crashes if a bull market follows a bear market. Consequently, hedging momentum exposure on systematic risk could eliminate potential crashes. Nevertheless, this hedging strategy has been proved inefficient because the ex-ante calculation of loser portfolio beta is incorrect when a bull market succeeds a bear market. More precisely, the ex-ante forecast of loser portfolio beta underestimates the magnitude of its real beta at the time that momentum crashes occur. Hence, it is not possible to avoid momentum crashes based on the ex-ante beta immunization of momentum strategy. On the other hand, the second way to avoid momentum crashes is based on the risk management of the total risk of momentum. Contrary to systematic risk of momentum, the total risk of momentum strategy is easily forecastable. In this case, the strategy consists of the momentum portfolio itself and an investment in risk-free rate. The wealth invested in risk-free rate equals the bet on momentum. By shifting the size of the bet on momentum while keeping the position in risk-free stable, it is possible to manage the total risk of the combined portfolio. Subsequently, when ex-ante momentum volatility is high, the bet on momentum declines accordingly, in order to maintain a stable level of volatility in portfolio. The opposite happens when ex-ante momentum volatility is low. The terms high/low volatility refer to the states when ex-ante momentum volatility is higher/ lower than the desirable level of volatility that the investor wants to maintain. This target level of volatility is set arbitrarily. Managing the risk of momentum improves dramatically momentum Sharpe ratio in US equities. Furthermore, it improves the higher moments of momentum and dampens momentum crashes. However, momentum performance is not improved solely during turbulent times but also in times that the unmanaged momentum realizes positive 6

8 returns. In my research, this strategy is also implemented in other international stock markets except for US equities. The results of the risk managed momentum imply that momentum returns are negatively correlated with momentum volatility and the latter can forecast the former. Hence, it is of great interest to examine whether instead of shifting the bet on momentum in order to keep the volatility stable, it is better to shift it in a different way. In a way that the exposure in momentum would be zero during really turbulent times on the one hand, and on the other hand momentum strong performance would be exploited during good times. Note that turbulent and good times of momentum can be forecasted by its own volatility. This thesis investigates the construction of a strategy with the aforementioned characteristics. In order to achieve that, every month I derive the bet on momentum strategy, using a function of volatility that leads to smaller/larger bet on momentum when the volatility is high/low than the aforementioned risk managed strategy. My results show that weighting momentum based on a more convex function of volatility, improves further the performance of risk managed momentum, in terms of Sharpe ratio and crash risk. The remainder of this thesis is organized as follows: In Section 2, I review the literature and formulate the hypotheses, Section 3 describes the dataset and the construction of momentum strategy, Section 4 presents the empirical methodology followed in my study, Section 5 presents the results of my research and Section 6 concludes. 2. Literature Review and Hypothesis Development 2.1 In general about momentum Despite the fact that the momentum phenomenon has been observed much earlier, the first paper focused on the returns of momentum strategy was published in 1993 by Jegadeesh and Titman (henceforth JT). 7

9 During that period, the academic literature was focused on contrarian strategies 1. However, JT, motivated from the fact that successful mutual funds are implementing relative strength strategies 2, examine in their seminal paper the performance of relative strength strategies in US common stocks. They study the performance of momentum strategies for several ranking and holding periods and they find positive and statistically strong returns of momentum which are unexplained by systematic risk. However, JT also report significant negative returns of momentum strategy in pre- World War II data. In addition, they note that relative strength strategies experience large loses when the market index rebounds from a big decline and the market volatility is high. Jegadeesh and Titman (2001) report the efficacy of momentum strategies in US stock market in the years after the publication of their initial paper in Carhart (1997) adds the momentum strategy as an additional risk factor in the 3- factor model of Fama and French (1992), since the 3-factor model cannot explain the variation of momentum returns. He forms the momentum portfolio based on the preceding one year returns of US equities, eliminating the last month before the formation date. Carhart also uses his 4-factor model to explain Hendricks, Patel and Zeckhauser s (1993) hot hands effect in mutual fund performance. 2.2 Momentum in International Stock Markets Apart from the efficacy of momentum strategy in US stock market, many academic papers indicate strong momentum effects in international stock markets outside US. Rouwenhorst (1998) examines the presence of momentum effect in developed markets, using a sample of 12 European markets covering the period 1980 to He finds that the return of an international diversified momentum portfolio is about 1 percent per month. In addition, Rouwenhorst in the same paper presents evidence that momentum returns are also strong, after controlling for country and size. Moreover, he notes the correlation between momentum returns in international stock markets and US market. 1 De Bondt and Thaler (1985, 1987) show that contrarian strategies achieve abnormal returns. 2 Most of the funds studied by Grinblatt and Titman (1989, 1991) tend to invest in stocks with recent strong performance. 8

10 Rouwenhorst (1999) documents momentum in emerging markets. More precisely, he observes strong momentum effect in the 17 out of the 20 markets in his sample. Apart from individual equities, momentum is also present in equities indices. For example, Asness, Liew, and Stevens (1997) note significantly positive returns in a strategy that buys a country index portfolio when the country performed well in the past and sells the indices of countries with poor past performance. Momentum effect is also documented in other asset classes except for equities. For example, Okunev and White (2003) document momentum in currencies and Asness, Moskowitz and Pedersen (2008) examine momentum in equities, currencies, government bonds and commodities futures. Moskowitz, Ooi and Pedersen (2010) find positive returns of time series momentum in exchange traded future contracts. Nevertheless, the empirical study of momentum in asset classes other than equities is beyond the scope of my research. 2.3 Momentum Risk and Momentum Crashes In the academic literature, it is well documented that momentum strategy in equities apart from high returns experienced also severe crashes. Cooper, Gutierrez and Hameed (2004) find that momentum returns are positively related to past market states. Their sample consists of all NYSE and AMEX firms and covers the period 1929 to 1995.They report that when market past 3-year return is negative, the momentum return is not significantly different from zero. On the other hand, when past 3-year market performance is non-negative, the monthly return of momentum strategy is 0.93 percent. Moreover, they regress momentum returns on lagged market performance and the square of lagged market performance. The coefficients of this regression indicate a positive but nonlinear relationship between momentum and lagged market performance. They also conduct their analysis using momentum returns adjusted for market, size and value factors and they are led to similar results. According to Grundy and Martin (2001), momentum downward risk can be reduced by the dynamic hedging of market and size factors. They argue that despite the fact that 9

11 common factors cannot explain the mean return of momentum, they can capture a quite big part of its return variability. First of all, they indicate the time-varying exposure of momentum to common risk factors. Whether momentum strategy has positive or negative loads on factors and their magnitude, depends on the performance of these factors during the ranking period. For a single factor model like CAPM this mechanism is straightforward. For example, if the market experienced high returns during the ranking period, the beta of the winner portfolio would be high and the beta of the loser portfolio would be low. Hence, the market beta of momentum portfolio would be positive. On the other hand, momentum beta will be negative after a bear market. Grundy and Martin use monthly data from US stock market from 1926 to 1995 in their analysis. As reported in their paper, hedging momentum exposure to market and size reduced the volatility without sacrificing momentum returns at the same time, which has a beneficiary effect on the Sharpe ratio of the strategy. Furthermore, the negative results of the strategy in Januaries and the poor performance in pre-world War II data are eliminated. They show similar results in the case of hedging the momentum against Fama French risk factors for the period 1966 to The time-varying factor loadings of momentum strategy are calculated by Grundy and Martin using a window of the future six months after the formation date. Due to the fact that this version of the hedging strategy is not implementable, since the coefficients of the factors are calculated based on the six months after the formation date, Grundy and Martin also present a feasible strategy. In this version, they calculate the factor loadings using a 60-month window, ending one month before the formation date (t-61 to t-1). Although the feasible hedging strategy mitigates the volatility of momentum and improves momentum performance before 1945, it does not have as impressive results as the first non-implementable version. In addition, the window of 60 months seems too large to capture the dynamic nature of momentum systematic risk. The authors also argue that a momentum strategy that ranks the stocks according to their firm specific component of returns, performs better than the classic momentum strategy. 10

12 Daniel and Moskowitz (2011) study momentum crashes in equities and other asset classes, which they define as infrequent but enduring series of big in magnitude negative returns. They argue that momentum crashes in equities occur when market volatility is high and the market rejuvenates from a significant decline. They also mention that the most severe momentum crashes are triggered because of an upward crash of losers. Namely, momentum worst crashes have been observed in times when the loser decile experienced extremely higher performance than the winner decile. In consistency with the results of Grundy and Martin, Daniel and Moskowitz show that the market beta of momentum is significantly lower following bear markets than after bull markets. The criterion that indicates a bear/bull market is whether the CRSP value weighted index cumulative returns of the past 24 months is negative/positive. However, they show that the results of Grundy and Martin could not be achieved in real world because they are primarily based on a non-implementable strategy that uses expost calculated betas. On the other hand, hedging momentum for market risk based on ex-ante betas actually leads to worse performance than the unhedged version of the strategy. According to Daniel and Moskowitz, the reason that ex-post betas lead to misguided results is that market beta of momentum portfolio is negatively correlated with the contemporaneous performance of the market if the latter turns positive after a decline. In order to show this, they use a methodology similar to the one used by Henriksson and Merton (1981) regarding the evaluation of market timing ability of fund managers. They run regressions using indicators for bear/bull and up markets 3. They find that after bear markets, momentum market beta is significantly more negative in up markets than in down markets. Responsible for this asymmetry is the short side of momentum strategy, since loser portfolio has a much higher market beta when the market performs well after a decline. On the other hand, this asymmetry is not present after bull markets. Similar results are presented for the stock markets outside US as well as for currencies and commodities markets. As a result, it is straightforward that because of this asymmetry, momentum systematic risk cannot be hedged based on exante betas. 3 The indicator for up markets is 1 when the excess CRSP VW index return is positive during the holding period. 11

13 Daniel and Moskowitz propose Merton s (1990) model as a possible explanation for the optionality in loser portfolio returns. According to the aforementioned model, a stock of a firm with debt in its capital structure is a call option for common shareholders, having as underlying asset the firm value. In turbulent times, which are indicated by bear markets, loser firms are probably financially distressed firms. So, loser stocks are out-of-the money options and possibly their option convexity is very high. Hence, if the market rises and the economy is improved, the past loser firms will survive and their stock price will rise dramatically. However, Daniel and Moskowitz note that this theory cannot explain similar patterns in market betas of currencies and commodities. In the same paper, a strong negative relationship between market lagged volatility and momentum return is documented in all international equities markets but not in other asset classes. Barroso and Santa-Clara (2012), using the same sample for US equities as Daniel and Moskowitz, show that momentum crashes make the strategy unattractive to investors that are not risk lovers. In addition, they focus on the total risk of momentum instead of systematic risk. The risk of momentum is time varying and according to the authors, managing the total risk of momentum leads to double Sharpe ratio for the strategy. They also argue that the total risk of momentum is highly predictable. According to Barroso and Santa-Clara, R square of an autoregressive process with one lag for momentum realized variance 4 is greater than the one for the market, indicating that momentum risk is even more predictable than the highly predictable market risk. They also support their conclusion with the results of an out of sample R square estimator. Moreover, they use CAPM to decompose total risk in systematic and firm specific risk. Their findings suggest that firm specific risk counts for the 80% of momentum total risk and is more predictable than systematic risk. For these reasons, beta hedging does not work. In the updated version of their paper (2013), they present the results of a predictive regression of monthly momentum returns on the realized variance. These results show that realized variance can forecast momentum future returns. 4 Total variance is calculated by the daily returns of past six months. 12

14 Barroso and Santa-Clara argue that stabilizing the risk of momentum results in higher returns, lower volatility and substantially improved skewness and kurtosis. The proposed methodology is to weight momentum portfolio by the ratio of a constant to the estimation for momentum volatility. Since momentum is a zero investment strategy, the choice of the constant is arbitrary and expresses the desired level of steady volatility that someone wants to maintain through time. Thus, it is called target volatility. Barroso and Santa-Clara arbitrarily set annualized target volatility equal to 12%. The results of risk managed momentum are quite impressive, since the yearly Sharpe ratio roughly doubles from 0.53 to 0.97 and the crashes are mitigated. In order to examine how attractive momentum is for a reasonably risk averse investor, Barroso and Santa-Clara use a power utility function, as it takes into consideration also the higher moments. Based on the certain equivalent of raw momentum, they conclude that the strategy is not appealing for the average investor. On the other hand, risk managed momentum is more attractive than an investment in market index. The improvement in certain equivalent is due to the reduction in crash risk of momentum. 2.4 Hypothesis development Barroso and Santa-Clara (2012) draw their conclusions by examining only US stock market. In their updated version of their paper (2013) they also examine equities in France, Germany, Japan and UK. However there is evidence in the literature 5 which suggests that a common factor drives the momentum returns around international stock markets, since they are correlated. Due to this fact, it is reasonable to conclude that a momentum hedging strategy that is effective in US stock market may also be efficient in other stock markets outside US. Hence, my hypothesis is: H1: Risk-managed version of momentum could reduce the crash risk of momentum in international stock markets. 6 Barroso and Santa-Clara results suggest that momentum crashes can be forecasted based on the ex-ante volatility of momentum. In order to shift down the bet 5 Asness, Moskowitz and Pedersen (2012) 6 The sample of international stock markets consists of 22 countries apart from USA. 13

15 on momentum in turbulent times, they use as weight the ratio of target volatility to exante estimated future volatility. Their results indicate improvement on momentum performance not only in turbulent but also in good times. Consequently, not only momentum crashes, but also momentum returns in general can be forecasted based on their realized volatility. Hence, if a more convex function of volatility than the aforementioned simple ratio is used to calculate the weight on momentum, it may lead to an even better performance. H2: Momentum can be timed using an ex-ante estimation of the volatility of the strategy. 3. Data and portfolio formation 3.1 Portfolio formation In this subsection, the general case of constructing the momentum portfolio is described since the specific occasions will be analyzed later. In my research, I form 12-2 momentum portfolios, which means that I use as ranking period the past twelve months before the formation date excluding the last month. Consistent with the literature, the final month is excluded in order to avoid short term reversals in returns 7. The holding period of the strategy is one month and follows the formation date as Figure 1 highlights. All firms are sorted based on their cumulative returns during the eleven-month ranking period. The top 10% of the firms with the highest performance form the winner portfolio and the bottom 10% form the loser portfolio. The momentum strategy (WML) is a zero cost strategy that consists of a long position in the value weighted winner portfolio and a short position of the same amount of money in the value weighted loser portfolio. In addition, momentum is a buy-and-hold strategy which means that except for delistings, dividends and cash payouts portfolio weights remain stable during the holding period. 7 Jegadeesh (1990) and Lehmann (1990) present evidence of short-term reversals in stock returns. 14

16 Figure 1: Construction of Momentum Portfolio The figure below illustrates the ranking and the holding period of momentum strategy as well as the omitted month before the portfolio formation date. Formation Date t-12 t-2 t 3.2 Momentum data in US stock market Ranking Period Excluded Last Month Holding Period In order to conduct my research regarding momentum in US equities, I use Kenneth French s data library as data source. More precisely, I obtain monthly returns for the market portfolio, the high-minus-low (HML) and the small-minus-big (SMB) portfolios as well as the risk free rate. The market portfolio is a value weighted portfolio of all NYSE, AMEX and NASDAQ firms with CRSP share code of 10 or 11. SMB and HML portfolios are constructed using stocks from the same universe based on size and value respectively and the risk free rate is the one-month Treasury bill rate. Concerning the construction of momentum portfolio I use both daily and monthly returns of the ten portfolios formed on momentum from Kenneth French s data library. The universe includes all NYSE, AMEX and NASDAQ equities which are filtered every month before the construction of the portfolios according to some criteria 8. As I described earlier, in order to construct the momentum portfolio, all stocks are sorted based on their cumulative returns in the ranking period which starts 12 months before the portfolio formation date and ends 1 month before the formation date (12-2). Despite the fact that the sample includes NYSE, AMEX and NASDAQ firms, the formation of each decile is based on the NYSE firms only. Hence, each one of the ten portfolios does not include the same number of firms but the same number of NYSE firms. Actually, most of the times the extreme portfolios consist of more stocks because NYSE firms are on average less volatile than AMEX and NASDAQ firms. The firms with the highest cumulative returns during the ranking period compose portfolio 10 8 According to the description in Kenneth French s website To be included in a portfolio for month t (formed at the end of month t-1), a stock must have a price for the end of month t-13 and a good return for t-2. Each included stock also must have ME for the end of month t-1. 15

17 (winners) and the ones with the lowest compose portfolio 1 (losers). The momentum portfolio is constructed by a long position on winners and a short position on losers as I showed above. My sample consists of monthly observations; covering 86 years from 1/1927 until 12/2012. The sample of daily returns for the 10 portfolios sorted on momentum, covers a slightly larger period; from 30/11/1926 to 31/12/ International momentum data The dataset of international equities covers a sample of twenty-three developed countries 9 for the period 1/1/ /12/2012. I acquired daily market value and return data 10 for international stocks through my fellow students Stergios Axiotis, Cosmin- Ionut Mazilu, Erik Pedersen and Daan Rottger after the permission of their M.Sc. thesis supervisor professor Alberto Manconi. The source of the data is Thomson Reuters Datastream and the dataset is screened according to Ince and Porter (2006) methodology for the treatment of the individual equity return data from Thomson Datastream. Ince and Porter state in their paper that the use of raw equity returns from Thomson Datastream leads to ambiguous results regarding the calculation of returns for equity strategies like momentum. Therefore, the implementation of their two-level approach is very crucial in order to have a reliable sample. The dataset contains only common stocks of local firms of each country. Other listings than those on the primary exchange are excluded. In addition to this, all firms should have realistic returns. For this reason, any return larger than 300% that is reversed within a month, it is considered as missing value. For the purposes of this research, the twenty-three countries are distributed in seven regions; US, Canada, UK, Japan, Europe excluding UK, Asia-Pacific excluding Japan and Global. In this way, I have the opportunity to control for the robustness of my results outside the US stock market. Before I proceed to the construction of the momentum portfolio, I convert the daily returns to monthly returns. In order to achieve that, I use the cumulative daily log- 9 Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Hong Kong, Ireland, Italy, Japan, Netherlands, New Zealand, Norway, Portugal, Singapore, Spain, Switzerland, Sweden, United Kingdom, United States. 10 In order to have comparable data across countries, dollar returns and market value are used. 16

18 returns of each month and then I transform this sum back to arithmetic return as the following formula indicates: T ln(1 Rt ) t1 R Monthly exp 1 where R t is the daily return of day t and T are the trading days of the specific month. Regarding the construction of momentum portfolio, it is required that every stock has a valid return and market value at the formation date and 12 months of past performance before the formation date. The way that the momentum strategy in international equities is constructed, differentiates slightly from the one used by French in US equities. Using a similar approach to Asness, Moskowitz and Pedersen (2012), every month are selected the stocks with the higher market value that count for the 90% of the total market capitalization of each market at the formation date. As a result, the sample is reduced significantly, since on average the 16.4% of total firms represents the 90% of the total market capitalization. Table 1 presents more details about the size of the final sample for every region. Thereafter, I sort the remaining stocks based on their cumulative returns during the ranking period. The 30% of the stocks with the highest performance constitutes the winner portfolio while the 30% with the worst performance constitutes the loser portfolio instead of the strategy that is used in the first dataset for US. Note that in this case, the winner and the loser portfolios have exactly the same number of firms. Due to the fact that I restrict the sample to the biggest firms of each country, the formed portfolios are extremely liquid and treatable with minimal transaction costs. As follows, the strategy is fully implementable and because the transaction costs are very low, they do not have a strong impact on the momentum portfolio returns. My results are expected to be conservative since small firms, which in general perform better than big firms, are excluded from the sample. Finally, according to Ince and Porter, Thomson Datastream data for smaller firms are the most problematic. Therefore, by using the biggest firms the sample becomes even more reliable. 17

19 Table 1 The table below reports the maximum and the minimum number of stocks of the final sample per region and also what percentage of the total firms per region represents the 90% of the total market capitalization on average. Number of Stocks Large Firms/Total Max Min Firms US ,13% UK ,35% Japan ,00% Europe (ex UK) ,38% Asia-Pacific (ex Japan) ,92% Canada ,31% Global ,42% It is straightforward that because of the slightly different approach in the construction of the momentum portfolio and the shorter time period, the results for the US stock market are expected to be different in this part of the study. 4. Empirical Methodology This section discusses the way that the ex-ante forecasted volatility of momentum is calculated. Next, it is explained how forecasted volatility is used in order to scale momentum portfolio. Furthermore, I show the derivation of statistical tests for changes in skewness, kurtosis and Sharpe ratio. Moreover, the construction of an implementable momentum strategy is explained. 4.1 Ex-ante calculation of volatility Since my intention is to weight momentum strategy dynamically using its risk, momentum volatility has to be estimated. In my research, I calculate the forecasted volatility of momentum, based on past data, following Barroso and Santa-Clara (2013) methodology. They show in their paper that momentum volatility is accurately predicted based on past data. More precisely, each month, momentum variance is calculated by the daily returns of momentum during the six months leading up to the formation date. The estimator is the following: t rt i i1 ˆ

20 where t is the first trading day of each month, hence the formation date. Moreover, rt 1 is the daily arithmetic return of the corresponding trading day. In consistency with the literature, the mean is not deducted during the calculation of the variance, since daily returns are used. By convention, the trading days of each month are considered equal to 21. Therefore, the multiplication with 21 is necessary in order to convert the daily variance into monthly. 4.2 Risk-managed momentum In this case, the methodology of Barroso and Santa-Clara is followed precisely. Therefore, in order to stabilize the risk of momentum, I scale momentum portfolio by the ratio of target volatility to forecasted volatility. Hence, the return of risk-managed momentum at month t is: r RM, t ˆ t arg et t r Raw, t where Raw,t r is the return of raw momentum portfolio at month t. In addition, t arget is the level of volatility that I attempt to maintain through time and ˆ t is the square root of the aforementioned forecasted variance of momentum for month t. Barroso and Santa-Clara arbitrarily set the annualized target volatility equal to 12%. It is straightforward that when the forecasted annualized volatility of momentum is 0,12, the weight on the strategy is one and when the forecasted volatility is higher/lower than the target, the weight is lower/higher than one. In this way, I try to stabilize the volatility of momentum. In turbulent times, when the danger of momentum (volatility) is high, the strategy is underweighted, dampening a potential crash. For example, if the annualized forecast of momentum volatility is 24%, the weight on momentum portfolio will become 0,5. So, the holdings of the implementable portfolio will be 0,5 in winners, 0,5 in losers and 1 in risk free rate. 4.3 Timing Momentum As I mentioned in hypothesis development, Barroso and Santa-Clara s results suggest that realized volatility of momentum can forecast its future returns. In order to time 19

21 momentum, I use a more convex function of volatility as weight on the strategy. The function that I propose is the following: wt exp( parameter1 parameter2 ˆ t ) if exp( parameter1 parameter2 ˆ t ) 1% 0 if exp( parameter1 parameter2 ˆ t ) 1% where parameter1 1, 8 and parameter2 15 I choose an exponential function in order to be more convex than the simple ratio that is used in risk-managed momentum strategy. The selection of parameter 1 and parameter 2 are subject only to one restriction; w 1 when ˆ 12%. The restriction is defined so as to maintain the benchmark of 12% as in risk-managed momentum 11. Apart from this restriction, I chose these values for the parameters using also intuition in order to achieve the desirable steepness and convexity in the weight function 12. As it is depicted in Figure 2 the function for timed momentum weight is more convex than the corresponding function for time-managed momentum. However, I construct the function in a way that avoids extreme weights. Even for the unrealistic volatility of 0% the weight is 6,05. Furthermore, the strategy is more conservative than risk-managed momentum in turbulent times, because the weight becomes zero when the annual volatility of momentum is slightly below 46%. 4.4 Implementable momentum portfolio Momentum is a zero cost strategy since at the inception of the strategy the inflows of the short leg are used to purchase the stocks of the long leg. In this way, the weights in winner and loser portfolios are 1 and -1 respectively. Hence, the total weight of the strategy is zero. In this study and in consistency with the recent literature 13, I use an implementable version of momentum, in which apart from the investment in long and short portfolios there is also an equal investment in risk free rate. t t 11 Changing the parameters in order to fulfill the restriction of wt 1 when ˆ t 11% or when ˆ t 13% has a similar effect on the Sharpe ratio of the strategy. 12 Defining the value of the parameters by using an in-sample optimization of the Sharpe ratio and allowing for extreme values does not improve the SR significantly, relative to the parameters values used in this study. 13 Daniel and Moskowitz(2011), Barroso and Santa-Clara (2013) 20

22 Weight How can momentum crashes be dampened? Figure 2 The following figure depicts the weights on momentum of timed and risk managed versions for different levels of forecasted volatility Timed Weight Risk Managed Weight Realized Volatility The investment in risk free rate serves as marginal account. So, a 1$ investment in implementable momentum portfolio means that an investor deposits 1$ in marginal account and earns the risk free rate, goes short 1$ in the loser portfolio and long 1$ in the winner portfolio. This is also aligned with Reg-T requirements. The return of this portfolio in month t is the following: R R R R t f, t W, t L, t where R ft, is the risk free rate for month t, R Wt, and RL,t winner and loser portfolios during t. are the arithmetic returns of Every month, the gains of the portfolio are reinvested in the strategy. Hence, the gains are added to marginal account and the bets on winner and loser portfolio grow accordingly. I assume that there are no margin calls during the holding period and transaction costs are considered zero for the purposes of this research. In risk managed and timed version of momentum, the investor can overweight or underweight the momentum strategy. If the weight in momentum strategy is greater than one, the investor borrows the extra money in order to cover the marginal account. In this case, it is assumed that the investor can borrow in risk free rate. In this way, the 21

23 earnings from the marginal account offset the borrowing costs. If the weight is under one, the long/short positions are smaller than the invested wealth in risk free rate. Note that for the calculation of forecasted volatility, the excess returns of implementable momentum portfolio are used due to the fact that the variance of the one month risk free spot rate and its correlation with momentum returns are zero. 4.5 Statistical Tests Regarding the evaluation of momentum performance improvement, Sharpe ratio, skewness and kurtosis are used. However, mean, standard deviation, skewness and kurtosis are unknown quantities that must be estimated. Consequently, their estimations are accompanied with estimation error. Therefore, the use of statistical tests is necessary, in order to eliminate the possibility that the observed changes in momentum statistics are just estimation errors. I follow the methodology of De Roon et al. (2012), which is based on Lo s (2002) approach when assuming non-i.id. returns, in order to derive statistical tests for the pairwise changes in skewness, kurtosis and Sharpe ratio between the three versions of momentum. I do not test for changes in volatility, since the choice of target volatility has a straight impact on the volatility of the modified momentum strategies. As the empirical results show, the distribution of momentum is far from normal. For this reason, I use statistical tests that are valid also in the case of non-normality. These statistical tests rely only on the assumption of stationarity 14 in returns but apart from that, allow for several phenomena observed in time series returns like serial correlation, time-varying conditional volatilities and jumps. Sharpe Ratio The Sharpe ratio is a widely used statistic in financial analysis, which loosely speaking counts the excess return per unit of risk. It is the ratio of the excess expected return of an investment to its returns volatility. Most of the times the excess expected return is relative to the risk free rate. The formula of Sharpe Ratio is the following: 14 All samples are tested for stationarity using the augmented Dickey-Fuller test for a unit root in a time series sample. 22

24 Rf SR The components of Sharpe ratio, which are the mean of the return distribution and the standard deviation, are unobservable and must be estimated from past data. Obviously, that causes estimation error also in the estimation of the Sharpe ratio. The estimator of the SR is: ˆ Rf SR ˆ where T 1 ˆ Rt and T t1 1 ˆ T T t1 ( R ˆ ) t 2 In our case R t is the monthly realized return of the momentum portfolios. The most commonly used statistical test for differences in Sharpe ratio is the one proposed by Jobson and Korkie (1989). However, Jobson and Korkie assume independent and identical return distribution, which for sure is not the case for momentum. On the other hand, Lo (2002) proposes a robust estimator for the variance of the Sharpe ratio, which is effective under a wide variety of assumptions, and using it derives the asymptotic distribution of Sharpe ratio. The mean and the variance are estimated by the use of generalized method of moments estimators (GMM): T 1 ( R ˆ t ) 0 is the estimator for the mean and T t1 1 is the estimator for the variance T T 2 2 [( R ˆ) ˆ t ] 0 t1 Based on the results of Hansen (1982) the following distribution for the Sharpe ratio is obtained: T (SR SR) N(0,V(SR)) 23

25 For the differences between Sharpe ratios of two portfolios A and B, if the true difference between them is δ, the following asymptotic distribution is derived: T (( SR SR ) ) (0,V(SR SR )) A B A B where V(SR) and V(SR SR ) are the limiting variance of the Sharpe ratio and the A B limiting variance of the difference in Sharpe ratios respectively. Both variances depend on the first derivatives of the Sharpe ratio with respect to mean and variance. The derivations of the expressions for both variances are shown in Appendix A. Subsequently, the standard error of the Sharpe ratio can be calculated as follows: SE( SR) V(SR) T And for the difference SE( SR) V(SR SR ) A B T Using the above formula and since the distribution of the difference in Sharpe ratio is known, I can test the statistical significance of the pairwise differences in portfolios Sharpe ratios. Higher moments To derive a statistical test for the skewness, the methodology is similar as for the Sharpe ratio. The skewness of the returns can be expressed in terms of the first three noncentral moments as follows: E[( Rt ) ] m 3m m 2m S /2 ( m2 m1) where m i is the i th noncentral moment estimated as before by the method of moments: T 1 i ( Rt mi) 0 T t 1 The expression for the variance-covariance matrix of ˆm is derived in Appendix A. Similar to the expression for the variance of Sharpe ratio, skewness variance is a 24

26 function of the first derivatives of skewness with respect to the first three noncentral moments and their joint variance-covariance matrix. Consistent with the approach of De Roon et al., kurtosis is expressed as a function of the first four noncentral moments: E[( R )] m 4m m 6m m 3m K t /2 ( m2 m1) Regarding the standard error for kurtosis, it is derived following the same procedure as before (see appendix A). 5. Empirical Results 5.1 Momentum portfolio performance in US equities In this subsection, I discuss the characteristics of momentum portfolio returns using my first large sample for the US equities. In consistency with the literature, our results show that momentum strategy experienced extremely large returns in US equities during the period 1/1927 to 12/2012. As Table 2 highlights, momentum portfolio has experienced not only the highest average return in comparison with the market, size and value portfolios but also the highest Sharpe ratio. This indicates that momentum portfolio has a better return- risk tradeoff relative to the other factors. Note that the average return of momentum portfolio is 1,48% per month, substantially higher than that of the market, which is 0,91%. Moreover, the monthly Sharpe ratios are 0,15 and 0,11 for the momentum and the market respectively. Table 2 This table presents the characteristics of momentum returns in comparison with Fama-French factors for the period from 1/1927 to 12/2012. Market Size Value Momentum Mean 0,91 0,53 0,68 1,48 SD 5,42 3,30 3,57 7,93 Skewness 0,14 2,16 1,75-2,48 Kurtosis 10,26 24,76 17,99 21,32 SR 0,11 0,07 0,11 0,15 MAX 37,87 39,08 35,51 26,61 MIN -28,95-15,92-13,43-78,93 25

27 Observations How can momentum crashes be dampened? In addition, as it is shown in Table 3 momentum portfolio has a statistically significant alpha of 1,84% per month and negative weights on the risk factors. The strong alpha and the high Sharpe ratio make momentum very appealing; it actually looks like a free lunch. Table 3 The following table presents the results of the regression of momentum excess returns on Fama- French factors. The sample consists of monthly returns and covers the period from 1/1927 to 12/2012. T-statistics are reported in brackets. α Mkt-Rf SMB HML 1,75-0,38-0,26-0,67 (8,00) (-8,86) (-3,68) (-10,75) R 2 0,23 Observations However, momentum returns have a very high kurtosis of 21,32 and a significant negative skewness of -2,48. These values for skewness and kurtosis indicate that momentum distribution of returns has a very fat left tail, which is also illustrated in Figure 3. This fat left tail is attributed to the extremely negative returns that momentum has experienced through the years. Therefore, the risk of a potential crash for the momentum portfolio is very high. Figure 3 The following figure depicts the histogram of momentum returns and the fitted normal distribution. The fitted normal distribution is a normal distribution with the same mean and variance as momentum returns Momentum Returns Fitted Normal Distribution % Returns 26

28 $ Value of investments (log scale) How can momentum crashes be dampened? Figure 4 shows the cumulative gains of a one dollar investment from 1/1927 to 12/2012 in momentum portfolio in comparison with a one dollar investment in market, value and size portfolios. It is straightforward that momentum outperforms the Fama- French factors, since its cumulative gains at the end of 2012 is $ compared to only 2.648$ for the market. Figure 4 The following figure depicts the cumulative gains of a one dollar investment in implementable momentum strategy, market portfolio, implementable size and value portfolios for the period 1/ /2012. The gains of every strategy are reinvested every month in each strategy Momentum Market Size Value Cumulative gains from investments $ $2.648 $ $ Date On the other hand, Figure 4 also highlights the two biggest crashes of momentum during the summer of 1932 and the spring of The most severe crash for the strategy took place during the summer of 1932 when momentum portfolio lost 91,59% of its value within only two months, July and August. As Figure 5 shows, this crash occurred after the great depression of and when the market rebounded after its own crash. Note that momentum strategy continued to have positive returns during the recession and momentum crash took place when market had already started to rise again. These results are fully aligned with the existing literature. The second feature of momentum returns that is observed in Figure 4 and well documented in literature, is strategy poor performance in pre-world War II era. 27

29 $ Value of investment (log scale) $ Value of investment (log scale) How can momentum crashes be dampened? Figure 5 The following figure depicts the cumulative gains of a one dollar investment in implementable momentum strategy and in market portfolio for the period 1/ /1944. The gains of every strategy are reinvested every month in each strategy. Cumulative gains from investments Momentum Market $2, $0, Date Figure 6 depicts the performance of momentum during the turbulent period of 00 s. In the figure, it is illustrated a smaller in magnitude crash or momentum in 2001 after the dot com bubble and the big crash of 2009, which took place after the recent financial crisis of From March to May 2009, momentum experienced cumulative losses of 73,41%. The crash of 2009 as that of 1932 occurred after a big crash in the stock market and when the contemporaneous market return was positive. Figure 6 The following figure depicts the cumulative gains of a one dollar investment in implementable momentum strategy and in market portfolio for the period 1/ /2009. The gains of every strategy are reinvested every month in each strategy. Momentum Market Cumulative gains from investments 10 0 $0,95 $0, Date 28