Expected Idiosyncratic Skewness and Price Momentum: Does Lottery-Like Return Structure Affect Momentum Profits?

Transcription

1 Expected Idiosyncratic Skewness and Price Momentum: Does Lottery-Like Return Structure Affect Momentum Profits? Hongrui Feng Oklahoma State University Yuecheng Jia* Oklahoma State University * Correspondent Author, Doctoral Student of Finance, Oklahoma State University, Stillwater, OK Tel: (405) ; yuecheng.jia@okstate.edu. We thank Alex Boquist and Yuzhao Zhang for continuous guidance and great help. 1

2 Abstract This paper examines the relationship between expected idiosyncratic skewness and momentum returns. We document a significant negative effect of expected idiosyncratic skewness on momentum returns. This negative relationship is driven by the lottery-like return structure of high expected idiosyncratic skewness winner stocks. This relationship is robust after controlling idiosyncratic volatility in cross sectional and time series regressions. The relationship is robust to different specifications of momentum portfolio formation and holding periods. Our findings demonstrate that higher moments of returns have significant effect on momentum returns and provide new source of the momentum phenomenon. 2

3 1. Introduction 1.1 Literature on momentum strategy The seminal paper Jegadeesh and Titman (1993) explored the trading strategy that buying stocks with good performance in the past and selling stocks that don t do well previously could generate significant positive returns over a 3 to 12 month holding period. This is known as the momentum strategy, one of the most robust anomalies. Even though the persistence of momentum profits is widely accepted, there is no consensus on the sources of the profits. Two groups of literature explore the determination of momentum profits. The first group demonstrates that momentum profits are risk based: Chordia and Shivakumar (2002) show that momentum profits can be explained by lagged macroeconomic variables. Liu and Zhang (2008) argue that the growth rate of industrial production can explain more than half of momentum profits. Using an investmentbased model framework, Liu and Zhang (2013) find that winner stocks have higher expected growth and expected marginal productivity, which explains why winners earn higher expected stock returns than losers. In contrast, the second group of literature provides behavioral explanations for momentum returns, since many papers fail to provide evidence of a risk-based explanation of momentum profits (Jegadeesh and Titman (1993); Fama and French (1996); Moskowitz (2003)). These papers consequently argue that momentum profits come from an underreaction to firm-specific information. Some other papers provide direct evidence to support a behavioral explanation of momentum. Hong, Lim and Stein (2002) argue that the slow diffusion of negative information explains momentum profits. Jegadeesh and Titman (2001) also support the behavioral story. 3

4 This paper provides a new explanation of momentum profits by exploring the relation between expected return skewness and price momentum. Using the framework of Boyer, Mitton and Vorkink (2010) to estimate expected idiosyncratic skewness, we document a negative relationship between the expected idiosyncratic skewness and momentum profits. We sort stocks into quintiles first by past returns and then by the magnitude of expected idiosyncratic skewness (EIS). Higher expected EIS quintiles yield lower momentum profits. The time series behavior of EIS is consistent with its cross sectional pattern. EIS is negatively related to subsequent momentum profits. But why skewness affects momentum return? In the following two subsections, 1.2 and 1.3, we survey the literature on return skewness and provide an explanation on the relationship between momentum and expected idiosyncratic skewness. 1.2 Return skewness There is a huge literature implying higher moments of returns matter in the cross section of stock returns. The origin of the literature could be traced back to Kraus and Litzenberger (1976) which extend CAPM model to incorporate skewness preference. The most straightforward estimate of true ex ante skewness is to use historical data. Following Kraus and Litzenberger (1976), papers are interested in systematic skewness. Harvey and Siddique (1999, 2000) show how higher moments could affect the stochastic discount factor. They estimate skewness using historical data and find that co-skewness is priced in the market. However, it is hard to estimate the true ex ante skewness from historical data. Harvey and Siddique (1999) estimate a time varying systematic skewness but find that the lagged measure of skewness is not adequate as a predictor of skewness. 4

5 More recent papers estimate return skewness by incorporating option information: Chang, Christoffersen and Jacobs (2013) estimate market-wide higher moments from daily Standard & Poor s 500 index option data employing methods of Bakshi, Kapadia and Madan (2003). They find that stocks with high exposure to innovations in estimated market skewness have low returns on average. Conrad, Dittmar and Ghysels (2013) use option price information of individual stocks to back out ex ante skewness for individual stocks and document a negative relationship between ex ante risk neutral skewness and subsequent returns. Besides looking at systematic skewness, idiosyncratic skewness generates great interest for researchers. Barberis and Huang (2007) demonstrate that if investors have prospect preference, stocks with greater idiosyncratic skewness should have lower returns. Mitton and Vorkink (2007) introduce a rational model of investors that have heterogeneous preferences for skewness and show that idiosyncratic skewness can negatively impact prices. Boyer, Mitton and Vorkink (2010) estimate idiosyncratic skewness and document a negative correlation between expected idiosyncratic skewness and returns. Consistent with Boyer, Mitton and Vorkink (2010), Conrad, Dittmar and Ghysels (2013) find evidence that idiosyncratic skewness matters even after controlling for the systematic skewness. 1.3 Expected Idiosyncratic skewness and momentum Even though many works document negative relation between idiosyncratic skewness and returns, the effect of idiosyncratic return skewness on momentum returns is not clear. By our humble knowledge, we are the first to explain and test the effect of expected idiosyncratic skewness on momentum returns. We argue that idiosyncratic skewness of a stock is an indicator 5

6 of the stock s lotto behavior likelihood. Idiosyncratic skewness of stocks may affect momentum profits due to a lottery-like return structure. A lottery-like stock typically has low price and larger risk (variance) than normal stocks. Most importantly, lottery stock has an extremely small probability of a huge reward, i.e. positively skewed returns. Stock with higher idiosyncratic skewness has a higher propensity to behave as a lottery than those with low idiosyncratic skewness. To explore how expected idiosyncratic skewness (EIS) affects momentum returns, we could analyze how EIS influences winner and loser portfolio returns. Winner stocks with relatively high expected idiosyncratic skewness have a larger propensity to behave like a lottery than those with low expected idiosyncratic skewness. In other words, high EIS winner stocks have relatively low probability than low EIS winner stocks to get huge reward. The same argument could also be applied to loser stocks: high EIS loser stocks have relatively low probability than low EIS loser stocks to get huge reward. From the previous literature such as Jegadeesh and Titman (1993, 2001), we know that the momentum anomaly is largely drven by winner stocks. The ability of winner stocks to generate significant subsequent positive abnormal returns could be constrained by stocks with high idiosyncratic skewness, i.e. lotto behavior. The underperformance of high idiosyncratic skewness winner stocks consequently leads to lower profits. Thus, we expect a negative relationship between expected idiosyncratic skewness and expected stock returns. 6

7 The rest of the paper is organized as follows. Section 2 introduces the sample selection rules and the construction of variables. Section 3 explores the relationship between expected skewness and momentum profits. Section 4 reports robustness checks. Section 5 concludes. 2. Data and Method 2.1 Sample construction rule We use CRSP database to generate momentum returns, including in our sample traded common stocks on the NYSE, AMEX and NASDAQ with share code 10 or 11 from January 1970 to December To be consistent with Jegadeesh and Titman (2001), we exclude penny stocks with price less than $5. We also exclude stocks with less than 24 months of continuous observations in CRSP for portfolio formation periods. 2.2 Momentum Following Jegadeesh and Titman (1993), we rank each stock at the beginning of each month by its past six month cumulative returns and we sort all stocks into 10 equally weighted portfolios. The past six month period is the portfolio formation period. The portfolio with the highest return (Q10) during the portfolio formation period is the winner portfolio. In contrast, the portfolio which has the lowest return (Q1) in the portfolio formation period is the loser portfolio. After defining the winner and loser portfolios, we track the two portfolios for the next six months. The difference between the post portfolio formation six month aggregated returns of winner (P10) and loser (Q1) portfolios is the momentum profit. 7

8 2.3 Expected idiosyncratic skewness and idiosyncratic volatility We follow the framework of Boyer, Mitton and Vorkink (2010) to construct expected idiosyncratic skewness. First, we need to define idiosyncratic skewness and idiosyncratic volatility. We estimate the residual from the following equation: (1) t represents the current month, T denotes the forecasting horizon. B(t) is the set of all trading days from month t-t+1 to month t. N(t) denote the trading days from month t-t+1 to the end of month t. stands for return of security i on day d. MKTRF, SMB and HML are Fama French three factors from Fama and French s website. is the residual from the regression on day d for stock i. The coefficient to calculate the residual is estimated using all daily data in B(t). Once we have the residual, we define the idiosyncratic skewness (is i, t ) and idiosyncratic volatility (ivol i,t ) as follows: ( ) ( ) (2) ( ) (3) If we replace with demeaned returns, we get a historical estimate of total volatility and total skewness. Based on Boyer, Mitton and Vorkink (2010), we assume investors time t expected idiosyncratic skewness is a conditional expectation formed on firm characteristics. And we first estimate cross-sectional regressions separately at the end of each month t, 8

9 (4) stands for user defined firm specific variables at month t-t. We then save the regression coefficients from regression (4) and use the information observable at the end of each month t, to generate the expected idiosyncratic skewness at firm level as follows: (5) The same method can also be applied to estimate expected total skewness. To avoid time varying parameter changes of the forecast regression, we use a shorter forecast horizon T=36 (i.e.3 years) than the horizon (5 years) used in Boyer, Mitton and Vorkink (2010). The firm specific variables we include in vector are momentum (mom i, t-t ), turnover (turn i,t-t ), leverage (lever i, t-t ) and dummy variables. We include three groups of dummy variables which are the same as those in Boyer, Mitton and Vorkink (2010): dummy variables for small and medium size firms; industry dummy variables using Fama-French 12 industry classification; and a NASDAQ dummy variable. The time span for our skewness data is from 1970 to Results 3.1 Summary statistics Table 1 reports the momentum profits for different time periods ( , , ). For the whole sample period from 1970 to 2012, the monthly average momentum profit is 9

10 0.57%. The period 1970 to 1999 is the period of highest momentum profits and the momentum crashes after 2000 (Daniel and Moskowitz (2012), Bhattacharya, Kumar and Sonaer (2013)). Our results confirm the booming and fade pattern of momentum. The monthly average momentum return from 1970 to 1999 is around 0.78% (t=2.99) which is much larger than that of the whole sample period. However, from 2000 to 2012, the average monthly momentum profit is an insignificant 0.07%. <Insert Table 1 Here> In panel A of table 2, we present the summary statistics for skewness related variables and variables with possible relation to momentum profits. A strong U-shaped pattern exists across deciles in expected idiosyncratic skewness, expected total skewness and idiosyncratic volatility. The similar patterns across quantiles for idiosyncratic volatility and two expected skewness variables imply that these variables have strong correlations. We also find that the winner and loser portfolios have lower market capitalization and price level but higher turnover. The winner and lower portfolios have an average market capitalization of $ million and $ million respectively. Meanwhile, the middle quintile (Q5) has a market capitalization of $ million. <Insert Table 2 Here> Panel B of table 2 presents the correlation matrix of all variables considered in panel A. We confirm the high correlations among expected idiosyncratic and total skewness and idiosyncratic volatility. The correlation between two skewness variables is surprisingly high at The correlations of each skewness variables with idiosyncratic volatility are all high and surpass 0.3. This is not surprising since expected skewness variables are partly generated from idiosyncratic volatility. The skewness variables do not have high correlations with price level, turnover and 10

11 market capitalization. This means the return skewness variables may contain different information. 3.2 Expected idiosyncratic skewness, expected total skewness and momentum To explore the relation between expected idiosyncratic skewness (EIS) and momentum, we divide the sample into three portfolios based on their magnitudes of expected idiosyncratic skewness. This method is similar to that of Lee and Swaminathan (2000). IS1 portfolio has the lowest skewness and IS3 portfolio has the highest skewness. Compared to stocks in low and medium skewness portfolio, stocks in portfolio IS3 perform most like lottery stocks. We then sort the whole sample of stocks based on past returns into ten momentum deciles and calculate momentum returns for each EIS portfolio, resulting in an independent double sort on EIS and momentum. Panel A of table 3 presents the results. Within each decile, EIS has a mostly negative effect on the cross section of returns. This is consistent with the basic results of Boyer et al. (2010). The average monthly momentum returns for low, medium and high EIS quintile is 0.65%, 0.8% and 0.05% respectively. The difference of momentum returns across EIS portfolios (EIS3-EIS1) is -0.6% monthly (-5.69). This shows that expected idiosyncratic skewness has a negative cross-sectional effect on momentum returns. <Insert Table 3 Here> From Panel A of table 3, we see that the decrease of momentum profits moving from low EIS to high EIS portfolio is driven primarily by winner stocks. The winner returns decrease from 2.03% (low EIS) to 1.64% (medium EIS), and finally to 0.81% (high EIS). The returns for the loser portfolio across EIS portfolios do not move that much. This phenomenon is consistent with our hypothesis outlined in the introduction: winner stocks in the high EIS group are very much 11

12 lottery-like. The lottery-like behavior of high EIS winner stocks leads the stocks to have low subsequent returns. We report Fama-French and Carhart four factor regression results for each EIS portfolios. We first run the four factor regression for all stocks ignoring the momentum quintiles. Then, we regress momentum returns for each EIS portfolio on the four factors. For the four factor regressions on different EIS portfolios ignoring momentum quintiles, the intercepts are all significant and decreasing by EIS. This phenomenon indicates that Fama-French and Carhart four factors do not explain the effect of EIS on cross sectional stock returns. Moreover, we regress the momentum returns for each EIS portfolio on the four factors. The intercepts for the low and high EIS portfolios are significant. The intercept of the high EIS portfolio is smaller than that of the low EIS portfolio. These results indicate that the effect of EIS on momentum returns cannot be explained by the four factor model and also confirm the negative relation between EIS and momentum returns. <Insert Table 4 Here> We then repeat the same procedure for expected total skewness. Panel A of table 4 reports the independent sort on expected total skewness (ETS) and momentum. We find ETS also has a negative effect on momentum returns. However, in panel B of table 4, the momentum returns are explained by Fama-French and Carhart four factor models, as evidenced by the statistically insignificant intercepts for the Q10-Q1 returns. 3.3 Control for idiosyncratic volatility 12

13 From panel B of table 2, we find expected skewness has high correlation with idiosyncratic volatility (IVOL). Moreover, IVOL has very strong explanatory power for expected idiosyncratic and total skewness (Boyer, Mitton and Vorkink (2010)). Arena, Haggard and Yan (2008) document a positive relation between IVOL and momentum returns. These findings give rise to a possibility that the explanatory power of expected skewness could be absorbed by IVOL. To test the incremental explanatory power of expected skewness on momentum return, we do a threeway sort on momentum, IVOL and EIS. <Insert Table 5 Here> Panel A of table 5 reports the results of conditional triple sort on momentum, IVOL and EIS. We first sort the sample by momentum to pick out winner and loser portfolios. Then, we divide the winner and loser portfolios each into 5 quintiles by magnitude of IVOL. In table 5, IVOL1 quintile corresponds to the portfolio with lowest IVOL. IVOL5 is the portfolio with the highest IVOL. In each IVOL quintile, we subdivide the quintile into 5 quintiles based on EIS. We generate momentum returns for each of these 25 portfolios. In panel A, momentum profits monotonically decrease by EIS. For example, in IVOL5 quintile, the momentum returns for low, medium and high EIS groups are 1.04%, 0.45%, -0.36%, -1.07% and -1.57% respectively. The momentum return differences between high and low EIS groups are all significant. So, controlling IVOL does not affect the explanatory power of EIS. On the other hand, in each EIS group except EIS group 1, momentum returns have a hump-shaped pattern across the idiosyncratic volatility quintiles. Panel B of table 5 reports results of triple sort on momentum, idiosyncratic volatility and expected total skewness (ETS). The results are not as strong as those for EIS. The momentum 13

14 return differences for high and low ETS are significant only in IVOL quintile 2, 4 and 5. Taking the results shown in panel A and B, it is safe to argue that expected idiosyncratic skewness and expected total skewness have incremental explanatory power on momentum returns even controlling for IVOL. The effect of IVOL on momentum and that of expected skewness on momentum cannot absorb each other. 3.4 Time series regression In this section, we explore the time series behavior of expected idiosyncratic skewness. To filter out noise, we do this analysis at annual frequency. We estimate the following regression: (6) is the aggregated momentum profit calculated by winner returns minus loser returns for year t., and are value weighted aggregated expected idiosyncratic and total skewness, and idiosyncratic volatility, respectively. is the lagged aggregated momentum return. We include this variable to capture the autocorrelation in the regression. A group of macro variables are also included. is the lagged term spread which is the difference between ten year T-bond yields and three month T-bill rates. The bond yields data is from Federal Reserve Bank of St. Louis website. is the lagged price earning ratio which is from Robert Shiller s website. is the lagged default spread which is calculated as the spread between Baa corporate bond yields and Aaa corporate yields. is the consumption-wealth ratio which is proposed by Lettau and Ludvigson (2001). We include 14

15 CAY because Lettau and Ludvigson argue that CAY helps to predict returns. CAY is obtained from Lettau s website. <Insert Table 6 Here> Table 6 summarizes the regressions including different sets of variables. In all combinations of explanatory variables, expected idiosyncratic skewness has a significant negative relation with subsequent momentum return. On contrary, expected total skewness has a very strong positive relation with subsequent momentum returns. The opposite sign of idiosyncratic and total skewness is surprising since they have a high positive correlation of Moreover, in all regressions, lagged idiosyncratic volatility has very strong explanatory power. We find a negative sign for the coefficient on lagged idiosyncratic volatility. Arena, Haggard and Yan (2008) report a positive sign in the time series regression of momentum return on idiosyncratic volatility using sample period 1965 to Since our construction of IVOL is based on Ang, Hodrick, Xing and Zhang (2006) but Arena, Haggard and Yan (2008) use methods by Ali, Hwang and Trombley (2003), we argue that different estimations of IVOL could be the reason for this contradiction. Similar to the conclusion of cross sectional test, the explanatory power of expected idiosyncratic skewness and idiosyncratic volatility cannot absorb each other. 4. Robust Check To confirm that our results are not constrained to one specific setting of portfolio formation and holding period, we change the formation and holding periods from (6,6) to others such as (9,3) and (12,3). Our cross sectional results still hold, i.e. expected idiosyncratic skewness has a negative effect on momentum returns. The results (unreported) are available from the authors upon request. The time series pattern also does not change. 15

16 <Insert Table 7 Here> Table 7 reports the time series pattern for different portfolio formation and holding settings. J stands for the months of portfolio formation. K stands for the months of portfolio holding. We see that the sign and significance level of expected idiosyncratic skewness, total skewness, and idiosyncratic volatility is stable and statistically significant. 5. Conclusion This paper examines the relationship between expected idiosyncratic skewness and momentum returns. We document a negative effect of expected idiosyncratic skewness on momentum returns. This negative relationship is driven by the lottery-like behavior of high expected idiosyncratic skewness winner stocks. This relationship is robust after controlling for idiosyncratic volatility in both cross sectional and time series regressions. The relationship also holds under different specifications of momentum portfolio formation and holding periods. We also find that expected total skewness has a surprising positive relationship with momentum profits. These results confirm that higher moments of returns have an effect on momentum returns and provide new insights into the momentum phenomenon. 16

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20 Table 1: Momentum Profits for Different Time Periods This table reports the monthly average momentum returns for different time periods. We include in our sample traded common stocks on the NYSE, AMEX and NASDAQ with share code 10 or 11 in the CRSP from January, 1970 to December, We exclude the stocks with price lower than $5 ( penny stocks ) following Jegadeesh and Titman (2001). We also exclude stocks with less than 24 months continuous observations in CRSP for portfolio formation periods. J=6, K= Loser 0.92% % % Q2 0.97% % % Q3 1.06% % % Q4 1.06% % % Q5 1.07% % % Q6 1.10% % % Q7 1.11% % % Q8 1.13% % % Q9 1.21% % % Winner 1.49% % % Q10-Q1 0.57% % %

21 Table 2: Summary Statistics for Momentum Portfolios This table reports the characteristics related to momentum returns for each momentum decile. Idio skew stands for expected idiosyncratic skewness. Total skew represents expected total skewness. IVOL is idiosyncratic volatility which is constructed as in Ang, Hodrick, Xing and Zhang (2006). Market Cap is the market capitalization. Turnover is defined as trading volume divided by shares outstanding. Panel A: Time Series of Cross-sectional Averages Portfolio Idio skew Total Skew IVOL Price Market Cap Turnover Loser Q Q Q Q Q Q Q Q Winner Idio Skew Panel B: Correlation Matrix Idio skew Total Skew IVOL Price Market Cap Turnover 1 Total Skew IVOL Price Market Cap Turnover

22 Table 3: Cross Sectional Test for Expected Idiosyncratic Skewness This table reports the cross sectional results for expected idiosyncratic skewness. In panel A, we do an independent sort on momentum and expected idiosyncratic skewness. In panel B, we report the Fama, French and Carhart four factor regressions for momentum returns of each EIS portfolio. Panel A: Momentum Profits by EIS EIS1(Low) EIS2(Medium) EIS3(High) EIS3-EIS1 Loser 1.38 (21.56) 0.83 (10.83) 0.76 (6.65) Q (29.75) 1.31 (24.45) 0.71 (9.66) Q (32.32) 1.38 (28.82) 1.11 (15.9) Q (32.89) 1.39 (30.93) 1.17 (18.24) Q (32.10) 1.38 (31.98) 1.13 (18.66) Q (33.94) 1.32 (31.7) 1.15 (19.79) Q (33.70) 1.31 (31.21) 1.12 (19.9) Q (34.39) 1.31 (31.55) 1.04 (18.01) Q (34.53) 1.40 (31.18) 0.99 (16.15) Winner 2.03 (35.11) 1.64 (26.95) 0.81 (9.77) Q10-Q (2.96) 0.80 (2.33) 0.05 (1.52) Panel B: Fama and French Regressions for EIS quantiles (-5.69) Intercept MKTRF SMB HML UMD Adj R square EIS1(Low) All Quantiles (25.16) (341.17) (79.76) (77) (-34.13) Q10-Q (-3.51) (0.4) (-0.43) (-0.12) (30.49) EIS2(Medium) All Quantiles (13.77) (318.38) (124.06) (84.19) (-50.67) Q10-Q (-1.56) (0.18) (-7.31) (-1.42) (26.31) EIS3(High) All Quantiles (-8.04) (193.93) (153.15) (74.61) (-41.37) Q10-Q (-4.17) (0.91) (-6.35) (0.16) (16.33) EIS3-EIS1 All Quantiles (-7.84) (-2.96) (18.32) (4.47) (-4.09) 22

23 Table 4: Cross Sectional Test for Expected Total Skewness This table reports the cross sectional results for expected total skewness. In panel A, we do an independent sort on momentum and expected total skewness. In panel B, we report the Fama, French and Carhart four factor regressions for momentum returns of each ETS portfolio. Panel A: Momentum Profits by ETS ETS1(Low) ETS2(Medium) ETS3(High) ETS3-ETS1 Loser Q Q Q Q Q Q Q Q Winner Q10-Q Panel B: Fama and French Regressions for ETS quantiles Intercept MKTRF SMB HML UMD Adj R square ETS1(Low) All Quantiles (45.45) (37.78) (14) (0.58) (-25.09) Q10-Q (1.12) (-2.45) (0.53) (-1.44) (1.08) ETS2(Medium) All Quantiles (40.21) (55.18) (14.01) (1.82) (-19.47) Q10-Q (1.38) (-3.36) (-0.06) (-0.58) (0.86) ETS3(High) All Quantiles (31.88) (68.06) (26.34) (10.63) (-11.05) Q10-Q (-1.16) (-4.1) (0.39) (-0.24) (-0.58) ETS3-ETS1 All Quantiles (-2.15) (6.71) (2.7) (1.29) (0.37) 23

24 Table 5: Control For Idiosyncratic Volatility This table reports the results of triple sort on momentum, idiosyncratic volatility and expected skewness. After sorting the sample by momentum, for each month, we sort the winner portfolio first into 5 quantiles based on magnitude of idiosyncratic volatility and then sort each idiosyncratic volatility quantiles into 5 sub-quantiles based on expected skewness. Winner portfolio is divided into 25 portfolios. We then do the same thing to the loser portfolio. By taking difference of winner and loser average returns for each of the 25 portfolio, we form the following two way table reporting the momentum returns for each IVOL-EIS or IVOL-ETS portfolio. Panel A: Triple Sort on Momentum, Idiosyncratic Volatility and Expected Idiosyncratic Skewness EIS1(Low) EIS2 EIS3(Medium) EIS4 EIS5(High) EIS5-EIS1 IVOL1 0.32% (1.07) 0.28% (0.94) 0.23% (0.74) 0.05% (0.16) 0.26% (0.81) -0.06% (-1.63) IVOL2 0.66% (1.89) 1.18% (3.23) 0.54% (1.39) 0.81% (2.09) 0.40% (1.05) -0.27% (-1.97) IVOL3 0.80% (2.05) 1.27% (2.93) 1.02% (2.56) 0.91% (2.19) 0.45% (1.11) -0.35% (-2.14) IVOL4 0.90% (1.77) 1.31% (2.81) 0.72% (1.49) 0.63% (1.35) 0.65% (1.45) -0.24% (-1.83) IVOL5 1.04% (1.99) 0.45% (0.81) -0.36% (-0.53) -1.07% (-1.39) -1.57% (-2.52) -2.61% (-4.55) Panel B: Triple Sort on Momentum, Idiosyncratic Volatility and Expected Total Skewness ETS1(Low) ETS2 ETS3(Medium) ETS4 ETS5(High) ETS5-ETS1 IVOL1 0.41% (1.39) 0.27% (0.88) 0.30% (1.02) -0.12% (-0.39) 0.30% (0.90) -0.12% (-1.54) IVOL2 0.75% (2.11) 1.05% (2.95) 0.70% (1.79) 0.88% (2.25) 0.24% (0.63) -0.51% (-1.72) IVOL3 0.69% (1.74) 1.25% (2.87) 1.15% (2.66) 0.82% (2.13) 0.54% (1.32) -0.15% (-1.38) IVOL4 1.07% (2.03) 1.01% (2.26) 1.26% (2.69) 0.44% (0.92) 0.45% (1.02) -0.61% (-1.95) IVOL5 1.18% (2.28) 0.59% (1.10) -0.81% (-1.15) -0.77% (-1.14) -1.70% (-2.67) -2.87% (-4.76) 24

25 Table 6: Time Series Regression of annual momentum returns This table reports the results of a time series regression of annualized momentum returns on expected skewness, idiosyncratic volatility and other variables. AGGEIS is the aggregated annual expected idiosyncratic skewness. EtotSkew is aggregated annual expected total skewness. Momeret is the lagged one year momentum return. AGGIVOL is the aggregated idiosyncratic volatility. is the lagged term spread which is the difference between ten year T-bond yields and three month T- bill rates. The bond yields data is from the Federal Reserve Bank of St. Louis website. is the lagged price earning ratio which is from Robert Shiller s website. is the lagged default spread which is calculated as the spread between Baa corporate bond yields and Aaa corporate yields. is the consumption-wealth ratio which is proposed by Lettau and Ludvigson (2001). The momentum return is constructed based on J=6 months portfolio formation period and K=6 month portfolio holding period. J=6, K=6 (1) (2) (3) (4) (5) Intercept *** *** *** (-0.39) (1.13) (3.36) (3.82) (3.21) AggEISt ** * * ** (-2.02) (-1.71) (-1.74) (-1.98) ETotSkewt ** *** *** *** (2.08) (3.03) (2.88) (2.77) MomRet t * * (0.88) (-0.61) (-1.88) (-1.46) (-1.72) AggIVOLt *** *** *** (-3.63) (-3.56) (-3.24) TERMt *** (-0.46) (-0.5) (0.19) (-0.58) PEt ** (-2.27) (-0.32) (1.25) (-0.38) Defaultt ** *** * *** (-2.28) (-3.21) (-1.81) (-2.78) CAYt *** (-0.34) (1.54) (2.57) (1.54) Adj R Square

26 Table 7: Time Series Robust Check This table provides the results of a robustness check of the time series regression in table 6. We vary the portfolio formation periods and portfolio holding periods of the momentum strategies. The results indicate that the significance level and the magnitude of expected idiosyncratic skewness is quite stable across different momentum strategies. J=12, K=3 J=12, K=6 J=9, K=3 J=9, K=6 J=6, K=3 Intercept *** *** 0.085*** *** *** (3.64) (3.58) (3.34) (3.33) (3.23) AggISt * * ** ** * (-1.78) (-1.82) (-2.15) (-2.29) (-1.71) TotSkewt ** ** *** *** ** (2.21) (2.54) (2.81) (2.56) (2.22) MomRett ** ** * ** (-2.37) (-2.38) (-1.95) (-2.32) (-0.68) AggIVOLt *** *** *** *** *** (-2.81) (-2.62) (-3.34) (-2.67) (-3.62) TERMt (-0.98) (-1.11) (-0.72) (-0.71) (-1.35) PEt (-1.12) (-1.41) (-0.55) (-1.11) ((0.32) Defaultt *** *** *** *** ** (-3.13) (-3.36) (-3.1) (-3.25) (-1.91) CAYt (1.1) (0.95) (1.36) (1.07) (1.54) Adj R Square