Higher Moment Gaps in Mutual Funds

Transcription

1 Higher Moment Gaps in Mutual Funds Yun Ling Abstract Mutual fund returns are affected by both unobserved actions of fund managers and tail risks of fund returns. This empirical exercise reviews the return gap documented by Kacperczyk, Sialm, and Zheng (2007) and quantifies the higher moment gaps. We study higher standardized moment gaps and find substantial cross-sectional variations in volatility gap, skewness gap and kurtosis gap, and strong persistence in the standardized higher moment gaps. We also study higher central moment gaps and find that the past central thrid and fourth moment gaps help to predict fund performance. As documented by Kacperczyk, Sialm, and Zheng (2007), the return gap, defined as the difference between the reported fund return and the return on a portfolio that invests in the most recently disclosed holdings, helps to measure the unobserved actions of fund managers and predicts fund performance. Since it is well known that mutual fund returns are subject to substantial tail risks, which can be characterized by higher moments of returns, one natural way to study the impact of unobserved actions on tail risks of fund returns is to examine the higher moment gaps, which are defined by analogy, as the difference between the higher moments of the reported returns and the corresponding higher moments of the portfolio that invests in the most recently disclosed holdings. Recent studies have developed intuition in modeling stochastic discount factor with higher moments of returns, and demonstrated the empirical relevance of incorporating volatility, sknewness, and kurtosis in analyzing investment fund returns. Krauss and Litzenberger (1976) document the importance of considering the skewness of returns, and later extended by Emmanuel and Bertrand (2000) to a three-moment CAPM framework. Robert Dittmar (2002) investigates nonlinear pricing kernels in which the stochastic discount factor takes a polynomial form with respect to return and studies higher moment preference up to the fourth moment (kurtosis). There are also numerous literature studying the value of higher moments of return capturing tail risks in investment fund performance evaluation, such as Vikas, Gurdip, and Joop (2008), David and Rosa (2009), and Jerchern Lin (2011). This empirical exercise is ERO #1 (substitued for the summer empirical exercise 5) Address: Yun.Ling.2015@marshall.usc.edu 1

2 Although the importance of higher moments in studying investment fund returns has been largely documented, there are limited studies focusing on the internal factors that contribute to fund tail risks. This empirical exercise, in particular, studies the role of unobserved actions of fund managers in generating tail risks in mutual fund. There are three higher moment gaps togerther with the return gap examined here the volatility gap, the skewness gap, and the kurtosis gap, and their corresponding central version. Higher than the fourth moment is traditionally not studied because of lack of economic interpretation or statistical significance. We find substantial corsssectional variation in volatiltiy gap, skewness gap and kurtosis gap, and strong persistence in the standardized mement gaps. We also show that the past central third and fourth moment gaps help to predict fund performance by forming momentun trading strategies. The remainder of this report is organized as follows. Section 1 defines the higher moment gaps and motivates the use of these gaps in assessing the effect of unobserved actions on higher moments of returns. Section 2 describes data sources. Section 3 quantifies the higher moment gaps by examining the significance and persistence. Section 4 studies the relationship between the past higher moment gaps and future fund performance. Section 5 concludes and extends this empirical exercise. 1 The higher moment gaps To evaluate the impact of unobserved actions, Kacperczyk, Sialm, and Zheng (2007) define the return gap as follows, which is based on the comparison of the net investor return and the net return of the fund s holdings: RG f t = (RF f t + EXP f t ) RH f t where RG f t, RF f t, EXP f t and RH f t are the return gap, the net investor return, the expense ratio, and the hypothetical holding return of fund f at time t, respectively. Taking rret f t RF f t + EXP f t as the raw return, and hret f t = RH f t as the holding return, we can define the second, third and fourth central moment gaps as: V ar C Gap f t [ ( κ 2 (rret f t ) κ 2 (hret f t ) = E rret f t E(rret f t ) Skew C Gap f t [ ( κ 3 (rret f t ) κ 3 (hret f t ) = E rret f t E(rret f t ) Kurt C Gap f t κ 4 (rret f t ) κ 4 (hret f t ) = E ) 2 ] ) 3 ] [ ( ) ] 4 rret f t E(rret f t ) E [ ( ) ] 2 E hret f t E(hret f t ) [ ( ) ] 3 E hret f t E(hret f t ) [ ( ) ] 4 hret f t E(hret f t ) 2

3 and the first moment gap, corresponding to the return gap, can be redefined as: [( )] [( )] Mean C Gap f t κ(rret f t ) κ(hret f t ) = E rret f t E(rret f t ) E hret f t E(hret f t ) where κ n (.) denote the central n th moment and the expectation is taken as the backward moving average for the past J months. Similarly, we can define the standardized moments (mean gap, volatility gap, skewness gap and kurtosis gap) as: MeanGap f t κ(rret f t ) κ(hret f t ) = Mean C Gap f t V argap f t κ 2 (rret f t ) κ 2 (hret f t ) = V ar C Gap f t SkewGap f t κ 3(rret f t ) κ 3/2 f t ) 2 (rret f t ) κ 3(hret κ 3/2 2 (hret f t ) KurtGap f t κ 4(rret f t ) κ 2 2 (rretf t ) κ 4(hrett ) κ 2 2 (hretf t ) Note that the standandized and central first and second moment gaps are the same, while the standardized third and fourth moment gaps are the central third and fourth moment gaps of standardized returns. As the return gap captures the fund s unobserved actions, which include hidden benefits and hidden costs, the higher moment gaps characterize the effect of fund s unobserved actions on return tail distribution. While hidden costs, which includes trading costs, commissions, agency costs and investor externalities, might have little impact on higher moment gaps, the hidden benefits, among which an important component is fund s interim trades, should play a major role in determining higher moment return risks. If these interim trades are able to decrease volatility, then the volatility gap should decrease since κ 2 (rret f t ) will decrease and κ 2 (hret f t ) will remain the same. Similarly, interim trades which succeed in increasing (decreasing) skewness or kurtosis should lead to increase (decrease) in skewness or kurtosis gaps. f 2 Data As Kacperczyk, Sialm, and Zheng (2007), we merge the CRSP Survivorship Bias Free Mutual Fund Database, the Thompson Financial CDA/Spectrum holdings database, and the CRSP monthly stock return data. Our sample covers the time period between 1986 and 2010, while the previous paper covers time period between 1984 and We also focus our analysis on open-end domestic equity mutual funds and exclude funds that hold fewer than 10 stocks and those which in the previous month managed less than $5 million. There are 3139 distinct funds and 380,863 fund- 3

4 month observations. The number of funds ranges from 79 to 1977 and there are 1032 funds per month on average. 3 Quantifying the higher moment gaps In this section, we quantify the aggregate higher moment gaps between 1986 and 2010 and discuss significance and persistence. 3.1 Significance of the higher moment gaps Table 1 and 2 present the equal- and value- weighted averages of the standardized and central MeanGap, VarGap, SkewGap and KurtGap. We obtain the higher moment gaps by calculating the differences between the 12 months backward higher moments (both standardized and central) of the net investor return and the net return of the fund s holdings, and obtain the statistics by first computing the cross-sectional means in each month and then reporting the time-series means along with the corresponding standard errors. The raw higher moment gaps are calculated using raw excess return (R i,t R F,t ), and the abnormal higher moment gaps are calculated using excess returns over market excess return ((R i,t R F,t ) (R M,t R F,t )). Similar to Kacperczyk, Sialm, and Zheng (2007), we also report the higher moment gaps based on abnormal returns after adjusting for the factor loading using the one-factor CAPM, Fama French three-factor model, and the Carhart four-factor model by running rolling regressions with a window size of two years: R i,t R F,t = α i + β i,m (R M,t R F,t )e i,t R i,t R F,t = α i + β i,m (R M,t R F,t ) + β i,smb SMB t + β i,hml HML t + e i,t R i,t R F,t = α i + β i,m (R M,t R F,t ) + β i,smb SMB t + β i,hml HML t + β i,mom MOM t + e i,t Using equal-weighted average, the raw MeanGap and VarGap are about and per month and the t-values are and 0.303, both statistically insignificant. The raw standardized and central SkewGap are about and per month with t-values and , still insignificant. The raw standardized and central KurtGap are about and per month with t-values and Consistent with the results in Kacperczyk, Sialm, and Zheng (2007), the mean return gap is not significantly different from zero, and the same conclusion extends to all central and standardized higher moment gaps. Similarly, using the value-weighted average, all higher moment gaps are again not significantly different from zero. The same result stands using risk-adjusted net investor return and risk-adjusted net return of the fund s holdings. To summarize, we find that, in the aggregate sample, all higher moment gaps are very small, 4

5 indicating that fund unobserved actions generate insignificant tail risks of fund returns on average, or, fund interim trades proceed without significantly changing the higher moments of net investor return in aggregate, given that the impact of hidden costs, including trading costs, commissions, agency costs and investor externalities, is neglectable. 3.2 Persistence of the higher moment gaps As documented in Kacperczyk, Sialm, and Zheng (2007), many features of the unobserved actions indicate that such actions should be persistent, especially,steady interim trading patterns persistent higher moment gaps. In this subsection, we study the short-term and long-term persistence of the higher moment gaps Short-Term Persistence To test whether the higher moment gaps are persistent in short run, we sort all funds in our sample into 10 deciles in an ascending order according to their one-month lagged higher moment gaps during the previous 12, 36, and 60 months, and compute the average higher moment gaps during the subsequent month by weighting all funds in each decile equally. Table 3 reports the raw and Carhart-adjusted standardized (or central) 1 st and 2 nd moment gaps of the decile portfolios. Table 3 and 4 report the raw and Carhart adjusted standardized and central 3 rd and 4 th moment gaps. All raw standardized moment gaps display strong short-term persistence, with Spearman correlation equal to 1. With Carhart-adjusted standardized moments, the lowest Spearman correlation equals 0.865, occuring at the VarGap calculated during the previous 12 months (1 year). Thus, all standardized moment gaps, both raw and Carhart-adjusted, are significantly persistent in the short term. As for the central higher moment gaps, raw gaps still display strong persistence, with all Spearman correlations above With Carhart-adjusted central moments, the lowest Spearman correlation is 0.722, at the KurtGap calculated during the previous 60 months (5 years). Furthermore, all standardized and central average moment gaps of the 10-1 and second half - first half portfolios are significantly positive at 5-percent level with only two exceptions, occuring at the central KurtGap calculated during the previous 12 months and 36 months (1 year and 3 years). Note that even though here we only explore the marginal ranking of each moment gap unconditionally, the short-term persistence of all higher moment gaps is obvious Long-Term Persistence We also study the long-term stability of the above observed patterns by tracking the average higher moment gaps of each decile over the subsequent 5 years, where the experiment is only performed on higher moment gaps calculated during the pervious 1 year with raw and Carhart-adjusted returns. 5

6 Figure 1 depicts the future standardized (central) MeanGap and VarGap (using raw and Carhartadjusted returns), and Figure 2 and 3 depict the future SkewGap and KurtGap. All higher moment gaps are over the subsequent 60 months (5 years). Consistent with the findings in Kacperczyk, Sialm, and Zheng (2007), MeanGap is persistent over a subsequent 60-months period using both raw and Carhart-adjusted abnormal returns. Moreover, the VarGap also display strong long-term persistence using raw returns and the pattern dimishes while using Carhart-adjusted abnormal returns. On ther other hand, the previous rankings of the standardized SkewGap and KurtGap only last for about 12 months for raw returns and 6 months for Carhart-adjusted returns, after which the average gaps of all deciles collapse together. Compared with the apparent patterns of standardized higher moment gaps, the patterns of the central SkewGap and KurtGap are less obvious. Although the short-term persistence in the subsequent one month can be detected from the figure, the pattern disappears quickly beyond one month and the order of higher moment gaps of different decile portfolios is completely independent of the original ranking. All pattern diminish more quickly using Carhart-adjusted returns, and there s no subsequent difference of higher moment gaps across different decile portfolios. To sum up, the MeanGap exhibits strong long-term persistence using both raw and Carhartadjusted returns; the VarGap is also persistent in the long run and using Carhart-adjusted returns doesn t change the ordering of subsequent decile gaps substantially; additionaly, standardized SkewGap and KurtGap only exhibit short-term persistence over 12 months using raw returns and 6 months using Carhart-adjusted returns, beyond which period there s no difference in higher moment gaps across different decile portfolios. Contrary to the persistent patterns of standardized higher moment gaps, there s no clear pattern for the central SkewGap and KurtGap, though the gaps do differ among distinct decile portfolios when using raw returns, they all diminish to zero when using Carhart-adjusted returns. 4 Trading strategies based on the higher moment gaps In this section, we study whether higher moment gaps can predict fund performance. First, we study the future returns of decile portfolios formed based on the marginal unconditional rankings of each higher moment gap. We then examine two extensions, one based on the marginal unconditional ranking of some higher moment gap while controlling others, and the other based on the ranking of some higher moment gaps conditioning on the marginal rankings of others. 6

7 4.1 Strategies based on the marginal unconditional rankings of higher moment gaps Our first experiment tests whether marginal information of each higher moment gap can predict fund performance. We sort all funds in our sample into 10 deciles according to their past moment gaps, respectively, calculated during the previous 12 months. We then compute for each month the average subsequent return by weighting all funds in a decile equally. Similar to Kacperczyk, Sialm, and Zheng (2007), we introduce a 3-month lag in the higher moment gaps before implementing the trading strategy, which allows for at least a 4-month window for the holdings information to become public. Table 6 reports the raw, abnormal and risk-adjusted abnormal future returns using the rankings of MeanGap and VarGap. Table 7 and 8 report the results for standard and central SkewGap and KurtGap. Figure 4 and 5 plot the portfolio returns of the 10 deciles using different risk-adjusted models and higher moment gaps. Similar to the persistence test, the returns of 10-1 and second half - first half portfolios are calculated, along with the Spearman correlation statistics for the decile portfolios. The MeanGap is reexamined and confirmed to contain valuable information predicting future fund performance, with all 10-1 and second half - first half portfolios exerting significant returns, and lowest Spearman correlation is equal to among all risk-adjustment models. Furthermore, although the patterns of the portfolio returns based on all other three standardized higher moment gaps are not very clear from the graph, past central SkewGap (KurtGap) seems to be postively (negatively) correlated with future returns, while past VarGap seems to contain no information predicting future returns, as shown by the Spearman correlation statistics. The returns of second half - first half portfolio based on central SkewGap are even significant at 5-percent level, with a lowest t-value equal to 2.73 among all risk-adjusted models. However, the observed correlation between past higher moment gaps and future returns could be spurious because the past higher moment gaps could contain no information related to the future fund performance but only related to the MeanGap. Thus, we continue to explore the role of past higher moment gaps in predicting future returns in absence of the effect of MeanGap, by forming strategies based on unconditional rankings of higher moment gaps when controlling MeanGap, and on rankings of higher moment gaps conditioning on the marginal ranking of MeanGap. 4.2 Strategies based on the joint unconditional rankings of higher moment gaps Controlling the ranking of MeanGap To control the effect of past MeanGap on future returns when studying other higher moment gaps, we divide all funds in our sample into at most 100 cells according to their joint unconditional 7

8 ranking of MeanGap and the higher moment gap we want to study. We then form the equalweighted portfolio and generate risk-adjusted abnormal returns for each cell. Lastly, we weight the abnormal returns equally across the cells with the same ranking of the target higher moment gap. Since the unconditional rankings of past MeanGap and the target moment gap don t depend on each other, it is possible that there s no fund in one specific group with a rank of m for MeanGap and n for the target moment gap, thus, the actual number of groups with positive numbers of funds could be less than 100. Also, funds are not evenly distributed for all cells. Compared with the method using conditional ranking explored in the next subsection, this method using joint unconditional ranking has some advantages and disadvantages. One advantage is that we can take care of the correlations of any two past moment gaps fairly, rather than hiding all interaction effect into the moment gap that we sort last. Besides, by controling MeanGap, we can focus on the predictability of other moment gaps, without the disturbance of the effect from MeanGap. However, one problem is that we might have far less groups with positive numbers of funds if the correlations between moment gaps are strong. The worst case is that we could only have 10 non-empty cells. For this reason, the experiment based on conditional rankings of higher moment gaps will follow next. Table 9 and 10 report the portfolio returns based on the unconditional rankings of standardized and central VarGap, SkewGap, and KurtGap controlling MeanGap. Compared with the previous results based on marginal unconditional rankings of higher moment gaps, the sign of Spearman correlation doesn t change, and all results for the central moment gaps are relatively more significant than for the standardized moment gaps. Still, among all higher moment gaps, the VarGap is least correlated to future returns. However, there are two changes here. First, the returns of 10-1 and second half - first half portfolios are not significant for the central SkewGap after controlling MeanGap, which implies that some predictability of marginal unconditional ranking of central SkewGap comes from its positive correlation with MeanGap. Second, the absolute value of Spearman correlation statistics increases for the central KurtGap using all risk-adjusted returns, which implies that some predictability of marginal unconditional ranking of central KurtGap is weakened by its negative correlation with MeanGap. This can also be seen from Figure 6 and 7, where the graph for central KurtGap display clear downward pattern on average More on controlling higher moment gaps To explore more hidden correlations among higher moment gaps and the predicting power of them on future fund performance, we repeat the above strategy after controlling different combinations of unconditional rankings of moment gaps. Figure 7 gives the plots for controlling MeanGap and VarGap, and Figure 8 gives the plots for controlling all other three moment gaps. Table 13 also summarizes the Spearman correlation, for abnormal return ((R i,t R F,t ) (R M,t R F,t ), defined as the raw model) and Carhart-adjusted abnormal return. Past central SkewGap and KurtGap continue to exhibit positive and negative correlations with future fund performance after controlling 8

9 the first two moment gaps. However, one thing to notice is that the results for controlling three moment gaps are not reliable, since there are far less groups containing positive numbers of funds than supposed. (There are 10,000 cells to be filled in but only 1032 funds per month on average in our sample, and the actual non-empty cells are much less than 1000.) Thus, the Spearman correlations for the central KurtGap and SkewGap flip signs unexpectedly in Table 13. It can also be easily detected from the flat pattern of abnormal returns across different deciles in Figure 8. To alleviate this problem and to make funds in our sample more evenly distributed across different groups, we explore the trading strategies based on the conditional rankings of higher moment gaps next. 4.3 Strategies based on the conditional rankings of higher moment gaps Conditional rankings of higher moment gaps on the ranking of MeanGap We obtain the conditional rankings of higher moment gaps by first sort the MeanGap unconditionally as before and then sort the higher moment gap we want to study within each deciles of MeanGap. This method guarantees that there are almost equal numbers of funds within each group (100 groups in this case), however, the drawback is that the predictability resulted from the correlation cannot be disentangled from the predictability of the target higher moment gap. Therefore, we expect to see that the Spearman correlation of the central SkewGap blow up because of its positive correlation with MeanGap, and similarly the Spearman correlation of the central KurtGap will shrink. Fortunately, since we can difference out the fixed effect of MeanGap for each deciles with the same unconditional ranking of MeanGap, the results for the 10-1 and second half - first half portfolios should remain more credible. Table 11 and 12 report the portfolio returns based on the conditional rankings of standardized and central VarGap, SkewGap, and KurtGap, conditioning on the ranking of MeanGap. As expected, the t values for the returns of the 10-1 and second half - first half portfolios increase, which indicates that the conditioning method, after differencing out fixed effect, should be more reliable than using the controlling method. Table 13 also report the Spearman correlations for the raw and Carhart models, and the signs remain the same. Also as predicted, the absolute values for SkewGap (KurtGap) are larger (small) than using the controlling method. Figure 9 gives the graphs of risk-adjusted returns for the 10 deciles of higher moment gaps. Same as before, we still cannot detect any predictability of VarGap from the figure More on the conditional rankings of higher moment gaps Similarly, we explore the strategies based on the rankings of SkewGap and KurtGap conditioning on MeanGap and VarGap, and the strategies based on one target moment gap conditioning on all other three moment gaps. Figure 10 and 11 give the plots for those results, and Table 13 gives the 9

10 Spearman correlation statistics. Note first that the rankings of gaps to be conditioned on are all unconditional rankings, which is to guarantee that the order of those base rankings doesn t affect the result. 1 Furthermore, the conditioning method seems to alleviate the problem of controlling method by generating enough fluctuations in returns across different groups conditioning on multiple moment gaps, as shown in Figure 11. To sum up, all results for trading strategies based on reliable unconditional or conditional rankings of higher moment gaps (after controlling other moment gaps) support that past central SkewGap (KurtGap) is positively (negatively) correlated with future fund performance, while the standardized version displays much weaker correlation, and there seems no predictability of past VarGap for future fund returns. 5 Conclusions and extensions In this empirical exercise, we extend the study of Kacperczyk, Sialm, and Zheng (2007) and analyze the impact of unobserved actions on tail risks of fund returns by examining higher moment gaps, defined as the difference between higher moments of the net investor return and the net return of the fund s holdings. We find that there is substantial cross-sectional variation and short-term persistence among all standardized and central higher moment gaps. While the first two moment gaps exhibit strong long-term persistence, the standardized third and Fourth moment gaps exhibit strong median-term persistence. We also find that central third moment gap predicts future positive abnormal returns while central fourth moment gap predicts future negative abnormal returns by exploring trading strategies based on conditional and unconditional rankings of those moment gaps (after controlling other moment gaps). What remains to do is to analyze the determinants of higher moment gaps, and divide our sample into groups pf fund styles to carry on a more detailed study. Intuitively, the impact of unobserved actions on tail risks should vary substantially across different fund styles. One can also perform the same experiment for higher co-moments (eg: covariance, coskewness, and cokurtosis) between the raw return and the holding return. References [1] Campbell R. Harvey and Akhtar Siddique, 2000, Conditional skewness in asset pricing tests, Journal of Finance 55, If the base rankings are not unconditional, it could be the case that, for example, the ranking of SkewGap conditioning on MeanGap and VarGap depends on whether MeanGap or VarGap is sorted first, since the moment gap that is sorted next will be the conditional ranking based on the first moment gap. Therefore, the conditioning method cannot exclude empty-cell problem arising using the controlling method, but can at least restrict it to a second-order effect. 10

11 [2] Robert F. Dittmar, 2002, Nonlinear pricing kernels, kurtosis preference, and evidence from the cross section of equity returns, Journal of Finance 57, [3] Marcin Kacperczyk, Clemens Sialm, and Lu Zheng, 2007, Unobserved actions of mutual funds, Review of Financial Studies 21, [4] David Moreno and Rosa Rodríguez, 2009, The value of coskewness in mutual fund performance evaluation, Journal of Banking & Finance 33, [5] Jerchern Lin, 2011, Tail risks across investment funds, working paper. [6] Vikas Agarwal, Gurdip Bakshi, and Joop Huij, 2008, Implications of volatiltiy, skewness, and kurtosis risks for hedge fund performance, working paper. 11

12 Figure 1: Persistence of 1 st and 2 nd moment gaps 1

13 Figure 2: Persistence of standardized and central higher 3 rd moment gaps 2

14 Figure 3: Persistence of standardized and central higher 4 th moment gaps 3

15 Figure 4: Portfolio Returns based on the marginal unconditional rankings of higher moment gaps (standardized) 4

16 Figure 5: Portfolio Returns based on the marginal unconditional rankings of higher moment gaps (central) 5

17 Figure 6: Portfolio Returns based on the joint unconditional rankings of higher moment gaps (control for MeanGap) 6

18 Figure 7: Portfolio Returns based on the joint unconditional rankings of higher moment gaps (control for MeanGap and VarGap) 7

19 Figure 8: Portfolio Returns based on the joint unconditional rankings of higher moment gaps (control for the other three higher moment gaps) 8

20 Figure 9: Portfolio Returns based on the conditional rankings of higher moment gaps (conditioning on MeanGap) 9

21 Figure 10: Portfolio Returns based on the conditional rankings of higher moment gaps (conditioning on MeanGap and VarGap) 10

22 Figure 11: Portfolio Returns based on the conditional rankings of higher moment gaps (conditioning on the other three higher moment gaps) 11

23 Table 1: Significance of the higher moment gaps (equal weighted) MeanGap VarGap SkewGap KurtGap standardized central standardized central standardized central standardized central Raw Raw Raw Raw Mean Std t Abnormal Abnormal Abnormal Abnormal Mean Std t CAPM CAPM CAPM CAPM Mean Std t FF3 FF3 FF3 FF3 Mean Std t Carhart Carhart Carhart Carhart Mean Std t

24 Table 2: Significance of the higher moment gaps (value weighted) MeanGap VarGap SkewGap KurtGap standardized central standardized central standardized central standardized central Raw Raw Raw Raw Mean Std t Abnormal Abnormal Abnormal Abnormal Mean Std t CAPM CAPM CAPM CAPM Mean Std t FF3 FF3 FF3 FF3 Mean Std t Carhart Carhart Carhart Carhart Mean Std t

25 Table 3: Persistence of the higher moment gaps (1 st and 2 nd moments) Portfolios second half Spearman first half correlation MeanGap - Raw (10 3 ) 1Y (0.174) (0.0845) (0.0645) (0.0541) (0.0481) (0.0471) (0.0516) (0.0616) (0.0810) (0.165) (0.282) (0.467) 3Y (0.0923) (0.0522) (0.0409) (0.0341) (0.0307) (0.0288) (0.0294) (0.0346) (0.0490) (0.0963) (0.158) (0.282) 5Y (0.0682) (0.0446) (0.0367) (0.0309) (0.0272) (0.0245) (0.0241) (0.0281) (0.0402) (0.0699) (0.113) (0.214) MeanGap - Carhart (10 3 ) 1Y (0.0715) (0.0316) (0.0231) (0.0189) (0.0167) (0.0142) (0.0152) (0.0194) (0.0315) (0.105) (0.160) (0.234) 3Y (0.0463) (0.0237) (0.0181) (0.0156) (0.0140) (0.0134) (0.0137) (0.0191) (0.0305) (0.0885) (0.122) (0.180) 5Y (0.0332) (0.0197) (0.0142) (0.0138) (0.0132) (0.0126) (0.0147) (0.0195) (0.0303) (0.0792) (0.0978) (0.149) VarGap - Raw (10 4 ) 1Y (1.10) (0.519) (0.386) (0.304) (0.244) (0.189) (0.152) (0.112) (0.111) (0.435) (1.40) (2.32) 3Y (0.761) (0.334) (0.247) (0.201) (0.159) (0.135) (0.113) (0.0831) (0.0651) (0.311) (0.904) (1.48) 5Y (0.634) (0.285) (0.212) (0.169) (0.138) (0.119) (0.108) (0.0843) (0.0661) (0.312) (0.729) (1.21) VarGap - Carhart (10 6 ) 1Y (1.13) (0.236) (0.102) (0.0929) (0.320) (0.549) (0.207) (0.289) (0.872) (0.468) (1.40) (1.87) 3Y (0.884) (0.0739) (0.110) (0.0940) (0.137) (0.0614) (0.140) (0.110) (0.210) (0.648) (1.38) (1.43) 5Y (0.619) (0.133) (0.101) (0.151) (0.0996) (0.102) (0.0785) (0.101) (0.0952) (0.525) (1.06) (1.13) 3

26 Table 4: Persistence of the higher moment gaps (3 rd moment) Portfolios second half Spearman first half correlation Standardized SkewGap - Raw (10 2 ) 1Y (0.822) (0.411) (0.277) (0.214) (0.163) (0.157) (0.163) (0.211) (0.324) (0.789) (1.49) (2.67) 3Y ) (0.516) (0.338) (0.249) (0.171) (0.126) (0.116) (0.131) (0.259) (0.997) (1.97) (3.35) 5Y (1.04) (0.498) (0.357) (0.277) (0.221) (0.190) (0.100) (0.110) (0.189) (1.06) (1.94) (3.15) Standardized SkewGap - Carhart (10 2 ) 1Y (0.972) (0.442) (0.283) (0.195) (0.148) (0.142) (0.175) (0.285) (0.444) (0.991) (1.87) (3.32) 3Y (1.15) (0.436) (0.267) (0.182) (0.151) (0.163) (0.219) (0.332) (0.569) (1.22) (2.26) (3.88) 5Y (1.28) (0.467) (0.287) (0.189) (0.167) (0.221) (0.316) (0.474) (0.699) (1.56) (2.67) (4.57) Central SkewGap - Raw (10 5 ) 1Y (1.99) (0.564) (0.233) (0.250) (1.12) (0.468) (0.614) (0.783) (1.08) (3.91) (4.81) (6.60) 3Y (1.31) (0.167) (0.339) (0.104) (0.173) (0.221) (0.276) (0.362) (0.502) (3.36) (3.94) (4.41) 5Y (0.807) (0.111) (0.0642) ) (0.605) (0.141) (0.177) (0.232) (0.303) (3.22) (3.44) (3.78) Central SkewGap - Carhart (10 8 ) 1Y (4.90) (2.84) (3.16) (3.03) (2.40) (8.26) (25.1) (2.73) (13.4) (9.02) (11.2) (41.6) 3Y (6.80) (0.897) (1.14) (3.37) (2.81) (0.864) (3.02) (1.87) (1.62) (6.39) (9.82) (11.1) 5Y (3.48) (2.81) (0.255) (0.280) (1.06) (2.55) (1.72) (3.89) (1.24) (4.77) (7.31) (9.51) 4

27 Table 5: Persistence of the higher moment gaps (4 th moment) Portfolios second half Spearman first half correlation Standardized KurtGap - Raw (10 1 ) 1Y (0.262) (0.0933) (0.0609) (0.0427) (0.0335) (0.0398) (0.0570) (0.0813) (0.128) (0.260) (0.497) (0.873) 3Y (0.487) (0.174) (0.0859) (0.0602) (0.0483) (0.0809) (0.118) (0.183) (0.282) (0.610) (1.06) (1.92) 5Y (0.392) (0.137) (0.0736) (0.0569) (0.0752) (0.101) (0.149) (0.211) (0.307) (0.803) (1.12) (1.90) Standardized KurtGap - Carhart (10 1 ) 1Y (0.204) (0.0935) (0.0585) (0.0452) (0.0362) (0.0324) (0.0403) (0.0583) (0.0920) (0.200) (0.377) (0.689) 3Y (0.347) (0.145) (0.0876) (0.0619) (0.0450) (0.0406) (0.0443) (0.0676) (0.126) (0.290) (0.617) (1.06) 5Y (0.508) (0.241) (0.156) (0.102) (0.0636) (0.0619) (0.0785) (0.0872) (0.150) (0.383) (0.850) (1.52) Central KurtGap - Raw (10 5 ) 1Y (2.05) (0.605) (0.725) (0.323) (0.247) (0.192) (0.833) (1.52) (0.0902) (6.13) (6.37) (6.92) 3Y (1.64) (0.249) (0.177) (0.138) (0.109) (0.0876) (0.345) (0.0385) (0.0740) (4.76) (5.34) (5.44) 5Y (0.938) (0.170) (0.977) (0.0938) (0.0698) (0.0846) (0.0387) (0.0329) (0.0236) (4.19) (4.34) (4.60) Central KurtGap - Carhart (10 8 ) 1Y (1.98) (0.178) (1.67) (0.495) (0.535) (0.227) (2.06) (0.765) (9.85) (7.93) (8.18) (13.3) 3Y (1.87) (0.0679) (0.207) (0.513) (0.0834) (1.10) (0.120) (0.261) (1.93) (0.762) (2.11) (2.14) 5Y (0.806) (0.149) (0.0946) (0.910) (1.09) (0.296) (0.0602) (1.02) (0.0870) (0.425) (1.04) (1.41) 5

28 Table 6: Portfolio Returns based on the marginal unconditional rankings of 1 st and 2 nd moment gaps (in percentage) Portfolio second half spearman first half correlation MeanGap excess return (0.080) (0.051) (0.044) (0.043) (0.044) (0.045) (0.047) (0.052) (0.067) (0.114) (0.083) (0.161) CAPM alpha (0.081) (0.051) (0.044) (0.043) (0.043) (0.044) (0.047) (0.052) (0.068) (0.113) (0.081) (0.157) FF alpha (0.062) (0.040) (0.040) (0.040) (0.039) (0.041) (0.042) (0.045) (0.051) (0.065) (0.069) (0.145) Carhart alpha (0.063) (0.041) (0.040) (0.040) (0.039) (0.042) (0.043) (0.046) (0.052) (0.063) (0.070) (0.148) VarGap excess return (0.116) (0.071) (0.059) (0.051) (0.050) (0.048) (0.045) (0.043) (0.042) (0.089) (0.087) (0.193) CAPM alpha (0.117) (0.072) (0.059) (0.050) (0.048) (0.047) (0.044) (0.043) (0.042) (0.080) (0.077) (0.189) FF alpha (0.080) (0.047) (0.043) (0.044) (0.043) (0.043) (0.040) (0.038) (0.038) (0.052) (0.071) (0.137) Carhart alpha (0.081) (0.048) (0.043) (0.045) (0.043) (0.044) (0.041) (0.039) (0.039) (0.052) (0.072) (0.140) 6

29 Table 7: Portfolio Returns based on the marginal unconditional rankings of 3 rd moment gaps Portfolio second half spearman first half correlation Standardized SkewGap excess return (0.067) (0.056) (0.050) (0.048) (0.046) (0.043) (0.047) (0.049) (0.056) (0.070) (0.047) (0.107) CAPM alpha (0.067) (0.056) (0.050) (0.049) (0.047) (0.043) (0.047) (0.050) (0.056) (0.069) (0.047) (0.107) FF alpha (0.049) (0.043) (0.041) (0.043) (0.043) (0.039) (0.042) (0.039) (0.043) (0.052) (0.047) (0.108) Carhart alpha (0.048) (0.043) (0.042) (0.044) (0.044) (0.039) (0.042) (0.039) (0.043) (0.051) (0.048) (0.110) Central SkewGap excess return (0.122) (0.061) (0.048) (0.050) (0.046) (0.044) (0.043) (0.054) (0.063) (0.116) (0.065) (0.143) CAPM alpha (0.117) (0.061) (0.047) (0.048) (0.044) (0.041) (0.041) (0.053) (0.063) (-0.027) (0.064) (0.140) FF alpha (0.077) (0.046) (0.043) (0.044) (0.039) (0.036) (0.037) (0.046) (0.044) (0.072) (0.064) (0.139) Carhart alpha (0.078) (0.046) (0.043) (0.045) (0.040) (0.037) (0.037) (0.046) (0.044) (0.072) (0.065) (0.140) 7

30 Table 8: Portfolio Returns based on the marginal unconditional rankings of 4 th moment gaps Portfolio second half spearman first half correlation Standardized KurtGap excess return (0.070) (0.053) (0.050) (0.048) (0.044) (0.044) (0.048) (0.051) (0.058) (0.069) (0.047) (0.097) CAPM alpha (0.070) (0.053) (0.051) (0.048) (0.044) (0.045) (0.048) (0.052) (0.059) (0.068) (0.047) (0.097) FF alpha (0.051) (0.039) (0.041) (0.043) (0.041) (0.042) (0.043) (0.042) (0.043) (0.054) (0.047) (0.098) Carhart alpha (0.050) (0.040) (0.041) (0.043) (0.042) (0.042) (0.043) (0.042) (0.043) (0.052) (0.048) (0.099) Central KurtGap excess return (0.138) (0.084) (0.061) (0.054) (0.050) (0.049) (0.048) (0.045) (0.043) (0.104) (0.081) (0.226) CAPM alpha (0.137) (0.085) (0.061) (0.052) (0.048) (0.045) (0.045) (0.043) (0.043) (0.095) (0.080) (0.227) FF alpha (0.089) (0.053) (0.048) (0.045) (0.043) (0.040) (0.039) (0.036) (0.038) (0.061) (0.073) (0.156) Carhart alpha (0.090) (0.053) (0.048) (0.046) (0.044) (0.041) (0.040) (0.037) (0.039) (0.061) (0.074) (0.158) 8

31 Table 9: Portfolio Returns based on the joint marginal rankings of 2 nd, 3 rd, and 4 th moment gaps control for MeanGap (standardized, in percentage) Portfolio second half spearman first half correlation Standardized VarGap excess return (0.176) (0.107) (0.090) (0.084) (0.083) (0.092) (0.086) (0.080) (0.081) (0.116) (0.178) (0.314) CAPM alpha (0.176) (0.108) (0.090) (0.083) (0.081) (0.092) (0.086) (0.080) (0.082) (0.112) (0.175) (0.329) FF alpha (0.152) (0.091) (0.081) (0.076) (0.075) (0.086) (0.079) (0.074) (0.077) (0.098) (0.0170) (0.303) Carhart alpha (0.154) (0.092) (0.081) (0.076) (0.076) (0.086) (0.080) (0.074) (0.078) (0.099) (0.172) (0.306) Standardized SkewGap excess return (0.110) (0.089) (0.086) (0.087) (0.086) (0.086) (0.086) (0.085) (0.088) (0.101) (0.123) (0.242) CAPM alpha (0.110) (0.089) (0.086) (0.087) (0.086) (0.086) (0.086) (0.085) (0.088) (0.100) (0.123) (0.249) FF alpha (0.101) (0.081) (0.078) (0.078) (0.080) (0.081) (0.080) (0.077) (0.078) (0.091) (0.124) (0.250) Carhart alpha (0.102) (0.082) (0.079) (0.079) (0.080) (0.082) (0.081) (0.077) (0.079) (0.091) (0.125) (0.253) Standardized KurtGap excess return (0.100) (0.087) (0.086) (0.088) (0.082) (0.087) (0.086) (0.093) (0.093) (0.100) (0.111) (0.232) CAPM alpha (0.100) (0.087) (0.086) (0.088) (0.081) (0.088) (0.086) (0.093) (0.093) (0.099) (0.11) (0.245) FF alpha (0.089) (0.078) (0.077) (0.080) (0.077) (0.083) (0.082) (0.085) (0.082) (0.090) (0.111) (0.245) Carhart alpha (0.090) (0.079) (0.077) (0.081) (0.078) (0.084) (0.083) (0.086) (0.082) (0.090) (0.113) (0.248) 9

32 Table 10: Portfolio Returns based on the joint marginal rankings of 2 nd, 3 rd, and 4 th moment gaps control for MeanGap (central, in percentage) Portfolio second half spearman first half correlation Central VarGap excess return (0.176) (0.107) (0.090) (0.084) (0.083) (0.092) (0.086) (0.080) (0.081) (0.116) (0.178) (0.314) CAPM alpha (0.176) (0.108) (0.090) (0.083) (0.081) (0.092) (0.086) (0.080) (0.082) (0.112) (0.175) (0.329) FF alpha (0.152) (0.091) (0.081) (0.076) (0.075) (0.086) (0.079) (0.074) (0.077) (0.098) (0.0170) (0.303) Carhart alpha (0.154) (0.092) (0.081) (0.076) (0.076) (0.086) (0.080) (0.074) (0.078) (0.099) (0.172) (0.306) Central SkewGap excess return (0.162) (0.092) (0.080) (0.086) (0.082) (0.076) (0.076) (0.077) (0.088) (0.143) (0.152) (0.267) CAPM alpha (0.157) (0.092) (0.079) (0.085) (0.081) (0.074) (0.075) (0.077) (0.088) (0.142) (0.152) (0.284) FF alpha (0.132) (0.083) (0.075) (0.082) (0.076) (0.070) (0.070) (0.071) (0.076) (0.116) (0.152) (0.283) Carhart alpha (0.134) (0.084) (0.076) (0.083) (0.077) (0.071) (0.071) (0.072) (0.077) (0.117) (0.154) (0.286) Central KurtGap excess return (0.179) (0.116) (0.088) (0.085) (0.075) (0.084) (0.078) (0.076) (0.078) (0.128) (0.163) (0.317) CAPM alpha (0.176) (0.116) (0.088) (0.083) (0.074) (0.082) (0.075) (0.075) (0.078) (0.122) (0.162) (0.330) FF alpha (0.145) (0.097) (0.079) (0.078) (0.069) (0.079) (0.071) (0.070) (0.075) (0.103) (0.158) (0.293) Carhart alpha (0.147) (0.097) (0.080) (0.079) (0.070) (0.080) (0.072) (0.070) (0.076) (0.104) (0.160) (0.296) 10

33 Table 11: Portfolio Returns based on the conditional rankings of 2 nd, 3 rd, and 4 th moment gaps on MeanGap (standardized, in percentage) Portfolio second half spearman first half correlation Standardized VarGap excess return (0.104) (0.067) (0.055) (0.048) (0.046) (0.041) (0.041) (0.040) (0.043) (0.075) (0.076) (0.159) CAPM alpha (0.105) (0.068) (0.055) (0.047) (0.045) (0.041) (0.040) (0.040) (0.042) (0.069) (0.071) (0.152) FF alpha (0.072) (0.051) (0.041) (0.039) (0.040) (0.037) (0.036) (0.036) (0.036) (0.051) (0.064) (0.118) Carhart alpha (0.074) (0.051) (0.041) (0.040) (0.041) (0.037) (0.036) (0.036) (0.036) (0.052) (0.065) (0.119) Standardized SkewGap excess return (0.059) (0.054) (0.049) (0.048) (0.049) (0.047) (0.048) (0.050) (0.053) (0.062) (0.041) (0.103) CAPM alpha (0.058) (0.055) (0.050) (0.048) (0.050) (0.047) (0.048) (0.051) (0.053) (0.060) (0.041) (0.102) FF alpha (0.046) (0.043) (0.041) (0.042) (0.042) (0.041) (0.038) (0.038) (0.042) (0.047) (0.041) (0.103) Carhart alpha (0.046) (0.044) (0.041) (0.042) (0.043) (0.041) (0.039) (0.038) (0.042) (0.047) (0.042) (0.104) Standardized KurtGap excess return (0.059) (0.052) (0.051) (0.046) (0.046) (0.048) ( (0.054) (0.054) (0.064) (0.042) (0.096) CAPM alpha (0.049) (0.052) (0.051) (0.047) (0.046) (0.049) (0.047) (0.054) (0.054) (0.063) (0.042) (0.096) FF alpha (0.046) (0.041) (0.041) (0.039) (0.039) (0.042) (0.041) (0.042) (0.042) (0.051) (0.043) (0.097) Carhart alpha (0.046) (0.041) (0.042) (0.040) (0.040) (0.043) (0.042) (0.042) (0.042) (0.049) (0.043) (0.098) 11