Alternative Sorting and Portfolio Schemes

In all previous tests, I use independent sorting for IV, momentum and reversal. There are some benefits of independent sorting since it includes a firm in a portfolio irrespective of its ranking on the other variable. However, using independent sorting sometimes leaves a small number of stocks in one portfolio and large number of stocks in another portfolio. In contrast, dependent sorting ensures that the results are not driven by a portfolio with a small number of stocks because all portfolios have an equal number of stocks over the same holding period. Therefore, in this section (Panels A and B, Table 4.6), I use dependent sorting as a robustness test.50

Arena et al. (2008) report greater momentum returns for high IV stocks using tercile portfolios for IV and decile portfolios for momentum. The advantage of tercile portfolios is the availability of a sufficiently large number of stocks in each portfolio but the disadvantage is that it cannot provide a clear picture about the tail values compared with quintile or decile portfolios. However, in this section (Panels C and D, Table 4.6), I sort stocks into tercile portfolios based on IV and decile portfolios based on past returns t-6 to t-1, for momentum and t-36 to t-7 for reversal, to test whether my results are robust to the portfolio scheme used by Arena et al. (2008).

In Panels E and F of Table 4.6, I report the results with stocks sorted into IV terciles and momentum and reversal quintiles. The advantage of this is to ensure that there are enough stocks in each portfolio; this scheme results in only 15 portfolios compared with 25 portfolios (30 portfolios) for the IV and momentum or reversal quintiles (IV terciles and momentum or reversal deciles), respectively.

Panel A, Table 4.6, shows that momentum returns of each IV quintile are quite close to those reported in Panel A, Table 4.3, when I double sort stocks, first on IV and then each IV

quintile on past 6-month returns. The difference in momentum returns between the high and low IV quintiles is very similar to those shown in Panel A, Table 4.3.

Panel B, Table 4.6, reports the reversal returns for quintiles sorted on IV with dependent sorts. The results in Panel B are inconsistent with the results shown in Panel B, Table 4.3. The reversal return of the lowest IV quintile becomes insignificant in dependent sorts but the returns in other quintiles remain significant. More importantly, the difference in reversal returns between high- and low-IV quintiles is positive and significant, which is inconsistent with the results reported in Table 4.3. This highlights a shortcoming of the portfolio-sorting technique especially in emerging markets which typically have relatively small numbers of stocks. In such markets, the choice of the sorting procedure could play a critical role in determining the relationship between IV (and possibly other firm characteristics) and cross- sectional stock returns.

In the next four Panels, I test the effect of changing the number of portfolios used in the analysis. Panels C and D, Table 4.6, report the average monthly momentum and reversal returns for IV terciles and the momentum and reversal deciles. At the beginning of each month, stocks were sorted into IV terciles and past return deciles. The past returns and IV were sorted independently. In Panels E and F, I use IV terciles and momentum or reversal quintiles.

Panel C, Table 4.6, shows that momentum returns for the low IV tercile double (1.63% per month) compared with the 6-month holding period returns of the low IV quintile reported in Panel A, Table 4.3 (0.79% per month). The momentum returns of the second (0.69% per month) and high IV (0.93% per month) terciles are also large and statistically significant. The large increase in momentum returns with IV terciles and momentum deciles appears to be related to the decile sort of momentum since I find that momentum returns are large in the decile sort compared with the quintile sort (see Table 2.2). Interestingly, the difference in momentum returns between the high and low IV terciles decreases to -0.70 % per month (t- statistic = -3.54) compared with the insignificant -0.07% per month (t-statistic = -0.34) between the high and low IV quintiles (6-month holding period) reported in Panel A, Table 4.3. Therefore, these results indicate that momentum returns become larger and statistically more significant with decile sorts on past 6-month returns. The larger momentum returns of the low IV tercile might be related to the size effect instead of low IV since I find that the average market capitalization for low IV stocks is CNY6078 million compared with

CNY5665 million for medium IV and CNY4363 for high IV stocks. This argument is consistent with the momentum returns of size-based terciles where the momentum returns of

big size terciles are larger than small and medium size terciles (see Table 2.6). These results also suggest that the existence and significance of a relationship between IV and momentum depends on the breakpoints used to sort stocks into portfolios since I find an insignificant difference between high and low IV quintiles (Panel A, Table 4.3) when I sort stocks into IV and momentum quintiles but a significant difference between high and low IV terciles when I sort stocks into IV terciles and momentum deciles.

Panel D, Table 4.6, shows that reversal returns of the high IV tercile increases to1.34% per month when I sort stocks into IV terciles and reversal deciles compared with the reversal returns of the high IV quintile (0.94% per month) reported in Panel B, Table 4.3. Reversal returns of the second (0.44% per month) and low IV (0.51% per month) terciles are insignificant compared with the significant results for all quintiles shown in Panel B, Table 4.3 Surprisingly, the difference in reversal returns between the high and low IV terciles increases to 0.83 % per month (t-statistic = 2.48) compared with the insignificant 0.35% per month (t-statistic = 1.57) between the high and low IV quintiles reported in Panel B, Table 4.3. These results indicate that the reversal returns of the high IV tercile becomes larger and statistically significant than the reversal returns of high IV portfolio reported in Panel B, Table 4.3. However, there is a small decrease in reversal returns in the low and second IV terciles. These results also indicate that the significance of a relationship between IV and reversal depends on the breakpoints used to sort stocks since I find an insignificant difference between high and low IV quintiles (Panel B of Table 4.3) but a significant difference between high and low IV terciles when I sort stocks into IV terciles and reversal deciles. The results in Panels C and D, Table 4.6 are consistent with the findings of Bali and Cakici (2008) who find that the relationship between IV and expected returns is not robust to different breakpoints used to sort stocks into portfolios.

Panels E and F, Table 4.6, report average monthly momentum and reversal returns based on IV terciles and momentum and reversal quintiles instead of momentum and the reversal deciles used in Panels C and D. At the beginning of each month, stocks are independently sorted into terciles based on their IV ranking and into quintiles based on the past returns of momentum (t-6 to t-1) and reversal (t-36 to t-7).

Panel E, Table 4.6, shows that momentum returns for the low IV tercile (0.98% per month) are somewhat higher than the momentum returns of the low IV quintile (0.79% per month) reported in Panel A, Table 4.3 (6-month holding period). However, the momentum returns of the high IV tercile falls to 0.55% per month, when I sort stocks into IV terciles, from 0.72% per month reported in Panel A, Table 4.3. Interestingly, the difference in momentum returns

between the high and low IV terciles decreased to -0.44 % per month (t-statistic = -3.81) from the insignificant -0.07% per month (t-statistic = -0.34) for the 6-month holding period reported in Panel A, Table 4.3. These results also indicate that the existence and significance of a relationship between IV and momentum depends on the different breakpoints used to sort stocks into portfolios since I find an insignificant difference between high and low IV quintiles (Panel A, Table 4.3) but a significant difference between the high and low IV terciles. This argument is also consistent with the findings of Bali and Cakici (2008), who find that the relationship between IV and expected returns is not robust to different breakpoints used to sort stocks into portfolios.

The reversal returns of the IV terciles in Panel F, Table 4.6, become larger than the returns of the IV quintiles reported in Panel B, Table 4.3. The increases in reversal returns suggest that a change in the number of IV portfolios affects the reversal returns.

The results reported in Table 4.3 show that IV is not related to momentum and reversal. In the robustness tests, I find that the existence and significance of a cross-sectional relationship between IV and momentum or reversal changes with the different IV measures, sorting methods and different number of IV portfolios (tercile versus quintile). On balance, these results suggest that IV is not related to momentum and reversal. These results are consistent with the findings of Bali and Cakici (2008), who find that the relationship between IV and expected returns is not robust to different portfolio-weighting schemes, different data frequency to estimate IV and breakpoints used to sort stocks into portfolios.