< Table 3 is inserted about here >
< Table 4 is inserted about here >
< Table 5 is inserted about here >
< Table 6 is inserted about here >
Table 3-6 present the empirical results of the estimation with the DCC model over the sample period from 07/13/2006 to 03/18/2008. Because of the procedure for parameters which are
estimated under the setting of standard DCC mode, we divide these tables into two parts consistent with the two steps in the DCC estimation. In Panel A of each table, we apply the GARCH model to individual assets to obtain the standardized residuals. Then, these standardized residuals series are brought into the second stage for dynamic conditional correlation estimating, and we show the estimated parameters of DCC model in Panel B of each table.
Furthermore, in panel A of Table 3-6, we can find that most of the coefficients estimated in the univariate GARCH (1,1) models are significant under 5% level excluding some coefficients of constant parameters in the conditional variance equations. The results reveal that very strong time-varying conditional heteroskedasticity is shown by the large t-statistics of the coefficients of the lagged squared residuals (α) and the lagged conditional variance terms (β). Besides, the sums of α+β for all series are near to one, and this is the evidence that there exists strong persistence in the conditional variances.
Finally, in Panel B of Table 3-6, the results show that almost all of the estimated coefficients (b) are significant at 5% level. These outcomes indicate that the correlations are significantly dynamic, and we can conclude that current dynamic conditional correlations are significantly affected by previous dynamic conditional correlations.
Based on the results above, we will focus on the tracking errors of each ETF in this section. First, the comparison of the tracking error between Short ETF and UltraShort ETF related to the same benchmark will be delivered. Furthermore, we try to observe whether the less perfect conditional correlation leads to the larger tracking error. Although the conditional correlations between returns of stock market indices and returns of their short ETF products are negative, we modify these numbers to positive for intuitive understanding. Then, we will also make the comparison of tracking error across the different stock market indices. Besides,
we will investigate the relationship between tracking error and trading volume of each short ETF as well as the relationship between tracking error of ETF and the return volatilities of its corresponding index futures.
< Table 7 is inserted about here >
< Table 8 is inserted about here >
Table 7 show that all of the coefficients estimated in the univariate GARCH (1,1) models are significant under 5% level. The results reveal very strong time-varying conditional heteroskedasticity. The sums of α+β for are near to one, and this is the evidence that there exists strong persistence in the conditional variances. Table 8 presents the descriptive statistics of the tracking error (TE) between the Dow Jones Industrial Average index (DJIA) and its corresponding ETFs. As we can see, the TE also presents the fat-tail distributions and is found to reject the null hypothesis of a normal distribution. This indicates that the TE also has the same characteristics like most of the financial data. Also, in Table 8, the mean of TE between DJIA and its related Short ETF (DOG) is smaller than that between DJIA and UltraShort product (DXD). This means, on average, the DXD has the larger TE than that of DOG while the standard deviation between DOG and DXD does not have large differences.
Based on these results, a conclusion can be made that the TE of DOG is smaller than TE of DXD.
< Figure 5 is inserted about here >
Figure 5 shows the comparison between the dynamic conditional correlation and the tracking error of short ETFs related to the DJIA. Although we can’t observe the perfect relationship between these two series, the less perfect conditional correlation seems to produce the larger tracking error. This phenomenon exists in both DOG and DXD. The
unconditional correlation between the TE and the dynamic conditional correlation for DOG is -0.557 while that for DXD is -0.377, which is much lower.
< Table 9 is inserted about here >
< Table 10 is inserted about here >
Table 9 reveals that all of the coefficients estimated in the univariate GARCH (1,1) models are significant under 5% level except one coefficients of constant parameter. Table 10 shows the descriptive statistics of the tracking error (TE) between the S&P500 index and its corresponding Short (SH)/UltraShort (SDS) EFTs. The TE series of the SH presents the fat-tail distributions, whereas that of SDS is not. Both series are found to reject the null hypothesis of a normal distribution. Moreover, as shown in Table 10, both the mean and the standard deviation of TE between S&P500 and its related UltraShort ETF (SDS) outperform that between S&P500 and its Short product (SH). As a result, we can conclude that the SDS is better on the tracking ability than SH.
< Figure 6 is inserted about here >
Figure 6 shows the comparison between the dynamic conditional correlation and the tracking error of short ETFs related to the S&P500. In this figure, the relationship between these two series is not obvious. However, we still find the positive unconditional correlations between these two series. The value for SH is -0.529, and the value is only -0.112.
Consequently, the less perfect conditional correlation will cause the TE to be larger.
< Table 11 is inserted about here >
< Table 12 is inserted about here >
Table 11 reveals that all of the coefficients estimated in the univariate GARCH (1,1) models are significant under 5% level. Table 12 shows the descriptive statistics of the tracking error (TE) between the S&P400 MidCap index and its corresponding Short (MYY)/UltraShort (MZZ) EFTs. The TE series of the MYY and MZZ present the fat-tail distributions, and both series are found to reject the null hypothesis of a normal distribution.
Furthermore, Table 12 also exhibit that both the mean and the standard deviation of TE between S&P400 MidCap and its related Short ETF (MYY) outperform that between S&P400 MidCap and its UltraShort product (MZZ). Based on these results, we can conclude that the MYY is better on the tracking performance than MZZ.
< Figure 7 is inserted about here >
Figure 7 reveals the comparison between the dynamic conditional correlation and the tracking error of short ETFs related to the S&P400.MidCap. We can see the strong positive relationship between these two series. The unconditional correlation between these two series of Short ETF (MYY) is -0.726, and that of UltraShort (MZZ) is -0.598. The less perfect conditional correlation between S&P400 MidCap and its corresponding short ETFs also leads to larger TE for these two ETFs.
< Table 13 is inserted about here >
< Table 14 is inserted about here >
Table 13 reveals that all of the coefficients estimated in the univariate GARCH (1,1) models are significant under 5% level. Table 14 shows the descriptive statistics of the tracking error (TE) between the NASDAQ100 index and its corresponding Short (PSQ) EFTs. The TE series of the PSQ shows the fat-tail distributions, and are found to reject the null hypothesis of a normal distribution. Furthermore, Table 12 exhibits the mean of TE is 0.317, and the
standard deviation is 0.247.
< Figure 8 is inserted about here >
Figure 8 reveals the comparison between the dynamic conditional correlation and the tracking error of PSQ. We can see the comovement between the two series is very obvious.
The unconditional correlation between the two series of PSQ is -0.893.
After we investigate the TEs of each Short/UltraShort ETF related to the same stock market index, here we try to compare the TE of Short/UltraShort ETF related to the different stock market index. Specifically, we try to show which Short/UltraShort ETF, across the different stock market indices, has the smallest TE.
< Table 15 is inserted about here >
Table 15 shows the statistics of tracking error of Short/UltraShort ETFs across different market indices. As we can see in Table 15, the PSQ is the worst on tracking performance in the group of short ETFs because it has the largest mean and standard deviation of tracking error while in the group of UltraShort ETFs, the MZZ is the worst on tracking performance due to the same reason.
In this section, we will investigate the relationship between tracking error and trading volumes of each ETF.
< Figure 9 is inserted about here >
The relationsip between tracking error and trading volumes of the “Short”(DOG)/
“UltraShort”(DXD) ETF of Dow Jones Industrial Average index seems vague. The unconditional correlation of these two series is 0.538 for DOG and 0.621 for DXD. This result shows that the larger the volumes, the larger the tracking error.
< Figure 10 is inserted about here >
In figure 10, we can’t not see clearly the relatioship between tracking error and trading volumes of the “Short”(SH)/ “UltraShort”(SDS) ETF of S&P500. We report that the unconditional correlation of these two series is 0.526 for SH and 0.857 for SDS. This result also shows that the larger the volumes, the larger the tracking error.
< Figure 11 is inserted about here >
In figure 11, the relatioship between tracking error and trading volumes of the
“UltraShort”(MZZ) ETF is easier to observe than that of “UltraShort”(MZZ) ETF . The unconditional correlation of these two series is 0.439 for MZZ, and the value is mush smaller for MYY (0.007). This result reveals that the larger the volumes, the larger the tracking error.
< Figure 12 is inserted about here >
In figure 12, it’s hard for us to tell whether there is any relation between tracking error and trading volumes of the “Short”(PSQ) ETF. The unconditional correlation of these two series is only 0.191. This result also reveals that there is weak positive relation between the tracking error and trading volumes.
< Table 16 is inserted about here >
We utilize Table 16 to discuss the positive relationship between tracking error and trading volumes of each ETF. As we can see in Table 16, the higher correlation between tracking errors and trading volumes accompanies higher trading volumes. One possible reason for this phenomenon is that when the trading volumes go too large, the large trading volumes themselves generate the large tracking errors. This implies that when investors throng to market to buy these products, the over-trading will produce tracking errors. We can see the trading volumes of MYY and MZZ.
When the trading volumes are six times more for MZZ than MYY, the unconditional correlation goes from 0.007 for MYY to o.439 for MZZ
< Table 17 is inserted about here >
We use Table 17 to confirm our conjecture. In Table 17, we can find that the more the trading volumes, the bigger the tracking errors of each short ETF except MYY. This table also shows that the tracking errors come from the quarter which contains the larger trading volumes, are larger than the average tracking errors. Because of this reason, we conclude that the over-trading will lead to larger tracking error.
We will investigate the relationship between tracking error and volatilities of index futures in this section. ProShares uses index futures to rebalance its UltraShort ETFs daily to keep leverage consistent with each ETF’s daily investment objective so ProShares uses more index futures on UltraShort ETFs than on Short ETFs. When the volatilities of index futures go up, it may cause ProShares to miss the target prices and lead to tracking errors of ETFs.
Because of this reason, we presume that volatilities of index futures have more influences on tracking errors of the UltraShort ETF than on those of Short ETF.
< Table 18 is inserted about here >
Table 18 reveals that all of the coefficients estimated in the univariate GARCH (1,1) models are significant under 5% level except one coefficients of constant parameter. The results show very strong time-varying conditional heteroskedasticity. The sums of α+β for are near to one, and this is the evidence that there exists strong persistence in the conditional variances.
After the the estimations of GARCH (1,1) for each index futures are made, we can difine the conditional standard deviations form GARCH (1,1) as the volatilities of index futures.
Now we can use these results to discuss the relationship between the volatilities of index
futures and the tracking errors.
< Figure 13 is inserted about here >
Figure 13 shows that the relationsip between the tracking error of DOG/DXD and the volatilities of DJIA index futures seems to correlate positively. The unconditional correlation of these two series is 0.688 for DOG and 0.680 for DXD. This result shows that the tracking errors of “Short” (DOG)/”UltraShort” (DXD) ETF of DJIA are affected almost equally by the volatilities of DJIA index futures.
< Figure 14 is inserted about here >
The relationsip between the tracking error of SH/SDS and the volatilities of S&P500 index futures shown in Figure 14 also seems to correlate positively. The unconditional correlation of these two series is 0.676 for SH and 0.882 for SDS. This result shows that the volatilities of S&P500 index futures have more influences on the tracking errors of the UltraShort (SDS) ETF than on those of Short (SH) ETF.
< Figure 15 is inserted about here >
The relationsip between the tracking error of MYY/MZZ and the volatilities of S&P400 MidCap index futures shown in Figure 15 is obscure. The unconditional correlation of these two series is 0.388 for MYY and 0.556 for MZZ. This result shows that the volatilities of S&P400 MidCap index futures have more influences on tracking errors of the UltraShort (MZZ) ETF than on those of Short (MYY) ETF.
According to the daily holdings of short ETFs revealed by ProShares, this company uses more index futures on the UltreaShort ETFs than on the Short ETFs. Coinciding with this fact, our results show that the volatilities of S&P500 and S&P400 MidCap index futures have more influences on tracking errors of the UltraShort ETFs than on those of Short ETFs.
However, the tracking errors of “Short” (DOG)/”UltraShort” (DXD) ETF of DJIA are affected almost equally by the volatilities of DJIA index futures.
Finally, the hedge performance of short ETFs will be shown, and the comparison will be made.
Based on the results of performing DCC, we can use the conditional covariance and variance to calculate the hedge ratios.
< Figure 16 is inserted about here >
< Figure 17 is inserted about here >
Figures 16 and 17 show that the minimum-variance hedge ratios (MVHRs) for stock market indices using their related Short ETF are all close to 1 and the values are close to 0.5 using their related UltraShort ETF. We can conclude that the basic functions of the Short ETF and the UltraShort ETF exist.
< Table 19 is inserted about here >
Table 19 reports that there is no certain answer that which knid of ETF outperform the other kind when it comes to hedging performance. For ETFs relate to DJIA, the Short (DOG) ETF has the better hedging performance than the UltraShort (DXD) ETF. The Short ETF (MYY) of S&P400 MidCap also outperform MZZ in hedging performance while the UltraShort (SDS) ETF of S&P500 outperform SH in hedging performance. Futhermore, for the comparison across different market indices, the MYY has the best hedging performance among the Short ETFs. SDS has the best hedging effectiveness among the UltraShort ETFs.
Ⅴ. Conclusion
We investigate the tracking errors and the hedging effectiveness of each short ETF. This article shows that when it comes to tracking errors of Short/UltraShort ETFs related to the same benchmark, the Short ETFs of DJIA and S&P400 MidCap outperform the UltraShort ETFs of these two indices. On the contrary, the UltraShort ETF of S&P500 has the better tracking ability than the Short ETF of the S&P500. As for the cross indices comparison, the Short ETF of NASDAQ100 is the worst on tracking performance in the group of Short ETFs while the MZZ has the worst tracking ability in the group of UltraShort ETFs. Still, after the time-varying correlations between ETFs and their benchmark are estimated from the DCC model, we report the negtive unconditional correlation between tracking errors and these time-varying correlations. This result corroborates that the unperfect correlation between ETFs and their benchmarks will lead to tracking errors as mentioned by Hehn (2005).
Furthermore, we also examine the relationship between tracking errors and volatilities of their related index futures as well as that between tracking errors and trading volumes. We conclude that the tracking errors of DOG and DXD are affected almost equally by the volatilities of DJIA index futures while the volatilities of S&P500 (S&P400 MidCap) index futures have more influences on tracking errors of SDS (MZZ) than on those of SH (MYY).
The results, except for the short ETFs of DJIA, coincide with the facts that ProShares uses more index futures on UltraShort ETFs than on Short ETFs. ProShares uses index futures to rebalance its UltraShort ETFs daily to keep leverage consistent with each ETF’s daily investment objective. When the volatilities of index futures go up, it may cause ProShares to miss the target prices and lead to tracking errors of ETFs. We also find that over-trading on the shot ETFs may lead to larger tracking errors, and this effect is quite obvious regarding MYY and MZZ.
Finally, we research the hedging performance of each short ETFs. We find that Short
ETFs outperform UltraShort ETF when DJIA and S&P400 MidCap are concerned while the UltraShort (SDS) ETF of S&P500 has the better hedging performance than SH. Besides, the MYY has the best hedging performance among the Short ETFs when SDS has the best hedging effectiveness among the UltraShort ETFs.
As to the further research, one may try to evaluate the hedging performance using short ETFs to hedge the portfolio not matching a broad market index (as Graham and Jennings (1987), Butterworth and Holmes (2001)). After all, not everyone has the ability to create such a portfolio so much like a benchmark unless one buys another ETF. More recently, ProShares also launches Short international ETFs, and therefore one can research if there are more tracking errors for these Short international ETFs than Short domestic ETFs as suggested by Engle and Sarkar (2006) as far as ordinary ETFs are concerned.
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