4. Empirical Results
4.1 Overlapping return
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there is no difference in the transaction cost between the two strategies and calculate the return difference by the below formula.
Return difference= Ret. (LETF) – multiple2 *Ret. (ETF)
Besides, we can examine whether there is a statistically significant relationship between the return difference and volatility by regression analysis. Finally, we attempt to infer that the major reason why the realized multiples are not equal to the fund multiples is volatility. Eventually, we may be able to develop a strategy in terms of the empirical test of this research.
𝑦holding period = α0 + 𝛾 σ holding period + 𝑢𝑡
Where 𝑦holding period is the return difference, the 𝛼0 is intercept, and 𝛾 is the coefficient of σholding period and 𝑢𝑡 is the error term.
4. Empirical Results
4.1 Overlapping return
As mentioned in Chapter 3, for example, a portfolio consisting of two dollars invested in Taiwan 50 (symbol: 0050 ) and short one dollar can be compared with an investment of one dollar in Yuanta Daily Taiwan 50 Bull 2X. From Figure 2 , it compares the returns of Taiwan 50 Bull 2X and a twice-leveraged buy-and- hold strategy and spots all data points. We calculate all 30-day (overlapping) return from October 31, 2014 to November 18, 2016. The concentrated plie of points near the 45-degree line means its relatively lower tracking error. The remaining points occurred not only below the 45-degree line, but also above it. As what we discussed, even in longer holding period, the leveraged ETF is possible to outperform. Next, we show returns in logarithmic scales in Figure 5 so that the difference between two strategies would be clearer.
2 It is the multiple required on daily basis.
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Figure 2. 30-day returns for Bull 2X (symbol: 00631L) versus leveraged 30-day return of Taiwan 50(symbol: 0050). (X = 2 Ret.(0050);Y = Ret.(00631L)).
Also, from Figure 3, it compares the returns of Taiwan 50 Bull 2X and a twice-leveraged buy-and- hold strategy and spots all 60-day return points. There are some 60-day return points above the 45-degree line. Even so, we show returns in logarithmic scales in Figure 6 and we see that the returns of the LETFs have predominantly underperformed a twice-leveraged buy-and- hold strategy.
Figure 3. 60-day returns for Bull 2X (symbol: 00631L) versus leveraged 60-day return of Taiwan 50(symbol: 0050). (X = 2 Ret.(0050);Y = Ret.(00631L)).
-0.40
-0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40
-0.50
-0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50
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and a twice-leveraged buy-and- hold strategy. Apparently, there are some 120-day return points above the 45-degree line. Within longer investment period, the leveraged ETF is possible to outperform. Next, we show returns in logarithmic scales in Figure 7 so that the difference between two strategies would be clearer. However, the most lie below the 45-degree line.Figure 4. 120-day returns for Bull 2X (symbol: 00631L) versus leveraged 120-day return of Taiwan 50(symbol: 0050). (X = 2 Ret.(0050);Y = Ret.(00631L)).
Figure 5. Same as in Figure 2, but returns are logarithmic, i.e., X = 2 ln((0050) t
/(0050)t−30 ), Y = ln((00631L) t / (00631L)t−30).
-0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50
-0.40
-0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30
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Figure 6. Same as in Figure 3, but returns are logarithmic, i.e., X = 2 ln((0050) t
/(0050)t−60 ), Y = ln((00631L) t / (00631L)t−60).
Figure 7. Same as in Figure 4, but returns are logarithmic, i.e., X = 2 ln((0050) t
/(0050)t−120 ), Y = ln((00631L) t / (00631L)t−120).
-0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40
-0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40
-0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50
-0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50
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Taiwan 50 Bull 2X. From Figures 8, we mark the returns of Taiwan 50 Bear -1X and a static leveraged strategy with Taiwan 50 since October 31, 2014. We calculate all 30-day (overlapping) return from October 31, 2014 to November 18, 2016. As what we metioned, even in longer holding period, the inverse ETF is possible to outperform. Next, we also show returns in logarithmic scales in Figure 8 so that the difference between two strategies. We should figure out that the return of the Taiwan 50 Bull 2X was not worse than that of the Taiwan 50 at all.A.
B.
Figure 8. The graph A shows 30-day returns for Bear -1X (symbol: 00632R) versus leveraged 30-day return of Taiwan 50(symbol: 0050). (X = 2 Ret.(0050);Y =
Ret.(00632R)). The graph B presents returns in logarithmic, i.e., X = 2 ln((0050) t
/(0050)t−30 ), Y = ln((00632R) t / (00632R)t−30).
-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30
-0.15
-0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20
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holding period, the inverse ETF is possible to outperform. Next, we also show returns in logarithmic scales in Figure 9 so that the difference between two strategies. We should figure out that the return of the Taiwan 50 Bull 2X was not worse than that of the Taiwan 50 at all.A.
B.
Figure 9. The graph A shows 60-day returns for Bear -1X (symbol: 00632R) versus leveraged 60-day return of Taiwan 50(symbol: 0050). (X = 2 Ret.(0050);Y =
Ret.(00632R)). The graph B presents returns in logarithmic, i.e., X = 2 ln((0050) t
/(0050)t−60 ), Y = ln((00632R) t / (00632R)t−60).
-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30
-0.20
-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30
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Finally, from Figures 10, we mark the all 120-day returns of Taiwan 50 Bear -1X and a static leveraged strategy with Taiwan 50 since October 31, 2014. Even in long holding period, the inverse ETF is possible to outperform. Next, we also show returns in logarithmic scales in Figure 10 so that the difference between two strategies. However, we find that the points do not fall on the 45-degree line; the most of them are below it.
This effect is due to volatility A.
B.
Figure 10. The graph A shows 120-day returns for Bear -1X (symbol: 00632R) versus leveraged 120-day return of Taiwan 50(symbol: 0050). (X = 2 Ret.(0050);Y = Ret.(00632R)). The graph B presents returns in logarithmic, i.e., X = 2 ln((0050) t
/(0050)t−120 ), Y = ln((00632R) t / (00632R)t−120).
-0.30 -0.20 -0.10 0.00 0.10 0.20 0.30
-0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20
-0.30 -0.20 -0.10 0.00 0.10 0.20 0.30
-0.30 -0.20 -0.10 0.00 0.10 0.20 0.30
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Observing from Figure 2 to 10, these charts can be partially interpreted by the magnitude of the compounding effect, which is caused by the mismatch between the investment horizon and the daily rebalancing frequency; yet there are some points we can observe from these. First, due to the daily rebalancing of LETFs, the geometric return which is continuously compounded is more appropriate than the arithmetic one which is simply compounded. In order to make the difference between two strategies much clearer, we convert geometric returns into logarithmic returns. The return of the LETFs is not always less than that of leveraged buy-and- hold strategy with Taiwan 50. However, the effect is caused by the volatility.
ret.(LETF) ≈ β ret.(ETF), β = +2/-1 log ret.(LETF) ≈ β log ret.(ETF), β = +2/-1
4.2 Return Difference
As seen on the charts, there is no obvious difference among different holding period.
We may suggest that the investment horizon be not the major factor, influencing the return of the LETFs. Still, some data points do not fall on the 45-degree line; therefore, we calculate the difference between the returns of two strategies and then split them into ten deciles. The relationship between each decile and volatility demonstrates below. We use the median of return difference and annualized standard deviation of each decile as proxies. Obviously, the return differences are mainly caused by volatility. No matter how long the holding periods is, the return difference seems to follow the pattern of the volatility.
From Figure 11-A, we can find that the return difference between the Taiwan 50 Bear -1X and a corresponding portfolio in 30-day holding period comes smaller or even positive when the standard deviation of the ETF is getting smaller and smaller. It means the volatility and the performance of Taiwan 50 Bear -1X may be in the inverse relationship. If the realized variance is not moderate, the realized multiple would be away from the target one.
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Figure 11- A. the Relationship between the Return Difference and Volatility within 30-Day Holding Period A: -1x Taiwan 50 Daily Strategy Returns vs. -1 times ETF Return
Also, form Figure 11-B, we can see the 60-day return difference becomes bigger when the standard deviation of the ETF is smaller. It means the volatility and the performance of Taiwan 50 Bear -1X may be in the inverse relationship.
Figure 11- B. the Relationship between the Return Difference and Volatility within 60-Day Holding Period A: -1x Taiwan 50 Daily Strategy Returns vs. -1 times ETF Return
0-10th 10-20th 20-30th 30-40th 40-50th 50-60th 60-70th 70-80th 80-90th 90-100th diff in return -2.02% -1.21% -0.87% -0.58% -0.18% 0.08% 0.42% 0.74% 1.18% 1.83%
std 19.65% 15.38% 15.40% 14.27% 14.92% 14.90% 14.65% 14.05% 14.93% 13.15%
12.00%
0-10th 10-20th 20-30th 30-40th 40-50th 50-60th 60-70th 70-80th 80-90th 90-100th diff in return -3.08% -2.12% -1.13% -0.76% -0.40% 0.11% 0.48% 1.00% 1.90% 2.42%
std 21.24% 16.57% 15.89% 14.42% 14.33% 13.68% 13.43% 13.48% 13.46% 14.14%
10.00%
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Similarly, form Figure 11-C, we can notice that the 120-day return differences turn from negative to positive when the standard deviation of the ETF is smaller. It means the volatility and the performance of Taiwan 50 Bear -1X may be in the inverse relationship.
As a result, we examine whether the relationship is statistically significant in Chapter 4.4.
Figure 11- C. the Relationship between the Return Difference and Volatility within 120-Day Holding Period A: -1x Taiwan 50 Daily Strategy Returns vs. -1 times ETF Return
On the contrary, from Figure 12-A, the return difference between the Taiwan 50 Bull 2X and a corresponding portfolio in 30-day holding period comes greater or even positive when the underlying ETF is more volatile. However, there is no obvious difference in standard deviation among each group.
Figure 12- A. the Relationship between the Return Difference and Volatility within 30-Day Holding Period A: 2x Taiwan 50 Daily Strategy Returns vs. 2 times ETF Return
0-10th 10-20th 20-30th 30-40th 40-50th 50-60th 60-70th 70-80th 80-90th 90-100th diff in return -5.03% -3.70% -2.90% -1.86% -0.33% 0.38% 1.30% 2.06% 2.79% 4.01%
std 18.98% 18.89% 15.48% 18.07% 14.68% 14.94% 14.85% 13.59% 13.22% 14.45%
12.00%
0-10th 10-20th 20-30th 30-40th 40-50th 50-60th 60-70th 70-80th 80-90th 90-100th diff in return -2.57% -1.71% -1.04% -0.58% -0.30% 0.04% 0.40% 0.70% 1.12% 2.38%
std 14.19% 14.12% 14.65% 14.56% 15.40% 15.72% 15.26% 14.29% 15.18% 15.47%
13.00%
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Apparently, from Figure 12-B, the 60-day return difference comes greater or even positive when the underlying ETF is more volatile. It means the volatility and the performance of Taiwan 50 Bull 2X may be in the positive relationship. However, if the realized variance is moderate, the realized multiple would be close to the target one.
Figure 12- B. the Relationship between the Return Difference and Volatility within 60-Day Holding Period A: 2x Taiwan 50 Daily Strategy Returns vs. 2 times ETF Return
Similarly, form Figure 12-C, we can observe that the 120-day return differences turn from negative to positive when the standard deviation of the ETF is bigger. It means the volatility and the performance of Taiwan 50 Bear 2X may be in the positive relationship.
As a result, we examine whether the relationship is statistically significant in Chapter 4.4.
0-10th 10-20th 20-30th 30-40th 40-50th 50-60th 60-70th 70-80th 80-90th 90-100th diff in return -3.70% -2.26% -1.59% -0.90% -0.56% -0.09% 0.38% 0.80% 1.26% 1.53%
std 13.54% 14.19% 13.55% 13.65% 13.64% 14.03% 14.84% 15.75% 16.46% 16.75%
12.00%
12.50%
13.00%
13.50%
14.00%
14.50%
15.00%
15.50%
16.00%
16.50%
17.00%
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Figure 12- C. the Relationship between the Return Difference and Volatility within 120-Day Holding Period A: 2x Taiwan 50 Daily Strategy Returns vs. 2 times ETF Return
4.3 Paired t-test
Next, we examine whether the return difference is statistically significant.
According to Table 2, the mean of the return difference is bigger in the longer holding period, and it can be mainly explained by the compounding effect. Although the means of the return differences are all negative, as what we discussed, not all of the data points are below 45-degree line; still, it is caused by volatility.
Moreover, we can notice from Table 2, the return differences in the 30-day holding period, the shortest one, are not statistically significant in both LETFs. Yet, semiannually, in the 120-day holding period, it is statistically significant that the both of return differences are not zero. Therefore, buy-and-hold strategy’s investors must be much more carefully of the market volatility, especially in the longer investment horizon. The difficulty of investors is to predict volatility. If we are able to do so and expect it would be a down market, we could invest bearish LETFs when low volatility market; in the contrast, if we expect it would be a bull market, we should invest bullish LETFs rather than holding twice-leveraged buy-and-hold Taiwan 50 when high volatility market.
0-10th 10-20th 20-30th 30-40th 40-50th 50-60th 60-70th 70-80th 80-90th 90-100th diff in return -5.77% -3.59% -2.28% -1.81% -1.38% -0.71% -0.02% 0.61% 1.45% 3.26%
std 12.91% 15.98% 15.30% 15.08% 14.86% 14.88% 15.36% 15.28% 19.15% 18.74%
12.00%
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Based on the above strategy, investors may be able to outperformance a required multiple times the underlying ETF.
Table 2 P-value of return differences among 30-Day, 60-Day, 120-Day Holding Period
Bear -1X Bull 2X
Return diff. 30-Day 60-Day 120-Day 30-Day 60-Day 120-Day mean -0.06% -0.09% -0.34%** -0.13% -0.43*** -0.74%***
p-value -0.3178 -0.3354 -0.0232 -0.0528 (<0.001) (<0.001)
4.4 Regression
As we can see from Table 3, the results are the same as the charts shown in Chapter 4.2. The relationship between the volatility of the underlying index and the return differences in two strategies is statistically significant, but the effect of the Bear -1X is opposite to the Bull 2X; yet, there are two points that deserve attention. First, with the longer holding period, the absolute value of the coefficient of volatility, gamma γ, becomes greater in both. It means the effect of volatility comes bigger. Second, obviously, the explanatory power of the variable, the volatility of the underlying index, increases sharply when the investment horizon is longer. The most important is all of the coefficients are statistically significant.
While low volatility market, if investors expect a bear market, no matter how long investors plan to hold, it is more appropriate to buy a bearish LETF because it may outperform than shorting the underlying ETF. On the contrary, as high volatility market, if investors plan to magnify the market return, it is more appropriate to buy a bullish LETF because it may outperform than loaning cash to hold more the underlying ETF.
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Table 3 Results of Regression Model: X=σ holding period (0050); Y=return difference
Bear -1X Bull 2X
Return diff. 30-Day 60-Day 120-Day 30-Day 60-Day 120-Day γ -1.6705 *** -5.2215 *** -12.7082 *** 1.4267 *** 4.3446 *** 10.1182 ***
p-value <0.001 <0.001 <0.001 <0.001 <0.001 <0.001
N 474 444 384 474 444 384 required leverage ratio, over time. However, owing to compounding effect, we all know it is highly likely to occur tracking errors. Therefore, we calculate the realized multiples for every investment horizon so as to infer the volatility is the key factor rather than the holding period.
In Table 4A and 4B, due to compounding effect, either Bear -1X or Bull 2X is not in the proper leverage ratio, which should be respectively close to -1 and 2. Accordingly, they usually are overexposed or underexposed if investors hold them more than one day.
For example, from the Table 4A, within 30-, 60-, 120-day, respectively 12%, 18% and 14% of realized multiples of the Taiwan 50 Bear -1X are out of the range of reasonable multiples from 0 to -2. Next, from the Table 4B, we can notice that 5%, 13% and 4% of realized multiples of the Taiwan 50 Bull 2X are out of the range from 0 to 4, respectively within 30-, 60-, 120-day.
Realized Multiple =𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑓 𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒𝑑 𝐸𝑇𝐹 𝑖𝑛 𝑊−𝐷𝑎𝑦 𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑃𝑒𝑟𝑖𝑜𝑑
𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑓 𝑈𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 𝐸𝑇𝐹 𝑖𝑛 𝑊−𝐷𝑎𝑦 𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑃𝑒𝑟𝑖𝑜𝑑, W=30, 60 or 120 As discussed, the volatility of the underlying index is the major factor to the performance of the LETFs because no matter how long we invested, the compounding effect is a few differences among the different holding period. Furthermore, the gap from the required leverage ratio did not larger due to the longer investment horizons. That’s
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why we attempt to defend time is not the key factor and try to infer the volatility leads the realized multiples are far away from the required beta.
Table 4- A Frequency of Realized Multiples of Taiwan 50 Bear -1X for 30-, 60-, 120-Day Holding Periods
-1x 30-Day 60-Day 120-Day
Multiple Ranges N=474 N=444 N=384
-0.75 to -1.25 53% 55% 50%
-0.50 to -1.50 72% 68% 70%
-0.25 to -1.75 81% 77% 80%
0.00 to -2.00 88% 82% 86%
others 12% 18% 14%
Table 4- B Frequency of Realized Multiples of Taiwan 50 Bull 2X for 30-, 60-, 120-Day Holding Periods
2x 30-Day 60-Day 120-Day
Multiple Ranges N=474 N=444 N=384
1.75 to 2.25 50% 53% 57%
1.50 to 2.50 70% 71% 77%
1.25 to 2.75 78% 77% 83%
1.00 to 3.00 86% 79% 87%
0.75 to 3.25 89% 81% 90%
0.50 to 3.50 91% 84% 92%
0.25 to 3.75 93% 86% 95%
0.00 to 4.00 95% 87% 96%
others 5% 13% 4%
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5. Conclusion and Suggestions
As mentioned in Chapter 1, this research attempts to demonstrate the main factor of the performance of the leveraged and inverse ETF is not the length of the holding period, but the volatility of the underlying index or ETF. Observing the empirical test, no matter how long the investment horizon is, the coefficient of the variance of the Taiwan 50 is statistically significant both in the Taiwan 50 Bear -1X and the Taiwan 50 Bull.
However, its effect on the Bear -1X is opposite to that on the Bull 2X. First, in 30-, 60-, 120-day holding period, the relationship between the volatility and the return difference of Taiwan 50 and the Bear -1X is all negative. On the contrary, the relationship between the volatility and the return difference of Taiwan 50 and the Bull 2X is all positive. However, in accordance with the distribution of the realized multiples, either the Bear -1X or Bull 2X was no obvious difference within 30-, 60-, 120-day. (See in Table 4-A and Table 4-B)
In summary, the results of our research show the variance has a significant effect on both, no matter how long investors hold. If the volatility is moderate, the return difference may be close to zero; then the LETFs would be a convenient way to investors who desire to magnify the market return. Moreover, due to the increasing explanatory power of the volatility, we may make a further inference that whether the compounding effect is positive or negative depends on the volatility, especially within longer holding period. That is, without the great ability to forecast the variance, the LETFs are not recommended to the long-term investors.
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Table A. Change in Trading Volume of the Taiwan Exchange Market and OTC Market
Panel A. Change in Trading Volume Unit: (NTD) Panel B. Change in Trading Percentage
Unit: Percentage (%) 2009 2010 2011 2012 2013 2014 2015
Source:2015 annual report from financial Supervisory commission (FSC) , R.O.C
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6. Reference
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Financial Supervisory Commission (FSC), (2015). 2015 Annual Report R.O.C.
Joanne M.Hill and George O.Foster, (2009). Understanding Returns of Leveraged and Inverse Funds and Examining Performance over Time. IEOR Columbia, 10-16.
Lu, Lei, Jun Wang and Ge Zhang, (2009). Long-term Performance of Leveraged ETFs.
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Little, P.K., (2010). Inverse and Leveraged ETFs: Not Your Father’s ETF. The Journal of Index Investing, 1, pp. 83-89.
Marco Avellaneda, Stanley Zhang, (2009). Path-Dependence of Leveraged ETF Returns.
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Richard Co, (2009). Leveraged ETFs vs. Futures: Where Is the Missing Performance?
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Performance of Commodity-Based Leveraged ETFs. Journal of Index Investing, 3, 14-23 Trainor, W.J., and E. Baryla, (2008). Leveraged ETFs: A Risky Double That Doesn’t Multiply by Two. Journal of Financial Planning, 21 , 48-55.
Trainor, W.J., (2011). Solving the Leveraged ETF Compounding Problem. Journal of Index Investing, 21, 1-9.