大 國 槓桿型與反向型 ETF 之理論乘數與實際表現 國立政治大學財務管理學系研究所碩士學位論文 指導教授 : 周冠男博士 研究生 : 江怡婷撰 中華民國 106 年 02 月. Performance of the Leveraged and Inverse ETFs and their

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1 財務管理系研究所碩士位論文 槓桿型與反向型 ETF 之理論乘數與實際表現 Performance of the Leveraged and Inverse ETFs and their Multiples 指導教授 : 周冠男博士 研究生 : 江怡婷撰 中華民 106 年 02 月

2 摘要 自從槓桿型指數基金於各股票市場發行後, 各主管機關皆紛紛發出聲明表示, 該商品並不適合長期持有 ; 因此, 該類型投資商品的公開說明書皆會註明不宜長期投資 然而, 本研究實證結果發現, 持有期間長短並非主要風險來源 雖然, 如家所知, 槓桿型指數基金多是以 日 為單位追蹤指數, 而導致複利效果 (Compounding Effect) 使基金長期報酬與槓桿倍數不同 根據算出上述的報酬差異 (Return Difference) 可以發現不論是正向 2 倍或是反向 1 倍皆與台灣 50 報酬率的標準差有統計上顯著關係 反向 1 倍皆與台灣 50 報酬率的標準 差有顯著負相關 ; 反之, 正向 2 倍與台灣 50 報酬率的標準差有顯著正向相關 然而, 從已實現乘數 (Realized Multiple) 的分佈中可發現, 不合理值並不隨投資期間越長而 越多 意即儘管投資期間越長, 並不一定會導致複利效果越, 而與目標槓桿倍數 脫節 再者, 隨著投資期間越長, 波動度 (volatility) 的對於報酬差異的解釋力越強 ; 因此, 若想長期投資槓桿型指數資金, 預測標的波動度的能力更顯為重要 i

3 Abstract When we browse the reports about the inverse and leveraged ETF, most of them emphasize that the LETF is not appropriate to long-term investors. However, in this research, we attempt to demonstrate the main factor of the performance of the leveraged and inverse ETF is not how long the LETF we hold, 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, the relationship between the volatility and the return difference of Taiwan 50 and the Bear -1X is negative. In contrast, the relationship between the volatility and the return difference of Taiwan 50 and the Bull 2X is positive. However, in accordance with the distribution of the realized multiples, the frequency of either the Bear -1X or Bull 2X was not more and more when the holding period is longer. As a result, 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. Therefore, without the great ability to forecast the variance, the LETFs are not recommended to the long-term investors. ii

4 Table of Contents 1. Introduction Motivation of the study Chapter outlines Literature review Data and Methodology Data Methodology Empirical Results Overlapping return the Return Difference Paired t-test Regression Realized Multiples Conclusion and Suggestions Reference iii

5 List of Tables Table 1 Double-leveraged and inverse ETFs considered in the study Table 2 P-value of return differences among 30-Day, 60-Day, 120-Day Holding Period Table 3 Results of Regression Model: X=σ holding period (0050); Y=return difference Table 4- A Frequency of Realized Multiples of Taiwan 50 Bear -1X for 30-, 60-, 120-Day Holding Periods Table 4- B Frequency of Realized Multiples of Taiwan 50 Bull 2X for 30-, 60-, 120-Day Holding Periods iv

6 List of Figures Figure 1. Change in Trading Volume of the Taiwan Exchange Market... 2 Figure day returns for Bull 2X (symbol: 00631L) versus leveraged 30-day return of Taiwan 50(symbol: 0050). (X = 2 Ret.(0050);Y = Ret.(00631L)) Figure day returns for Bull 2X (symbol: 00631L) versus leveraged 60-day return of Taiwan 50(symbol: 0050). (X = 2 Ret.(0050);Y = Ret.(00631L)) Figure 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 )... 7 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 )... 8 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 ) 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 )... 9 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 ) 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 ) 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 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 v

7 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 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 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 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 vi

8 1. Introduction 1.1 Motivation of the study Since the financial crisis, obviously, the trading volume in Taiwan has decreased sharply in both the Exchange market and the OTC market. From Figure 1, we see that the annual trading volume has plunged. In the past 7 years, it reduced from NT$ 301,188 hundred million to NT$ 225,051 hundred million, about 25% of the 2009 trading volume. Nevertheless, there is one in the opposite direction. It is exchanged traded fund (ETF). Although all kinds of securities were diminishing, apparently, after 2014, the trading volume of ETF increased quickly. In 2015, it s up to 7% of total trading volume. The reason why it grew rapidly is the launch of inverse and leveraged exchanged traded fund (LETF). They could respectively deliver the inverse of the performance of its stated index and twice of it.on October 31, 2014, Yuanta Securities Investment Trust Co., Taiwan's largest ETF (exchange-traded fund) provider, launched first leveraged and inverse ETFs. These are the Yuanta Daily Taiwan 50 Bull 2X (symbol: 00631L) and Yuanta Daily Taiwan 50 Bear -1X (symbol: 00632R), listed the Taiwan Stock Exchange. They are two of the oldest; therefore, the data we analyzed in this research shows in Table 1. We can see that the realized betas calculated by daily returns are similar to the required betas. Table 1 Double-leveraged and inverse ETFs considered in the study. Yuanta 2X bull Yuanta -1X bear Underlying ETF Symbol 00631L 00632R Taiwan 50 Required Beta β = 2 β = -1 Taiwan 50 Realized Beta Βr = 1.9*** Βr = -0.9*** Taiwan 50 However, leveraged and inverse Exchange Traded Funds (ETFs) are designed to earn a multiple return of the stated index on a daily basis; therefore, owing to compounding, the return of a leveraged or inverse ETFs may be greater or less than that of the underlying index return times the multiple. That s why some reports, even the one from Taiwan Stock Exchange, have mentioned that leveraged an inverse ETFs are not appropriate to hold over time because of compounding problem which may 1

9 lead to destroy the portfolio values. Instead, this research attempts to defend time is not the key risk factor but others, which means LETFs could hold for extended periods of time if the volatility is moderate. Certainly, the compounding effect would be bigger in the longer holding period; nevertheless, when the volatility is moderate, the effect would be smaller, or even zero. It means if investors could forecast the market volatility, it is a convenient way to magnify the market return by holding the leveraged ETFs; however, if we cannot forecast it, the realized multiples may be far away from what we expect. Percntage of trading volume (%) 100% 98% 96% 94% 92% 90% 88% 301, , , , , , , , , , , ,000 unit: one hundred million (NTD) 86% c 84% 0% 100, Stock ETF Closed fund Beneficiary certificate Warrant TDR Convertible Bond Exchange Market LETFs Figure 1. Change in Trading Volume of the Taiwan Exchange Market 1 Source:2015 annual report from financial Supervisory commission (FSC), R.O.C 1 The actual figures show in appendix Table A. 2

10 1.2 Chapter outlines The remainder of this research is structured as follows. Chapter 2 delineates the relevant literatures of the long-term performance of leveraged and inverse ETFs and also focuses on the appropriate investment strategy of leveraged and inverse ETFs. Chapter 3 demonstrates the data we collect and the methods we utilize. Chapter 4 shows the empirical tests we analyze. Finally, in Chapter 5, we make the conclusions and provide some suggestions. 2. Literature review It is well known, because of compounding, leveraged ETFs do not provide the corresponding multiple return of the underlying index over extended investment horizons. Actually, all investments encounter compounding problem; however, on a daily basis, LETFs must rebalance daily to keep the fund multiple close to leverage ratio; nevertheless, people concern it would destroy the portfolio value. When the underlying ETF or index goes up, managers need to long more so as to increase the fund exposure, trying to make the multiple constant, and vice versa. Based on the above, it points out problem that the LETF manager must necessarily buy high and sell low in order to meet the target leverage requirement. However, it s not exactly at all. Hill and Foster (2009) suggests that the effect of compounding be almost neutral over time for strategies set up to provide a multiple return of an index or ETF. The magnitude of the compounding effect is related to market conditions. That s whether they are in a trending market or a stable market during the investment holding period. Moreover, Trainor and Carroll (2013) asserts that extended holding periods can be justified depending on volatility levels up to an investor's willingness to accept. Avellaneda and Zhang (2010) presents a formula for the value of an LETF linking with an index or ETF. This formula delivers the fact that LETFs will underperform the nominal returns due to the realized volatility of the underlying ETF. Also, we can clearly understand what elements directly influence the return of LETFs. These are not only the holding period, but also the funding rate, the expense ratio, the cost of borrowing shares in the case of short LETFs, the leverage ratio β, and the realized variance of the underlying ETF. It means the return of LETFs are not always worse than 3

11 a multiple of the underlying ETF s return at appropriate conditions or by a suitable investment strategy. Besides, Cheng and Madhavan (2009) states that the unsuitability of LETFs for longer-term investors is reinforced by the drag on returns from high transaction costs and tax inefficiency. It s the same as the formula in Avellaneda and Zhang (2010). It has been widely discussed by scholars for many years that how to utilize LETFs well. Avellaneda and Zhang (2010) also demonstrates that LETFs can be used for hedging and replicating unleveraged ETFs, provided that traders engage in dynamic hedging. Trainor (2011) suggests that if investors are able to predict low volatility markets so that leveraged and inverse ETFs are more likely to outperform or not diminish portfolio values. 3. Data and Methodology 3.1 Data In 2006, the first inverse and leveraged ETFs were introduced by ProShares, but in Taiwan, we launched the first LETFs by Yuanta on October 31, Apparently, we introduced the LETFs much later and Yuanta Daily Taiwan 50 Bull 2X and Yuanta Daily Taiwan 50 Bear -1X are two of the oldest. Considering the sample size; as a result, the data we analyze shows in Table 1 from October 31, 2014 to November 18, Methodology In this paper, we make an effort to state investment horizons is not the major factor which leads the underperformance of LETFs. Thus, we present the relationship between the LETF returns and the returns of the corresponding leveraged buy-and-hold portfolios composed of index ETFs and cash in different holding period. Next, by paired t-test, we can clearly know if there is a significant return difference between the LETF returns and the returns of the corresponding leveraged buy-and-hold portfolios. However, it is usual difficult to short the Taiwan 50 ETF in the spot market, which makes the high transaction cost; therefore, we assume that the corresponding strategy be based on the derivatives market, such the future market. By so doing, we can consider 4

12 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) multiple 2 *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, 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. 5

13 Figure 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 day returns for Bull 2X (symbol: 00631L) versus leveraged 60-day return of Taiwan 50(symbol: 0050). (X = 2 Ret.(0050);Y = Ret.(00631L)). 6

14 Furthermore, from Figure 4, it marks all the 120-day returns of Taiwan 50 Bull 2X 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 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). 7

15 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). 8

16 Moreover, the result of the inverse ETF, the Taiwan 50 Bear -1X, is similar with the 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, We calculate all 30-day (overlapping) return from October 31, 2014 to November 18, 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). 9

17 In addition, from Figures 9, we mark the all 60-day returns of Taiwan 50 Bear -1X and a static leveraged strategy with Taiwan 50 since October 31, Even in longer 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). 10

18 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, 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). 11

19 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. 4.2 Return Difference ret.(letf) β ret.(etf), β = +2/-1 log ret.(letf) β log ret.(etf), β = +2/-1 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. 12

20 21.00% 20.00% 19.00% 18.00% 17.00% 16.00% 15.00% 14.00% 13.00% 12.00% 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 0-10th 10-20th 20-30th 30-40th 40-50th 50-60th 60-70th 70-80th 80-90th th 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% performance of Taiwan 50 Bear -1X may be in the inverse relationship % 20.00% 18.00% 16.00% 14.00% 12.00% 10.00% 0-10th 10-20th 20-30th 30-40th 40-50th 50-60th 60-70th 70-80th 80-90th th 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% 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 13

21 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 % 19.00% 18.00% 17.00% 16.00% 15.00% 14.00% 13.00% 12.00% 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. 0-10th 10-20th 20-30th 30-40th 40-50th 50-60th 60-70th 70-80th 80-90th th 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% 16.00% 15.50% 15.00% 14.50% 14.00% 13.50% 13.00% 0-10th 10-20th 20-30th 30-40th 40-50th 50-60th 60-70th 70-80th 80-90th th 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% 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 14

22 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 % 16.50% 16.00% 15.50% 15.00% 14.50% 14.00% 13.50% 13.00% 12.50% 12.00% 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 0-10th 10-20th 20-30th 30-40th 40-50th 50-60th 60-70th 70-80th 80-90th th 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% 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

23 20.00% 19.00% 18.00% 17.00% 16.00% 15.00% 14.00% 13.00% 12.00% 0-10th 10-20th 20-30th 30-40th 40-50th 50-60th 60-70th 70-80th 80-90th th 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% 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. 16

24 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 (<0.001) (<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. 17

25 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 *** *** *** *** *** *** p-value <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 N R-square 9.93% 25.54% 36.37% 4.63% 19.49% 28.91% adjusted R-square 9.74% 25.37% 36.20% 4.43% 19.31% 28.72% 4.5 Realized Multiples Some investors may need a portfolio with a realized multiple, which is close to the 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 18

26 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= to % 55% 50% to % 68% 70% to % 77% 80% 0.00 to % 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= to % 53% 57% 1.50 to % 71% 77% 1.25 to % 77% 83% 1.00 to % 79% 87% 0.75 to % 81% 90% 0.50 to % 84% 92% 0.25 to % 86% 95% 0.00 to % 87% 96% others 5% 13% 4% 19

27 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. 20

28 Appendix Table A. Change in Trading Volume of the Taiwan Exchange Market and OTC Market Panel A. Change in Trading Volume Unit: (NTD) One hundred million Exchange Market 301, , , , , , ,051 Stock 296, , , , , , ,915 ETF 1,978 1,996 3,608 2,692 2,824 4,333 16,344 Closed fund Beneficiary certificate Warrant 1,064 2,050 2,847 2,313 3,400 6,789 6,450 TDR 1,285 2,619 1, Convertible Bond OTC 770, , , , , , ,848 Stock 52,390 56,336 39,930 29,519 40,309 63,559 56,892 Warrant ,624 1,566 Bond 717, , , , , , ,390 Panel B. Change in Trading Percentage Unit: Percentage (%) Exchange Market 100% 100% 100% 100% 100% 100% 100% Stock 99% 98% 97% 97% 97% 95% 90% ETF 1% 1% 1% 1% 1% 2% 7% Closed fund 0% 0% 0% 0% 0% 0% 0% Beneficiary certificate 0% 0% 0% 0% 0% 0% 0% Warrant 0% 1% 1% 1% 2% 3% 3% TDR 0% 1% 1% 0% 0% 0% 0% Convertible Bond 0% 0% 0% 0% 0% 0% 0% OTC 100% 100% 100% 100% 100% 100% 100% Stock 7% 7% 5% 4% 7% 11% 10% Warrant 0% 0% 0% 0% 0% 0% 0% Bond 93% 93% 95% 96% 93% 89% 90% Source:2015 annual report from financial Supervisory commission (FSC), R.O.C 21

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