ETFs provide the benefits of diversification, liquidity, transparency, easy access and tax efficiency [21]. As a subset of ETFs, LETFs intend to deliver a given multiple, such as 2×, 3×, −1×, −2×, or −3×, of underlying index within a certain period. This multiple is also referred to as fund multiple, stated multiple, promised multiple, or product multiple. The most popular LETFs are the ones to achieve an intended daily multiple. To deliver a constant leverage on the return of underlying index, fund managers rebalance the exposure on the index at the end of each trading day⁶. More specifically, making a deal with counterparties in the swaps market or using the futures to attain a certain level of exposure are the two major ways. The final return received by an LETF investor is the remaining after deducting the financing costs to deliver the leverage effect and the management fees paid to fund management companies from the net earnings of all the assets of LETFs during each trading day. Apparently, the multiple return on the index is not the original “cake” that is distributed to the participants related to LETF investing, but is the perfect mold of the cake. When it comes to multiple holding days, the actual return that investors can receive from their holding LETFs will be determined by the path of index return, which is highly related to market characteristics. Therefore, the stock indices from different markets may contribute to the return behaviors of LETFs in a substantially different way.

Some institutional background pertinent to Chinese A-share market can make the performance of Chinese LETFs more complicated than the performance of LETFs leveraging other markets. Note that the impact of institutional backgrounds is not directly exerted to LETF performance but is through the performance of the indices underlying LETFs, and these market-specific features of indices could not be reflected by the simulated returns of hypothetical indices with the normal distribution⁷. In the context of Chinese A-share market, there are two evident features relevant to the index performance. First, although the A-share market is on the way of being more mature and effective, the market maturity and effectiveness of the market is still at a relatively low level. This feature could be taken along with few years in the future, especially under the limitations pertaining to the short selling which could impede the price efficiency in Chinese market⁸. The other feature relates to the structures and behaviors of investors in Chinese financial market. From the shareholding structure based on the circulation market value, the individual investors have accounted for more than 75% of the total investors in Chinese A-share market, resulting in a market with highly speculative atmosphere. Such a phenomenon will be more evident due to the frequent policy intervention imposed on A-share market. All these mentioned features contribute to the complicated behaviors of securities trading in Chinese A-share market, such as high volatility and strong comovement. As a result, there is likely more uncertainty in the performance of LETFs leveraging Chinese A-share market, which couldn’t be explained by the extant literature nor be disclosed by any simulation based on a given index return distribution, such as a log-Normal distribution.

In the academic literature, performance is one of the most notable issues of LETFs. The payoff function of an LETF is a multiple of the underlying asset’s return over a certain rebalancing period. Since there are often multiple rebalancing periods within a holding period, LETF performance is path dependent. Therefore, pricing LETFs and the relationship between returns of LETFs and the returns of underlying index can be analyzed mathematically [12] [16] [20].

There are two popular simulation methods in the literature used to examine LETF performance. One method is to simulate using the actual historical index returns [4] and the other is to use the Monte Carlo simulation based on an assumed distribution, such as log normal returns [30]. The historical index return method has the advantage to be objective as the historical returns are unique. This method also has the advantage that it does not need any distribution assumption of returns. It is well known that stock returns have fatter tails than typical log-normal distribution, affecting LETF performances. For these reasons, we adopt the simulation method by using historical index returns.

Simulating the performance of LETFs by employing the return data of the entire history of Dow Jones Industry Average, [4] confirmed that the general perception regarding the value destruction arising from index return’s path is not substantiated. Similarly, [5] find bull LETFs benchmarked to the S&P 500 Index produced much higher return multiple than their given leverage over long investment horizon. As shown in our later empirical tests, there is much to say about the performance properties of Chinese LETFs, which are different from either the performance enhancement documented in [4] and [5] or the value destruction warned by the U.S. Securities and Exchange Commission (SEC) and financial media.

Since LETF performance is highly path dependent, the prior studies on the performance of LETFs, mainly focusing on the developed markets, could not reflect the performance properties of LETFs leveraging the emerging markets⁹. This is particularly relevant as the stock indices of emerging markets perform with more risks and uncertainty than those from developed markets. Even without taking into account the delivery of leveraged effect of index return, the index performance can still significantly affect the return deviation of LETFs over any holding period different from the rebalancing period. Therefore, given the high possibility of being more volatile for stock indices from emerging markets than the stock indices from developed markets, the investigation on the performance of Chinese LETFs can produce different results, which are not documented in the extant literature. In addition, due to the relatively short history of LETFs and the increasing availability of LETFs leveraging different markets, the new evidence of LETF performance based on a rarely explored market can provide new insight for regulators and potential investors.

The data for this study are obtained from Wind database and Bloomberg. The prices of stock indices in the mainland of China are collected from Wind database, and the American indices data are collected from Bloomberg.

Because the CSI 300 Index is the most popular stock index of Chinese mainland and the most commonly used index of Chinese ETFs, we use it as the underlying index to simulate a set of hypothetical LETFs. When it comes to the U.S. market, due to the similar reasons and for comparison purposes, we use the S&P 500 Index as the underlying index of LETFs. In addition, we also use some other representative stock indices of Chinese market and the U.S. market, including the Shanghai Composite Index, the Shenzhen Component Index, and the CSI 1000 Index from Chinese market, as well as the Dow Jones Industrial Average Index, the NASDAQ Composite Index, and the Russell 2000 Index from the U.S. market.

We use the return data of indices underlying LETFs of Chinese and the U.S. markets. To measure LETF performance, we adopt returns, return deviations, and leverage of LETFs, following previous literature.

We compute the target return, the naively expected return and the return deviation of LETFs following [3] ¹⁰. As for the leverage, we calculate the effective multiple following [4]. In particular, the daily target return is the product between the target multiple of the LETF and the return on the underlying index. As shown in Equation (1), we compound the daily target return geometrically to obtain the target return over multiple holding days.

Target Return = ∏ k = 1 T ( 1 + m r k ) − 1 , (1)

where r_k is the index return on day k, T is the number of trading days during the holding period, and m is the target multiple of an LETF.

The naively expected return of an LETF is calculated as the product between the daily target multiple of the LETF and the cumulative return of the underlying index for any investment horizon, which is given as:

Expected Return = m × ( Index Return ) = m × ( ∏ k = 1 T ( 1 + r k ) − 1 ) . (2)

The return deviation is computed by using the target return less the naively expected return, and the effective multiple is the ratio of target return to underlying index return. Equations (3) and (4) specify the calculation of return deviation and effective multiple, respectively.

Return Deviation = Target Return − Expected Return . (3)

Effective Multiple = Target Return Underlying Index Return . (4)

To clearly show the calculation of the above performance metrics of LETFs, we give a simple example as follows. To simplify the expression, we use prefix of 2×, 3×, −1×, −2× and −3× to indicate the promised multiple of an LETF, corresponding to the return multiple of 2, 3, −1, −2 and −3, respectively. If the underlying index of an LETF experiences a return of 10% in a trading day and then a return of −10% in the next trading day, the index cumulative return is (1 + 10%) × (1 − 10%) − 1 = −1%, the daily target returns of the 2× LETF for the first day and the next day are 20% and −20%, respectively, and the daily expected return is the same as the daily target return. However, the target returns over the two trading days is (1 + 2 × 10%) × (1 − 2 × 10%) − 1 = −4%, differing from the expected return, 2 × ((1 + 10%) × (1 − 10%) − 1) = −2%. Consequently, the return deviation for the fund during this holding period is −4% − (−2%) = −2%, and the effective multiple is −4% ÷ (−1%) = 4, totally different from the stated multiple.

Another alternative way to measure the performance of LETFs is to regress the fund returns on index returns. In this study, LETF return is approximated by the target return, so the regression equation is expressed by:

Target Return k = α + β r k + ε . (5)

Based on naïve expectation, an LETF should deliver the target multiple of underlying index return in any investment horizon. If the expectation is true, then α will be zero, and β will equal the promised return multiple of the LETF. In this study, since the fund returns are over multiple days during which the index performance will distort the fund performance, there may be discrepancy between constant and zero, and also between β and the promised multiple. How well an LETF performs relative to the expectation is measured by the significance of α and the closeness of β to the promised multiple. For each LETF, we run the regression as shown in Equation (5).

Table 1 shows the summary statistics for the performance of hypothetical LETFs tracking the CSI 300 Index. We compute the index return, the target return, the naively expected return, the return deviation and the effective multiple of LETFs. From April 11, 2005 to June 30, 2020, there are 3700 daily return observations. As shown in Panel A, the average daily return of the CSI 300 Index is 5.3 basis points. The standard deviation of the daily return of the CSI 300 Index is 1.719%, showing a high volatility over the history, which is also confirmed by the wide range of the index returns, from the minimum of −9.24% to the maximum of 9.342%. The median return is 9 basis points, showing that the CSI 300 Index experiences positive returns in a majority of trading days. We find a positive return in 1981 out of the 3700 observations or 53.5% of the time. Due to the positive mean and median of the index return, the bull funds, on average and often, experience positive returns in various holding periods, while the bear funds experience negative average and median returns. It should be noted that, due to the daily rebalancing nature of LETFs, the mean (median) of an LETF’s daily return is closely associated with the mean (median) of underlying index’s daily return, but is not affected by the volatility of the index. However, as we will explain later,

Note: This table reports the summary statistics for the performance of a set of hypothetical LETFs tracking the CSI 300 Index within the holding period of 1 trading day (Panel A), 5 trading days (Panel B), 22 trading days (Panel C) and 250 trading days (Panel D). The sample period is from the inception date of the CSI 300 Index (April 11, 2005), to June 30, 2020. R_index, R_target, and R_expected are the index’s return, the target return and the expected return of investors, respectively. Deviation is the return deviation defined by Equation (3), and multiple is the effective multiple defined by Equation (4). N is the number of observations, and N > 0 (%) is the ratio of the number of positive observations to N. All returns and return deviations are denoted in percentage terms.

for a holding period longer than a trading day, the case will be different and complicated.

In Panels B, C and D, we extend the holding period to 5, 22 and 250 trading days. As shown in Panel B, the returns of underlying index return range from −22.142% to 17.275%, with the average return of 0.285% and the median return of 0.582% during 5 trading days. The standard deviation of the index’s return is 4.042%, displaying a high volatility. As shown in the second row of Panel B, the target returns of the 2× LETF averages at 0.583%, ranging from −40.493% to 35.607%. At the same time, the expected return is 0.57%. These results show that the target return is actually different from what a naïve investor would expect, which leads to a return deviation. As shown in the fourth row of Panel B, the return deviation has a positive mean of 0.013%, but a negative median of −0.07%. These results show that the return deviation, in terms of frequency, is more likely to be detrimental to the performance of the 2× LETF during the sample period. As the holding period is not too longer than a trading day, the effective multiples of all the five LETFs in Panel B are close to their stated multiples. The rest of Panel B illustrates that the positive mean and negative median of return deviations also apply to the 3×, −1×, −2× and −3× LETFs. For LETFs in Panels C and D, a similar conclusion applies to their performance.

Comparing across the panels, as the number of holding days increases, the returns of bull LETFs increase and those of bear funds decrease. For example, for 3× fund, the average target return increases from 0.16% to 0.895%, 4.75%, and 213.8% during 1, 5, 22, and 250 trading days respectively. Without leverage, the average returns of 1× fund would be 0.053%, 0.285%, 1.35%, 24.8% during 1, 5, 22, and 250 trading days respectively. Based on these findings, it is clear that the gap between the 3× fund and the 1× fund gets larger when the holding period lengthens. The existence of this gap is due to the fact that the LETFs have a daily rebalancing. It is a bit surprising to observe such a large gap between the average return of 24.8% for 1× fund and 213.8% for 3× fund. This gap is much larger than the product multiple of 3 times would predict. To accurately measure this gap, as previously mentioned, we define the variable of return deviation between the cumulative return of 3× fund and 3× of the cumulative index returns of the index. When this return deviation is positive, there is a positive surprise to the LETF fund holder. For example, as shown in Panel D, the highest return deviation during the 250-trading-day periods is 4115.9%. This return deviation can also be detrimental to the LETF returns. Actually, the worst return deviation is −58.2%, showing that the return of 3× fund is much lower than its naively expected counterpart during the 250 trading days. Across Panels B, C and D, we find that the return deviation increases as the holding period lengthens. For example, the average return deviations of −3× fund change from 0.073% to 0.967%, and further to 57.41% for 5, 22, and 250 holding days respectively, showing that, on average, the LETFs perform better than the naive expectation during these holding periods respectively.

When comparing the effective multiple of LETFs across Panels B, C, and D, we find that the discrepancy between the effective multiple and stated multiple is increasing with the holding period length. For example, the average effective multiple for −3× fund changes from −2.988, to −2.907, and further to −1.808 for 5, 22, and 250 holding days respectively, suggesting a larger deviation from the target multiple of −3× as the holding period lengthens.

Collectively, bull Chinese LETFs deliver positive returns during various holding periods on average. These returns increase as the holding period lengthens. On the other hand, bear LETFs deliver negative returns on average. Moreover, both bull and bear LETFs perform better than their naively expected counterparts on average during the holding period of a week, a month, and a year. However, in terms of frequency, both bull and bear funds are likely to underperform, relative to their naively expected counterparts, more often for holding periods of 5 trading days or longer. These results indicate that the performance of CSI 300 LETFs based on the entire history of index is complicated and cannot be outlined directly by the performance enhancement documented in [4] and [5] nor the potential value destruction often warned by regulators and financial media.

To provide some context to the Chinese LETFs, in this subsection, we investigate the relative performance of CSI 300 LETFs benchmarked on the S&P 500 LETFs, as the U.S. LETFs are mostly studied in the literature. To ease the inconsistency concern of sample period between the two indices, we match the two samples to the same period of January 2006 to June 2020 and present the comparison in Table 2¹¹.

Across the two markets, the results of Table 2 show that, the average target returns and return deviations of all the bull funds written on the CSI 300 Index are always higher than their counterparts leveraging the S&P 500 Index. However, we should not conclude that these Chinese LETFs perform better than U.S. LETFs, because the means of returns and deviations are easily affected by the extreme observations. In this case, the medians could be more instructive for investors when judging the investment decisions on the leveraged funds. As shown in Panels A and B, Chinese bull LETFs deliver higher target returns than the U.S. counterparts when the holding period is one day or 5 trading days. For example, the 5-day target returns for 3× Chinese LETF have a median of 1.554%, which is higher than the median of 1.239% for 3× U.S. LETF counterpart. In contrast, as shown in Panels C and D, when the holding period lengthens to 22 or 250 trading days, U.S. bull LETFs actually deliver higher median target returns than their Chinese counterparts. For example, the 250-day target returns for 3× Chinese LETF have a median of −5.03%, which is lower than the median of 31.476% for 3× U.S. LETF counterpart. These findings suggest that a 3× Chinese LETF holder

Note: This table reports the mean, standard deviation and the median of the performance metrics of hypothetical LETFs tracking the CSI 300 Index (columns (1) - (3)) and the S&P 500 Index (columns (4) - (6)) within the holding period of 1 trading day (Panel A), 5 trading days (Panel B), 22 trading days (Panel C) and 250 trading days (Panel D). We also conduct the difference tests of mean and medians of LETF performance between the CSI 300 LETFs and S&P 500 LETFs (see columns (7) and (8)). The sample period is from January 2006 to June 2020. For CSI 300 LETFs, the number of observations for Panels A, B, C and D are 3521, 3341, 3325 and 3108, respectively. For S&P 500 LETFs, the number of observations for Panels A, B, C and D are 3647, 3461, 3445 and 3228, respectively. R_index, R_target, and R_expected are the index’s return, the target return and the expected return of investors, respectively. Deviation is the return deviation defined by Equation (3), and multiple is the effective multiple defined by Equation (4). All returns and return deviations are denoted in percentage terms. *significant at the 10% level; **significant at the 5% level; ***significant at the 1% level.

is more likely to experience a negative cumulative return for a holding period of one year. In contrary, a 3× U.S. LETF holder is more likely to experience a positive cumulative return for a holding period of one year.

Examining the medians of return deviations in Table 2, we find that all the median return deviations are negative for all bull and bear Chinese LETFs, showing that the target returns are more likely to be less than the naively expected returns for all Chinese LETFs. In addition, the median return deviation of Chinese funds is always more negative than that of the corresponding U.S. funds, indicating that, from January 2006 to June 2020, Chinese LETFs are more likely to suffer losses from the path of underlying index return, resulting in poor performance over a long holding period. For the −3× funds, the median return deviations are −9.335% for Chinese LETF and −0.124% for the U.S. LETF, differing by more than nine percentage points, as shown in Panel D. Consequently, the effective multiples of Chinese LETFs display larger deviations to the stated multiples than those of U.S. LETFs.

To formally test the difference of performance between CSI 300 LETFs and S&P 500 LETFs, we conduct the mean-difference tests and median-difference tests of performance metrics of LETFs between the two markets. The results are shown in the last two columns of Table 2. As shown in Panel A, the difference tests show that, no matter on means or medians, all the target returns of Chinese funds are higher than those of U.S. funds, suggesting a better performance in Chinese funds than in U.S. funds over short holding periods. However, these results do not hold any more as the holding period lengthens to 22 and 250 trading days. As for the return deviations, almost all the means (medians) of return deviations of CSI 300 LETFs are significantly more positive (negative) than those of S&P 500 LETFs, within a holding period of 5, 22 and 250 trading days. Collectively, the difference test in LETF performance shows that there are indeed many differences in LETF performance across the two markets. In particular, Chinese leveraged funds are more appropriate for short-term investment than their U.S. counterparts but suffer more distortions in fund performance over long holding periods.

Observing that the results of average return deviations and median return deviations are opposite, we conduct further analysis to compare the performance across CSI 300 LETFs and S&P 500 LETFs. Specifically, we regress the target returns of LETFs on the cumulative return of index over LETFs’ holding period of 5, 22 and 250 trading days, respectively. The results are presented in Table 3. As shown in Panel A, over a holding period of 5 trading days, all ten funds have superior performance, indicated by a coefficient very close to the promised return multiple and an R-squared very close to 1. However, when it comes to the constant, there is a significant difference between each CSI 300 LETF and its counterpart tracking the S&P 500 Index. The constants of all the five CSI 300 LETFs are positive and significant at 10% level, and the results of S&P 500 LETFs are significantly negative at the level of 1%. Collectively, these findings suggest that, over a short holding period, the LETFs tracking the CSI 300 Index can deliver

Note: This table presents the regression results of LETF target returns on index returns over 5 trading days (Panel A), 22 trading days (Panel B) and 250 trading days (Panel C). The sample period is from January 2006 to June 2020. R_index is the return of LETFs’ underlying index. Newey-West robust t-statistics are given in parentheses; *significant at the 10% level; **significant at the 5% level; ***significant at the 1% level.

positive excess returns, while the LETFs leveraging the S&P 500 Index deliver negative excess returns. When holding period lengthens to 22 or 250 trading days, the coefficients of LETFs seem to deviate from their respective stated multiples, especially for CSI 300 LETFs. Therefore, since the results in Panels B and C don’t have the intended tracking multiple, the differences in the constants between CSI 300 LETFs and S&P 500 LETFs cannot be regarded as the excess returns like the case in Panel A. Overall, the regression results in Table 3 are consistent with the findings in Table 2.

To test whether the results are robust, we compute the Sharpe ratios of LETFs, one of the risk-adjusted performance metrics of funds, and then again compare between CSI 300 LETFs and S&P 500 LETFs. The results are presented in Table 4. For the purpose of comparison across different holding periods, we annualize

Note: This table presents the annual Sharpe ratio of hypothetical LETFs tracking the CSI 300 Index (columns (1) - (4)) and the S&P 500 Index (columns (5) - (8)) within the holding period of 1, 5, 22 and 250 trading day/days. The sample period is from January 2006 to June 2020. Panel A is based on rolling observations and Panel B is based on non-rolling observations. The risk-free rate of Chinese market is attained from RESSET database and for U.S. market, the risk-free rate is collected from Kenneth French’s web site at Dartmouth.

all the ratios. As shown in the table, the results computed by rolling returns are very close to those based on non-rolling results. The results in Panel B show that, only the bull LETFs have positive Sharpe ratios and all the bear funds have negative Sharpe ratios. Across the two types of funds leveraging different market indices, the bull Chinese funds seem to have higher positive risk-adjusted returns than their U.S. counterparts over the holding periods of 1, 5, and 22 trading day/days. Interestingly, this result reverses when the holding period lengthens to 250 trading days. Nevertheless, these results are consistent with our previous analyses, which support that Chinese LETFs tend to perform better than U.S. LETFs in short holding periods while the result reverses in long holding periods.

The better performance of CSI 300 LETFs over short holding period is due to the higher possibility of extreme observations in the returns of the CSI 300 Index relative to those of the S&P 500 Index. As documented in [4], compared to the normal distribution, the more leptokurtic distribution of actual index results in more performance enhancement of LETFs¹². Applying this proof to our cases, since the observations of kurtosis of daily returns on the CSI 300 Index during 5-trading-day holding period are more likely to be higher than that those on the S&P 500 Index, it is reasonable to expect a better performance of CSI 300 LETFs over that 5 holding period¹³. Similarly, it is natural to observe an inversed result over the holding period of 250 trading days since we do observe a lower mean kurtosis of the daily returns for CSI 300 in this long holding period.

Based on the empirical analysis in this subsection, we draw the following conclusions On Chinese LETFs. A high volatility of underlying index does not necessarily correspond to poor LETF performance, and the cumulative return of index can also play a key role on the net effect on LETF performance. Therefore, estimating the net effect of index on LETF performance is not a simple nor easy task. In our performance comparison, although the volatility of the CSI 300 Index is higher than that of the S&P 500 Index, CSI 300 LETFs perform better than S&P 500 LETFs for short holding period. In contrast, S&P 500 LETFs perform better than CSI 300 LETFs for longer holding time periods. On one hand, these findings alleviate the concern that high volatility of index is necessarily linked with LETF poor performance, and LETFs leveraging the Chinese A-share market seem to display a complicated performance due to the characteristics of stock market index. On the other hand, it inspires us to investigate the explicit quantitative relationship between index return and index volatility, which is instructive to market participants.

As the demand to trade LETFs often increases when there could be large changes in the index returns, we examine the performance of Chinese LETFs during crisis periods. In particular, the entire history of the underlying indices allows us to investigate the 2008 U.S. financial crisis period, the 2015 Chinese stock market crash period, and the 2020 COVID-19 pandemic period¹⁴. Unlike the performance of LETFs based on the entire history of index, the tests based on a short interval don’t always allow for the holding period of 250 trading days during the crisis period. Therefore, in this subsection, we only conduct the tests on fund performance over the holding periods of 1, 5, and 22 trading day/days.

The results during the 2008 financial crisis period are shown in columns (1) - (3) of Table 5. Following [4], this crisis period is set from October 2007 to March 2009. The standard deviation of daily return for the CSI 300 Index is 2.816%, higher than that based on a long history of index (as shown in Table 1: 1.719%). This higher volatility of index is also associated with a more negative return deviation of LETFs in the same holding period. For example, except for the 3× fund during a holding period of 22 trading days, all the other medians are negative, suggesting that index performance is more likely to be detrimental to the performance of these LETFs during the financial crisis of 2008.

The results during the 2015 Chinese market crash period (June 2015 to February 2016) are similar to those during the 2008 financial crisis period. As shown in columns (4) - (6), the median daily return of the CSI 300 Index is positive. In most cases, the bull (bear) LETFs on the CSI 300 Index register negative (positive) returns. As for the return deviation, all the median values are negative but seven of the ten average values are positive, similar to the results based on the history of index.

When it comes to the 2020 COVID-19 pandemic period (June 2015 to February 2016), the results seem to be quite different with those during the 2008 financial crisis period and the 2015 Chinese stock market crash period. As shown columns (7) - (9), the CSI 300 Index cumulative returns over 1, 5 and 22 trading day/days are positive on average and in terms of frequency, resulting in positive mean and median returns of LETFs. However, the positive return does not mean positive return deviations. Actually, due to a high volatility of index return, all the median return deviations of CSI 300 LETFs are negative. In our untabulated results related to the performance of the S&P 500 Index LETFs during the same period, we observe that the mean return of each bull (bear) fund is negative (positive), against the results of CSI 300 LETFs. It suggests that Chinese market indices are not affected as severely as the U.S. market indices by the pandemic.

As shown in Table 6, our further tests on the Sharpe ratios of Chinese LETFs during the three financial crisis periods reveal that it is highly profitable for investors to substitute the bull LETFs with bear LETFs during the 2008 financial crisis period and the 2015 Chinese market crash period. Very high Sharpe ratios occur to the bear funds during the two periods. Specifically, the highest Sharpe ratio during a long history of the CSI 300 Index is 0.518 (3× CSI 300 funds in Table 4 over the holding period 22 trading days based on the non-rolling observations), while the three bear funds during the two crises registered Sharpe ratios around 1. However, the results during the 2020 COVID-19 pandemic period are

Note: This table presents the annual Sharpe ratio of hypothetical LETFs tracking the CSI 300 Index within the holding period of 1, 5, and 22 trading day/days during three typical crisis periods. Columns (1) - (3), (4) - (6), and (7) - (9) present the results during the 2008 financial crisis, 2015 Chinese stock market crash, and 2020 COVID-19 pandemic, respectively. Panel A is based on rolling observations and Panel B is based on non-rolling observations. The risk-free rate of Chinese market is attained from RESSET database and for U.S. market, the risk-free rate is collected from Kenneth French’s web site at Dartmouth.

quite different, with positive Sharpe ratios in bull funds and negative Sharpe ratios in bear funds. These results are consistent with the evidence shown in Table 5 implying that the performance of Chinese market indices does not experience too many changes during the pandemic.

Collectively, the tests of LETF performance during the three crisis periods show that LETF performance will be distorted severely if the underlying index is highly fluctuating. Moreover, contrary to the results based on a long history of index, bull funds tend to have negative returns and bear funds tend to have positive returns during the crisis periods. Among the three crisis periods, the returns of CSI 300 LETFs seem to be dragged down most during the 2015 Chinese stock market crash period.

In the previous analyses, we find that the sample LETFs perform better than naïve expectation sometimes, but worse in other time, due to the compounding effect. This compounding effect is also important to the comparison across the two countries. In this subsection, to provide some guidance to investors, we further explore the explicit conditions of index returns for them to benefit the fund performance in general.

Before carrying out the further investigation on the association between LETF performance and index performance, we wonder whether the impact of index performance accounts for a significant part of the total return deviation in the actual LETFs. To address this issue, we compute the absolute ratio of compounding deviations to the NAV return deviation of a batch of the earliest listed LETFs in the world. The sample period is from 6/20/2006 to 6/30/2020. The NAV deviation is the difference between NAV return and target return of LETFs. Therefore, the ratio we computed can tell which deviation dominates in the total return deviation. We compute the ratios in 5, 22, 250 holding periods, respectively, and all results are winsorized at the 1% tails. The results are shown in Table 7.

As shown in Table 7, all the mean ratios are greater than one, indicating that the compounding deviation over multiple days is the main component of return deviation. As the holding periods lengthen, the compounding deviations seem to account for more of the total return deviations. With that in mind, it is likely that the compounding deviations in Chinese potential LETFs are likely to be more pronounced given the higher volatility of stock indices in Chinese A-share market.

LETF performance vitally depends on the return path of the underlying index [20]. When the target return exceeds the expected return, or when the return deviation is positive, the index return’s path benefits the fund performance and vice versa. Building on the analyses of [12], we model the relationship between LETF performance and index performance, and present the detailed derivation in Appendix.

According to the derivation in Appendix, there must be just two critical values of index returns corresponding to the zero return deviation¹⁵. In order to figure out these two critical values, we let Equation (A6) be zero:

( 1 + R I ) m exp ( 1 2 ( m − m 2 ) σ 2 t ) − 1 − m R I = 0 , (6)

where R_I and σ₂ are the cumulative return and the variance of the underlying index return during time t, respectively.

The approximate values of the two critical returns¹⁶, which are denoted as R_c1 and R_c2, are given as the following two expressions, respectively:

R c 1 = − m ( a − 1 ) + m ( a − 1 ) [ m ( a − 1 ) − 2 a ( m − 1 ) ] a m ( m − 1 ) , (7)

R c 2 = − m ( a − 1 ) − m ( a − 1 ) [ m ( a − 1 ) − 2 a ( m − 1 ) ] a m ( m − 1 ) , (8)

Note: This table presents the summary statistics of the ratio of absolute compounding deviation to absolute NAV deviation of a batch of the earliest listed LETFs in the world. All these eight LETFs are listed in the U.S. market. The sample period is from 6/20/2006 to 6/30/2020. The compounding deviation is the discrepancy between target return and the expected return of naïve investors, and the NAV deviation is the difference between NAV return and target return of LETFs. Therefore, the ratio we computed can tell which deviation dominates in the total return deviation computed by the difference between NAV return and expected return. We compute the ratios in 5, 22, 250 holding periods, respectively, and all results are winsorized in at the 1% tails. All deviations are denoted in percentage terms.

where a is given as:

a = exp ( 1 2 ( m − m 2 ) t σ 2 ) . (9)

Given that m > 1 or m < 0, Equation (7) is always positive and Equation (8) is always negative, implying that there are always two critical returns corresponding to the zero-value of LETF return deviation, one of which is positive and the other is negative. It also means that, in order to benefit LETF performance, the underlying index has to exceed the positive critical return (R_c1) for bull funds, or accumulate a return more negative than the negative critical return (R_c2) for bear funds.

Let m equal 2 or −2, we find that our formulas are equivalent to the expressions of break-even levels of index proposed by [20], indicating that our formulas of critical returns are applicable and general, and the two given numerical expressions of [20] can be included in our extensive formulas.

As shown by the derivation and analysis above, now we can assume that the graph function of return deviation with respect to the index return is approximately U-shaped when the index returns range from −1 to 1¹⁷. This property of return deviation makes it very easy to understand the clear quantitative relationship between index performance and LETF performance.

Figure 1 helps us intuitively understand the impact of index performance on LETF performance. In this figure, we set the variances of the index daily returns to 4 in the left three graphs and 2 in the right three graphs. Indeed, as shown in Figure 1, each line is U-shaped and crosses the zero-line of return deviation twice when the return of index ranges from −1 to 1. The two critical returns corresponding to the break-even level of return deviation divide the graph of return deviation into three parts. These geometrical characteristics reveal that the magnitude of return deviation depends on the return difference between the index return during the holding period and the apex return of the graph, and the direction of return deviation depends on the range where the return of underlying index lies and which is specified by the two critical returns. In addition, the lines in Figure 1 are not symmetric around the zero-line of index return, indicating that the two critical returns are not the inverse to each other. More importantly, Figure 1 intuitively shows that a bull (bear) fund has to accumulate a positive (negative) return higher (less) than the positive (negative) critical return to benefit the fund performance.

To provide market participants general cases about the impact of index on LETF performance for references, given a set of variances of index daily return (0.5, 1, …, 9.5, 10) during the holding period of 5, 22 and 250 trading days, we compute the corresponding values of return that index needs to accumulate to offset the negative effect of these given variances. Since most of the variances of

index daily returns are always in the range from 0 to 10, we compute the critical returns based on the variances in this range. Because investors who wish to trade on the upward (downward) movement of index will buy bull (bear) LETFs, we only report the positive (negative) critical returns for bull (bear) LETFs. These critical returns are computed via Equations (7) or (8). The results are presented in Table 8. In Panel A, for the 2× funds, the critical returns for variances of 0.5 and 10 are 1.61% and 7.6%, respectively, with a difference of 5.99%. Comparing the critical returns across 2×, 3× and −3× funds with the same volatility during the same holding period, such as 5 in Panel C, the absolute values increase from 28.34% to 32.69%, and then to 34.43%. Given the same target leverage and the same volatility, such as 3× fund with a variance of 5, when the holding periods

Note: This table presents the positive (negative) critical return of the underlying index for bull (bear) LETFs, given the target multiple, the holding period of fund and the expected daily return volatility of the underlying index. The results based on a holding period of 5, 22 and 250 trading days are shown in Panels A, B and C, respectively. The expected variance of index return is given at twenty different level, from 0.5 to 10. The critical return is the extent of return that an index underlying an LETF should reach in order to benefit the fund performance. All returns are denoted in percentage terms.

change from 5 to 22, and then to 250 trading days, the critical returns change from 5.40% to 12.38%, and then to 32.69%¹⁸.

Collectively, based on the above analyses in subsection 4.4, we can draw the following conclusions. First, to benefit LETF investors and LETF performance, a bull (bear) fund has to accumulate a return higher (lower) than the positive (negative) critical return. Second, to offset the negative effect of index volatility during the holding period of LETFs, ceteris paribus: 1) LETFs with higher leverage require more cumulative return of index; 2) With the same magnitude of promised multiple, bear funds require more absolute return of index than the corresponding bull funds; 3) LETFs over longer holding periods require more cumulative return of index. Last, within a long holding period, even a small difference between index volatility can cause a substantial negative effect on LETF performance.

We also simulate the performance of LETFs on some other representative stock indices in Chinese A-share market, including the Shanghai Composite Index, the Shenzhen Component Index, and the CSI 1000 Index. The results, which are available upon request, are consistent with the results based on the CSI 300 Index. In addition, the chart patterns of the historical prices of these four Chinese stock indices are highly similar to each other. We thus conjecture that there may be some market-specific features of Chinese markets affecting the performance of Chinese LETFs through the channel of index performance.

We also examine the performance of LETFs on the S&P 500 Index during these crisis periods and compare the results between CSI 300 LETFs and S&P 500 LETFs. The results are available upon request. For S&P 500 LETFs, the impact of index path during the 2020 COVID-19 crisis period on LETF return is no less than the impact during the 2008 financial crisis period, which can be explained by the substantial turbulence in U.S. market during the first half of 2020.

As cumulative index return and the variance of daily returns are the two most important factors to determine whether an LETF will outperformance its native expectation during a multiple-day holding period, we present the key summary statistics for these two variables on the most prevailing stock indices in the two countries. In particular, these indices include the CSI 300 Index, the Shanghai Composite Index, the Shenzhen Component Index, and the CSI 1000 Index from Chinese market, as well as the S&P 500 Index, the Dow Jones Industrial Average Index, the NASDAQ Composite Index, and the Russell 2000 Index from the U.S. market. The results are available upon request. The most important result that is critical to multiple-day LETF performance is that all four Chinese stock indices experience a much higher volatility than the four U.S. stock indices within 5, 22, and 250 trading days.