The Levered ETF Trick: Can the Compounding Deviation Phenomenon Be Arbitraged Through Short-Trading Leveraged Exchange Traded Funds?

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The Levered ETF Trick: Can the Compounding Deviation

Phenomenon Be Arbitraged Through Short-Trading Leveraged

Exchange Traded Funds?

Igor Quint University of Groningen

S3464040 Faculty of Economics and Business

I.Quint@student.rug.nl Master of Finance

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Abstract

Multiple studies demonstrate how the investment returns of Leveraged Exchange Traded Funds (LETFs) can be worse than expected for longer investment horizons, due to the costs associated with being exposed to leverage continuously. Therefore, shorting LETFs and inverse LETFs could provide the investor with additional return compared to regular index Exchange Traded Funds (ETFs). This paper examines short portfolios of LETF pairs and verifies empirically that such portfolios can outperform the underlying index. These results are robust for various levels of transaction costs. The costs associated with shorting stocks could, however, diminish outperformance. Furthermore, the relationship between LETF portfolio outperformance and underlying index return and volatility is established, giving investors better understanding of these dynamics.

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Key words: Exchange Traded Funds, Pairs trading, Quantitative finance, Algorithmic trading, Leverage JEL Classification: G11, G12

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1. Introduction

In the field of finance, new innovations emerge on a frequent basis. Financial engineers regularly invent new products to meet specific industry needs. For instance, a product that provides risk seeking investors with opportunities for increased returns. Leveraged ETFs (LETFs) are an example of such an invention and seek to amplify the daily return of an underlying index by a given multiplier. Related to LETFs are inverse exchange traded funds, which aim to deliver the inverse of the daily returns of the underlying index. Many of these funds carry leverage as well, aiming to return a negative multiple of the underlying index returns. In the subsequent sections of this paper, LETF refers to both inverse and regular LETFs. LETF pairs refer to combinations of inverse and regular LETFs with the same leverage multiple, based on the same underlying index. The LETF aims to return the product of the leverage multiple and the daily return of the underlying index. Hence, naïve investors holding a LETF position might expect to see the returns of the underlying index multiplied by a leverage multiple ranging from, on average, minus three to three. This expectation is generally realistic for short time periods such as a day or a week (Lu, Wang & Zhang, 2009). However, over time horizons longer than a month, investors often find that these funds are unable to deliver the so-called “naïve expected return”, a product of the leverage multiple and the cumulative underlying returns (Tang & Xu, 2013). The reason why the cumulative return of the LETF tends to deviate from the underlying index return over longer investment horizons is due to an effect dubbed as leverage decay, beta slippage or, in more formal literature, as the compounding deviation (Tang & Xu, 2013).

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Table 1. Fictional example of the compounding deviation effect

Demonstration of the compounding deviation effect by comparing ETF and LETF performance. Numbers are chosen such that cumulative returns of S&P500 investment equal zero.

t % S&P500

1-period linear return

$1 invested in S&P500 ETF % LETF linear return (3x multiple) $1 invested in S&P500 LETF 1 5.00 1.05 15.00 1.15 2 -4.80 1.00 -14.40 0.98 3 4.00 1.04 12.00 1.10 4 -3.80 1.00 -11.40 0.98 5 7.00 1.07 21.00 1.18 6 -6.54 1.00 -19.62 0.95 7 -6.00 0.94 -18.00 0.78 8 6.37 1.00 19.12 0.93 Cumulative returns 0.00 -7.20

The return characteristics of LETFs have been thoroughly detailed by various authors, including Avellaneda and Zhang (2010), Cheng and Madhavan (2009) and Tang and Xu (2013). One thing to note is that, although often depicted as negative, the compounding deviation effect does not necessarily imply the longer-term returns are less than the leverage multiple times the underlying index return over that period. In fact, in certain circumstances the total LETF return divided by the ETF return may well exceed the leverage multiple. As explained by Avellaneda and Zhang (2010), the compounding deviation effect is highly path dependent. The direction and magnitude of the long-term return deviations compared to the underlying index vary with volatility levels of the underlying index, holding periods of the LETF and the tracking error of the LETF fund manager (Tang & Xu, 2013).

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As noted, multiple studies have shown how the leverage decay effect, measured through compounding deviation, is detrimental to long-term LETF performance. The aim of this research is to empirically determine how investment strategies that are net short in the compounding deviation factor can outperform their underlying indices. This is relevant, as consistently outperforming a leading market index without incurring additional risk is generally considered to be very hard, if not impossible.

This research builds upon the research done by Jiang and Peterburgsky (2017) by providing an empirical evaluation of the proposed LETF pair shorting strategies and identifying the sources of potential excess returns across various LETF pairs with actual market data. This thesis continues with a literature review in Section two. Section three describes the methodology and data used for both the backtesting and the regression models, of which the results are presented in Section four. Finally, Section five summarizes the results and gives new pointers towards future research.

2. Literature review

To date, several studies have investigated the longer-term performance of (inverse) leveraged exchange traded funds (LETFs) compared to the performance of the underlying benchmark and/or the performance of unlevered ETF’s tracking the same index. In recent years, research has gradually expanded knowledge about the LETF performance characteristics.

Lu, Wang and Zhang (2012) were among the first to provide empirical evidence on the deviation of long-term (inverse) LETF returns versus their benchmark. Focusing only on double leveraged funds, for shorter investment horizons up until one month, they find that cumulative LETF returns remain reasonably close to the product of the 2x or -2x multiple and the underlying benchmark cumulative returns. For longer periods, the returns of the LETF start to deviate significantly from the underlying benchmark. The authors show both analytically and empirically how based on the continuous time framework, assuming that underlying returns follow a geometric Brownian motion, quadratic variation is negatively related to LETF returns. In this context, the authors define the quadratic variation 𝑄𝑉 as the variance of n-day returns times the number of days, where 𝜎 denotes the daily return volatility and 𝑇 the time horizon covering n days:

𝑄𝑉 = 𝜎2𝑇 (1)

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applying Itô’s lemma. The full derivation proving this result will be shown in the subsequent paragraphs.

The result found by Lu, Wang and Zhang (2012) confirms those of earlier studies by Avellaneda and Zhang (2010) and Cheng and Madhavan (2009). Both of these papers derive a formula that links the return of an LETF over longer time horizons to the return of the underlying index and its volatility in a continuous time setting. Assuming the underlying index 𝑆𝑡 follows a

geometric Brownian motion:

𝑑𝑆_𝑡= 𝜇𝑆_𝑡𝑑𝑡 + 𝜎𝑆_𝑡𝑑𝑊_𝑡 (2)

with 𝑆_𝑡 as index price level at time 𝑡, 𝜇 as the drift rate of the index, 𝜎 as the volatility of the index and 𝑊_𝑡 as a standard Wiener process with a zero mean and variance 𝑡. In a continuous time setting, the target return of the LETF is linked to the underlying index return with a factor of 𝑥. Cheng and Madhavan (2009) show:

𝑑𝐴_𝑡 𝐴_𝑡 = 𝑥

𝑑𝑆_𝑡 𝑆_𝑡

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with 𝑥 as the leverage multiple and 𝐴_𝑡 as the LETF level at time 𝑡. As the underlying index is assumed to follow a geometric Brownian motion, the LETF follows this as well:

𝑑𝐴𝑡= 𝑥𝜇𝐴𝑡𝑑𝑡 + 𝑥𝜎𝐴𝑡𝑑𝑊𝑡 (4)

Applying Itô’s lemma with 𝑍_𝑡= ln⁡(𝐴_𝑡):

𝑑𝑍_𝑡= 𝜕𝑍 𝜕𝑡𝑑𝑡 + 𝜕𝑍 𝜕𝐴𝑑𝐴𝑡+ 1 2 𝜕2_𝑍 𝜕𝐴2(𝑑𝐴𝑡)2 (5)

This formula reduces into the following:

𝑑𝑍𝑡 = 1 𝐴_𝑡𝑑𝐴𝑡− 1 2 1 𝐴_𝑡2(𝑑𝐴𝑡)2 (6)

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7 𝑑𝑍_𝑡= (𝑥𝜇 −1 2𝑥 2_𝜎2_{) 𝑑𝑡 + 𝑥𝜎𝑑𝑊} 𝑡 (7)

Then, substituting 𝑍_𝑡= ln⁡(𝐴_𝑡) after integration of (7) on [𝐴₀, 𝐴_𝑡] results in:

ln 𝐴𝑡− ln 𝐴0 = (𝑥𝜇 − 1 2𝑥 2_𝜎2_{) 𝑡 + 𝑥𝜎𝑊} 𝑡 (8)

where 𝑊𝑡 is a normally distributed Wiener process with mean 0 and standard deviation

√𝑇. Through further reduction and by applying eq. 5 through 8 on eq. 2, it can be shown that:

ln (𝐴𝑡𝑁 𝐴₀) = 𝑥𝑙𝑛 ( 𝑆_𝑇 𝑆₀) + (𝑥 − 𝑥2_)𝜎2_𝑇 2 (9)

Tang and Xu (2013) built upon this work and report extensively on the possible determinants of the deviation between long-term returns of LETFs compared to their underlying indices. The authors develop a framework that is able to assess all sources of return deviations. They find that the deviations are mainly driven by the effect of compounding through daily rebalancing, tracking errors and market inefficiencies. The authors split the deviations between the compounding and non-compounding deviations, with the latter consisting of residual and Net Asset Value (NAV) deviations.

The authors derive a formula to approximate the compounding deviation between returns on the LETF and the underlying index. The following lines explain their derivation in more detail.

The second order Taylor polynomial of the natural log of 𝑥 is:

ln(𝑥) ≃ (𝑥 − 1) −(𝑥 − 1)

2

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The authors use this approximation and proceed by defining 𝑅𝑀 as the cumulative index

return ( 𝑆𝑇

𝑆0 − 1) and 𝑅𝐿 as the LETF cumulative return (

𝐴𝑇

𝐴0− 1), such that from eq. 9 the following equation can be derived:

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As the LETF return is very close to the leverage multiple times the underlying index return for small values of 𝑅_𝑀, using first order linear approximation with R_𝐿 = 𝑥𝑅_𝑀, this can be further reduced to:

R_𝐿 − 𝑥𝑅_𝑀 =(𝑥 2_{− 𝑥)} 2 𝑅𝑀 2₋ (𝑥2− 𝑥)𝜎2𝑇 2 (12)

The first order derivative with respect to 𝑅_𝑀 is: 𝑑 𝑑𝑅𝑀 ((𝑥 2_{− 𝑥)} 2 𝑅𝑀 2 ₋(𝑥 2_{− 𝑥)𝑇𝜎}2 2 ) = 𝑅𝑀(𝑥 2 _{− 𝑥)} (13)

and the first order derivative of this function with respect to 𝜎 is: 𝑑 𝑑𝜎( (𝑥2_{− 𝑥)} 2 𝑅𝑀 2 ₋(𝑥2− 𝑥)𝑇𝜎2 2 ) = −𝑇𝜎(𝑥 2_{− 𝑥)} (14)

Compounding deviation, approximated by eq. 12, influences long term LETF results in either a positive or a negative way, depending on the magnitude of the squared underlying index return. The absolute index return needs to be sufficient enough to compensate for the negative volatility and the time dependent term. As can be seen from eq. 13 and eq. 14, compounding deviation increases with the return of the underlying index and decreases with the volatility of the underlying index. Assuming both are constant, a longer holding period will lower the compounding deviation. For example, if an index returns seven percent over a 10-day holding period with a 25% volatility, the compounding deviation for a triple leveraged ETF will be approximately 0.96%. When the same index return and volatility were to be achieved over a 30-day period, the compounding deviation would be slightly negative at approximately -0.07%.

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the authors find that shorting the triple-leveraged S&P500 Bear ETF (i.e. the three times inverse LETF), and the three times levered S&P500 Bull ETF in a 2:1 ratio delivers a superior Sharpe ratio compared to a buy-and-hold S&P500 ETF position over the same period.

On first impression, Jiang’s and Peterburgsky’s (2017) finding appears to provide investors with an opportunity to outperform the market on a risk-adjusted basis. This would be surprising, as this is not consistent with the weak form of the Efficient Market Hypothesis by Fama (1970). Investors would be able to use patterns and historical data of LETFs to their advantage to achieve superior returns. However, it is arguably too early to draw this conclusion. The authors note that some LETFs might be hard to borrow, implicating difficulties when implementing such a strategy in practice due to potential liquidity issues. Another difficulty that may arise are the high borrowing costs that must be paid to short the shares. Dobi and Avellaneda (2012) show that the high shorting fees could offset the potential gains from poor LETF performance relative to the underlying index.

Besides these practical implications, it is important to verify the simulation results of Jiang and Peterburgsky (2017) empirically. This leads to the first hypothesis: Using empirical market data, the shorted LETF pairs designed to replicate the underlying index outperform the underlying index on a risk-adjusted basis (H1).

Continuing with the approximation for the compounding deviation (Eq. 12), substituting 𝑥 with 3 and 3 for respectively the triple leveraged bull and bear fund, the expected returns of a -33/-67 portfolio, assuming continuous rebalancing, should roughly follow:

R𝐿𝐸𝑇𝐹⁡𝑝𝑜𝑟𝑡−33,−67 = −33%(3𝑅𝑀2− 3𝜎2𝑇 + 3𝑅𝑀) − 67%(6𝑅𝑀2− 6𝜎2𝑇 − 3𝑅𝑀) (15)

Resulting in the expected outperformance through compounding deviation:

R_{𝐿𝐸𝑇𝐹⁡𝑝𝑜𝑟𝑡−33,−67}− 𝑅_𝑀 = −5𝑅_𝑀2+ 5𝜎2_𝑇 ₍₁₆₎

A similar exercise for double leveraged funds with a -25%/-75% portfolio yields:

R_{𝐿𝐸𝑇𝐹⁡𝑝𝑜𝑟𝑡−25,−75}− 𝑅_𝑀 = −5 2𝑅𝑀

2₊ 5

2𝜎

2_𝑇 (17)

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outperformance on the LETF portfolio and the squared cumulative return and volatility of the underlying index. The regression results by Tang and Xu (2013) confirm this relationship on the level of the LETF returns themselves, but this relationship has not yet been empirically confirmed for short LETF portfolios. Hence resulting in the following hypothesis: the squared cumulative index return will impact the LETF short portfolio performance negatively whereas the underlying index variance will positively impact the returns of the LETF short portfolio (H2).

3. Research methods and data collection

To empirically establish the viability of combined short positions in LETF pairs to outperform the underlying index, the strategies are formulated and backtested thoroughly in accordance with the strategies proposed by Jiang and Peterburgsky (2017). This section describes the methods used and the assumptions made, to assess the main research question.

3.1. Data

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Table 2: Relevant information regarding LETF strategy configurations

Underlying index Ticker symbols LETF pairs Bull / Bear Bench mark ETF Start date test sample End date test sample Multi ple

Avg. spread Issuer Bull / Bear

S&P500 SXPL/SPXS SPY 06-11-2008 31-12-2018 3x/-3x 0.02% / 0.05% Direxion S&P500 SSO/SDS SPY 03-01-2007 31-12-2018 2x/-2x 0.02% / 0.03% Proshares NASDAQ-100 TQQQ/SQQQ QQQ 01-03-2010 31-12-2018 3x/-3x 0.02% / 0.11% Proshares NASDAQ-100 QLD/QID QQQ 03-01-2007 31-12-2018 2x/-2x 0.02% / 0.03% Proshares NYSE Arca Gold Miners NUGT/DUST GDX 03-01-2011 31-12-2018 3x/-3x 0.06% / 0.05% Direxion NYSE Arca Gold Miners GDXX/GDXS GDX 02-03-2015 31-12-2018 2x/-2x 0.27% / 0.42% Proshares MSCI Emerging markets

EDC/EDZ EEM 02-01-2009 31-12-2018 3x/-3x 0.09% / 0.06% Direxion MSCI Emerging

markets

EET/EEV EEM 01-07-2009 31-12-2018 2x/-2x 0.63% / 0.07% Proshares Sources: finance.yahoo.com (available sample size), Proshares.com (Proshares LETF multiples & tickers),

direxioninvestments.com (Direxion LETF multiples & tickers), etf.com (average spreads), accessed May 25th_{, 2019}

3.2. Method

To determine whether combinations of the LETFs specified in Table 2 can be traded to outperform the benchmark ETF, the portfolios are first defined and backtested over the sample period. Next, the portfolio daily return data is aggregated into quarterly data. As the compounding deviation effect is visible over holding periods longer than a month, quarterly data provide a good balance between sample size and holding period length (Avellaneda & Zhang, 2010). Using this quarterly data, a two-tailed independent sample t-test is used to determine if the returns are significantly higher than a simple buy-and-hold investment in the benchmark ETF. The proposed shorting strategy itself, as described by Jiang and Peterburgsky (2017) is best explained by summarizing the steps that are involved:

1. A triple LETF portfolio is initialized by accumulating a position in a given bull LETF and its inverse counterpart (bear LETF).

2. If the total short balance relative to the portfolio value exceeds the set rebalance thresholds, inflows or outflows take place. Inflows occur when the portfolio value relative to the short balance is under the lower threshold. Cash is then used to cover short positions (decreasing exposure). This happens with a maximum of once a day.

3. If the position size exceeds the set rebalance thresholds, the portfolio will be rebalanced to the initial weights. This happens with a maximum of once a day.

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discussed will be backtested using Quantopian (see quantopian.com). This online platform is designed to enable users to backtest investment strategies written in the Python programming language. The platform contains per-minute pricing data of many securities in the United States (US), including the relevant LETFs for this research. All data are survivorship bias free.

Quantopian feeds the algorithm asset pricing data and keeps track of all opened positions. By default, a five basis point spread surrounds the midpoint price for every asset. Half of this spread is incurred as transaction costs for every trade. As there is no parameter optimization or fitting involved, all the tests performed are out-of-sample. After backtesting, Quantopian provides the user with detailed performance and risk metrics including Sharpe ratio and detailed position & transaction analysis. The backtesting module in Quantopian is based on the open-source Zipline library for Python, which has over 100 contributors1. Appendix B contains an overview of the Quantopian editor, enabling the reproduction of the backtest results.

Over the available data period, short positions are initiated in both the inverse and the regular LETF. These positions are rebalanced at different frequencies. The backtest is run multiple times using various allocations between the inverse and regular LETF. In the research by Jiang and Peterburgsky (2017), various combinations of weights were used. The authors find that the strategy designed to replicate a theoretical unlevered exposure to the underlying index performs best.

For this study, a portfolio was constructed in such a way that, in theory, it equals the underlying index, given that the main research question revolves around beating the underlying index. For triple leveraged ETFs, this can be done by shorting ⅔ of the inverse LETF and ⅓ of the long LETF. This replicates a single dollar invested in a non-leveraged ETF. For double leveraged ETFs, ¼ and ¾ short exposure are needed for the bull and bear LETF respectively. Within this research, the used weightings are limited to ¼th_{, ¾}th_{for the double leveraged funds and ⅓}th_{, ⅔}th

for the triple leveraged funds.

Jiang and Peterburgsky (2017) discuss two types of rebalancing: inflows and outflows into the short subsection of the portfolio and actual rebalancing between the two short positions. The first type entails rebalancing whenever the ratio between the portfolio value and the absolute value of the short portfolio breaches an upper or lower boundary. These boundaries can be defined as:

𝐵_𝑢 = 1 + 𝐵,⁡⁡⁡𝐵_𝑙 = 1 − 𝐵 (18)

With 𝐵𝑢, 𝐵𝑙 as the upper and lower boundary respectively. 𝐵 is the band width, indicating

the threshold level. Whenever the ratio drops below the lower boundary, this requires rebalancing as the margin requirements may soon be reached. Whenever the ratio rises above the upper

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boundary, funds may be taken out of the short portfolio to realize profits. The authors use a bandwidth of -10%/+10%, based on the margin requirements for shorting triple leveraged LETFs. In this study, the same bandwidths as in Jiang and Peterburgsky (2017) are used. The rebalancing between positions occurs whenever the actual weights deviate from the strategic weights of a given position in the short portfolio by more than the allowed bandwidth. This is specified as follows:

𝑤_𝑖,𝑡 < 𝑤_𝑖,𝑡− 𝐵_𝑙⁡⁡𝑜𝑟⁡⁡𝑤_𝑖,𝑡 > 𝑤_𝑖,𝑡+ 𝐵_𝑢 (19) with 𝑤𝑖,𝑡 as the position weight of security 𝑖 at time 𝑡. The backtest is run with various

combinations of thresholds (see Table 3). This table shows the lower and upper thresholds used within the algorithm to initiate a rebalancing to the initial portfolio weights.

Table 3: Rebalancing thresholds for LETF strategies (%)

Lower threshold Upper threshold

-5 +5

-10 +10

-20 +20

A commonly used measure to compare portfolio performance is the risk-adjusted return. The Sharpe ratio is a metric designed to easily capture risk-adjusted returns. In order to compare the short LETF strategy to the underlying index buy and hold strategy, the Sharpe ratio of both strategies will be determined as follows:

𝑆𝑅_𝑠,𝑡 =𝐶𝑅𝑠,𝑡 𝜎_𝑠,𝑡

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with 𝑆𝑅𝑠,𝑡 as the Sharpe ratio of strategy 𝑠 at time 𝑡, 𝐶𝑅𝑠,𝑡 as the cumulative return of

strategy 𝑠 at time 𝑡 and 𝜎𝑠,𝑡 as the volatility of daily returns of strategy 𝑠 over time period 𝑡. Note

that the Sharpe ratio is calculated using the cumulative returns from the LETF portfolios. Eq. 16 and 17 only express the compounding deviation of total returns. Comparing investments on a risk-adjusted basis requires the total returns of both investments. Therefore, the observed cumulative return is used.

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3.3. Market frictions

When implementing the strategies described above, there are several drawbacks that investors may encounter. Although the aim of this research is not to produce a ready-to-go implementable strategy, this section will list potential drawbacks, the potential impact on observed results and how these frictions were dealt with.

The costs for engaging in a trade consist of two parts. The bid-ask spread, which is earned by the market maker, and the commissions to the broker. Brokers often charge a fixed minimum fee for buying ETFs, whereas a variable component kicks in when ordering in large volumes. The impact of the bid-ask spread on the backtest results can be modelled directly into the backtest. As noted earlier, the backtesting platform Quantopian uses a default slippage model that incorporates a five basis point spread across asset prices. Every trade the algorithm makes will be executed at a price either half the spread above, or half the spread below, the midpoint price for respectively buying and selling. Hence, by default, transaction costs are accounted for within the backtesting results. Observing the average spreads over the past 60 days from Table 2, this five basis point spread appears to be a prudent approximation. Some of the used ETFs are less liquid and have a higher average spread. As a robustness check, the effect of higher spreads are measured by running the backtest again with a custom spread. The backtest is run for the triple leveraged S&P500 LETF portfolio with 10% rebalancing thresholds (base case). The results are presented as a sensitivity analysis with spreads up to 40 basis points, using increments of five basis points (see Section 4.3 and Appendix C).

Commissions are highly dependent on trading size and may vary across brokerages. Certain parties enjoy direct market access which allows them to circumvent broker commissions entirely. As a simplifying assumption, commissions to the broker will not be considered in the backtest. To approximate the potential impact, the average number of yearly rebalances must be multiplied by two and by the required broker commissions. As a proxy, the commissions of an average priced Dutch broker ($10 per stock transaction, Lynx early 2019) can be used. This total number can be divided by the portfolio value, to indicate the yearly percentage lost to broker commissions. This exercise indicates that commission costs diminish the larger the portfolio grows.

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various factors, it can be safely assumed to be more costly to short LETFs than it is to short their ETF counterparts. In Table 4, for informational purposes, the lending fees for LETFs tracking the S&P500 are compared with the lending fee for a regular S&P500 ETF (SPY) on April the 26th_,

2019. In the table, the rebate rates and expense ratios are presented as well. The rebate rate is the lending fee minus the current fed funds rate. This is the effective fee, as the funds obtained by selling can be reinvested at the fed funds rate. Jiang and Peterburgsky (2017) designed the portfolio such that the short sale proceeds were invested in T-bills, achieving a similar effect. Note how the rebate rates for the double leveraged and non-leveraged S&P500 ETFs are currently positive. Shorting these funds yields the investor a positive return, but rates are subject to change.

Table 4: S&P500 LETF shorting fees

This table shows the observed securities lending fees, rebate rates and expense ratios for the S&P500 ETF and LETFs used within this research. The rebate rate is the securities lending fee minus the

interest rate received on proceedings from the short sale.

Sources: Interactive Brokers (lending fee & rebate rate), etfdb.com (expense ratio); accessed 26th_{of April} 2019.

As a robustness check, the impact of the rebate rates on the shorting strategies is approximated. Unlike Jiang and Peterburgsky (2017), this will not be done with a position in T-bills, but by discounting the rebate rates. First, the rates in Table 4 are converted to daily compounding rates, using the target weight for both LETFs to compile an aggregate daily rate. Next, these rates are discounted from the daily returns for the double and triple leveraged S&P500 portfolios. In Section 4.2, the annualized returns are presented with and without the discounted rebate fee. This method involves several assumptions. First, the exposure to both LETFs is always assumed to be equal to the target weight. Furthermore, the rebate rate is fixed, which is an important limiting assumption. Investors could, however, fix the shorting fee for longer periods of time using derivatives. This could decrease the shifts in the effective rebate rates.

The expense ratio is mostly concerned with the funds’ management fee. Most managed funds charge their investors a given management fee. This is also the case for ETFs and LETFs. As the management fund is extracted from the funds NAV, ceteris paribus, the fund value decreases over time. The investor pays this fee indirectly. When shorting such a fund, however,

Target Symbol Sec. lending fee Rebate rate Expense ratio

3x S&P500 SPXL 3.77% -1.35% 1.02%

-3x S&P500 SPXS 4.79% -2.37% 1.08%

2x S&P500 SSO 0.59% 1.83% 0.90%

-2x S&P500 SDS 1.40% 1.02% 0.90%

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the investor will profit from the decrease of the stock price due to the management fee being incurred. This means that the securities lending fee is likely to be at least as high as the management fee. Hence, if lending fees would be consistently lower than the management fee, arbitrageurs may construct positions that are risk free or almost risk free, earning the management fee in the process. As investors short in ETFs earn the management fee but pay the securities lending fee, the true cost of shorting may be approximated as the management fee subtracted from the rebate rate.

In certain circumstances, the investor may be forced to close its short position earlier than desired. This phenomenon, known as short squeeze, may occur when many investors decide to sell their shares at the same time. Their shares may have been lent out to short sellers. The broker may require the short seller to return the borrowed shares prematurely to enable the owners to sell their shares. The modelling of liquidity is out of the scope of this research. The risks of short squeezes could potentially be avoided by using derivative contracts, as holders of these contracts cannot be forced to close their position when enough margin is held.

In order to solve some of the issues raised above, an option strategy known as synthetic short stock or reversal could potentially be employed. This strategy involves buying an at-the-money (ATM) put option and selling an ATM call option. The payoff of a synthetic short stock strategy replicates the payoff from an actual short position, but has a few key advantages. Once established, the holder cannot be forced to close his position due to the possible event of a short squeeze. Also, by engaging in this strategy, no dividends or borrowing cost have to be paid - these mostly cancel out between the put and the call option. There are a few downsides, however, involving exposure to volatility (vega), rollover costs (carry) and wider bid-ask spreads. The rollover costs, which involve the investor paying for the implied volatility over the option’s lifetime, could decrease the advantages of this strategy compared to shorting stocks directly. This thesis will not address the synthetic short stock strategy. For future research, however, it could potentially be an interesting alternative, as this strategy could mitigate the risks and costs of shorting the stock directly.

3.4. Regression

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It is important to decompose the return deviations in the components described above, as the hypothesized relationship between the outperformance of the LETF portfolio and the squared cumulative index returns and the index return variance is only derived analytically for the compounding return component (eq. 13 & 14). Therefore, total return deviation may not be properly explained by these independent variables.

The deviations between strategy and index returns are decomposed in a similar way as done by Tang and Xu (2013). First, the target return is defined as the daily index return times the leverage multiple compounded over a one-month period. For this research, these returns are weighted with the short LETF portfolio weights, resulting in a sign change. The naïve return is defined as the one-month index return times the leverage multiple and the actual return as the LETF strategy return over the one-month periods.

Secondly, the compounding return deviation is defined as the difference between the target return and the naïve expected return. The non-compounding return deviations are measured as the difference between the actual observed strategy return and the target return. Both components make up the total deviation between the index return and the strategy return. For the compounding return deviation, the compounding deviation of the short LETF portfolio is regressed onto the squared cumulative monthly returns and variance of the underlying index. In the previous section, Hypothesis 2 was formed on the expected relationship between the compounding deviation and the squared underlying index return and variance. This was based on the formula for approximation of compounding deviation. Equation 13 and equation 14 then proved through derivation that the compounding deviation is related to the squared index return and index variance. To measure the impact of squared cumulative index returns and the index variance, the following regression equation is formulated (model 1):

𝐶𝐷_𝑡 = 𝛼 + 𝛽₁𝐶𝑅𝐼_𝑡2+ 𝛽₂𝑉𝐴𝑅_𝑡+ 𝜀_𝑡 (21)

where 𝐶𝐷_𝑡 is the compounding return deviation between the short LETF portfolio and the underlying index, 𝐶𝑅𝐼𝑡 is the cumulative return of the underlying index and 𝑉𝐴𝑅𝑡 is the variance

of the index. All variables are measured for month 𝑡. Using ordinary least squares regression (OLS), the results of the regressions for each of the LETF strategies with a -10/+10 rebalancing regime will be presented in Section 4.

To indicate the effect of the independent variables from equation 21 on the total return deviation, the following additional regression model is performed:

𝑇𝐷_𝑡 = 𝛼 + 𝛽₁𝐶𝑅𝐼_𝑡2_{+ 𝛽}

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where 𝑇𝐷_𝑡 is the total return deviation between the short LETF portfolio and the underlying index for month 𝑡. This includes both the compounding and the non-compounding deviations. Therefore, it is likely that the independent variables do not fully explain the dependent variable. Note how, unlike the comparative results of the backtests, the actual underlying index returns will be used as opposed to the benchmark ETF returns. This originates from the expected LETF return equations detailed in Section 2. Additionally, the argument can be made that the benchmark ETF does not drive the LETF returns, but that the underlying index does. If the benchmark ETF has a sizeable tracking error or management fee, this might skew the regression results. Therefore, using the underlying index itself makes for a better comparison.

4. Results

This section will first present the results from the backtests of the LETF strategies described in Section 3. Following, the results of the regressions involving the short LETF portfolios and index returns are presented.

4.1.1. Backtest of S&P500 LETF strategies

The main takeaway from the research by Jiang and Peterburgsky (2017) is that a portfolio consisting of short positions in triple leveraged S&P tracking ETFs in a 33:67 ratio can outperform the underlying index on a risk-adjusted basis. Table 5 shows the average yearly returns of this portfolio based on actual observed market data over the backtested period. The first row contains the linear average annualized return, the second contains the geometrical average annual return. Furthermore, the average annualized standard deviation, Sharpe ratio and yearly rebalances are shown. The backtest results of the triple leveraged S&P500 ETF strategy are consistent with the observations made in the study by Jiang and Peterburgsky (2017). On a risk-adjusted basis, for all rebalancing thresholds, the Sharpe ratio is significantly higher over the backtested period.

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Table 5: Summary backtest results of triple LETF S&P500 strategies

Within this table, the results from running backtests with triple leveraged S&P500 LETF portfolios are shown. The rebalance thresholds indicate the three different configurations run. The SPY

column contains the average returns from a buy and hold investment in the benchmark ETF.

Triple LETF S&P500 strategy SPXL/SPXS 33:67 SPY

Rebalance threshold 20/20 10/10 5/5

Average annual return 14.7%

[3.44***]

14.7% [3.6***]

14.6%

[3.56***] 10.5% Geometric average annual return 13.9% 13.9% 13.8% 10.0% Average annual standard deviation 19.4% 19.5% 19.5% 17.6%

Average annual Sharpe ratio 1.16

[3.13***]

1.16 [2.99***]

1.17

[3.28***] 1.00 Average yearly number of position rebalances 11.7 11.7 18.5 0.0 Note: 2-tailed t-statistics between brackets, * p<0.1 ** p<0.05 *** p<0.01, n=41.

Table 6: Summary backtest results of double LETF S&P500 strategies

Within this table, the results from running backtests with double leveraged S&P500 LETF portfolios are shown. The rebalance thresholds indicate the three different configurations run. The SPY

Double LETF S&P500 strategy SSO/SDS 25:75 SPY

[1.82*]

9.3% [1.82*]

9.4%

[1.77*] 7.6%

Geometric average annual return 7.8% 7.8% 8.0% 6.1%

Average annual standard deviation 17.5% 17.5% 17.6% 16.1%

[2.01**]

0.92 [2.01**]

0.92

[1.65*] 0.87 Average yearly number of position rebalances 13.3 13.3 12.8 0.0 Note: 2-tailed t-statistics between brackets, * p<0.1 ** p<0.05 *** p<0.01, n=41.

4.1.2. Backtest of Nasdaq-100 LETF strategies

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Table 7: Backtest results of triple LETF Nasdaq-100 strategies

Within this table, the results from running backtests with triple leveraged Nasdaq-100 LETF portfolios are shown. The rebalance thresholds indicate the three different configurations run. The QQQ

Triple LETF Nasdaq-100 strategy TQQQ/SQQQ 33:67 QQQ

[0.23]

16.1% [0.12]

16.6%

[0.51] 15.6% Geometric average annual return 15.8% 15.4% 15.9% 15.1% Average annual standard deviation 16.8% 17.0% 16.9% 16.2% Average annual Sharpe ratio 1.25

[2.52**]

1.23 [2.55**]

1.28

Table 8: Backtest results of double LETF Nasdaq-100 strategies

Within this table, the results from running backtests with double leveraged Nasdaq-100 LETF portfolios are shown. The rebalance thresholds indicate the three different configurations run. The QQQ

Double LETF Nasdaq-100 strategy QLD/QID 25:75 QQQ

[2.02**]

16.9% [2.02**]

16.5%

[1.72*] 14.3% Geometric average annual return 14.5% 14.5% 14.1% 11.7% Average annual standard deviation 19.8% 19.8% 19.8% 19.2%

[3.13***]

1.19 [3.13***]

1.17

4.1.3. Backtest of NYSE Arca Gold Miners LETF strategies

This section presents the results from the backtest of strategies designed to replicate an exposure to the NYSE Arca Gold Miners index, using short positions in opposing LETF pairs. First, table 9 shows the results for triple leveraged ETFs, ticker symbols NUGT and DUST for the bull and bear funds respectively. In the backtested period, the LETF strategies appear to perform significantly better than a buy-and-hold position in the standard, non-leveraged benchmark ETF. The simple average yearly return of the ETF was negative whereas the LETF strategies performed positively over the sampled investment horizon. Due to the very marginal increase in volatility and a high increase in return, the Sharpe ratio for these strategies is

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Table 10 shows the results for the double leveraged ETFs, ticker symbols GDXX and GDXS. In the backtested period, the results are consistent with the results observed in the triple LETF backtest. Note that the average benchmark ETF (GDX) return, as shown in Table 10, is positive as opposed to the benchmark ETF return shown in Table 9. This is most likely due to the smaller sample over which the backtest is run.

Table 9: Backtest results of triple LETF NYSE Arca Gold Miners Index strategies

Within this table, the results from backtests with triple leveraged NYSE Arca Gold Miners Index LETF portfolios are shown. The rebalance thresholds indicate the three different configurations run. The

GDX column contains the average returns from a buy and hold investment in the benchmark ETF. Triple LETF NYSE Arca Gold Miners index NUGT/DUST 33:67 GDX

[3.61***]

6.5% [3.7***]

5.3%

[4.46***] -7.8% Geometric average annual return 2.4% 2.7% 1.0% -12.0% Average annual standard deviation 37.3% 37.3% 36.5% 36.2%

[3.75***]

0.13 [3.8***]

0.09

[4.31***] -0.21 Average yearly number of position rebalances 42.1 41.9 64.9 0.0 Note: 2-tailed t-statistics between brackets, * p<0.1 ** p<0.05 *** p<0.01, n=41.

Table 10: Backtest results of double LETF NYSE Arca Gold Miners Index strategy

Within this table, the results from the backtests of double leveraged NYSE Arca Gold Miners Index LETF portfolios are shown. The rebalance thresholds indicate the different configurations run. The

GDX column contains the average returns from a buy and hold investment in the benchmark ETF. Double LETF NYSE Arca Gold Miners index GDXX/GDXS 25:75 GDX

[2.7**]

12.9% [2.7**]

12%

[2.74**] 7.4%

Average annual standard deviation 37.4% 37.4% 37.1% 35.2%

[2.39**]

0.27 [2.39**]

0.26

[2.76**] 0.15 Average yearly number of position rebalances 59.0 59.0 59.5 0.0 Note: 2-tailed t-statistics between brackets, * p<0.1 ** p<0.05 *** p<0.01, n=41.

4.1.4. Backtest of MSCI Emerging Markets index LETF strategies

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the triple LETF strategies, although the latter seem to perform better with a moderate increase in risk. Across rebalance thresholds, the strategy with a -10/+10 bandwidth for asset allocation performs slightly better in the backtest as the -5/+5 strategy. The results of the -20/+20 and -10/+10 thresholds are exactly equal, as most rebalances are triggered by a margin call or inflow of additional funds in the short portfolio. Therefore, the exact same trades were made.

Table 11: Backtest results of triple LETF MSCI Emerging Markets Index strategies

Within this table, the results from backtests with triple leveraged MSCI Emerging Market Index LETF portfolios are shown. The rebalance thresholds indicate the three different configurations run. The

EEM column contains the average returns from a buy and hold investment in the benchmark ETF. Triple LETF MSCI Emerging Markets index EDC/EDZ 33:67 EEM

[2.99***] 16.2% [2.99***] 15.8% [3.13***] 8.8% Geometric average annual return 12.3% 12.3% 12.0% 6.3% Average annual standard deviation 23.5% 23.5% 23.1% 21.3% Average annual Sharpe ratio 0.7

[5.11***] 0.7 [5.11***] 0.69 [5.00***] 0.41 Average yearly number of position rebalances 20.8 20.8 32.6 0.0 Note: 2-tailed t-statistics between brackets, * p<0.1 ** p<0.05 *** p<0.01, n=41.

Table 12: Backtest results of double LETF MSCI Emerging Markets Index strategy

Within this table, the results from backtests with double leveraged MSCI Emerging Market Index LETF portfolios are shown. The rebalance thresholds indicate the three different configurations run. The

EEM column contains the average returns from a buy and hold investment in the benchmark ETF. Double LETF MSCI Emerging markets index EET/EEV 25:75 EEM

[2.29**]

7.6% [2.29**]

7.5%

[2.16**] 5.1%

Average annual standard deviation 21.7% 21.7% 21.8% 20.1% Average annual Sharpe ratio 0.48

[3.6***]

0.48 [3.6***]

0.46

4.2. Robustness checks

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be found in Appendix C and Appendix D for respectively the transactional costs and shorting costs analysis.

To account for transaction costs, a sensitivity analysis is performed. The transaction costs, assumed to be constant, are expressed in the bid/ask spread. As discussed in Section 3, additional (fixed) commissions can be charged by brokers but are not applicable to all market participants. Therefore, this analysis only involves the bid/ask spread. As described, within the current backtesting results, a five basis point spread is already modeled into the asset pricing. As the average 60-day spread (Table 2) for most LETFs is lower, this is a prudent approximation. From Table 2 it can be seen, however, that the spread for less liquid LETFs is considerably higher. Also, in times of market distress, liquidity could potentially dry up, widening the bid/ask spreads. Therefore, the bid/ask spread sensitivity analysis on the base case portfolio inhibits spreads up to 40 basis points.

Table C.1 shows how various levels of bid/ask spreads impact the average yearly returns of the triple leveraged base case portfolio. Although average performance decreases with increased spreads, the returns remain significantly higher than the average S&P500 returns in the same period. Due to small changes in the number of shares that can be afforded or must be sold, the Sharpe ratio may increase even though the spread increases. The results above indicate that even with high spreads, the performance remains significantly higher than the benchmark ETF performance over the same period.

For the shorting fee, the impact on the LETF portfolio returns is approximated using the rebate fees observed on April 26th, 2019; see Table 4 from Section 3. As described earlier, the annual rebate fees are converted to a daily compounding rate and subsequently discounted from the daily returns of the LETF portfolios. Table D.1 shows how the current rebate fees impact the average yearly returns of the triple leveraged S&P500 LETF portfolio. The results following the two-tailed t-test indicate that, even though there is a higher average annualized return, the outperformance relative to the SPY ETF is no longer statistically significant when discounting the fixed rebate rate from the daily portfolio returns.

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4.3. Regression estimations

In this section, the results from the regressions run on the (compounding) return deviations between the LETF strategies are presented. As described in Section 3.2, the regression equation uses the underlying index returns as opposed to the benchmark ETF returns.

In Table 13, the estimated OLS coefficients are shown for the portfolios based on double and triple leveraged S&P500 ETFs. As described earlier, the regressions are performed with both the total return deviation and the compounding return deviation as dependent variables. The results between round brackets indicate the standard errors. The coefficient values with a *, ** or *** indicate a 10%, 5% or 1% statistical significance respectively.

For both regression models, the estimated coefficients for the squared cumulative return and the variance of the underlying index are highly significant. Also, the estimated coefficient for the squared cumulative return is negative, whereas this coefficient is positive for the underlying index variance. This confirms the theoretical relationship between portfolio outperformance and these factors as described in Section 2. When correcting the variance coefficient for the number of trading days within the measured one-month period, note how the coefficients are reasonably close to the expected values for these portfolios (eq. 16 & 17). In line with these equations, the estimated coefficients can be interpreted as the change in compounding deviation or total return deviation when the independent variables increase or decrease with one hundred basis points. The regression estimates from Model 2 only explain a small portion of the total return deviation (adjusted R2=0.153). For the SPXL-SPXS portfolio, the adjusted R2 is only 0.05 for Model 2, as opposed to 0.99 for Model 1. This was to be expected, as total return deviation is also explained by NAV deviation, tracking errors and interest rates (Tang & Xu, 2013). Those factors are left out of Model 2 as described in Section 3. In the portfolio case, rebalancing thresholds may also impact the extent to which total returns deviate from the naïve expected returns.

Table 14 shows the estimated OLS coefficients for the portfolios based on double and triple leveraged NASDAQ-100 ETFs. Similar to the S&P500 LETF portfolio case, the estimated coefficients for the squared cumulative return and the variance of the underlying index are statistically significant. Analogous with Table 13, the high adjusted R2 for Model 1 indicates that the mathematical relationship derived by various authors (Cheng & Madhavan, 2009, Avellaneda & Zhang, 2010 and Tang & Xu, 2013) explains the compounding return deviation, not only for the individual LETFs, but also for the short LETF portfolio case (eq. 16 & eq. 17).

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Table 13: Regression estimates S&P500 LETF portfolio

This table shows the estimation results from the regression equations described in Section 3. Model (1) refers to the compounding deviation from equation 21, whereas model (2) refers to the total deviations from equation 22. Coefficient estimates for the different models are denoted behind the variables with the standard error within parentheses. The coefficient values with a *, ** or *** indicate a 10%, 5% or 1% statistical

significance respectively. The adjusted R2_{indicates the model fit.}

Regression results S&P500 SDS-SSO 25:75 portfolio, n=144 SPXL-SPXS 33:67 portfolio, n=122

Model (1)/CD Model (2)/TD Model (1)/CD Model (2)/TD

Intercept; α -0.001*** 0.001 0.000 0.002

(0.000) (0.001) (0.000) (0.001)

Squared cumulative index return; 𝐶𝑅𝐼_𝑡2 _-1.941*** _-2.252*** _-4.661*** _-1.153**

(0.082) (0.446) (0.074) (0.540)

Index variance; 𝑉𝐴𝑅_𝑡 52.969*** 21.288*** 96.765*** 14.87***

(0.846) (4.606) (0.75) (5.433)

Adjusted R2 _0.970 _0.153 _0.993 _0.05

Table 14: Regression estimates NASDAQ-100 LETF portfolio

Regression results NASDAQ-100 QLD-QID 25:75 portfolio, n=144 TQQQ-SQQQ 33:67 portfolio, n=106

Intercept; α -0.001*** 0.001 -0.001*** 0.002

(0.000) (0.001) (0.000) (0.002)

Squared cumulative index return; 𝐶𝑅𝐼_𝑡2 -2.168*** -1.437*** -4.398*** -2.258***

(0.06) (0.326) (0.052) (0.429)

Index variance; 𝑉𝐴𝑅_𝑡 55.317*** 24.986*** 104.947*** 15.841*

(0.818) (4.471) (1.094) (9.087)

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Table 15 presents the estimated OLS coefficients for regressions of the compounding and total return deviations on the squared cumulative index return and the index variance, for the NYSE Arca Gold Miners index LETF portfolios. Equivalent to the coefficient estimates obtained for the S&P500 and Nasdaq-100 LETF cases, both independent variables are statistically significant in the regression on the compounding deviation dependent variable (model 1). Unlike earlier, for Model 2, the squared cumulative index return coefficient is not statistically significant. The underlying index variance variable in this model is only significant for the triple leveraged ETF portfolio. Observable by the higher adjusted R2 for the triple leveraged case, the independent variables of Model 1 explain more of the total return deviation than of Model 2. As indicated by the coefficients in eq. 16 and eq.17, the higher leverage multiple is expected to cause a larger compounding return deviation when the underlying index return is low, but the return variance is high. Therefore, the share of compounding deviation within the total return deviation is higher for the triple leveraged LETF portfolios, all else being equal. As can be observed from Tables 11 and 12, the gold index benchmark ETF return is highly volatile over the sample periods for both the double and triple leveraged portfolios. The index return is low, compared to the S&P500 benchmark ETF returns. These figures support the theory why Model 2 explains more of the total return deviation for the triple leveraged ETF portfolios.

Table 16 displays the estimated coefficients for the MSCI Emerging Markets LETF portfolios. The results are comparable to the gold index case. When correcting the index variance coefficient for the average number of trading days within the 1-month period, the coefficients are reasonably close to the expected values for these portfolios following eq. 16 & 17. Furthermore, the factor between the coefficients from eq.16 and 17 (factor of 2) is roughly observable between the double and triple leveraged ETF return and variance coefficients.

As described, the high adjusted R2_values2_{for Model 1 indicates that the mathematical}

relationship derived by various authors (Cheng & Madhavan, 2009, Avellaneda & Zhang, 2010 and Tang & Xu, 2013) explains the compounding return deviation. The fact that the adjusted R2 values are close but not equal to 1, is likely due to the assumptions made in the derivation of this relationship. Among these factors is the use of Taylor approximations.

2_{Again, similar as described before, the augmented Dickey-Fuller test was used to test for presence of a unit root.}

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Table 15: Regression estimates NYSE Arca Gold Miners Index LETF portfolio

Regression results NYSE Arca Gold Miners Index GDXX-GDXS 25:75 portfolio, n=46 NUGT-DUST 33:67 portfolio, n=96

Intercept; α -0.001 0.002 -0.003 -0.010**

(0.002) (0.005) (0.003) (0.004)

Squared cumulative index return; 𝐶𝑅𝐼_𝑡2 -1.937*** -0.079 -4.015*** -0.230

(0.051) (0.132) (0.085) (0.15)

Index variance; 𝑉𝐴𝑅𝑡 45.217*** 15.809 92.533*** 54.117***

(3.6) (9.406) (4.36) (7.737)

Adjusted R2 _0.970 _0.018 _0.960 _0.332

Table 16: Regression estimates MSCI Emerging Markets LETF portfolio

Regression results MSCI Emerging Markets Index EET-EEV 25:75 portfolio, n=114 EDC-EDZ 33:67 portfolio, n=120

Intercept; α 0 0 0 0

(0.000) (0.002) (0.000) (0.003)

Squared cumulative index return; 𝐶𝑅𝐼_𝑡2 _-2.268*** _-0.519 _-4.372*** _0.388

(0.023) (0.423) (0.038) (0.426)

Index variance; 𝑉𝐴𝑅_𝑡 46.97*** 22.901 95.038*** 16.764

(0.994) (18.694) (1.798) (19.977)

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5. Conclusions

Prior research has shown that LETF returns are often worse than expected due to the effect of compounding deviation (Tang & Xu, 2013). Jiang and Peterburgsky (2017) propose portfolios consisting of short positions in LETFs to take advantage of the compounding deviation effect to outperform the underlying index over longer time horizons. In this study, these results were verified using backtests with market data, confirming Hypothesis 1. By means of these backtests, empirical evidence has been found in support of the findings by Jiang and Peterburgsky (2017). Not accounting for market frictions, investors can outperform the underlying index by holding a portfolio with short positions in LETFs. This result not only holds for the S&P500 LETF portfolios that Jiang and Peterburgsky examined, but especially for other indices, such as the MSCI Emerging Markets index. The results are generally consistent across various rebalancing thresholds.

From the regression estimates shown in Section 4.3, the mathematical relationship between the compounding returns deviations and both the squared underlying index return and index variance is confirmed empirically for the short LETF portfolios. This evidence is in support of Hypothesis 2 and gives a better understanding of the implications of the compounding deviation effect. The coefficients found in the regressions for explaining the compounding deviation indicate that environments with low absolute index returns and high volatility positively impact the compounding return deviation for these short portfolios. The total return deviation, however, is not explained by these factors alone. Further research is necessary to explain the total return deviations of index replicating short LETF portfolios.

From a theoretical perspective the results from Hypothesis 1 are surprising as they are not in line with the expectations following the weak form of the Efficient Market Hypothesis by Fama (1970). Dobi and Avellaneda (2012) noted that the fee required to engage in short positions may be sized sufficiently large to obstruct any potential excess returns. Robustness tests point out that the rebate rate diminishes outperformance for the triple leveraged S&P500 ETF portfolio. For its double leveraged counterpart, however, the result is opposite as the assumed rebate rate is positive. The stochastic nature of shorting fees and rebate rates is not considered in this research. There is a way to potentially circumvent or fix these fees, by using the synthetic short stock option strategy. Future research could examine the viability of using the synthetic short stock strategy with LETF options. This would help to further understand the pricing dynamics of shorting fees and options on LETFs. The various tested levels of transaction costs have only a marginal impact on the portfolio returns, assuming the portfolio size to be reasonably large and no presence of fixed trading commissions.

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opposing LETF pairs. When both positions hedge out the exposure of one another entirely, the portfolio would benefit from increased variance and low absolute return of the underlying index, without making the investor vulnerable to directional exposure. Due to traditional no-arbitrage theory, such portfolios shouldn’t be able to generate any returns when controlled for the squared index return and variance.

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A. Algorithm base code

The annotated code used to backtest the algorithm can be found underneath:

1. # Import required Quantopian Python libraries

2. from quantopian.algorithm import order_optimal_portfolio 3. import quantopian.optimize as opt

4.

5. #global variables, used for counting within various functions

6. rebalance_days = 0 7. initially_bought = 0

8. rebalanced_bool = 0

9.

10. #This function sets up the algorithm and initializes variables

11. def initialize(context): 12.

13. #Benchmark ETF is set and the bull and bear funds are defined using a SID code

14. set_benchmark(symbol('SPY')) 15. context.bull = sid(32270) 16. context.bear = sid(32382)

17. #Setting the default shorting weight of the bull LETF

18. context.bull_amount = 0.25

19. context.bear_amount = 1-context.bull_amount 20. #The rebalancing threshold is defined next

21. band_amount = 0.05

22. # boundaries for determining if inflows or outflows to the short account must take place

23. context.upper_band_outflow = (1+0.1) 24. context.lower_band_inflow = (1-0.1)

25. log.info("Upper band outflow= "+str(context.upper_band_outflow)) 26. log.info("Lower band inflow= "+str(context.lower_band_inflow)) 27. # boundaries for rebalancing between bull & bear positions

28. context.upper_band_allocation = (1 /((context.bear_amount-band_amount)/(context.bull_amount+band_amount)))

29. context.lower_band_allocation =

(1 /((context.bear_amount+band_amount)/(context.bull_amount-band_amount)))

30. log.info("Upper band allocation=

"+str(context.upper_band_allocation))

31. log.info("Lower band allocation=

"+str(context.lower_band_allocation))

32. # Schedule a function which is called every day just after the market opens

33. schedule_function(rebalance_limiter, date_rules.every_day(), time_rules.market_open(minutes = 1))

34.

35. # This function is continously run as long as the market is opened

36. def handle_data(context, data):

37. #get global variables for use in rebalancing functions

38. global initially_bought 39. global rebalanced_bool

40. # If bull and bear funds have been bought and are in the portfolio, then..

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42. (context.bear in context.portfolio.positions)): 43. # Define bull exposure as a percentage

44. bull = context.portfolio.positions[context.bull] 45. bull_value = bull.amount * bull.last_sale_price

46. bull_percent = bull_value/context.portfolio.portfolio_value 47. # Analogously for the bear fund exposure

48. bear = context.portfolio.positions[context.bear] 49. bear_value = bear.amount * bear.last_sale_price

50. bear_percent = bear_value/context.portfolio.portfolio_value 51. # calculate the short ratio, as the size of the portfolio

divided by the value of short positions

52. ratio_short = abs(context.portfolio.portfolio_value / (context.portfolio.positions_value))

53. # calculate bullbear ratio, the ratio of exposure to both funds within the short portfolio

54. ratio_bullbear = abs(bull_value / bear_value)

55. #record these variables to access them after running the backtest

56. record(ratio_short=ratio_short)

57. record(ratio_bullbear=ratio_bullbear) 58. record(bull_percent=bull_percent) 59. record(bear_percent=bear_percent)

60. #first threshold check. If short ratio > upperband or < lower band (-10%/+10%), rebalance

61. if ((ratio_short > context.upper_band_outflow) or (ratio_short < context.lower_band_inflow)):

62. # only do this if so far there has not been and rebalances today

63. if (rebalanced_bool == 0): 64. # Note to user via log

65. log.info("Rebalancing due to outflow/inflow") 66. log.info("Ratio = "+str(ratio_short))

67. # Call function to count a extra rebalancing day

68. count_days()

69. # Call the rebalancing function

70. rebalance(context, data)

71. # Note there has been rebalanced for today

73. #If the allocation between funds is crossing the allowed bandwidth, rebalance

74. elif ((ratio_bullbear > context.upper_band_allocation) or

(ratio_bullbear <

75. context.lower_band_allocation)):

76. #Again, only if there have been no rebalances on this business day

77. if (rebalanced_bool == 0):

78. log.info("Rebalancing for allocation breach") 79. log.info("Ratio = "+str(ratio_bullbear)) 80. log.info("Bull value = "+str(bull_value)) 81. log.info("Bear value = "+str(bear_value)) 82. count_days()

83. # Call the rebalancing function

84. rebalance(context, data)

85. # Set variable to indicate rebalancing has occurred today

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87. # If there is not yet any position in both funds, buy them initially.

88. elif (initially_bought == 0 & rebalanced_bool == 0): 89. log.info("Buying initially")

90. # Set indicator so that the algorithm knows not to exercise this function again within backtest

91. initially_bought = 1

92. # Call rebalancing function as this buys initially as well

93. rebalance(context, data) 94.

95. # This is the rebalancing function which actually places orders

96. def rebalance(context, data): 97. global rebalanced_bool

98. # If called, rebalanced bool will be set to one

100. # Log info for user before rebalance

101. log.info("Value before: "+str(context.portfolio.positions_value)) 102. # Ordering bull and bear funds in the assigned negative exposure

values defined in initialize function

103. order_target_percent(context.bull, -context.bull_amount) 104. order_target_percent(context.bear, -context.bear_amount) 105. log.info("Holdings after: "+str(context.portfolio.positions)) 106.

107. # Function to output additional information to log

108. def summary(context, data):

109. log.info("Holdings eod: "+str(context.portfolio.positions)) 110. log.info("Value eod: "+str(context.portfolio.positions_value)) 111. log.info("Cash eod: "+str(context.portfolio.cash))

112. log.info("Portfolio value eod:

"+str(context.portfolio.portfolio_value))

113. log.info("Portfolio pnl eod: "+str(context.portfolio.pnl)) 114. log.info("Portfolio positions value eod:

"+str(context.portfolio.positions_value))

115.

116. # Function to count the number of days on which a rebalance occurred

117. def count_days():

118. global rebalance_days

119. rebalance_days = rebalance_days+1

120. record(rebalance_days=rebalance_days) 121.

122. # Function which sets the rebalanced indicator back to 0. Called immediately after market open.

123. def rebalance_limiter(context, data): 124. global rebalanced_bool

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B. How to use the Quantopian platform

The code in Appendix A can be used to backtest various strategies on the Quantopian platform. In this Appendix, a brief guide into the Quantopian platform is given to enable

reproduction of the observed and reported backtesting results. Note: the analysis is done partially within a spreadsheet program as well. After collecting the backtest results, the data is moved outside of Quantopian. The following steps will explain how to use the code in Appendix A to initiate a backtest.

First, navigate to https://www.quantopian.com. Hover over research and click “Algorithms” (See Figure 1).

Figure 1: Quantopian menu interface

Click “New Algorithm” and give the algorithm a name.

Figure 2: Create new algorithm naming form

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Figure 3: Quantopian developing environment

One can tweak various variables by changing the variables in the initialize function. For example, the allocation between bull & bear funds can changed with the variables:

- Context.bull_amount - Context.bear_amount

The funds themselves can be replaced by other funds easily. Quantopian contains backtesting data for many US securities including LETFs. To replace the funds and/or

benchmark, erase the current sid code at the context.bull and context.bear variables respectively. When erased, a white pop-up box will appear which allows to quickly name search the desired fund. After hitting enter, the corresponding sid code will be filled-in.

Figure 4: Changing the LETF portfolio holdings