Leveraged ETF pairs: An empirical evaluation of portfolio ...

Leveraged ETF pairs: An empirical evaluation of portfolio performance Stanley Peterburgsky Brooklyn College, 2900 Bedford Ave, New York, NY 11210

E-mail: phinance@

Abstract This paper examines the performance of investment strategies involving leveraged and inverse leveraged ETF pairs. As in Jiang and Peterburgsky's (2017) simulation analysis, the empirical analysis in this paper indicates that simple portfolios of bull/bear short positions constructed to approximately mimic an underlying index (such as the S&P 500) outperform the index on a riskadjusted basis. The outperformance asymptotically diminishes as the investor's rebalance bounds tighten. Additionally, the results support the theoretical positive relationship between outperformance and index standard deviation and negative relationship between outperformance and index absolute return.

JEL classification: G11 Keywords: Exchange traded funds, ETFs, pairs trading, constant leverage trap, portfolio management

This version: 10/8/18

I thank Charles Jones for very helpful comments. All errors are mine.

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I. Introduction Leveraged and inverse leveraged exchange traded funds (ETFs) are high-risk financial

instruments that have been trading in the U.S. since 2006. Their goal is to generate specific multiples of the return on an underlying index, such as the S&P 500 or the Russell 2000, on (usually) a daily basis.1 For example, the triple-leveraged ETF ProShares UltraPro S&P 500 (UPRO) seeks daily investment results corresponding to triple the daily performance of the S&P 500, while the inverse triple-leveraged ETF ProShares UltraPro Short S&P 500 (SPXU) seeks daily investment results corresponding to triple the inverse (i.e., negative) of the daily performance of the S&P 500. Likewise, the triple-leveraged ProShares UltraPro Russell2000 (URTY) attempts to provide daily returns equal to triple the returns of the Russell 2000, while the inverse triple-leveraged ProShares UltraPro Short Russell2000 (SRTY) attempts to provide daily returns equal to triple the inverse of the daily returns of the Russell 2000. To achieve the target leverage, these funds enter into futures and/or swap contracts tied to the underlying index,2 and adjust their exposure to the index by rebalancing their holdings on a daily basis.

Among academics and practitioners specializing in leveraged and inverse leveraged exchange traded funds (henceforth, collectively, LEFTs), it is well known that these funds usually achieve their stated daily objectives but surprise inexperienced investors with holding periods of more than a day. Specifically, LETFs come reasonably close to producing the expected daily returns of leverage ? daily index return, but longer-term returns generally deviate from leverage ? multiday index return (see, for example, Cheng and Madhavan (2009), Lauricella (2009), Avellaneda and Zhang (2010), Jarrow (2010), Charupat and Miu (2011), and

1 The benchmark returns are price returns, not total returns. That is, dividends are not included. 2 The details of the process involved in constructing leveraged and inverse leveraged positions are described in

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Tang and Xu (2013)) because of the so-called "constant leverage trap". The intuition behind the

constant leverage trap can be illustrated with the following example. Suppose that on day 0 (after

markets close) investors A, B, and C invest $1 in the S&P 500, UPRO, and SPXU, respectively.

Further, suppose that the S&P 500 return is 25% on days 1, 3, 5, 7, and 9, and -20% on days 2, 4,

6, 8, and 10. As Panel A of Table 1 shows, investor A will have $1 after 10 days. On the other

hand, if the LETFs achieve their target daily returns of 3? and -3? the S&P 500 return, investor

B will have $0.17 and investor C will have $0.01. Although UPRO and SPXU have opposite

objectives at daily frequency, both vastly underperform the S&P 500 in the long run in this

particular example. Of course, the order of the 25% and -20% days doesn't matter; the end result

would be the same if, say, the S&P 500 return were 25% on days 1, 2, 3, 4, and 5, and -20% on

days 6, 7, 8, 9, and 10. Panel B of Table 1 illustrates the same idea with the double-leveraged

ProShares Ultra S&P500 (SSO) and the inverse double-leveraged ProShares UltraShort S&P500

(SDS), while panel C considers the inverse ProShares Short S&P500 (SH).

In reality, LETFs can either underperform or outperform the na?ve expectation of long-

term returns. Tang and Xu (2013) and Loviscek, et al. (2014) show that, under mild assumptions,

the return dynamics of LETFs can be modeled as

rLETF rM

(2 ) 2

rM

(2 )T 2

2 rM

,

where rLETF is the cumulative return on the LETF, rM is the cumulative return on the index, is

the leverage multiple, T is the length of the holding period, and 2 is the index variance.

The above equation implies that a fund's "effective long-term leverage" increases with the index

cumulative return and decreases with the index standard deviation during bull markets but

decreases with the absolute value of index cumulative return and increases with the index

standard deviation during bear markets. Consequently, a trending market benefits long-term

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investors (as compared to na?ve expectations) while a volatile market hurts them.3 This tension between trend and volatility ultimately determines whether inexperienced investors are pleasantly or unpleasantly surprised.

The following set of examples of actual S&P 500 5-day return patterns can illustrate this point. During 6/27/11 ? 7/1/11, the daily (total) returns on the S&P 500 were 0.92%, 1.32%, 0.84%, 1.01%, and 1.45%, an uptrend (Table 2, left side). The compounded return during this window was therefore 5.67%, and the na?ve expectations for UPRO and SPXU were 3?5.67% = 17.00% and -3?5.67% = -17.00%, respectively. The actual returns on the two LETFs, however, were 17.49% and -15.39%, producing positive residuals. Next, consider the downtrend of 8/19/15 ? 8/25/15. Daily returns on the index were -0.82%, -2.11%, -3.17%, -3.94%, and -1.35% (Table 2, middle). The compounded return during this window was therefore -10.91%, and the na?ve expectations for UPRO and SPXU were 3?(-10.91%) = -32.73% and -3?(-10.91%) = 32.73%, respectively. The actual returns, however, were -30.06% and 38.80%, producing positive residuals once again. Finally, consider a volatile period without any significant trend in either direction. During the week of 8/8/11 ? 8/12/11, daily S&P 500 returns were -6.65%, 4.74%, -4.38%, 4.65%, and 0.53% (Table 2, right side). The compounded return was an unspectacular -1.64%, but the annualized volatility rose to 82.3%. The na?ve expectations for UPRO and SPXU were 3?(-1.64%) = -4.91% and -3?(-1.64%) = 4.91%, respectively. The actual returns, however, were -8.02% and -1.74%, very disappointing for unsuspecting investors. To summarize, positive and negative trends usually result in positive residuals, while volatile markets without a clear trend typically produce negative ones.

3 It should be noted that actual and na?ve long-term returns can differ for reasons other than trends and volatility. For example, fund managers may be unable to achieve the daily targets, management fees can erode fund value, etc.

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Empirically, the volatility effect dominated during the 2008-2009 financial crisis, leading to several class action lawsuits filed against ProShares and rival fund family Direxion by disappointed investors (see Michaels (2016)). On the other hand, more recently there have been periods when the trend effect dominated, and in May 2017 the SEC approved the first quadrupleleveraged products (see Hunnicutt (2017) and Zweig (2017)). In light of this, it would be interesting to explore potential investment strategies involving bull, bear, and twin funds. In particular, if the volatility effect dominates on a more or less consistent basis, various buy-andhold investment strategies that include short positions should perform well in the long run.

My contributions to the literature can be summarized as follows. First, I document that the volatility effect prevails over long horizons, and therefore bull LETFs are not profitable longterm investments. Second, consistent with Jiang and Peterburgsky's (2017) simulation analysis, simple portfolios of short bull/bear positions constructed to approximately mimic an underlying index (such as the S&P 500 or the Russell 2000) outperform the index on a risk-adjusted basis. Third, I find that the outperformance asymptotically diminishes as the investor's rebalance bounds tighten. Finally, I present evidence supporting the theoretical positive relationship between portfolio outperformance and index standard deviation and negative relationship between portfolio outperformance and index absolute return.

The rest of this paper is organized as follows. In Section II, I summarize the relevant literature pertaining to leveraged ETFs. In Section III, I discuss the data used in subsequent analysis. Section IV examines the profitability of long bull leveraged ETF positions. Section V presents the main findings on the relative performance of portfolios constructed using bull/bear leveraged ETF pairs and the relationships between outperformance and index absolute return and standard deviation. In section VI, I examine the effect of modifying the rebalance bounds on the

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