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Susan Weerts
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I'm working as a consultant for the Chinese companies in their deals with Private Equity Firms. I'm considered as a pattern day trader. I have been actively managing my personal portfolio since 2007. My trading strategies stemmed from my researches and my dissertations on Behavioral Finance as a... More
• ##### A Long Term Market Neutral Arbitrage Trading Strategy for Leveraged ETFs 0 comments
Mar 22, 2010 9:22 PM | about stocks: SPY, SSO, SDS, IWM, UWM, TWM, XLF, UYG, SKF
Leveraged ETFs (LETFs) are designed for a short term trading. Using leverage, the daily returns of LETFs are set at a LX daily returns of underlying index. On a daily basis, both Proshares and Direxions do a relatively good job in replicating the underlying index.
Many traders/investors mistakenly expect that LETFs will deliver a LX return of the underlying index over the time, too. They cry foul when the returns of both the long/short LETFs over a year appear to be far less of multiple (NYSE:L) than index retrun. The decay on the return of LETFs over a long period can be seen in their return dynamics, and is caused by the time decay of options used to achieve the leverage.  A long-term market-neutral (β=0) arbitrage strategy can be built to capture this decay regardless of the directional movement of the underlying index.

Return Dynamics

Let It be the process of underlying index. It follows a geometric Browian Motion, N~(μi, σi2

Let L be the process of LX LETFs. Since it is designed to replicate LX daily returns of the underlying index and then follows the Brownian Motion, N~(Lμi, L2 σi2):

The expected compounding returns RT,L of LETFs over the time period T and RT,i is the expected return of the underlying Index over the time period T:

The second term in the equation is the decay factors due to the effects of time, leverage, and volatility. The Short (inverse) LETFs suffers more decay than their regular counterparts. The expected return of LETFs is convex in term of leverage L and volatility σ.
The holding-period returns of LETFs can be explicitly expressed as follows:

Market-neutral strategy

Simple market-neutral arbitrage trading strategies can implemented by pairing the LETFs with different leverage that track the same underlying Index. A  market-neutral portfolio consisting of a L1X LETF and a L2X LETF can be constructed as shown in table 2. The number of shares ( (λ=L1 /L2) of  L2X LETF  is derived by dividing the leverage factor of LTF1 by leverage factor LTF2.

 ETFs ETFs Example L1>L2>0 Short 1 share Long λshares Short 1 Share SSO(2X   SP500), Long 2 Shares SPY (1x SP500) L1< L2<0 short 1 share Long λ shares Short 1 Share FAZ (-3X Financial), Long 1.5 Shares SKF ( -2X Financial) L1>0,L2<0 Short 1 share short λ shares 1 Share SSO (2X SP500), Short 1 Share SDS (-2X SP500)

the expected return RT,P of this portfolio over a holding period of T will be

Hence, the expected return of this trading strategy should be always positive regardless of the market movement and increasing with time.
Empirical Results

The empirical studies over the past 3 years (1/2007-1/2010) show very rewarding results with one exception, Financial ETFs.  The reason to choose this 3-year time period is that all the 2x/-2x LETFs were launched sometime in January, 2007. Furthermore, it encompasses both bull and bear market.

 ETF return from 1/2007-1/2010 Market Neutral Strategy Return from 1/2007-1/2008 1X 2X -2X Short 2X long 2*1X Short 2X short-2x Short -2X short -2*1x SP 500 -20.3% -56.8% -16.6% 16.2% 28.6% 73.4% Russul2000 -21.7% -62.6% -44.3% 19.1% 43.9% 106.8% Financial -56.9% -91.5% -65.8% -22.3% 89.8% 157.3%

A study by IndexUniverse show that the decay starts to be noticable after a month. The following tables list the portfolio monthly returns in both up and down market of the underlying index during the same time period.

 Monthly returns of the underlying index <0 SP 500 Mean Monthly Return Standard Deviation Kurtosis Skewness Short 2X long 2*1X 0.0072 0.0079 3.6690 1.4357 Short 2X short-2x 0.0223 0.0448 4.3810 2.2035 Short -2X short -2*1x 0.0151 0.0388 4.8555 2.2471 Russel 2000 Mean Monthly Return Standard Deviation Kurtosis Skewness Short 2X long 2*1X 0.0054 0.0088 -1.2712 0.1017 Short 2X short-2x 0.0270 0.0464 1.9667 1.4234 Short -2X short -2*1x 0.0216 0.0441 2.3785 1.5694 Financial Mean Monthly Return Standard Deviation Kurtosis Skewness Short 2X long 2*1X -0.0057 0.0244 0.0301 -0.3425 Short 2X short-2x 0.1644 0.2101 2.1721 1.0190 Short -2X short -2*1x 0.1701 0.2180 3.0435 1.2170

 Monthly returns of the underlying index >0 SP 500 (SPY, SSO,SDS) Mean Monthly Return Standard Deviation Kurtosis Skewness Short 2X long 2*1X 0.0050 0.0087 3.0762 1.3860 Short 2X short-2x 0.0081 0.0225 7.0750 2.5431 Short -2X short -2*1x 0.0031 0.0154 6.3429 2.2617 Russel 2000(IWM, RWM, TWM) Mean Monthly Return Standard Deviation Kurtosis Skewness Short 2X long 2*1X 0.0086 0.0164 7.8593 2.6883 Short 2X short-2x 0.0195 0.0411 5.7996 2.3849 Short -2X short -2*1x 0.0109 0.0258 3.9786 1.9952 Financial(XLF, UYG, SKF) Mean Monthly Return Standard Deviation Kurtosis Skewness Short 2X long 2*1X 0.0336 0.0434 1.5724 1.4650 Short 2X short-2x -0.0860 0.1950 0.8106 -0.5188 Short -2X short -2*1x -0.1196 0.2050 1.3160 -1.1345

The monthly returns of portfolios following SP 500 and Russel 2000 ETFS confirm the hypothesis that the trading strategies will earn positive returns regardless the directional movement of the market and reduce volatility as measured by standard deviation. However, the results on Financial ETF’s are not consistent with the hypothesis. Many factors can contribute to this contradiction. The prime reason may be due to the fact that XLF tracks selected SP financial stocks while UYG and SKF track Dow Jones US Financials. XLF and UYG/SKF are offered by two different vendors, which construct their ETF’s using different methods. However, it doesn't explain the negative monthly return on the Double-short Strategy.

Conclusion
Theoretically, a market-neutral trading strategy can be implemented by pairing two LETFs that track the same underlying index to capture the return decay regardless the market movement. The empirical studies on SP 500 and Russel 2000 ETFs are promising. However, the results on Financial ETFs are less than satisfactory. The empirical studies confirm that the above market neutral strategy reduce the volatility, which is a very attractive feature.
The above trading strategies are built on a simplified theoretical model. In reality, μ(mean), σ(volatility) of index are not constant and vary with time. In addition, there are refinance costs and borrowing costs in ETF fund construction.  Hence, λ might not be exactly equal to. To build a more robust trading strategy, more sophisticate models, such as a stochastic volatility model, are needed to analysis the combination of leverage effects, volatility effects, and fat tail.

/************/

Many thanks to Exchange-traded Funds Investing Summit 2010 by IGlobal Forum for the idea of arbitrage strategy

Disclosure: Long UYG
Stocks: SPY, SSO, SDS, IWM, UWM, TWM, XLF, UYG, SKF
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