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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 (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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