Guedj, Ilan, Guohua Li, and Craig McCann. 2010. Leveraged and Inverse
ETFs, Holding Periods, and Investment Shortfalls. Journal of Index
Investing 1, no. 3 (winter): 45–57.
Hougan, M. 2009. How Long Can You Hold Leveraged ETFs? Journal of
Indexes 12, no. 2 (March/April): 36–41.
hunnicutt, trevor. 2017. U.S. Sec Approves Request to List quadruple-
Leveraged ETFs. Reuters (May 2). http://www.reuters.com/article/
hunnicutt, trevor, and John mccrank. 2017. more quadruple-Leveraged
ETFs Proposed Despite SEC Review. Reuters (June 20). http://www.
Justice, Paul. 2009. Warning: Leveraged and Inverse ETFs Kill Portfolios.
Morningstar Investing Specialists (January 22). http://news.
Lau, Ashley. 2014. Leveraged ETFs Not Just for Day Traders Anymore.
Reuters (February 12). http://www.reuters.com/article/us-
Loehr, Ross, and Reinhold Lamb. 2013. Long-Term Investing with
Leveraged Exchange Traded Funds. International Journal of Arts
and Commerce 2, no. 4 (April): 27–34.
Maxey, Daisy. 2009. ProShare Draws Suit over a Leveraged ETF.
Wall Street Journal (August 7). https://www.wsj.com/articles/
Opdyke, John Douglas. 2007. Comparing Sharpe Ratios: So Where Are the
p-values? Journal of Asset Management 8, no. 5 (December): 308–336.
Trainor, Jr., William J. 2011a. Daily vs. Monthly Rebalanced Leveraged
Funds. Journal of Finance and Accountancy 6, no. 1 (March): 1–14.
———. 2011b. Solving the Leveraged ETF Compounding Problem. Journal
of Index Investing 1, no. 4 (spring): 66–74.
Trainor, Jr., William J., and Edward A. Baryla, Jr. 2008. Leveraged ETFs:
A Risky Double That Doesn’t Multiply by Two. Journal of Financial
Planning 21, no. 5 (May): 48–55.
Trainor, Jr., William J., and Mark G. Carroll. 2013. Forecasting Holding
Periods for Leveraged ETFs Using Decay Thresholds. Journal of
Financial Studies & Research, 2013: 179–190.
Zweig, Jason. 2009. how managing Risk with etFs can Backfire.
Wall Street Journal (February 27). https://www.wsj.com/articles/
———. 2017. Are you Really crazy enough to Buy a quadruple-Leveraged
ETF? Wall Street Journal (May 19). https://www.wsj.com/articles/
1. Trainor (2011a) shows daily and monthly rebalancing have the same
type of drawbacks for long-term investors because the difference in
the realized leveraged ratio between daily and monthly rebalancing
becomes fairly blurred after six months. hougan (2009) finds roughly
the same type of result.
2. Equation ( 1) approximates the decay in leverage because higher ordered
terms along with the sequence of returns is ignored. Empirically, the
realized volatility along with the sequence of the returns are important
because leveraged funds have positively skewed returns. See Cheng
and Madhavan (2009), Avellaneda and Zhang (2010), or Trainor and
Carroll (2013) for more details in the derivation of Equation ( 1).
3. ProShares 2.0x SPRO and 3.0x UPRO S&P leveraged funds have expense ratios of 0.89 percent and 0.94 percent, respectively. 1.0 percent
is used in this study to account for additional transaction costs that are
associated with daily rebalancing.
4. Sampling five days at a time to create return series also was tested
to determine if some daily momentum or reversion occurs with no
significant changes in the results.
5. In any given day the 4.0x fund will supposedly be limited to a
96-percent loss with the use of put options.
6. With Vanguard now offering this portfolio as an etF (BNd), one easily
could replicate these portfolios with just two ETFs.
Annaert, Jan, Sofieke Van osselaer, and Bert Verstraete. 2009. Performance Evaluation of Portfolio Insurance Strategies using Stochastic
Dominance Criteria. Journal of Banking and Finance 33, no. 2: 272–280.
Avellaneda, Marco, and Stanley Zhang. 2010. Path-Dependence of
Leveraged ETF Returns. Society for Industrial and Applied Mathematics
Journal on Financial Mathematics 1, no. 1: 586–603.
Brown, Stan. 2008. Measures of Shape: Skewness and Kurtosis. Down-loaded from https://brownmath.com/stat/shape.htm.
Bulmer, M. G. 1979. Principles of Statistics. Dover.
Carver, Andrew. 2009. Do Leveraged and Inverse ETFs Converge to Zero?
Journal of Derivatives 2009, no. 1 (fall): 144–149.
Cheng, Minder, and Ananth Madhavan. 2009. The Dynamics of Leveraged
and Inverse Exchange-Traded Funds. Journal of Investment Management 7, no. 4 (fourth quarter): 43–62.
DiLellio, James, Rick Hesse, and Darrol Stanley. 2014. Portfolio Performance with Inverse and Leveraged ETFs. Financial Services Review 23,
no. 2 (summer): 123–149.