Understanding Returns Of Leveraged And Inverse Funds

August 25, 2009


Analysis Of Returns Of Leveraged And Inverse Funds Held Longer Than One Day

Commentary regarding leveraged and inverse ETFs has frequently cited examples in extremely volatile market conditions to illustrate the effects of compounding on leveraged and inverse fund performance over holding periods longer than one day. To more completely understand how these ETFs behave over holding periods beyond a day, it is important to look at a long history of returns for leveraged and inverse index strategies. To do this, we analyzed returns of hypothetical funds with a daily target of 2x and -2x S&P 500 returns over a 50-year time frame. (We performed a similar analysis for the NASDAQ-100 along with the Dow Jones financial and energy sector indexes for somewhat shorter time frames based on availability of historical data.) In our analyses, we ignored fees, expenses, financing and transaction costs. We present findings for two-, seven- and 30-day holding periods on 2x and -2x S&P 500 Index strategy returns relating to:

• The average and median difference of returns for hypothetical 2x and -2x funds compared with the returns of the index times the one-day target multiple (period target returns). Included are selected percentiles from the distribution of these return differences.

• Realized betas (leverage ratios) of the daily target (2x or -2x) hypothetical index returns for two-, seven-, 30-, 91- and 183-day holding periods versus the S&P 500 Index. These are compared with the target multiple of 2x or -2x that some investors may want to achieve over a longer holding period.

Looking at comparative results for this long time frame, we see that the impact of compounding has historically been virtually neutral, with an average effect close to zero and medians close to zero or slightly negative. The overall potential for compounding to lead to positive versus negative effects is approximately equal. We also see a high percentage of periods in which hypothetical S&P 500 daily-target strategies approximate a 2x or -2x leverage ratio over holding periods of a week and a month. The probabilities of achieving a range around a target leverage ratio fall as the holding period lengthens. Daily target leveraged and inverse index returns for seven- and 30-day holding periods were at times the opposite sign to the period target (a “flipped” return), but this was very rare.


Our study compares the returns of hypothetical S&P 500 Index funds having a 2x and -2x daily leverage target (daily target return) with a period target return (defined as the period index return times this same daily leverage target) for holding periods of two, seven, 30, 91 and 183 calendar days. Our sample contains all two-day, weekly, monthly, quarterly and semiannual holding periods possible within the last 50 years (1959 through 2008) for the S&P 500 Index. This gives us a large sample to compare all possible end-of-day entry and exit points, but it also leads to overlapping observations. The benefit of this approach is that it removes any potential bias of starting a holding period on a particular day of the week or month.

To focus on the compounding effect, we note some additional assumptions used in this analysis:

• For the period target return, the leverage ratio/beta is set at the beginning of each period and not changed for the duration of that period. For the hypothetical 2x and -2x funds, the leverage is reset daily to either 2x or -2x. Therefore, the hypothetical daily target return leveraged or inverse funds achieve is exactly the daily multiple times the daily index return each and every day of the holding periods.

• Index price return is the basis for the analysis.

• Hypothetical leverage and inverse fund return calculations for both the daily target and period target returns exclude fees, financing, interest and expenses.

• Period target returns are not constrained by capital (i.e., losses can exceed -100 percent).



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