We discussed the goal of leveraged ETFs previously – to provide daily returns that match the desired ratio over the underlying index (https://www.cumber.com/margin-trading-vs-leveraged-etfs/). These ETFs rebalance daily to maintain the proportional leverage through derivatives such as futures, forwards, and swaps. We will demonstrate that this daily rebalancing feature dictates the long-term returns of leveraged ETFs, deviating from the multiple of the underlying index over the same period due to compounding.
One of our earlier commentaries compared the long-term returns of a leveraged ETF and an unleveraged index that suffers from a lack of compounding (http://www.cumber.com/leveraged-etfs/). We will revisit the issue with a simple example. If an index returned 30% in one year, then the arithmetic average daily return would be 0.1190%, using 252 trading days a year (1); however, the geometric average would be 0.1042% (2):
30%/252 ≈ 0.1190% (1)
(1+30%)1/252 – 1 ≈ 0.1042% (2)
Apparently, requiring both daily and long-term returns of leveraged ETFs to match the underlying index is not realistic. Hence, given that leveraged ETFs’ target is to track the daily multiple returns, we recommend focusing on the daily tracking error.* We continue with our previous choice of the ETF SPXL as our example. We use one of the largest ETFs, SPY, as our comparison. First, we notice that the correlation between SPX and SPY has been lower than the correlation between SPX and SPXL (Table 1) since their inceptions. Moreover, the average daily tracking errors of SPY and SPXL are both very small, around 0.01%. The spread between the tracking errors is only 0.0034% (Table 2), net of expenses. On the other hand, we also compare the absolute values of these daily tracking errors. Interestingly, even if both the absolute values are greater than before, the spread between the absolute averages is still relatively trivial – 0.0335%. From the daily return perspective, leveraged ETFs do not provide significantly higher tracking errors than their counterparts do.
is the leverage ratio, n is the number of holding days, 
is the index average return, S is the standard deviation of the daily returns, and 
is the compounded return over the period, ignoring expenses. The main takeaway from equation (3) is that the higher the volatility S and the leverage ratio , the lower the compounded return 
over the holding period. The return can be very poor in a sideways environment or rather pleasing in a bull market.Leo Chen, Ph.D.
Portfolio Manager & Quantitative Strategist
Email | Bio
*We define daily tracking error as the difference between an ETF's daily NAV return and the underlying index's daily return.
**Data from Bloomberg.
***We adapt the derivation from the Richard Co and John Labuszewski paper "Leveraged ETFs: Where Is the Missing Performance?" (2009 ). We derive the approximation by applying Taylor series.
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