Leveraged ETF Tracking Error

Author: Leo Chen, Ph.D., Post Date: December 14, 2018

Cumberland Advisors Market Commentary - Leo Chen, Ph.D.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.

Next, we will demonstrate that the  compounding effect accounts mathematically for most of the long-term performance discrepancy.*** The approximation below shows that the compounded return of a leveraged ETF over a long period can deviate from the underlying index even without any tracking error.
where  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.
We can thus conclude that the so-called tracking error of a leveraged ETF over the long run is not really the same as the traditional tracking error; instead, it reflects the compounding effect. This effect is more pronounced with high leverage ratios and volatility. Alternatively, some traders may refer to the effect as “time decay” or “volatility decay.” From the mathematical perspective, we can identify the compounding effect as the driving factor.

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.

Links to other websites or electronic media controlled or offered by Third-Parties (non-affiliates of Cumberland Advisors) are provided only as a reference and courtesy to our users. Cumberland Advisors has no control over such websites, does not recommend or endorse any opinions, ideas, products, information, or content of such sites, and makes no warranties as to the accuracy, completeness, reliability or suitability of their content. Cumberland Advisors hereby disclaims liability for any information, materials, products or services posted or offered at any of the Third-Party websites. The Third-Party may have a privacy and/or security policy different from that of Cumberland Advisors. Therefore, please refer to the specific privacy and security policies of the Third-Party when accessing their websites.

Sign up for our FREE Cumberland Market Commentaries

Cumberland Advisors Market Commentaries offer insights and analysis on upcoming, important economic issues that potentially impact global financial markets. Our team shares their thinking on global economic developments, market news and other factors that often influence investment opportunities and strategies.

cumber map
Cumberland Advisors® is registered with the SEC under the Investment Advisers Act of 1940. All information contained herein is for informational purposes only and does not constitute a solicitation or offer to sell securities or investment advisory services. Such an offer can only be made in the states where Cumberland Advisors is either registered or is a Notice Filer or where an exemption from such registration or filing is available. New accounts will not be accepted unless and until all local regulations have been satisfied. This presentation does not purport to be a complete description of our performance or investment services. Please feel free to forward our commentaries (with proper attribution) to others who may be interested. It is not our intention to state or imply in any manner that past results and profitability is an indication of future performance. All material presented is compiled from sources believed to be reliable. However, accuracy cannot be guaranteed.