Leveraged ETF Tracking Error

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.


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Margin Trading vs. Leveraged ETFs

Our previous commentary on leveraged ETFs (www.cumber.com/leveraged-etfs/) received many comments from our readers. We would like to thank you for sharing your thoughts and address one common question among many investors: Which is better, margin trading or leveraged ETFs?

Market Commentary - Cumberland Advisors - Margin Trading vs. Leveraged ETFs

First, what is the difference between these two methods of investing? We walked through leveraged ETFs last time, so we will focus mainly on margin trading today. Margin trading refers to borrowing funds from a broker in order to trade an asset such as stocks. The purpose of the loan is to amplify the gains on the invested position. However, margin trading is inevitably subject to the risk of substantial losses in the meantime. Hence, the purchased asset is also the collateral for the loan. Oftentimes, a broker allows an account to borrow up to 50% of the purchased price of a stock. The portion that is not borrowed funds is called the initial margin. A broker usually requires a minimum balance to be maintained in the account. This restriction is called the maintenance margin. If the account balance falls below the maintenance margin, your broker will send you a margin call and ask you to deposit funds to meet the requirement.

In a worst-case scenario, if an investor fails to deposit funds to meet the margin requirement, the broker can force-sell the investor’s stocks without notice. Margin liquidation is the main risk embedded in margin trading. Investors could lose 100% of their investment and possibly more by using margin. To demonstrate, we will use the example summarized in the table below. Assume we borrow 50% to purchase 100 shares of a stock at $100 a share, with a 6% interest rate, and ignore transaction costs. Hypothetically, the stock price goes down to $50 three months later. We are then left with zero equity and 100% debt. We also need to pay the interest on the loan to the broker, who will likely close the position if we fail to add any funds. If we don’t, we will lose 100% of our investment plus roughly 1.5% interest on the $10,000, for a total -101.5% return.


Table 1. Margin trading example – downside risk

What if we used a leveraged ETF and had the same stock movement? Although there is no ETF tracking just one stock, we will assume that there is, for the sake of simplicity. The short answer is that we would lose the 100% investment as the ETF fell to $0 per share. But we wouldn’t be subject to more than a 100% loss, as all ETFs incorporate the management fees into their NAVs.

After comparing the downside risk, now let’s take a look at the upside potential. Let’s assume the stock goes up to $150 after three months. The net return from margin trading would be 98.50%, as demonstrated below.


Table 2. Margin trading example – upside potential

If we used a leveraged ETF, the net return would be 2X the 50% return minus the 0.25% fee–about 99.75%. Hence, as long as the ETF fee is below the margin cost, leveraged ETFs have a competitive advantage over margin trading. Please note that for the purpose of demonstration the above example ignores the daily rebalancing effect, or “tracking error,” embedded in ETFs, as the performance of leveraged ETFs is path-dependent.

Beyond performance, there is something crucial yet often overlooked by investors. In the example shown in Table 2, we started by having 50% margin. However, as the stock price went up, our margin eventually fell to 1/3. Of course, we could continuously borrow more to maintain a 50% margin, but that would require constant rebalancing. In contrast to margin trading, leveraged ETFs maintain the targeted ratio continuously regardless of price movement. This effect is more pronounced in a bear market, as example 1 depicts. When stock prices plummet, investors tend to sell quickly to minimize their losses, exacerbating price declines. Meanwhile, investors are left owing more than the value of their stock positions and are likely to default on their margin loans. Moreover, lenders depend on the value of stocks as collateral. The cost of borrowing rises in a tumbling market, which in turn adds more pressure to margin trading.

Therefore, leveraged ETFs have the ability to achieve the same goal as margin trading while eliminating the margin call risk at potentially a lower cost. Compared to traditional margin, leveraged ETFs make it easier to maintain the targeted ratio. Leveraged ETFs are also better for short-term trading than margin, because it is easy to move in and out of a position. Last but not least, many leveraged ETFs have little liquidity risk, since this industry has grown to $80 billion in size in 2018 (source: https://www.reuters.com/article/investment-etf/leveraged-inverse-etfs-grew-to-record-in-january-etfgi-idUSL8N1QH4D0).

Leo Chen, Ph.D.
Portfolio Manager & Quantitative Strategist
Email | Bio


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.

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Leveraged ETFs

Our quantitative strategy at Cumberland Advisors is a trading model that combines fundamental indicators and quantitative analysis into a binary output – either fully invested or all in cash. The strategy trades the S&P 500 in two versions: unleveraged and leveraged. Specifically, the leveraged portfolio uses a leveraged ETF as our vehicle to track 3X the market movement. As many may wonder whether one should use a leveraged ETF, we would like to express our opinions on leveraged ETFs today.

Market Commentary - Cumberland Advisors - Leveraged ETFs

First, what is a leveraged ETF? It is simply an ETF using derivatives and debt to track and amplify the return of an index. However, a leveraged ETF does not expose investors to traditional margin risk; rather, investors just pay the ETF cost. A leveraged ETF resets each day and targets to track an index’s daily movement. A leveraged ETF is usually considered a trading tool. It is typically held for less than a week at most, and oftentimes just daily. Investors are generally told not to buy and hold this type of security due to “time decay,” a term that is often misused when applied to leveraged ETFs. Time decay is a term used to describe the loss of value of an option as time approaches the expiration date. However, leveraged ETFs are not subject to option expirations. What “time decay” really refers to, in connection with leveraged ETFs, is the compounding effect. For example, if the market went up 10% on day 1 and went down 10% on day 2, one would lose 1% at the end of day 2; with 3X leverage, one would go up 30% on day 1 and down 30% on day 2, being left with a 9% loss at the end:

1 – (1+0.1) x (1-0.1) = 0.01     (1)
1 – (1+0.3) x (1-0.3) = 0.09     (2)

Time has nothing to do with the math above. It is compounding that magnifies the leveraged number. In other words, anything that increased 30% and then decreased 30% would have the same outcome regardless of leverage. Imagine that equation (2) represented a scenario where the market went up 30% on day 1 and down 30% on day 2 – one would be left with 9% loss without any leverage or so-called “time decay” effect. Another example: If the market dropped 1% a day for 10 consecutive days, one would suffer a 9.56% loss, while the loss would be 26.26% with 3X leverage:

1 – (1-0.01)10 ≈ 0.0956     (3)
1 – (1-0.03)10 ≈ 0.2626     (4)

Again, the math demonstrates that time is not the reason for the significant difference; compounded return is the master behind the scene. To understand the power of compounding, let’s take a look at a famous motivational poster that some people have as their desktop – if you improve just 1% a day, you will be much better in a year:

Chart 1. 1% a day difference
Now that we have clarified the mathematical misconception, let’s dig into the more important question: Should one buy and hold leveraged ETFs? The chart below compares the 3X leveraged ETF SPXL against the benchmark S&P 500.

Chart 2. S&P 500 vs. SPXL, 11/5/2008–11/9/2018. Data source: Bloomberg

Clearly, the 3X ETF has substantially outperformed the S&P 500 since its inception on November 5, 2008. However, not every investor would have had the stomach for the volatility that was experienced along the way. Leverage can be a powerful tool to take advantage of a bull market if used properly, but one must possess extensive risk management skills. At Cumberland Advisors, we prioritize risk control by keeping our focus on risk-adjusted returns.

If you are interested in obtaining information about our quantitative strategy, please email me.

Leo Chen, Ph.D.
Portfolio Manager & Quantitative Strategist
Email | Bio


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.

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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.