Leveraged ETF: Analysis and Estimation of CL2 return from 1928 to 2024
There is 3 parts here are the links for the oterh sections:
Introduction
First, I want to clarify that this is not investment advice. The information provided does not constitute a recommendation or an offer to buy or sell any securities or to adopt any strategy. The reader should verify the accuracy of the information, and I make no commitments. Readers must form their own opinion through more in-depth research. I am not a financial professional, just an enthusiast. This text does not replace the advice that a financial advisor can provide. It is possible that the assumptions made do not reflect reality. Readers should consult multiple sources of information before making any decisions. Furthermore, I remind you that past performance is not indicative of future performance.
English is not my native language therefore I apology if I make some mistakes.
There is lot of different opinions both positive and negative, about leveraged ETFs that replicate major stock indexes such as the S&P500 and NASDAQ for long-term investing. Here are some quotes:
Negatives:
· “These ETFs are not meant to be held for a long time because they quickly lose their value”
· “It is recommended to use these ETFs for a short period”
· “Leveraged ETFs embody the worst of modern finance”
Positives:
· “200k€ […] , almost entirely in LQQ” (LQQ = 2x leverage on Nasdaq)
· “As for the beta slippage introduced by the daily reset, I bet […]on a positive beta slippage”
· ” €65,000 invested 70% in CW8 and 30% in CL2” (CL2=2*leveraged on MSCI USA)
All these opinions provide either optimistic or pessimistic views on leveraged ETFs, which can leave one uncertain about the judgment to adopt. In this study, we will analyze leveraged ETFs to better understand how they perform over the long term.
First, we will briefly present leveraged ETFs or LETFs. Then, we will analyze what volatility drag is and how to quantify it. Next, we will look into the “hidden” borrowing cost fees. With these aspects covered, we will attempt to model the theoretical performance of a leveraged ETF since 1928. Finally, we will discuss a strategy for using leveraged ETFs.
Generality on leveraged ETF
A daily leveraged ETF, or LETF for (Leveraged ETF), replicates the daily performance of the underlying index with a lever factor L (often x2 or x3). For example, if the underlying index loses 2% in a day, a three time leveraged ETF would return -2% x 3 = -6%. There are various leverage factor, both positive and negative.
However, to achieve this lever, ETF issuers use financial instruments or issue loans. These two methods of obtaining leverage incur significant costs, such as loan interest or fees on financial instruments. Moreover, there is daily reset fees for synthetic replication ETFs. During the study, we will quantify these two types of fees for a leveraged ETF.
Finally, these borrowing and replication costs do not appear in the KIID (in Europe I am not sure for the US) of the ETF issuer because they are either integrated into the calculation of the replicated index (for borrowing costs) or into the tracking error (for replication costs). Only the management fees appear in the KIID.
Volatility drag or Performance Gap
An Initial Understanding of Volatility drag
At first glance, one might think that these ETFs outperform because they multiply the performance of indices that increases over the long term. Therefore, if the S&P 500 increases over 10 years, a leveraged ETF should perform better since a lever has been applied to it. However, it is a bit more complicated due to what is known as volatility drag (an intimidating term for a simple concept).
Since the lever has been applied daily, there is a daily reset. Thus, for a single day, the performance, without the borrowing costs is doubled for a 2-time lever. However, over a long period, the result is not necessarily doubled. This phenomenon is called volatility drag.
For example, if an index starts at 100, loses 10% on the first day, and then gains 11.1% the second day, its value will be 100 * 0.9 * 1.111 = 100 after two trading days. So, 0% over two days.
However, with a 2-time leverage, the ETF starts at 100, loses 20%, and then gains 22.2%. Its value would be 100 * 0.8 * 1.222 = 97.7 after two days. Therefore, an underperformance of 2.3%.
The consensus is that this volatility drag effect will inevitably ruin long-term investment. This is a misconception because even without leverage the volatility drag remains. In our example, the index, after losing 10%, needs to gain 11.1% to return to 100, not just 10%. This performance gap to recover what was lost is called volatility drag. The main issue with leveraged ETF is that this difference is more significant: 22.2% – 20% = 2.22% versus 11.1% – 10% = 1.11%. Moreover, volatility drag also applies to indexes without leveraged (or with a leverage of 1). Therefore, asserting that leveraged ETFs underperform in the long term solely due to volatility drag is incorrect, as every ETF also experience volatility drag even without any lever.
Quantifying volatility drag over the Long Term
Thinking further, this difference is related to the gap between geometric and arithmetic averages. The arithmetic average is the mean of the daily performances, while the geometric average is the average gain needed each day to achieve our final gain.
For the unleveraged index in our example:
If the index starts at 100, loses 10% on the first day (ending at 90), and then gains 11.1% on the second day (returning to 100), the arithmetic average of the daily returns is
However, the geometric average is calculated over the entire period to reflect the actual compounded return. In this case, the total return over two days is:
For the leveraged ETF:
The daily arithmetic performance is calculated as follows:
Meanwhile, the true performance (geometric average) is:
The difference between these two averages is the volatility drag:
· In the first case, the volatility drag is : 0.55%-0%=0.55%
· In the second case, it is 1.11%-(-1.11%)=2.22%
We notice that for the LETF the volatility drag is more than twice the one without any leveraged. Indeed, the volatility drag is proportional to the square of the lever. Essentially, this is the risk with leveraged ETF: it amplifies volatility drag by the square of the lever.
We need to go a bit further into the concept of averages. Let xi denote the performance on day i and n be the investment period in days.
Do not worry about the formulas; they are easy. Here is the arithmetic average:
And the geometric average:
To summarize, the arithmetic average is average of daily performance values. The geometric average, however, is more relevant for our purposes because when raised to the power of the investment duration, it represents our overall gain.
A leveraged ETF multiplies the arithmetic mean by a factor L, thus each xi is now Lxi, and therefore we have:
However, the LETF do not necessarily multiplies our gains.
The risk of leveraged ETFs is that they multiply the arithmetic mean by the leveraged factor but not the geometric mean, which represents the actual gain.
If we take the assumption that xi follows a normal distribution with µ as the average (the arithmetic average of daily returns) and a standard deviation σ (we will verify the normality assumption later), the relationship between the arithmetic Ma and the geometric mean Mg is:
However, in order to get a leveraged factor of L, ETF issuers must take out a loan and pay interest on that loan, which results in a decrease in daily performance. These borrowing costs are also included in the index’s yield replicated by the LETF. In conclusion, the daily performance of a leveraged index is not Lµ , but Lµ-r with r being the daily interest cost of the loan.
In the case of the MSCI USA Leveraged 2 Index, according to MSCI documentation, the daily leveraged is not Lµ, but is instead:
With:
𝑇: the number of calendar days between two successive trading dates.
𝑅: the overnight risk-free borrowing cost (€STR since 2021 and EONIA before). €STR and EONIA are European borrowing cost, since the index is replicated in Europe
Henceforth, I denote:
The €STR and EONIA rates are the interest paid by a bank which are borrowing euros for a one-day duration from another bank. EONIA was replaced by €STR since 2021 and both rates are closely linked to the European Central Bank (ECB) rates. The €STR is annualized, so it must be divided by 360 to know the interest paid for one day (mathematically, the 360th root should be taken, but bankers calculate by dividing by 360). Today, on April 29, 2024, the €STR rate is 3.90%.
For a lever factor L , µ becomes Lµ – r and σ becomes L²σ², so we have:
volatility drag is the second term in the formula in front of the minus sign
One can notice that volatility drag is proportional to the square of the leverage factor, as mentioned earlier in our example. This formula indicates a fundamental relationship: our gains will not be necessary boosted even if the daily average performance µ is good; the cost of borrowing should also be low.
However, the higher the volatility, the more the gains are eroded, in proportion to the square of the lever factor. Therefore, it is better to use lever on indexes with a high average daily performance, low volatility, and a low borrowing cost.
In reality, lever should be applied to ETFs that maximize the ratio µ- r / σ². This ratio provides something more than the Sharpe ratio µ-r /𝜎 (which measures the gain µ – r relatively to the risk σ ) because it gives the best possible gain for a taken risk. This explains why leveraging individual stocks is a poor idea, even if µ – r might be higher; the volatility σ of a single stock is too great. It is better to focus on indices.
Lastly, it is important to note that holding a portfolio consisting of half an unleveraged index and half an index with 2x leverage will not simulate a performance equivalent to a 1.5x leverage. This is because the volatility drag term is proportional to the square of the leverage, not the leverage itself. In the previous case, we have multiplied the volatility drag by 2²+1²/2=2.5 for the 50-50 portfolio, whereas for a portfolio entirely with 1.5x leverage, it is 1.5²=2.25. The only way to achieve a true 1.5x leverage is to rebalance the portfolio daily to maintain the 1.5 ratio. This rebalancing incurs significant costs (such as brokerage fees and the difference between the bid and ask prices).
It is also observed that the leverage that maximizes Mg seems to be L =µ – r / σ². Thus, we conclude that the higher the volatility σ of a reference index, the poorer the performance (geometric mean) is. Finally, contrary to what I have read, there is no beneficial volatility drag, as the value of volatility drag is necessarily negative.
Normal Law hypothesis
Before looking at practical aspects of leveraged stock indices, we will check if our assumption of a normal distribution and the equality mentioned above are correct in practice. In the graph below, the borrowing cost r has been neglected and is set to 0. This approximation is justified because the goal of the graph is to validate the assumption of normality, not to estimate the gain of such an index. Ignoring the borrowing cost will not invalidate the following proof.
In the graph, the data for the S&P 500 price return (without reinvested dividends) has been considered, and a leverage of two (with borrowing cost r=0) has been applied. It's does not matter if the index taken is total or price return, indeed in this section the goal is to prove that normal distribution assumption is verify.
The averages shown in the graph are rolling averages over 10 years, meaning that the average for 1978 reflects the mean value between 1968 and 1978.
In blue is the beta slippage, given by
where, µ and σ are estimated by taking the rolling average over 10 years preceding the date. In gray it is the actual annualized average gain over 10 years, and in orange is the estimation using the formula.
In other word the gray curve is the real annualized gain over 10 years (without borrowing cost) find thanks to S&P500 data. The orange curve use the formula above to find the annualized gain. One can notice that both curves are closed which means that formula given is trustable. If the formula is trustable the hypothesis behind it are also trustable, thus the normal distribution assumption is a good one.
Dont get me wrong the normal distribution hypothesis is trustable for this formula of volatility drag, however for other formulas or analysis this assumption may not be trustable.
It is observed that the gray and orange curves are overlapping, which allows us to verify the accuracy of our assumptions and the theory behind them. We conclude that the normality assumption in the quantification of volatility drag is correct for a market such as the S&P 500. For other markets, this normality assumption may not be accurate and should be verified.
The equation
is therefore an accurate estimation of reality (neglected the borrowing cost r does not invalidate the normality assumption). One might then think the solution is simple, we only have to set the lever factor to L= µ- r / σ² to maximize our gains. Yes, this is true, if the future values of 𝜇, 𝜎 and 𝑟 for the S&P 500 are known. However, here is the graph showing the average 𝜇 and 𝜎 of the S&P 500 without leverage over rolling 10-year periods.
And here is the graph of the Federal Reserve interest rates, which r is closely related to, over the period from 1954 to 2024.
One can notice that predicting 𝜇, 𝜎 and 𝑟 in advance is challenging, despite apparent cycles. Therefore, even if the theory provides a good estimate of the optimal leverage one should have if 𝜇 𝜎 and 𝑟 were known, it cannot predict in advance the best lever factor over time.
Summary of the volatility drag
In summary, volatility drag is easy to estimate for a given period and is given by
volatility drag is also present when L=1. Therefore, volatility drag is not proof that a leveraged ETF (LETF) will necessarily lose value over time, as it is present in all stock indices. Finally, there is no beneficial volatility drag, contrary to what may be read, as the value is necessarily negative. The goal is to offset the negative volatility drag by increasing the average performance this average performance (Lµ-r) is reduced by the borrowing costs. We will explore their impact of borrowing cost on a LETF in the next section.
here it is for the first part
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