Leveraged ETFs and volatility decay revisited. With more math.

This is a follow-up to an earlier article about leveraged ETFs, where I discussed how volatility decay makes leveraged ETFs unsuitable for long-term investment. But it’s no secret that many leveraged ETFs have significantly outperformed their non-leveraged index fund counterparts in the past few years. Let’s take a deeper dive into this topic today.

Is it actually OK to hold leveraged ETFs?

The prevailing wisdom is that leveraged ETFs are unsuitable as long-term “buy and hold” investments, due to both high fees (expense ratios often exceed 1%) as well as volatility decay. Leveraged ETFs employ daily rebalancing, so while they do multiply the daily returns of an index, this does not equate to providing the same multiple of long-term returns, such as annual returns. In fact, the higher the leverage multiple, the worse the volatility decay becomes. My original article talks about this in more detail. Similar sentiments are echoed elsewhere online, such as here and here.

But there’s a growing body of evidence to suggest that this is a myth. I recently stumbled upon this fascinating paper by Tony Cooper titled Alpha Generation and Risk Smoothing Using Managed Volatility and related materials at Double Digit Numerics. I’ll lean heavily on Mr. Cooper’s work for the mathematics, as I didn’t do any of the math or come up with any of these arguments myself.

This paper points out that leverage is a continuum. The traditional definition of leverage is to borrow money in order to invest in the market. In this sense, you can make a binary distinction between no leverage (not borrowing money) versus using leverage (borrowing money).

A more precise definition of leverage, however, would be to define it as a ratio or multiple that describes how much exposure you have to the market compared to the total investable assets that you own.

Leverage = Amount invested in the market / Total investable assets owned

Therefore, if you have $100 to your name and invest all $100 into the market, your leverage ratio or multiple is 1 (although you are not using any leverage in a binary definition). For example, this can represent any traditional index fund or ETF investment held in a long position. If, on the other hand, you borrow another $100 and invest a total of $200 into the market, your leverage multiple is 2, making you 2x leveraged. And of course, if you have $100 and decide not to invest, your leverage multiple is 0, meaning you have no exposure to the market at all. Investing in a 2x leveraged ETF gives you 2x leverage in the market, which is, in theory, the same leverage and market exposure as borrowing money to reach 2x leverage instead.

To re-iterate, in traditional investing with zero leverage, the leverage multiple is not 0, but in fact 1x. And the kicker is that daily fluctuations will always hurt overall returns, regardless of how much leverage is used. We refer to this as volatility decay and the paper use the term volatility drag. But the point is that this occurs with any degree of market exposure, even without using leverage!

The math is simple. If the market goes down by x one day and up by the same x the next day, the net returns after the 2 days are given by:

Returns = (1 + x)*(1 - x) = 1 - x²

As the author points out, because x² is positive but the sign in front is negative, your position will always suffer a loss of x². The example in the paper is that the market goes down 5% one day, then up by 5%. The net result is (1-0.05)*(1+0.05)= 0.9975; a loss of 0.05², which is 0.0025 or 0.25%.

Thus, any daily loss requires a greater percentage daily gain in the future to return to original value. Increasing leverage increases the value of x, and therefore x², causing greater volatility decay. Decreasing leverage decreases the value of x, and therefore x², causing less volatility decay. However, volatility decay is not unique to leveraged ETFs, and occurs with any degree of market exposure.

The leverage ratio multiplies x. The only way to avoid volatility decay is to only invest in non-volatile assets (such as CDs) or to not invest in the market at all. Suppose we reduce leverage to 0.5x by leaving half of our money un-invested. Although we’ve reduced our market exposure by 1/2, we’ve reduced our susceptibility to volatility decay by 1/4! However, most investors would balk at this despite the promise of less volatility decay, as they would rightly point out that leaving half your money un-invested seems suboptimal!

If, on the other hand, we go beyond 1x leverage to say, 2x, we increase the effect of volatility decay by 4x, but also increase our market exposure. Yet most investors balk at any leverage > 1x because volatility decay is bad! For some reason, conventional wisdom has decided that employing no leverage, which is in fact a 1x multiple, might strike the optimal balance between volatility decay and capturing market returns. Yet there’s nothing special about a 1x multiple, nor any inherent reason that 1x leverage is “optimal”. The “optimal” amount of leverage, under historical market conditions, might be 0.8x, 1.2x or 1.5x or some such. It is very unlikely to be neatly 1x.

In fact, it is no secret that at least in the past decade, if an investor used a “buy and hold” strategy on many popular 2x or 3x leveraged ETFs, they would have significantly outperformed a 1x “buy and hold” strategy of the underlying index:

Of course, the past decade has also seen an almost uninterrupted bull run except for the brief market crash in 2020 from COVID. There’s no question that when the markets go up rapidly, leverage amplifies the gains. The argument against leveraged ETFs always boils down to underperformance when the markets are volatile.

The actual stock market, at least in the United States, experiences daily fluctuations, cyclic crashes and downturns which last from months to years, and overall positive returns over longer time horizons (decades). This is the general expectation for most investors and it also forms the foundation for long-term, buy and hold index investing.

In this type of market, leveraged ETFs may or may not outperform the index, depending on three factors:

The magnitude of daily volatility.
The speed and magnitude of overall market returns.
The leverage multiple (as multiple increases market exposure linearly but increases volatility decay exponentially).

So in the real world, leveraged ETFs behave thusly over any particular time period:

If volatility is low, leverage is not too high, or markets rise quickly, leveraged ETFs will outperform the index despite volatility decay.
If volatility is high, leverage is too high, or markets rise slowly, leveraged ETFs will not outperform the index due to volatility decay.

Again, in the past decade, the first scenario was realized: we’ve had essentially an uninterrupted bull run, with the S&P 500 achieving an annualized return of almost 15% from 2010 to 2021 with dividends reinvested. This was sufficient for 2x and 3x leveraged ETFs to outperform the index, and for 3x to outperform 2x. The real question is whether the past decade was an anomaly, or whether leveraged ETFs can continue this behavior under historical market conditions besides just the past decade. The other question is whether there is an “ideal” or “optimal” amount of leverage to use, given the actual historical behavior of the markets.

Figure 1 of the aforementioned paper is very telling. The author backtested leverage versus returns using 135 years (1885 to 2009) of historical U.S. market data, although note that this is without dividend reinvestment:

Image source: Cooper, Tony. Alpha Generation and Risk Smoothing Using Managed Volatility.

As you can see, returns increase as leverage increases, up to a point. The “optimal” amount of leverage to maximize returns, for this set of market conditions over this time period, was not 1x nor 3x, but in fact close to 2x. We also see that as the leverage continues to increase beyond 2x, the returns trail again, due to exponentially increasing volatility decay. But as the paper point out, “There is nothing magic about the leverage value 1. There is no mathematical reason for returns to suddenly level off at that leverage.” This is despite most of the sensible investing world espousing that any leverage (going beyond 1x) is bad.

After some mathematical analysis, the author states that the maximum leverage k is approximated by the following formula:

k = μ / σ²

Where μ is the mean daily return of the market or index, and σ is the volatility of the market or index (defined as the standard deviation of μ).

Therefore, we can see that if either μ is high, or if σ is low, the optimal leverage k will increase, and vice-versa. This remains consistent with our intuitive understanding of how leverage and volatility decay works.

The author’s comments in the next section are also quite illuminating: “Leveraged ETFs can be held long term provided the market has enough return to overcome volatility drag. It usually does. For most markets in recent times the optimal leverage is about 2. But some markets and time frames will reward a leverage of up to 3. No markets will reward a leverage of 4.” (Emphasis mine).

We can in fact fine-tune leverage to any reasonable multiple we desire. In order to achieve a leverage in between 1x and 2x (such as 1.5x) in the S&P 500, for example, we can invest some our assets into a S&P 500 index fund or ETF (at 1x leverage), and some of our assets into a 2x leveraged ETF such as SSO. In order to achieve leverage in between 2x and 3x, we can do the same using a combination of a 2x leveraged ETF such as SSO, and a 3x leveraged ETF such as UPRO. As far as I am aware, there are no leveraged ETFs that employ greater than 3x leverage. Exceeding this number would be unwise, but could be done through alternative methods (i.e., by using options, or through old-fashioned borrowing money).

The same is true if we wish to decrease leverage below 1x. We can decrease our exposure to the S&P 500 by partially investing our assets and holding the rest in cash. In reality, most investors are actually investing at < 1x leverage, because the entirety of their assets are not invested into the market. So, for the sake of both practicality as well as risk mitigation, the vast majority of passive, index fund investors are probably underleveraged in the market and their returns are far from optimal.

The rest of the paper goes on to look at the optimal leverage for some other historical markets apart from the U.S. stock market, and also explores some portfolio strategies that use volatility prediction. The discussion gets quite advanced and I’ll leave the rest to interested readers.

Does modeling stack up in the real world?

One aspect of leveraged ETFs that cause trepidation is a relative lack of data under real-world conditions. Most leveraged ETFs have only been around for the past 10 years, and while the March 2020 COVID market crash did provide a glimpse into their behavior during a market downturn and subsequent recovery, many people remain unconvinced about how they would behave in a “black-swan" event.

The oldest leveraged ETF in existence is the Profund UltraBull, ULPIX, introduced in 1997. It is a 2x leveraged S&P 500 ETF. It provides the most data that we have and it has not been kind to the investor since inception:

As you can see, from 1997 up until present day (August 2021), ULPIX continues to underperform the S&P 500, reinforcing the idea that even a 2x leveraged ETF should not be held long-term. The results speak for themselves. Lump-sum investors in 1997 would have been better off in the S&P 500.

This real-world data is hard to argue against. In defense of leveraged ETFs, however, I’ll present two counter-arguments. First, this time period was marked by the 2000 dot-com crash followed shortly by the 2007 - 2009 global financial crisis. A lump-sum investment into ULPIX at its inception in 1997 is just about the worst timing possible for a leveraged ETF. Any leveraged portfolio would take a beating from these back-to-back market crashes and quickly fall behind the index, requiring years of future market gains to catch back up.

If, on the other hand, an investor made ongoing investments into ULPIX versus the S&P 500 (for example, $100 per month) instead of a lump-sum in 1997, they started outperforming the S&P 500 by around 2016 all the way to present day (although the 2020 COVID crash almost brought the two portfolios back to parity). As of August 2021, the ULPIX portfolio has a significant advantage:

The reason for this is due to the interaction between ongoing cashflows and sequence of returns. The S&P 500 had better annualized returns (CAGR) than ULPIX from December 1997 to August 2021, but a different sequence of returns to get there. If this concept is unfamiliar, read my article on sequence of returns for a quick reference. Thus, ULPIX actually had better MWRR (money-weighted rate of return) with monthly contributions instead of lump-sum. I should point out that most investors actually invest this way rather than lump-sum, simply due to the periodic nature of their paychecks coming in.

Second, I’m not convinced that ULPIX does a good job of providing 2x returns. If we compare it to another 2x S&P 500 leveraged ETF, such as SSO, we see that ULPIX lags behind significantly (data only available since 2006 as that was the inception of SSO). I don’t know what’s going on “behind-the-scenes”, so to speak, but SSO is pulling further and further ahead of ULPIX. I should point out that ULPIX has a higher expense ratio than SSO (1.6% vs. 0.91%), but I haven’t done the math on whether this alone explains the difference.

This goes to show the inadequacy of merely simulating a leverage multiple, as real-world leveraged ETFs do not perfectly replicate a multiple of the index’s returns. I suspect that if SSO was conceived in 1997, it probably does beat the S&P 500 as of August 2021 even with lump-sum, albeit maybe only slightly. The main takeaways for me is that 1) modeling does not always translate into the real world, as the actual products are not perfect, 2) not all leveraged ETFs are created equal, and 3) lump-sum investing into a leveraged ETF is risky because your timing might be horrible.

Conclusion

Personally, I am not yet ready to engage in any highly leveraged strategies, although I am increasingly convinced that 1x leverage being somehow optimal is a myth. It appears that leveraged ETFs which do not have aggressive multiples can indeed be successfully held in the long-term, at least according to mathematical models. For the historical U.S. stock market, investing with a multiple of 1x is probably quite conservative, although the optimal leverage for any given time period fluctuates. For most time periods, it is probably somewhere in-between 1x and 2x, although for certain time periods (such as the past 10 years), it was as high as 3x. Unfortunately, the optimal leverage for the future is unknown, as future returns and future volatility is unknowable. However, investors can fine-tune their overall leverage multiple to any number they like, as described previously. A list of leveraged ETFs can be found here and elsewhere on the internet.

With all of that said, I think any investors using leverage need to be exceptionally careful. All too often, leverage has resulted in complete disaster for both individuals and institutions. We rarely make life decisions based on what is purely mathematically optimal, and using leverage to chase returns takes that concept to an extreme. Thanks for reading, and happy investing!

Leveraged ETFs and volatility decay revisited. With more math.