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Across a variety of sources, I often see the following argument (paraphrased):

Leveraged ETFs are a bad investment, because if the stock goes up x%, then down x%, an ETF leveraged n-times has a value of (1-nx)(1+nx) = 1-n^2x^2. So, increased leverage makes you hurt more from an (up,down) or (down,up) move.

While this argument (or something similar) is often given, it strikes me as incorrect, or at least improperly defended. Sure, Leveraged ETFs do worse when the market makes two moves in opposite directions, but it does better when the market makes two moves in the same direction. I.e. (1-nx)(1-nx) > (1-x)(1-x) and (1+nx)(1+nx) > (1+x)(1+x). In fact, the four cases (UP, UP), (UP, DOWN), (DOWN,UP), (DOWN,DOWN), when summed together, mutually cancel, so that leverage doesn't have an effect on the expected value. (**EDIT: I've now explained this at the bottom of the post, in a section labeled (EDIT))


But, stocks aren't symmetric and tend to increase over time, so the leverage actually works in your favor.

Running an analysis of the buy-and-hold strategy everyday from 2010 to now, TQQQ almost always outperforms the QQQ over long time horizons. enter image description here

With all of this in mind, I wonder what real arguments can be given against long holding leveraged ETFs. I'm especially interested in arguments that would apply to a risk neutral investor (obviously to a risk-averse investor that -81% as the worst return is very significant).

I'm by no means an experienced or knowledgeable trader, so let me know if I'm missing something obvious and significant in my highly simplified model.

(EDIT)

To explain this in more detail, assume we model the market in the following way.

  1. 50% chance the market goes up by 1%
  2. 50% chance the market goes down by 1%

Start with $100, and hold for two days. Then, the expected return is

  1. 25% chance of (UP,UP) = (1.01)^2 return = 1.0201
  2. 25% chance of (DOWN,UP) = (.99)(1.01) return = 0.9999
  3. 25% chance of (UP,DOWN) = (1.01)(.99) return = 0.9999
  4. 25% chance of (DOWN,DOWN) = (.99)(.99) = return = 0.9801

Each of these has a 1/4 chance of occuring, so add all these numbers together and then dividing by 4 gives 1. So, if we start with $100, we should expect to end with $100.

Now, let us generalize. Assume the market is symmetric, so something like this

  1. 1/8 chance for 10% return
  2. 1/8 chance for 2% return
  3. 1/4 chance for 1% return
  4. 1/4 chance for -1% return
  5. 1/8 chance for -2% return
  6. 1/8 chance for -10% return

Or even

  1. 1/8 chance for 100% return
  2. 1/8 chance for 10% return
  3. 1/4 chance for 1% return
  4. 1/2 chance for -28% return

Anything where the expected value for a single day is zero. Then the expected value over n periods is is also zero (irrespective of leverage). If the expected value over a single day is postive, then the expected value of multiple periods is also postive (and increases exponentially with leverage).

For instance, here is the expected value over n days for the following market

  1. 50% chance the market goes up by x%
  2. 50% chance the market goes down by x%

Then the expected return over n time periods is enter image description here


Comments on D Stanley's answer

Can you expand a bit more on what your view of risk is (especially, what does it mean to risk 3X the amount?). I'm not sure I see exactly why you are viewing the risk as, for lack of a better term, 'annualized'.

As you can see in the data, over a 1 yr time horizon, the annualized average returns are almost exactly 3X. Over longer time horizon, you are right that the annualized returns are less than 3X, I hadn't thought about looking at the annualized returns benchmark. Here is the data on that

enter image description here

However, I think thats not the right way to look at risk. Let me take an example. Let's take my baseline as investing $100 in QQQ. If I decide to invest 3x that money (i.e. $300) in QQQ, then I'd say my risk is 3x as much. All losses are multiplied by a factor of 3. In order to make this increased investment worth it, I should expect 3x as much returns. This is not the same as expecting 3x annualized returns. If I want to get 3x the return over 10 years (i.e. if were to make $20, then I want to make $60 instead), I need much less than a 3x annualized return.

For instance, over a 10 year time horizon, the average annualized returns of TQQQ are only about 2.5x QQQ. But, that corresponds to a ~10x return for TQQQ over QQQ in the 10 year period.

This is not to say I don't believe your statement that one is risking more than they expect to gain is ultimately true, but rather I don't see how your example of the annualized returns being less than 3x actually implies that the risk is 3x and the returns are less than 3x.

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  • The better argument is simply that leverage increases potential risk as much as it increases potential reward. (Maybe a bit more, since there's the cost of the loan to account for.) I'm the long term, that makes it a bad choice because you simply don't want an investment where your wealth may vanish at any moment
    – keshlam
    Commented Jun 3 at 14:12
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    I wonder, based on the title of the question, if you full appreciate what "risk" means in this context. Risk doesn't just mean "didn't make an optimal amount of money" or even "you lost some money" it can very easily mean you lose all of your money. The long time horizon only matters if you don't get wiped out in a downturn. Don't forget that "exogenous" life circumstances can wipe you out too, not just the investment going to zero. Commented Jun 4 at 13:44
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    Model your expected return after 100 periods for the 50-50 case. But don't look at the expected value, look at the chance of having a value >1. A lottery with an expected value > 1 still doesn't earn you money if you can't buy all tickets.
    – DonQuiKong
    Commented Jun 5 at 17:03
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    I suspect that if there is a good case against LETFs, it would come from the Kelly Criterion, not the marginal effects of vol decay. Love your question.
    – Fadeway
    Commented Jun 29 at 16:58
  • The statement (1-n x) (1- nx) > (1-x) (1-x) is not true in general but it is neither true for the specific situation envisioned in your question. In your case, 1/n > x>0 and n>1, so 1-n x < 1-x, which implies that (1- nx)^2 < (1-x)^2, in contrast to your claim. The leveraged ETF only does better when the market goes up twice. Commented Jul 1 at 21:38

6 Answers 6

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The Basic Problem:

Leveraged Exchange-Traded Funds (ETFs), like TQQQ, are designed to amplify the daily returns of an underlying index. In this case of TQQQ, the Nasdaq-100. However, it's crucial to understand that their performance can deviate significantly from the stated leverage over periods longer than a single day. This is explicitly mentioned on the ProShares TQQQ webpage, underscoring the importance of comprehending how these financial instruments operate.

That's why there is a warning on the main page of tqqq. enter image description here

The Volatility Trap:

The inherent volatility of leveraged ETFs often leads to returns that diverge markedly from their daily targets. As illustrated on the ProShares website,

"These differences may be significant. Smaller index gains/losses and higher index volatility contribute to returns worse than the Daily Target. Larger index gains/losses and lower index volatility contribute to returns better than the Daily Target."

This means that, over time, volatility can erode the value of these ETFs, a phenomenon known as volatility decay.

Real-World Examples:

Consider the period from February 2020 to January 2023: while the QQQ, which tracks the Nasdaq-100, appreciated by approximately 30.76%, the TQQQ, a 3x leveraged version of the same index, actually declined by 11.37% according to Yahoo Finance.

enter image description here

A similar pattern emerges from January 2018 to January 2023. During this time, the QQQ rose by 64%, whereas the TQQQ managed only a 16% gain.

enter image description here

The Long-Term Reality:

While it is theoretically possible to achieve substantial gains over very long periods, such as 20 or 30 years, the reality is that the last years have been statistical outliers. Moreover, many investors have shorter time horizons for their investments. For instance, individuals might be saving for a house or university expenses within a few years. Imagine a scenario where you invested your hard-earned savings in the TQQQ in mid-2018 with the intention of purchasing a house. Unfortunately, by the end of that year, your investment could have plummeted by 55%.

enter image description here

The situation was even more dire in 2022, with TQQQ falling by 80%. enter image description here

Understanding Volatility Decay:

The concept of volatility decay is well-illustrated with a basic example. Suppose the underlying index loses 10% on the first day and then gains 12% on the second. While this results in a net positive for the index, the TQQQ, due to its leveraged nature, would suffer a significant loss. You can explore a detailed two-day horizon example here.

The example shows a nice gain on QQQ, but a hefty loss on TQQQ. enter image description here

Risk-Adjusted Returns:

From a risk-adjusted returns perspective, the QQQ consistently outperforms the TQQQ when evaluated using standard financial metrics. This is corroborated by data from Portfolioslab.com

enter image description here

This disparity in performance is a significant reason for the cautionary statements on the main page of leveraged ETF providers and in financial news outlets like Bloomberg: SEC may regret the day it allowed leveraged ETFs.

The Bull Market Effect:

As mentioned, it's important to note that the impressive returns seen in TQQQ over recent years have been largely driven by an exceptional bull market (bar Covid-19), particularly in the technology sector. This period of strong growth may not be sustainable, and the unique conditions that have favored leveraged ETFs may not persist indefinitely. Drawing a conclusion based on this period is akin to asserting that Alaska isn't particularly cold after visiting in August.

Historical Context and Future Prospects:

Comparing the Nasdaq-100 to the more established S&P 500 (SPX), which has a longer historical record, reveals the volatility and risks associated with leveraged investments. The SPX data, downloaded from Bloomberg into Julia, highlights the variations in returns over different time spans.

enter image description here

Examining logarithmic prices over extended periods provides a clearer view of these variations and underscores the misleading nature of exponential growth over long horizons.

enter image description here

I computed the daily returns of the index, as well as several "leveraged ETFs" (simply assuming it is an exact daily leverage).Here are some time spans. The one you reference looks very good, for the reasons I mentioned above.

enter image description here

Looking at the time span I provided in my first example you see enter image description here

Now, assume we have 1930, and you dump a large sum you inherited from your wealthy grandparents into a leveraged ETF on SPX. 48 years later, you look at the following returns.

enter image description here

Now, assume you start in 1929, and hold all the way to 2024. That's almost a century!

enter image description here

Holding a 3x leveraged ETF would have given you similar overall returns compared to just holding SPX, which was a lot less volatile. Therefore, risk adjusted returns for SPX are significantly better. Holding a 5x leveraged ETF would mea you would have missed out on a LOT of upside. Now, that is actual SPX data, but QQQ is a lot more volatile. So Vol Decay is even more pronounced.

  • SPX lost ~ 20% in 2022. enter image description here


  • QQQ lost ~33% enter image description here

Optimal Leverage and Risk Management:

Calculating returns for various leveraged ETFs over different time spans reveals that there is ougth to be an optimal level of leverage, particularly in volatile markets. This optimal leverage can be less than one, especially when volatility is high and average returns are low. Since this SE doesnt allow mathjax, I am just adding a screenshot of the code.

enter image description here

In general, the more volatile, and the less the average return.

enter image description here

Conclusion:

In summary, volatility decay is a critical issue for investors holding leveraged ETFs over extended periods. Without consistent upward trends and low volatility, these investments can result in substantial losses or subpar returns. While the past decade has seen favorable conditions for such instruments, relying on this trend to continue indefinitely is a risky proposition. Investing in leveraged ETFs requires careful consideration of the associated risks and a thorough understanding of market dynamics.

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  • 3
    As I mentioned in the post, I largely don't buy the volatility decay argument as being relevant, at least not a posed. As I mentioned, an easy mathematical argument shows that in a symmetric random walk, the effect of volatility decay is precisely canceled by the increased compounding returns from consecutive up days, and the decreased loss from consecutive down days. Using the historical data over TQQQ's lifespan (testing buy-and-hold on each day, starting from 2010), it averages a return ABOVE the 3x benchmark over essentially every timeframe one holds it for. Commented Jun 3 at 21:52
  • The high drawdown I take as a good point. Thanks for sharing the link, I'll look more into those risk-adjusted performance ratios you shared to see how they quantify the risk. Commented Jun 3 at 21:55
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    As mentioned, 2010 until now was a very unusual time that favoured the performance of TQQQ in all aspects. Vol decay is really the only argument against these ETFs. The large drawdowns are in fact a direct result of it as well.
    – AKdemy
    Commented Jun 3 at 22:14
  • 1
    Total layman here, but obviously, with respect to charts 1 and 2, many time periods (in particular, all that end in 2021 or before April 2022) show a much better performance for TQQQ. Both indexes show largely parallel developments, with the leveraged one at larger amplitude, unsurprisingly. Consequently, the difference between buying and selling point will likely be larger with TQQQ (either way). All that is to be expected. What exactly is your point? If you want low volatility, don't leverage? Agreed. Commented Jun 4 at 20:58
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    @Peter-ReinstateMonica No, that's not the only thing going on. Getting personal leverage & investing with it, is not the same as investing in TQQQ, which rebalances daily, and therefore warps the returns in a way you might not expect [eg: suffer a large loss, and a replicating re-gain does not actually make you 'whole', because after the large loss, TQQQ rebalances, compared with personal leverage]. Commented Jul 11 at 17:29
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The main argument against is that when things go bad, they go really bad, and bad days are much harder to recover from.

Suppose the Nasdaq drops 5% in one day. That means TQQQ would drop 15%. Now you might think that a 5.26% gain (1/0.95) in QQQ the next day that brings QQQ back to even would also cause TQQQ to fully recover, but that's not the case. A 5.26% gain in QQQ would result (ideally) in a 15.78% gain in TQQQ, which is not enough to recover from the 15% loss. You'll end up with about a 1.5% loss overall for the two days. That might not seem too bad, but it times of high volatility this drag compounds, resulting in worse performance than you'd expect.

You can see this in your data if you annualize the average return over each horizon - the average annual return for TQQQ in each case in less than 3X the benchmark.

The feature that the leverage is designed to be applied daily means that losses hurt worse over the long run, and the investment is not expected to earn exactly 3X the baseline over the long run. Bad days will drag it down more than equivalent good days will bring it up.

So financially you're getting less than 3X return in exchange for 3X the risk.

I don't see how your example of the annualized returns being less than 3x actually implies that the risk is 3x and the returns are less than 3x.

In finance, "risk" is typically defined as "volatility", or "how much can my investment change in value". When you use leverage (say X:1), by definition you multiply expected returns by X and the risk by X (the investment changes by X times the benchmark by definition. But these investments are designed to multiply daily returns, which means that they change their composition so that the return for one day is 3X the benchmark, which, as shown, means that it's harder to recover from losses.

To be fair, it may also mean that the risk is slightly lower than 3X as well, but I'm not as certain in the math to try and quantify it.

Take the 1-year worst return (-81%) as an example. I'd be curious to see how long it took TQQQ to recover from that loss versus QQQ. I would bet it took significantly longer, but after it did recover it performed about 3X better than QQQ. Note that in the worst cases, you had to stick it out with TQQQ for over 5 years to catch back up to QQQ.

Instead, I would go back to my main point of the danger of these investments is that when times are good, they are really good, but when times are bad they are really bad. They're not bad investments, per se, but one needs to completely understand the risks involved and have proper risk management strategies in place to avoid catastrophic losses.

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  • I've written a message that was a bit too long for a comment as an edit to my question. Commented Jun 3 at 18:49
  • "A 5.26% gain in QQQ would result (ideally) in a 15.78% gain in QQQ," Missing a T from the second QQQ? Commented Jul 11 at 4:19
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TL;DR: the simple argument for "don't hold leveraged ETFs for multiple periods" is not really correct, but your argument that it's not correct is flawed to the point that it's more or less wrong.

In fact, the four cases (UP, UP), (UP, DOWN), (DOWN,UP), (DOWN,DOWN), when summed together, mutually cancel, so that leverage doesn't have an effect on the expected value.

This is absolutely true. But it's also irrelevant to the strategy you propose of buy-and-hold. The arithmetic expected value of returns from a 2 day holding period is only completely relevant if the strategy is something like: "On day 0, I will buy $10k worth of TQQQ. On day 2, if it's worth more than $10k, I'll sell down to $10k; if it's worth less than $10k, I'll buy enough to bring me back to $10k". The more the actual strategy differs from this, the less relevant the arithmetic expected value is (note that any time you didn't have the money to buy enough to get you to $10k would be a major deviation from the strategy ("major" in the sense that not being able to buy a dip will remove a lot of your final return)).

In the case of buy-and-hold for multiple periods, the relevant expected value is the logarithmic expected value. Thanks to the magic ability of logarithms to "multiply by adding", we don't even really have to look at the 2-day periods. Let's say that QQQ boils down to a daily coin-flip: heads it goes up 2.6% (a little bit more than the daily CGR for its best year in your sample), tails it goes down 0.1% (rounding up the daily CGR for its worst year in your sample). Then the expected log for QQQ is (log(1.026) + log(0.999))/2, or (after exponentiating back) +1.24%. For TQQQ, the expected log would be (log(1.078) + log(0.997)) / 2, or (again after exponentiating back) +3.67%.

3.67% is still a dramatically greater daily CGR than 1.24%. Who cares that it's not quite 3x? Typically this is when quant-finance folks would introduce something about volatility (and define risk as volatility...). But I'm going to get more philosophical.

Why limit us to 3x? Why not 400x or 800x?

The daily CGR of 400x would be +162%. But the daily CGR of 800x would, oddly enough, only be 109% (doubling the leverage reduced our compound growth rate?). At 960x, the daily CGR becomes +1.90% (which isn't much greater than the CGR with 1x leverage). At 961x the daily CGR becomes +0.67%. And at 962x leverage, we get the curious result that the daily CGR is negative (-0.58%).

Clearly, then, there's a point beyond which increasing leverage decreases our compound growth rate. In the "heads +2.6%, tails -0.1%" situation, it's exactly like a casino offering you 26:1 odds on the coin-flip. You get to choose how much to bet. The log return gets maximized if you wager about 48% of your wealth on every flip (or 480x leverage: it's convenient (and not intentional...) that the max daily loss is a power of 10), and at twice that, the log return is negative (in which case, it is a near certainty thatif you play this game long enough betting 96% of your wealth every flip, you will go broke (assuming there's a minimum bet you can make) making a sequence of absurdly high-EV bets!).

Perhaps more realistically, we can model the post-2010 QQQ daily return as being something like (with fatter tails than were observed; the 1/16 chances are best and worst post-2010 daily returns ignoring 2020)

  • 1/16 chance of +7.4%
  • 7/16 chance of +2.6%
  • 7/16 chance of -0.1%
  • 1/16 chance of -6.9%

With that profile, 9x leverage is about where increasing leverage decreases the rate of compounding (and 15x is where ruin is assured). Assuming that this profile is accurate, TQQQ probably will compound faster than QQQ.

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  • Great answer, I'll have to think about this one. My first impression is that it shouldn't have such a huge impact except over significant time horizons, since locally ln(1+x)≈x, but I'd like to do an analysis of how much the expected loss is later. Commented Jun 4 at 19:57
  • On second blush, I've written my question in a confusing way. My argument is not that the arithmetic mean upon reinvestment cancels out, my argument is that the expected value actually also cancels to zero.This correspond to investing by buying at the first time frame, and then holding.Here is the computation for 2 days (notice it never involves reinvesting), with an x% gain, and n times leverage (UP,UP) = (1+nx)(1+nx) = 1+2nx + n^2x^2 (DOWN,UP),(UP,DOWN) = (1-nx)(1+nx) = 1-n^2x^2 (DOWN,DOWN) = (1-nx)(1-nx) = 1-2nx+n^2x^2 Each has a 1/4 chance of occurring and summed together they equal 1 Commented Jun 4 at 20:06
  • The calculation you have around the coin toss is incorrect. The point where the expected returns are zero comes when the odds look like this 1. 50% chance of +100% 2. 50% chance of -100% (i.e. lose all money) Now, an argument can be made (especially in a case like this), that expected value is a bad measurement. I.e. after 20 iterations of this game, 1,048,575 people end up with 0$, and one guy ends up with $1,048,576, so that on average everyone has $1, but I'd surely never take that bet. Commented Jun 4 at 20:48
  • Holding the investment from one period to the next is reinvesting it (ignoring things like tax implications for gains and transaction fees). As noted, the arithmetic EV only really matters to the extent that a) a positive arithmetic EV is a precondition for a positive log EV or b) you are not aiming for any compounding (in which case, the compound growth rate will asymptotically approach zero). Commented Jun 4 at 21:13
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    @CalebBriggs I think you will see why the compound growth rate is relevant if you consider the long-term behavior, when the number of rounds goes to infinity. For example, look at the symmetric case, starting with a value of 1 and with a 1/2 chance to increase or decrease by 10% every day. As you calculate, the expected value at each day is exactly 1. But if you simulate this, you will invariably see that the value WILL go to zero as time goes on. (Indeed this is a sequence of random variables, each of which has expected value 1, but which still converges to 0 almost surely!)
    – hgmath
    Commented Jun 5 at 10:45
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Imagine you hold the underlying fund and it goes up 8% and then down 7%. If you do the math, you'll see that you will be up .44%.

Now imagine you hold instead a 3X leveraged fund. When the underlyuing goes up 8%, you go up 24%. When the underlying goes down 7%, you go down 21%. Going up 24% and then down 21% leaves you down 2%.

So if the underlying is volatile, it can go up 0.44% and you can be down 2%. If this happens 20 times, the underlying will be up 9% but you will be down 33%.

The problem with holding leveraged funds long term is that if the underlying is volatile, even if it gradually wanders up, the leveraged fund can crash downwards. To hold a leveraged fund over a long period of time (without rebalancing), you must not only accurately predict the direction of the underlying but also the volatility and timing of its movements.

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Leveraged ETFs follow a buy high, sell low strategy.

Let's say the fund starts with $100 and 3x leverage. To do this, the fund effectively borrows $200 to behave as if it had $300 invested.

The stock market goes up 8% on the first day. So $300 becomes $324. The fund is now worth $124. To maintain the 3x leverage, the fund must borrow an additional $48 so that 3 * $124 = $372 is invested the next day. => Buy high

On the second day, the index falls 7%. So $372 becomes $345.96. The fund is worth $97.96 because $248 was borrowed. Now the fund is overleveraged. To adjust for the next day, $52.08 of the invested capital must be repaid, leaving only 3 * $97.96 = $293.88 invested. => Sell low

This is my intuitive understanding of why the volatility tax exists.

0

Each of these has a 1/4 chance of occuring, so add all these numbers together and then dividing by 4 gives 1.

That's the arithmetic mean. Investing generally uses the geometric mean, and the market is generally modelled as having a lognormal distribution, in which the log return, not the return, is symmetric.

In your model, as time goes on, the expected value gets concentrated more and more into the "market always goes up" scenario; you have an extremely unlikely, but highly profitable, scenario that bring the arithmetic mean up, while the geometric mean, median, and mode all sink lower and lower. Suppose you have $100k to invest, and I tell you that I have an investment opportunity that has a 99.99% chance of losing all your money, and a 0.01% chance of getting you a billion dollars. Is that a wise retirement plan?

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