Are 3x leveraged ETFs the long-term winning strategy?

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OohLaLa
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by OohLaLa »

tradri wrote: Fri Mar 26, 2021 11:50 am Sorry, I am not that familiar with the culture of this forum.

How should I respond to individual posts instead?
When you use the quote function, you can:
- simply trim the quoted paragraphs, to only have the specific portions you are responding to.
- remove entire quote blocks if they are not necessary (ex: I quote you just now, but I removed the previous quote that you quoted yourself).
Hydromod
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by Hydromod »

JackoC wrote: Fri Mar 26, 2021 11:13 am 2. Second one first, the borrowing cost is not included in the ER. So we'd be confusing ourselves to speak of higher ER when rates are higher. The drag of financing (taken in isolation) is higher when rates are higher, but the ER is presumably constant (or falling/rising with actual costs and competitive pressures on the fund sponsor, not directly related to rates).

1. You're missing my explanation of why it is absolutely incorrect to assume your return being long 3X fund does not include the *implied* financing cost on L-1, 2 times for a 3X fund, as well as the dividend portion of return on all 3X, on every dollar of the fund, regardless of whether the fund sponsor runs a -3X fund. A 3X fund without a -3X sister does a total return swap (TRS) with a derivatives dealer where the dealer pays the fund 'Total Return of S&P,' fund pays the dealer 'short term borrowing index+spread', for 3 times the notional of the fund. These funds rarely if ever actually borrow to buy cash stocks. Now, a fund with a -3X counterpart can save a little *bid offer* cost by not having to do two TRS in opposite directions for the whole amount. But the *pricing* of the TRS and fund still reflects the implied borrowing cost, -(L-1)*r to investors in the 3X fund, +(L-1)*r to investors in the -3X fund* and dividends 3X to the 3X holder, -3X to the -3X holder. If either fund's return deviated consistently in one direction from the abritrage free price which includes implied financing cost and implied dividends, you would make free money holding/shorting one fund or other v long/short the cash stock basket (or the futures whose pricing manifestly closely obeys the same arbitrage condition at all times).

If you can't see this in looking at particular historical sequences of returns, it's noise confusing the picture. It's harder to see in noisy data when rates are well below 1% but it's still there. It would be blindly obvious at rates of several %. This is Derivatives Pricing 101 not an opinion or theory.

*of course that's shortcutting. The TRS is 3*(S&P Total Return vs. short term borrowing index+spread), then there's the return on the 1X in cash the investors put in the fund, so -(L-1)*r net assuming the fund invests the cash at the same r as the TRS. The r embedded in the TRS price is probably a little higher in practice.
I will be the first to say that I know nothing about how the financing aspects work. So I plead ignorance on these aspects. Be easy on me :)

The whole reason I brought this up in the first place is because I did a little test of holding 50/50 UPRO/SPXU equivalents, using 3x/-3x GSPC as a proxy. Unrelated to this discussion. You are absolutely right, holding this basket would have made a bunch of money back in the day, with or without ER, with or without the borrowing costs in the formulas. This was not seen in the actual UPRO/SPXU basket starting in 2009 even though the 3x/-3x GSPC equivalent says it should be there. This has been puzzling me greatly.

I agree that the market should not let such arbitrage occur, and it didn't with UPRO/SPXU.

So what I'm trying to get at is that the actual behavior of the LETFs must differ somehow from the model in order to take out the arbitrage effect.
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tradri
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

DMoogle wrote: Fri Mar 26, 2021 8:35 am
tradri wrote: Fri Mar 26, 2021 7:33 amI had a look at the Wikipedia article of the Kelly Criterion, but from I understood, it mostly applies to situations where you either gain or lose money. Since (I think) we all assume that "stonks only go up" in the long-run, I don't see how I can calculate the probability of losing money. It's not like I have to realize my losses when investing in a leveraged ETF, so I should be able to sit out any market crash.
This is not a correct interpretation of the Kelly Criterion. It applies to any situation with positive expected value and risk. Doesn't matter if it's a stock, or you're betting on heads on a weighted coin that wins 51% of the time. Either way, you have a limited lifespan, and if you take on too much risk, there is a point where a more conservative strategy is straight up more likely (median, not mean) to come out ahead of a max-bet-everytime strategy (e.g. huge leverage). Quickly googling, this article seems to address how the concept can be applied to stocks: https://blogs.cfainstitute.org/investor ... alf-of-it/. Here's a Bogleheads thread I found on how it applies to leveraged portfolios: viewtopic.php?t=273154.

One another topic, it seems like one of the biggest issues we keep bumping into here is which leveraged ETF backtest is accurate. I haven't had a chance to read through the links you posted (yet), but the claim that 5.8x leverage is optimal likely only applies if there are no (or minimal) borrowing costs. In fact, it seems like MANY of the theoretically leveraged ETF "backtests" that have been posted are really flawed simulations that don't adequately account for historical borrowing costs.
There is a lot of talk here that higher interest rates would negatively impact the efficiency of such funds.

But we did have LIBOR rates of 1-3% a couple of years back, and the negative impact on leveraged ETFs can't really be seen, from what I can tell.

Something seems off to me...
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OohLaLa
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by OohLaLa »

I think veering into talks about borrowing rates is not a good idea in this case. There are fundamental misconceptions and lack of knowledge that have been pointed out and should be resolved first, as they are much bigger obstacles to success here.
Hydromod
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by Hydromod »

OohLaLa wrote: Fri Mar 26, 2021 12:19 pm I think veering into talks about borrowing rates is not a good idea in this case. There are fundamental misconceptions and lack of knowledge that have been pointed out and should be resolved first, as they are much bigger obstacles to success here.
Probably that would be better done in the simulating leveraged funds thread, I agree.
DMoogle
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by DMoogle »

Hydromod wrote: Fri Mar 26, 2021 12:24 pm
OohLaLa wrote: Fri Mar 26, 2021 12:19 pm I think veering into talks about borrowing rates is not a good idea in this case. There are fundamental misconceptions and lack of knowledge that have been pointed out and should be resolved first, as they are much bigger obstacles to success here.
Probably that would be better done in the simulating leveraged funds thread, I agree.
I'm reading through that thread and might revive it afterward, but being able to provide an accurate simulation of the past hypothetical performance of a UPRO-like instrument seems like the crux of the discussion.

Some simulations show it killed, some simulations show it barely outperformed. I don't think we'll ever reach a mutual agreement until we can agree on a "correct" hypothetical backtest.
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by JackoC »

Hydromod wrote: Fri Mar 26, 2021 12:03 pm
JackoC wrote: Fri Mar 26, 2021 11:13 am 2. Second one first, the borrowing cost is not included in the ER. So we'd be confusing ourselves to speak of higher ER when rates are higher. The drag of financing (taken in isolation) is higher when rates are higher, but the ER is presumably constant (or falling/rising with actual costs and competitive pressures on the fund sponsor, not directly related to rates).

1. You're missing my explanation of why it is absolutely incorrect to assume your return being long 3X fund does not include the *implied* financing cost on L-1, 2 times for a 3X fund, as well as the dividend portion of return on all 3X, on every dollar of the fund, regardless of whether the fund sponsor runs a -3X fund.

If you can't see this in looking at particular historical sequences of returns, it's noise confusing the picture. It's harder to see in noisy data when rates are well below 1% but it's still there. It would be blindly obvious at rates of several %. This is Derivatives Pricing 101 not an opinion or theory.
I will be the first to say that I know nothing about how the financing aspects work. So I plead ignorance on these aspects. Be easy on me :)

The whole reason I brought this up in the first place is because I did a little test of holding 50/50 UPRO/SPXU equivalents, using 3x/-3x GSPC as a proxy. Unrelated to this discussion. You are absolutely right, holding this basket would have made a bunch of money back in the day, with or without ER, with or without the borrowing costs in the formulas. This was not seen in the actual UPRO/SPXU basket starting in 2009 even though the 3x/-3x GSPC equivalent says it should be there. This has been puzzling me greatly.

I agree that the market should not let such arbitrage occur, and it didn't with UPRO/SPXU.

So what I'm trying to get at is that the actual behavior of the LETFs must differ somehow from the model in order to take out the arbitrage effect.
The formula given above is correct over any short period, LETF E[r]=L*mu-(L-1)r-0.5*L(L-1)*vol^2-ER/L. I've called this 'theory' but it's 'theory' like Newton's theories of motion: manifestly true under given realistic assumptions. If you look at any even few day period the realized mu and vol (by which the fund rebalances itself daily) is all over the map. So that's a serious issue tying in past results day by day with the *average* mu and vol over the period. But looking forward and not having any idea what the every zig and zag and sequence of mu and vol will be, we're back to the formula as to best prediction using today's expected mu and vol. And r. It is absolutely in there. Again if we pull up right now's Jun S&P futures price, detailed info on basically wholly known dividend payments of all S&P stocks from now to the Jun contract date, as well as their prices we'll see an implied financing cost of around 0.5% is also built into that futures price. Because there are people jumping on the arbitrage any time it gets even *slightly* out of line with that. Same they'd do with 3X LETF's if they consistently gave return that didn't include finance drag.

Again though the other issue looking forward is that we don't necessarily expect stock expected return to be constant if rates go up. The more conventional assumption would be mu-r is the thing that tends more to be constant on average than mu. So there isn't actually a reason to think these funds will do worse with higher r. They'll do badly with lower mu-r unless vol is lower, so again as in my example if expected mu=5%, r=0.5%, vol ~19%, E[r] of 3X is 3.09% v 4.97% for 1X. Why would anyone do that, given the huge std dev of return of 3X on top? If they believed expected mu-r is much higher, and/or vol would be considerably lower. In 2009>now there have been periods of especially high mu-r and sometimes historic low VIX. Replicate that and 3X will do great. That's what to focus on not r per se, but the first paragraph is again guaranteed.

Also I don't know about you but I've made mistakes sometimes backtesting stuff. It would be important to get to the bottom of it if my foregoing statements were not true. The fate of 3X v 1X depends very heavily on expected ERP and vol. Looking at what did happen is even less relevant for 3X than 1X (where it's given *way* too much attention here IMO) because vol is directly affecting return.
Last edited by JackoC on Fri Mar 26, 2021 12:55 pm, edited 1 time in total.
DMoogle
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by DMoogle »

JackoC wrote: Fri Mar 26, 2021 10:29 am Differences in perceived expected return are what make a market. :happy However back on the theoretical plane there's no fundamental reason to think stock total expected return is constant and so the ERP simply goes up when interest rates are low and down when they are high. The conventional assumption is that the ERP is a function of overall market risk appetite which tends to be persistent so it's the ERP that tends more toward being constant than the absolute return. IOW assuming the expected ERP now is the same as past is the more conventional assumption. Assuming it's higher is an out to lunch assumption IMO, but to each his own. Assuming it's lower (than past) is the more likely IMO because one would figure that central banks creating a 'TINA' (there is no alternative) attraction for stocks by making 'riskless' rates so low are compressing risk premia. There's some evidence of that if you look at similar more directly visible risk premia like risky bond spreads. Also I didn't pull 5% nominal expected return of the S&P entirely out of the air. There's a fundamental relationship between earnings yield and expected real return under somewhat realistic assumptions, 1/CAPE is a more stable expression of that than 1/spot PE, and 1/CAPE is sub 3% now, add back inflation expectation of 2% and you get around 5% nominal expected return. But mainly I'm illustrating the theory of leveraged return v expected return and expected vol. You can plug in whatever you want for either one, doesn't depend on me thinking it's realistic. :happy
You make good points, and honestly I don't have a lot to add, except I think that CAPE might not as be as good of a predictor as it has been in the past, due to a fundamental shift in business models at a macro level. Specifically, I'm referring to tech companies focusing on customer base growth over revenue and profitability. I believe it's a relatively new concept/trend that's at least partially responsible for the compression of P/E.

One other thing I've been meaning to bring up, totally unrelated: a LOT of people come into these threads and ask "is this really in line with Bogleheads philosophy"? And honestly, I think it's tough to argue that it is. The problem is, IMO, is that this is the probably the largest and best forum for finance enthusiasts on the internet. There's a lot of really smart, knowledgeable people here.

Not to say smart folks don't exist elsewhere, but there generally isn't as much organization. I'm glad the admins have been flexible to allow this kind of discussion.
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tradri
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

DMoogle wrote: Fri Mar 26, 2021 8:35 am
tradri wrote: Fri Mar 26, 2021 7:33 amI had a look at the Wikipedia article of the Kelly Criterion, but from I understood, it mostly applies to situations where you either gain or lose money. Since (I think) we all assume that "stonks only go up" in the long-run, I don't see how I can calculate the probability of losing money. It's not like I have to realize my losses when investing in a leveraged ETF, so I should be able to sit out any market crash.
This is not a correct interpretation of the Kelly Criterion. It applies to any situation with positive expected value and risk. Doesn't matter if it's a stock, or you're betting on heads on a weighted coin that wins 51% of the time. Either way, you have a limited lifespan, and if you take on too much risk, there is a point where a more conservative strategy is straight up more likely (median, not mean) to come out ahead of a max-bet-everytime strategy (e.g. huge leverage). Quickly googling, this article seems to address how the concept can be applied to stocks: https://blogs.cfainstitute.org/investor ... alf-of-it/. Here's a Bogleheads thread I found on how it applies to leveraged portfolios: viewtopic.php?t=273154.

One another topic, it seems like one of the biggest issues we keep bumping into here is which leveraged ETF backtest is accurate. I haven't had a chance to read through the links you posted (yet), but the claim that 5.8x leverage is optimal likely only applies if there are no (or minimal) borrowing costs. In fact, it seems like MANY of the theoretically leveraged ETF "backtests" that have been posted are really flawed simulations that don't adequately account for historical borrowing costs.
Correct me if I'm wrong, but as far as I understand it, the Kelly Criterion is about the strategy that will maximize your return in the long-run.

Since the variance in returns should decrease the longer you hold onto an investment, the longer the time-frame you are planning to hold your investment, the more you can afford to increase volatility, as that volatility shouldn't matter that much in the long-run.

Of course, investing in something so volatile that you wouldn't recover in 50 or 100 years from it makes little sense, even for the investor playing the longest game possible. The 5.8x optimum is only theoretical and assuming the fund's expenses remained the same over the whole time period.

While additional costs surely would massively hurt leveraged ETF returns, I don't see the effect of borrowing costs when comparing the simulations(without borrowing costs) and the actual performance of the leveraged ETF, even though LIBOR rates were higher than 0% over the last 10 years.
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

watchnerd wrote: Fri Mar 26, 2021 9:20 am
tradri wrote: Fri Mar 26, 2021 7:33 am I can't think of a logical argument why stock market returns should decrease in the future.
In the near-medium term?

Valuations.

Medium-long term?

'Meh' GDP growth, aging societies, declining fertility.
Immigration?
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

Jacotus wrote: Fri Mar 26, 2021 9:38 am tradri, it seems to me that implicit in all of your posts is that you equate "risk" with "volatility", and that you hold the belief that there is truly no risk in 3x leveraged ETFs, as long as you hold for long enough time.

You must disabuse yourself of this notion.
Can you please explain to me the other risks of holding leveraged ETFs other than volatility?
Do you think these funds are "less safe"? If so, why?
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

Impatience wrote: Fri Mar 26, 2021 9:45 am No matter how amazing leverage looks on paper, the behavioral impact of holding it day to day is just too hard for most people. The deep swings, the long stretches of underperformance, it creates fear and invites you to either tinker or bail out. Makes it very difficult for your real performance to match the math. Just look in the Hedgefundie thread every time the market tanks and you’ll see page upon page of people questioning the approach, trying to make it into an active trading strategy, etc.
That's true. I am assuming that investors aren't emotional for the sake of argument. :wink:
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by JackoC »

DMoogle wrote: Fri Mar 26, 2021 12:52 pm
JackoC wrote: Fri Mar 26, 2021 10:29 am There's a fundamental relationship between earnings yield and expected real return under somewhat realistic assumptions, 1/CAPE is a more stable expression of that than 1/spot PE, and 1/CAPE is sub 3% now, add back inflation expectation of 2% and you get around 5% nominal expected return. But mainly I'm illustrating the theory of leveraged return v expected return and expected vol. You can plug in whatever you want for either one, doesn't depend on me thinking it's realistic. :happy
You make good points, and honestly I don't have a lot to add, except I think that CAPE might not as be as good of a predictor as it has been in the past, due to a fundamental shift in business models at a macro level. Specifically, I'm referring to tech companies focusing on customer base growth over revenue and profitability. I believe it's a relatively new concept/trend that's at least partially responsible for the compression of P/E.
You're talking about CAPE as predictor of future valuation (high CAPE now means lower CAPE later). I'm talking about earnings yield's fundamental relationship to expected return at constant valuation. The latter has nothing directly to do with business models etc. and is harder to argue with as estimate, see derivation*. It just assumes companies reinvest at their cost of capital and inflation corrections for balance sheet items tend to average out (research shows that's not so unrealistic). No reference is being made to previous or future CAPE's, 'mean reversion', etc. which is where differences in nature of companies and accounting enter in.

*
https://www.investopedia.com/articles/04/012104.asp
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by OohLaLa »

DMoogle wrote: Fri Mar 26, 2021 12:52 pm [...]
One other thing I've been meaning to bring up, totally unrelated: a LOT of people come into these threads and ask "is this really in line with Bogleheads philosophy"? And honestly, I think it's tough to argue that it is. The problem is, IMO, is that this is the probably the largest and best forum for finance enthusiasts on the internet. There's a lot of really smart, knowledgeable people here.

Not to say smart folks don't exist elsewhere, but there generally isn't as much organization. I'm glad the admins have been flexible to allow this kind of discussion.
I don't think a fixed AA with leveraged versions of common index and bond funds is necessarily diametrically opposed to the BH philosophy, but I admit the thread often veered into more "creative" versions, sometimes going into pure market timing (which I do not apply).

I definitely agree with your assessment of the BH forum membership :sharebeer ; I haven't joined an online forum for something like a decade, before BH, and it's precisely this concentration of people that got me from year-long lurker to poster. The HFEA thread is the black sheep of the forum, without a doubt, but it has helped me (and others, I imagine) immensely... and I am not talking about $$$ but overall investment knowledge.
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tradri
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

Hydromod wrote: Fri Mar 26, 2021 9:55 am All of these calculations related to borrowing assume that each LETF is being run in isolation.

The providers run positive and inverse versions against each other. There is no borrowing needed for the fraction of the assets that pair up; gains in one exactly match losses in the other, just transfer over.

It's only the imbalance that counts. So borrowing costs are lower than the naive assumption.
This theory is more sound with historical backtesting, as tracking differences of leveraged ETFs didn't increase meaningfully from 2017 to 2020, when LIBOR rates were higher.
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by Jacotus »

tradri wrote: Fri Mar 26, 2021 1:02 pm
Jacotus wrote: Fri Mar 26, 2021 9:38 am tradri, it seems to me that implicit in all of your posts is that you equate "risk" with "volatility", and that you hold the belief that there is truly no risk in 3x leveraged ETFs, as long as you hold for long enough time.

You must disabuse yourself of this notion.
Can you please explain to me the other risks of holding leveraged ETFs other than volatility?
Do you think these funds are "less safe"? If so, why?
Two words: unknown unknowns. As pointed out earlier, we have no real record of a 3x leveraged ETF through a large market crash like 2008-2009. Simulated backtests are not the same thing as reality. Complex financial products have complex risks and downsides. The nature of black swans is that it is difficult to know what form they may take before they occur.

More fundamentally, if it were the case that 3x leveraged ETFs were a sure winning bet, if only one had the audacity to hold on through the volatility, then one has to ask: Why doesn't ALL of the smart money accumulate in these funds? I do not think anyone will be much smarter than the market as a whole. Therefore, if the market doesn't think of these as a sure thing, I conclude they must not be. There may be a pretty good chance you'll beat the market, but that comes with a very nontrivial chance of large losses.
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Re: Are 3x leveraged ETFs the long-term winning strategy?

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JackoC wrote: Fri Mar 26, 2021 10:15 am
Hydromod wrote: Fri Mar 26, 2021 9:55 am All of these calculations related to borrowing assume that each LETF is being run in isolation.

The providers run positive and inverse versions against each other. There is no borrowing needed for the fraction of the assets that pair up; gains in one exactly match losses in the other, just transfer over.

It's only the imbalance that counts. So borrowing costs are lower than the naive assumption.
No, the interest rate subtracts from the return of the person buying the 3X fund, but adds to the return of the person buying the -3X fund. This is an arbitrage relationship not dependent on their being any actually borrowing. It's just like with the index futures. Every long position must be offset by a short position, but the futures trade at premium to the cash stock basket to the degree of what the borrowing cost *would be* to finance that basket (and at a discount to the degree of dividends the cash basket holder will receive over the futures contract period that the long in the futures will not receive, a net discount at the moment). If being long the 3X fund didn't cost you the implied financing rate (and assuming it still benefited you the missed dividends, which it also does) you could arbitrage that by being long the 3X fund and shorting 3X the cash stock basket. That would hedge your equity position and you could just keep whatever interest your earn on the proceeds of the short sale (ie if a professional did that, which they could and would if the 3X fund was mispriced to not include the implied financing cost).

Including the financing cost in expected return of 3X fund is not a 'naive assumption'. :happy
Correct me if I'm wrong, but even if you didn't have to pay any financing rate, the 3x leveraged ETFs are still pretty expensive, even if you just consider the TER. So even if you offset your trades, you would still have to pay the TER of both funds, right?
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by OohLaLa »

tradri wrote: Fri Mar 26, 2021 1:02 pm
Jacotus wrote: Fri Mar 26, 2021 9:38 am tradri, it seems to me that implicit in all of your posts is that you equate "risk" with "volatility", and that you hold the belief that there is truly no risk in 3x leveraged ETFs, as long as you hold for long enough time.

You must disabuse yourself of this notion.
Can you please explain to me the other risks of holding leveraged ETFs other than volatility?
Do you think these funds are "less safe"? If so, why?
If you want the boring, official list, starting at page 5: https://www.proshares.com/funds/prospec ... icker=UPRO

This "volatility risk" has many other associated risks such as "behavioral risk" and something I would call "total failure risk". It's not just slightly more discomfort along the way and it's not that these have 0 chance of appearing for unleveraged funds, but their importance is greatly amplified the more leverage you add on.

Throughout the thread you seem to simplify that risk and understate it. Other posters presented periods where this fund would have lost 75-100% of its value, and you seemed to be OK with that notion. I don't think anybody would actually stick it out in such situations, with significant funds invested, unless they were simply paralyzed from fear. This is precisely why I wrote that you should start investing in this and find out for yourself. I genuinely mean this. Some people that were 100% sure they were OK with the volatility were scrambling over the last year... and this is with diversification... not 100% UPRO.
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

nisiprius wrote: Fri Mar 26, 2021 10:46 am
tradri wrote: Fri Mar 26, 2021 7:33 amI had a look at the Wikipedia article of the Kelly Criterion, but from I understood, it mostly applies to situations where you either gain or lose money. Since (I think) we all assume that "stonks only go up" in the long-run, I don't see how I can calculate the probability of losing money. It's not like I have to realize my losses when investing in a leveraged ETF, so I should be able to sit out any market crash.
1) If you want to engage in long-term, high risk investing, you really owe it to yourself to understand the Kelly Criterion. (As in, understand it better than I do. I don't need to). If you are actually interested in learning, not confirmation of your pre-existing ideas, you need to learn about it. You can reject the practical conclusions if you like, but only after you understand the ideas. 2) If you want to understand the Kelly Criterion, I suggest taking the time to read William Poundstone's book, Fortune's Formula--which, incidentally, does not endorse the idea uncritically. It's a thoughtful and intelligent analysis, including discussions of the difference in opinion between traditional financial economists and practitioners of the Kelly Criterion in investing.

The intellectual problem is not simple. Briefly, traditional analysis suggests that the highest long-term return will be obtained by using as much leverage as possible.

I believe that's what you think. That you should use as much leverage as possible, limited only by personal risk tolerance, and that this will "obviously" give the highest long-term return. And that you are only setting a 3X limit for practical reasons... and that you also believe that financial ruin with 3X leveraged stocks and daily rebalancing is impossible because you personally cannot envision a scenario of the S&P 500 losing more than 33.34% in a day, so you set the limit at 3X, not because you can actually tolerate it necessarily, but because you think that at 3X it is impossible for your intolerable risk ever to show up.

The Kelly criterion writing turns on goal definition. The goal is assumed to be to maximize the long-term return in the sense of compound average growth rate (CAGR). The key word here is "compound," and the key insight is that maximizing the average annual return (which is increased without limit by using more and more leverage) does not maximize the long-term multi-year compounded return.

The big problem in applying the Kelly criterion is that despite Poundstone's phrase "fortune's formula," the only formulas for applying the Kelly criterion require idealized models in which you accurately know probabilities a priori--for example, gambling games like roulette or dice where you know the probabilities from the physical design of the equipment, but not horse races or the stock market, where you do not.

It is made worse by the fact that financial data is ill-behaved, with extreme events occurring much more often than they do in statistical models based on the normal distribution.

Various stock market estimates based on the Kelly criterion come up with values of 1.2X or 1.4X leverage. Even there, there is a problem, because I have read that people who really use the Kelly criterion think it is too due to inability to know the true odds. Therefore, the conventional rule-of-thumb is to go no more than farther than "half Kelly."

That would rationally suggest that a risk-tolerant stock investor might go beyond 100% stocks and consider 10% or 20% leverage.

The point is that there are rational, numerical, computational reasons for limiting leverage, and to much less than 2X. Of course there are other people who assert the opposite, but it's important to see there is a cold hard argument for limiting leverage.

The second point is that even if we are not talking about personal financial ruin, if your hardnosed goal is to "maximize return" in the sense of longterm CAGR, using more than a small amount of leverage probably will not do that.

A detail that's always argued in discussions of the Kelly criterion is whether it contains an assumption of a logarithmic utility function. That is, does it assume that $10,000 to someone with $10 million than to someone with $100,000? Some say that it absolutely does not. Some say that this assumption has surely been smuggled in. Even if it has been, I don't think it's all that crazy an assumption.

2) And of course even the appropriateness of the Kelly criterion has been challenged. The late economist Paul Samuelson had very strong views, and ultimately expressed them in a paper entitled Why We Should Not Make Mean Log of Wealth Big Though Years to Act are Long. His point is that the goal of maximizing long-term average return (CAGR) is, in his opinion, wildly inappropriate. The odd title is that eventually he got so irritated that he wrote this paper literally in words of one syllable. Although it's a droll idea, unfortunately in this case using words of one syllable doesn't actually make the argument particularly clear, but at least when the topic came up he could say "I've explained this in words of one syllable."
Thanks for the book recommendation. I will take a look at it.

Please explain it to me in case I'm missing something, but I really don't understand how there can be a fixed "Kelly" like "half kelly" that is the optimum when considering stock market leverage.

Doesn't it all depend on the investor's time horizon and personal preferences? Isn't the general recommendation that one should start with a stock market exposure of 1x and decrease it to something like 0.4x as you enter retirement? Why not start higher than 1x?

How can there be a fixed kelly, when the "optimum" exposure you should/can have to the stock market is dependent on so many (personal) factors?

As for the utility of using leveraged investments, I would argue the utility decreases over time, as you become more concerned about maintaining/living off your investments, rather than growing them.

As for the paper by Paul Samuelson, I really didn't understand what he was trying to communicate. :? (English isn't my first language)
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by Jacotus »

nisiprius wrote: Fri Mar 26, 2021 10:46 am 2) And of course even the appropriateness of the Kelly criterion has been challenged. The late economist Paul Samuelson had very strong views, and ultimately expressed them in a paper entitled Why We Should Not Make Mean Log of Wealth Big Though Years to Act are Long. His point is that the goal of maximizing long-term average return (CAGR) is, in his opinion, wildly inappropriate. The odd title is that eventually he got so irritated that he wrote this paper literally in words of one syllable. Although it's a droll idea, unfortunately in this case using words of one syllable doesn't actually make the argument particularly clear, but at least when the topic came up he could say "I've explained this in words of one syllable."
I just read that Samuelson paper, and I agree the point is terribly made. Do you happen to know of another source where Samuelson's point is better explained?
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

OohLaLa wrote: Fri Mar 26, 2021 11:58 am - Going in with a preconceived notion as being right.
- Dismissing actual data with proof of work all while privileging an article missing comments about important aspects of the studied strategy.
- Implying that one can just brush off a 97% drawdown and push onward, without ******** the proverbial bed.
- Believing that failure of an investing strategy must necessarily include a drop to 0% and you living out of a cardboard box, instead of simply understanding that lagging behind for decades will do the trick just fine.

At what point can we just come out and say that the main thing fueling this thread right now is hubris?

OP, putting all your money into 100% UPRO (or equivalent) is ridiculous and if all the info provided to that effect is not enough to make you go back to the workshop and reassess then just have at it! Dive in and enjoy the swim.

If you are truly open to rethinking leverage as part of a more diversified portfolio, then you are a perfect candidate for reading the whole HFEA thread. You'll be armed with plenty of options to improve your odds of survival and of actually getting those higher returns you wanted (all with higher risk, of course). The most important part is to actually start putting considerable money into the chosen approach... you will start feeling the burn and you'll see if you have the testicular fortitude you think you do.
I'm sorry if I came across a bit confrontational.

I am not necessarily dismissing any data. Just because other users have tried to incorporate the LIBOR rates into their simulations, doesn't mean that these reflect the actual costs of a leveraged ETF. Again, I don't see the tracking difference that these leveraged ETFs should have produced, even though the LIBOR rates were 1-2.5% from 2017 to 2020. But please show it to me, if they actually suffered because of the higher LIBOR rates.

The strategy has a very big behavioral risk, I am not denying that.

Underperforming for 1-2 decades is OK in my opinion, if you are planning to hold these leveraged ETFs for longer than 1-2 decades.

I am not denying that risk parity is a sound strategy. I just want to push an edge case here and take risk parity to its extreme, by not leveraging a well-balanced portfolio, but by leveraging the best performing asset in that portfolio.

I don't want this to turn into hubris, as you put it, but this strategy definitely requires massive testicular fortitude.
Last edited by tradri on Fri Mar 26, 2021 1:54 pm, edited 1 time in total.
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by DMoogle »

tradri wrote: Fri Mar 26, 2021 12:59 pmCorrect me if I'm wrong, but as far as I understand it, the Kelly Criterion is about the strategy that will maximize your return in the long-run.
This is wrong. It's about maximizing a return with a limited bankroll or time horizon.
tradri wrote: Fri Mar 26, 2021 12:59 pmSince the variance in returns should decrease the longer you hold onto an investment,
This is also (mostly) incorrect. The longer you hold your investment, the *higher* the variance in returns, not lower. Simple example: if you compare today's UPRO's price to tomorrow's vs. the end of the year's, there's a much higher chance that it'll close down 10% at the end of the year than it will tomorrow.

It's the concept of heteroskedasticity, if I'm not mistaken (been a while since I've studied it). The graph on this page illustrates it well: https://www.investopedia.com/terms/h/he ... ticity.asp
tradri wrote: Fri Mar 26, 2021 12:59 pmthe longer the time-frame you are planning to hold your investment, the more you can afford to increase volatility, as that volatility shouldn't matter that much in the long-run.

Of course, investing in something so volatile that you wouldn't recover in 50 or 100 years from it makes little sense, even for the investor playing the longest game possible. The 5.8x optimum is only theoretical and assuming the fund's expenses remained the same over the whole time period.

While additional costs surely would massively hurt leveraged ETF returns, I don't see the effect of borrowing costs when comparing the simulations(without borrowing costs) and the actual performance of the leveraged ETF, even though LIBOR rates were higher than 0% over the last 10 years.
I agree with your point here, and it seems you're seeing that we can't remove risk from the equation entirely. Another illustrative example: would you bet all the money you have on a 1000:1 shot, if your return when you "hit" the 1000:1 is 10000000x? Your average return is enormous, but a rational investor would still say no - this is the kind of question that the Kelly Criterion aims to tackle (how much money should you bet?). However, the "5.8x optimum" to maximize expected return REALLY needs to be pressure-tested. It was based on one guy's hypothetical backtest. As you've seen others post, other hypothetical backtests are showing the "optimum" is closer to 2x.

As I said before, we're not going to align unless we can agree on what the correct hypothetical backtest is on a leverage ETF.
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

OohLaLa wrote: Fri Mar 26, 2021 12:01 pm When you use the quote function, you can:
- simply trim the quoted paragraphs, to only have the specific portions you are responding to.
- remove entire quote blocks if they are not necessary (ex: I quote you just now, but I removed the previous quote that you quoted yourself).
Ok, thanks :sharebeer
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

DMoogle wrote: Fri Mar 26, 2021 12:41 pm
Hydromod wrote: Fri Mar 26, 2021 12:24 pm
OohLaLa wrote: Fri Mar 26, 2021 12:19 pm I think veering into talks about borrowing rates is not a good idea in this case. There are fundamental misconceptions and lack of knowledge that have been pointed out and should be resolved first, as they are much bigger obstacles to success here.
Probably that would be better done in the simulating leveraged funds thread, I agree.
I'm reading through that thread and might revive it afterward, but being able to provide an accurate simulation of the past hypothetical performance of a UPRO-like instrument seems like the crux of the discussion.

Some simulations show it killed, some simulations show it barely outperformed. I don't think we'll ever reach a mutual agreement until we can agree on a "correct" hypothetical backtest.
I agree.

It's kind of unfortunate that leveraged ETF providers aren't more transparent with the exact costs that their funds incur. (and how these costs might change in the future)
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by OohLaLa »

tradri wrote: Fri Mar 26, 2021 1:48 pm I'm sorry if I came across a bit confrontational.

I am not necessarily dismissing any data. Just because other users have tried to incorporate the LIBOR rates into their simulations, doesn't mean that these reflect the actual costs of a leveraged ETF. Again, I don't see the tracking difference that these leveraged ETFs should have produced, even though the LIBOR rates were 1-2.5% from 2017 to 2020. But please show it to me, if they actually suffered because of the higher LIBOR rates.
I don't think you were confrontational, just to argue. I just think you were really minimizing some of the data + risks others stated. Throughout the thread, I detected genuine concern from folks and not just a smarta** attitude (to win some sort of debate).

I wrote what I wrote as someone who is mostly putting funds into 70% TQQQ/ 20% TMF/ 10% VXZ, in 2021. I am definitely not someone who will try to dissuade you from using leveraged funds, in general. You just have to make sure it's really a good fit, by understanding the different dynamics and potential issues, and actually start putting $$$ into it.

P.S. Like I mentioned, LIBOR rates should be a secondary concern, IMO. First of all, like you and some others mentioned, it doesn't seem to be a simple "oh LIBOR went up 1%, you now pay/ lose 1%" scenario. Secondly, it's not the thing that will cause failure. I would look at leverage %, other assets to be held with the leveraged equities, your resulting AA, your age/ income stability, how long you want to keep this up, whether you have a de-risking plan with time etc.
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

DMoogle wrote: Fri Mar 26, 2021 12:52 pm You make good points, and honestly I don't have a lot to add, except I think that CAPE might not as be as good of a predictor as it has been in the past, due to a fundamental shift in business models at a macro level. Specifically, I'm referring to tech companies focusing on customer base growth over revenue and profitability. I believe it's a relatively new concept/trend that's at least partially responsible for the compression of P/E.

One other thing I've been meaning to bring up, totally unrelated: a LOT of people come into these threads and ask "is this really in line with Bogleheads philosophy"? And honestly, I think it's tough to argue that it is. The problem is, IMO, is that this is the probably the largest and best forum for finance enthusiasts on the internet. There's a lot of really smart, knowledgeable people here.

Not to say smart folks don't exist elsewhere, but there generally isn't as much organization. I'm glad the admins have been flexible to allow this kind of discussion.
I agree that the folks here are amazing.

I tried discussing these things on some subreddits, but the level of depth that is "normal" here is really cool.
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

Jacotus wrote: Fri Mar 26, 2021 1:13 pm Two words: unknown unknowns. As pointed out earlier, we have no real record of a 3x leveraged ETF through a large market crash like 2008-2009. Simulated backtests are not the same thing as reality. Complex financial products have complex risks and downsides. The nature of black swans is that it is difficult to know what form they may take before they occur.

More fundamentally, if it were the case that 3x leveraged ETFs were a sure winning bet, if only one had the audacity to hold on through the volatility, then one has to ask: Why doesn't ALL of the smart money accumulate in these funds? I do not think anyone will be much smarter than the market as a whole. Therefore, if the market doesn't think of these as a sure thing, I conclude they must not be. There may be a pretty good chance you'll beat the market, but that comes with a very nontrivial chance of large losses.
Of course, there are black swan events everywhere.

I do agree that there is a non-zero chance that a leveraged ETF will close down at some point in the future. My only reassurance for that kind of scenario is, that the leveraged ETF is supposed to hold "safe" (hopefully) assets that will act as collateral in the event that the fund closes down. Obviously getting liquidated would suck if the fund is down -98% at that point, but the only hope then is that there is a different leveraged ETF still around that I can immediately reinvest the cash into.

But I do acknowledge this risk.
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by UsualLine »

I'm starting to feel like tradri is trolling this thread. 226 posts and not a single poster agrees it makes sense under any circumstance. Thinks he would hold onto his 100% UPRO portfolio through an 8 year 97% drawdown? Come on.

Just go for it already. Let us know how it goes.
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by OohLaLa »

tradri wrote: Fri Mar 26, 2021 2:02 pm
DMoogle wrote: Fri Mar 26, 2021 12:41 pm
Hydromod wrote: Fri Mar 26, 2021 12:24 pm
OohLaLa wrote: Fri Mar 26, 2021 12:19 pm I think veering into talks about borrowing rates is not a good idea in this case. There are fundamental misconceptions and lack of knowledge that have been pointed out and should be resolved first, as they are much bigger obstacles to success here.
Probably that would be better done in the simulating leveraged funds thread, I agree.
I'm reading through that thread and might revive it afterward, but being able to provide an accurate simulation of the past hypothetical performance of a UPRO-like instrument seems like the crux of the discussion.

Some simulations show it killed, some simulations show it barely outperformed. I don't think we'll ever reach a mutual agreement until we can agree on a "correct" hypothetical backtest.
I agree.

It's kind of unfortunate that leveraged ETF providers aren't more transparent with the exact costs that their funds incur. (and how these costs might change in the future)
Just to make sure... Why is the simulation data from the HFEA thread being questioned, as far as validity goes? From what I remember when I read through it, the group of geniuses :beer were able to replicate what the fund managers seem to be doing with the equity swaps + fees. They overlapped the chart with the data that is actually available for the LETFs and it aligned. I would be surprised that it's significantly off in the simulated years.
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

OohLaLa wrote: Fri Mar 26, 2021 1:31 pm If you want the boring, official list, starting at page 5: https://www.proshares.com/funds/prospec ... icker=UPRO

This "volatility risk" has many other associated risks such as "behavioral risk" and something I would call "total failure risk". It's not just slightly more discomfort along the way and it's not that these have 0 chance of appearing for unleveraged funds, but their importance is greatly amplified the more leverage you add on.

Throughout the thread you seem to simplify that risk and understate it. Other posters presented periods where this fund would have lost 75-100% of its value, and you seemed to be OK with that notion. I don't think anybody would actually stick it out in such situations, with significant funds invested, unless they were simply paralyzed from fear. This is precisely why I wrote that you should start investing in this and find out for yourself. I genuinely mean this. Some people that were 100% sure they were OK with the volatility were scrambling over the last year... and this is with diversification... not 100% UPRO.
I agree that the only way to see whether these funds are appropriate for me is to try it out.
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

DMoogle wrote: Fri Mar 26, 2021 1:49 pm
tradri wrote: Fri Mar 26, 2021 12:59 pmCorrect me if I'm wrong, but as far as I understand it, the Kelly Criterion is about the strategy that will maximize your return in the long-run.
This is wrong. It's about maximizing a return with a limited bankroll or time horizon.
tradri wrote: Fri Mar 26, 2021 12:59 pmSince the variance in returns should decrease the longer you hold onto an investment,
This is also (mostly) incorrect. The longer you hold your investment, the *higher* the variance in returns, not lower. Simple example: if you compare today's UPRO's price to tomorrow's vs. the end of the year's, there's a much higher chance that it'll close down 10% at the end of the year than it will tomorrow.

It's the concept of heteroskedasticity, if I'm not mistaken (been a while since I've studied it). The graph on this page illustrates it well: https://www.investopedia.com/terms/h/he ... ticity.asp
tradri wrote: Fri Mar 26, 2021 12:59 pmthe longer the time-frame you are planning to hold your investment, the more you can afford to increase volatility, as that volatility shouldn't matter that much in the long-run.

Of course, investing in something so volatile that you wouldn't recover in 50 or 100 years from it makes little sense, even for the investor playing the longest game possible. The 5.8x optimum is only theoretical and assuming the fund's expenses remained the same over the whole time period.

While additional costs surely would massively hurt leveraged ETF returns, I don't see the effect of borrowing costs when comparing the simulations(without borrowing costs) and the actual performance of the leveraged ETF, even though LIBOR rates were higher than 0% over the last 10 years.
I agree with your point here, and it seems you're seeing that we can't remove risk from the equation entirely. Another illustrative example: would you bet all the money you have on a 1000:1 shot, if your return when you "hit" the 1000:1 is 10000000x? Your average return is enormous, but a rational investor would still say no - this is the kind of question that the Kelly Criterion aims to tackle (how much money should you bet?). However, the "5.8x optimum" to maximize expected return REALLY needs to be pressure-tested. It was based on one guy's hypothetical backtest. As you've seen others post, other hypothetical backtests are showing the "optimum" is closer to 2x.

As I said before, we're not going to align unless we can agree on what the correct hypothetical backtest is on a leverage ETF.
I probably used the wrong terminology, but I didn't mean that the variance in return decreases with time, I meant that the chance of underperforming decreases with time, as the stock market tends to go up over time.

Getting back to the Kelly Criterion (assuming I roughly understood what it is about), leveraged ETFs would be optimal under the Kelly Criterion if there is more than a 50% chance that the leveraged ETF will outperform over a certain time period, right?
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by Hydromod »

OohLaLa wrote: Fri Mar 26, 2021 2:24 pm Just to make sure... Why is the simulation data from the HFEA thread being questioned, as far as validity goes? From what I remember when I read through it, the group of geniuses :beer were able to replicate what the fund managers seem to be doing with the equity swaps + fees. They overlapped the chart with the data that is actually available for the LETFs and it aligned. I would be surprised that it's significantly off in the simulated years.
I'm questioning it because I saw a potential arbitration effect when I applied the leveraging formulas to simulate UPRO and SPXU, but the effect doesn't show up in the actual LETFS. In the deeper past with higher rates, this would have given me a 30-fold increase in investment over a decade or two. Which implies that either I am screwing up something that I strongly want to get right or there is something missing in simulating leveraged ETFs that we should be able to figure out.
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

OohLaLa wrote: Fri Mar 26, 2021 2:07 pm I don't think you were confrontational, just to argue. I just think you were really minimizing some of the data + risks others stated. Throughout the thread, I detected genuine concern from folks and not just a smarta** attitude (to win some sort of debate).

I wrote what I wrote as someone who is mostly putting funds into 70% TQQQ/ 20% TMF/ 10% VXZ, in 2021. I am definitely not someone who will try to dissuade you from using leveraged funds, in general. You just have to make sure it's really a good fit, by understanding the different dynamics and potential issues, and actually start putting $$$ into it.

P.S. Like I mentioned, LIBOR rates should be a secondary concern, IMO. First of all, like you and some others mentioned, it doesn't seem to be a simple "oh LIBOR went up 1%, you now pay/ lose 1%" scenario. Secondly, it's not the thing that will cause failure. I would look at leverage %, other assets to be held with the leveraged equities, your resulting AA, your age/ income stability, how long you want to keep this up, whether you have a de-risking plan with time etc.
Thanks for presenting other very important concerns that should be taken into account.

Following such a strategy is only really an option (in my opinion) if you don't care about seeing massive losses (although unrealized) in your account, when you are still fairly young and want to follow this strategy for at least 20 years, when you don't need the money for anything else over the next decades, and when you have the discipline to step off the gas at a reasonable time. (maybe around 10 years before you want to start "securing" the money)

I realize that these criteria would only suit a very specific type of person/circumstance.
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

UsualLine wrote: Fri Mar 26, 2021 2:21 pm I'm starting to feel like tradri is trolling this thread. 226 posts and not a single poster agrees it makes sense under any circumstance. Thinks he would hold onto his 100% UPRO portfolio through an 8 year 97% drawdown? Come on.

Just go for it already. Let us know how it goes.
Ok, I will make sure to post updates from time to time. :wink:
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

OohLaLa wrote: Fri Mar 26, 2021 2:24 pm Just to make sure... Why is the simulation data from the HFEA thread being questioned, as far as validity goes? From what I remember when I read through it, the group of geniuses :beer were able to replicate what the fund managers seem to be doing with the equity swaps + fees. They overlapped the chart with the data that is actually available for the LETFs and it aligned. I would be surprised that it's significantly off in the simulated years.
I haven't read through the entire "Simulating Returns of Leveraged ETFs" thread yet, but if they used the formula from the starting post Daily Return = [Underlying Index Adjusted Daily Return (including dividend) x Leverage] - (12 month LIBOR)/250 - (Expense Ratio)/250 then they are basically subtracting the total amount of the 12-month LIBOR rate from the ETF return each year.

Since the 12-month LIBOR rate wasn't zero over the last decade, it should be easy to check, whether the Tracking Difference of the funds increased by the 12-month LIBOR rate in the time period 2016-2020, where the 12-month LIBOR rate was anywhere between 1% to 3%.

I ran some simulations in Python (using the code from this article: https://teddykoker.com/2019/04/simulati ... in-python/) to simulate the UPRO with different expense ratios over the time period 2016, 1, 1 and 2020, 1, 1 and then compared the CAGR of the simulated UPRO to the CAGR of the actual UPRO during that time period. It turned out, that in that time period the simulated UPRO had almost the exact same CAGR as the actual UPRO when I calculated it with a 1.5% expense ratio.

So while the expenses (including the cost of borrowing) were higher than the 0.95% TER in that time period, they did incur an extra tracking difference of only 0.55% annually over that time period, while the 12 month LIBOR rates were anywhere between 1%-3%.

Please let me know if they actually used a different formula in the end. :sharebeer
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by RubyTuesday »

nisiprius wrote: Fri Mar 26, 2021 10:46 am
tradri wrote: Fri Mar 26, 2021 7:33 amI had a look at the Wikipedia article of the Kelly Criterion, but from I understood, it mostly applies to situations where you either gain or lose money. Since (I think) we all assume that "stonks only go up" in the long-run, I don't see how I can calculate the probability of losing money. It's not like I have to realize my losses when investing in a leveraged ETF, so I should be able to sit out any market crash.
1) If you want to engage in long-term, high risk investing, you really owe it to yourself to understand the Kelly Criterion. (As in, understand it better than I do. I don't need to). If you are actually interested in learning, not confirmation of your pre-existing ideas, you need to learn about it. You can reject the practical conclusions if you like, but only after you understand the ideas. 2) If you want to understand the Kelly Criterion, I suggest taking the time to read William Poundstone's book, Fortune's Formula--which, incidentally, does not endorse the idea uncritically. It's a thoughtful and intelligent analysis, including discussions of the difference in opinion between traditional financial economists and practitioners of the Kelly Criterion in investing.

The intellectual problem is not simple. Briefly, traditional analysis suggests that the highest long-term return will be obtained by using as much leverage as possible.

I believe that's what you think. That you should use as much leverage as possible, limited only by personal risk tolerance, and that this will "obviously" give the highest long-term return. And that you are only setting a 3X limit for practical reasons... and that you also believe that financial ruin with 3X leveraged stocks and daily rebalancing is impossible because you personally cannot envision a scenario of the S&P 500 losing more than 33.34% in a day, so you set the limit at 3X, not because you can actually tolerate it necessarily, but because you think that at 3X it is impossible for your intolerable risk ever to show up.

The Kelly criterion writing turns on goal definition. The goal is assumed to be to maximize the long-term return in the sense of compound average growth rate (CAGR). The key word here is "compound," and the key insight is that maximizing the average annual return (which is increased without limit by using more and more leverage) does not maximize the long-term multi-year compounded return.

The big problem in applying the Kelly criterion is that despite Poundstone's phrase "fortune's formula," the only formulas for applying the Kelly criterion require idealized models in which you accurately know probabilities a priori--for example, gambling games like roulette or dice where you know the probabilities from the physical design of the equipment, but not horse races or the stock market, where you do not.

It is made worse by the fact that financial data is ill-behaved, with extreme events occurring much more often than they do in statistical models based on the normal distribution.

Various stock market estimates based on the Kelly criterion come up with values of 1.2X or 1.4X leverage. Even there, there is a problem, because I have read that people who really use the Kelly criterion think it is too due to inability to know the true odds. Therefore, the conventional rule-of-thumb is to go no more than farther than "half Kelly."

That would rationally suggest that a risk-tolerant stock investor might go beyond 100% stocks and consider 10% or 20% leverage.

The point is that there are rational, numerical, computational reasons for limiting leverage, and to much less than 2X. Of course there are other people who assert the opposite, but it's important to see there is a cold hard argument for limiting leverage.

The second point is that even if we are not talking about personal financial ruin, if your hardnosed goal is to "maximize return" in the sense of longterm CAGR, using more than a small amount of leverage probably will not do that.

A detail that's always argued in discussions of the Kelly criterion is whether it contains an assumption of a logarithmic utility function. That is, does it assume that $10,000 to someone with $10 million than to someone with $100,000? Some say that it absolutely does not. Some say that this assumption has surely been smuggled in. Even if it has been, I don't think it's all that crazy an assumption.

2) And of course even the appropriateness of the Kelly criterion has been challenged. The late economist Paul Samuelson had very strong views, and ultimately expressed them in a paper entitled Why We Should Not Make Mean Log of Wealth Big Though Years to Act are Long. His point is that the goal of maximizing long-term average return (CAGR) is, in his opinion, wildly inappropriate. The odd title is that eventually he got so irritated that he wrote this paper literally in words of one syllable. Although it's a droll idea, unfortunately in this case using words of one syllable doesn't actually make the argument particularly clear, but at least when the topic came up he could say "I've explained this in words of one syllable."
Quoting this in its entirety in hopes OP will give it due attention.

Nisi, thanks for this contribution
“Doing nothing is better than being busy doing nothing.” – Lao Tzu
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by cos »

tradri wrote: Fri Mar 26, 2021 7:33 am The main article that I have seen talking about the optimal leverage being at 2x is the first article by ddnum (http://www.ddnum.com/articles/leveragedETFs.php). However, there they only use the price index, so it isn't very meaningful. In a follow-up article they actually use the Total Return index and found that in theory the optimal leverage is at 5.8x. (http://www.ddnum.com/articles/leveragedETFsandDCA.php)

I had a look at the Wikipedia article of the Kelly Criterion, but from I understood, it mostly applies to situations where you either gain or lose money. Since (I think) we all assume that "stonks only go up" in the long-run, I don't see how I can calculate the probability of losing money. It's not like I have to realize my losses when investing in a leveraged ETF, so I should be able to sit out any market crash.
The Kelly criterion absolutely applies to investing, especially with leverage, and it's literally equivalent to maximizing CAGR. Using real total return data and a tool like Portfolio Visualizer, we can trivially replicate ddnum's findings in that first article. Check out the optimal leverage in these backtests, but keep in mind that these assume you can borrow at the risk-free rate on top of assuming that past is precedent. They're also releveraged monthly rather than daily, and they're rounded to the nearest whole multiple.

S&P 500: 3x

NASDAQ 100: 2x

Russell 2000: 2x
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typical.investor
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by typical.investor »

tradri wrote: Wed Mar 24, 2021 5:06 pm
MotoTrojan wrote: Wed Mar 24, 2021 4:51 pm
tradri wrote: Wed Mar 24, 2021 4:50 pm
jarjarM wrote: Wed Mar 24, 2021 4:23 pm
tradri wrote: Wed Mar 24, 2021 4:16 pm

Yeah, I will definitely dig deeper into it and understand how these backtests were constructed.

Just a few decades ago, no-one was advocating for index funds either, so I would be carefully with using that argument.
Sure, index fund didn't really come into form until Mr Bogle came about. But there's many here that are not conventional BH, myself included, still none of us will accept buy and hold 3x LETF in a long term portfolio. This is from someone who's currently holding well north of 7 figure in 3xLETF.
Well, if you already have a 7-figure portfolio I don't blame you. Wealth creation and wealth preservation are two totally different things.
Think you read that wrong, sounds like jarjarM has 7 figures in 3x ETFs alone :twisted:.
I understood that correctly. My argument was, that if you already have a 7-figure portfolio you are much more worried about market downturns, and will therefore invest in a risk parity strategy like the one proposed by Hedgefundie. That will limit your downturns, but potentially also limits your upside compared to a 300% stock investment.
I think you are woefully mistaken and don't really understand the risks here. 3X leverage is going to amplify your risks.

You simply are not looking at risk management correctly to assume that 1) losses amplified by leverage won't happen together at the same time in bonds and stocks and 2) that your portfolio will recover by the time you need to start spending it down.

I put $100k in and am at $265k, so it's not like I don't approve of the strategy. Still, while pre-Volker Fed policies will never be seen again [meaning we won't have high inflation for the reasons we had before], there are still elements we don't know about and the Fed can't control. I mean the Phillips curve broke when Chinese labor entered the market. So what happens as China (and most developed nations) ages and workers leave, or what happens if China and the West decouple economically? Or what happens if everyone and their uncle bob implements financial stimulus at the same time and global growth picks up? I mean it'll be tough to offshore rising labor costs if they are rising everywhere.

And then there is the ghost of climate change. Maybe it's science fiction in our lifetimes or maybe it'll have economic costs.

There is just so much unknown here (and where is QE, globalization unwind, and climate change in the backtests?) that I don't think you can completely count on long bonds always saving equity. And leverage is potentially going increase the volatility.

So maybe 3x wins and maybe it doesn't. I am afraid there is no way to know what the optimal strategy is in advance.

If inflation and higher rates does hit TMF and UPRO crumbles along with it, and volatility is high -- the only path to recovery might be to put more money in. If you aren't prepared to do that, then I think you need to pare back.
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tradri
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

RubyTuesday wrote: Fri Mar 26, 2021 6:58 pm
nisiprius wrote: Fri Mar 26, 2021 10:46 am
tradri wrote: Fri Mar 26, 2021 7:33 amI had a look at the Wikipedia article of the Kelly Criterion, but from I understood, it mostly applies to situations where you either gain or lose money. Since (I think) we all assume that "stonks only go up" in the long-run, I don't see how I can calculate the probability of losing money. It's not like I have to realize my losses when investing in a leveraged ETF, so I should be able to sit out any market crash.
1) If you want to engage in long-term, high risk investing, you really owe it to yourself to understand the Kelly Criterion. (As in, understand it better than I do. I don't need to). If you are actually interested in learning, not confirmation of your pre-existing ideas, you need to learn about it. You can reject the practical conclusions if you like, but only after you understand the ideas. 2) If you want to understand the Kelly Criterion, I suggest taking the time to read William Poundstone's book, Fortune's Formula--which, incidentally, does not endorse the idea uncritically. It's a thoughtful and intelligent analysis, including discussions of the difference in opinion between traditional financial economists and practitioners of the Kelly Criterion in investing.

The intellectual problem is not simple. Briefly, traditional analysis suggests that the highest long-term return will be obtained by using as much leverage as possible.

I believe that's what you think. That you should use as much leverage as possible, limited only by personal risk tolerance, and that this will "obviously" give the highest long-term return. And that you are only setting a 3X limit for practical reasons... and that you also believe that financial ruin with 3X leveraged stocks and daily rebalancing is impossible because you personally cannot envision a scenario of the S&P 500 losing more than 33.34% in a day, so you set the limit at 3X, not because you can actually tolerate it necessarily, but because you think that at 3X it is impossible for your intolerable risk ever to show up.

The Kelly criterion writing turns on goal definition. The goal is assumed to be to maximize the long-term return in the sense of compound average growth rate (CAGR). The key word here is "compound," and the key insight is that maximizing the average annual return (which is increased without limit by using more and more leverage) does not maximize the long-term multi-year compounded return.

The big problem in applying the Kelly criterion is that despite Poundstone's phrase "fortune's formula," the only formulas for applying the Kelly criterion require idealized models in which you accurately know probabilities a priori--for example, gambling games like roulette or dice where you know the probabilities from the physical design of the equipment, but not horse races or the stock market, where you do not.

It is made worse by the fact that financial data is ill-behaved, with extreme events occurring much more often than they do in statistical models based on the normal distribution.

Various stock market estimates based on the Kelly criterion come up with values of 1.2X or 1.4X leverage. Even there, there is a problem, because I have read that people who really use the Kelly criterion think it is too due to inability to know the true odds. Therefore, the conventional rule-of-thumb is to go no more than farther than "half Kelly."

That would rationally suggest that a risk-tolerant stock investor might go beyond 100% stocks and consider 10% or 20% leverage.

The point is that there are rational, numerical, computational reasons for limiting leverage, and to much less than 2X. Of course there are other people who assert the opposite, but it's important to see there is a cold hard argument for limiting leverage.

The second point is that even if we are not talking about personal financial ruin, if your hardnosed goal is to "maximize return" in the sense of longterm CAGR, using more than a small amount of leverage probably will not do that.

A detail that's always argued in discussions of the Kelly criterion is whether it contains an assumption of a logarithmic utility function. That is, does it assume that $10,000 to someone with $10 million than to someone with $100,000? Some say that it absolutely does not. Some say that this assumption has surely been smuggled in. Even if it has been, I don't think it's all that crazy an assumption.

2) And of course even the appropriateness of the Kelly criterion has been challenged. The late economist Paul Samuelson had very strong views, and ultimately expressed them in a paper entitled Why We Should Not Make Mean Log of Wealth Big Though Years to Act are Long. His point is that the goal of maximizing long-term average return (CAGR) is, in his opinion, wildly inappropriate. The odd title is that eventually he got so irritated that he wrote this paper literally in words of one syllable. Although it's a droll idea, unfortunately in this case using words of one syllable doesn't actually make the argument particularly clear, but at least when the topic came up he could say "I've explained this in words of one syllable."
Quoting this in its entirety in hopes OP will give it due attention.

Nisi, thanks for this contribution
I haven't gotten around to reading the book Fortune's Formula yet, but from what I could understand the big idea in this book is the formula f=2p-1, where f is the fraction of wealth you should bet and p is the probability of a gain. (please correct me if I'm wrong)

I'm actually curious how one would apply that formula to stock market investing. In theory, the longer you are exposed to the stock market, the lower the chance of loss (the higher the chance of gain). So, assuming we know the probability of the stock market (or leveraged investments for that matter) outperforming over a year, a decade and so on, we can calculate what the optimal fraction of wealth we should bet on the strategy should be.

Common sense would tell us, that the longer we are planning to stay invested, the higher the fraction we should bet will be, as the probability of gain increases the longer we are exposed to the stock market.

Also, I don't think that it's best to use as much leverage as possible, especially when dealing with leveraged ETFs. There is a turning point, at which increasing leverage actually decreases returns because of volatility decay.
Last edited by tradri on Sat Mar 27, 2021 7:49 am, edited 3 times in total.
Laurizas
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by Laurizas »

tradri wrote: Wed Mar 24, 2021 3:04 pm
MotoTrojan wrote: Wed Mar 24, 2021 2:48 pm Be careful. I don't know much about ETPs but ETNs are much more dangerous than ETFs as you don't own the underlying financial assets, it is a pure derivative. ETP sounds similar.
I was also very skeptical at first, but after digging deeper into the WisdomTree ETPs I can say that they are pretty much the same as regular swap ETFs.

The only reason why they are called "ETP" and not "ETF" is because under European law a "fund" can't exceed a leverage of 2x. But the WisdomTree ETPs are fully collateralized, so in the event of default, a trustee actually holds securities that will be paid out to me.
I think you have this product in mind - 3USL WisdomTree S&P 500 3x Daily Leveraged.https://www.wisdomtree.eu/en-gb/etps/eq ... -leveraged
Here it says that:
wrote:What is this product?
An English law governed, uncertificated, registered, collateralised exchanged-traded note linked to the S&P 500 Net Total Return
through swap arrangements (“Swaps”) entered into with eligible swap providers (“Swap Providers”).
wrote:What happens if WisdomTree Multi Asset Issuer PLC is unable to pay out?
The product is not protected by the Irish Deposit Guarantee Scheme or any other investor compensation or guarantee scheme. This means that if WisdomTree Multi Asset Issuer PLC is unable to pay out, you may lose all of your investment. Since the ability of WisdomTree Multi Asset Issuer PLC to pay out depends on receiving the amounts due under the product from the Swap Providers under the Swaps, it receives from the Swap Providers daily assets as collateral (the "Collateral") for such obligations with a value equal to or in excess of the value of the daily price of the product. If the Swap Providers are unable to pay WisdomTree Multi Asseet Issuer PLC, its ability to pay out will be limited to the amounts realised from the Collateral, as further explained in the Prospectus
https://www.wisdomtree.eu/en-gb/-/media ... l---en.pdf

Also on page 8 here it says that ETN's are usually entirely reliant on the creditworthiness of the issuing entity.
https://www.wisdomtree.eu/en-gb/-/media ... tpedia.pdf

I do not know much about this but to my mind it introduces additional risk (creditworthiness of the issuing entity) that ETF's do not contain.
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tradri
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

cos wrote: Fri Mar 26, 2021 11:00 pm The Kelly criterion absolutely applies to investing, especially with leverage, and it's literally equivalent to maximizing CAGR. Using real total return data and a tool like Portfolio Visualizer, we can trivially replicate ddnum's findings in that first article. Check out the optimal leverage in these backtests, but keep in mind that these assume you can borrow at the risk-free rate on top of assuming that past is precedent. They're also releveraged monthly rather than daily, and they're rounded to the nearest whole multiple.

S&P 500: 3x

NASDAQ 100: 2x

Russell 2000: 2x
I think it's more useful to run the simulations with daily compounding instead of monthly compounding.

Using an estimated expense ratio of 2% over the entire time period from 1977 to 2021 (which I think is a reasonable average, annual expense) we can see how leveraged ETFs would have performed at different levels of leverage.

CAGRs
SPY: 10.64%
2x SPY simulated: 14.00%
3x SPY simulated: 15.91%
4x SPY simulated: 13.36%

Assuming that the expenses that each of these leveraged ETFs had to incur are the same for all 3 funds, a 3x leveraged ETF would have produced the highest absolute return in that time period.
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

typical.investor wrote: Sat Mar 27, 2021 2:58 am I think you are woefully mistaken and don't really understand the risks here. 3X leverage is going to amplify your risks.

You simply are not looking at risk management correctly to assume that 1) losses amplified by leverage won't happen together at the same time in bonds and stocks and 2) that your portfolio will recover by the time you need to start spending it down.

I put $100k in and am at $265k, so it's not like I don't approve of the strategy. Still, while pre-Volker Fed policies will never be seen again [meaning we won't have high inflation for the reasons we had before], there are still elements we don't know about and the Fed can't control. I mean the Phillips curve broke when Chinese labor entered the market. So what happens as China (and most developed nations) ages and workers leave, or what happens if China and the West decouple economically? Or what happens if everyone and their uncle bob implements financial stimulus at the same time and global growth picks up? I mean it'll be tough to offshore rising labor costs if they are rising everywhere.

And then there is the ghost of climate change. Maybe it's science fiction in our lifetimes or maybe it'll have economic costs.

There is just so much unknown here (and where is QE, globalization unwind, and climate change in the backtests?) that I don't think you can completely count on long bonds always saving equity. And leverage is potentially going increase the volatility.

So maybe 3x wins and maybe it doesn't. I am afraid there is no way to know what the optimal strategy is in advance.

If inflation and higher rates does hit TMF and UPRO crumbles along with it, and volatility is high -- the only path to recovery might be to put more money in. If you aren't prepared to do that, then I think you need to pare back.
I do agree that there are a lot of unknowns about the future. But I don't think this is anything new. In the past there were fears around world wars, communism and global pandemics, but the stock market didn't care in the long run.

I am not saying that the past is a reliable indicator of the future, but I think it isn't very productive to base your investing decisions (or life decisions) only around such fears.

I also think that risk parity (like the strategy proposed by Hedgefundie) has its problems. Most of the backtests start after 1980, when interest rates were at an all-time high and were steadily falling since. This has been very good for bonds. While interest rates may stay low for quite some time to come, I don't see the exceptional performance that long-term bonds had in the past necessarily repeating in the future.
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tradri
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

Laurizas wrote: Sat Mar 27, 2021 6:42 am
tradri wrote: Wed Mar 24, 2021 3:04 pm
MotoTrojan wrote: Wed Mar 24, 2021 2:48 pm Be careful. I don't know much about ETPs but ETNs are much more dangerous than ETFs as you don't own the underlying financial assets, it is a pure derivative. ETP sounds similar.
I was also very skeptical at first, but after digging deeper into the WisdomTree ETPs I can say that they are pretty much the same as regular swap ETFs.

The only reason why they are called "ETP" and not "ETF" is because under European law a "fund" can't exceed a leverage of 2x. But the WisdomTree ETPs are fully collateralized, so in the event of default, a trustee actually holds securities that will be paid out to me.
I think you have this product in mind - 3USL WisdomTree S&P 500 3x Daily Leveraged.https://www.wisdomtree.eu/en-gb/etps/eq ... -leveraged
Here it says that:
wrote:What is this product?
An English law governed, uncertificated, registered, collateralised exchanged-traded note linked to the S&P 500 Net Total Return
through swap arrangements (“Swaps”) entered into with eligible swap providers (“Swap Providers”).
wrote:What happens if WisdomTree Multi Asset Issuer PLC is unable to pay out?
The product is not protected by the Irish Deposit Guarantee Scheme or any other investor compensation or guarantee scheme. This means that if WisdomTree Multi Asset Issuer PLC is unable to pay out, you may lose all of your investment. Since the ability of WisdomTree Multi Asset Issuer PLC to pay out depends on receiving the amounts due under the product from the Swap Providers under the Swaps, it receives from the Swap Providers daily assets as collateral (the "Collateral") for such obligations with a value equal to or in excess of the value of the daily price of the product. If the Swap Providers are unable to pay WisdomTree Multi Asseet Issuer PLC, its ability to pay out will be limited to the amounts realised from the Collateral, as further explained in the Prospectus
https://www.wisdomtree.eu/en-gb/-/media ... l---en.pdf

Also on page 8 here it says that ETN's are usually entirely reliant on the creditworthiness of the issuing entity.
https://www.wisdomtree.eu/en-gb/-/media ... tpedia.pdf

I do not know much about this but to my mind it introduces additional risk (creditworthiness of the issuing entity) that ETF's do not contain.
Yes, that's the WisdomTree ETP I am talking about.

From what I can tell there isn't really a credit risk with these products, since there is a trustee that should pay me out in the case of default.
Here is a diagram from WisdomTree: https://www.wisdomtree.eu/de-at/-/media ... ucture.jpg

ETNs on the other hand aren't collateralized, so there is a credit risk when investing in those.

The closest definition that I could find for those WisdomTree ETPs is this definition from Wikipedia:

Exchange-traded instruments (ETIs) are derivative securities repackaging the value of an index or even actively managed portfolio issued by financial institutions and listed at a stock exchange. ETIs are known in the European market where several investment strategies cannot be replicated within a mutual fund and ETIs are set up as an alternative investment vehicle to overcome these restrictions. The distribution and marketing of ETCs thus is not regulated by mutual fund laws but by the Prospectus Directive. These investment vehicles are sometimes also marketed as Exchange-traded certificates or (if unlisted) actively managed certificates (AMC)

Since they are fully collateralized, I don't see a difference between these WisdomTree ETPs and swap ETFs.
JackoC
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by JackoC »

tradri wrote: Fri Mar 26, 2021 1:14 pm
JackoC wrote: Fri Mar 26, 2021 10:15 am
Hydromod wrote: Fri Mar 26, 2021 9:55 am All of these calculations related to borrowing assume that each LETF is being run in isolation.
The providers run positive and inverse versions against each other. There is no borrowing needed for the fraction of the assets that pair up; gains in one exactly match losses in the other, just transfer over.

It's only the imbalance that counts. So borrowing costs are lower than the naive assumption.
No, the interest rate subtracts from the return of the person buying the 3X fund, but adds to the return of the person buying the -3X fund. This is an arbitrage relationship not dependent on their being any actually borrowing.
Including the financing cost in expected return of 3X fund is not a 'naive assumption'. :happy
Correct me if I'm wrong, but even if you didn't have to pay any financing rate, the 3x leveraged ETFs are still pretty expensive, even if you just consider the TER. So even if you offset your trades, you would still have to pay the TER of both funds, right?
Yes you have to pay the ER of the fund, but the important point which you still seem to resist in later posts is there is no 'even if' about paying the financing. You absolutely do. You're looking at something where the clear reality is one thing, and asking others to delve into and correct whatever error you are making to 'empirically' find an answer other than this reality. When leveraging you have to pay financing cost, whether via futures, direct doing an equity Total Return Swap yourself, LETF (whose principal 'asset' is often TRS's), margin loan or any other way. Assuming you don't have to is incorrect, backtest schmacktest. :happy

Now again this doesn't mean '3X funds do worse when rates are higher', because the non vol terms in the return equation depend largely on the *difference* between stock return and financing cost. If now you expect equity return is 5% (humoring my estimate) and r is 0.5% (pretty clear from futures prices), in a possible future where r=10%, the expected stock return will not still be 5%: stock valuations would almost surely be lower and stock expected return higher. Investors, even non-leveraged, are attracted to risk assets by a premium in expected return over 'the risk free rate', IOW mu-r is going to tend, *relatively* toward being more constant than either mu or r (though obviously not literally constant or even highly predictable).

However going further into the equation for LETF return (which is again 'theoretical' only like Newton's laws are 'theoretical'), E[r]=L*mu-(L-1)r-0.5*L(L-1)*vol^2-ER/L, in case of 3X ETF volatility is not just something some people are 'afraid of' and others 'aren't afraid of', it's directly subtracting from your return.

This whole thread is an interesting case IMO of commonly held BH beliefs which are not quite correct but may lead naive investors away from harming themselves. But when you adopt a wrong concept, then stretch it into a new domain, you can be more seriously wrong. The wrong concept in this case, commonly stated here, is 'volatility doesn't matter to long term investors'. In a certain sense that statement has merit, especially considering the various ways the converse statement might be misinterpreted by beginners. But it's not technically true*, and the error in it becomes large at 3X leverage. At 3X, higher than expected vol can devastate your return. It's not a matter of how you 'feel' about vol, -0.5*L*(L-1)*vol^2 is a real subtraction as the fund rebalances itself daily as the market bounces around x%, (1-x)*(1+x)<(1-x)+(1+x). The vol term in the equation is the integration of that daily expected loss. Which is why the *expected* return of 3X is actually lower than 1X if you estimate mu-r at 4.5% and assume now's vol of around 19%, as I would, and consider the (actual not including finance cost) ER also. With more optimistic assumptions about mu-r (higher) and vol (lower) 3X can have higher expected return, but you must forecast both ERP and vol to estimate expected return of 3X. This is perhaps the most basic point about high leverage (once everyone gets on the same page that financing cost is real, not something whose existence we argue about based on 'backtests').

*the equation still holds at L<1, but at L<1 the sign of the volatility term flips and the 'volatility drag' becomes the 'rebalancing bonus'. Most are familiar with the latter, but sometimes still contradict themselves saying 'vol doesn't matter', though expected vol determines the expected size of that bonus...IOW it does matter even at L<1, just not as dramatically as it does at L>>1.
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by langlands »

tradri wrote: Sat Mar 27, 2021 5:57 am I haven't gotten around to reading the book Fortune's Formula yet, but from what I could understand the big idea in this book is the formula f=2p-1, where f is the fraction of wealth you should bet and p is the probability of a gain. (please correct me if I'm wrong)

I'm actually curious how one would apply that formula to stock market investing. In theory, the longer you are exposed to the stock market, the lower the chance of loss (the higher the chance of gain). So, assuming we know the probability of the stock market (or leveraged investments for that matter) outperforming over a year, a decade and so on, we can calculate what the optimal fraction of wealth we should bet on the strategy should be.

Common sense would tell us, that the longer we are planning to stay invested, the higher the fraction we should bet will be, as the probability of gain increases the longer we are exposed to the stock market.

Also, I don't think that it's best to use as much leverage as possible, especially when dealing with leveraged ETFs. There is a turning point, at which increasing leverage actually decreases returns because of volatility decay.
The analogous formula to f=2p-1 for stock market investing is the formula JackoC has been posting (numerous times :happy). Namely (μ-r)/σ^2 where μ is the expected arithmetic return, r is the risk free rate, and σ is the volatility. Assuming you want to maximize CAGR (which is equivalent to maximizing utility under log wealth utility), the amount of leverage you should use is proportional to the equity risk premium (ERP=μ-r) and inversely proportional to variance. The notion of half-Kelly applies here as well. Because you can't measure either the ERP or the volatility with high precision, you should be conservative or you will surely eventually go bust. Hence, if you thought 3x was optimal under perfect estimate of parameters, it might be better to use 1.5x or 2x to take into account measurement error.

As you mention, volatility decay is the key concept. In your posts, you should really avoid referring to "returns" without saying exactly what you mean though. In regards to leverage, the distinction between the expected arithmetic return, expected geometric return, realized arithmetic return, and realized geometric return become very important. Again, from your posts I get the strong sense you are mainly focused on CAGR, but you should explicitly say this to avoid misunderstandings. Leverage is really a mathematical concept and I think getting into the trenches and really grappling with the appropriate formulas and equations is the most effective way to understand it.
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tradri
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

JackoC wrote: Sat Mar 27, 2021 9:37 am Yes you have to pay the ER of the fund, but the important point which you still seem to resist in later posts is there is no 'even if' about paying the financing. You absolutely do. You're looking at something where the clear reality is one thing, and asking others to delve into and correct whatever error you are making to 'empirically' find an answer other than this reality. When leveraging you have to pay financing cost, whether via futures, direct doing an equity Total Return Swap yourself, LETF (whose principal 'asset' is often TRS's), margin loan or any other way. Assuming you don't have to is incorrect, backtest schmacktest. :happy

Now again this doesn't mean '3X funds do worse when rates are higher', because the non vol terms in the return equation depend largely on the *difference* between stock return and financing cost. If now you expect equity return is 5% (humoring my estimate) and r is 0.5% (pretty clear from futures prices), in a possible future where r=10%, the expected stock return will not still be 5%: stock valuations would almost surely be lower and stock expected return higher. Investors, even non-leveraged, are attracted to risk assets by a premium in expected return over 'the risk free rate', IOW mu-r is going to tend, *relatively* toward being more constant than either mu or r (though obviously not literally constant or even highly predictable).

However going further into the equation for LETF return (which is again 'theoretical' only like Newton's laws are 'theoretical'), E[r]=L*mu-(L-1)r-0.5*L(L-1)*vol^2-ER/L, in case of 3X ETF volatility is not just something some people are 'afraid of' and others 'aren't afraid of', it's directly subtracting from your return.

This whole thread is an interesting case IMO of commonly held BH beliefs which are not quite correct but may lead naive investors away from harming themselves. But when you adopt a wrong concept, then stretch it into a new domain, you can be more seriously wrong. The wrong concept in this case, commonly stated here, is 'volatility doesn't matter to long term investors'. In a certain sense that statement has merit, especially considering the various ways the converse statement might be misinterpreted by beginners. But it's not technically true*, and the error in it becomes large at 3X leverage. At 3X, higher than expected vol can devastate your return. It's not a matter of how you 'feel' about vol, -0.5*L*(L-1)*vol^2 is a real subtraction as the fund rebalances itself daily as the market bounces around x%, (1-x)*(1+x)<(1-x)+(1+x). The vol term in the equation is the integration of that daily expected loss. Which is why the *expected* return of 3X is actually lower than 1X if you estimate mu-r at 4.5% and assume now's vol of around 19%, as I would, and consider the (actual not including finance cost) ER also. With more optimistic assumptions about mu-r (higher) and vol (lower) 3X can have higher expected return, but you must forecast both ERP and vol to estimate expected return of 3X. This is perhaps the most basic point about high leverage (once everyone gets on the same page that financing cost is real, not something whose existence we argue about based on 'backtests').

*the equation still holds at L<1, but at L<1 the sign of the volatility term flips and the 'volatility drag' becomes the 'rebalancing bonus'. Most are familiar with the latter, but sometimes still contradict themselves saying 'vol doesn't matter', though expected vol determines the expected size of that bonus...IOW it does matter even at L<1, just not as dramatically as it does at L>>1.
I do acknowledge that financing costs also have to be subtracted from the leveraged ETF returns, but I don't think it's as easy as simply subtracting the 12-month LIBOR rate from the yearly leveraged ETF return, as it has been proposed in the "Simulating Returns of Leveraged ETFs" thread.

By checking the Tracking Difference of leveraged ETFs over the time period 2016-2020 when LIBOR rates were above 1%, it should be easy to see whether these can be easily added to the LETFs tracking difference or not. From the backtests I have done, the UPRO only experienced additional expenses of about 0.55% annually over that time period, even though the LIBOR rates were 1-3%.

So, I am not saying financing costs don't matter, I am saying that using a simple formula that subtracts the LIBOR rates from the returns isn't how it played out in 2016-2020. On top of that, I have read articles that talk about how the LIBOR is going to be discontinued and replaced by the SOFR, which according to its Wikipedia article should reduce borrowing costs.

I only talked about "volatility doesn't matter" in terms of meeting your investment goals in the long term. Of course there is a point at which increasing the leverage of leveraged ETFs will reduce returns. From backtesting I strongly assume that 3x is still below (or at) the optimum, but I might be wrong. It all depends on the formula and assumptions you make.
JackoC
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by JackoC »

tradri wrote: Sat Mar 27, 2021 10:11 am
JackoC wrote: Sat Mar 27, 2021 9:37 am E[r]=L*mu-(L-1)r-0.5*L(L-1)*vol^2-ER/L
I do acknowledge that financing costs also have to be subtracted from the leveraged ETF returns, but I don't think it's as easy as simply subtracting the 12-month LIBOR rate from the yearly leveraged ETF return, as it has been proposed in the "Simulating Returns of Leveraged ETFs" thread.

By checking the Tracking Difference of leveraged ETFs over the time period 2016-2020 when LIBOR rates were above 1%, it should be easy to see whether these can be easily added to the LETFs tracking difference or not. From the backtests I have done, the UPRO only experienced additional expenses of about 0.55% annually over that time period, even though the LIBOR rates were 1-3%.

So, I am not saying financing costs don't matter, I am saying that using a simple formula that subtracts the LIBOR rates from the returns isn't how it played out in 2016-2020. On top of that, I have read articles that talk about how the LIBOR is going to be discontinued and replaced by the SOFR, which according to its Wikipedia article should reduce borrowing costs.

I only talked about "volatility doesn't matter" in terms of meeting your investment goals in the long term. Of course there is a point at which increasing the leverage of leveraged ETFs will reduce returns. From backtesting I strongly assume that 3x is still below (or at) the optimum, but I might be wrong. It all depends on the formula and assumptions you make.
Remember that equation, and forget or greatly downgrade your emphasis on 'backtesting' is my closing advice to you. If your 'backtesting' shows 3% short term financing rate isn't visible in the returns on 3X, vs just getting 3X the S&P return, you're probably doing something wrong, or the data is corrupted. But anyway what happened in the fairly recent past never matters as much as many people here seem to think, but even less so for a product like this. See if you can come up with reasonable assumptions for each of the parameters in that equation *now* which result in 3X having a much higher expected return *now* than 1X. If you can, and assuming you ignore standard deviation (ie risk adjusted return in classic terms, Sharpe Ratio etc) you're good to go with 3X as a long term investment not a tool for trading or a tool to achieve <3 leverage combined with 1X. But I would highly recommend against coming up with estimated parameters that show a pretty poor expected return on 3X per the equation but then assuming your 'backtests' show some big hole in the equation, a portal to a magic domain where 3X funds give 3 times the return not subtracting finance cost and vol drag. But I have a feeling you'll do what you want to do, so good luck.
Topic Author
tradri
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

JackoC wrote: Sat Mar 27, 2021 10:30 am
tradri wrote: Sat Mar 27, 2021 10:11 am
JackoC wrote: Sat Mar 27, 2021 9:37 am E[r]=L*mu-(L-1)r-0.5*L(L-1)*vol^2-ER/L
I do acknowledge that financing costs also have to be subtracted from the leveraged ETF returns, but I don't think it's as easy as simply subtracting the 12-month LIBOR rate from the yearly leveraged ETF return, as it has been proposed in the "Simulating Returns of Leveraged ETFs" thread.

By checking the Tracking Difference of leveraged ETFs over the time period 2016-2020 when LIBOR rates were above 1%, it should be easy to see whether these can be easily added to the LETFs tracking difference or not. From the backtests I have done, the UPRO only experienced additional expenses of about 0.55% annually over that time period, even though the LIBOR rates were 1-3%.

So, I am not saying financing costs don't matter, I am saying that using a simple formula that subtracts the LIBOR rates from the returns isn't how it played out in 2016-2020. On top of that, I have read articles that talk about how the LIBOR is going to be discontinued and replaced by the SOFR, which according to its Wikipedia article should reduce borrowing costs.

I only talked about "volatility doesn't matter" in terms of meeting your investment goals in the long term. Of course there is a point at which increasing the leverage of leveraged ETFs will reduce returns. From backtesting I strongly assume that 3x is still below (or at) the optimum, but I might be wrong. It all depends on the formula and assumptions you make.
Remember that equation, and forget or greatly downgrade your emphasis on 'backtesting' is my closing advice to you. If your 'backtesting' shows 3% short term financing rate isn't visible in the returns on 3X, vs just getting 3X the S&P return, you're probably doing something wrong, or the data is corrupted. But anyway what happened in the fairly recent past never matters as much as many people here seem to think, but even less so for a product like this. See if you can come up with reasonable assumptions for each of the parameters in that equation *now* which result in 3X having a much higher expected return *now* than 1X. If you can, and assuming you ignore standard deviation (ie risk adjusted return in classic terms, Sharpe Ratio etc) you're good to go with 3X as a long term investment not a tool for trading or a tool to achieve <3 leverage combined with 1X. But I would highly recommend against coming up with estimated parameters that show a pretty poor expected return on 3X per the equation but then assuming your 'backtests' show some big hole in the equation, a portal to a magic domain where 3X funds give 3 times the return not subtracting finance cost and vol drag. But I have a feeling you'll do what you want to do, so good luck.
I encourage you to check the backtest as well.
Here is the Python code I used (taken from this article: https://teddykoker.com/2019/04/simulati ... in-python/): https://pastebin.com/KUyVK1Fs

You can change the start/end date, the leverage ratio, the expense ratio, as well as uncomment the data for the real URPO to compare the simulated UPRO to the performance of the actual UPRO.

Would you consider the data from Yahoo wrong?

I don't know exactly what I should do with an equation like the one you presented. Should I enter the historical averages for the variables of your equation? I feel more comfortable looking at the past data and drawing conclusions from there. (I assume long-run past returns are a pretty good indicator of what is to come in the future)
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steve r
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by steve r »

Understand how these things work. They rebalancing daily. Also understand the math. If $100 loses 20 percent down to $80. You need a 25 percent positive return to breakeven (25 percent of $80).

Now imagine over a stretch of years you have 200 days up 1 percent and 200 days down 1 percent. In this scenario you lose 2 percent. Excel command "=(1+.01)^200*(1-.01)^200" or "=1.01^200*0.99^200"

Now triple this, with 200 days up 3 percent and 200 days down 3 percent. In this scenario you lose 16.5 percent. (=1.03^200*0.97^200).

Worse, imagine you have 200 days up and down 2 percent over a decade or so but 3X to 6 percent. In this scenario you lose 51 percent. (=1.06^200*0.94^200).

Now you can play with the numbers and assume more big up days than down days, and great. But what if you are wrong. What if you have 210 days down 6 percent and 190 days up 6 percent. Well, you will be down 85 percent.

These vehicles are meant for day traders only, not buy and hold. If you want leverage, straight margin is the way to go. (I do not suggest this, but the math works out better).
"Owning the stock market over the long term is a winner's game. Attempting to beat the market is a loser's game. ..Don't look for the needle in the haystack. Just buy the haystack." Jack Bogle
Topic Author
tradri
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Re: Are 3x leveraged ETFs the long-term winning strategy?

Post by tradri »

steve r wrote: Sat Mar 27, 2021 10:59 am Now you can play with the numbers and assume more big up days than down days, and great. But what if you are wrong. What if you have 210 days down 6 percent and 190 days up 6 percent. Well, you will be down 85 percent.
I assume the long-term past performance is a good reflection of what is to be expected in the future.
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