Why are leveraged funds (2x/3x) so expensive?

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skierincolorado
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by skierincolorado »

nisiprius wrote: Sun Sep 19, 2021 8:14 am
skierincolorado wrote: Sun Sep 19, 2021 7:51 am...After factoring in #1 you should get close to 2x for ULPIX anyways. I did in my calculations. Any remaining difference, if any, would be because this is a terrible start date. Lots of analyses blow up when a late 90s start date is picked because it's right before two crashes...
I believe "since inception"--all available data, for real-world funds running real money with real expenses is the closest thing to unbiased one can be.

But my whole point is that ULPIX

1) most emphatically did not double the return of the S&P 500, and
2) it was not primarily due to fund expenses, although they probably were responsible for it literally underperforming the S&P 500.

As for "adding back in borrowing cost" I am darned if I can think of any justification for doing that.

Anyone who thinks they can double the return of the S&P 500 by buying and holding a 2X leveraged fund is likely to be severely disappointed. Their expectation will only be met if they can time the purchase for the start of a long, steady bull market.

Why not believe Direxion itself?
Q. Are Direxion ETFs Right for You?

A...Definitely not if you are a conservative investor who:
  • ...
  • Is unfamiliar with the unique nature and performance characteristics of funds which seek leveraged daily investment results
  • Is unable to manage your portfolio actively and make changes as market conditions and fund performance dictate.
    ...
"The unique nature and performance characteristics" is exactly what we are discussing, and they are unique, and are obviously hard to understand since they are as debated as the Monty Hall problem. Direxion says flatly that a leveraged ETF investor needs to manage their portfolio "actively." That means you have to believe you can time the market to make it work, you can't expect mechanical buying-and-holding to deliver that hoped-for 2X.

They also say:
Q. If the target index is up 10% for a month, shouldn't I expect to have a 30% gain in my Direxion Bull 3X ETF?

A. No, not typically.
and
Q. Are Direxion Shares ETFs appropriate for buy and hold investing?

A. No, this is not recommended. Leveraged ETFs seek daily investment results and should therefore be considered primarily for short-term trading purposes. [And special-case exceptions]
This is an argument against leverage generally because it incurs borrowing costs and is not specific to LETF. Any leveraged strategy, with any rebalance schedule or market timing strategy, would incur these costs. The point of the analysis is whether or not volatility decay occurs in addition to fees and financing costs. Your post amounts to a concession that volatility decay does not occur and that the underperformance of LeTFs is due to the same factors that cause all leveraged strategies to underperform: fees and borrowing costs.

Proving that LETFs, and all leveraged strategies, underperform their gearing ratio due to borrowing costs is a trivial question and trivial to prove. That wasn't the point of this discussion. The point is does volatility decay occur over long time horizons in efficient markets? And the answer remains no.

I repeat: volatility decay does not exist in the long term in efficient markets. Any underperformance is due to fees and financing costs which are inherent in all leveraged positions, including the proposed solution by typical.investor of fighting the daily rebalance. Your partial analysis has illustrated the point nicely. When we remove fees, we get much closer to the gearing ratio. When we remove borrowing costs, we'll get even closer. The same would be true for literally any levered strategy - not just LETF with daily rebalancing.

It's also worth noting that the proposed solution - fighting the daily rebalance - does nothing to improve our risk adjusted long term return compared to daily rebalancing, and incurs these same borrowing costs. If the excess funds are held in cash, the risk adjusted return of fighting the daily rebalance will be the same as not fighting it. The only benefit is coming from the diversification benefit if the excess funds are held in bonds. Again, obviously diversifying into bonds will improve risk-adjusted return - the benefits of diversification were proved many decades ago and we need not reprove it.

typical.investor has proposed that fighting the daily rebalance would guarantee 2x returns. That certainly is not true after borrowing costs, unless significantly more risk is being incurred, such as taking much more than 2x leverage.
Last edited by skierincolorado on Sun Sep 19, 2021 1:38 pm, edited 3 times in total.
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by skierincolorado »

typical.investor wrote: Sat Sep 18, 2021 6:12 pm

Very nice work Nisi!!!!

More empirical evidence that as expected under theory, that the long term returns of a leveraged fund doesn't necessarily approximate a return of the corresponding multiple of the index.

To ensure an approximation, one must adjust the amount of exposure via cashflow (by removing exposure when the fund out performs and adding exposure when it under performs).
The above analysis does not subtract borrowing costs. Your proposed solution of fighting the daily rebalance by adding to the position when it goes down, would also not come close to a 2x return after financing costs (or fees).

Could you provide detail on how your proposed solution would work that does not include investing in other asset classes such as bonds? Obviously diversifying into bonds will improve returns and risk-adjusted returns due to the diversification benefit. But how do you propose moving between cash and S&P500 leverage? I would expect that whatever strategy you propose will be very near the 1.4x return of ULPIX since 1998 after financing costs. You may be able to do slightly better due to the excess volatility since 1998 above the historical mean and above market expectations. It will be well under 2x though after financing costs, unless you take significantly more risk overall. And of course going forward we should not expect that volatility will be as high as 1998-present. Whatever strategy you propose would do worse than daily rebalancing in a low volatility environment.


For example would you start at 2x and simply not rebalance if the market went down? This would lead to margin calls. Would you have start at 1.5x and have rebalancing bands to keep leverage between 1.5x and 2.5x? Again don't use bonds because then you'll be reaping a benefit from diversification, not avoiding volatility decay. If bonds correlate with stocks, any benefit you may have had would dissapear entirely.
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by typical.investor »

skierincolorado wrote: Sun Sep 19, 2021 10:51 am
typical.investor wrote: Sat Sep 18, 2021 6:12 pm

Very nice work Nisi!!!!

More empirical evidence that as expected under theory, that the long term returns of a leveraged fund doesn't necessarily approximate a return of the corresponding multiple of the index.

To ensure an approximation, one must adjust the amount of exposure via cashflow (by removing exposure when the fund out performs and adding exposure when it under performs).
The above analysis does not subtract borrowing costs. Your proposed solution of fighting the daily rebalance by adding to the position when it goes down, would also not come close to a 2x return after financing costs (or fees).
Hello? Maybe if you read what I actually wrote ...sigh. Let's rewind, shall we.
skierincolorado wrote: Fri Sep 17, 2021 6:04 pm
typical.investor wrote: Fri Sep 17, 2021 5:56 pm
Because people need to know that the only way to offset volatility drag is by taking money out when it does well and putting more in when it doesn’t? That’s the only way you can earn 3X (minus costs).
Not really true. In theory, volatility drag shouldn't exist in the long-run.
And then to recap, you asked for theoretical and empirical proof of what I wrote which I provided.
skierincolorado wrote: Sun Sep 19, 2021 10:14 am typical.investor has proposed that fighting the daily rebalance would guarantee 2x returns. That certainly is not true after borrowing costs, unless significantly more risk is being incurred, such as taking much more than 2x leverage.
Have I said that though? I encourage you to read more accurately and request you not misquote me. I have not proposed what you claim I proposed, and to show that above have quoted your quote of my quote proving you knew, or should have known that I have said no such thing. Never thought it. Never wrote it.
skierincolorado wrote: Sun Sep 19, 2021 10:14 am I repeat: volatility decay does not exist in the long term in efficient markets. Any underperformance is due to fees and financing costs which are inherent in all leveraged positions
Guess what. Saying the same false thing over and over won't make it true.

Here is what Ivan Ivanov from the Board of Governors of the Federal Reserve System says in his paper:
holding the ETF for more than one period without rebalancing his portfolio generally will not provide the investor with a return that equals m-times the performance of the index over the longer holding period. This mismatch of returns is due to the fact that the return process for the ETF differs from the index return process.
It's beyond financing costs and fees. It's a difference in the return process and a result of the leverage reset.

Again, no matter how many times you claim that I claim cashflows will make up for costs, I have not made that claim. All I have said is that they can be used to keep you on target of m-times the performance of the index (minus costs as I previously posted and which you repeatedly ignore).

And no matter how many times you claim the leverage reset is not expected to have an effect long term, Ivan Ivanov from the Board of Governors of the Federal Reserve System is not agreeing with you from a theoretical and empirical perspective. Me too!
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by skierincolorado »

typical.investor wrote: Mon Sep 20, 2021 12:08 am All I have said is that they can be used to keep you on target of m-times the performance of the index.
Without getting into a pissing match of who said what, I will simply say: So do LETFs, long-term, in an efficient market (minus costs).

Nisi's analysis proves this to be the case for the longest running 2x LETF since 1998. Subtracting the fees take ULPIX from 0.9x to 1.4x. Subtracting financing costs will take us very close to 2x. And theory also proves that this must be the case in the long-run, in an efficient market, or else there would be risk-free return for hedge funds to arbitrage.

Markets may be temporarily irrational, temporarily inefficient. So over short and medium time horizons there is some additional risk of LETFs deviating from the gearing ratio. But if there was a long-term statistically significant premium or cost to daily rebalancing, hedge funds would arbitrage it away.
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by 000 »

skierincolorado wrote: Mon Sep 20, 2021 12:14 am Markets may be temporarily irrational, temporarily inefficient. So over short and medium time horizons there is some additional risk of LETFs deviating from the gearing ratio. But if there was a long-term statistically significant premium or cost to daily rebalancing, hedge funds would arbitrage it away.
Deviation from the gearing ratio can occur even in a mundane sideways market.
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by Mirakel23 »

skierincolorado wrote: Mon Sep 20, 2021 12:14 am
typical.investor wrote: Mon Sep 20, 2021 12:08 am All I have said is that they can be used to keep you on target of m-times the performance of the index.
Without getting into a pissing match of who said what, I will simply say: So do LETFs, long-term, in an efficient market (minus costs).

Nisi's analysis proves this to be the case for the longest running 2x LETF since 1998. Subtracting the fees take ULPIX from 0.9x to 1.4x. Subtracting financing costs will take us very close to 2x. And theory also proves that this must be the case in the long-run, in an efficient market, or else there would be risk-free return for hedge funds to arbitrage.

Markets may be temporarily irrational, temporarily inefficient. So over short and medium time horizons there is some additional risk of LETFs deviating from the gearing ratio. But if there was a long-term statistically significant premium or cost to daily rebalancing, hedge funds would arbitrage it away.
This could be checked using Simba's spreadsheet. It contains data starting from 1955. I am pretty sure that even with no borrowing costs and a TER of zero, the returns of a 2x strategy would not have amounted to 2x the return of the underlying. This is also pretty clear for the 3x ETF simulation, which did not significantly outperform the 1x S&P on a CAGR basis.

The reason for that is the fact that the growth rate of a leveraged strategy is proportional to the return of the underlying times leverage ratio, which, however, is reduced by a volatility factor that is quadratic w.r.t. to leverage ratio, see Eq.7 in this paper:
https://papers.ssrn.com/sol3/papers.cfm ... id=1510344
Otherwise there would be no optimal leverage ratio, but there is and it is a compromise between return and (ex-ante unidentifiable) volatility decay.

Note that the path dependence is not inherent to LETFs, it is simply a result of constant leverage ratio. It doesn't matter whether you use options, futures, a margin loan.. keeping your leverage ratio and thereby your risk aversion constant (i.e., daily rebalancing) will result in volatility decay, which reduces the growth rate as described in the above-referenced paper.
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by nisiprius »

As best I can figure at the moment, the disagreement might be this:

Looking at total, cumulative return over periods of much more than a day, comparing a LETF with a gearing ratio of G to the index X it's tracking,

Both sides agree that the return of LETF will not be exactly G·X.

Both sides agree that the return of LETF usually will differ in amount, and sometimes even in direction, from G·X.

Both sides agree that there have been times with real funds in the real world running real money when the differences from G·X would have been consequential, even catastrophic, for the investor if they really expected G·X.

Both sides agree that this would be true even if LETF had a 0% expense ratio.

The question is whether, over many averages of time periods--far more than have existed historically and therefore not studiable except in theoretical simulations--looking at the average of alternate universes--

--the simple long-term buy-and-hold total return of LETF will be systematically biased and less than G·X

or

--the simple long-term buy-and-hold total return of LETF will actually be G·X.

I guess my reaction is "who cares?" Even if the second is true, it doesn't make it reasonable to buy and hold an individual LETF as an easy way to boost returns (as recommended by the authors of "Lifecycle Investing!") The issue isn't return, it is the amount of risk involved for the long-term buy-and-hold investor.

Let me add a third question to the list.

Will the Sharpe ratio of the LETF be the same as, or systematically less than that of the unleveraged index?
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by Lee_WSP »

Regardless of rebalancing time periods, it will never be simply G * X, it will always be G * X - expenses because leverage is never free.
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by skierincolorado »

Mirakel23 wrote: Mon Sep 20, 2021 4:46 am

This could be checked using Simba's spreadsheet. It contains data starting from 1955. I am pretty sure that even with no borrowing costs and a TER of zero, the returns of a 2x strategy would not have amounted to 2x the return of the underlying. This is also pretty clear for the 3x ETF simulation, which did not significantly outperform the 1x S&P on a CAGR basis.

The reason for that is the fact that the growth rate of a leveraged strategy is proportional to the return of the underlying times leverage ratio, which, however, is reduced by a volatility factor that is quadratic w.r.t. to leverage ratio, see Eq.7 in this paper:
https://papers.ssrn.com/sol3/papers.cfm ... id=1510344
Otherwise there would be no optimal leverage ratio, but there is and it is a compromise between return and (ex-ante unidentifiable) volatility decay.

Note that the path dependence is not inherent to LETFs, it is simply a result of constant leverage ratio. It doesn't matter whether you use options, futures, a margin loan.. keeping your leverage ratio and thereby your risk aversion constant (i.e., daily rebalancing) will result in volatility decay, which reduces the growth rate as described in the above-referenced paper.
How do you remove borrowing cost in Simba's spreadsheet? I see how to set the ER to 0, but it's cearly also got borrowing costs factored in. This seems to be present in the raw data before any calculations are made, so I am not sure it is removable.

The limit to leverage doesn't come from expected volatility decay. It comes from expenses, borrowing costs, and limits to risk (including the possibility of volatility decay especially over shorter horizons).
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by skierincolorado »

nisiprius wrote: Mon Sep 20, 2021 5:36 am As best I can figure at the moment, the disagreement might be this:

Looking at total, cumulative return over periods of much more than a day, comparing a LETF with a gearing ratio of G to the index X it's tracking,

Both sides agree that the return of LETF will not be exactly G·X.

Both sides agree that the return of LETF usually will differ in amount, and sometimes even in direction, from G·X.

Both sides agree that there have been times with real funds in the real world running real money when the differences from G·X would have been consequential, even catastrophic, for the investor if they really expected G·X.

Both sides agree that this would be true even if LETF had a 0% expense ratio.

The question is whether, over many averages of time periods--far more than have existed historically and therefore not studiable except in theoretical simulations--looking at the average of alternate universes--

--the simple long-term buy-and-hold total return of LETF will be systematically biased and less than G·X

or

--the simple long-term buy-and-hold total return of LETF will actually be G·X.

I guess my reaction is "who cares?" Even if the second is true, it doesn't make it reasonable to buy and hold an individual LETF as an easy way to boost returns (as recommended by the authors of "Lifecycle Investing!") The issue isn't return, it is the amount of risk involved for the long-term buy-and-hold investor.

Let me add a third question to the list.

Will the Sharpe ratio of the LETF be the same as, or systematically less than that of the unleveraged index?
This is a good summary. Instead of:

"the simple long-term buy-and-hold total return of LETF will actually be G·X."

I would further clarify

the simple long-term buy-and-hold total return of LETF will actually be *near* G·X, in an efficient market, before fees and borrowing costs.


You ask what the sharpe ratio will be. Sharpe ratio should be *near* G*X, in an efficient market, before fees - but not before borrowing costs. Note that the sharpe ratios should be similar after subtracting fees but not borrowing costs, because sharpe ratios are performance over the RFR and the borrowing cost should be near the RFR. For ULPIX, I still expect that the sharpe ratio would be a little lower since 1998 simply because this was such an abnormally turbulent period for the market. If we can extend a simulation back to 1955, I would expect the sharpe ratio, pre-fees, to be similar - assuming that borrowing is at the RFR. If borrowing costs exceed the RFR (which in reality they would by a hair, especially before the advent of futures), then this will lower the sharpe ratio of geared returns.


You also ask, "who cares?" because the limiting factor is risk. If you are referring to the risk of LETF deviating from the gearing ratio over medium horizons, I agree this is an additional risk. But this is a risk with *all* leveraged strategies that must periodically rebalance the leverage ratio. Leveraging into or out of the stock market may occur at inoportune times. Diversifying across stocks and bonds, as in HFEA, doesn't reduce this risk but rather compensates for it through the benefits of diversification. Holding SSO exposes us to the same type of rebalancing risk as holding 2x leverage on margin. While with SSO the risk is that daily volatility is temporarily high, leading to volatility decay, the risk with 2x on margin with wider rebalancing bands is that quarterly volatility is high. The expectation with both, is that pre-fees pre-financing we return 2x. But over short and medium horizons it could be more or less. Over long enough time horizons though the expectation is 2x pre-fees&financing. The "Lifecycle Investing" proved that this leverage always pays off for long-term investors. For investors with long-time horizons, leverage can be highly desirable then. The choice then is does one do LETF or margin. The difference really comes down to the fees. As we see in the HFEA thread, many investors are happy to pay these fees for the convenience. Personally, I agree, for significant sums of money the fees are excessive so I would not invest significant sums of money using a LETF. Even for smaller sums of money, I generally recommend younger investors use futures just so they can start learning and use them when the sum of money grows.
Last edited by skierincolorado on Mon Sep 20, 2021 11:58 am, edited 1 time in total.
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by JackoC »

skierincolorado wrote: Sun Sep 19, 2021 10:14 am 1. Proving that LETFs, and all leveraged strategies, underperform their gearing ratio due to borrowing costs is a trivial question and trivial to prove.

2. I repeat: volatility decay does not exist in the long term in efficient markets. Any underperformance is due to fees and financing costs which are inherent in all leveraged positions, including the proposed solution by typical.investor of fighting the daily rebalance. Your partial analysis has illustrated the point nicely. When we remove fees, we get much closer to the gearing ratio. When we remove borrowing costs, we'll get even closer. The same would be true for literally any levered strategy - not just LETF with daily rebalancing.
See link for a simple derivation approximating the return of a rebalanced position under standard assumptions:
Return=L*mu -(L-1)*r -.5*L*(L-1)*vol^2, with L the leverage ratio, mu the stock return, r the borrowing cost or riskless return as the case may be (the equation is applicable for L > and < 1), assumed to have to zero correlation to mu for simplicity, and vol the annualized standard deviation of daily log stock returns.
https://link.springer.com/article/10.1057/jam.2010.16

1. True, the analysis you refer to leaves out the whole second and third terms of the equation.

2. But if I interpret your statement correctly to say the third term doesn't exist, it does also exist. I would go back to basics on this, not look at limited and perhaps inaccurate info on actual LETF's. Because indeed it's nothing exclusive to LETF's but applies to rebalanced portfolio's in general, with L > *or* < 1. I did the following. I took the daily value of the S&P 500 Total Return Index and proxy of a short term borrowing rate (the latter doesn't need to be exactly accurate, it barely affects the comparative result we're looking for) from 6/1/1988 (the date of inception of VXO/VIX, just data I already had handy) to last Friday. I took the V(Day 0) for simplicity be the starting value of SPTR. Then V(Day 1)=L*V(0)*daily S&P total return-(L-1)*V(0)*daily borrowing return, V(Day 2)=L*V(1)*daily SPTR return-(L-1)*daily borrowing return, etc. I copied down 100's of rows over 33 yrs of daily data, then compared total CAGR using the starting and ending values of the portfolio to the 'zero vol drag' estimate of L*mu-(L-1)*r with mu and r taken for whole period.
Results:
SPTR CAGR 11.19% pa. short term proxy CAGR 3.43% pa.
L=.6, 60%*11.19%+40%*3.43%=8.08% The actual daily rebalanced return was 8.44%: 'rebalancing bonus' 0.35% pa.
L=2, 2X position. 2*SPTR-money market return=18.95%, actual portfolio return 15.68%: 'vol drag' 3.27% pa.
L=3, 3*SPTR-2*money market=26.71%, actual return 16.41%, vol drag 10.3% pa.

What volatility would you have to plug into the equation to give those three answers? 17.2%, 18.1% and 18.5% respectively. Whereas the actual annualized vol of the data series was 18.0%. Pretty close, likewise plugging in the realized vol, would have estimated the third term as +.4%, -3.2% and -9.7% respectively. This is a typical result AFAIK: approximation of drag as proportional to .5*vol^2 is pretty good over extended periods for various asset classes. Also typical that drag is greater than whole period standard deviation predicts as L grows because deviation from lognormality of returns tends to be more damaging to high leverage. Sometimes when people say 'vol drag is a myth' they might mean different things than 'the third term doesn't exist', but it does exist. If direct study of LETF's doesn't show it, first guess should be something wrong with the data, analysis or too short a period to be meaningful. Anyway I'd strongly recommend doing it more hands on with the building blocks of stock daily return and financing rates because indeed it's not an artifact of LETF's, it's a feature of rebalanced portfolios.
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by skierincolorado »

JackoC wrote: Mon Sep 20, 2021 11:49 am
skierincolorado wrote: Sun Sep 19, 2021 10:14 am 1. Proving that LETFs, and all leveraged strategies, underperform their gearing ratio due to borrowing costs is a trivial question and trivial to prove.

2. I repeat: volatility decay does not exist in the long term in efficient markets. Any underperformance is due to fees and financing costs which are inherent in all leveraged positions, including the proposed solution by typical.investor of fighting the daily rebalance. Your partial analysis has illustrated the point nicely. When we remove fees, we get much closer to the gearing ratio. When we remove borrowing costs, we'll get even closer. The same would be true for literally any levered strategy - not just LETF with daily rebalancing.
See link for a simple derivation approximating the return of a rebalanced position under standard assumptions:
Return=L*mu -(L-1)*r -.5*L*(L-1)*vol^2, with L the leverage ratio, mu the stock return, r the borrowing cost or riskless return as the case may be (the equation is applicable for L > and < 1), assumed to have to zero correlation to mu for simplicity, and vol the annualized standard deviation of daily log stock returns.
https://link.springer.com/article/10.1057/jam.2010.16

1. True, the analysis you refer to leaves out the whole second and third terms of the equation.

2. But if I interpret your statement correctly to say the third term doesn't exist, it does also exist. I would go back to basics on this, not look at limited and perhaps inaccurate info on actual LETF's. Because indeed it's nothing exclusive to LETF's but applies to rebalanced portfolio's in general, with L > *or* < 1. I did the following. I took the daily value of the S&P 500 Total Return Index and proxy of a short term borrowing rate (the latter doesn't need to be exactly accurate, it barely affects the comparative result we're looking for) from 6/1/1988 (the date of inception of VXO/VIX, just data I already had handy) to last Friday. I took the V(Day 0) for simplicity be the starting value of SPTR. Then V(Day 1)=L*V(0)*daily S&P total return-(L-1)*V(0)*daily borrowing return, V(Day 2)=L*V(1)*daily SPTR return-(L-1)*daily borrowing return, etc. I copied down 100's of rows over 33 yrs of daily data, then compared total CAGR using the starting and ending values of the portfolio to the 'zero vol drag' estimate of L*mu-(L-1)*r with mu and r taken for whole period.
Results:
SPTR CAGR 11.19% pa. short term proxy CAGR 3.43% pa.
L=.6, 60%*11.19%+40%*3.43%=8.08% The actual daily rebalanced return was 8.44%: 'rebalancing bonus' 0.35% pa.
L=2, 2X position. 2*SPTR-money market return=18.95%, actual portfolio return 15.68%: 'vol drag' 3.27% pa.
L=3, 3*SPTR-2*money market=26.71%, actual return 16.41%, vol drag 10.3% pa.

What volatility would you have to plug into the equation to give those three answers? 17.2%, 18.1% and 18.5% respectively. Whereas the actual annualized vol of the data series was 18.0%. Pretty close, likewise plugging in the realized vol, would have estimated the third term as +.4%, -3.2% and -9.7% respectively. This is a typical result AFAIK: approximation of drag as proportional to .5*vol^2 is pretty good over extended periods for various asset classes. Also typical that drag is greater than whole period standard deviation predicts as L grows because deviation from lognormality of returns tends to be more damaging to high leverage. Sometimes when people say 'vol drag is a myth' they might mean different things than 'the third term doesn't exist', but it does exist. If direct study of LETF's doesn't show it, first guess should be something wrong with the data, analysis or too short a period to be meaningful. Anyway I'd strongly recommend doing it more hands on with the building blocks of stock daily return and financing rates because indeed it's not an artifact of LETF's, it's a feature of rebalanced portfolios.
This is great. We're actually talking about two very different things however. You're talking about CAGR. Volatility drag definitely happens on CAGR.. the CAGR will not be anywhere close to the gearing/leverage ratio. I'm talking about total return.

Using your numbers for the 2x position, you say you got a CAGR of 15.68% over 33.25 years vs 11.19% for 1x.

That means that the total return for 1x was a 34x return and the total return for the 2x position was a 127x return.

Thus the total return of a 2x daily rebalanced strategy was nearly 4x (127/34) compared to the unleveraged, and this is after borrowing costs.

This seems high to me, but shows there is no volatility drag in total return. We exceeded the gearing ratio by a lot. But this seems high to me, am I making a mistake? Or perhaps it's just because 1988-present is an exceptionally good period for the market. Nevertheless, this is a very long period of 33+ years during which the returns of daily rebalancing exceeded the leverage ratio. Hopefully that puts to rest the idea that volatility decay is an inherent feature of daily rebalancing total returns.


*IF* these numbers are accurate, I think it should put a definitive end to the idea that volatility drag *on total return* should be expected over long time horizons. Obviously, CAGR will not correspond to the gearing ratio. But total return should be close.
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by JackoC »

skierincolorado wrote: Mon Sep 20, 2021 12:15 pm
JackoC wrote: Mon Sep 20, 2021 11:49 am
skierincolorado wrote: Sun Sep 19, 2021 10:14 am 1. Proving that LETFs, and all leveraged strategies, underperform their gearing ratio due to borrowing costs is a trivial question and trivial to prove.

2. I repeat: volatility decay does not exist in the long term in efficient markets.
See link for a simple derivation approximating the return of a rebalanced position under standard assumptions:
Return=L*mu -(L-1)*r -.5*L*(L-1)*vol^2, with L the leverage ratio, mu the stock return, r the borrowing cost or riskless return as the case may be (the equation is applicable for L > and < 1), assumed to have to zero correlation to mu for simplicity, and vol the annualized standard deviation of daily log stock returns.
https://link.springer.com/article/10.1057/jam.2010.16

1. True, the analysis you refer to leaves out the whole second and third terms of the equation.

2. But if I interpret your statement correctly to say the third term doesn't exist, it does also exist.
This is great. We're actually talking about two very different things however. You're talking about CAGR. Volatility drag definitely happens on CAGR.. the CAGR will not be anywhere close to the gearing/leverage ratio. I'm talking about total return.

Using your numbers for the 2x position, you say you got a CAGR of 15.68% over 33.25 years vs 11.19% for 1x.

That means that the total return for 1x was a 34x return and the total return for the 2x position was a 127x return.

Thus the total return of a 2x daily rebalanced strategy was nearly 4x (127/34) compared to the unleveraged, and this is after borrowing costs.

This seems high to me, but shows there is no volatility drag in total return. We exceeded the gearing ratio by a lot.
You mean cumulative return. 'Total return' means annualized return of price plus reinvested dividends, if anyone else gets confused by that. The relevant measure of vol drag is in terms of annualized total return. Let's assume we've demonstrated the basic correctness of the equation Return=L*mu -(L-1)*r -.5*L*(L-1)*vol^2 where all terms are stated on an annualized basis. We can play with it substituting different values of mu assuming the same r (3.43% pa) and vol (18% pa), realizing the vol drag is 3.24% pa if L=2 and vol=18% (mu and r don't appear in that term) and calculating cumulative return over 33 yrs.

mu=11.19%, L=1 annualized return 11.19%, cumulative return 33X. L=2, annualized return 15.71% cumulative return 123X, ratio 3.7:1

mu=5%, L=1 return 5%, cumulative 5X; L=2, annualized return 3.33%, cumulative return 2.9X, 0.59:1.

mu=15%, L=1 return 15%, cumulative 100.7X; L=2 annualized return 40.8%, cumulative 1012.2X, 10.1:1.

Cumulative return is an inherently non-linear function of annual return depending on time. That's why we so seldom look at it for comparative purposes. All three of those cases have the same volatility drag (properly defined) but show vastly different cumulative return relationships of L=2 v L=1 depending on mu, as they also would if we changed the number of years. It's not meaningful to say the third case 'smashes the myth of vol drag' nor the second 'proves the devastating effect of vol drag'. Cumulative return is driven by the 'magic of compounding' over the period assumed. That's *why* it's categorically less confusing to look at return relationships on a per annum basis.

Your expected cumulative return L=2 vs. L=1 over a long enough period, assuming something like a historical level of mu-r and vol will be more than twice as big. But it's meaningless to infer from that 'therefore there's no volatility drag'. The driver of that result is simply high enough absolute return mu-r over a long enough period. Again, same goes for the 80% stock person over the 60% stock one, this relationship isn't even limited to leverage let alone just to LETF's. There's no implication that the cumulative return for the 80% stock investor will be 33% more than for the 60% stock investor. It all depends on mu, r (the 'riskless' return of the other 20-40%) and for how long, with the third term (vol anti-drag aka rebalancing bonus) being pretty minor in those cases.
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by Mirakel23 »

skierincolorado wrote: Mon Sep 20, 2021 11:34 am I would further clarify

the simple long-term buy-and-hold total return of LETF will actually be *near* G·X, in an efficient market, before fees and borrowing costs.
I did some more thinking and I would like to add that volatility drag does not specifically apply to LETFs, it is simply the result of constant leverage ratio (i.e., rebalancing). Therefore, SPY leveraged 3x on margin, rebalanced appropriately to maintain the leverage ratio of 3, will have identical returns to a 3x S&P500 LETF (assuming identical total costs, of course). There is no arbitrage opportunity here.
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by nisiprius »

To answer my own question about risk-adjusted return... from inception to 8/31/2021, the risk-adjusted return of ULPIX (2X daily-leveraged S&P 500 fund) was much lower than that of VFINX (unleveraged straight S&P 500 index fund), 0.36 and 0.50 respectively.

In order for the risk-adjusted return of ULPIX to have equaled that of VFINX, it would have needed to have an annual return 4.32% higher than it did.

I had PortfolioVisualizer show me past monthly returns for ULPIX (2X daily leveraged S&P 500 fund), VFINX ("straight" unleveraged S&P 500 index fund), and CASHX (PortfolioVisualizer's symbol for a risk-free cashlike return). I used all available data (12/31/1997 through 8/31/2021).

Data source

I copied and pasted the monthly returns data into a spreadsheet.

I did my own calculations of standard deviation and Sharpe ratio, and they matched those shown by PortvolioVisualizer. (For anyone trying to duplicate my results, note that the Sharpe ratio is based on the arithmetic average return, not the CAGR).

I then created a set of monthly returns equal to the sum of the ULPIX monthly return and the contents of a "booster" cell, and used Excel's Goal Seek to adjust the booster cell so as to raise the Sharpe ratio of the ULPIX data to match that of the VFINX data. It was necessary to add 0.360% to the monthly return = 4.32% annually.

I think it is reasonable to state this by saying that since inception, the return of ULPIX has fallen -4.32% per year short of matching the risk-adjusted return of an unleveraged fund.
Last edited by nisiprius on Mon Sep 20, 2021 4:47 pm, edited 1 time in total.
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by skierincolorado »

JackoC wrote: Mon Sep 20, 2021 2:26 pm
skierincolorado wrote: Mon Sep 20, 2021 12:15 pm
JackoC wrote: Mon Sep 20, 2021 11:49 am
skierincolorado wrote: Sun Sep 19, 2021 10:14 am 1. Proving that LETFs, and all leveraged strategies, underperform their gearing ratio due to borrowing costs is a trivial question and trivial to prove.

2. I repeat: volatility decay does not exist in the long term in efficient markets.
See link for a simple derivation approximating the return of a rebalanced position under standard assumptions:
Return=L*mu -(L-1)*r -.5*L*(L-1)*vol^2, with L the leverage ratio, mu the stock return, r the borrowing cost or riskless return as the case may be (the equation is applicable for L > and < 1), assumed to have to zero correlation to mu for simplicity, and vol the annualized standard deviation of daily log stock returns.
https://link.springer.com/article/10.1057/jam.2010.16

1. True, the analysis you refer to leaves out the whole second and third terms of the equation.

2. But if I interpret your statement correctly to say the third term doesn't exist, it does also exist.
This is great. We're actually talking about two very different things however. You're talking about CAGR. Volatility drag definitely happens on CAGR.. the CAGR will not be anywhere close to the gearing/leverage ratio. I'm talking about total return.

Using your numbers for the 2x position, you say you got a CAGR of 15.68% over 33.25 years vs 11.19% for 1x.

That means that the total return for 1x was a 34x return and the total return for the 2x position was a 127x return.

Thus the total return of a 2x daily rebalanced strategy was nearly 4x (127/34) compared to the unleveraged, and this is after borrowing costs.

This seems high to me, but shows there is no volatility drag in total return. We exceeded the gearing ratio by a lot.
You mean cumulative return. 'Total return' means annualized return of price plus reinvested dividends, if anyone else gets confused by that. The relevant measure of vol drag is in terms of annualized total return. Let's assume we've demonstrated the basic correctness of the equation Return=L*mu -(L-1)*r -.5*L*(L-1)*vol^2 where all terms are stated on an annualized basis. We can play with it substituting different values of mu assuming the same r (3.43% pa) and vol (18% pa), realizing the vol drag is 3.24% pa if L=2 and vol=18% (mu and r don't appear in that term) and calculating cumulative return over 33 yrs.

mu=11.19%, L=1 annualized return 11.19%, cumulative return 33X. L=2, annualized return 15.71% cumulative return 123X, ratio 3.7:1

mu=5%, L=1 return 5%, cumulative 5X; L=2, annualized return 3.33%, cumulative return 2.9X, 0.59:1.

mu=15%, L=1 return 15%, cumulative 100.7X; L=2 annualized return 40.8%, cumulative 1012.2X, 10.1:1.

Cumulative return is an inherently non-linear function of annual return depending on time. That's why we so seldom look at it for comparative purposes. All three of those cases have the same volatility drag (properly defined) but show vastly different cumulative return relationships of L=2 v L=1 depending on mu, as they also would if we changed the number of years. It's not meaningful to say the third case 'smashes the myth of vol drag' nor the second 'proves the devastating effect of vol drag'. Cumulative return is driven by the 'magic of compounding' over the period assumed. That's *why* it's categorically less confusing to look at return relationships on a per annum basis.

Your expected cumulative return L=2 vs. L=1 over a long enough period, assuming something like a historical level of mu-r and vol will be more than twice as big. But it's meaningless to infer from that 'therefore there's no volatility drag'. The driver of that result is simply high enough absolute return mu-r over a long enough period. Again, same goes for the 80% stock person over the 60% stock one, this relationship isn't even limited to leverage let alone just to LETF's. There's no implication that the cumulative return for the 80% stock investor will be 33% more than for the 60% stock investor. It all depends on mu, r (the 'riskless' return of the other 20-40%) and for how long, with the third term (vol anti-drag aka rebalancing bonus) being pretty minor in those cases.
The point is that over very long time horizons these values should converge so that leverage ratios return approximately the return of their leverage ratio minus fees and borrowing costs. If there were highly statistically significant tendencies for vol to be high and return lower than what would cause convergence, then hedge funds could arbitrage and make risk free profit off of what is essential irrational volatility. Simply do the opposite of daily rebalancing, or short daily rebalancing within an overall hedged position. This provides some measure of assurance that over long time horizons cumulative return will come close to leverage ratio. It’s not a coincidence that we see this actually happening in the data. The cumulative return resulting from your equation is highly sensitive on the volatile parameter. Volatility just a few % higher would have resulted in cumulative returns much lower than the leverage ratio, or lower than not leveraging at all. Volatility could have been much higher or lower. But coincidentally it is close to the value that causes convergence of cumulative return ratios to leverage ratios. Over longer periods, or in more efficient markets, we would expect this convergence to be stronger.

Why is it that we see cagr of 11 and vol of 18? There are fundamental theoretical reasons for this relationship and when we deviate too far from them risk free arbitrage opportunities arise.

See my next post for an example.
Last edited by skierincolorado on Mon Sep 20, 2021 3:40 pm, edited 3 times in total.
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by MIretired »

Mirakel23 wrote: Mon Sep 20, 2021 2:57 pm
skierincolorado wrote: Mon Sep 20, 2021 11:34 am I would further clarify

the simple long-term buy-and-hold total return of LETF will actually be *near* G·X, in an efficient market, before fees and borrowing costs.
I did some more thinking and I would like to add that volatility drag does not specifically apply to LETFs, it is simply the result of constant leverage ratio (i.e., rebalancing). Therefore, SPY leveraged 3x on margin, rebalanced appropriately to maintain the leverage ratio of 3, will have identical returns to a 3x S&P500 LETF (assuming identical total costs, of course). There is no arbitrage opportunity here.
+1. Not for you the leveraged one; just for the market as a whole. Ie. arbitrage.
Last edited by MIretired on Mon Sep 20, 2021 3:29 pm, edited 1 time in total.
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by skierincolorado »

:sharebeer
Mirakel23 wrote: Mon Sep 20, 2021 2:57 pm
skierincolorado wrote: Mon Sep 20, 2021 11:34 am I would further clarify

the simple long-term buy-and-hold total return of LETF will actually be *near* G·X, in an efficient market, before fees and borrowing costs.
I did some more thinking and I would like to add that volatility drag does not specifically apply to LETFs, it is simply the result of constant leverage ratio (i.e., rebalancing). Therefore, SPY leveraged 3x on margin, rebalanced appropriately to maintain the leverage ratio of 3, will have identical returns to a 3x S&P500 LETF (assuming identical total costs, of course). There is no arbitrage opportunity here.
The arbitrage opportunity is not margin vs letf. The arbitrage opportunity is shorting letf within an overall approximately hedged position. If volatility decay were a statistically significant feature of long term stock returns, risk free profit could be made in this way. I could simply buy 10M of spy today and short 5M of SSO. There could be some ups and downs, but if the phenomenon is large and statistically significant, I would be guaranteed a risk free return after 10 years. I will have made more on the 10m of spy than I lost shorting 5M of sso. Since this is a mostly hedged position with very little net risk, I could apply very high leverage ratios typical of hedge funds of 5x or 10x leverage. My risk free return is even more cherished because it is not correlated with market returns. As a hedge fund I can market this non-correlated risk free return to clients that have extremely high demand for uncorrelated returns.

Why don’t hedge funds do this? Because they expect cumulative returns of sso to approximately match 2x spy - with uncertainty on either side of 2x. Over longer time horizons, in more efficient markets, there is greater confidence in this 2x.
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by MIretired »

skierincolorado wrote: Mon Sep 20, 2021 3:28 pm :sharebeer
Mirakel23 wrote: Mon Sep 20, 2021 2:57 pm
skierincolorado wrote: Mon Sep 20, 2021 11:34 am I would further clarify

the simple long-term buy-and-hold total return of LETF will actually be *near* G·X, in an efficient market, before fees and borrowing costs.
I did some more thinking and I would like to add that volatility drag does not specifically apply to LETFs, it is simply the result of constant leverage ratio (i.e., rebalancing). Therefore, SPY leveraged 3x on margin, rebalanced appropriately to maintain the leverage ratio of 3, will have identical returns to a 3x S&P500 LETF (assuming identical total costs, of course). There is no arbitrage opportunity here.
The arbitrage opportunity is not margin vs letf. The arbitrage opportunity is shorting letf within an overall approximately hedged position. If volatility decay were a statistically significant feature of long term stock returns, risk free profit could be made in this way. I could simply buy 10M of spy today and short 5M of SSO. There could be some ups and downs, but if the phenomenon is large and statistically significant, I would be guaranteed a risk free return after 10 years. I will have made more on the 10m of spy than I lost shorting 5M of sso. Since this is a mostly hedged position with very little net risk, I could apply very high leverage ratios typical of hedge funds of 5x or 10x leverage. My risk free return is even more cherished because it is not correlated with market returns. As a hedge fund I can market this non-correlated risk free return to clients that have extremely high demand for uncorrelated returns.

Why don’t hedge funds do this? Because they expect cumulative returns of sso to approximately match 2x spy - with uncertainty on either side of 2x. Over longer time horizons, in more efficient markets, there is greater confidence in this 2x.
OK. I give you that much, rehashing. But it's not perfect re volatily of the trade.
But you definitely rely on reversion to mean+. That is you get more than 2x on a bull market. But also in a bear. Compound leverage!

Edit: I mistakenly responded thinking you were replying to me. See my above post. Somewhat applies.
Last edited by MIretired on Mon Sep 20, 2021 3:47 pm, edited 1 time in total.
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by skierincolorado »

MIretired wrote: Mon Sep 20, 2021 3:43 pm
skierincolorado wrote: Mon Sep 20, 2021 3:28 pm :sharebeer
Mirakel23 wrote: Mon Sep 20, 2021 2:57 pm
skierincolorado wrote: Mon Sep 20, 2021 11:34 am I would further clarify

the simple long-term buy-and-hold total return of LETF will actually be *near* G·X, in an efficient market, before fees and borrowing costs.
I did some more thinking and I would like to add that volatility drag does not specifically apply to LETFs, it is simply the result of constant leverage ratio (i.e., rebalancing). Therefore, SPY leveraged 3x on margin, rebalanced appropriately to maintain the leverage ratio of 3, will have identical returns to a 3x S&P500 LETF (assuming identical total costs, of course). There is no arbitrage opportunity here.
The arbitrage opportunity is not margin vs letf. The arbitrage opportunity is shorting letf within an overall approximately hedged position. If volatility decay were a statistically significant feature of long term stock returns, risk free profit could be made in this way. I could simply buy 10M of spy today and short 5M of SSO. There could be some ups and downs, but if the phenomenon is large and statistically significant, I would be guaranteed a risk free return after 10 years. I will have made more on the 10m of spy than I lost shorting 5M of sso. Since this is a mostly hedged position with very little net risk, I could apply very high leverage ratios typical of hedge funds of 5x or 10x leverage. My risk free return is even more cherished because it is not correlated with market returns. As a hedge fund I can market this non-correlated risk free return to clients that have extremely high demand for uncorrelated returns.

Why don’t hedge funds do this? Because they expect cumulative returns of sso to approximately match 2x spy - with uncertainty on either side of 2x. Over longer time horizons, in more efficient markets, there is greater confidence in this 2x.
OK. I give you that much, rehashing. But it's not perfect re volatily of the trade.
But you definitely rely on reversion to mean+. That is you get more than 2x on a bull market. But also in a bear. Compound leverage!
Agreed! There is some definite risk over shorter horizons. And the risk doesn’t disappear for the longest horizons, even if the mean expectation should be that cumulative returns match the leverage ratio. Unless there is a premium for this risk, it’s one of the reasons I avoid letf (the other being fees). Although based on what Jacko posted it seems like maybe there is a premium for this risk since 1988?

Anyways, glad somebody else gets it!
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by MIretired »

Not exactly sure how you think compounding risk is more beneficial than linear expansion of risk. Other than it's really a momentum of risk. On the upside and downside in a mean reverting market.
But I see that only in a mean reverting market. But such is risk. I got a headache.
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by skierincolorado »

MIretired wrote: Mon Sep 20, 2021 4:10 pm Not exactly sure how you think compounding risk is more beneficial than linear expansion of risk. Other than it's really a momentum of risk. On the upside and downside in a mean reverting market.
But I see that only in a mean reverting market. But such is risk. I got a headache.
It’s not compounding the risk. The returns are compounded. Above normal volatility is the risk. But at least we agree the expected mean should be the leverage ratio, with some risk around it especially on shorter horizons.

And I’m not advocating for anything. Nothing is “more beneficial”. It’s just an academic point that cumulative returns should be expected to be near leverage ratios. And it’s inherent in literally all leveraged strategies including hfea. We can compensate for the risk of volatility with diversification, but it does not reduce it. Diversification is a benefit with or without leverage.

It’s an important point. If volatility decay were inherent, owning letf would make sense under no circumstances and honestly they should be banned to retail investors. If volatility decay and boost with expected but not guaranteed mean reversion is the reality, then owning letf may make sense to some if you can swallow the fees.
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by MIretired »

No. Yes it is strictly compounding the leverage (2X to 3X). Not the investment.
By the way. Is arbitrage to THE MARKET compounding? Or arithmetic, daily?
You are not not benefiting from arbitrage for the market. There is a loser and a winner every period.
You are taking momentum risk. Your markets MUST outperform 0%
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by nisiprius »

skierincolorado wrote: Mon Sep 20, 2021 4:20 pm...Above normal volatility is the risk. But at least we agree the expected mean should be the leverage ratio, with some risk around it especially on shorter horizons...
Only if you consider 24 years to be an example of "shorter horizons."

And that having your investment cut to a fifth of its value instead of a third (10/31/2007 to 2/28/2009, ULPIX compared to normally-2X-leveraged VFINX) is just "above normal volatility."
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by skierincolorado »

nisiprius wrote: Mon Sep 20, 2021 5:17 pm
skierincolorado wrote: Mon Sep 20, 2021 4:20 pm...Above normal volatility is the risk. But at least we agree the expected mean should be the leverage ratio, with some risk around it especially on shorter horizons...
Only if you consider 24 years to be an example of "shorter horizons."

And that having your investment cut to a fifth of its value instead of a third (10/31/2007 to 2/28/2009, ULPIX compared to normally-2X-leveraged VFINX) is just "above normal volatility."
What do you mean by 24 years? pre fees and pre borrowing costs that any leverage would incur, including hfea, the return was near the leverage ratio. As jacko proved, over 33 years, the returns exceeded the leverage ratio.

What do you mean by normally leveraged? Daily rebalancing reduces losses in down markets so I doubt that any other leverage plan would fair any better excluding fees and borrowing costs. Id ulpix did worse in 2009 it’s because of its very high fees and borrowing cost. I find that 2x vfinx was also cut to a fifth.
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by JackoC »

skierincolorado wrote: Mon Sep 20, 2021 3:17 pm
JackoC wrote: Mon Sep 20, 2021 2:26 pm
skierincolorado wrote: Mon Sep 20, 2021 12:15 pm
JackoC wrote: Mon Sep 20, 2021 11:49 am
skierincolorado wrote: Sun Sep 19, 2021 10:14 am 1. Proving that LETFs, and all leveraged strategies, underperform their gearing ratio due to borrowing costs is a trivial question and trivial to prove.

2. I repeat: volatility decay does not exist in the long term in efficient markets.
See link for a simple derivation approximating the return of a rebalanced position under standard assumptions:
Return=L*mu -(L-1)*r -.5*L*(L-1)*vol^2, with L the leverage ratio, mu the stock return, r the borrowing cost or riskless return as the case may be (the equation is applicable for L > and < 1), assumed to have to zero correlation to mu for simplicity, and vol the annualized standard deviation of daily log stock returns.
https://link.springer.com/article/10.1057/jam.2010.16
1. True, the analysis you refer to leaves out the whole second and third terms of the equation.

2. But if I interpret your statement correctly to say the third term doesn't exist, it does also exist.
This is great. We're actually talking about two very different things however. You're talking about CAGR. Volatility drag definitely happens on CAGR.. the CAGR will not be anywhere close to the gearing/leverage ratio. I'm talking about total return.
You mean cumulative return.
mu=11.19%, L=1 annualized return 11.19%, cumulative return 33X. L=2, annualized return 15.71% cumulative return 123X, ratio 3.7:1

mu=5%, L=1 return 5%, cumulative 5X; L=2, annualized return 3.33%, cumulative return 2.9X, 0.59:1.

mu=15%, L=1 return 15%, cumulative 100.7X; L=2 annualized return 40.8%, cumulative 1012.2X, 10.1:1.

Cumulative return is an inherently non-linear function of annual return depending on time.
1. The point is that over very long time horizons these values should converge so that leverage ratios return approximately the return of their leverage ratio minus fees and borrowing costs.

2. If there were highly statistically significant tendencies for vol to be high and return lower than what would cause convergence, then hedge funds could arbitrage and make risk free profit off of what is essential irrational volatility. Simply do the opposite of daily rebalancing, or short daily rebalancing within an overall hedged position. This provides some measure of assurance that over long time horizons cumulative return will come close to leverage ratio.

3. It’s not a coincidence that we see this actually happening in the data. The cumulative return resulting from your equation is highly sensitive on the volatile parameter. Volatility just a few % higher would have resulted in cumulative returns much lower than the leverage ratio, or lower than not leveraging at all. Volatility could have been much higher or lower. But coincidentally it is close to the value that causes convergence of cumulative return ratios to leverage ratios. Over longer periods, or in more efficient markets, we would expect this convergence to be stronger.

4. Why is it that we see cagr of 11 and vol of 18? There are fundamental theoretical reasons for this relationship and when we deviate too far from them risk free arbitrage opportunities arise.

5. See my next post for an example.
1. Demonstrably untrue in the example we just went through. The cumulative return of L=2 in 1988 to present was 3.7 times as much as L=1. But if we'd simply chosen a shorter period with the same annual returns the ratio would have been lower. Same returns for a longer period it would have been higher. There is no 'convergence' of the relative cumulative returns to anything. It's a completely mistaken concept I'm afraid. The annualized return of L=2 is less than twice the return of L=1 by a) the borrowing cost on one of the L's, b) the vol drag .5*L*(L-1)*vol^2 or just vol^2 in the L=2 case. What the ratio of cumulative return is over any 'long run' depends how long and what stock return, borrowing rate and vol turn out to be.

2. There is likewise no way to arbitrage something which has no requirement whatsoever to converge to anything.

3. The very risk you take in leveraging is that future returns will be unfavorable also considering future volatilities and hence vol drag. At L=1 you don't have to directly consider future vol in your estimate of return, at L<1 higher vol is slightly favorable to return (the 3rd term flips sign to be 'the rebalancing bonus'). At L>1 a higher vol directly subtracts from your return, if vol is higher than expected all else equal return lower than expected.

4. And there is no fixed relationship between vol and return. There are historical periods of significantly lower realized return than 1988-present with similar realized vol. And now's expected return is by any reasonable estimate much lower than 11% for stock (expected riskless borrowing also lower, but expected 2*mu-r still lower now than 1988-today realized), but implied vol on the longest term ATM SPX options is relatively similar to 18% (higher but not greatly). Opinions can vary on those expected values, but there is no arbitrageable relationship between them.

5. Your SPY v SSO example doesn't show anything. Let's assume hypothetical 0 ER SPY and SSO (otherwise it would actually make a little money to short high expense SSO and delta hedge it with lower cost instruments). At time zero $5mil short SSO is a perfect hedge for $10mil long SPY. But let's stop a minute there. If I bought the position with my own capital I'd now be set to earn the riskless rate on $5mil of my capital ($10mil purchase price of SPY minus $5mil proceeds of the short of SSO), not very hedge fund like. If instead I'm financing the whole position, I have to borrow that $5mil. So borrow $5mil to buy $10mil SPY...kind of sounds like an L=2 position, which it is! Now on day 2 the market moves a little. The SSO position is no longer $10mil but something slightly different. To remain hedged I have to shift my SPY position the same amount. Over time all I'm doing is being long L=2 that I rebalance myself vs. short L=2 the LETF rebalances for me, net nothing, whether vol drag exists or not. If there was some concept of 'long term convergence' where I could *not* rebalance the SPY v SSO position and come back in a long time and it would still be hedged...but there is again no basis for that assumption. In a long time the two positions would likely have become massively mismatched in one direction or the other depending on market returns. To actually hedge I must continuously rebalance and experience the same vol drag, whatever it is, that the LETF experiences.
skierincolorado
Posts: 2377
Joined: Sat Mar 21, 2020 10:56 am

Re: Why are leveraged funds (2x/3x) so expensive?

Post by skierincolorado »

JackoC wrote: Tue Sep 21, 2021 9:23 am

5. Your SPY v SSO example doesn't show anything. Let's assume hypothetical 0 ER SPY and SSO (otherwise it would actually make a little money to short high expense SSO and delta hedge it with lower cost instruments). At time zero $5mil short SSO is a perfect hedge for $10mil long SPY. But let's stop a minute there. If I bought the position with my own capital I'd now be set to earn the riskless rate on $5mil of my capital ($10mil purchase price of SPY minus $5mil proceeds of the short of SSO), not very hedge fund like. If instead I'm financing the whole position, I have to borrow that $5mil. So borrow $5mil to buy $10mil SPY...kind of sounds like an L=2 position, which it is! Now on day 2 the market moves a little. The SSO position is no longer $10mil but something slightly different. To remain hedged I have to shift my SPY position the same amount. Over time all I'm doing is being long L=2 that I rebalance myself vs. short L=2 the LETF rebalances for me, net nothing, whether vol drag exists or not. If there was some concept of 'long term convergence' where I could *not* rebalance the SPY v SSO position and come back in a long time and it would still be hedged...but there is again no basis for that assumption. In a long time the two positions would likely have become massively mismatched in one direction or the other depending on market returns. To actually hedge I must continuously rebalance, and experience the same vol drag, whatever it is, that the LETF experiences.
I'm going to focus on the last point because this is key. You've correctly pointed out that it's not possible to remain perfectly hedged the whole time. This is true. However, it would be possible to remain mostly hedged - especially if volatility drag on cumulative return were a real highly statistically signficiant phenomenon. And even if you have to post cash to cover the whole position and don't consider it hedged at all (although it *is* clearly perfectly hedged initially, the hedging may be less than perfect over time), it's still a guaranteed long-term return that is not correlated to market returns, which is extremely valuable.



Let's say that volatility drag on cumulative return were hypothetically a real highly statistically significant phenenon (I say hypothetical, because it's not in reality). Over multiple historical periods, with high statistical significance, 2x leverage with daily rebalancing returned substantially less than 2x. Let's take it to the extreme to illustrate the point. The volatility drag is so bad that over many decades the return of daily rebalancing is less than 1x the unleveraged position. If we buy 10M of SPY and short 5M of SSO and the market doubles, the 10M SPY doubles to 20M, and the 5M SSO doesn't even double - it goes to 8M. We just made 7M net. Maybe the position would not be fully hedged. So we have to post 2M cash to maintain the position (a hedge fund could probably buy 10M of SPY naked with 2M cash, without shorting the SSO to hedge). We just made 7M net on 2M of capital. If the market traded sideways, the 10M SPY maintains its value, and the SSO 'decays' to 1M. We just made 4M net on 2M capital in a sideways market. I guarantee you there would be an army of hedge funds shorting SSO and buying SPY. If the opposite was true, they would be shorting SPY and buying SSO.

If we expect volatility decay of leveraged cumulative returns, we can clearly arbitrage them by shorting SSO and buying SPY. This is true whether we consider the position hedged or not, although it *is* perfectly hedged initially and would require very little capital.
JackoC
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Re: Why are leveraged funds (2x/3x) so expensive?

Post by JackoC »

skierincolorado wrote: Tue Sep 21, 2021 9:40 am
JackoC wrote: Tue Sep 21, 2021 9:23 am

5. Your SPY v SSO example doesn't show anything. Let's assume hypothetical 0 ER SPY and SSO (otherwise it would actually make a little money to short high expense SSO and delta hedge it with lower cost instruments). At time zero $5mil short SSO is a perfect hedge for $10mil long SPY. But let's stop a minute there. If I bought the position with my own capital I'd now be set to earn the riskless rate on $5mil of my capital ($10mil purchase price of SPY minus $5mil proceeds of the short of SSO), not very hedge fund like. If instead I'm financing the whole position, I have to borrow that $5mil. So borrow $5mil to buy $10mil SPY...kind of sounds like an L=2 position, which it is! Now on day 2 the market moves a little. The SSO position is no longer $10mil but something slightly different. To remain hedged I have to shift my SPY position the same amount. Over time all I'm doing is being long L=2 that I rebalance myself vs. short L=2 the LETF rebalances for me, net nothing, whether vol drag exists or not. If there was some concept of 'long term convergence' where I could *not* rebalance the SPY v SSO position and come back in a long time and it would still be hedged...but there is again no basis for that assumption. In a long time the two positions would likely have become massively mismatched in one direction or the other depending on market returns. To actually hedge I must continuously rebalance, and experience the same vol drag, whatever it is, that the LETF experiences.
I'm going to focus on the last point because this is key. You've correctly pointed out that it's not possible to remain perfectly hedged the whole time. This is true. However, it would be possible to remain mostly hedged - especially if volatility drag on cumulative return were a real highly statistically signficiant phenomenon. And even if you have to post cash to cover the whole position and don't consider it hedged at all (although it *is* clearly perfectly hedged initially, the hedging may be less than perfect over time), it's still a guaranteed long-term return that is not correlated to market returns, which is extremely valuable.

Let's say that volatility drag on cumulative return were hypothetically a real highly statistically significant phenenon (I say hypothetical, because it's not in reality). Over multiple historical periods, with high statistical significance, 2x leverage with daily rebalancing returned substantially less than 2x. Let's take it to the extreme to illustrate the point. The volatility drag is so bad that over many decades the return of daily rebalancing is less than 1x the unleveraged position. If we buy 10M of SPY and short 5M of SSO and the market doubles, the 10M SPY doubles to 20M, and the 5M SSO doesn't even double - it goes to 8M. We just made 7M net. Maybe the position would not be fully hedged. So we have to post 2M cash to maintain the position (a hedge fund could probably buy 10M of SPY naked with 2M cash, without shorting the SSO to hedge). We just made 7M net on 2M of capital. If the market traded sideways, the 10M SPY maintains its value, and the SSO 'decays' to 1M. We just made 4M net on 2M capital in a sideways market. I guarantee you there would be an army of hedge funds shorting SSO and buying SPY. If the opposite was true, they would be shorting SPY and buying SSO.

If we expect volatility decay of leveraged cumulative returns, we can clearly arbitrage them by shorting SSO and buying SPY. This is true whether we consider the position hedged or not, although it *is* perfectly hedged initially and would require very little capital.
We're going around in a circle now, on your refusal to accept the fallacy of a proposition you keep repeating with no evidence or analytical justification: 'in the long run, the cumulative return of a L=2 position will tend to be twice as much as the return of an L=1 position'. Not true. I've shown and you haven't contested how the annualized return of L=2 is not 2*mu or 2*mu-r, it's 2*mu-r-vol^2. It's quite close in historical examples using the realized vol. The annual return will never approximate or converge to 2*mu in any year where there's any r or any vol. And the ratio of cumulative returns doesn't even have a linear relationship to the ratio of annual returns, the most puzzling part of your theory. If I earn 4% or 2% for 10 yrs the ratio of cumulative return is ~1.2:1. If I do it for 1000 yrs the ratio is 271,000:1. It does not 'converge' to anything. Likewise right now I'd say mu=6% and r=1%, and vol=19%. Therefore costless L=1 has expected return 6%, costless L=2 has expected return 2*6%-1%-.19^2=7.4%. If return met my expectation for 10 yrs the ratio of cumulative return would be 1.14:1, for 1,000 yrs it would be 499,000:1. The idea it would converge to a particular integer over an indefinite time interval makes no sense, that I can see.

Likewise my analysis of the SPV v SSO trade is correct, and your repetition 'we can clearly arbitrage' vol drag that way is baseless. If I just borrowed $5mil to set $10 mil SPY against -$5 mil SSO, and walked away for 30 yrs, there would be no tendency for those two positions to end up approximately hedged when I got back. The position would be a random outcome of not only the CAGR but *path* of returns of the S&P over that long period, therefore the net profit or loss of the mismatched position random. To keep the position hedged I must rebalance the SPY position, therefore experiencing the same vol drag as SSO does in the other direction. And if a smart trader could figure out a more optimal rebalancing interval than daily, that's not an arbitrage but alpha.
skierincolorado
Posts: 2377
Joined: Sat Mar 21, 2020 10:56 am

Re: Why are leveraged funds (2x/3x) so expensive?

Post by skierincolorado »

JackoC wrote: Tue Sep 21, 2021 11:11 am
skierincolorado wrote: Tue Sep 21, 2021 9:40 am
JackoC wrote: Tue Sep 21, 2021 9:23 am

5. Your SPY v SSO example doesn't show anything. Let's assume hypothetical 0 ER SPY and SSO (otherwise it would actually make a little money to short high expense SSO and delta hedge it with lower cost instruments). At time zero $5mil short SSO is a perfect hedge for $10mil long SPY. But let's stop a minute there. If I bought the position with my own capital I'd now be set to earn the riskless rate on $5mil of my capital ($10mil purchase price of SPY minus $5mil proceeds of the short of SSO), not very hedge fund like. If instead I'm financing the whole position, I have to borrow that $5mil. So borrow $5mil to buy $10mil SPY...kind of sounds like an L=2 position, which it is! Now on day 2 the market moves a little. The SSO position is no longer $10mil but something slightly different. To remain hedged I have to shift my SPY position the same amount. Over time all I'm doing is being long L=2 that I rebalance myself vs. short L=2 the LETF rebalances for me, net nothing, whether vol drag exists or not. If there was some concept of 'long term convergence' where I could *not* rebalance the SPY v SSO position and come back in a long time and it would still be hedged...but there is again no basis for that assumption. In a long time the two positions would likely have become massively mismatched in one direction or the other depending on market returns. To actually hedge I must continuously rebalance, and experience the same vol drag, whatever it is, that the LETF experiences.
I'm going to focus on the last point because this is key. You've correctly pointed out that it's not possible to remain perfectly hedged the whole time. This is true. However, it would be possible to remain mostly hedged - especially if volatility drag on cumulative return were a real highly statistically signficiant phenomenon. And even if you have to post cash to cover the whole position and don't consider it hedged at all (although it *is* clearly perfectly hedged initially, the hedging may be less than perfect over time), it's still a guaranteed long-term return that is not correlated to market returns, which is extremely valuable.

Let's say that volatility drag on cumulative return were hypothetically a real highly statistically significant phenenon (I say hypothetical, because it's not in reality). Over multiple historical periods, with high statistical significance, 2x leverage with daily rebalancing returned substantially less than 2x. Let's take it to the extreme to illustrate the point. The volatility drag is so bad that over many decades the return of daily rebalancing is less than 1x the unleveraged position. If we buy 10M of SPY and short 5M of SSO and the market doubles, the 10M SPY doubles to 20M, and the 5M SSO doesn't even double - it goes to 8M. We just made 7M net. Maybe the position would not be fully hedged. So we have to post 2M cash to maintain the position (a hedge fund could probably buy 10M of SPY naked with 2M cash, without shorting the SSO to hedge). We just made 7M net on 2M of capital. If the market traded sideways, the 10M SPY maintains its value, and the SSO 'decays' to 1M. We just made 4M net on 2M capital in a sideways market. I guarantee you there would be an army of hedge funds shorting SSO and buying SPY. If the opposite was true, they would be shorting SPY and buying SSO.

If we expect volatility decay of leveraged cumulative returns, we can clearly arbitrage them by shorting SSO and buying SPY. This is true whether we consider the position hedged or not, although it *is* perfectly hedged initially and would require very little capital.

Likewise my analysis of the SPV v SSO trade is correct, and your repetition 'we can clearly arbitrage' vol drag that way is baseless. If I just borrowed $5mil to set $10 mil SPY against -$5 mil SSO, and walked away for 30 yrs, there would be no tendency for those two positions to end up approximately hedged when I got back. The position would be a random outcome of not only the CAGR but *path* of returns of the S&P over that long period, therefore the net profit or loss of the mismatched position random. To keep the position hedged I must rebalance the SPY position, therefore experiencing the same vol drag as SSO does in the other direction. And if a smart trader could figure out a more optimal rebalancing interval than daily, that's not an arbitrage but alpha.
The position does not have to be perfectly hedged. It doesn't even have to be hedged at all (although it is perfectly hedged initially). Buy 10M SPY, short 5M SSO. Walk away for 10 or 20 years. If volatility decay is real, this is a *guaranteed* return. You can do this with 100% cash and consider it completely unhedged. Or you can consider it hedged, which it is, and which will lower the capital requirement and amplify the return. Initially, the capital requirement would be nearly zero since it is perfectly hedged to start. Over time it may not be as hedged as it was initially, and the capital requirement might increase. But even if we do it in 100% cash, it's still a *guaranteed* return if volatility decay is real. And this *guaranteed* return is not correlated with market returns which makes it all the more valuable.
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