## Simulating Returns of Leveraged ETFs

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no simpler
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### Re: Simulating Returns of Leveraged ETFs

Here is my simple function that calculates a levered return series, net of daily trading costs:

Code: Select all

``````simulate_LETF <- function(daily_return_series,
daily_borrowing_costs,
leverage_ratio,
##assumes all inputs are vectors the same length as the source return series

magnified_source_return <- daily_return_series + (leverage_ratio-1)*(daily_return_series-daily_borrowing_costs)
return(net_returns) #returns vector of daily returns
}``````
All arguments are assumed to be vectors. Calculating compound return and geometric mean is trivial from the returned vector, as you would with any other return series.

Once nice thing is that by including leverage_ratio as a vector, you can simulate target volatility strategies that have dynamic leverage (eg, vol is fixed but leverage varies).
All inputs are straightforward to obtain, except for trading costs. I know there was some discussion back and forth - did we ever determine a good way to estimate trading cost? I will dig into the appendix of the aforementioned paper to see if I can get something.

siamond
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Joined: Mon May 28, 2012 5:50 am

### Re: Simulating Returns of Leveraged ETFs

no simpler wrote:
Sat Jul 20, 2019 5:20 pm
Here is my simple function that calculates a levered return series, net of daily trading costs:

Code: Select all

``````simulate_LETF <- function(daily_return_series,
daily_borrowing_costs,
leverage_ratio,
##assumes all inputs are vectors the same length as the source return series

magnified_source_return <- daily_return_series + (leverage_ratio-1)*(daily_return_series-daily_borrowing_costs)
return(net_returns) #returns vector of daily returns
}``````
Yup, we used an equivalent formula WHEN daily returns are available. That's the easy part. The trickier part is to develop a fairly reliable model in the absence of daily data... Also note that the expense ratio math has to added to such formula when computing total returns.
no simpler wrote:
Sat Jul 20, 2019 5:20 pm
All inputs are straightforward to obtain, except for trading costs. I know there was some discussion back and forth - did we ever determine a good way to estimate trading cost?
We discussed the matter and we did not find any such way. I'd love to solve this problem in a reasonably clean manner. For now, we compared the theoretical formula (without trading costs) to the reality of known leveraged ETFs, observed the difference over time and inferred friction costs (trading costs and more) that seem reasonable to apply to a lesser known past. Check all the Telltale charts I posted in this thread. Results are inconsistent though and such ad hoc approach is clearly NOT satisfying...

no simpler
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Joined: Sat Jul 13, 2019 4:54 pm

### Re: Simulating Returns of Leveraged ETFs

siamond wrote:
Sat Jul 20, 2019 8:29 pm

We discussed the matter and we did not find any such way. I'd love to solve this problem in a reasonably clean manner. For now, we compared the theoretical formula (without trading costs) to the reality of known leveraged ETFs, observed the difference over time and inferred friction costs (trading costs and more) that seem reasonable to apply to a lesser known past. Check all the Telltale charts I posted in this thread. Results are inconsistent though and such ad hoc approach is clearly NOT satisfying...
I think estimating trading costs empirically makes a lot of sense, and is a lot better than a bad theoretical model. In the paper I shared earlier, they do have a theoretical model of trading costs in the appendix, but it's so complex it's unrealistic (it takes into account changes in weights to all the assets inside of an index). I think the key is to at least model trading costs as a function of leverage amount and underlying index costs. The other thing the paper points out is that trading costs are non-stationary and declining over time. So it's probably impossible to get a precise model of future trading costs anyhow.

By leaving trading costs as a variable, we can simulate effect of changes in trading cost assumptions. We can also use a prior from the paper which assumes average annualized trading costs of 0.96% due to monthly leverage rebalancing, and 0.07% for the source portfolio, from 1929 - 2013. But even the paper admits they due not have a truly precise model.

siamond
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### Re: Simulating Returns of Leveraged ETFs

no simpler wrote:
Sun Jul 21, 2019 11:37 am
I think estimating trading costs empirically makes a lot of sense, and is a lot better than a bad theoretical model. In the paper I shared earlier, they do have a theoretical model of trading costs in the appendix, but it's so complex it's unrealistic (it takes into account changes in weights to all the assets inside of an index). I think the key is to at least model trading costs as a function of leverage amount and underlying index costs. The other thing the paper points out is that trading costs are non-stationary and declining over time. So it's probably impossible to get a precise model of future trading costs anyhow.

By leaving trading costs as a variable, we can simulate effect of changes in trading cost assumptions. We can also use a prior from the paper which assumes average annualized trading costs of 0.96% due to monthly leverage rebalancing, and 0.07% for the source portfolio, from 1929 - 2013. But even the paper admits they due not have a truly precise model.
If we could see a fairly consistent pattern across all the empirical tests we ran, I would be much more comfortable, but this is unfortunately not the case. So we're clearly still missing something, I'm afraid.

Thanks for sharing this new paper, I am traveling now, but I'll look at it when I get home.

Lee_WSP
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Location: Arizona

### Re: Simulating Returns of Leveraged ETFs

I'm throwing in the towel. Has anyone done a TYD simulation (3x 10yr treasury) and want to share the data?

MotoTrojan
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### Re: Simulating Returns of Leveraged ETFs

Lee_WSP wrote:
Tue Aug 06, 2019 12:53 pm
I'm throwing in the towel. Has anyone done a TYD simulation (3x 10yr treasury) and want to share the data?
The data is out there (Siamond has 3x 10 year returns) but TYD in general seems to be a pretty poor fund (doesn't track well, large spreads). My backtesting has shown there is not much improvement to mixing different duration bonds other than a change in effective duration.

For example a blend of TYD & TMF would have similar overall performance as a blend of EDV & TMF, TLT & TMF, TLT & EDV, IEF & TMF, etc... which has the same cumulative volatility (duration exposure). So if you want less overall exposure to interest rate changes but don't want to change your total allocation % to bonds then I would recommend dosing TMF with EDV, which has the maximum duration exposure without any leverage drag or high expenses (thus minimizing the amount of TMF you need). In other words, TMF & EDV with your desired duration is better than TMF & IEF with the same duration since that would have more TMF overall.

Lee_WSP
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Location: Arizona

### Re: Simulating Returns of Leveraged ETFs

forgot that extended duration was available as am ETF. I'm so used to trading in mutual funds only.

But that makes a lot of sense. My current idea is to use bl as a backbone against volatility. But the returns aren't great.

MotoTrojan
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### Re: Simulating Returns of Leveraged ETFs

Lee_WSP wrote:
Tue Aug 06, 2019 1:33 pm
forgot that extended duration was available as am ETF. I'm so used to trading in mutual funds only.

But that makes a lot of sense. My current idea is to use bl as a backbone against volatility. But the returns aren't great.
BL?

Lee_WSP
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Location: Arizona

### Re: Simulating Returns of Leveraged ETFs

Blv. Stupid auto correct. Vanguard long term bonds fund.

MotoTrojan
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### Re: Simulating Returns of Leveraged ETFs

Lee_WSP wrote:
Tue Aug 06, 2019 2:08 pm
Blv. Stupid auto correct. Vanguard long term bonds fund.
Gotcha. If you have any TMF involved then I’d use EDV instead and reduce TMF, win-win.

Lee_WSP
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Location: Arizona

### Re: Simulating Returns of Leveraged ETFs

MotoTrojan wrote:
Tue Aug 06, 2019 2:12 pm
Lee_WSP wrote:
Tue Aug 06, 2019 2:08 pm
Blv. Stupid auto correct. Vanguard long term bonds fund.
Gotcha. If you have any TMF involved then I’d use EDV instead and reduce TMF, win-win.
Exactly. I was using blv to reduce the volatility and smooth out the ride to make it more palatable.

Lee_WSP
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Location: Arizona

### Re: Simulating Returns of Leveraged ETFs

Has anyone done UBT and/or SSO simulated data? Is a shortcut just 2/3 of UPRO & TMF?

I think I can figure out how to do an excel spreadsheet if I can get the following information. Is the daily LIBOR rate available in a spreadsheet?

MotoTrojan
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### Re: Simulating Returns of Leveraged ETFs

Lee_WSP wrote:
Thu Aug 22, 2019 3:27 pm
Has anyone done UBT and/or SSO simulated data? Is a shortcut just 2/3 of UPRO & TMF?

I think I can figure out how to do an excel spreadsheet if I can get the following information. Is the daily LIBOR rate available in a spreadsheet?
The Simba spreadsheet has the 2x offerings but I would be curious how you plan to utilize them. I would think a combo of EDV/TMF or UPRO/VOO would be more efficient (less daily rebalancing decay and expenses) than holding a 2x fund. I can't really see any justifiable application for using them.

Lee_WSP
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Location: Arizona

### Re: Simulating Returns of Leveraged ETFs

MotoTrojan wrote:
Thu Aug 22, 2019 3:53 pm
Lee_WSP wrote:
Thu Aug 22, 2019 3:27 pm
Has anyone done UBT and/or SSO simulated data? Is a shortcut just 2/3 of UPRO & TMF?

I think I can figure out how to do an excel spreadsheet if I can get the following information. Is the daily LIBOR rate available in a spreadsheet?
The Simba spreadsheet has the 2x offerings but I would be curious how you plan to utilize them. I would think a combo of EDV/TMF or UPRO/VOO would be more efficient (less daily rebalancing decay and expenses) than holding a 2x fund. I can't really see any justifiable application for using them.
Lol, I was coming back to delete the post as I found the dataset.

I'm interested in ways to de lever from 3x. Hmm, interesting. I'll have to test that as well, I'll report back on any interesting findings.

MotoTrojan
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### Re: Simulating Returns of Leveraged ETFs

Lee_WSP wrote:
Thu Aug 22, 2019 4:11 pm
MotoTrojan wrote:
Thu Aug 22, 2019 3:53 pm
Lee_WSP wrote:
Thu Aug 22, 2019 3:27 pm
Has anyone done UBT and/or SSO simulated data? Is a shortcut just 2/3 of UPRO & TMF?

I think I can figure out how to do an excel spreadsheet if I can get the following information. Is the daily LIBOR rate available in a spreadsheet?
The Simba spreadsheet has the 2x offerings but I would be curious how you plan to utilize them. I would think a combo of EDV/TMF or UPRO/VOO would be more efficient (less daily rebalancing decay and expenses) than holding a 2x fund. I can't really see any justifiable application for using them.
Lol, I was coming back to delete the post as I found the dataset.

I'm interested in ways to de lever from 3x. Hmm, interesting. I'll have to test that as well, I'll report back on any interesting findings.
Sure thing, curious to hear. I settled on 43/57 UPRO/EDV for a larger holding (on top of some 55/45 UPRO/TMF on the side for the camaraderie).

siamond
Posts: 5136
Joined: Mon May 28, 2012 5:50 am

### Re: Simulating Returns of Leveraged ETFs

Lee_WSP wrote:
Thu Aug 22, 2019 4:11 pm
Lol, I was coming back to delete the post as I found the dataset.

Lee_WSP
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### Re: Simulating Returns of Leveraged ETFs

Hmm, you're right, nothing new was discovered. Slightly more deviation using 2x leveraged funds vs cutting the 3x funds with unleveraged funds. However, if you're constantly rebalancing, cutting down the number of funds to juggle may be worth it.

MotoTrojan
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### Re: Simulating Returns of Leveraged ETFs

Lee_WSP wrote:
Thu Aug 22, 2019 4:44 pm
Hmm, you're right, nothing new was discovered. Slightly more deviation using 2x leveraged funds vs cutting the 3x funds with unleveraged funds. However, if you're constantly rebalancing, cutting down the number of funds to juggle may be worth it.
My option only had 2 funds; how much are you trying to cut the leverage? If using M1 I don't think it is much effort to juggle 4.

Lee_WSP
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Location: Arizona

### Re: Simulating Returns of Leveraged ETFs

MotoTrojan wrote:
Thu Aug 22, 2019 5:03 pm
Lee_WSP wrote:
Thu Aug 22, 2019 4:44 pm
Hmm, you're right, nothing new was discovered. Slightly more deviation using 2x leveraged funds vs cutting the 3x funds with unleveraged funds. However, if you're constantly rebalancing, cutting down the number of funds to juggle may be worth it.
My option only had 2 funds; how much are you trying to cut the leverage? If using M1 I don't think it is much effort to juggle 4.
Splitting 25/25/25/25

3x SP500/SP500/3xLTT/LTT

yielded a less volatile portfolio with slightly less yield. The volatility improvement was nice, but if it comes at the cost of a lot of trading fees, it may not workout.

Using the 2x leveraged ETF's is certainly simpler. But the 3x ETF's still had bigger returns. Probably just easy enough to limit the leveraged risk you take on.

MotoTrojan
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### Re: Simulating Returns of Leveraged ETFs

Lee_WSP wrote:
Thu Aug 22, 2019 5:10 pm
MotoTrojan wrote:
Thu Aug 22, 2019 5:03 pm
Lee_WSP wrote:
Thu Aug 22, 2019 4:44 pm
Hmm, you're right, nothing new was discovered. Slightly more deviation using 2x leveraged funds vs cutting the 3x funds with unleveraged funds. However, if you're constantly rebalancing, cutting down the number of funds to juggle may be worth it.
My option only had 2 funds; how much are you trying to cut the leverage? If using M1 I don't think it is much effort to juggle 4.
Splitting 25/25/25/25

3x SP500/SP500/3xLTT/LTT

yielded a less volatile portfolio with slightly less yield. The volatility improvement was nice, but if it comes at the cost of a lot of trading fees, it may not workout.

Using the 2x leveraged ETF's is certainly simpler. But the 3x ETF's still had bigger returns. Probably just easy enough to limit the leveraged risk you take on.
Replacing your 50/50 3xLTT/LTT with just EDV will be close, it is more like a 1.5xLTT.

Lee_WSP
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### Re: Simulating Returns of Leveraged ETFs

MotoTrojan wrote:
Thu Aug 22, 2019 5:29 pm
Lee_WSP wrote:
Thu Aug 22, 2019 5:10 pm
MotoTrojan wrote:
Thu Aug 22, 2019 5:03 pm
Lee_WSP wrote:
Thu Aug 22, 2019 4:44 pm
Hmm, you're right, nothing new was discovered. Slightly more deviation using 2x leveraged funds vs cutting the 3x funds with unleveraged funds. However, if you're constantly rebalancing, cutting down the number of funds to juggle may be worth it.
My option only had 2 funds; how much are you trying to cut the leverage? If using M1 I don't think it is much effort to juggle 4.
Splitting 25/25/25/25

3x SP500/SP500/3xLTT/LTT

yielded a less volatile portfolio with slightly less yield. The volatility improvement was nice, but if it comes at the cost of a lot of trading fees, it may not workout.

Using the 2x leveraged ETF's is certainly simpler. But the 3x ETF's still had bigger returns. Probably just easy enough to limit the leveraged risk you take on.
Replacing your 50/50 3xLTT/LTT with just EDV will be close, it is more like a 1.5xLTT.
I think the expense ratio savings would be worth it. But I think just reducing the total leveraged position is a better way to reduce the volatility & max drawdown problem as the problem is purely psychological.

MotoTrojan
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### Re: Simulating Returns of Leveraged ETFs

Lee_WSP wrote:
Thu Aug 22, 2019 5:32 pm

I think the expense ratio savings would be worth it. But I think just reducing the total leveraged position is a better way to reduce the volatility & max drawdown problem as the problem is purely psychological.
It is not purely psychological; there is a point where more leverage is a bad thing.

InclusivInstitutions
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Thanks for all the work Siamond. How hard would it be to synthesize the returns of JPNL (Direxion Daily Japan Bull 3X)? The purpose would be to test Hedgefundie's portfolio on a worst case scenario with a prolonged bear/stagnating market.

Keep it up!

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### Re: Simulating Returns of Leveraged ETFs

InclusivInstitutions, Welcome! I moved your post into the relevant discussion.
To some, the glass is half full. To others, the glass is half empty. To an engineer, it's twice the size it needs to be.

siamond
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InclusivInstitutions wrote:
Sat Sep 07, 2019 5:02 pm
Thanks for all the work Siamond. How hard would it be to synthesize the returns of JPNL (Direxion Daily Japan Bull 3X)? The purpose would be to test Hedgefundie's portfolio on a worst case scenario with a prolonged bear/stagnating market.
Hi there. This is an interesting idea. There are a few challenges though:
- JPNL tracks the MSCI Japan index; monthly returns are available since 1970, but daily returns aren't available for this index until end of 2000. While the period of main interest is the 90s (Japanese asset bubble blowing up).
- I don't think we can reasonably use the daily volatility of the post-2000 Japanese market as a proxy for the very troubled asset bubble period. If we were to do that, we could make the simulation work based on monthly returns + volatility (we have a very good formula for that), but this doesn't seem a proper proxy.
- There is a Nikkei Price-only index which starts in the 50s, now that could provide a good volatility proxy, but unfortunately the daily values start mid-1995, which is better, but not good enough.
- I used the Russell/Nomura Japan Equity Indexes for another project a while ago, they start in 1980, but I am pretty sure this is monthly-only as well (I didn't check though).
- Then to truly validate Hedgefundie's ideas, we would need to ALSO synthesize the (leveraged) returns of Japanese LT Treasuries. And that is another ball of wax, which I would need to investigate...

I am traveling for a couple of weeks. This is the extent of my thinking for now. Will have to come back to it and think harder... Good question!

Lee_WSP
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### Re: Simulating Returns of Leveraged ETFs

Does anyone know why Simba's LETF spreadsheet's EDV returns over the 1965 - 1985 period are so much lower than 20 year treasuries? Is it because of zero coupon?

MotoTrojan
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### Re: Simulating Returns of Leveraged ETFs

Lee_WSP wrote:
Fri Sep 20, 2019 5:37 pm
Does anyone know why Simba's LETF spreadsheet's EDV returns over the 1965 - 1985 period are so much lower than 20 year treasuries? Is it because of zero coupon?
It did seem to be odd how much lower it was (drawdown worse than 60% compared to something in the 20’s ic I recall right), but it should certainly be lower by some amount due to higher duration.

Lee_WSP
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### Re: Simulating Returns of Leveraged ETFs

MotoTrojan wrote:
Sat Sep 21, 2019 10:30 am
Lee_WSP wrote:
Fri Sep 20, 2019 5:37 pm
Does anyone know why Simba's LETF spreadsheet's EDV returns over the 1965 - 1985 period are so much lower than 20 year treasuries? Is it because of zero coupon?
It did seem to be odd how much lower it was (drawdown worse than 60% compared to something in the 20’s ic I recall right), but it should certainly be lower by some amount due to higher duration.
It also never bounces back. It barely resembles the LTT growth chart.

MotoTrojan
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### Re: Simulating Returns of Leveraged ETFs

Lee_WSP wrote:
Sat Sep 21, 2019 10:59 am
MotoTrojan wrote:
Sat Sep 21, 2019 10:30 am
Lee_WSP wrote:
Fri Sep 20, 2019 5:37 pm
Does anyone know why Simba's LETF spreadsheet's EDV returns over the 1965 - 1985 period are so much lower than 20 year treasuries? Is it because of zero coupon?
It did seem to be odd how much lower it was (drawdown worse than 60% compared to something in the 20’s ic I recall right), but it should certainly be lower by some amount due to higher duration.
It also never bounces back. It barely resembles the LTT growth chart.
Worth noting that these are simulated returns, not real. These assets were not created that early.

Lee_WSP
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### Re: Simulating Returns of Leveraged ETFs

MotoTrojan wrote:
Sun Sep 22, 2019 9:40 am
Lee_WSP wrote:
Sat Sep 21, 2019 10:59 am
MotoTrojan wrote:
Sat Sep 21, 2019 10:30 am
Lee_WSP wrote:
Fri Sep 20, 2019 5:37 pm
Does anyone know why Simba's LETF spreadsheet's EDV returns over the 1965 - 1985 period are so much lower than 20 year treasuries? Is it because of zero coupon?
It did seem to be odd how much lower it was (drawdown worse than 60% compared to something in the 20’s ic I recall right), but it should certainly be lower by some amount due to higher duration.
It also never bounces back. It barely resembles the LTT growth chart.
Worth noting that these are simulated returns, not real. These assets were not created that early.
Hence my question. Is this a problem with the model, or is this a reasonable approximation of what happens to EDV during rising interest rate regimes?

siamond
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### Re: Simulating Returns of Leveraged ETFs

Lee_WSP wrote:
Fri Sep 20, 2019 5:37 pm
Does anyone know why Simba's LETF spreadsheet's EDV returns over the 1965 - 1985 period are so much lower than 20 year treasuries? Is it because of zero coupon?
I posted your question on this thread. Longinvest developed the model for both LT Treasuries and LT Strips, using the bond fund model discussed in the corresponding thread.

Lee_WSP
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### Re: Simulating Returns of Leveraged ETFs

siamond wrote:
Sun Sep 22, 2019 11:29 pm
Lee_WSP wrote:
Fri Sep 20, 2019 5:37 pm
Does anyone know why Simba's LETF spreadsheet's EDV returns over the 1965 - 1985 period are so much lower than 20 year treasuries? Is it because of zero coupon?
I posted your question on this thread. Longinvest developed the model for both LT Treasuries and LT Strips, using the bond fund model discussed in the corresponding thread.
I knew there was an origin thread, but I couldn't find it. Thanks!

Hydromod
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### Re: Simulating Returns of Leveraged ETFs

I'm curious about how to properly account for dividends when simulating leveraged ETFs. Excuse me if this is a bit naive.

This came up from RayKeynes repeated challenges regarding a simulating a sequence of monthly inputs to SPY/SSO/UPRO. There seems to be some discrepancy in the predicted results.

I have been trying to see what both RayKeynes and Siamond were doing. As far as I can tell, both methods are accounting for expense ratios and Libor costs. The major difference is that Siamond incorporated dividends and RayKeynes omitted dividends. Another is that Siamond worked up to monthly returns while RayKeynes stayed with daily calculations.

I started by stepping through the process with daily returns starting from 12/31/1954. In one case I used Shiller's averaged dividends and the other I used Siamond's painstakingly obtained dividends. The dividends are not the same, but the long-term CAGR works out to be very similar when I reproduce Siamond's SPY equivalent with zero expenses.

The question is how to incorporate the dividends for SSO and UPRO simulated funds. If I was tracking a portfolio, I'd increase the number of shares each time the dividends are provided, based on the current number of shares. The dividends are expressed as a price returned per share. So after a dividend event with reinvestment, I have N*d/P new shares, where N is the original number of shares, d is the return per share, and P is the current price.

For an unlevered fund and tracking an initial investment without additions, the total return is equivalent to adding the dividend return to the price return while keeping the same number of shares.

For a levered fund (neglecting expenses), the returned dividend is a multiple of the S&P dividend on the current number of shares, or N * m * d / P1, where m is the leverage and P1 is the unlevered S&P price. This is not the same as applying the dividend working directly with the equivalent SSO or UPRO price.

This formula is the only way I could get the same ending portfolio value, working with daily prices and actual shares, for the same series of investments as I arbitrarily changed the starting prices for SPY, SSO, and UPRO equivalents while preserving the relative price changes.

For a simple test case from 1955 through 2018 where
• Expenses are neglected
• Dividends are 3% annual, with 1/12 applied at the end of each month
• Unit starting price for SPY, SSO, and UPRO equivalents
• Unit starting investment into SPY, SSO, and UPRO equivalents
I end up with 1.6, 2.55, and 4.06 shares for SPY, SSO, and UPRO equivalents (monotonic increase in number of shares with leverage). If I turn off dividends but add the same fixed amount of 0.1 units each month, I end up with 19.7, 11.3, and 8.5 shares for SPY, SSO, and UPRO equivalents (monotonic decrease in number of shares with leverage).

I think Siamond may have calculated the dividend as an equivalent price change for the unlevered S&P, then the combined return was scaled by a volatility factor appropriate for price changes with leveraged ETFs.

Am I missing something here? If I'm right, it implies that the dividend calculation may need to be adjusted for the simulated UPRO and TMF funds.

I'm especially concerned that calculations with periodic additions may be misrepresented.

siamond
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### Re: Simulating Returns of Leveraged ETFs

Hydromod wrote:
Fri Nov 29, 2019 1:37 pm
I think Siamond may have calculated the dividend as an equivalent price change for the unlevered S&P, then the combined return was scaled by a volatility factor appropriate for price changes with leveraged ETFs.
Yes, this is correct. It's essentially a total-return kind of math. The research papers formulas that we used in our model seem to make the same assumption. Yet I am not 100% sure if this is correct or not.

When checking samples of daily returns between the levered fund and the underlying index, it became very clear that dividends of the underlying index were accounted for and that the leveraging multiplier had an impact. So there is no question that dividends have to be accounted for. Still, I had nagging doubts because equity levered funds always document that they track the index price series, not the total-return series (contrary to bond levered funds). But clearly the reality is that there is a form of leveraging happening with the equity index dividends. As a side note, UPRO and co returned very few dividends of their own and I couldn't figure out any rhyme or reason for the corresponding distributions.

Trouble is there is absolutely zero documentation on how this works in reality. I don't disagree with your doubts, but I am not quite sure your suggested improvement is quite right either. I really don't know. It may seem more logical to you, but is that really what's happening with a real-life fund? Anybody having a clue?

In any case, yes, it might be worth running a more in-depth empirical test on some of the well-known levered funds.

PS. note that I did use daily math when possible. That is, in most cases. But we do have to switch to monthly logic when going back in time (historical daily data is hard to find and often non-existent, notably when it comes to dividends & total-returns).

Hydromod
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### Re: Simulating Returns of Leveraged ETFs

Okay, a followup.

The ProShares web site clarified that the dividends provided by UPRO have nothing to do with the tracked S&P 500 index, they reflect profits from the daily swaps etc.

I found a statement buried in the gray matter at the end of the UPRO fact sheet.

ProShares may invest in equity securities and/or financial instruments (including derivatives) that, in combination, should have similar daily price return characteristics to the fund's benchmark. Derivative contracts are priced to include the underlying index yield and will not generate dividend income. Because ProShares invest in derivatives and other financial instruments, their dividend distributions may not reflect those of their applicable indexes.

Italics are mine.

This is the only place I was able to find some sort of explicit statement regarding daily price return versus total return. The first sentence implies matching the price return of the index. The italicized statement alters that to clarify that the underlying yields are priced in.

I'm not overly impressed with the clarity of the description, but it looks like my previous concern was moot and Siamond's interpretation of scaling total returns is confirmed.

I'll go back under my rock again.

typical.investor
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### Re: Simulating Returns of Leveraged ETFs

Hydromod wrote:
Fri Nov 29, 2019 8:37 pm
Okay, a followup.

...

I'll go back under my rock again.
Rather than your hiding away, perhaps you should simply conclude that Siamond rocks!

siamond
Posts: 5136
Joined: Mon May 28, 2012 5:50 am

### Re: Simulating Returns of Leveraged ETFs

Hydromod wrote:
Fri Nov 29, 2019 8:37 pm
I'm not overly impressed with the clarity of the description, but it looks like my previous concern was moot and Siamond's interpretation of scaling total returns is confirmed.

I'll go back under my rock again.
LOL, absolutely not, get out of there! You identified a couple of useful facts we had not spotted during our initial research. Thank you for doing that, this is progress. And to be honest, I do retain some doubts that that (index-level) dividends are processed in as simple a way as we modeled so far. We still have an imperfect mapping between the model and actual returns, and I've long suspected that part of it has to do with dividends. I just don't know how to proceed to go any further. It would be really great to interact with somebody who's been involved in the actual implementation of such a levered fund.

Uncorrelated
Posts: 261
Joined: Sun Oct 13, 2019 3:16 pm

### Re: Simulating Returns of Leveraged ETFs

I'm currently attempting to reproduce this effort for the interest of peer review. A few thoughts:

siamond mentioned before that we have access to S&P 500 monthly total return data series. Where can I find that data series?

I have simulated UPRO and SSO in my simulator with very similar results as siamond. But there appears to be a negative correlation between the S&P500 and simulation error. That is, when the S&P goes up UPRO seems to lag behind the simulation, and when the S&P goes down UPRO appears to gain ground relative to the simulation. When simulating with 2.85x (UPRO) or 1.95x (SSO) leverage the negative slopes disappear. This seems to indicate that the friction costs are higher on up days than on down days. Thoughts?

When regressing the difference between the simulation and UPRO/SSO for each day of the week, both ETF's display a positive alpha on monday (UPRO 5% annualized) and negative alpha on friday (UPRO -9.3% annualized). This effect appears to be unrelated to volatility differences between the weekdays.

Both effects persist when the borrow rate and ER of the simulated fund are set to zero, so I'm fairly confident that the effect isn't caused by the expense ratio or accrued interest/debt on the swap contract during the weekend. The magnitude of the errors also appears to be too large to be explained by these things.

The annual report of UPRO states that 77% of the market exposure is due securities, 215% in swap contracts and 8% in futures. We calculated earlier that the net exposure is 3x equities and -2x the borrow rate. But it appears that this fund pays the spread on the borrow rate on 215%+8% = 223% of it's assets. Has there been any effort in quantifying the size of the spread? As a data point, UPRO currently holds t-bills 2.31-2.37% and swap contracts 2.66%~3.11%, which seems to indicate the spread is significant.

The swap agreements from UPRO have the following note attached to it: "Reflects the floating financing rate, as of May 31, 2019, on the notional amount of the swap agreement paid to the counterparty or received from the counterparty, excluding any commissions. This amount is included as part of the unrealized appreciation/(depreciation).". How large are the commissions? What kinds of commissions are there? I did some digging in a few leveraged UCITS funds, but was unable to find any information (UTICS ETF's are required to disclose such costs, but nobody except vanguard actually does).

Finally, UPRO holds a large part of it's assets (15% in the annual report, 27% in semiannual) in repurchase agreements (various counterparties, rates 2.15% - 2.49%, dated 5/31/2019, due 6/3/2019, total to be received \$312,396,795 (Cost \$312,332,777)). Does anyone have a clue what these agreements are and how they impact the costs of the fund?
Last edited by Uncorrelated on Sun Dec 08, 2019 6:30 am, edited 1 time in total.

RayKeynes
Posts: 72
Joined: Mon Nov 11, 2019 2:14 am

### Re: Simulating Returns of Leveraged ETFs

Hey Uncorrelated,

Could you share with me your data series for UPRO and/or SSO on a daily basis since 1955 so I can verify the calculations so far?

Your thoughts are very interesting - however - I think for them to be answered we'd need to speak with someone who actually works in administrating a leveraged ETF. Somebody here vom Vanguard?

Uncorrelated
Posts: 261
Joined: Sun Oct 13, 2019 3:16 pm

### Re: Simulating Returns of Leveraged ETFs

I need access to a S&P 500 monthly total return dataset or better before I can simulate back to 1955. My current daily dataset (obtained from cboe) only goes back to 1988. So far, with a 1% curve fitting adjustment, my results appear to match siamond's bottom chart in this post exactly (although mine is a bit noisier).

Digging deeper, I noticed that the average swap contract rate from the annual report is 2.91%, around .55% above the t-bill rate. Although this is only one data point, it fits nicely with the 1% curve fitting adjustment. Also, SPXU (3x short) holds the same swap contracts as UPRO, but at a different interest rate (usually differs by ~0.2% depending on the bank). We might be able to use the difference between UPRO en SPXU as a proxy for swap spreads. The interesting part is that SPXU holds treasury bills (84% of net assets) at 2.4% and swap contracts at around minus 2.6%, implying that they get paid for these swap contracts at above market rates.

HEDGEFUNDIE
Posts: 3973
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### Re: Simulating Returns of Leveraged ETFs

Uncorrelated wrote:
Sun Dec 08, 2019 5:07 am
Finally, UPRO holds a large part of it's assets (15% in the annual report, 27% in semiannual) in repurchase agreements (various counterparties, rates 2.15% - 2.49%, dated 5/31/2019, due 6/3/2019, total to be received \$312,396,795 (Cost \$312,332,777)). Does anyone have a clue what these agreements are and how they impact the costs of the fund?
Repos are ultrashort term instruments similar to money market funds. They serve as cash management for the ETF.

siamond
Posts: 5136
Joined: Mon May 28, 2012 5:50 am

### Re: Simulating Returns of Leveraged ETFs

Uncorrelated wrote:
Sun Dec 08, 2019 7:45 am
Digging deeper, I noticed that the average swap contract rate from the annual report is 2.91%, around .55% above the t-bill rate.
Would somebody volunteer to document swap rates used for reasonably long-lived LETFs (2x and 3x) for every year of existence, tallying data from corresponding annual reports? This would seem an interesting data point to have. Then we can compare with historical LIBOR/EFFR rates (which we use in the model).

HEDGEFUNDIE
Posts: 3973
Joined: Sun Oct 22, 2017 2:06 pm

### Re: Simulating Returns of Leveraged ETFs

From UPRO’s annual report.
Compounding of Daily Returns and Volatility:Each fund seeks daily investment results, before fees and expenses, that correspond to the performance of a daily benchmark such as the multiple (i.e., 3x or 2x), the inverse (-1x) or an inverse multiple (i.e., -3x or -2x) of its underlying index for a single day only, not for any other period. For longer periods, performance may be greater than or less than a Fund’s one-day multiple times the index performance over the pe- riod, before fees and expenses. This is due to the effects of compounding, which exists in all investments, but has a more significant impact on geared funds. In general, during periods of higher index volatility, compounding will cause Fund performance for periods longer than a single day to be more or less than the multiple of the return of the index. This effect becomes more pronounced as volatility increases. Conversely, in periods of lower index volatility (particularly when combined with higher index returns), Fund returns over longer periods can be higher than the multiple of the return of the index.
This is the definitive explanation that has been vetted by fund lawyers and the SEC. So I would strongly suggest that the last italicized bit needs to be included in the wiki.

siamond
Posts: 5136
Joined: Mon May 28, 2012 5:50 am

### Re: Simulating Returns of Leveraged ETFs

HEDGEFUNDIE wrote:
Sun Dec 08, 2019 12:22 pm
From UPRO’s annual report.
Compounding of Daily Returns and Volatility:Each fund seeks daily investment results, before fees and expenses, that correspond to the performance of a daily benchmark such as the multiple (i.e., 3x or 2x), the inverse (-1x) or an inverse multiple (i.e., -3x or -2x) of its underlying index for a single day only, not for any other period. For longer periods, performance may be greater than or less than a Fund’s one-day multiple times the index performance over the pe- riod, before fees and expenses. This is due to the effects of compounding, which exists in all investments, but has a more significant impact on geared funds. In general, during periods of higher index volatility, compounding will cause Fund performance for periods longer than a single day to be more or less than the multiple of the return of the index. This effect becomes more pronounced as volatility increases. Conversely, in periods of lower index volatility (particularly when combined with higher index returns), Fund returns over longer periods can be higher than the multiple of the return of the index.
This is the definitive explanation that has been vetted by fund lawyers and the SEC. So I would strongly suggest that the last italicized bit needs to be included in the wiki.
Certainly something to consider, but then please post this material in the relevant thread and suggest where to insert such quote?

HEDGEFUNDIE
Posts: 3973
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### Re: Simulating Returns of Leveraged ETFs

siamond wrote:
Sun Dec 08, 2019 11:54 am
Uncorrelated wrote:
Sun Dec 08, 2019 7:45 am
Digging deeper, I noticed that the average swap contract rate from the annual report is 2.91%, around .55% above the t-bill rate.
Would somebody volunteer to document swap rates used for reasonably long-lived LETFs (2x and 3x) for every year of existence, tallying data from corresponding annual reports? This would seem an interesting data point to have. Then we can compare with historical LIBOR/EFFR rates (which we use in the model).
Here is where the historical reports can be found:

https://www.proshares.com/resources/xbrl_resources.html

Uncorrelated
Posts: 261
Joined: Sun Oct 13, 2019 3:16 pm

### Re: Simulating Returns of Leveraged ETFs

A fresh day, fresh idea's.

I attempted to estimate the borrow costs by doing some math with SSO and UPRO based on the following formula:

Code: Select all

``````rf = overnight libor as a proxy for risk free rate
(S&P500 - rf) * 2 = (SSO - rf) + borrow_spread
(S&P500 - rf) * 3 = (UPRO - rf) + 2 * borrow_spread

===>
3 * SSO - 2 * UPRO - rf = borrow_spread
``````
The borrow spread is equal to the cumulative borrow costs (spread over the risk free rate) since the last trading day, so we divide the borrow spread by the number of trading days since the last trading day to arrive at daily rates, and then we multiply by 360 again to get the annualized figure.

Here S&P500 is the inferred borrow spread based on SSO (2x S&P) and UPRO (3x S&P), and the lines represent the same calculations repeated for various other proshares 2x/3x funds. The data source for s&p500 midcap appears faulty.

Here is the same plot with a different filter.

Conclusions:
From 2010 to 2013 it appears that the different funds had substantially different borrow costs.
Since 2014 the different funds follow more or less the same trajectory.
The data appears to be too noisy to be of real use, but perhaps if we average enough funds together we can try to correlate it with spread data from the annual reports.

HEDGEFUNDIE wrote:
Sun Dec 08, 2019 12:43 pm
siamond wrote:
Sun Dec 08, 2019 11:54 am
Uncorrelated wrote:
Sun Dec 08, 2019 7:45 am
Digging deeper, I noticed that the average swap contract rate from the annual report is 2.91%, around .55% above the t-bill rate.
Would somebody volunteer to document swap rates used for reasonably long-lived LETFs (2x and 3x) for every year of existence, tallying data from corresponding annual reports? This would seem an interesting data point to have. Then we can compare with historical LIBOR/EFFR rates (which we use in the model).
Here is where the historical reports can be found:

https://www.proshares.com/resources/xbrl_resources.html
I can't find the data from UPRO/SSO in there. But with enough digging we might be able to find the reports on the SEC website: https://sec.report/CIK/0001174610/28#documents. I found one report from 2014 (t-bill rate 0.05% - 0.10%, swap rate: 0.20%~0.56%). If nobody volunteers I'll write a script to crawl the data later this week. Getting the data from the Quarterly Holding Report's as instead of the annual and semi-annual reports seems to give the highest resolution. Edit: currently working on it.

Uncorrelated
Posts: 261
Joined: Sun Oct 13, 2019 3:16 pm

### Re: Simulating Returns of Leveraged ETFs

I crawled the SEC reports to bring you these wonderful plots.

Exhibit 1: percentage of common stock holdings compared to the AUM:

Technically data back to 2005 is available, but I only went back to 2009 due to time constraints. The X axis on the plot represents the date the report was uploaded to the SEC, which does not necessarily match the date the holdings were measured.

Leveraged ETF's hold a portion of common stock, a portion of futures contracts and a portion of swap contracts. I ignored the futures contracts. I estimate that 2x ETF's have 72% common stock holdings and 3x ETF's have 58% common stock holdings. All inverse ETF's have 0% common stock holdings.

Why does this matter? A 2x ETF has 72% common stock holdings and 128% swap contracts. An 3x ETF has 58% common stock holdings and 242% swap contracts. An inverse 2x ETF has -200% swap contracts. The amount of swap contracts determine how many times the swap contract spread is paid.

Exhibit 2 weighted swap rate of proshares S&P500 over time:

There is no data prior to 2014 because the old annual reports don't contain that information. The X axis on the plot represents the date on which the swap rate was recorded (the reports explicitly state that this is the swap rate as of date X). Nowadays the swaps settle on the S&P index, but some older swaps settle on the SPDR S&P500 ETEF or some MSCI S&P500 ETF, which have additional costs (I didn't look further into this).

Exhibit 3 & 4: weighted swap rate of proshares S&P500 relative to the overnight LIBOR:

There are some oddballs, but in general:
The swap spread for 3x LETF's is about 0.04% to 0.05% higher than the swap spread for 2x LETF.
The swap spread for inverse LETF is around 0.2% lower than the swap spread for normal LETF.
The swap spread for large cap is around 0.4%.
The swap spread for mid cap is less than large cap.
The swap spread for small cap is less than mid cap. Any suggestions as to why this is the case?

Using this information we can compile a new equation for LETF:

Code: Select all

``(S&P500 - rf) * leverage = (LETF - rf) + (leverage - common_stock) * swap_spread``
Here rf is the overnight libor as a proxy for the risk free rate (I did not investigate other proxy's). Leverage is 2 or 3. Common stock is the amount of common stock holdings (.72 for 2x LETF, .58 for 3x LETF. Zero for inverse ETF's). Swap spread is the difference between the risk free rate and the rate on the swap contract, which can be observed on exhibit 3 & 4.

For Ultra S&P500, we find that (leverage - common_stock) * swap_spread is equal to (2 - .72) * .38% = 0.4864%. Siamond used a curve fitting method to arrive at the figure 0.5%.
For UltraPro S&P500, we find that (leverage - common stock) * swap_spread is equal to (2 - .58) * .42% = 1.0164%. Siamond used a curve fitting method to arrive at the figure 1%.

We still have to look into simulating LETF's with daily math and taking a closer look at the equations for inverse ETF's, but I think the friction cost mystery is solved.

Edit: it is worth noting that some of the 2x and 3x ETF's, such as ProShares Ultra MSCI Emerging Markets (2x), have no common stock holdings and are built entirely with swaps, sometimes below market rates.
Last edited by Uncorrelated on Wed Dec 11, 2019 5:56 pm, edited 3 times in total.

parval
Posts: 11
Joined: Tue Oct 22, 2019 9:23 pm

### Re: Simulating Returns of Leveraged ETFs

Not sure if right place but why even use swaps? Can't you get 3x or really any X exposure w/ just futures (ES or MES) + underlying + cash and it'd be cheaper to implement rolling yourself?

HEDGEFUNDIE
Posts: 3973
Joined: Sun Oct 22, 2017 2:06 pm

### Re: Simulating Returns of Leveraged ETFs

Uncorrelated wrote:
Wed Dec 11, 2019 12:30 pm

For Ultra S&P500, we find that (leverage - common_stock) * swap_spread is equal to (2 - .72) * .38% = 0.4864%. Siamond used a curve fitting method to arrive at the figure 0.5%.
For UltraPro S&P500, we find that (leverage - common stock) * swap_spread is equal to (2 - .58) * .42% = 1.0164%. Siamond used a curve fitting method to arrive at the figure 1%.

We still have to look into simulating LETF's with daily math and taking a closer look at the equations for inverse ETF's, but I think the friction cost mystery is solved.
Very nice work! So there are no more mystery costs to these LETFs. 1% ER + 1% Swap Spread + risk-free-rate is it.

Can I ask you to run a chart for TMF?

Uncorrelated
Posts: 261
Joined: Sun Oct 13, 2019 3:16 pm

### Re: Simulating Returns of Leveraged ETFs

Here are some more plots of treasuries and some odd lot ETF's:

Apparently you can get paid to leverage treasuries. Perhaps there's a lot of demand for shorting them?

The rate on the FTSE china may look attractive. But keep in mind the deliverable on the swap is the iShares China Large Cap ETF, which might have considerable tax inefficiencies. As a Dutch investor I might be able to leverage our superior tax treaties to take advantage of this situation.
parval wrote:
Wed Dec 11, 2019 3:04 pm
Not sure if right place but why even use swaps? Can't you get 3x or really any X exposure w/ just futures (ES or MES) + underlying + cash and it'd be cheaper to implement rolling yourself?
I don't know. I suspect that daily rebalancing is costly with futures. It's also possible that banks are able to offer superior swap rates by offsetting the contracts against other customers, something that might not be possible to do within the context of an ETF due to regulations.

But I really don't know.
HEDGEFUNDIE wrote:
Wed Dec 11, 2019 3:31 pm
Uncorrelated wrote:
Wed Dec 11, 2019 12:30 pm

For Ultra S&P500, we find that (leverage - common_stock) * swap_spread is equal to (2 - .72) * .38% = 0.4864%. Siamond used a curve fitting method to arrive at the figure 0.5%.
For UltraPro S&P500, we find that (leverage - common stock) * swap_spread is equal to (2 - .58) * .42% = 1.0164%. Siamond used a curve fitting method to arrive at the figure 1%.

We still have to look into simulating LETF's with daily math and taking a closer look at the equations for inverse ETF's, but I think the friction cost mystery is solved.
Very nice work! So there are no more mystery costs to these LETFs. 1% ER + 1% Swap Spread + risk-free-rate is it.

Can I ask you to run a chart for TMF?
If I want to make a chart for TMF then I'll have to crawl the SEC reports from Direxion, which will take a few hours of coding. I opened a random quarterly earnings report to inspect the rates on TMF and they appear to be distinct from proshares. I'm observing a spread of around 0.7% between the direxion 3x bull/bear in 2015 (SEC link). More data points are needed before we can draw conclusions.

Additionally, we don't know what the swap rates were during the great recession or dot-com bubble. We should look into some of the longer running LETF's to observe how rates behave when the markets are more volatile.