Simulating Returns of Leveraged ETFs

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EfficientInvestor
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Simulating Returns of Leveraged ETFs

Post by EfficientInvestor » Mon Feb 11, 2019 3:15 pm

By now, I hope many have had the chance to read through some or all of the thread (viewtopic.php?f=10&t=272007) discussing using leveraged ETFs (LETF) as part of a risk parity strategy. Thanks to HEDGEFUNDIE for getting the thread started. One aspect of the strategy that we have been discussing is the way in which these LETFs operate and how we can simulate performance prior to inception of the LETFs. As part of the thread, I provided a link (at bottom of post) to an article I had written about my crude way of estimating what performance would have been in the past. This methodology was generally based on determining a weighted average of price return and total return (w/ dividends), applying leverage, and then accounting for expense ratio. However, I believe there is a key piece that is missing and this methodology needs to be updated. I'm looking to the collective wisdom of the forum to get to the bottom of this.

After chewing on the Proshares and Direxion annual reports over the weekend (2nd and 3rd links below), I'm realizing that a pretty large portion of the market exposure of these funds are due to the swap contracts. For these contracts, the funds have to pay interest (based on LIBOR rate) to the bank for the privilege of having access to the contracts. The Direxion report shows that this cost is the "1 month LIBOR rate + spread". This borrowing cost hasn't been that big of a deal since 2009, when the 3X funds hit the street. However, 1 month LIBOR was as high as 10% in the late 80s, and this would have been quite the drag on the ETFs. Since I don't currently know what the "spread" needs to be, I'm currently using the higher 12 month LIBOR to account for the "spread".

I am still crunching some numbers, but I wanted to go ahead and get this thread started to provide an update on the appropriate way to simulate these funds start drawing in input from others. My current assumption on how to simulate these funds is as follows:

Daily Return = [Underlying Index Adjusted Daily Return (including dividend) x Leverage] - (12 month LIBOR)/250 - (Expense Ratio)/250

By applying this equation to the adjusted daily data of VFINX, I achieved results that were very similar to the actual results of ULPIX, the 2X S&P 500 that has been around since 1997. I will post actual results once I confirm a few things. In the meantime, please chime in on how you think this equation could be updated.

https://theleveragedindexer.com/2018/11 ... -backtest/
http://www.proshares.com/media/document ... 9832890959
http://direxioninvestments.onlineprospe ... Prospectus

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siamond
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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Mon Feb 11, 2019 3:39 pm

First off, let me applaud EfficientInvestor's openness and eagerness to refine his simulation model. Those things are touchy, the risk of doing a form of curve-fitting that would NOT properly apply to the past is high, and we really need careful thinking, peer reviews, constructive feedback, and refinements of such model before we can start developing some level of trust about those - or just decide to give up!

We had several similar endeavors in the past, some successful (longinvest's bond simulator; Tyler9000's 3x3 factors stock simulator) and some less successful (e.g. our attempt of simulating TIPS or the use of the IFA data sets). It's always worth giving it a solid try, while keeping a good dose of healthy skepticism.

Here the point would be to try to extend back in time Leveraged ETFs like UPRO (3x S&P 500), SSO and ULPIX (2x S&P 500), TMF (3x LT Treasuries), TYD (2x 7-10y Treasuries), etc. Most of those have been in existence for roughly a decade, with ULPIX being the grand-daddy of sorts (started end of 1997). Quite obviously, this is a short period of time, and we've seen in the past various deep crises for both stocks bonds and very different conditions (notably interest rates and inflation). It is therefore a legitimate (and fascinating) question to ponder how such leveraged funds would have fared by then.

For all of those ETFs, we do have fairly extensive index data going back in time in the 70s and 80s. Leveraged funds are basically trying to mimic daily returns of such indices, multiplied by the leverage factor (one could construe such process as 'passive', although this seems a bit of a stretch). The reality is more complicated though (the borrowing cost identified by EfficientInvestor being one example; the exact way dividends are processed being another; etc). Let's see if we can make a model stick...

EDIT: adding ProShares UBT (2x LT Treasuries) and ProShares UST (2x 7-10y Treasuries) to the list of interesting LETFs - both created early 2010.
Last edited by siamond on Mon Feb 11, 2019 11:54 pm, edited 1 time in total.

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hdas
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Re: Simulating Returns of Leveraged ETFs

Post by hdas » Mon Feb 11, 2019 3:44 pm

EfficientInvestor wrote:
Mon Feb 11, 2019 3:15 pm
By now, I hope many have had the chance to read through some or all of the thread (viewtopic.php?f=10&t=272007) discussing using leveraged ETFs (LETF) as part of a risk parity strategy. Thanks to HEDGEFUNDIE for getting the thread started. One aspect of the strategy that we have been discussing is the way in which these LETFs operate and how we can simulate performance prior to inception of the LETFs. As part of the thread, I provided a link (at bottom of post) to an article I had written about my crude way of estimating what performance would have been in the past. This methodology was generally based on determining a weighted average of leveraged price return and total return (w/ dividends) daily pricing and then accounting for expense ratio. However, I believe there is a key piece that is missing and this methodology needs to be updated. I'm looking to the collective wisdom of the forum to get to the bottom of this.

After chewing on the Proshares and Direxion annual reports over the weekend (2nd and 3rd links below), I'm realizing that a pretty large portion of the market exposure of these funds are due to the swap contracts. For these contracts, the funds have to pay interest (based on LIBOR rate) to the bank for the privilege of having access to the contracts. The Direxion report shows that this cost is the "1 month LIBOR rate + spread". This borrowing cost hasn't been that big of a deal since 2009, when the 3X funds hit the street. However, 1 month LIBOR was as high as 10% in the late 80s, and this would have been quite the drag on the ETFs. Since I don't currently know what the "spread" needs to be, I'm currently using the higher 12 month LIBOR to account for the "spread".

I am still crunching some numbers, but I wanted to go ahead and get this thread started to provide an update on the appropriate way to simulate these funds start drawing in input from others. My current assumption on how to simulate these funds is as follows:

Daily Return = [Underlying Index Adjusted Daily Return (including dividend) x Leverage] - (12 month LIBOR)/250 - (Expense Ratio)/250
Kudos for trying to add some sort of rigor to this. Another issue you have for the bond part is using VUSTX, since the adjusted price includes dividends. Why don't you get yields for constant maturities 20-30 years and create an index from yields. There should be some papers with the methodology for this.

My own simulations vs futures show that these products lose around 3% CAGR vs futures performance (since 2010)
"whenever there is a randomized way of doing something, then there is a nonrandomized way that delivers better performance but requires more thought" ET Jaynes

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siamond
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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Mon Feb 11, 2019 3:53 pm

hdas wrote:
Mon Feb 11, 2019 3:44 pm
Kudos for trying to add some sort of rigor to this. Another issue you have for the bond part is using VUSTX, since the adjusted price includes dividends. Why don't you get yields for constant maturities 20-30 years and create an index from yields. There should be some papers with the methodology for this.
We have Barclays (and much more recently ICE) 20+ treasuries index numbers since the mid 70s (PR and TR). More precisely:
* ICE U.S. Treasury 20+ Year Bond TR USD
* Bloomberg Barclays US Treasury 20+ Yr TR USD

We could also extend a couple of more years (critical ones, cf. the oil crisis) with this index, which started in Dec-72:
* Bloomberg Barclays US Treasury Long TR USD

I would suggest we focus on such time period to start with, to not add one more level of uncertainty in the model. If this goes well, then we can try to use AlohaJoe's excellent effort to go back to the 50s, but... one thing at a time! :wink:
Last edited by siamond on Mon Feb 11, 2019 4:00 pm, edited 1 time in total.

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siamond
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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Mon Feb 11, 2019 3:57 pm

Extracting one last critical piece of information which was posted on HedgeFundie's thread:
siamond wrote:
Sun Feb 10, 2019 5:32 pm
samsdad wrote:
Sun Feb 10, 2019 4:51 pm
I think you're gonna find some of what your looking for on page 124-125 (just over halfway into the humongous document), and then back in the back starting around page 234, then 266, and 282.
http://www.proshares.com/media/document ... 9832890959
Your eyeballs might fall out, however.

EDIT: PAGE LVII (close to the beginning) describes the underlying index thusly in note. 2:
2. The Index is a price return index. The total return and any graph or table reflect the theoretical reinvestment of dividends on securities in the Index. The impact of transaction costs and the deduction of expenses associated with an exchange traded fund such as investment management and accounting fees are not reflected in the Index calculation. It is not possible to invest directly in an Index.
Ah, excellent, great sleuthing! Yup, page LVII does refer to UPRO. Actually, all the stock-centric leveraged fund in there (incl. the EAFE one) have a similar note.

BUT if we check page LVIII, we find information about TTT, which is a 3x LT bond fund similar to TMF. And in this case, the footnote says:
2 The Index is a total return index. The total return and any graph or table reflect the theoretical reinvestment of dividends on securities in the Index. The impact of transaction costs and the deduction of expenses associated with an exchange traded fund such as investment management and accounting fees are not reflected in the Index calculation. It is not possible to invest directly in an Index.
Ok, that's cool, we're starting to see things a little more clearly now.
Note that TTT is a short LETF, a fund essentially trying to create a negative 3x leverage. I ran a quick test on their daily numbers, and yes, they do track the corresponding TR index, no question. While it is trickier to figure out if stock-centric LETFs track the daily price-return (PR) or total-return (TR) indices by just looking at the numbers, but the ProShares report above makes it clear.

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Re: Simulating Returns of Leveraged ETFs

Post by samsdad » Mon Feb 11, 2019 4:03 pm

EfficientInvestor wrote:
Mon Feb 11, 2019 3:15 pm
The Direxion report shows that this cost is the "1 month LIBOR rate + spread". This borrowing cost hasn't been that big of a deal since 2009, when the 3X funds hit the street. However, 1 month LIBOR was as high as 10% in the late 80s, and this would have been quite the drag on the ETFs. Since I don't currently know what the "spread" needs to be, I'm currently using the higher 12 month LIBOR to account for the "spread".

I am still crunching some numbers, but I wanted to go ahead and get this thread started to provide an update on the appropriate way to simulate these funds start drawing in input from others. My current assumption on how to simulate these funds is as follows:

Daily Return = [Underlying Index Adjusted Daily Return (including dividend) x Leverage] - (12 month LIBOR)/250 - (Expense Ratio)/250
As far as the backend of this goes, let's make the math easy for a moment.

Let's assume ER of 1% for UPRO.

Then lets look at the largest historical one-month Libor and make it 10%, again for ease of use. The highest FRED has is 10.0625 back in February/March of 1989. https://fred.stlouisfed.org/series/USD1MTD156N

Today it's 2.51313. So, let's say 2.5%.

So, the cost of the borrowing back in 1989 would have been 10/250-1/250 or 0.04-0.004, which equals 0.036 (basis points?).
Versus the current cost of the borrowing, which in model is 2.5/250-1/250 or 0.01-0.004, which equals 0.006 bp?.

Am I correct?

EDIT: maybe not correct. I think the 10 and the 1 and the 2.5 need some decimals in front.

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Re: Simulating Returns of Leveraged ETFs

Post by samsdad » Mon Feb 11, 2019 4:07 pm

Let's try this again.

.1/250-.01/250 is 0.0004-0.00004, which equals .00036 basis points?
Today it's therefore .00006 bp?

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EfficientInvestor
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Re: Simulating Returns of Leveraged ETFs

Post by EfficientInvestor » Mon Feb 11, 2019 4:13 pm

hdas wrote:
Mon Feb 11, 2019 3:44 pm
EfficientInvestor wrote:
Mon Feb 11, 2019 3:15 pm
By now, I hope many have had the chance to read through some or all of the thread (viewtopic.php?f=10&t=272007) discussing using leveraged ETFs (LETF) as part of a risk parity strategy. Thanks to HEDGEFUNDIE for getting the thread started. One aspect of the strategy that we have been discussing is the way in which these LETFs operate and how we can simulate performance prior to inception of the LETFs. As part of the thread, I provided a link (at bottom of post) to an article I had written about my crude way of estimating what performance would have been in the past. This methodology was generally based on determining a weighted average of leveraged price return and total return (w/ dividends) daily pricing and then accounting for expense ratio. However, I believe there is a key piece that is missing and this methodology needs to be updated. I'm looking to the collective wisdom of the forum to get to the bottom of this.

After chewing on the Proshares and Direxion annual reports over the weekend (2nd and 3rd links below), I'm realizing that a pretty large portion of the market exposure of these funds are due to the swap contracts. For these contracts, the funds have to pay interest (based on LIBOR rate) to the bank for the privilege of having access to the contracts. The Direxion report shows that this cost is the "1 month LIBOR rate + spread". This borrowing cost hasn't been that big of a deal since 2009, when the 3X funds hit the street. However, 1 month LIBOR was as high as 10% in the late 80s, and this would have been quite the drag on the ETFs. Since I don't currently know what the "spread" needs to be, I'm currently using the higher 12 month LIBOR to account for the "spread".

I am still crunching some numbers, but I wanted to go ahead and get this thread started to provide an update on the appropriate way to simulate these funds start drawing in input from others. My current assumption on how to simulate these funds is as follows:

Daily Return = [Underlying Index Adjusted Daily Return (including dividend) x Leverage] - (12 month LIBOR)/250 - (Expense Ratio)/250
Kudos for trying to add some sort of rigor to this. Another issue you have for the bond part is using VUSTX, since the adjusted price includes dividends. Why don't you get yields for constant maturities 20-30 years and create an index from yields. There should be some papers with the methodology for this.

My own simulations vs futures show that these products lose around 3% CAGR vs futures performance (since 2010)
I just pulled daily data for TLT (tracks exact index as TMF), and pulled daily Fed Fund rate data (close approx. for LIBOR). Using only the adjusted close data, I multiplied the daily return by 3, subtracted 1%/250 ER, subtracted the daily Fed fund rate/250, and subtracted 0.5%/250 per day to approximate the "spread". (Note that I think a higher spread is needed for stock funds due to volatility). The resulting .csv file is at the link below. Tracks extremely well with TMF. The only real assumption I'm making at this point is what the "spread" needs to be. I iterated a couple times to come up with the 0.5%/year.

Switching gears...the problem going beyond 2003 is that no other bond fund exactly tracks the same index ICE 20+ year index (not that I know of at least). VUSTX is our closest approximation. My current thought is to pair VUSTX daily data with WHOSX data and then apply my equation. Even if that works, that will only get us back to 1987. Therefore, I think an effort to get solid 20+ year bond index daily data beyond 1987 would be ideal. However, I think we will have to settle for monthly data, but I think that may get us close enough.

https://drive.google.com/open?id=1RILzm ... _5FjxK5tNz

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Re: Simulating Returns of Leveraged ETFs

Post by EfficientInvestor » Mon Feb 11, 2019 4:16 pm

samsdad wrote:
Mon Feb 11, 2019 4:07 pm
Let's try this again.

.1/250-.01/250 is 0.0004-0.00004, which equals .00036 basis points?
Today it's therefore .00006 bp?
My last post should clear things up a bit. However, here is a full equation for Friday for TMF, using TLT data from Yahoo, using Fed Rate instead of 1 month LIBOR, and assuming 0.5% for "spread":

Daily 3X Return = (Daily Adj %)*3 - ER/250 - (Fed Rate)/250 - Spread/250
Daily 3X Return = (122.35/121.83 - 1)*3 - .01/250 - .024/250 - .005/250 = 1.265%
Last edited by EfficientInvestor on Mon Feb 11, 2019 4:40 pm, edited 1 time in total.

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Re: Simulating Returns of Leveraged ETFs

Post by samsdad » Mon Feb 11, 2019 4:40 pm

So
122.35/121.83=1.00426824
1.00426824-1=0.00426824
0.0042684*3=0.01280472
So, yesterday's return before everybody got their cut was 1.280472%. After the cut at today's fed rate of 2.4% it was 1.265%, or an effective total daily cost of .015?

Correct?

Edit: I'm just trying to figure out what that translates into a yearly drag, and then what that means when rates go to 10%.

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Re: Simulating Returns of Leveraged ETFs

Post by EfficientInvestor » Mon Feb 11, 2019 4:46 pm

samsdad wrote:
Mon Feb 11, 2019 4:40 pm
So
122.35/121.83=1.00426824
1.00426824-1=0.00426824
0.0042684*3=0.01280472
So, yesterday's return before everybody got their cut was 1.280472%. After the cut at today's fed rate of 2.4% it was 1.265%, or an effective total daily cost of .015?

Correct?

Edit: I'm just trying to figure out what that translates into a yearly drag, and then what that means when rates go to 10%.
Yes. In other words, the expense ratio and cost of borrowing (1 month LIBOR + spread) added up to .015% for the day.

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hdas
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Re: Simulating Returns of Leveraged ETFs

Post by hdas » Mon Feb 11, 2019 4:49 pm

EfficientInvestor wrote:
Mon Feb 11, 2019 4:13 pm

Switching gears...the problem going beyond 2003 is that no other bond fund exactly tracks the same index ICE 20+ year index (not that I know of at least). VUSTX is our closest approximation. My current thought is to pair VUSTX daily data with WHOSX data and then apply my equation. Even if that works, that will only get us back to 1987. Therefore, I think an effort to get solid 20+ year bond index daily data beyond 1987 would be ideal. However, I think we will have to settle for monthly data, but I think that may get us close enough.

https://drive.google.com/open?id=1RILzm ... _5FjxK5tNz
If you are serious about this I would recommend you to look at ways that you can implement this with futures, you already need so many pieces to go right at the same time, and then add on top the cost, inefficiency and unpredictability element of these leveraged ETF's. You are spending a lot of time and effort in a sub-optimal solution. :greedy
"whenever there is a randomized way of doing something, then there is a nonrandomized way that delivers better performance but requires more thought" ET Jaynes

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Re: Simulating Returns of Leveraged ETFs

Post by samsdad » Mon Feb 11, 2019 4:57 pm

EfficientInvestor wrote:
Mon Feb 11, 2019 4:46 pm
samsdad wrote:
Mon Feb 11, 2019 4:40 pm
So
122.35/121.83=1.00426824
1.00426824-1=0.00426824
0.0042684*3=0.01280472
So, yesterday's return before everybody got their cut was 1.280472%. After the cut at today's fed rate of 2.4% it was 1.265%, or an effective total daily cost of .015?

Correct?

Edit: I'm just trying to figure out what that translates into a yearly drag, and then what that means when rates go to 10%.
Yes. In other words, the expense ratio and cost of borrowing (1 month LIBOR + spread) added up to .015% for the day.
Can you do the math for me at 10% Libor?

EDIT: Sorry EF! I've got my hands full with infant twins today!
Last edited by samsdad on Mon Feb 11, 2019 5:00 pm, edited 1 time in total.

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EfficientInvestor
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Re: Simulating Returns of Leveraged ETFs

Post by EfficientInvestor » Mon Feb 11, 2019 4:57 pm

hdas wrote:
Mon Feb 11, 2019 4:49 pm
EfficientInvestor wrote:
Mon Feb 11, 2019 4:13 pm

Switching gears...the problem going beyond 2003 is that no other bond fund exactly tracks the same index ICE 20+ year index (not that I know of at least). VUSTX is our closest approximation. My current thought is to pair VUSTX daily data with WHOSX data and then apply my equation. Even if that works, that will only get us back to 1987. Therefore, I think an effort to get solid 20+ year bond index daily data beyond 1987 would be ideal. However, I think we will have to settle for monthly data, but I think that may get us close enough.

https://drive.google.com/open?id=1RILzm ... _5FjxK5tNz
If you are serious about this I would recommend you to look at ways that you can implement this with futures, you already need so many pieces to go right at the same time, and then add on top the cost, inefficiency an unpredictability element of this leveraged ETF's. You are spending a lot of time and effort for a sub-optimal path. :greedy
I have to admit that I am not as up to speed on futures as I want to be. However, based on my general understanding, I don't think futures would work as well for this strategy. Since the futures contracts track the index price and aren't based on the total return price, you miss out on the dividends. The benefit of the leveraged ETFs is that they provide you with access to swap contracts that are based on the total return of the index. Are there futures contracts out there that track the total return of stock and/or bond indexes that I'm not aware of? Am I misunderstanding how futures contracts work?

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Re: Simulating Returns of Leveraged ETFs

Post by EfficientInvestor » Mon Feb 11, 2019 5:05 pm

samsdad wrote:
Mon Feb 11, 2019 4:57 pm
EfficientInvestor wrote:
Mon Feb 11, 2019 4:46 pm
samsdad wrote:
Mon Feb 11, 2019 4:40 pm
So
122.35/121.83=1.00426824
1.00426824-1=0.00426824
0.0042684*3=0.01280472
So, yesterday's return before everybody got their cut was 1.280472%. After the cut at today's fed rate of 2.4% it was 1.265%, or an effective total daily cost of .015?

Correct?

Edit: I'm just trying to figure out what that translates into a yearly drag, and then what that means when rates go to 10%.
Yes. In other words, the expense ratio and cost of borrowing (1 month LIBOR + spread) added up to .015% for the day.
Can you do the math for me at 10% Libor?

EDIT: Sorry EF! I've got my hands full with infant twins today!
No worries. I can relate. I have 2 under the age of 3. Here is what the equation and result would have been for Friday with a 10% LIBOR:

Daily 3X Return = (122.35/121.83 - 1)*3 - .01/250 - .10/250 - .005/250 = 1.234%

Expense ratio, LIBOR, and spread would have been 0.046% for the day.

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Re: Simulating Returns of Leveraged ETFs

Post by hdas » Mon Feb 11, 2019 5:31 pm

EfficientInvestor wrote:
Mon Feb 11, 2019 4:57 pm
hdas wrote:
Mon Feb 11, 2019 4:49 pm
EfficientInvestor wrote:
Mon Feb 11, 2019 4:13 pm

Switching gears...the problem going beyond 2003 is that no other bond fund exactly tracks the same index ICE 20+ year index (not that I know of at least). VUSTX is our closest approximation. My current thought is to pair VUSTX daily data with WHOSX data and then apply my equation. Even if that works, that will only get us back to 1987. Therefore, I think an effort to get solid 20+ year bond index daily data beyond 1987 would be ideal. However, I think we will have to settle for monthly data, but I think that may get us close enough.

https://drive.google.com/open?id=1RILzm ... _5FjxK5tNz
If you are serious about this I would recommend you to look at ways that you can implement this with futures, you already need so many pieces to go right at the same time, and then add on top the cost, inefficiency an unpredictability element of this leveraged ETF's. You are spending a lot of time and effort for a sub-optimal path. :greedy
I have to admit that I am not as up to speed on futures as I want to be. However, based on my general understanding, I don't think futures would work as well for this strategy. Since the futures contracts track the index price and aren't based on the total return price, you miss out on the dividends. The benefit of the leveraged ETFs is that they provide you with access to swap contracts that are based on the total return of the index. Are there futures contracts out there that track the total return of stock and/or bond indexes that I'm not aware of? Am I misunderstanding how futures contracts work?
Let’s say that it’s true for a second (it’s more complicated)....notice that since 2010 those ETF have lagged the futures 3x by 3% CARG. So, where did the swaps return go?

I posted those stats in the other thread. Good luck. :greedy
"whenever there is a randomized way of doing something, then there is a nonrandomized way that delivers better performance but requires more thought" ET Jaynes

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Re: Simulating Returns of Leveraged ETFs

Post by HEDGEFUNDIE » Mon Feb 11, 2019 5:35 pm

hdas wrote:
Mon Feb 11, 2019 5:31 pm
EfficientInvestor wrote:
Mon Feb 11, 2019 4:57 pm
hdas wrote:
Mon Feb 11, 2019 4:49 pm
EfficientInvestor wrote:
Mon Feb 11, 2019 4:13 pm

Switching gears...the problem going beyond 2003 is that no other bond fund exactly tracks the same index ICE 20+ year index (not that I know of at least). VUSTX is our closest approximation. My current thought is to pair VUSTX daily data with WHOSX data and then apply my equation. Even if that works, that will only get us back to 1987. Therefore, I think an effort to get solid 20+ year bond index daily data beyond 1987 would be ideal. However, I think we will have to settle for monthly data, but I think that may get us close enough.

https://drive.google.com/open?id=1RILzm ... _5FjxK5tNz
If you are serious about this I would recommend you to look at ways that you can implement this with futures, you already need so many pieces to go right at the same time, and then add on top the cost, inefficiency an unpredictability element of this leveraged ETF's. You are spending a lot of time and effort for a sub-optimal path. :greedy
I have to admit that I am not as up to speed on futures as I want to be. However, based on my general understanding, I don't think futures would work as well for this strategy. Since the futures contracts track the index price and aren't based on the total return price, you miss out on the dividends. The benefit of the leveraged ETFs is that they provide you with access to swap contracts that are based on the total return of the index. Are there futures contracts out there that track the total return of stock and/or bond indexes that I'm not aware of? Am I misunderstanding how futures contracts work?
Let’s say that it’s true for a second (it’s more complicated)....notice that since 2010 those ETF have lagged the futures 3x by 3% CARG. So, where did the swaps return go?

I posted those stats in the other thread. Good luck. :greedy
Yes, but taxes :wink:

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Re: Simulating Returns of Leveraged ETFs

Post by hdas » Mon Feb 11, 2019 5:38 pm

HEDGEFUNDIE wrote:
Mon Feb 11, 2019 5:35 pm
hdas wrote:
Mon Feb 11, 2019 5:31 pm
EfficientInvestor wrote:
Mon Feb 11, 2019 4:57 pm
hdas wrote:
Mon Feb 11, 2019 4:49 pm
EfficientInvestor wrote:
Mon Feb 11, 2019 4:13 pm

Switching gears...the problem going beyond 2003 is that no other bond fund exactly tracks the same index ICE 20+ year index (not that I know of at least). VUSTX is our closest approximation. My current thought is to pair VUSTX daily data with WHOSX data and then apply my equation. Even if that works, that will only get us back to 1987. Therefore, I think an effort to get solid 20+ year bond index daily data beyond 1987 would be ideal. However, I think we will have to settle for monthly data, but I think that may get us close enough.

https://drive.google.com/open?id=1RILzm ... _5FjxK5tNz
If you are serious about this I would recommend you to look at ways that you can implement this with futures, you already need so many pieces to go right at the same time, and then add on top the cost, inefficiency an unpredictability element of this leveraged ETF's. You are spending a lot of time and effort for a sub-optimal path. :greedy
I have to admit that I am not as up to speed on futures as I want to be. However, based on my general understanding, I don't think futures would work as well for this strategy. Since the futures contracts track the index price and aren't based on the total return price, you miss out on the dividends. The benefit of the leveraged ETFs is that they provide you with access to swap contracts that are based on the total return of the index. Are there futures contracts out there that track the total return of stock and/or bond indexes that I'm not aware of? Am I misunderstanding how futures contracts work?
Let’s say that it’s true for a second (it’s more complicated)....notice that since 2010 those ETF have lagged the futures 3x by 3% CARG. So, where did the swaps return go?

I posted those stats in the other thread. Good luck. :greedy
Yes, but taxes :wink:
Only in taxable. IB margin for futures in Roth or IRA shouldn’t be too bad. Even if DOUBLE the normal. :greedy
"whenever there is a randomized way of doing something, then there is a nonrandomized way that delivers better performance but requires more thought" ET Jaynes

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Re: Simulating Returns of Leveraged ETFs

Post by skeptic42 » Mon Feb 11, 2019 7:15 pm

The cost of borrowing for a 3x leveraged ETF should be double that of a 2x leveraged ETF, because it applies to 200% instead of 100%.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Mon Feb 11, 2019 8:32 pm

skeptic42 wrote:
Mon Feb 11, 2019 7:15 pm
The cost of borrowing for a 3x leveraged ETF should be double that of a 2x leveraged ETF, because it applies to 200% instead of 100%.
Yes. More generally, it seems to me that there should be a multiplier which isn't equal to 1. For a $1 invested in a given fund, there isn't $1 being borrowed. The trouble is I have no clue how to figure out the right multiplier, and whether it stays reasonably constant over the years.

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Re: Simulating Returns of Leveraged ETFs

Post by EfficientInvestor » Mon Feb 11, 2019 8:42 pm

siamond wrote:
Mon Feb 11, 2019 8:32 pm
skeptic42 wrote:
Mon Feb 11, 2019 7:15 pm
The cost of borrowing for a 3x leveraged ETF should be double that of a 2x leveraged ETF, because it applies to 200% instead of 100%.
Yes. More generally, it seems to me that there should be a multiplier which isn't equal to 1. For a $1 invested in a given fund, there isn't $1 being borrowed. The trouble is I have no clue how to figure out the right multiplier, and whether it stays reasonably constant over the years.
I agree with what y’all are saying. In addition to taking into account the multiplier, we also need to take into account the amount of cash that is earning interest. That effectively reduces the multiplier. Is it a good assumption that the portion in cash is earning the same rate that is being paid for the swaps (not including the “spread”)?

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Mon Feb 11, 2019 9:06 pm

Dumb question: are we 100% sure that the cost of borrowing isn't somehow included in the ER? Is there an explicit statement to this effect in the annual report of ProShares?

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Re: Simulating Returns of Leveraged ETFs

Post by samsdad » Mon Feb 11, 2019 9:27 pm

Interesting paper, still reading but on point:
https://poseidon01.ssrn.com/delivery.ph ... 90&EXT=pdf

See pages 5 and 6 where it talks about the mathematics of LETFs.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Mon Feb 11, 2019 10:46 pm

samsdad wrote:
Mon Feb 11, 2019 9:27 pm
Interesting paper, still reading but on point:
https://poseidon01.ssrn.com/delivery.ph ... 90&EXT=pdf
Well, check the following extract and compare to the OP's first post...
With limited historical LETF data, additional theoretical LETF returns are calculated for the period prior to the inception of the LETFs. Since LETFs attain their exposure using a variety of derivative assets including swaps, there are embedded financing costs increasing with leverage (Charupat & Miu, 2014). Based on the methodology of Scott & Watsun (2013), LETF returns are calculated based on data going back to 1946. This shows how LETF portfolios are likely to perform in a variety of market environments including the very high interest rate period during the early 1980s. The equation to calculate daily returns using the S&P 500 LETF as an example is:

RL = L*RS&P – Rexp – (L-1)*RB (6)

where RL is the daily return to the LETF with a daily leverage ratio of L, RS&P is the daily return of the S&P, Rexp is the daily expense ratio, and RB is the borrowing rate using the 90-day T-bill rate as a proxy. Strictly speaking, the one-week/month Libor rate should be used, but Libor data begins in 1986 and to remain consistent with sampled returns before this date, the 90-day T-bill rate is used. The 90-day T-bill has a 98% correlation with Libor and averages 0.2% less than Libor. Thus, the borrowing rate is set at the 90-day T-bill yield + 0.2%.

The logic behind Equation (6) is a 2x LETF increases exposure by borrowing $1 for every $1 invested. A 3x LETF borrows $2 for every $1 invested.
Note that equation (6) actually comes from other research papers referenced as Scott & Watsun (2013) and Ott & Zimmer (2016).

And then the authors proceeded with validating this model against empirical data, although this section is really short on details. Well, I'm going to run those numbers by myself...

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Re: Simulating Returns of Leveraged ETFs

Post by EfficientInvestor » Mon Feb 11, 2019 11:08 pm

siamond wrote:
Mon Feb 11, 2019 10:46 pm
samsdad wrote:
Mon Feb 11, 2019 9:27 pm
Interesting paper, still reading but on point:
https://poseidon01.ssrn.com/delivery.ph ... 90&EXT=pdf
Well, check the following extract and compare to the OP's first post...
With limited historical LETF data, additional theoretical LETF returns are calculated for the period prior to the inception of the LETFs. Since LETFs attain their exposure using a variety of derivative assets including swaps, there are embedded financing costs increasing with leverage (Charupat & Miu, 2014). Based on the methodology of Scott & Watsun (2013), LETF returns are calculated based on data going back to 1946. This shows how LETF portfolios are likely to perform in a variety of market environments including the very high interest rate period during the early 1980s. The equation to calculate daily returns using the S&P 500 LETF as an example is:

RL = L*RS&P – Rexp – (L-1)*RB (6)

where RL is the daily return to the LETF with a daily leverage ratio of L, RS&P is the daily return of the S&P, Rexp is the daily expense ratio, and RB is the borrowing rate using the 90-day T-bill rate as a proxy. Strictly speaking, the one-week/month Libor rate should be used, but Libor data begins in 1986 and to remain consistent with sampled returns before this date, the 90-day T-bill rate is used. The 90-day T-bill has a 98% correlation with Libor and averages 0.2% less than Libor. Thus, the borrowing rate is set at the 90-day T-bill yield + 0.2%.

The logic behind Equation (6) is a 2x LETF increases exposure by borrowing $1 for every $1 invested. A 3x LETF borrows $2 for every $1 invested.
Note that equation (6) actually comes from other research papers referenced as Scott & Watsun (2013) and Ott & Zimmer (2016).

And then the authors proceeded with validating this model against empirical data, although this section is really short on details. Well, I'm going to run those numbers by myself...
Sounds like we have been in the right ballpark. Thanks for digging into that paper.

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Re: Simulating Returns of Leveraged ETFs

Post by badapu » Tue Feb 12, 2019 12:11 am

If increasing LIBOR rates create a drag on the model, can adding a bull LIBOR ETN to the portfolio help negate a rising LIBOR?

https://www.velocityshares.com/etns/product/ulbr/

BTW - Spider Network is a fantastic read about the LIBOR rate fixing scandal.

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Re: Simulating Returns of Leveraged ETFs

Post by PlateVoltage » Tue Feb 12, 2019 7:34 am

EfficientInvestor wrote:
Mon Feb 11, 2019 4:57 pm
I don't think futures would work as well for this strategy. Since the futures contracts track the index price and aren't based on the total return price, you miss out on the dividends.
The expected dividends are subtracted from the cost of carry.

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Re: Simulating Returns of Leveraged ETFs

Post by EfficientInvestor » Tue Feb 12, 2019 9:03 am

PlateVoltage wrote:
Tue Feb 12, 2019 7:34 am
EfficientInvestor wrote:
Mon Feb 11, 2019 4:57 pm
I don't think futures would work as well for this strategy. Since the futures contracts track the index price and aren't based on the total return price, you miss out on the dividends.
The expected dividends are subtracted from the cost of carry.
Thanks. I just did some research on cost of carry and it's all making sense now.

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Re: Simulating Returns of Leveraged ETFs

Post by EfficientInvestor » Tue Feb 12, 2019 11:06 am

hdas wrote:
Mon Feb 11, 2019 5:31 pm
EfficientInvestor wrote:
Mon Feb 11, 2019 4:57 pm
hdas wrote:
Mon Feb 11, 2019 4:49 pm
EfficientInvestor wrote:
Mon Feb 11, 2019 4:13 pm

Switching gears...the problem going beyond 2003 is that no other bond fund exactly tracks the same index ICE 20+ year index (not that I know of at least). VUSTX is our closest approximation. My current thought is to pair VUSTX daily data with WHOSX data and then apply my equation. Even if that works, that will only get us back to 1987. Therefore, I think an effort to get solid 20+ year bond index daily data beyond 1987 would be ideal. However, I think we will have to settle for monthly data, but I think that may get us close enough.

https://drive.google.com/open?id=1RILzm ... _5FjxK5tNz
If you are serious about this I would recommend you to look at ways that you can implement this with futures, you already need so many pieces to go right at the same time, and then add on top the cost, inefficiency an unpredictability element of this leveraged ETF's. You are spending a lot of time and effort for a sub-optimal path. :greedy
I have to admit that I am not as up to speed on futures as I want to be. However, based on my general understanding, I don't think futures would work as well for this strategy. Since the futures contracts track the index price and aren't based on the total return price, you miss out on the dividends. The benefit of the leveraged ETFs is that they provide you with access to swap contracts that are based on the total return of the index. Are there futures contracts out there that track the total return of stock and/or bond indexes that I'm not aware of? Am I misunderstanding how futures contracts work?
Let’s say that it’s true for a second (it’s more complicated)....notice that since 2010 those ETF have lagged the futures 3x by 3% CARG. So, where did the swaps return go?

I posted those stats in the other thread. Good luck. :greedy
So I have chewed on this a bit and learned some more about how futures contracts work. Since the dividend is a discount of the futures price, a futures investor is still benefiting from the dividend. However, the risk free rate goes against you when you are long the futures contract. Therefore, this is somewhat similar to the leveraged ETFs. You get the benefit of the dividend because the swap contract is based on the total return index. However, to get access to the swap, you have to pay the risk free rate (LIBOR). Therefore, it looks like the costs and benefits are very similar between the two products. The main difference then is the daily rebalancing of the LETFs and the added expense ratio of 1%. Would you agree with that assessment?

Side note...the daily rebalance has caused the 40/60 LETF portfolio to underperform the futures version since inception of LETFs, as you showed in the results you posted. However, I believe the LETF portfolio would outperform the futures portfolio in years like 2008 because your losses will be capped on the stock side and the daily rebalancing on the bond side would have resulted in annual outperformance of the 3X. After we get better simulations in place for the LETFs, we will have to do a longer-term comparison. Regardless of which is best, I think the results will end up being relatively close. Even if the futures portfolio ends up with a slight edge, I think the average investor would choose to stick with the LETFs due to ease of use.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Tue Feb 12, 2019 4:08 pm

Let's stay focused on the simulation topic, per individual LETF...
EfficientInvestor wrote:
Mon Feb 11, 2019 3:15 pm
After chewing on the Proshares and Direxion annual reports over the weekend (2nd and 3rd links below), I'm realizing that a pretty large portion of the market exposure of these funds are due to the swap contracts. For these contracts, the funds have to pay interest (based on LIBOR rate) to the bank for the privilege of having access to the contracts. The Direxion report shows that this cost is the "1 month LIBOR rate + spread". This borrowing cost hasn't been that big of a deal since 2009, when the 3X funds hit the street. However, 1 month LIBOR was as high as 10% in the late 80s, and this would have been quite the drag on the ETFs. Since I don't currently know what the "spread" needs to be, I'm currently using the higher 12 month LIBOR to account for the "spread".
It turns out that the LIBOR rates are available in Morningstar, so I retrieved the 1m/3m/12m rates, available since 1986. Over this time period, there is a 0.2% CAGR difference between LIBOR/1m and LIBOR/3m, and another 0.2% CAGR difference between LIBOR/3m and LIBOR/12m. It seems to me that 0.4% would be hard to justify for the spread, plus I don't feel comfortable ignoring the LIBOR/1m indication from Direxion (nowadays, differences are small, but back in the 80s or around the financial crisis, this was an entirely different story). Maybe using LIBOR/3m would be a better compromise?

I ran numbers using this new model, using UBT (2x LT Treasuries) and ULPIX (2x S&P 500) and LIBOR/12m, tracking the TR index in both cases. Results were a little mixed, as the model underestimated the ProShares UBT Gross Returns (ER not included) by roughly half a point, while overestimating the ProFunds ULPIX Gross Returns by roughly half a point.

Then I ran the numbers with LIBOR/3m, and the UBT simulation gets better, but the ULPIX simulation got worse, the model overestimating by a full point over ULPIX history. Hmpf. The good news (kinda) is that the model doesn't break down with a 3x fund like UPRO (which it overestimates by 0.7%) - and yes, I multiplied the LIBOR numbers by 2x in this case.

Let me dig a little more...

PS. seems that ProShares uses very short-term borrowing too. Here is a quote from their report (linked to in the OP):

Financing Rates Associated with Derivatives:The performance of each Fund was impacted by the related financing costs. Financial instruments such as futures contracts carry implied financing costs. Swap financing rates are negotiated between the Funds and their counterparties, and are typically set at the one-week/one-month London Interbank Offered Rate (“LIBOR”) plus or minus a negotiated spread. The one-week LIBOR appreciated from 0.95% to 1.75% during the fiscal year. The one-month LIBOR also increased during the fiscal year from 1.06% to 2.00%. Each Fund with long exposure via derivatives was generally negatively affected by financing rates. Conversely, most Funds with short/inverse derivative exposure generally benefited from financing rates. However, in low interest rate environments, LIBOR adjusted by the spread may actually result in a Fund with short/inverse exposure also being negatively affected by financing rates.

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Re: Simulating Returns of Leveraged ETFs

Post by gtwhitegold » Tue Feb 12, 2019 6:35 pm

With the data presented here and HEDGEFUNDIE's 3x portfolio thread, it seems like it would still be preferable to manage your own futures in order to minimize costs. I personally would do that and use the other funds in my account as collateral.

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Re: Simulating Returns of Leveraged ETFs

Post by samsdad » Tue Feb 12, 2019 7:22 pm

Earlier today I tried my hand at using that fancy equation above to simulate UPRO with a 3x GSPC fund. Story after the pic:

Image

Looks pretty good, no?

Here's the bad news: it ain't related to that equation. It's just the GSPC returns multiplied by 3.27. :D

The bond fund is a mystery in an enigma wrapped in bacon. Couldn't get anything close.

Feeling defeated.

EDIT: moreover, this is junk now that we know that the ever-changing Libor rates, plus swap spread affect this differently through time. I think short of knowing what the rates were to borrow, it's going to be impossible to model anything in the distant past.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Tue Feb 12, 2019 7:41 pm

samsdad wrote:
Tue Feb 12, 2019 7:22 pm
The bond fund is a mystery in an enigma wrapped in bacon. Couldn't get anything close.
I actually put aside TMF for now when I figured out that they do a really shitty job of tracking the (TR) index on a daily basis. While UBT (which is the 2x equivalent from ProShares) does a very convincing job of it, same as the various S&P 500 LETFs I played with (ProShares and ProFunds). I also did a test with a short LETF from ProShares (TTT) and this was a solid match too when adding the appropriate '-' sign. The Direxion folks clearly have their own methodology, I don't know if this is justified or not, but I am not going to try any harder to simulate it...
samsdad wrote:
Tue Feb 12, 2019 7:22 pm
EDIT: moreover, this is junk now that we know that the ever-changing Libor rates, plus swap spread affect this differently through time. I think short of knowing what the rates were to borrow, it's going to be impossible to model anything in the distant past.
LIBOR rates may not be available before 1986, but some form of T-Bills might be a decent proxy, from what I read. This needs to be further validated though. No clue about the spread effect though, although I suspect we have bigger uncertainties than this one.

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Re: Simulating Returns of Leveraged ETFs

Post by samsdad » Tue Feb 12, 2019 9:00 pm

siamond wrote:
Tue Feb 12, 2019 7:41 pm
samsdad wrote:
Tue Feb 12, 2019 7:22 pm
The bond fund is a mystery in an enigma wrapped in bacon. Couldn't get anything close.
I actually put aside TMF for now when I figured out that they do a really shitty job of tracking the (TR) index on a daily basis. While UBT (which is the 2x equivalent from ProShares) does a very convincing job of it, same as the various S&P 500 LETFs I played with (ProShares and ProFunds). I also did a test with a short LETF from ProShares (TTT) and this was a solid match too when adding the appropriate '-' sign. The Direxion folks clearly have their own methodology, I don't know if this is justified or not, but I am not going to try any harder to simulate it...
samsdad wrote:
Tue Feb 12, 2019 7:22 pm
EDIT: moreover, this is junk now that we know that the ever-changing Libor rates, plus swap spread affect this differently through time. I think short of knowing what the rates were to borrow, it's going to be impossible to model anything in the distant past.
LIBOR rates may not be available before 1986, but some form of T-Bills might be a decent proxy, from what I read. This needs to be further validated though. No clue about the spread effect though, although I suspect we have bigger uncertainties than this one.
Thanks be the gods you were having problems with modeling TMF too. I thought I was surely stuck dumb earlier today going back and forth between the two.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Tue Feb 12, 2019 10:37 pm

On HedgeFundie's thread, there was an indirect reference to a 1996 article from Cliff Asness, who basically used the same idea to simulate leveraged funds: using one unit of borrowing per unit of extra leverage. And he used 1-month T-Bills in his model as the cost of borrowing. The article can be downloaded here.

The author really didn't say much more and it is quite clear that he just did some quick back-of-the-envelope math, but this is how he assembled a backtest starting in 1925, essentially making the same case as being discussed on the other thread (a mix of leveraged stocks and leveraged bonds). Which is consistent with the model we've been working on so far.

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Re: Simulating Returns of Leveraged ETFs

Post by samsdad » Tue Feb 12, 2019 11:21 pm

Here's the published methodology that ICE uses to calculate their daily returns. It includes leveraged returns as well. Of course that doesn't account for the aformentioned swap costs, etc. that TMF et al. has to use.

https://www.theice.com/publicdocs/data/ ... dology.pdf

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Re: Simulating Returns of Leveraged ETFs

Post by EfficientInvestor » Wed Feb 13, 2019 3:24 pm

samsdad wrote:
Tue Feb 12, 2019 11:21 pm
Here's the published methodology that ICE uses to calculate their daily returns. It includes leveraged returns as well. Of course that doesn't account for the aformentioned swap costs, etc. that TMF et al. has to use.

https://www.theice.com/publicdocs/data/ ... dology.pdf
I pulled daily TLT data (2003 - present) and used the exact equation from the article to create simulations for UBT (2X TLT) and TMF (3X TLT). I used the fed fund rate instead of LIBOR as my borrow cost. In addition to what the equation shows, I also applied a 0.95% ER over the 250 trading days per year. My hypothesis was that if the simulations were off, they would be off in the same direction for both. However, as siamond has already discovered, it appears that Proshares (UBT) is doing a better job than Direxion (TMF) at hitting the target returns. My simulation for UBT underperformed the actual UBT ( 9.0% vs 9.5% CAGR) from Jan 2011 - Jan 2019 while my simulation for TMF overperformed the actual TMF (12.3% vs 11.6% CAGR) from Jan 2010 - Jan 2019. I'm not sure what would cause this difference other than poor execution. They charge the same ER. The biggest difference I have noticed between the two funds so far is that Proshares tends to directly own individual bonds while Direxion just owns TLT (thus, they have to pay the ER for it). But I don't see how that could cause the difference since TLT only has an ER of 0.15%.

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Re: Simulating Returns of Leveraged ETFs

Post by JackoC » Wed Feb 13, 2019 4:41 pm

EfficientInvestor wrote:
Tue Feb 12, 2019 11:06 am

So I have chewed on this a bit and learned some more about how futures contracts work. Since the dividend is a discount of the futures price, a futures investor is still benefiting from the dividend. However, the risk free rate goes against you when you are long the futures contract. Therefore, this is somewhat similar to the leveraged ETFs. You get the benefit of the dividend because the swap contract is based on the total return index. However, to get access to the swap, you have to pay the risk free rate (LIBOR). Therefore, it looks like the costs and benefits are very similar between the two products. The main difference then is the daily rebalancing of the LETFs and the added expense ratio of 1%. Would you agree with that assessment?
True, being long/short equity index futures is the essentially the same thing as being receiver/payer of the total return swap (TRS) on the index. Some leveraged ETF's use futures in addition to or rather than total return swaps.

It's also true though that the leveraged ETF can easily reset the position as X times your notional investment every day, since it even if it's using futures it will be a large number of them for the whole fund and being long or short one more is a fine adjustment. And it can change the size of its TRS's with its counterparties by any increment. You OTOH as individual replicating a leveraged ETF with futures can only change your position by increments of 1 futures contract, which for example for S&P E-mini contracts is equivalent to around $137,500 right now, a quite coarse adjustment unless you have quite a large portfolio. But you're not paying the ER.

But leveraged ETF's based on using over-the-counter derivatives for which there are exchange trade equivalents (ie futures) usually don't make sense IMO. If the investor is up to speed enough to fully understand and deal properly with leverage and investing at enough scale that single contracts aren't much too big, the same investor is probably capable of dealing with futures trades and cutting out the ER. However in case of many indexes for which there are leveraged ETF's there aren't futures. Also it's not necessarily practical to trade futures on foreign indices especially in a tax deferred account. And it's noisier tax-wise typically to do futures than a leveraged ETF in a taxable account.

Back to the funding rate, this is IMO a big lack of transparency in leveraged ETF's. They know the spreads on the LIBOR legs of their TRS's, but they don't tell you, or fund literature I've even seen doesn't anyway. Although even if they did, there is no firm answer for what the spread would have been many years ago, nor what it will be in the future. Again though by reference to futures you can gather some more info on this. The CME posts this page of futures implied borrowing rates around contract 'roll' dates. In the week or so from 12/6-13 the futures implied borrowing rate was around 2.90%. Per St. Louis Fed's page 3 mo LIBOR was around 2.78% that week. You can see on the CME page that the implied rate tended to be a lower for indicies other than S&P, sometimes lower than 3mo LIBOR, as influenced by technical factors in the futures market. But again this is basically the same market as for total return swaps, the difference likely being just that a leveraged ETF manager probably pays up a little on LIBOR+ leg of the TRS compared to the average futures implied financing rate, for the convenience of the TRS, or for a TRS on an index for which there are no futures contracts (and the swap dealer has to find somebody with an opposite interest).
https://www.cmegroup.com/trading/equity ... /main.html

I would not recommend using a longer term of LIBOR to substitute for the spread. That's just introducing noise from the varying shape of the yield curve over time. I would recommend your analysis spreadsheet have an input for LIBOR spread and see how much the outcome of the strategy changes bumping it up or down.

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Re: Simulating Returns of Leveraged ETFs

Post by JackoC » Wed Feb 13, 2019 4:52 pm

Also one minor point about OP and subsequent posts. If want to simulate past daily data, I think you have to calculate the interest between dates, so 1/360*rate between weekdays, 3/360 over weekends, whatever the span between data points over holidays, not 1/250 between any two business days. By convention of money markets, rates like LIBOR are accrued at 1/360 (not 365 or 366) per calendar day, you get or pay 365/360's or 366/360's of that rate every year. The ER could be accrued at 1/365.25 per day and be more or less correct.

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Re: Simulating Returns of Leveraged ETFs

Post by EfficientInvestor » Wed Feb 13, 2019 4:54 pm

JackoC wrote:
Wed Feb 13, 2019 4:41 pm
EfficientInvestor wrote:
Tue Feb 12, 2019 11:06 am

So I have chewed on this a bit and learned some more about how futures contracts work. Since the dividend is a discount of the futures price, a futures investor is still benefiting from the dividend. However, the risk free rate goes against you when you are long the futures contract. Therefore, this is somewhat similar to the leveraged ETFs. You get the benefit of the dividend because the swap contract is based on the total return index. However, to get access to the swap, you have to pay the risk free rate (LIBOR). Therefore, it looks like the costs and benefits are very similar between the two products. The main difference then is the daily rebalancing of the LETFs and the added expense ratio of 1%. Would you agree with that assessment?
True, being long/short equity index futures is the essentially the same thing as being receiver/payer of the total return swap (TRS) on the index. Some leveraged ETF's use futures in addition to or rather than total return swaps.

It's also true though that the leveraged ETF can easily reset the position as X times your notional investment every day, since it even if it's using futures it will be a large number of them for the whole fund and being long or short one more is a fine adjustment. And it can change the size of its TRS's with its counterparties by any increment. You OTOH as individual replicating a leveraged ETF with futures can only change your position by increments of 1 futures contract, which for example for S&P E-mini contracts is equivalent to around $137,500 right now, a quite coarse adjustment unless you have quite a large portfolio. But you're not paying the ER.

But leveraged ETF's based on using over-the-counter derivatives for which there are exchange trade equivalents (ie futures) usually don't make sense IMO. If the investor is up to speed enough to fully understand and deal properly with leverage and investing at enough scale that single contracts aren't much too big, the same investor is probably capable of dealing with futures trades and cutting out the ER. However in case of many indexes for which there are leveraged ETF's there aren't futures. Also it's not necessarily practical to trade futures on foreign indices especially in a tax deferred account. And it's noisier tax-wise typically to do futures than a leveraged ETF in a taxable account.

Back to the funding rate, this is IMO a big lack of transparency in leveraged ETF's. They know the spreads on the LIBOR legs of their TRS's, but they don't tell you, or fund literature I've even seen doesn't anyway. Although even if they did, there is no firm answer for what the spread would have been many years ago, nor what it will be in the future. Again though by reference to futures you can gather some more info on this. The CME posts this page of futures implied borrowing rates around contract 'roll' dates. In the week or so from 12/6-13 the futures implied borrowing rate was around 2.90%. Per St. Louis Fed's page 3 mo LIBOR was around 2.78% that week. You can see on the CME page that the implied rate tended to be a lower for indicies other than S&P, sometimes lower than 3mo LIBOR, as influenced by technical factors in the futures market. But again this is basically the same market as for total return swaps, the difference likely being just that a leveraged ETF manager probably pays up a little on LIBOR+ leg of the TRS compared to the average futures implied financing rate, for the convenience of the TRS, or for a TRS on an index for which there are no futures contracts (and the swap dealer has to find somebody with an opposite interest).
https://www.cmegroup.com/trading/equity ... /main.html

I would not recommend using a longer term of LIBOR to substitute for the spread. That's just introducing noise from the varying shape of the yield curve over time. I would recommend your analysis spreadsheet have an input for LIBOR spread and see how much the outcome of the strategy changes bumping it up or down.
Thanks for the great input regarding futures contracts on regarding the LIBOR spread. I actually already have my simulation spreadsheet set up with a separate adjustment for spread, as you suggest. The intent being that I could iteratively solve for spread and then just assume that spread into the past. However, as indicated in my post about UBT vs TMF, I'm at a bit of an impasse. I would have to use a positive spread (negative effect on the performance) to get the TMF simulation to meet actual TMF performance and a negative spread (positive effect on performance) to get the UBT simulation to meet actual UBT performance. The negative spread wouldn't make sense, so I'm assuming there is something else at play causing UBT performance to be better, such as investment of cash in a money market or dividends received from bonds. But TMF appears to do the same thing, but hasn't achieved the same results. Perhaps it is because the amount of cash yield and dividends received are relatively constant between UBT (2X) and TMF (3X), so UBT receives a relatively larger benefit from them, thus causing their performance to be better than the theoretical equation.

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Re: Simulating Returns of Leveraged ETFs

Post by JackoC » Wed Feb 13, 2019 5:22 pm

EfficientInvestor wrote:
Wed Feb 13, 2019 4:54 pm
JackoC wrote:
Wed Feb 13, 2019 4:41 pm
EfficientInvestor wrote:
Tue Feb 12, 2019 11:06 am

So I have chewed on this a bit and learned some more about how futures contracts work.
True, being long/short equity index futures is the essentially the same thing as being receiver/payer of the total return swap (TRS) on the index.
Thanks for the great input regarding futures contracts on regarding the LIBOR spread. I actually already have my simulation spreadsheet set up with a separate adjustment for spread, as you suggest. The intent being that I could iteratively solve for spread and then just assume that spread into the past. However, as indicated in my post about UBT vs TMF, I'm at a bit of an impasse. I would have to use a positive spread (negative effect on the performance) to get the TMF simulation to meet actual TMF performance and a negative spread (positive effect on performance) to get the UBT simulation to meet actual UBT performance. The negative spread wouldn't make sense, so I'm assuming there is something else at play causing UBT performance to be better, such as investment of cash in a money market or dividends received from bonds. But TMF appears to do the same thing, but hasn't achieved the same results. Perhaps it is because the amount of cash yield and dividends received are relatively constant between UBT (2X) and TMF (3X), so UBT receives a relatively larger benefit from them, thus causing their performance to be better than the theoretical equation.
I hadn't fully paid attention to everything in the thread, was just focusing on the TRS/futures comparison and assuming equity ETF's. But to simulate any leverage ETF you have to explicitly account for the money earned on the cash you invest. The return of an X-levered ETF=(Index Total Return-funding rate*)*X+return on cash-ER. So yes it's a bigger relative add on if X is 2 rather than 3. But another thing which is not so transparent about LETF's is return on cash. If you look at the holdings of such funds it's probably T-bills, so earning the fund less than the funding rate on TRS or futures. Also now contrary to my initial impression you're talking about leverage treasury ETF's. The implied borrowing rate on those is more like the term repo rate so they might pay below LIBOR on that leg of the TRS, or the implied financing rate on those futures is more likely below LIBOR if they use futures.

*which could be a futures implied rate or the explicit LIBOR+ spread leg of a Total Return Swap.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Wed Feb 13, 2019 5:32 pm

JackoC wrote:
Wed Feb 13, 2019 4:41 pm
I would not recommend using a longer term of LIBOR to substitute for the spread. That's just introducing noise from the varying shape of the yield curve over time. I would recommend your analysis spreadsheet have an input for LIBOR spread and see how much the outcome of the strategy changes bumping it up or down.
Yes, totally agreed. We should really avoid doing any kind of curve-fitting while we're investigating. I came back to LIBOR 1m in my own models. This is what is described in ProShares and Direxion reports. And it happens to work remarkably well when modeling UBT (although this could be luck). The spread thing is just an unknown at this stage, let's keep it separate.

Thank you for bringing your expertise, JackoC. It sounds that you know this LETF topic quite well. It will be tremendously useful to have such extra opinion.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Wed Feb 13, 2019 6:21 pm

EfficientInvestor wrote:
Wed Feb 13, 2019 3:24 pm
However, as siamond has already discovered, it appears that Proshares (UBT) is doing a better job than Direxion (TMF) at hitting the target returns.
I gave a good run to TYD (Direxion Daily 7-10 Year Treasury Bull 3x Shares). Long story short:
a) they do NOT closely track the daily index values (times the leverage)
b) the overall trajectory ends up being fairly similar to the theoretical model we've been using so far, but with big caveats

To elaborate on the last point, the correlation is very strong (general shape of the trajectory is very similar), but there is a decay of roughly 1% a year. Knowing that UBT (ProShares Ultra 20+ Year Treasury 3x Shares) does NOT display such a decay, this reinforces my reservations about Direxion, they do seem to have a significant issue with execution.

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Re: Simulating Returns of Leveraged ETFs

Post by mrspock » Wed Feb 13, 2019 7:37 pm

siamond wrote:
Wed Feb 13, 2019 6:21 pm
EfficientInvestor wrote:
Wed Feb 13, 2019 3:24 pm
However, as siamond has already discovered, it appears that Proshares (UBT) is doing a better job than Direxion (TMF) at hitting the target returns.
I gave a good run to TYD (Direxion Daily 7-10 Year Treasury Bull 3x Shares). Long story short:
a) they do NOT closely track the daily index values (times the leverage)
b) the overall trajectory ends up being fairly similar to the theoretical model we've been using so far, but with big caveats

To elaborate on the last point, the correlation is very strong (general shape of the trajectory is very similar), but there is a decay of roughly 1% a year. Knowing that UBT (ProShares Ultra 20+ Year Treasury 3x Shares) does NOT display such a decay, this reinforces my reservations about Direxion, they do seem to have a significant issue with execution.
The chart between the two is pretty different as well (if you overlay the 10 year), TMF has quite a bit more volatility by my eyes.

Should folks dump TMF for UBT? The net assets for UBT is only 32M vs 107M for TMF. Volume is probably proportionately lower as well. ER is similar.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Wed Feb 13, 2019 7:52 pm

mrspock wrote:
Wed Feb 13, 2019 7:37 pm
The chart between the two is pretty different as well (if you overlay the 10 year), TMF has quite a bit more volatility by my eyes.

Should folks dump TMF for UBT? The net assets for UBT is only 32M vs 107M for TMF. Volume is probably proportionately lower as well. ER is similar.
Well, let's not jump to conclusions too fast. First, they are not directly comparable (UBT is 2x leverage, TMF is 3x). Next, I still have to assemble a full graph for TMF, and maybe Direxion is doing something subtle that ends up working ok.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Wed Feb 13, 2019 9:01 pm

Here is what I was up to. The simulation model I used is very simple, it takes the daily value of the appropriate index (in Total Return form), multiplies by the leverage factor (2x or 3x), then subtracts -geometrically- the LIBOR rate times leverage minus 1. This is the model we discussed so far, which was mentioned by the various articles that were quoted, starting by the 1996 paper from Assness. The LIBOR rate is the 1 month rate, to be consistent with what we read from ProShares and Direxion reports. I ignored any other effect for now (e.g. spread, etc).

I then compared to the daily values of the leveraged funds being investigated (while using the appropriate index dependent on the fund). Please note that I used a "gross return" daily data series (extracted from Morningstar), which is the total return series minus the expense ratio. Note that the ER varied a bit over time for the various funds, so it's good to take this daily factor out of the equation when comparing to the index-based model.

Let's start with UBT (ProShares Ultra 20+ Year Treasury, 2x). The first chart is the usual growth chart (over the life of the fund). The second chart is a Telltale chart (a chart popularized by John Bogle himself, which is very handy to analyze things over time). The mapping is remarkably good, the simulated model slightly underperforming the real-life fund. Please note though that the timeframe is quite short (starting in Jan-10) and this is a low-interest time period, so it is not impossible that the distorting effect of some significant factor is visually missing. The next couple of charts is UST (ProShares Ultra 7-10 Year Treasury, 2x), with an equally good result. The black line is the trajectory of the regular index of reference.

Image

Image

EDIT: made the vertical axis logarithmic and added the black line (regular index of reference).
Last edited by siamond on Thu Feb 14, 2019 7:34 pm, edited 1 time in total.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Wed Feb 13, 2019 9:20 pm

Those UBT/UST results are truly excellent, but what about leveraged funds tracking the S&P 500? And what about a 3x fund?

My next test is based on UPRO (ProShares UltraPro S&P500, 3x). Here are the charts. Although the trajectories are remarkably well correlated, something is somewhat amiss here. The simulated model overshoots the real-life fund, by roughly 1% annual. I am not quite sure why, I can only observe that the decay of the real-life fund seems pretty steady, as the Telltale illustrates. Please note that I kept using the TR form of the index, although the ProShares report speaks of tracking the price return index, but looking at the daily values, it is very clear that they do something smart with the dividends too.

Image

Now what about a fund with a longer lifetime, spanning the last two deep crises? Luckily, ULPIX (ProFunds UltraBull Fund, 2x) is exactly that. Note that this one is a mutual fund, not an ETF. Here are the corresponding charts. In this case, the simulated model overshoots by roughly 1.2% annual, and the compounded difference is much more visible due to the longer time period. Now as to the comparison with the regular S&P 500 index (the black line), well, this seems quite sobering... The 'rosy' aspect of the UPRO chart clearly suffers from recency bias, I'm afraid.

Image

I think we are still missing a friction cost of sorts in the model. Still, overall, this seems like a really good start for our simulation endeavor!

EDIT: made the vertical axis logarithmic and added the black line (regular index of reference).
Last edited by siamond on Thu Feb 14, 2019 6:26 pm, edited 1 time in total.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Wed Feb 13, 2019 11:24 pm

Back to bond funds, with the strange case of Direxion LETFs.

This test is based on TMF (Direxion Daily 20+ Year Treasury Bull, 3x). Here are the charts. Please note that the index being used is directly comparable to UBT (although TMF's leverage is 3x while UBT is 2x). As you can see, there are several issues. Not only does the real-life fund badly undershoots the model (by nearly 2% a year), but the Telltale chart shows a bunch of tremors that we didn't see in any of the previous charts. As was previously mentioned, a daily examination of TMF returns does NOT show the close leverage multiplier compared to its index that we see with the other funds.

This being said, since 2013, the Telltale chart stayed roughly flat on average, which means that TMF apparently did its job, albeit in a strange manner. I don't know though, using such leverage funds already requires quite the leap of faith, and this chart just doesn't inspire a lot of trust. Or maybe I am missing something...

Image

I assembled a similar chart for TYD (Direxion Daily 7-10 Year Treasury Bull, 3x) and I spare you the details, the outcome is rather strange too.

EDIT: made the vertical axis logarithmic and added the black line (regular index of reference).
Last edited by siamond on Thu Feb 14, 2019 9:11 pm, edited 1 time in total.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Thu Feb 14, 2019 9:48 am

JackoC wrote:
Wed Feb 13, 2019 4:52 pm
Also one minor point about OP and subsequent posts. If want to simulate past daily data, I think you have to calculate the interest between dates, so 1/360*rate between weekdays, 3/360 over weekends, whatever the span between data points over holidays, not 1/250 between any two business days. By convention of money markets, rates like LIBOR are accrued at 1/360 (not 365 or 366) per calendar day, you get or pay 365/360's or 366/360's of that rate every year.
Maybe we should clarify that, in the case of LIBOR, the 1/360 math for daily rates is straight arithmetic, it is really X%/360 instead of power(1+X%, 1/360)-1. I actually double-checked this assumption by comparing the daily LIBOR rates from FRED and the corresponding growth index from Morningstar (secId FOUSA06JCM), and it maps. Note that Morningstar does the hard work of mapping the rate to trading and closed days, so I prefer to use this series than doing the 1/360 daily math from FRED.

Now this begs a question. It appears that the best proxy for LIBOR rates before 1986 would be the Effective Federal Funds Rate (which can found on FRED, and starts mid-1954). This rate is a better proxy than T-Bills, because it does capture the panicky times where banks clamp down on interbank lending (there is even a TED spread metric capturing the difference, and some crazy folks do market timing on this basis!). And I can't find the corresponding growth index series on Morningstar, hmpf.

Question is, how should we compute daily rates for the Federal Funds Rate? Does it work like LIBOR? Or is it more regular compounding math?

EDIT: I found the answer to my own question (empirically, thanks to a Credit Suisse data series), same rule, arithmetic 1/360. And while I was at it, I assembled a spliced series (Fed Effective Rate; LIBOR 1m) and also compared the LIBOR rate to the Fed's rate trajectory. The LIBOR/1m rate was 0.2% higher (per year) than the Fed's rate since 1987. Which might be a simple duration thing, as the Fed's rate is for overnight loans if I understand well, not for a full month.

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Re: Simulating Returns of Leveraged ETFs

Post by pezblanco » Thu Feb 14, 2019 4:47 pm

I'm just wanting to ask a question from those doing the simulations. Presumably the 3x funds are more expensive than the 2x funds. How do these costs compare to just borrowing money on a margin account and leveraging up by 3x or 2x? Is it substantially cheaper to use the leveraged funds than just borrowing money from a brokerage account (I think you can borrow at the 1month T-bill rate + 1%)?

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