Ah, excellent, this is perfect! I'll start with equation (4) and see how it goes, but yes, this is exactly what I was looking for. Thank you!interestediniras wrote: ↑Wed Feb 27, 2019 8:42 pmPerhaps look at Avellaneda and Zhang (2010), equation 10, which models the full continuoustime evolution of the return of a leveraged ETF.siamond wrote: ↑Wed Feb 27, 2019 8:34 pmI believe (?!) that there is a formulaic way to combine monthly returns (which we have) times leverage with daily volatility (which we could proxy to the aggregate of modern times?) and get the monthly returns of a leveraged fund (and then do the LIBOR math/etc). We could test such approach against actuals (modern times), check if this seems reasonable, then apply it to the old times. Except that I can't recall which academic paper described such formulaic math? Anybody having an idea?
https://www.math.nyu.edu/faculty/avella ... FS.pdf.pdf
Simulating Returns of Leveraged ETFs
Re: Simulating Returns of Leveraged ETFs

 Posts: 139
 Joined: Thu Jan 10, 2019 2:06 am
Re: Simulating Returns of Leveraged ETFs
Tang and Xu (2013) may also be generally relevant.
The paper is not publicly downloadable but I have hosted it here:
https://www.dropbox.com/s/rqypuh1re95y7 ... 3.pdf?dl=0
It will be removed in several days, so please save it now.
The paper is not publicly downloadable but I have hosted it here:
https://www.dropbox.com/s/rqypuh1re95y7 ... 3.pdf?dl=0
It will be removed in several days, so please save it now.
Re: Simulating Returns of Leveraged ETFs
Got it. I just skimmed through it, and it seems to address a LOT of the issues we've been discussing on this thread. I had a clear inkling that we were kind of reinventing the wheel... Thank you, will read in more details tomorrow.interestediniras wrote: ↑Wed Feb 27, 2019 11:20 pm Tang and Xu (2013) may also be generally relevant.
The paper is not publicly downloadable but I have hosted it here:
https://www.dropbox.com/s/rqypuh1re95y7 ... 3.pdf?dl=0
It will be removed in several days, so please save it now.

 Posts: 139
 Joined: Thu Jan 10, 2019 2:06 am
Re: Simulating Returns of Leveraged ETFs
Here is yet another paper with relevant information: Ginley et al. (2015).
https://file.scirp.org/Html/41490368_61565.htm
The PDF is freely available.
On page 2, they describe previous studies of the returns of LETFs, including the paper by Avellaneda and Zhang (2010) previously mentioned. The paper in general presents a novel method for simulating LETF tracking errors.
https://file.scirp.org/Html/41490368_61565.htm
The PDF is freely available.
On page 2, they describe previous studies of the returns of LETFs, including the paper by Avellaneda and Zhang (2010) previously mentioned. The paper in general presents a novel method for simulating LETF tracking errors.

 Posts: 139
 Joined: Thu Jan 10, 2019 2:06 am
Re: Simulating Returns of Leveraged ETFs
Building off Tang and Xu (2013), Loviscek et al. (2014), largely by the same authors, present simulation results based on the entire history of the DJIA which suggest that the median return deviation is positive, i.e. it is more probable than not that deviation from the nominal leverage factor is beneficial to the investor. They obtain the same result using the actual sequence of historical returns. These results are also obtained using the S&P 500.
As before, the paper will be hosted temporarily:
https://www.dropbox.com/s/ardm8pronbefa ... 4.pdf?dl=0
They comment on previous literature which is disfavorable to the feasibility of longterm investment in leveraged ETFs and explain why their results deviate. I think this paper is a particularly useful guide to understanding the overall feasibility of the leveraged strategy and would welcome your comments.
As before, the paper will be hosted temporarily:
https://www.dropbox.com/s/ardm8pronbefa ... 4.pdf?dl=0
They comment on previous literature which is disfavorable to the feasibility of longterm investment in leveraged ETFs and explain why their results deviate. I think this paper is a particularly useful guide to understanding the overall feasibility of the leveraged strategy and would welcome your comments.
Re: Simulating Returns of Leveraged ETFs
I spent time today working on the periods of time where we do NOT have daily (index) returns, using the following assumptions:
a) there is a reliable formula that can take the monthly returns and intramonth (day to day) volatility of a given index, and derive the monthly returns of a corresponding leveraged fund (just capturing the volatility decay factor, setting aside considerations about costs and fees),
b) if we proxy the intramonth volatility to the overall daily volatility of the known times (when daily values are available), the distortion to the outcomes is acceptable. Which means in turn that we're still okish to go if we only have monthly returns for the old days (which is indeed the challenge we're often facing).
About the magical formula, the papers from Avellaneda and Zhang (2010) and Tang and Xu (2013) provided the answer (many thanks to interestediniras for the pointers). Let me quote the equations of relevance (click to see a bigger display):
In equation (4), the 'H', 'r' and 'f' factors are about borrowing costs and expense ratios, which we cover by other means in our model, so let's set that aside. Then equation (2) and equation (4) are strictly equivalent. I quoted equation (2) because it helped me realize that the stddeviation (volatility) factor has to be multiplied by the number of days in the month ('t'), a fact that is kind of hidden by the way the variance is expressed in equations (4) and (5).
As to equation (10), it is supposed to be more accurate, but I have to say I don't quite understand it. There is a subtle difference in the way the 'V' factor is computed and I don't get it. Anyhoo, based on the results I will show in the following posts, solving this little mathematical mystery doesn't seem necessary.
a) there is a reliable formula that can take the monthly returns and intramonth (day to day) volatility of a given index, and derive the monthly returns of a corresponding leveraged fund (just capturing the volatility decay factor, setting aside considerations about costs and fees),
b) if we proxy the intramonth volatility to the overall daily volatility of the known times (when daily values are available), the distortion to the outcomes is acceptable. Which means in turn that we're still okish to go if we only have monthly returns for the old days (which is indeed the challenge we're often facing).
About the magical formula, the papers from Avellaneda and Zhang (2010) and Tang and Xu (2013) provided the answer (many thanks to interestediniras for the pointers). Let me quote the equations of relevance (click to see a bigger display):
In equation (4), the 'H', 'r' and 'f' factors are about borrowing costs and expense ratios, which we cover by other means in our model, so let's set that aside. Then equation (2) and equation (4) are strictly equivalent. I quoted equation (2) because it helped me realize that the stddeviation (volatility) factor has to be multiplied by the number of days in the month ('t'), a fact that is kind of hidden by the way the variance is expressed in equations (4) and (5).
As to equation (10), it is supposed to be more accurate, but I have to say I don't quite understand it. There is a subtle difference in the way the 'V' factor is computed and I don't get it. Anyhoo, based on the results I will show in the following posts, solving this little mathematical mystery doesn't seem necessary.

 Posts: 139
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Re: Simulating Returns of Leveraged ETFs
siamond: Thank you for all of your work on getting this simulation to work. I believe this will be extremely valuable in understanding the dynamics of leveraged ETFs. If you can reproduce results similar to those in Loviscek et al., that would really give us confidence moving forward.
I agree that the differences between equations (4) and (10) are not substantial, in fact I should have pointed you to the former instead of the latter. The derivation of (10) seems relevant mainly as a weak confirmation of (4) in a slightly different (continuous time) setting.
Incidentally, if you would like to look at Cheng and Madhavan (2009), here it is:
https://www.dropbox.com/s/koft8z8zvj9q3 ... 9.pdf?dl=0
I agree that the differences between equations (4) and (10) are not substantial, in fact I should have pointed you to the former instead of the latter. The derivation of (10) seems relevant mainly as a weak confirmation of (4) in a slightly different (continuous time) setting.
Incidentally, if you would like to look at Cheng and Madhavan (2009), here it is:
https://www.dropbox.com/s/koft8z8zvj9q3 ... 9.pdf?dl=0
Re: Simulating Returns of Leveraged ETFs
To test all of that, I used the S&P 500 data to begin with, restricted to 1973+ for now. As explained in this post, although we do not have the daily (total) returns until the late 80s, we can use a little trick based on the daily price series and the monthly total returns series, and assemble a very credible daily series which captures both growth and (daily) volatility.
So I ran two tests. I expected the first one to go well, but wasn't so sure about the second one.
a) In a first simulation, I took the monthly returns and intramonth (day to day) volatility of the S&P 500, and used the magical formula to compute monthly returns of a 2x and 3x leveraged fund. Then I adjusted with the monthly values of the borrowing cost. And then I compared to the regular daily computation I was using so far.
b) In a second simulation, I replaced the historical volatility values by a constant, equal to the daily volatility of the known times (when reallife daily total returns are fully available, i.e. 1988+). And compared to the first simulation and to the regular daily computation.
Here is the corresponding growth chart. Click on it for a larger version and you will see that the outcome is quite remarkable. I mean, we don't even see the blue line in the 2x graph (because the red line is so close to it, it is *slightly* different though, I swear!). The 3x results are less satisfying (note that the vertical scale is logarithmic, small differences do matter) for the averaged model (2nd test), but remain fairly reasonable imho.
PS. to extend the S&P 500 data series to the mid50s, we will actually NOT need those techniques (cf. my little trick between price and totalreturn), but I wanted to test the methodology in a context of fairly high volatility, i.e. stocks.
So I ran two tests. I expected the first one to go well, but wasn't so sure about the second one.
a) In a first simulation, I took the monthly returns and intramonth (day to day) volatility of the S&P 500, and used the magical formula to compute monthly returns of a 2x and 3x leveraged fund. Then I adjusted with the monthly values of the borrowing cost. And then I compared to the regular daily computation I was using so far.
b) In a second simulation, I replaced the historical volatility values by a constant, equal to the daily volatility of the known times (when reallife daily total returns are fully available, i.e. 1988+). And compared to the first simulation and to the regular daily computation.
Here is the corresponding growth chart. Click on it for a larger version and you will see that the outcome is quite remarkable. I mean, we don't even see the blue line in the 2x graph (because the red line is so close to it, it is *slightly* different though, I swear!). The 3x results are less satisfying (note that the vertical scale is logarithmic, small differences do matter) for the averaged model (2nd test), but remain fairly reasonable imho.
PS. to extend the S&P 500 data series to the mid50s, we will actually NOT need those techniques (cf. my little trick between price and totalreturn), but I wanted to test the methodology in a context of fairly high volatility, i.e. stocks.
Last edited by siamond on Fri Mar 01, 2019 10:51 pm, edited 1 time in total.
Re: Simulating Returns of Leveraged ETFs
Now let's try with LongTerm Treasuries. Same methodology, except that we do NOT have any daily data until early 1998. Which is why you will only see 1998+ data in the charts below (1997 on the horizontal axis should be understood as 31Dec1997).
Again, we just can't see the blue line although it is ohsoslightly different (I had to triplecheck to be sure!). The 'magical formula' (red line) works really well, no question. The coarser model (with the constant/average volatility) is less satisfying. The green line ends up catching up with the other lines, but the mid2000s were not quite right.
This made me wonder how sensitive the coarser model is to the constant volatility parameter, and the answer is QUITE A LOT (if I change it from 0.66 to 0.62, the green line aligns with the red line in the first decade, then overshoots in the last decade). This planted some doubts in my head, because the interest rate patterns were clearly very different in the 70s/80s compared to more modern days (although the daily volatility may not have been that different, we just don't know). Still... given that the S&P 500 test was quite reasonable, I guess this is probably good enough (and obviously more realistic than simply using monthly numbers). Feedback?
Again, we just can't see the blue line although it is ohsoslightly different (I had to triplecheck to be sure!). The 'magical formula' (red line) works really well, no question. The coarser model (with the constant/average volatility) is less satisfying. The green line ends up catching up with the other lines, but the mid2000s were not quite right.
This made me wonder how sensitive the coarser model is to the constant volatility parameter, and the answer is QUITE A LOT (if I change it from 0.66 to 0.62, the green line aligns with the red line in the first decade, then overshoots in the last decade). This planted some doubts in my head, because the interest rate patterns were clearly very different in the 70s/80s compared to more modern days (although the daily volatility may not have been that different, we just don't know). Still... given that the S&P 500 test was quite reasonable, I guess this is probably good enough (and obviously more realistic than simply using monthly numbers). Feedback?
Last edited by siamond on Fri Mar 01, 2019 10:54 pm, edited 1 time in total.

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Re: Simulating Returns of Leveraged ETFs
Interesting stuff. Is there a location where you are maintaining various dailyreturn data sets? Curious if you have something that is on the more refined side going back further than the 1986 data I have currently.
Re: Simulating Returns of Leveraged ETFs
I shared my spreadsheets with a couple of individuals who are the most active in corresponding numbercrunching and modeling, but I am trying to proceed in a cautious manner and limit exposure for now, as we're learning every day in this process. Various assumptions remain quite questionable, we're clearly missing something significant (the ~1% CAGR disconnect we have on those charts) and we still have little clue about daily transaction costs. Once I feel that we are on a more stable ground, I'll be more open with interested parties. Fair enough?MotoTrojan wrote: ↑Fri Mar 01, 2019 8:30 pm Interesting stuff. Is there a location where you are maintaining various dailyreturn data sets? Curious if you have something that is on the more refined side going back further than the 1986 data I have currently.

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Re: Simulating Returns of Leveraged ETFs
Absolutely! Enjoying following along in the meantime. My plan is set in play and wouldn’t change based on further data anyways.siamond wrote: ↑Fri Mar 01, 2019 8:37 pmI shared my spreadsheets with a couple of individuals who are the most active in corresponding numbercrunching and modeling, but I am trying to proceed in a cautious manner and limit exposure for now, as we're learning every day in this process. Various assumptions remain quite questionable, we're clearly missing something significant (the ~1% CAGR disconnect we have on those charts) and we still have little clue about daily transaction costs. Once I feel that we are on a more stable ground, I'll be more open with interested parties. Fair enough?MotoTrojan wrote: ↑Fri Mar 01, 2019 8:30 pm Interesting stuff. Is there a location where you are maintaining various dailyreturn data sets? Curious if you have something that is on the more refined side going back further than the 1986 data I have currently.
Re: Simulating Returns of Leveraged ETFs
I was pondering about the green lines in the graphs I posted earlier today (i.e. the coarser model based on a constant daily volatility for each month). When they undershoot the daily model, this is due to a volatility assumption which is too high, making the model overly conservative. So I charted the reallife volatility numbers that we know of. The straight line in those graphs is a simple trend line.
Stocks were agitated for sure and went berserk a few times (Oct87 as the prototypical example!). LT Treasuries were a little different. Before 2008, the trajectory was fairly mild. After that, it became more agitated. And the trend line is definitely going up. Those variations easily explain the trajectory of the green line in this post.
In other words, IF the aggregate daily volatility of the past two decades was higher than what happened in the early days (e.g. 70s, 80s), then the coarse model (green line) is conservative (which is kind of ok). But if this is the reverse way around, the coarse model is too aggressive (which isn't good). And in any case, we may miss some important dynamics. Maybe we could check some daily interest rates series from FRED to run a sanity check? Let me investigate...
Stocks were agitated for sure and went berserk a few times (Oct87 as the prototypical example!). LT Treasuries were a little different. Before 2008, the trajectory was fairly mild. After that, it became more agitated. And the trend line is definitely going up. Those variations easily explain the trajectory of the green line in this post.
In other words, IF the aggregate daily volatility of the past two decades was higher than what happened in the early days (e.g. 70s, 80s), then the coarse model (green line) is conservative (which is kind of ok). But if this is the reverse way around, the coarse model is too aggressive (which isn't good). And in any case, we may miss some important dynamics. Maybe we could check some daily interest rates series from FRED to run a sanity check? Let me investigate...
Re: Simulating Returns of Leveraged ETFs
Honing a little more on bonds... I am a little wary about the assumptions of the 'coarser' model (green line in previous posts), as daily volatility does seem to vary over time in a nonnegligible manner, and the equations we've been using are quite sensitive to the volatility input parameter.
(as a side note, for S&P 500 stocks, I was able to go back to the mid50s by combining daily prices and quarterly dividends until the early 70s, then daily prices and monthly dividends as previously explained, followed by daily data. I think this is quite reasonable since stocks volatility is really centered on price vagaries, so the coarse model is only required for bonds  and other data series like MSCI EAFE).
There are multiple Barclays indices which go back to the mid70s (e.g. Barclays Aggregate Bond Treasury TR; Barclays US Aggregate Bond TR USD), but unfortunately daily returns are hard to come by, most series only have daily returns from the mid90s, while the main aggregate series starts daily data in 1989. And they are not longterm bonds anyway. So this really doesn't help.
The FRED daily data series for 20yrs treasuries interest rates only starts at the end of 1994 (I guess this isn't a coincidence). Strangely enough, the 30yrs treasuries daily data series is much better, going back to 1977. And the 10yrs treasuries data series is excellent, going back to to the early 60s, impressive. This being said, the relation between interest rates and LT bond funds total returns is indirect, and even more for volatility, so we really can't use a quick and dirty quantitative rule of thumb here, we can only get a sense of the scale of the issue.
Here is a comparison of daily (dayto day within a month) volatility for 10yrs and 30yrs interest rates. The strange flat section for 30yrs rates is due to a hole in FRED's corresponding data series for a few years...
Clearly, the 80s were a period of extraordinary volatility. While the years before were milder in nature and more comparable to modern times (my guess is that it was true too in the 50s?). I would suspect the 'coarse' model to overshoot (i.e. be too optimistic) in the 80s, in other words, while being okish in previous years. I have no idea how to get better daily volatility data (for LT bonds total returns)... Ideas, anyone?
EDIT: I just checked the FRED NBER Macro History Database, which has yields series going way back in time, including for LT bonds/treasuries, but unfortunately... all monthly. Daily data series are hard to come by!
(as a side note, for S&P 500 stocks, I was able to go back to the mid50s by combining daily prices and quarterly dividends until the early 70s, then daily prices and monthly dividends as previously explained, followed by daily data. I think this is quite reasonable since stocks volatility is really centered on price vagaries, so the coarse model is only required for bonds  and other data series like MSCI EAFE).
There are multiple Barclays indices which go back to the mid70s (e.g. Barclays Aggregate Bond Treasury TR; Barclays US Aggregate Bond TR USD), but unfortunately daily returns are hard to come by, most series only have daily returns from the mid90s, while the main aggregate series starts daily data in 1989. And they are not longterm bonds anyway. So this really doesn't help.
The FRED daily data series for 20yrs treasuries interest rates only starts at the end of 1994 (I guess this isn't a coincidence). Strangely enough, the 30yrs treasuries daily data series is much better, going back to 1977. And the 10yrs treasuries data series is excellent, going back to to the early 60s, impressive. This being said, the relation between interest rates and LT bond funds total returns is indirect, and even more for volatility, so we really can't use a quick and dirty quantitative rule of thumb here, we can only get a sense of the scale of the issue.
Here is a comparison of daily (dayto day within a month) volatility for 10yrs and 30yrs interest rates. The strange flat section for 30yrs rates is due to a hole in FRED's corresponding data series for a few years...
Clearly, the 80s were a period of extraordinary volatility. While the years before were milder in nature and more comparable to modern times (my guess is that it was true too in the 50s?). I would suspect the 'coarse' model to overshoot (i.e. be too optimistic) in the 80s, in other words, while being okish in previous years. I have no idea how to get better daily volatility data (for LT bonds total returns)... Ideas, anyone?
EDIT: I just checked the FRED NBER Macro History Database, which has yields series going way back in time, including for LT bonds/treasuries, but unfortunately... all monthly. Daily data series are hard to come by!
Re: Simulating Returns of Leveraged ETFs
Took me a while, but I finally went past the 'magical formula' and read more about their regression testing. This is a well researched paper, making a good use of regression testing for the right reasons. I regret that they didn't try to better isolate the factors going beyond LIBOR rate and expense ratio though. Maybe I missed something, but I didn't see a concrete suggestion to improve the model beyond those factors.interestediniras wrote: ↑Wed Feb 27, 2019 11:20 pm Tang and Xu (2013) may also be generally relevant.
The paper is not publicly downloadable but I have hosted it here:
https://www.dropbox.com/s/rqypuh1re95y7 ... 3.pdf?dl=0
It will be removed in several days, so please save it now.
I like their approach clearly distinguishing between daily NAV factors and compounding factors. I'll try to run some math myself in this respect, this is a good idea.
Re: Simulating Returns of Leveraged ETFs
Not exactly sure what you mean, but since the CMT yields (e.g., from FRED) are par bond yields, you can calculate price, and thus price change (volatility), from the yield, assuming initial price of 100, and coupon rate = initial yield. Can either use PRICE or PV function to calculate price at t+1 (whether daily, monthly, or whatever). Assuming flat yield curve for very short periods is reasonable, especially for longterm bonds.
Kevin
....... Suggested format for Asking Portfolio Questions (edit original post)
Re: Simulating Returns of Leveraged ETFs
Sidetracking from bonds for a few...
I stumbled upon the fact that both S&P and MSCI actually maintain indices for 2x leveraged funds:
https://us.spindices.com/indices/strate ... ailyindex
https://us.spindices.com/documents/meth ... xmath.pdf (start at page 44)
https://www.msci.com/eqb/methodology/me ... st2014.pdf
What is interesting is that:
a) they describe the same methodology as what we've been using
b) both make very explicit that the math has to apply to total returns (incl. dividends)
c) both refer to the LIBOR overnight rate (as opposed to LIBOR 1w or LIBOR 1m)
I can't find the corresponding S&P data series on Morningstar, but I was able to download it from the graph on the S&P Web page, I will run comparisons with our model for the known years. I did find an MSCI 2x data series, although it is about the USA at large, not the S&P 500, which skews things.
I stumbled upon the fact that both S&P and MSCI actually maintain indices for 2x leveraged funds:
https://us.spindices.com/indices/strate ... ailyindex
https://us.spindices.com/documents/meth ... xmath.pdf (start at page 44)
https://www.msci.com/eqb/methodology/me ... st2014.pdf
What is interesting is that:
a) they describe the same methodology as what we've been using
b) both make very explicit that the math has to apply to total returns (incl. dividends)
c) both refer to the LIBOR overnight rate (as opposed to LIBOR 1w or LIBOR 1m)
I can't find the corresponding S&P data series on Morningstar, but I was able to download it from the graph on the S&P Web page, I will run comparisons with our model for the known years. I did find an MSCI 2x data series, although it is about the USA at large, not the S&P 500, which skews things.
Re: Simulating Returns of Leveraged ETFs
Fantastic find!siamond wrote: ↑Mon Mar 04, 2019 12:55 am Sidetracking from bonds for a few...
I stumbled upon the fact that both S&P and MSCI actually maintain indices for 2x leveraged funds:
https://us.spindices.com/indices/strate ... ailyindex
https://us.spindices.com/documents/meth ... xmath.pdf (start at page 44)
https://www.msci.com/eqb/methodology/me ... st2014.pdf
What is interesting is that:
a) they describe the same methodology as what we've been using
b) both make very explicit that the math has to apply to total returns (incl. dividends)
c) both refer to the LIBOR overnight rate (as opposed to LIBOR 1w or LIBOR 1m)
I can't find the corresponding S&P data series on Morningstar, but I was able to download it from the graph on the S&P Web page, I will run comparisons with our model for the known years. I did find an MSCI 2x data series, although it is about the USA at large, not the S&P 500, which skews things.
Re: Simulating Returns of Leveraged ETFs
I think I finally got to the bottom of this. LETFs just do NOT resell their swaps (or futures) entire positions at the end of the day. I mean, this would be a giant move involving a lot of money, high transaction costs, this would serve no real purpose, and this would give a golden opportunity for highfrequency traders to go ahead of the move... What happens is that LETFs reposition their market exposure to the target leverage at the end of the day. This may involve some buying/selling of their regular index positions as well as some buying/selling of their swap/future positions, but this is an INCREMENTAL process.siamond wrote: ↑Mon Feb 25, 2019 10:44 am Switching gears for a minute... I had an offline exchange with EfficientInvestor about borrowing costs. In his model, he avoided accounting for such daily costs for nontrading days (e.g. weekends). Which makes sense if the LETF truly implements a full daily process of borrowing/leveraging/selling (e.g. via counterparties as discussed here), although I have a nagging feeling that this is a very simplified view, and actual operations might be a bit different (e.g. more incremental).
In my model, thanks to the LIBOR data series I extracted from Morningstar, the weekend rates are null anyway (and obviously the index returns are null), therefore we've been doing the same thing by different means. This being said, for pre1986 data, I use the Effective Federal Funds Rate, and when checking this data series, the values during weekends are NOT null. So I added some (optional) logic to force the weekends (and some fixed date holidays) rates to zero. This has a nonnegligible impact on the model for the mid70s to mid80s (making it more favorable by avoiding useless borrowing during nontrading days, notably in a time of high interest rates!). This will also apply to a possible extension of the model towards the mid50s and 60s if we come to that (EFF rates are available since 1954). I will update all my spreadsheets accordingly.
This is something we cannot easily test against actuals, so if anybody has reasons to doubt the corresponding logic, please speak up.
In other words, the daily process we decomposed in this post was a simple way to explain things, but reality is about daily repositioning towards market exposure (not my words, I read that several times in funds/leveraging literature, although I didn't keep the exact pointer).
The indirect implication of such incremental considerations is that, at the end of the week (or before a holiday), the LETF keeps its position with swaps/futures, according to the targeted market exposure. And then over the weekend/holidays, well, the borrowing costs keep adding up.
I am not making this assertion out of the blue (although this now makes perfect sense to me), I actually verified it by comparing the MSCI USA Leveraged 2X data series (see this post) to the theoretical 2x leveraged model we've been discussing (derived from the corresponding MSCI USA index). When using a LIBOR/overnight data series while forcing the nontrading days to zero borrowing costs, this just doesn't match. When using a LIBOR/overnight data series while leaving the rates as is for nontrading days, funny, it matches perfectly. Same applies to S&P 500 Leveraged/Inverse 2x data series (I'll discuss special considerations about the regular S&P 500 leveraged series in the next post). And as I'll explain later, this also better matches the LETF actuals. A solid reasoning confirmed by empirical data from multiple independent sources, I think we're on solid ground here.
Note for those of you interested in modeling this stuff yourself:
 using the FRED LIBOR overnight data series works fine, as long as you divide the rate by 360 to get a daily value that you can then use in regular compounding math. Same for the (US) Effective Federal Funds Rate, which has much more history. Both data series (LIBOR/overnight and EFFR) actually follow each other very closely for overlapping years.
 using the LIBOR data series found on Morningstar (e.g. this one) does exclude trading days, and leads to flawed results.
 practically speaking, I spliced the EFFR and the LIBOR overnight data series (derived from FRED) and this is the 1955+ daily data series I now use for the borrowing costs in my LETF modeling effort.
EDIT: one small thing to be careful about... The FRED LIBOR (and EFFR) number for a given day is the rate at the end of the day. Therefore, it is the rate which applies the day after (after dividing by 360). This is made clear on the FRED Web site, and I doublechecked by comparing various (trading day) values from FRED to the corresponding Morningstar growth indices.
Last edited by siamond on Wed Mar 06, 2019 9:20 am, edited 1 time in total.

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Re: Simulating Returns of Leveraged ETFs
Yes, they merely incrementally reposition, which is why capital flow can obviate the need for explicit rebalancing:
https://www.federalreserve.gov/econresd ... 106pap.pdf
https://www.federalreserve.gov/econresd ... 106pap.pdf

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Re: Simulating Returns of Leveraged ETFs
So are there updated daily returns for various assets, for those of us less active in this effort ? Would love some International equity data and/or S&P500/LTTs going back beyond 1986.
Re: Simulating Returns of Leveraged ETFs
Give me a day or two, grasshopper! Almost there, just want to work on a couple of details and make a couple more tests.MotoTrojan wrote: ↑Tue Mar 05, 2019 4:10 pmSo are there updated daily returns for various assets, for those of us less active in this effort ? Would love some International equity data and/or S&P500/LTTs going back beyond 1986.

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Re: Simulating Returns of Leveraged ETFs
Exciting stuff! Grasshopper shall relax for a few days.siamond wrote: ↑Tue Mar 05, 2019 5:17 pmGive me a day or two, grasshopper! Almost there, just want to work on a couple of details and make a couple more tests.MotoTrojan wrote: ↑Tue Mar 05, 2019 4:10 pmSo are there updated daily returns for various assets, for those of us less active in this effort ? Would love some International equity data and/or S&P500/LTTs going back beyond 1986.
 privatefarmer
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Re: Simulating Returns of Leveraged ETFs
My head is spinning. Thank God smart folks like yourselves are doing all this data crunching. What I think myself and most others want to know is, does the simulated UPROHF86/TMFHF86 data sets that HEDGEFUNDIE uploaded on his original thread still hold water? Those I believe came out with about a 16.7% CAGR over the ~30 years. Does that need to be adjusted more downward now or is that still the most accurate simulation we have? Thanks again

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Re: Simulating Returns of Leveraged ETFs
privatefarmer wrote: ↑Wed Mar 06, 2019 9:52 am My head is spinning. Thank God smart folks like yourselves are doing all this data crunching. What I think myself and most others want to know is, does the simulated UPROHF86/TMFHF86 data sets that HEDGEFUNDIE uploaded on his original thread still hold water?
Yes they do.
viewtopic.php?f=10&t=272007&p=4397070#p4397070

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Re: Simulating Returns of Leveraged ETFs
Doesn’t mean they’ll hold water in the next 30 though. I’m still looking forward to finding out.privatefarmer wrote: ↑Wed Mar 06, 2019 9:52 am My head is spinning. Thank God smart folks like yourselves are doing all this data crunching. What I think myself and most others want to know is, does the simulated UPROHF86/TMFHF86 data sets that HEDGEFUNDIE uploaded on his original thread still hold water? Those I believe came out with about a 16.7% CAGR over the ~30 years. Does that need to be adjusted more downward now or is that still the most accurate simulation we have? Thanks again
Re: Simulating Returns of Leveraged ETFs
I am going to issue multiple posts comparing our current simulation methodology (aka the model) with actual returns from leveraged funds. As a reminder, here are the latest assumptions:
 all comparisons are performed based on funds' gross returns, which are essentially total returns (price+dividends) BEFORE the expense ratio adjustment.
 indices of references are also based on gross returns (i.e. dividends included), hence directly compare to funds' gross returns.
 the leverage / market exposure is assumed to be reset ('repositioned') at the end of each trading day.
 every day, the leverage math for the model is simple: daily return = X times the index return, minus (X1) times the borrowing cost.
 the daily math applies to nontrading days as well (cf. borrowing costs keep going).
 borrowing costs are based on the Overnight US Dollar LIBOR interest rate (and before its inception, the US effective federal funds rate, aka EFFR).
Such modeling math can be performed in a spreadsheet in two ways which basically deliver the same results:
1) run the exact math with daily numbers, one day at a time, and compounds the results to obtain monthly (leveraged) returns.
2) use the 'magical formula' described in this post, and derive the model monthly returns from the leverage factor, the index monthly return and its daytoday volatility (or a solid proxy when needs be).
As we'll see, there are additional 'friction' costs that we do not know how to account for in the model (e.g. transaction costs for repositioning being an important one). This will lead us to introduce a rough adjustment factor, which is unfortunate as this is basically curvefitting and may not reliably apply to the past, but... we couldn't find any better working assumption (so far).
We will make a heavy use of Telltale charts in the following posts, such a chart essentially depicts the ratio between the aggregate growth of a data series and the aggregate growth of a benchmark (e.g. the leveraged model). Please check the Telltale chart wiki page to better understand the concept (a very powerful analytical tool that Jack Bogle himself popularized).
 all comparisons are performed based on funds' gross returns, which are essentially total returns (price+dividends) BEFORE the expense ratio adjustment.
 indices of references are also based on gross returns (i.e. dividends included), hence directly compare to funds' gross returns.
 the leverage / market exposure is assumed to be reset ('repositioned') at the end of each trading day.
 every day, the leverage math for the model is simple: daily return = X times the index return, minus (X1) times the borrowing cost.
 the daily math applies to nontrading days as well (cf. borrowing costs keep going).
 borrowing costs are based on the Overnight US Dollar LIBOR interest rate (and before its inception, the US effective federal funds rate, aka EFFR).
Such modeling math can be performed in a spreadsheet in two ways which basically deliver the same results:
1) run the exact math with daily numbers, one day at a time, and compounds the results to obtain monthly (leveraged) returns.
2) use the 'magical formula' described in this post, and derive the model monthly returns from the leverage factor, the index monthly return and its daytoday volatility (or a solid proxy when needs be).
As we'll see, there are additional 'friction' costs that we do not know how to account for in the model (e.g. transaction costs for repositioning being an important one). This will lead us to introduce a rough adjustment factor, which is unfortunate as this is basically curvefitting and may not reliably apply to the past, but... we couldn't find any better working assumption (so far).
We will make a heavy use of Telltale charts in the following posts, such a chart essentially depicts the ratio between the aggregate growth of a data series and the aggregate growth of a benchmark (e.g. the leveraged model). Please check the Telltale chart wiki page to better understand the concept (a very powerful analytical tool that Jack Bogle himself popularized).
Re: Simulating Returns of Leveraged ETFs
Let's start with the leveraged model and funds based on the S&P 500 index. Here is the Telltale chart of corresponding monthly returns BEFORE applying any adjustment factor, using our theoretical model as the benchmark:
 I used every leveraged fund I know of, from ProFunds, ProShares, Direxion and Guggenheim/Rydex, from their date of inception
 I also used the S&P leveraged index and the MSCI USA leveraged index described in this post
 Click on the charts for a larger display
Let's focus on the S&P leveraged index (bright blue line) and the MSCI USA leveraged index (bright red line) in the 2x chart to start with. You can see the MSCI leveraged index hovering around our model, this is due to the fact that its index of reference is actually MSCI USA (hence including midcaps and smallcaps). I ran a comparison using the MSCI USA exact base index and the model was a perfect fit. We cannot say the same of the S&P leveraged index, ahem, which finds a way to strongly underperform the reallife funds. This sent me in a spin of verifying everything, running numerous sanity checks, etc. I finally checked the model against the S&P leveraged inverse index (minus 2 leverage) and this was a perfect fit. I can only draw one logical conclusion, somebody at S&P seriously messed up and we should just ignore their regular 2x leveraged index.
Let's look at the reallife funds now. Clearly, friction costs are at play with a downward trajectory (which is steeper for the 3x funds). Also, we can observe that the older funds displayed some distinct trajectory for a few years, then settled on a fairly steady decay compared to the model. In the early years, I can hypothesize that leveraged funds actually tracked index funds instead of directly tracking indices, hence having to pay for corresponding expense ratios (which were higher by then than nowadays). Finally Direxion SPXL did some weird stuff to begin with that I cannot explain before finding its groove.
Now let's introduce a 0.5% annual 'adjustment factor' for 2x funds and 1% for 3x funds, and see what goes. Interesting, isn't it? Ignoring the first few years, all leveraged funds seem to have their returns modeled pretty well (the Telltale lines are mostly flat), except for ULPIX which is still losing a bit of ground.
 I used every leveraged fund I know of, from ProFunds, ProShares, Direxion and Guggenheim/Rydex, from their date of inception
 I also used the S&P leveraged index and the MSCI USA leveraged index described in this post
 Click on the charts for a larger display
Let's focus on the S&P leveraged index (bright blue line) and the MSCI USA leveraged index (bright red line) in the 2x chart to start with. You can see the MSCI leveraged index hovering around our model, this is due to the fact that its index of reference is actually MSCI USA (hence including midcaps and smallcaps). I ran a comparison using the MSCI USA exact base index and the model was a perfect fit. We cannot say the same of the S&P leveraged index, ahem, which finds a way to strongly underperform the reallife funds. This sent me in a spin of verifying everything, running numerous sanity checks, etc. I finally checked the model against the S&P leveraged inverse index (minus 2 leverage) and this was a perfect fit. I can only draw one logical conclusion, somebody at S&P seriously messed up and we should just ignore their regular 2x leveraged index.
Let's look at the reallife funds now. Clearly, friction costs are at play with a downward trajectory (which is steeper for the 3x funds). Also, we can observe that the older funds displayed some distinct trajectory for a few years, then settled on a fairly steady decay compared to the model. In the early years, I can hypothesize that leveraged funds actually tracked index funds instead of directly tracking indices, hence having to pay for corresponding expense ratios (which were higher by then than nowadays). Finally Direxion SPXL did some weird stuff to begin with that I cannot explain before finding its groove.
Now let's introduce a 0.5% annual 'adjustment factor' for 2x funds and 1% for 3x funds, and see what goes. Interesting, isn't it? Ignoring the first few years, all leveraged funds seem to have their returns modeled pretty well (the Telltale lines are mostly flat), except for ULPIX which is still losing a bit of ground.
Re: Simulating Returns of Leveraged ETFs
It was comforting to see that our model matches the MSCI leveraged index as well as the S&P (inverse) leveraged index. Having to introduce a curvefitting adjustment to match reallife funds is more concerning though. Clearly, there ARE additional friction costs, compounded by the fact that the S&P 500 index is a volatile one (and even more when multiplied by daily 2x or 3x factors), but curvefitting is always a risky endeavor. So I ran a test 'out of sample', using the S&P 400 MidCap index, for which we can find a couple of leveraged funds (2x and 3x). And I used the same 0.5% (2x) and 1% (3x) adjustment factors. And I was VERY impressed by the outcome. I mean, this is no proof by any mean, but it is certainly good to see those flat lines.
As a side note, in multiple cases, I also checked the accuracy of the 'magical (monthly) formula' against the detailed daily math, and the results were very impressive. To the point that frankly, bothering with the day to day math isn't terribly useful anymore.
Tomorrow, we'll discuss leveraged bond funds and then the strange case of MSCI EAFE. And finally wrap up with backwards projections starting in 1955 for US indices (and 1970 for International).
As a side note, in multiple cases, I also checked the accuracy of the 'magical (monthly) formula' against the detailed daily math, and the results were very impressive. To the point that frankly, bothering with the day to day math isn't terribly useful anymore.
Tomorrow, we'll discuss leveraged bond funds and then the strange case of MSCI EAFE. And finally wrap up with backwards projections starting in 1955 for US indices (and 1970 for International).

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Re: Simulating Returns of Leveraged ETFs
Sorry if I missed something obvious but these charts were done prior to accounting for expenses and then a 1% friction gave great results; what am I missing?siamond wrote: ↑Wed Mar 06, 2019 10:48 pm It was comforting to see that our model matches the MSCI leveraged index as well as the S&P (inverse) leveraged index. Having to introduce a curvefitting adjustment to match reallife funds is more concerning though. Clearly, there ARE additional friction costs, compounded by the fact that the S&P 500 index is a volatile one (and even more when multiplied by daily 2x or 3x factors), but curvefitting is always a risky endeavor. So I ran a test 'out of sample', using the S&P 400 MidCap index, for which we can find a couple of leveraged funds (2x and 3x). And I used the same 0.5% (2x) and 1% (3x) adjustment factors. And I was VERY impressed by the outcome. I mean, this is no proof by any mean, but it is certainly good to see those flat lines.
As a side note, in multiple cases, I also checked the accuracy of the 'magical (monthly) formula' against the detailed daily math, and the results were very impressive. To the point that frankly, bothering with the day to day math isn't terribly useful anymore.
Tomorrow, we'll discuss leveraged bond funds and then the strange case of MSCI EAFE. And finally wrap up with backwards projections starting in 1955 for US indices (and 1970 for International).
Quite excited to get that deeper data, especially EAFE.
Re: Simulating Returns of Leveraged ETFs
The trajectory of the reallife funds depicted on those charts is based on gross returns, not total returns. Gross returns are computed like an index, accounting for price variations and dividends, but NOT accounting for expense ratios (while the usual Total Return does account for the ER). Morningstar maintains such 'gross return' data series, which are very handy for such analysis.MotoTrojan wrote: ↑Wed Mar 06, 2019 10:52 pmSorry if I missed something obvious but these charts were done prior to accounting for expenses and then a 1% friction gave great results; what am I missing?
If the model was perfect and if the reallife funds didn't suffer from additional friction costs, all lines on those Telltale charts would be flat and no adjustment factor would be required.
In other words, I entirely removed the effect of expense ratios for the comparison, to minimize the number of moving parts. Of course, when using the model to generate realistic data series, we'll introduce an expense ratio of 1% or so. Which will come in ADDITION to the 'adjustment factor'.

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Re: Simulating Returns of Leveraged ETFs
So in other words, we should be knocking an additional 1% off of the current simulated data for example which hedgefundie is utilizing? Atleast per this result.siamond wrote: ↑Wed Mar 06, 2019 11:01 pmThe trajectory of the reallife funds depicted on those charts is based on gross returns, not total returns. Gross returns are computed like an index, accounting for price variations and dividends, but NOT accounting for expense ratios (while the usual Total Return does account for the ER). Morningstar maintains such 'gross return' data series, which are very handy for such analysis.MotoTrojan wrote: ↑Wed Mar 06, 2019 10:52 pmSorry if I missed something obvious but these charts were done prior to accounting for expenses and then a 1% friction gave great results; what am I missing?
If the model was perfect and if the reallife funds didn't suffer from additional friction costs, all lines on those Telltale charts would be flat and no adjustment factor would be required.
In other words, I entirely removed the effect of expense ratios for the comparison, to minimize the number of moving parts. Of course, when using the model to generate realistic data series, we'll introduce an expense ratio of 1% or so. Which will come in ADDITION to the 'adjustment factor'.
Re: Simulating Returns of Leveraged ETFs
This is a question for HedgeFundie, I am pretty sure that he accounted for the ER in his data, I am less sure that he added a 'curve fitting' extra factor. We didn't resync recently though, will do when I'm done with bonds and EAFE.MotoTrojan wrote: ↑Wed Mar 06, 2019 11:43 pmSo in other words, we should be knocking an additional 1% off of the current simulated data for example which hedgefundie is utilizing? Atleast per this result.
Last edited by siamond on Thu Mar 07, 2019 6:52 pm, edited 1 time in total.
Re: Simulating Returns of Leveraged ETFs
Digging around for EAFE data, I incidentally noticed that there are multiple leveraged funds for SmallCaps. Some track the S&P SmallCap 600 index, but most track the Russell 2000 index. So I thought this would be a good opportunity for another round of 'out of sample' testing.
Here is the outcome. Surprisingly enough, I only had to use a 0.20% annual adjustment (aka curvefitting!) factor to make most Telltale lines basically flat, for both the 2x and 3x leveraged models. This is quite surprising to me, as smallcaps are more volatile than midcaps or largecaps, so I expected more frictions (e.g. transaction costs, spread, etc). We are still missing something in the model... But overall, those are really good results.
As a side note, we have daily Russell 2000 data going back to 1979, so we can go back that far with the corresponding model without starting to make more dubious assumptions.
Here is the outcome. Surprisingly enough, I only had to use a 0.20% annual adjustment (aka curvefitting!) factor to make most Telltale lines basically flat, for both the 2x and 3x leveraged models. This is quite surprising to me, as smallcaps are more volatile than midcaps or largecaps, so I expected more frictions (e.g. transaction costs, spread, etc). We are still missing something in the model... But overall, those are really good results.
As a side note, we have daily Russell 2000 data going back to 1979, so we can go back that far with the corresponding model without starting to make more dubious assumptions.
Re: Simulating Returns of Leveraged ETFs
Moving to LongTerm Treasuries (LTT)... Unfortunately, there are only very few corresponding leveraged funds. Which makes any kind of 'curve fitting' adjustment rather dubious. In the following test, I did NOT adjust the 2x or 1.2x math whatsoever. I did adjust the 3x math by a factor of 0.5% annual, to make the TMF telltale line reasonably flat.
It is rather surprising that UBT matches so well the theoretical model, without apparently suffering from extra friction costs (transaction costs, spread, etc). LT Treasuries are certainly less volatile than regular stocks, but they are definitely NOT that steady either. There are much less individual (treasury bond) securities to track than for a stock index though, this probably helps. Maybe Proshares found a way to do some security lending of sorts or something like that to compensate for (relatively low) friction costs? Not sure.
Note that TMF has a couple of peculiar points to account for:
1) it actually doesn't follow the LTT index per se, it relies on iShares TLT shares (TLT is a very well implemented index fund tracking the LTT index). I assume this means that the TLT ER (0.15%) has to be accounted for somehow, but I'm not sure how (is it part of the TMF ER? or is it extra cost?). I actually suspect that multiple leveraged funds did something similar to begin with (buy shares from a regular index fund instead of tracking the index themselves; this might (?!) explain the RYGBX slippage in its early years).
2) its daily operation does NOT track well either the LTT index nor iShares TLT times the leverage. While Proshares UBT does a great job on a daily basis (and iShares TLT too), I checked. The wiggly line in the chart shows that even on monthly basis, things do not line up well with the index. It appears that the Direxion managers use some form of 'artistic license', which departs from a proper passive paradigm.
It is rather surprising that UBT matches so well the theoretical model, without apparently suffering from extra friction costs (transaction costs, spread, etc). LT Treasuries are certainly less volatile than regular stocks, but they are definitely NOT that steady either. There are much less individual (treasury bond) securities to track than for a stock index though, this probably helps. Maybe Proshares found a way to do some security lending of sorts or something like that to compensate for (relatively low) friction costs? Not sure.
Note that TMF has a couple of peculiar points to account for:
1) it actually doesn't follow the LTT index per se, it relies on iShares TLT shares (TLT is a very well implemented index fund tracking the LTT index). I assume this means that the TLT ER (0.15%) has to be accounted for somehow, but I'm not sure how (is it part of the TMF ER? or is it extra cost?). I actually suspect that multiple leveraged funds did something similar to begin with (buy shares from a regular index fund instead of tracking the index themselves; this might (?!) explain the RYGBX slippage in its early years).
2) its daily operation does NOT track well either the LTT index nor iShares TLT times the leverage. While Proshares UBT does a great job on a daily basis (and iShares TLT too), I checked. The wiggly line in the chart shows that even on monthly basis, things do not line up well with the index. It appears that the Direxion managers use some form of 'artistic license', which departs from a proper passive paradigm.

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Re: Simulating Returns of Leveraged ETFs
I certainly plan to dig deeper into these telltale plots, but am I interpreting TMF right that it deviated (had some drag) in the first few years, but then steadied out after that and has managed to track decently over the longterm (albeit with some shortterm oscillations/volatility)?siamond wrote: ↑Fri Mar 08, 2019 2:11 pm Moving to LongTerm Treasuries (LTT)... Unfortunately, there are only very few corresponding leveraged funds. Which makes any kind of 'curve fitting' adjustment rather dubious. In the following test, I did NOT adjust the 2x or 1.2x math whatsoever. I did adjust the 3x math by a factor of 0.5% annual, to make the TMF telltale line reasonably flat.
It is rather surprising that UBT matches so well the theoretical model, without apparently suffering from extra friction costs (transaction costs, spread, etc). LT Treasuries are certainly less volatile than regular stocks, but they are definitely NOT that steady either. There are much less individual (treasury bond) securities to track than for a stock index though, this probably helps. Maybe Proshares found a way to do some security lending of sorts or something like that to compensate for (relatively low) friction costs? Not sure.
Note that TMF has a couple of peculiar points to account for:
1) it actually doesn't follow the LTT index per se, it relies on iShares TLT shares (TLT is a very well implemented index fund tracking the LTT index). I assume this means that the TLT ER (0.15%) has to be accounted for somehow, but I'm not sure how (is it part of the TMF ER? or is it extra cost?). I actually suspect that multiple leveraged funds did something similar to begin with (buy shares from a regular index fund instead of tracking the index themselves; this might (?!) explain the RYGBX slippage in its early years).
2) its daily operation does NOT track well either the LTT index nor iShares TLT times the leverage. While Proshares UBT does a great job on a daily basis (and iShares TLT too), I checked. The wiggly line in the chart shows that even on monthly basis, things do not line up well with the index. It appears that the Direxion managers use some form of 'artistic license', which departs from a proper passive paradigm.
Ah I see now, it is reasonably flat with an additional 0.5% tossed in.
Re: Simulating Returns of Leveraged ETFs
Yup. Well, except that 10 years isn't exactly longterm...MotoTrojan wrote: ↑Fri Mar 08, 2019 3:38 pm[...] am I interpreting TMF right that it deviated (had some drag) in the first few years, but then steadied out after that and has managed to track decently over the longterm (albeit with some shortterm oscillations/volatility)?
Ah I see now, it is reasonably flat with an additional 0.5% tossed in.
Re: Simulating Returns of Leveraged ETFs
The only thing that can "kill" a 3x leveraged fund is a singleday 33.34% decline in the underlying index. To date, the single largest daily decline in the emerging markets index is about 12%.gtwhitegold wrote: ↑Wed Feb 20, 2019 1:18 pm From what I can tell, the great recession wouldn't have killed any of the imaginary 3X funds mentioned.
"Far more money has been lost by investors preparing for corrections than has been lost in corrections themselves." ~~ Peter Lynch
Re: Simulating Returns of Leveraged ETFs
Be aware that some of these LETFs track underlying ETFs and those underlying ETFs may be subject to limit down trading halts.vineviz wrote: ↑Fri Mar 08, 2019 5:01 pmThe only thing that can "kill" a 3x leveraged fund is a singleday 33.34% decline in the underlying index. To date, the single largest daily decline in the emerging markets index is about 12%.gtwhitegold wrote: ↑Wed Feb 20, 2019 1:18 pm From what I can tell, the great recession wouldn't have killed any of the imaginary 3X funds mentioned.
Re: Simulating Returns of Leveraged ETFs
I suspect also that they are using options for leverage (that's been discussed hasn't it?) ... and so when the option becomes worthless, it can just be discarded i.e. the option holder doesn't participate in the complete fall of the index.samsdad wrote: ↑Fri Mar 08, 2019 7:45 pmBe aware that some of these LETFs track underlying ETFs and those underlying ETFs may be subject to limit down trading halts.vineviz wrote: ↑Fri Mar 08, 2019 5:01 pmThe only thing that can "kill" a 3x leveraged fund is a singleday 33.34% decline in the underlying index. To date, the single largest daily decline in the emerging markets index is about 12%.gtwhitegold wrote: ↑Wed Feb 20, 2019 1:18 pm From what I can tell, the great recession wouldn't have killed any of the imaginary 3X funds mentioned.
Re: Simulating Returns of Leveraged ETFs
Back to our modeling vs. reality test... Intermediate Treasury bonds (more precisely 710 Year Treasuries) now, aka ITTs.
The commentary is similar to LT Treasuries as previously discussed. ProShares UST performed in remarkable alignment with the theoretical model (actually very slightly better). Direxion YTD is somewhat erratic (on a daily as well as monthly basis), but actually better than its LT counterpart (TMF) as I didn't need to introduce any 'curve fitting' adjustment whatsoever to get fairly flat Telltale lines.
Overall, although we really could benefit from having a few other reallife funds to test against, it seems to me that we can use the model as is, without any adjustment factor, to model leveraged treasury funds.
Two more thoughts, given the results of the ITT and LTT tests:
1) if there is an extra cost due to bid/ask spread for borrowing, it doesn't seem to be significant (at least in the last decade), otherwise we'd see more friction costs between the model and reality for the ProShares funds.
2) the assumption about LIBOR/overnight (as spelled out in the S&P and MSCI leveraged indices methodology documents) appears to be confirmed, as any other form of LIBOR rates (e.g. 1 week or 1 month) would be more expensive, and would generate more friction costs  same reasoning.
The commentary is similar to LT Treasuries as previously discussed. ProShares UST performed in remarkable alignment with the theoretical model (actually very slightly better). Direxion YTD is somewhat erratic (on a daily as well as monthly basis), but actually better than its LT counterpart (TMF) as I didn't need to introduce any 'curve fitting' adjustment whatsoever to get fairly flat Telltale lines.
Overall, although we really could benefit from having a few other reallife funds to test against, it seems to me that we can use the model as is, without any adjustment factor, to model leveraged treasury funds.
Two more thoughts, given the results of the ITT and LTT tests:
1) if there is an extra cost due to bid/ask spread for borrowing, it doesn't seem to be significant (at least in the last decade), otherwise we'd see more friction costs between the model and reality for the ProShares funds.
2) the assumption about LIBOR/overnight (as spelled out in the S&P and MSCI leveraged indices methodology documents) appears to be confirmed, as any other form of LIBOR rates (e.g. 1 week or 1 month) would be more expensive, and would generate more friction costs  same reasoning.
Re: Simulating Returns of Leveraged ETFs
And finally, International/Developed (MSCI EAFE). I've been sitting on those charts for a few days now, hoping to figure out what the heck is going on and... I don't have a clue, to be honest. The following charts compare the MSCI EAFE NR USD index to the corresponding leveraged funds I could find (2x and 3x). I sanitychecked the MSCI index numbers against iShares EFA gross returns (the corresponding index fund) and they fit perfectly. No 'curve fitting' adjustment factor has been applied.
If you pay attention to the scale of the vertical axis, you'll see that the relative loss against the theoretical model is STEEP. To get those Telltale lines to become somewhat flat, I would have to use an (annual) adjustment factor like 1.5% (!!) for 2x funds and 3% (!!!) for 3x funds, which makes no sense, there is no way friction costs should be that high (I think?).
Quite clearly, the aggregate growth trajectory of those funds against the benchmark is highly correlated. We see the same kind of wiggles between UNPIX and EFO (even if UNPIX degraded a little faster), they clearly use a similar approach. There is something going on with those International funds that our model doesn't take in account. One thing I did read is that repositioning at the end of the day is trickier for such international leveraged funds because of the different times at which the various international stock markets close. Whatever the explanation is for those charts, making a longterm passive investment in leveraged funds suffering from such steep degradation against the theoretical model seems a little hard to swallow.
If you pay attention to the scale of the vertical axis, you'll see that the relative loss against the theoretical model is STEEP. To get those Telltale lines to become somewhat flat, I would have to use an (annual) adjustment factor like 1.5% (!!) for 2x funds and 3% (!!!) for 3x funds, which makes no sense, there is no way friction costs should be that high (I think?).
Quite clearly, the aggregate growth trajectory of those funds against the benchmark is highly correlated. We see the same kind of wiggles between UNPIX and EFO (even if UNPIX degraded a little faster), they clearly use a similar approach. There is something going on with those International funds that our model doesn't take in account. One thing I did read is that repositioning at the end of the day is trickier for such international leveraged funds because of the different times at which the various international stock markets close. Whatever the explanation is for those charts, making a longterm passive investment in leveraged funds suffering from such steep degradation against the theoretical model seems a little hard to swallow.
 privatefarmer
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Re: Simulating Returns of Leveraged ETFs
Maybe it has to do with the currency being converted? That creates more volatility I believe. I guess we can just hope that globalization continues to make US/INTL highly correlated. I don’t think it’s the biggest bet in the world to be 100% s/p500 in your equities. Bogle and Buffett both recommend it. I think owning a bunch of huge multinational companies that do half their business overseas will ultimately give you pretty decent exposure.siamond wrote: ↑Sat Mar 09, 2019 12:32 am And finally, International/Developed (MSCI EAFE). I've been sitting on those charts for a few days now, hoping to figure out what the heck is going on and... I don't have a clue, to be honest. The following charts compare the MSCI EAFE NR USD index to the corresponding leveraged funds I could find (2x and 3x). I sanitychecked the MSCI index numbers against iShares EFA gross returns (the corresponding index fund) and they fit perfectly. No 'curve fitting' adjustment factor has been applied.
If you pay attention to the scale of the vertical axis, you'll see that the relative loss against the theoretical model is STEEP. To get those Telltale lines to become somewhat flat, I would have to use an (annual) adjustment factor like 1.5% (!!) for 2x funds and 3% (!!!) for 3x funds, which makes no sense, there is no way friction costs should be that high (I think?).
Quite clearly, the aggregate growth trajectory of those funds against the benchmark is highly correlated. We see the same kind of wiggles between UNPIX and EFO (even if UNPIX degraded a little faster), they clearly use a similar approach. There is something going on with those International funds that our model doesn't take in account. One thing I did read is that repositioning at the end of the day is trickier for such international leveraged funds because of the different times at which the various international stock markets close. Whatever the explanation is for those charts, making a longterm passive investment in leveraged funds suffering from such steep degradation against the theoretical model seems a little hard to swallow.
Re: Simulating Returns of Leveraged ETFs
The base index (MSCI EAFE) has a given volatility while expressed in USD, it is what it is. This is directly taken in account in my model. I don't see why this would explain the reallife leveraged funds divergence from the model. Unless currency conversion adds to the pain of daily repositioning with the timing issue of the international markets closing at different times? It seems to me that this should wash out over time though, not steadily lose ground against the model?privatefarmer wrote: ↑Sat Mar 09, 2019 4:00 amMaybe it has to do with the currency being converted? That creates more volatility I believe.
What MAY partly explain the issue is that the market cap of those funds is extremely small (as of today, UNPIX is $3.1M; EFO is $7.05M; DZK is $14.15M). It seems difficult to operate in any efficient manner at this limited scale. There is clearly more to it though.
In general, I am a strong proponent of international diversification, but in the case of leveraged funds, I wouldn't touch it with a long pole...

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 Joined: Fri Sep 21, 2012 1:55 pm
Re: Simulating Returns of Leveraged ETFs
My extremely uneducated guess is along the same line as what you have lead to. The frictions in maintaining this strategy require a large asset base to spread transaction costs over, and none of the current international leveraged funds have a sufficient asset base to make the costs more palatable. The fund management should be using futures contracts based in the United States, which should match the base index very well.siamond wrote: ↑Sat Mar 09, 2019 7:26 amThe base index (MSCI EAFE) has a given volatility while expressed in USD, it is what it is. This is directly taken in account in my model. I don't see why this would explain the reallife leveraged funds divergence from the model. Unless currency conversion adds to the pain of daily repositioning with the timing issue of the international markets closing at different times? It seems to me that this should wash out over time though, not steadily lose ground against the model?privatefarmer wrote: ↑Sat Mar 09, 2019 4:00 amMaybe it has to do with the currency being converted? That creates more volatility I believe.
What MAY partly explain the issue is that the market cap of those funds is extremely small (as of today, UNPIX is $3.1M; EFO is $7.05M; DZK is $14.15M). It seems difficult to operate in any efficient manner at this limited scale. There is clearly more to it though.
In general, I am a strong proponent of international diversification, but in the case of leveraged funds, I wouldn't touch it with a long pole...

 Posts: 4801
 Joined: Sun Oct 22, 2017 2:06 pm
Re: Simulating Returns of Leveraged ETFs
Siamond has generously provided me with his latest simulated data for UPRO and TMF going back to 1955.
This data includes proprietary information, so I have agreed not to share the data files publicly.
But I will be creating charts using this data, and posting the results in the other "master" thread.
Stay tuned everybody, it's gonna be a wild ride
This data includes proprietary information, so I have agreed not to share the data files publicly.
But I will be creating charts using this data, and posting the results in the other "master" thread.
Stay tuned everybody, it's gonna be a wild ride
Re: Simulating Returns of Leveraged ETFs
Yeah, 'wild ride' is the exact right way to put it... You'll see...
PS. I'm fine with you sharing the monthly leveraged numbers that you input to Portfolio Visualizer. Just not the underlying data that led to those numbers. I'll do something similar with a Simba derivative.
Re: Simulating Returns of Leveraged ETFs
Here is a Simba backtesting spreadsheet customized with annual LETF leveraged data (S&P 500, ITT, LTT; 2x and 3x; 1955+) coming from our modeling efforts. I assumed a 1% Expense Ratio for all leveraged funds, and used the various 'adjustment factors' we discussed in the past few posts. For more information about such backtesting spreadsheet, check the corresponding wiki page, check the README tab of the file, and if you have a point question, shoot me a private message.
https://drive.google.com/open?id=16ORud ... WfTP0FggA
https://drive.google.com/open?id=16ORud ... WfTP0FggA
Re: Simulating Returns of Leveraged ETFs
While it's fresh in my head, let me summarize a few considerations about extending the datasets to the years before the existence of reallife funds.
First off, let me repeat that those 'curve fitting' adjustment factors we used to make the Telltale lines reasonably flat are really a poor's man way to proceed. We need to do something like that because friction costs are a reality and the degradation of the (stocks) leveraged funds against the model was too steady to not reflect something recurring. It is also better to be conservative with those simulations instead of knowingly overoptimistic. On the other hand, it is really annoying to not have a solid explanation (with some level of quantification) for such adjustment and surprising to see that (leveraged) bond funds do not seem to have much friction. This means that extending such adjustment factors towards the past is really a halfeducated guess at best. Still, better than nothing and probably not overly impactful to the big picture.
About S&P 500, I think we have a pretty solid data set going back to the mid50s. We don't have daily TR series for the index until the late 80s, but we do have monthly total returns and daily prices, and that's good enough to assemble a derived daily TR series and capture intramonth volatility. I provided more details in this post. Note that for some of the very early years, we only have quarterly dividends, which gave me some grief, but I made it work.
About ITTs and LTTs, it is more difficult as we do not have any daily data of any sort until 1997. So I had to resort to use the monthly returns combined with the average intramonth volatility (of the years with daily data) and the 'magical formula' described here. More details in this post. This probably underestimates the hiccups of the 80s though, where intramonth daily bonds volatility was undoubtedly higher than usual... Hence probably making the model somewhat optimistic for this time period. On the other hand, it probably makes the model somewhat pessimistic for milder times. This is an area where we can probably find a way to improve, if motivated enough.
Finally... this is all pretty heavyduty Excel work. I tried to be careful and sanitycheck myself in various ways, but I would really appreciate somebody taking the time to peer review the whole thing. I know it is a lot to ask though...
First off, let me repeat that those 'curve fitting' adjustment factors we used to make the Telltale lines reasonably flat are really a poor's man way to proceed. We need to do something like that because friction costs are a reality and the degradation of the (stocks) leveraged funds against the model was too steady to not reflect something recurring. It is also better to be conservative with those simulations instead of knowingly overoptimistic. On the other hand, it is really annoying to not have a solid explanation (with some level of quantification) for such adjustment and surprising to see that (leveraged) bond funds do not seem to have much friction. This means that extending such adjustment factors towards the past is really a halfeducated guess at best. Still, better than nothing and probably not overly impactful to the big picture.
About S&P 500, I think we have a pretty solid data set going back to the mid50s. We don't have daily TR series for the index until the late 80s, but we do have monthly total returns and daily prices, and that's good enough to assemble a derived daily TR series and capture intramonth volatility. I provided more details in this post. Note that for some of the very early years, we only have quarterly dividends, which gave me some grief, but I made it work.
About ITTs and LTTs, it is more difficult as we do not have any daily data of any sort until 1997. So I had to resort to use the monthly returns combined with the average intramonth volatility (of the years with daily data) and the 'magical formula' described here. More details in this post. This probably underestimates the hiccups of the 80s though, where intramonth daily bonds volatility was undoubtedly higher than usual... Hence probably making the model somewhat optimistic for this time period. On the other hand, it probably makes the model somewhat pessimistic for milder times. This is an area where we can probably find a way to improve, if motivated enough.
Finally... this is all pretty heavyduty Excel work. I tried to be careful and sanitycheck myself in various ways, but I would really appreciate somebody taking the time to peer review the whole thing. I know it is a lot to ask though...
Re: Simulating Returns of Leveraged ETFs
We have daily 10year CMT from FRED starting in 1962. We have daily 30year CMT starting 1977, and daily 20year starting 1993, but of course with some gaps.
We have daily FRB 10year and 9year yields starting 1971, daily FRB 20year and 19year starting in 1981, and daily 30/29year starting in 1985.
All of these are par yields, so price return and income return can easily be calculated from the yields.
Kevin
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