## Simulating Returns of Leveraged ETFs

Discuss all general (i.e. non-personal) investing questions and issues, investing news, and theory.
siamond
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### Re: Simulating Returns of Leveraged ETFs

interestediniras wrote: Wed Feb 27, 2019 8:42 pm
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?
Perhaps look at Avellaneda and Zhang (2010), equation 10, which models the full continuous-time evolution of the return of a leveraged ETF.

https://www.math.nyu.edu/faculty/avella ... FS.pdf.pdf
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
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### 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.
siamond
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### Re: Simulating Returns of Leveraged ETFs

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.
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
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### Re: Simulating Returns of Leveraged ETFs

Here is yet another paper with relevant information: Ginley et al. (2015).
https://file.scirp.org/Html/4-1490368_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.
interestediniras
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### 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 long-term 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.
siamond
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### 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 intra-month (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 intra-month 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 ok-ish 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 std-deviation (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.
interestediniras
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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
siamond
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### 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 intra-month (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 real-life 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 mid-50s, we will actually NOT need those techniques (cf. my little trick between price and total-return), 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.
siamond
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### Re: Simulating Returns of Leveraged ETFs

Now let's try with Long-Term 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 31-Dec-1997).

Again, we just can't see the blue line although it is oh-so-slightly different (I had to triple-check 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 mid-2000s 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.
MotoTrojan
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### Re: Simulating Returns of Leveraged ETFs

Interesting stuff. Is there a location where you are maintaining various daily-return data sets? Curious if you have something that is on the more refined side going back further than the 1986 data I have currently.
siamond
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### Re: Simulating Returns of Leveraged ETFs

MotoTrojan wrote: Fri Mar 01, 2019 8:30 pm Interesting stuff. Is there a location where you are maintaining various daily-return data sets? Curious if you have something that is on the more refined side going back further than the 1986 data I have currently.
I shared my spreadsheets with a couple of individuals who are the most active in corresponding number-crunching 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
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### Re: Simulating Returns of Leveraged ETFs

siamond wrote: Fri Mar 01, 2019 8:37 pm
MotoTrojan wrote: Fri Mar 01, 2019 8:30 pm Interesting stuff. Is there a location where you are maintaining various daily-return data sets? Curious if you have something that is on the more refined side going back further than the 1986 data I have currently.
I shared my spreadsheets with a couple of individuals who are the most active in corresponding number-crunching 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?
Absolutely! Enjoying following along in the meantime. My plan is set in play and wouldn’t change based on further data anyways.
siamond
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### 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 real-life 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 (Oct-87 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...
siamond
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### 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 non-negligible 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 mid-50s 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 mid-70s (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 mid-90s, while the main aggregate series starts daily data in 1989. And they are not long-term bonds anyway. So this really doesn't help.

The FRED daily data series for 20-yrs treasuries interest rates only starts at the end of 1994 (I guess this isn't a coincidence). Strangely enough, the 30-yrs treasuries daily data series is much better, going back to 1977. And the 10-yrs 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 (day-to day within a month) volatility for 10-yrs and 30-yrs interest rates. The strange flat section for 30-yrs 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 ok-ish 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!
siamond
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### Re: Simulating Returns of Leveraged ETFs

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.
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.

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.
Kevin M
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### Re: Simulating Returns of Leveraged ETFs

siamond wrote: Sun Mar 03, 2019 11:03 am 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.
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 long-term bonds.

Kevin
||.......|| Suggested format for Asking Portfolio Questions (edit original post)
siamond
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### 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 ... aily-index
https://us.spindices.com/documents/meth ... x-math.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.
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### Re: Simulating Returns of Leveraged ETFs

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 ... aily-index
https://us.spindices.com/documents/meth ... x-math.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.
Fantastic find!
siamond
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### Re: Simulating Returns of Leveraged ETFs

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 non-trading days (e.g. week-ends). 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 week-end 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 pre-1986 data, I use the Effective Federal Funds Rate, and when checking this data series, the values during week-ends are NOT null. So I added some (optional) logic to force the week-ends (and some fixed date holidays) rates to zero. This has a non-negligible impact on the model for the mid-70s to mid-80s (making it more favorable by avoiding useless borrowing during non-trading days, notably in a time of high interest rates!). This will also apply to a possible extension of the model towards the mid-50s 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.
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 high-frequency 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.

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 week-end/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 non-trading days to zero borrowing costs, this just doesn't match. When using a LIBOR/overnight data series while leaving the rates as is for non-trading 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 double-checked 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.
interestediniras
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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
MotoTrojan
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### Re: Simulating Returns of Leveraged ETFs

siamond wrote: Tue Mar 05, 2019 2:59 pm

- 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.
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.
siamond
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### Re: Simulating Returns of Leveraged ETFs

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.
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
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### Re: Simulating Returns of Leveraged ETFs

siamond wrote: Tue Mar 05, 2019 5:17 pm
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.
Give me a day or two, grasshopper! Almost there, just want to work on a couple of details and make a couple more tests.
Exciting stuff! Grasshopper shall relax for a few days.
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
HEDGEFUNDIE
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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
MotoTrojan
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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? 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
Doesn’t mean they’ll hold water in the next 30 though. I’m still looking forward to finding out.
siamond
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### 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 (X-1) times the borrowing cost.
- the daily math applies to non-trading 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 day-to-day 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 curve-fitting 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).
siamond
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### 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 mid-caps and small-caps). 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 real-life 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 real-life 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.

siamond
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### 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 curve-fitting adjustment to match real-life 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 curve-fitting is always a risky endeavor. So I ran a test 'out of sample', using the S&P 400 Mid-Cap 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).
MotoTrojan
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### Re: Simulating Returns of Leveraged ETFs

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 curve-fitting adjustment to match real-life 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 curve-fitting is always a risky endeavor. So I ran a test 'out of sample', using the S&P 400 Mid-Cap 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).
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?

Quite excited to get that deeper data, especially EAFE.
siamond
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### Re: Simulating Returns of Leveraged ETFs

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?
The trajectory of the real-life 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.

If the model was perfect and if the real-life 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'.
MotoTrojan
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### Re: Simulating Returns of Leveraged ETFs

siamond wrote: Wed Mar 06, 2019 11:01 pm
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?
The trajectory of the real-life 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.

If the model was perfect and if the real-life 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'.
So in other words, we should be knocking an additional 1% off of the current simulated data for example which hedgefundie is utilizing? At-least per this result.
siamond
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### Re: Simulating Returns of Leveraged ETFs

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? At-least per this result.
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.
Last edited by siamond on Thu Mar 07, 2019 6:52 pm, edited 1 time in total.
siamond
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### Re: Simulating Returns of Leveraged ETFs

Digging around for EAFE data, I incidentally noticed that there are multiple leveraged funds for Small-Caps. 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 curve-fitting!) factor to make most Telltale lines basically flat, for both the 2x and 3x leveraged models. This is quite surprising to me, as small-caps are more volatile than mid-caps or large-caps, 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.
siamond
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### Re: Simulating Returns of Leveraged ETFs

Moving to Long-Term 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.

MotoTrojan
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### Re: Simulating Returns of Leveraged ETFs

siamond wrote: Fri Mar 08, 2019 2:11 pm Moving to Long-Term 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.

I certainly plan to dig deeper into these tell-tale 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 long-term (albeit with some short-term oscillations/volatility)?

Ah I see now, it is reasonably flat with an additional 0.5% tossed in.
siamond
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### Re: Simulating Returns of Leveraged ETFs

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 long-term (albeit with some short-term oscillations/volatility)?

Ah I see now, it is reasonably flat with an additional 0.5% tossed in.
Yup. Well, except that 10 years isn't exactly long-term...
vineviz
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### Re: Simulating Returns of Leveraged ETFs

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.
The only thing that can "kill" a 3x leveraged fund is a single-day 33.34% decline in the underlying index. To date, the single largest daily decline in the emerging markets index is about 12%.
"Far more money has been lost by investors preparing for corrections than has been lost in corrections themselves." ~~ Peter Lynch
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### Re: Simulating Returns of Leveraged ETFs

vineviz wrote: Fri Mar 08, 2019 5:01 pm
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.
The only thing that can "kill" a 3x leveraged fund is a single-day 33.34% decline in the underlying index. To date, the single largest daily decline in the emerging markets index is about 12%.
Be aware that some of these LETFs track underlying ETFs and those underlying ETFs may be subject to limit down trading halts.
pezblanco
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### Re: Simulating Returns of Leveraged ETFs

samsdad wrote: Fri Mar 08, 2019 7:45 pm
vineviz wrote: Fri Mar 08, 2019 5:01 pm
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.
The only thing that can "kill" a 3x leveraged fund is a single-day 33.34% decline in the underlying index. To date, the single largest daily decline in the emerging markets index is about 12%.
Be aware that some of these LETFs track underlying ETFs and those underlying ETFs may be subject to limit down trading halts.
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.
siamond
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### Re: Simulating Returns of Leveraged ETFs

Back to our modeling vs. reality test... Intermediate Treasury bonds (more precisely 7-10 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 real-life 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.
siamond
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### 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 sanity-checked 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 long-term 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

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 sanity-checked 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 long-term passive investment in leveraged funds suffering from such steep degradation against the theoretical model seems a little hard to swallow.
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
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### Re: Simulating Returns of Leveraged ETFs

privatefarmer wrote: Sat Mar 09, 2019 4:00 amMaybe it has to do with the currency being converted? That creates more volatility I believe.
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 real-life 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?

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...
gtwhitegold
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### Re: Simulating Returns of Leveraged ETFs

siamond wrote: Sat Mar 09, 2019 7:26 am
privatefarmer wrote: Sat Mar 09, 2019 4:00 amMaybe it has to do with the currency being converted? That creates more volatility I believe.
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 real-life 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?

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...
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.
HEDGEFUNDIE
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### 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
siamond
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### Re: Simulating Returns of Leveraged ETFs

HEDGEFUNDIE wrote: Sat Mar 09, 2019 1:39 pmStay tuned everybody, it's gonna be a wild ride
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.
siamond
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### 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.

siamond
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### 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 real-life 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 over-optimistic. 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 half-educated 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 mid-50s. 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 intra-month 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 intra-month 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 intra-month 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 heavy-duty Excel work. I tried to be careful and sanity-check 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...
Kevin M
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### Re: Simulating Returns of Leveraged ETFs

siamond wrote: Sat Mar 09, 2019 3:31 pm About ITTs and LTTs, it is more difficult as we do not have any daily data of any sort until 1997.
We have daily 10-year CMT from FRED starting in 1962. We have daily 30-year CMT starting 1977, and daily 20-year starting 1993, but of course with some gaps.

We have daily FRB 10-year and 9-year yields starting 1971, daily FRB 20-year and 19-year starting in 1981, and daily 30/29-year 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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