Simulating Returns of Leveraged ETFs

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Re: Simulating Returns of Leveraged ETFs
siamond: I would be happy to try to reproduce some of your results and play around with the data myself. Can you write a post or send me a PM with links to all of the raw and simulated data? It would be helpful to have all of that consolidated in one place.
Re: Simulating Returns of Leveraged ETFs
Yes, sorry, I didn't express myself clearly. I meant that we do not have any daily price or total return data series readily available until 1997. I did take note of the suggestions you expressed in an earlier post, this is exactly what I meant when I said that "This is an area where we can probably find a way to improve, if motivated enough". Do you want to give it a try generating such data for IT (7 to 10 years) and LT (20+ years) treasuries? What we really need is a solid proxy for daytoday volatility for each month, so I don't think we need the whole 'bond fund' model.Kevin M wrote: ↑Sat Mar 09, 2019 6:23 pmWe 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.
Re: Simulating Returns of Leveraged ETFs
This would be fantastic. Please check your PMs.interestediniras wrote: ↑Sat Mar 09, 2019 7:26 pmsiamond: I would be happy to try to reproduce some of your results and play around with the data myself. Can you write a post or send me a PM with links to all of the raw and simulated data? It would be helpful to have all of that consolidated in one place.
Re: Simulating Returns of Leveraged ETFs
Hm, silly me, I didn't apply the ER properly when I assembled the spreadsheet earlier yesterday. As a bonus for downloading an updated version, you'll get the midcap and smallcap leveraged numbers (extended back in the 50s).siamond wrote: ↑Sat Mar 09, 2019 2:35 pmHere 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
Re: Simulating Returns of Leveraged ETFs
See PM!
Kevin
....... Suggested format for Asking Portfolio Questions (edit original post)
Re: Simulating Returns of Leveraged ETFs
I added Gold (price, not miners) leveraged data series, starting in 1969 (all daily), based on the LBMA Gold Price PM USD index.
And I ran my usual test against actuals. Not many leveraged funds tracking Gold besides ProShares (note that Direxion used to, then closed its 'BARS' fund, and now focuses on Gold Miners which is a completely different thing). I had to run a Total Return comparison instead of a Gross Return comparison, so that I could include DGP, a Velocity leveraged ETN, in addition to ProShares UGL. For whatever reason, I can't get Gross Returns data series for ETNs... Anyhoo, I used an ER averaging the ER of both funds, then had to add a 1% curvefitting 'adjustment' factor to make the telltale lines reasonably flat. I was surprised to find high friction costs in this case, as tracking the price of gold sounds much simpler than tracking entire segments of the stock/bond markets.
I updated all my spreadsheets (incl. the custom Simba), those of you I shared a link with only have to download again the file(s), same link. Enjoy.
And I ran my usual test against actuals. Not many leveraged funds tracking Gold besides ProShares (note that Direxion used to, then closed its 'BARS' fund, and now focuses on Gold Miners which is a completely different thing). I had to run a Total Return comparison instead of a Gross Return comparison, so that I could include DGP, a Velocity leveraged ETN, in addition to ProShares UGL. For whatever reason, I can't get Gross Returns data series for ETNs... Anyhoo, I used an ER averaging the ER of both funds, then had to add a 1% curvefitting 'adjustment' factor to make the telltale lines reasonably flat. I was surprised to find high friction costs in this case, as tracking the price of gold sounds much simpler than tracking entire segments of the stock/bond markets.
I updated all my spreadsheets (incl. the custom Simba), those of you I shared a link with only have to download again the file(s), same link. Enjoy.

 Posts: 436
 Joined: Fri Sep 21, 2012 1:55 pm
Re: Simulating Returns of Leveraged ETFs
It's noisy, but it doesn't look horrible. Can you tell what the average negative alpha is for DGP?siamond wrote: ↑Mon Mar 11, 2019 2:58 pmI added Gold (price, not miners) leveraged data series, starting in 1969 (all daily), based on the LBMA Gold Price PM USD index.
And I ran my usual test against actuals. Not many leveraged funds tracking Gold besides ProShares (note that Direxion used to, then closed its 'BARS' fund, and now focuses on Gold Miners which is a completely different thing). I had to run a Total Return comparison instead of a Gross Return comparison, so that I could include DGP, a Velocity leveraged ETN, in addition to ProShares UGL. For whatever reason, I can't get Gross Returns data series for ETNs... Anyhoo, I used an ER averaging the ER of both funds, then had to add a 1% curvefitting 'adjustment' factor to make the telltale lines reasonably flat. I was surprised to find high friction costs in this case, as tracking the price of gold sounds much simpler than tracking entire segments of the stock/bond markets.
I updated all my spreadsheets (incl. the custom Simba), those of you I shared a link with only have to download again the file(s), same link. Enjoy.
Re: Simulating Returns of Leveraged ETFs
DGP is noisy, but UGL is not. But then, this is ProShares and those folks seem to be the best around, from what I could observe in this modeling endeavor. On the other hand, an ETN isn't supposed to have much tracking error (that is, besides the ER)? Not that I know much about ETNs, to be honest.gtwhitegold wrote: ↑Mon Mar 11, 2019 4:46 pmIt's noisy, but it doesn't look horrible. Can you tell what the average negative alpha is for DGP?
Negative alpha? Well... around 1% as I already mentioned!
I did a more precise calculation, and the CAGR difference between the model (without an adjustment factor and using the DGP ER) and DGP was 1.5% (annualized). This actually triggered me to refine a bit the 'adjustment factor' in my model and now the chart looks like that, which is more satisfying. Shesh, that's a lot of friction costs...
Note that both funds have approximately $80M of assets as of today, which isn't too bad for this narrow market of leveraged funds. They were significantly bigger a few years ago though (in Jan12, ~$400M for UGL and ~$600M for DGP), the price of gold was higher by then, but still, they've been bleeding investment$ pretty bad.

 Posts: 436
 Joined: Fri Sep 21, 2012 1:55 pm
Re: Simulating Returns of Leveraged ETFs
I agree that it is ugly and I'm pretty sure that I don't want to take counterparty risk by investing in ETNs. I do appreciate the work you and others have put into this though.siamond wrote: ↑Mon Mar 11, 2019 5:13 pmDGP is noisy, but UGL is not. But then, this is ProShares and those folks seem to be the best around, from what I could observe in this modeling endeavor. On the other hand, an ETN isn't supposed to have much tracking error (that is, besides the ER)? Not that I know much about ETNs, to be honest.gtwhitegold wrote: ↑Mon Mar 11, 2019 4:46 pmIt's noisy, but it doesn't look horrible. Can you tell what the average negative alpha is for DGP?
Negative alpha? Well... around 1% as I already mentioned!
I did a more precise calculation, and the CAGR difference between the model (without an adjustment factor and using the DGP ER) and DGP was 1.5% (annualized). This actually triggered me to refine a bit the 'adjustment factor' in my model and now the chart looks like that, which is more satisfying. Shesh, that's a lot of friction costs...
Note that both funds have approximately $80M of assets as of today, which isn't too bad for this narrow market of leveraged funds. They were significantly bigger a few years ago though (in Jan12, ~$400M for UGL and ~$600M for DGP), the price of gold was higher by then, but still, they've been bleeding investment$ pretty bad.

 Posts: 436
 Joined: Fri Sep 21, 2012 1:55 pm
Re: Simulating Returns of Leveraged ETFs
Maybe the next thing to work towards is what level of leverage is the most efficient. I've seen mentions of 1.25x and 1.5x, but I don't see where the numbers came from. Any thoughts, or did I just miss it?
Re: Simulating Returns of Leveraged ETFs
There are multiple leveraged funds from Guggenheim, the Rydex family, which have been in existence for quite a while, and use this kind of odd multipliers, so I used some of them for my reality check testing. Check here:gtwhitegold wrote: ↑Mon Mar 11, 2019 7:59 pmMaybe the next thing to work towards is what level of leverage is the most efficient. I've seen mentions of 1.25x and 1.5x, but I don't see where the numbers came from. Any thoughts, or did I just miss it?
https://www.guggenheiminvestments.com/m ... leveraged
Re: Simulating Returns of Leveraged ETFs
Hi siamond  i was playing around with my own naive simulations using linear regression and feeling good till I stumbled upon this discussion. Amazing work by you and other on this thread to make simulations so close to the reality. Can you share daily/monthly simulated prices to play around with the real stuff?
Re: Simulating Returns of Leveraged ETFs
Tx. Please check your private messages.levitate wrote: ↑Tue Mar 12, 2019 6:03 pmHi siamond  i was playing around with my own naive simulations using linear regression and feeling good till I stumbled upon this discussion. Amazing work by you and other on this thread to make simulations so close to the reality. Can you share daily/monthly simulated prices to play around with the real stuff?
Re: Simulating Returns of Leveraged ETFs
I had two independent discussions today, which both reminded me that I had a lingering doubt about what I described here:
An alternate approach would be to simply use the intramonth volatility of the S&P 500 price series (until Dec87) and simply avoid to inject dividends. After all, price changes clearly drive volatility, while dividends flows tend to be much more stable. And since the 'magical formula' described here only requires monthly CAGR (of the index) and intramonth volatility, we could use a solid proxy for the latter and still have proper (and possibly better) results.
I gave it a quick try for the 1988+ time period (for which we have all daily data, hence no risk of distorted math). I used price volatility and the magical formula, computed leveraged returns and I can hardly notice the difference with the exact math.
Then I did the same for the 19551987 time period, and compared my previous approach to the new one. The differences are then noticeable, notably for the 3x leverage series. I mean, it's not huge, but it is visible (predictably making the leveraged returns a tad better). And the more I think about it, the more I convince myself that the approach using priceonly volatility is more straightforward AND should be more accurate than my previous approach with the spacedout dividends.
I will hold on making corresponding changes for now, as I'm having a parallel discussion with Kevin about bonds where I suspect we'll end up reaching a similar conclusion, relying on pricing data (derived from daily yields) as a more accurate volatility proxy for the old days.
What has been nagging me is that the 'derived' data series (needed until Dec87) displays a bit of a sudden 'hiccup' at the end of the month (or quarter) when dividends come in (because this is all we have with the corresponding historical data). And this 'hiccup' may have some unfortunate sideeffects on the intramonth (day to day) volatility math. While in reallife, dividends of the S&P 500 underlying components come in a much more distributed (over time) manner. I mean, this should be a fairly minor effect, but when the leveraging math applies its magnifying effect, I was wondering if this might create some non negligible distortion.siamond wrote: ↑Sat Mar 09, 2019 3:31 pmAbout 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.
An alternate approach would be to simply use the intramonth volatility of the S&P 500 price series (until Dec87) and simply avoid to inject dividends. After all, price changes clearly drive volatility, while dividends flows tend to be much more stable. And since the 'magical formula' described here only requires monthly CAGR (of the index) and intramonth volatility, we could use a solid proxy for the latter and still have proper (and possibly better) results.
I gave it a quick try for the 1988+ time period (for which we have all daily data, hence no risk of distorted math). I used price volatility and the magical formula, computed leveraged returns and I can hardly notice the difference with the exact math.
Then I did the same for the 19551987 time period, and compared my previous approach to the new one. The differences are then noticeable, notably for the 3x leverage series. I mean, it's not huge, but it is visible (predictably making the leveraged returns a tad better). And the more I think about it, the more I convince myself that the approach using priceonly volatility is more straightforward AND should be more accurate than my previous approach with the spacedout dividends.
I will hold on making corresponding changes for now, as I'm having a parallel discussion with Kevin about bonds where I suspect we'll end up reaching a similar conclusion, relying on pricing data (derived from daily yields) as a more accurate volatility proxy for the old days.
Re: Simulating Returns of Leveraged ETFs
Kevin and I had a very productive exchange about this topic in the past couple of days. As a refresher, historical bond index data suffers from a severe lack of daily numbers, and all we have for index returns prior to mid97 are monthly total returns (for both IT treasuries and LT treasuries). In the current leveraged model, we made a very coarse assumption that average daily volatility of the known days (mid97 till now) is a halfdecent proxy for the older times (then we use the 'magic formula' described here). I was quite suspicious though that we didn't properly capture the dynamics of the late 70s and 80s (wild changes of interest rates) and of previous decades (mild changes of interest rates), possibly making the leveraged model overshoot or undershoot for corresponding decades.Kevin M wrote: ↑Sat Mar 09, 2019 6:23 pmWe 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.
What Kevin pointed out is that we can use daily yield data to fill that gap, at least partly. The idea is that we have daily 5yrs, 7yrs and 10yrs CMT yield data, helpful for IT Treasuries. And 10yrs, 20yrs and 30yrs yield data, helpful for LT treasuries. For each CMT series, we computed daily income (dividends) and capital (price) returns, based on a simple bond model. Let me quote Kevin:
The underlying assumption is that a bond is bought at par value (100 in standard bond pricing terms, which means 100% of face value), sold the next day at the calculated price, and then another par bond is bought at 100; rinse and repeat.
Such simple bond model maps to very simple spreadsheet formulas, the income return is the yield from the previous day, and the price return is easy to compute based on a Present Value formula (as typical for bond pricing) based on the yield of the day and the yield of the previous day. Once we have the daily outcome for each CMT series, it's a simple matter of combining the return series to mimic the average maturity of an IT treasury fund (~8.5yrs) and an LT treasury fund (~25yrs). The fact that there are some holes in the CMT series from FRED gave us a bit of grief, but we found simple ways around that.
I confess that I was a little skeptical, but when running such bond model and comparing against actuals for the 1998+ time period, the results are very impressive, notably when it comes to intramonth (day to day) volatility, which is what we were looking for. Computing the RMSE between monthly volatility actuals and Kevin's model for 19982018, we get 0.03% for IT treasuries and 0.05% for LT treasuries, which is remarkably low (average volatility is around 0.3% for ITTs and 0.6% for LTTs by then). Comparing individual values as well as overall averages showed that this is indeed a close match. Based on such validation, we can reasonably infer that this accuracy should extend to the past, and therefore get a solid volatility proxy for IT treasuries (1962+) and for LT treasuries (1977+).
Re: Simulating Returns of Leveraged ETFs
IT treasuries (1962+) and LT treasuries (1977+) improved volatility values are great, but what about the years before (goal is to start in Jan55)? For IT treasuries, we can observe that the trajectory of interest rates in the late 50s didn't seem overly eventful, and it seems ok to use a coarse volatility value equal to the average of the next 15 years (19621976), which, interestingly enough was really quite low (0.16% compared to the 0.33% for the last two decades).
For LT treasuries, there is just no good answer, we still have 20+ years missing. Given the observation about IT treasury volatility (lower by then than more recently), we really shouldn't use a recent measure of LT treasury volatility as proxy. Both Kevin and myself independently came up with the same answer: LTT volatility when known has been consistently roughly double the ITT volatility. So why not use the ITT results, simply multiply by 2, and fill the 19621976 LTT gap in such a way. This is less than ideal, of course, but at least we'd capture the ups and downs of this time period, while using mild volatility values which seemed to be prevalent at the time. And then fill the late 50s gap with the same approach as for IT bonds, use the average of the next 15 years.
I'm sure that some of you are cringing while reading those lines, and yes, it IS less than ideal. But this seems WAY better than we had before. Also, there is a point where further refinements may not move the needle very much, and after running a couple of sensitivity analysis experiments, I believe we're pretty close to that point.
I'll share the updated leveraged model tomorrow (to give myself a chance to sleep on it!). The big picture won't change, but the (rather dire) leveraged returns of the 50s to 70s will get a bit better, thanks to the milder volatility of the corresponding times. Feedback welcome.
PS. confusingly enough, treasury yields need to be divided by 365 to get to daily yields  while EFFR/LIBOR rates need to be divided by 360. We did exactly that, after doublechecking with multiple sources.
For LT treasuries, there is just no good answer, we still have 20+ years missing. Given the observation about IT treasury volatility (lower by then than more recently), we really shouldn't use a recent measure of LT treasury volatility as proxy. Both Kevin and myself independently came up with the same answer: LTT volatility when known has been consistently roughly double the ITT volatility. So why not use the ITT results, simply multiply by 2, and fill the 19621976 LTT gap in such a way. This is less than ideal, of course, but at least we'd capture the ups and downs of this time period, while using mild volatility values which seemed to be prevalent at the time. And then fill the late 50s gap with the same approach as for IT bonds, use the average of the next 15 years.
I'm sure that some of you are cringing while reading those lines, and yes, it IS less than ideal. But this seems WAY better than we had before. Also, there is a point where further refinements may not move the needle very much, and after running a couple of sensitivity analysis experiments, I believe we're pretty close to that point.
I'll share the updated leveraged model tomorrow (to give myself a chance to sleep on it!). The big picture won't change, but the (rather dire) leveraged returns of the 50s to 70s will get a bit better, thanks to the milder volatility of the corresponding times. Feedback welcome.
PS. confusingly enough, treasury yields need to be divided by 365 to get to daily yields  while EFFR/LIBOR rates need to be divided by 360. We did exactly that, after doublechecking with multiple sources.
Re: Simulating Returns of Leveraged ETFs
Hi. Using Libor for the assumed interest rate is not really correct, although it is close. Equity market funding is not libor + 0bp. There is a spread to this rate that is mostly based on specific supply and demand for equity leverage. For instance, if you price a futures contract on SPX (which is presumably the method through which SSO, UPRO, etc. achieve leverage), the implicit interest rate might be libor+10bp or so today. This value can trade in say a range of negative to positive 2%, and the average value is around 0.30%.
Anyway, borrowing at 3m libor flat for investing in equities sounds pretty good to me at 2.60%, plus a bit, and was a no brainer at 1% not too long ago
Anyway, borrowing at 3m libor flat for investing in equities sounds pretty good to me at 2.60%, plus a bit, and was a no brainer at 1% not too long ago
Re: Simulating Returns of Leveraged ETFs
Hi there. Thanks for the feedback, appreciated. We did discuss the spread issue a while ago, but couldn't find a way to properly quantify it. We ended up using 'adjustment' empirical factors to cover for friction costs at large (incl. spread), but this is of course not quite satisfying, and I'd be happy to make progress on the spread topic. Mind sharing where you got your numbers (e.g. range, average value, etc)?ohai wrote: ↑Thu Mar 14, 2019 4:37 pmHi. Using Libor for the assumed interest rate is not really correct, although it is close. Equity market funding is not libor + 0bp. There is a spread to this rate that is mostly based on specific supply and demand for equity leverage. For instance, if you price a futures contract on SPX (which is presumably the method through which SSO, UPRO, etc. achieve leverage), the implicit interest rate might be libor+10bp or so today. This value can trade in say a range of negative to positive 2%, and the average value is around 0.30%.
Re: Simulating Returns of Leveraged ETFs
It's not a bid/ask spread if that's what you mean (sorry, the thread is quite long), but this value for SPX can be assumed for simplification to be 0.125 points of the index (half the futures tick size). What I meant, in case was unclear, is the fair value of equity funding that is usually above libor: you can sell, as well as buy, at the same price.
Anyway, data on equity funding levels is not published anywhere. For the most part, only institutional trading desks (i.e. banks and specifically, delta one desks) really track this explicitly. I do have historical data for this, but unfortunately cannot share it due to proprietary reasons.
However, there is a really easy way to simulate leveraged ETFs returns, which is to track SPX front month futures prices, not SPX closing levels. When you buy a future on SPX, the price includes: 1) "risk free" interest rate, 2) equity funding spread, and 3) predicted or known cash dividends. 3) cancels out, since you are buying the future value of the index after dividends are paid. This leaves 1) and 2), which together show the total cost of leverage for the index. You don't need any further data for interest rates or equity funding if you have tradable futures prices.
Anyway, data on equity funding levels is not published anywhere. For the most part, only institutional trading desks (i.e. banks and specifically, delta one desks) really track this explicitly. I do have historical data for this, but unfortunately cannot share it due to proprietary reasons.
However, there is a really easy way to simulate leveraged ETFs returns, which is to track SPX front month futures prices, not SPX closing levels. When you buy a future on SPX, the price includes: 1) "risk free" interest rate, 2) equity funding spread, and 3) predicted or known cash dividends. 3) cancels out, since you are buying the future value of the index after dividends are paid. This leaves 1) and 2), which together show the total cost of leverage for the index. You don't need any further data for interest rates or equity funding if you have tradable futures prices.
Re: Simulating Returns of Leveraged ETFs
Yeah, I suspect I did get mixed up in some earlier posts, but your definition of spread as fair value of equity funding (above LIBOR) is well worded. Thank you for clarifying.ohai wrote: ↑Thu Mar 14, 2019 9:24 pmIt's not a bid/ask spread if that's what you mean (sorry, the thread is quite long), but this value for SPX can be assumed for simplification to be 0.125 points of the index (half the futures tick size). What I meant, in case was unclear, is the fair value of equity funding that is usually above libor: you can sell, as well as buy, at the same price.
Just out of curiosity, may I ask you a few indirect questions:ohai wrote: ↑Thu Mar 14, 2019 9:24 pmAnyway, data on equity funding levels is not published anywhere. For the most part, only institutional trading desks (i.e. banks and specifically, delta one desks) really track this explicitly. I do have historical data for this, but unfortunately cannot share it due to proprietary reasons.
a) how far back does this proprietary data history go? Probably not a lot, I would guess?
b) does it make sense to you that such spread would be near zero for bond/treasury funding (i.e. for a leveraged fund like UBT or TMF)?
c) does it make sense to you that such spread would be smaller for mid/smallcaps than for S&P 500? And higher for Int'l (e.g. EAFE)?
d) what are the fundamental drivers of a lower or higher spread?
That's clever. Unfortunately, such 'futures' data series are not publicly available, I am afraid. And even with your proprietary access, I suspect you can't go very far back in time, am I correct? We've been trying to develop a model that extends back in the 50s, so that we can fully capture the 3 major crises of the US stock market while backtesting.ohai wrote: ↑Thu Mar 14, 2019 9:24 pmHowever, there is a really easy way to simulate leveraged ETFs returns, which is to track SPX front month futures prices, not SPX closing levels. When you buy a future on SPX, the price includes: 1) "risk free" interest rate, 2) equity funding spread, and 3) predicted or known cash dividends. 3) cancels out, since you are buying the future value of the index after dividends are paid. This leaves 1) and 2), which together show the total cost of leverage for the index. You don't need any further data for interest rates or equity funding if you have tradable futures prices.
Re: Simulating Returns of Leveraged ETFs
I am still sitting on this. The whole topic of simulating bond funds (with daily and/or monthly inputs) sent me in a bit of a spin, Kevin's help is extremely valuable, but I didn't fully converge yet. The change will NOT be terribly impactful anyway, so I'd rather do it right and be done with it than providing incremental updates.siamond wrote: ↑Thu Mar 14, 2019 4:28 pmIT treasuries (1962+) and LT treasuries (1977+) improved volatility values are great, but what about the years before (goal is to start in Jan55)? [...]
I'll share the updated leveraged model tomorrow (to give myself a chance to sleep on it!). The big picture won't change, but the (rather dire) leveraged returns of the 50s to 70s will get a bit better, thanks to the milder volatility of the corresponding times. [...]
Shifting gears, I am hoping that ohai comes back to this thread and enlightens us a bit more about borrowing spreads... Something that crossed my mind is that US treasuries are the most liquid investment vehicle of all, if I am not mistaken. Maybe this explains why we didn't find much 'friction costs' when testing the model again leveraged bond funds actuals, while we found much more friction for leveraged stock funds.
Re: Simulating Returns of Leveraged ETFs
Back to volatility in the early days... I finalized an update of the leveraged model, generating numbers which seem more realistic for the first few decades than the coarse assumptions I made before.
For the S&P 500, the net effect is that the intramonth volatility numbers up to 1987 have been updated (see discussion here) and this made the 2x and 3x leverage numbers go up a bit in the model.
For IT and LT Treasuries, the net effect is that the intramonth volatility numbers up to 1997 have been updated (see discussion here and here), and this also made the 2x and 3x leveraged numbers go up (treasuries volatility in those older decades was significantly lower than in more recent decades). I ended up switching to FRB rates as input (instead of the CMT data series) for two reasons, first the CMT series have more 'holes' notably in the 60s, next the FRBbased model matches the actuals (the corresponding bonds index) better for 1998+. With Kevin's precious help, I also explored a more complicated model based on the ideas expressed here, using an M1 rung on a daily basis when available, but this wasn't convincing enough to be worth the change.
Overall, the exact 2x and 3x leverage numbers did change in a nonnegligible manner, but the big picture didn't change much, the 50s/60s/70s would still have been a really difficult time for an investor using a good dose of (leveraged) treasury funds.
The usual number crunchers (HedgeFundie, EfficientInvestor, Samsdad, etc) can use the same links I shared before to download the updated model. I also updated the corresponding customized Simba spreadsheet (check this post, same link).
For the S&P 500, the net effect is that the intramonth volatility numbers up to 1987 have been updated (see discussion here) and this made the 2x and 3x leverage numbers go up a bit in the model.
For IT and LT Treasuries, the net effect is that the intramonth volatility numbers up to 1997 have been updated (see discussion here and here), and this also made the 2x and 3x leveraged numbers go up (treasuries volatility in those older decades was significantly lower than in more recent decades). I ended up switching to FRB rates as input (instead of the CMT data series) for two reasons, first the CMT series have more 'holes' notably in the 60s, next the FRBbased model matches the actuals (the corresponding bonds index) better for 1998+. With Kevin's precious help, I also explored a more complicated model based on the ideas expressed here, using an M1 rung on a daily basis when available, but this wasn't convincing enough to be worth the change.
Overall, the exact 2x and 3x leverage numbers did change in a nonnegligible manner, but the big picture didn't change much, the 50s/60s/70s would still have been a really difficult time for an investor using a good dose of (leveraged) treasury funds.
The usual number crunchers (HedgeFundie, EfficientInvestor, Samsdad, etc) can use the same links I shared before to download the updated model. I also updated the corresponding customized Simba spreadsheet (check this post, same link).

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Re: Simulating Returns of Leveraged ETFs
Awesome. Thanks Siamond. The new data is indeed more optimistic.siamond wrote: ↑Thu Mar 21, 2019 2:19 pmBack to volatility in the early days... I finalized an update of the leveraged model, generating numbers which seem more realistic for the first few decades than the coarse assumptions I made before.
For the S&P 500, the net effect is that the intramonth volatility numbers up to 1987 have been updated (see discussion here) and this made the 2x and 3x leverage numbers go up a bit in the model.
For IT and LT Treasuries, the net effect is that the intramonth volatility numbers up to 1997 have been updated (see discussion here and here), and this also made the 2x and 3x leveraged numbers go up (treasuries volatility in those older decades was significantly lower than in more recent decades). I ended up switching to FRB rates as input (instead of the CMT data series) for two reasons, first the CMT series have more 'holes' notably in the 60s, next the FRBbased model matches the actuals (the corresponding bonds index) better for 1998+. With Kevin's precious help, I also explored a more complicated model based on the ideas expressed here, using an M1 rung on a daily basis when available, but this wasn't convincing enough to be worth the change.
Overall, the exact 2x and 3x leverage numbers did change in a nonnegligible manner, but the big picture didn't change much, the 50s/60s/70s would still have been a really difficult time for an investor using a good dose of (leveraged) treasury funds.
The usual number crunchers (HedgeFundie, EfficientInvestor, Samsdad, etc) can use the same links I shared before to download the updated model. I also updated the corresponding customized Simba spreadsheet (check this post, same link).
Re: Simulating Returns of Leveraged ETFs
Thank you very much, Siamond, it's great to have someone like you here to help the community. Thanks to all involved as well.siamond wrote: ↑Thu Mar 21, 2019 2:19 pmBack to volatility in the early days... I finalized an update of the leveraged model, generating numbers which seem more realistic for the first few decades than the coarse assumptions I made before.
For the S&P 500, the net effect is that the intramonth volatility numbers up to 1987 have been updated (see discussion here) and this made the 2x and 3x leverage numbers go up a bit in the model.
For IT and LT Treasuries, the net effect is that the intramonth volatility numbers up to 1997 have been updated (see discussion here and here), and this also made the 2x and 3x leveraged numbers go up (treasuries volatility in those older decades was significantly lower than in more recent decades). I ended up switching to FRB rates as input (instead of the CMT data series) for two reasons, first the CMT series have more 'holes' notably in the 60s, next the FRBbased model matches the actuals (the corresponding bonds index) better for 1998+. With Kevin's precious help, I also explored a more complicated model based on the ideas expressed here, using an M1 rung on a daily basis when available, but this wasn't convincing enough to be worth the change.
Overall, the exact 2x and 3x leverage numbers did change in a nonnegligible manner, but the big picture didn't change much, the 50s/60s/70s would still have been a really difficult time for an investor using a good dose of (leveraged) treasury funds.
The usual number crunchers (HedgeFundie, EfficientInvestor, Samsdad, etc) can use the same links I shared before to download the updated model. I also updated the corresponding customized Simba spreadsheet (check this post, same link).
Re: Simulating Returns of Leveraged ETFs
Quoting an interesting post from HedgeFundie's thread:
Now if you could be kind enough to share where to find such a long Stoxx 50 data series (either via a public post or via private message), I would appreciate it and would be happy to derive a corresponding 2x and 3x data series (from the US perspective though) using the model we've been working on.
As to the formula, from a European perspective, this sounds good. Just do not forget to add a 'friction costs' factor, which can easily go in the 1% range as you can see with the various experiments we performed in this thread.
About MSCI, you can find corresponding data series here. GR series are the ones of interest, GR means Gross Returns (i.e. Total Returns without any tax nor expense ratio considerations).celerity wrote: ↑Fri Apr 19, 2019 10:51 pmThanks, I have daily data for Euro Stoxx 50 2x (and 3x) all the way back to 1992! I'm missing:siamond wrote: ↑Fri Apr 19, 2019 4:24 amYes, you need intramonth daily volatility to do any proper modeling of leveraged funds, as was explained in this thread. Having daily (total) returns is obviously the easiest way to get to that.
SPDR® EURO STOXX 50 ETF has daily data back to Nov02. iShares STOXX Europe 50 ETF EUR Dist has daily data back to Apr2000 and iShares EURO STOXX 50 (DE) goes back to Jan01. The index itself was introduced in Feb98. More generally, historical daily data is really hard to find when it comes to nonUS asset classes. MSCI, as a case in point, doesn't provide daily data in its indices (incl. MSCI EM) until Jan2001.
PS. don't know if that was already discussed, but here is a 3x leveraged ETF tracking the Stoxx 50 (in Euros): WisdomTree EU3L.
MSCI USA Net Daily
MSCI EM Net Daily
Then I can calculate leveraged returns using formula:Code: Select all
Daily return = (Leverage * MSCI USA)  Daily TER  (Leverage  1) * EONIA rate
Now if you could be kind enough to share where to find such a long Stoxx 50 data series (either via a public post or via private message), I would appreciate it and would be happy to derive a corresponding 2x and 3x data series (from the US perspective though) using the model we've been working on.
As to the formula, from a European perspective, this sounds good. Just do not forget to add a 'friction costs' factor, which can easily go in the 1% range as you can see with the various experiments we performed in this thread.
Re: Simulating Returns of Leveraged ETFs
Thanks, I know of MSCI. They offer only monthly return history for nonsubscribers though. (They actually do have daily but only the last 5 years).siamond wrote: ↑Sat Apr 20, 2019 9:22 pmAbout MSCI, you can find corresponding data series here. GR series are the ones of interest, GR means Gross Returns (i.e. Total Returns without any tax nor expense ratio considerations).
Now if you could be kind enough to share where to find such a long Stoxx 50 data series (either via a public post or via private message), I would appreciate it and would be happy to derive a corresponding 2x and 3x data series (from the US perspective though) using the model we've been working on.
As to the formula, from a European perspective, this sounds good. Just do not forget to add a 'friction costs' factor, which can easily go in the 1% range as you can see with the various experiments we performed in this thread.
As for return type, we Europeans need Net, not Gross. Gross is for local (i.e. US) citizens. For example ComStage tracks the Net index https://www.justetf.com/servlet/downloa ... DE&lang=en.
Euro Stoxx 50 2x: https://www.stoxx.com/indexdetails?symbol=SX5DLG (Data —> Historical Data)
Euro Stoxx 50 3x: https://www.stoxx.com/indexdetails?symbol=SX5GTDL3
Re: Simulating Returns of Leveraged ETFs
I’ve never heard of friction costs. I need to read about it first.
Re: Simulating Returns of Leveraged ETFs
Btw, could someone please confirm:
A leveraged EU domiciled fund which tracks an American index e.g. MSCI USA uses EONIA rate and NOT LIBOR for borrowing, right?
A leveraged EU domiciled fund which tracks an American index e.g. MSCI USA uses EONIA rate and NOT LIBOR for borrowing, right?

 Posts: 23
 Joined: Wed Jan 22, 2014 2:12 am
Re: Simulating Returns of Leveraged ETFs
I know we changed some PMs, but why do you need Net instead of Gross?celerity wrote: ↑Sun Apr 21, 2019 11:29 amAs for return type, we Europeans need Net, not Gross. Gross is for local (i.e. US) citizens. For example ComStage tracks the Net index https://www.justetf.com/servlet/downloa ... DE&lang=en.
In fact, in Europe, since these kind of leveraged ETFs do know distribute any dividends, and surely we should be using the Gross index, right?
BTW, I have compiled some daily information for the following indexes and corresponding European ETF, which you can find at the following link:
https://ufile.io/e56kkdve
MSCI World 2x EUR (I am not sure about this data)
MSCI EM 2x EUR and corresponding Comstage MSCI EM 2x (which is similar)
MSCI USA 2x EUR (both from MSCI.com and Investing.com) and corresponding Amundi MSCI USA 2x (definitately the information from MSCI.com does not seem correct )
Stoxx 50 and corresponding Comstage Stoxx 50 2x (which is similar)
SGI Bund 2x and corresponding Comstage Bund 2x (which is similar)
Now I will try to use the average daily differences between the Indexes and corresponding ETFs to test this starting 2002, which is how far back I can go with the Stoxx 50 index.
Re: Simulating Returns of Leveraged ETFs
Pretty much every ETF in Europe uses Net Return as benchmark. The fund still has to pay taxes on dividends, distributing or not AFAIK. Actually one problem with distribution is the investor may end up paying taxes TWICE (hence all those double taxation agreements). Perhaps something else applies to LETF because they’re synthetically replicated using swaps, but I don’t think they are tax exempt.Mouro_Emprestado wrote: ↑Sun Apr 21, 2019 11:51 amI know we changed some PMs, but why do you need Net instead of Gross?
In fact, in Europe, since these kind of leveraged ETFs do know distribute any dividends, and surely we should be using the Gross index, right?
Basically it’s like this, if you live in the eurozone (not just EU though) your benchmarks should be:
Euro Stoxx 50 Gross
MSCI USA Net
MSCI EM Net
For US citizens:
Euro Stoxx 50 Net
MSCI USA Gross
MSCI EM Net
Great! I’ll have a look at it when I’m back home!BTW, I have compiled some daily information for the following indexes and corresponding European ETF, which you can find at the following link:
https://ufile.io/e56kkdve
MSCI World 2x EUR (I am not sure about this data)
MSCI EM 2x EUR and corresponding Comstage MSCI EM 2x (which is similar)
MSCI USA 2x EUR (both from MSCI.com and Investing.com) and corresponding Amundi MSCI USA 2x (definitately the information from MSCI.com does not seem correct )
Stoxx 50 and corresponding Comstage Stoxx 50 2x (which is similar)
SGI Bund 2x and corresponding Comstage Bund 2x (which is similar)
Now I will try to use the average daily differences between the Indexes and corresponding ETFs to test this starting 2002, which is how far back I can go with the Stoxx 50 index.

 Posts: 23
 Joined: Wed Jan 22, 2014 2:12 am
Re: Simulating Returns of Leveraged ETFs
Ahhhh, it makes sense.
I did not remember the issue of the dividends within the ETF
I did not remember the issue of the dividends within the ETF

 Posts: 23
 Joined: Wed Jan 22, 2014 2:12 am
Re: Simulating Returns of Leveraged ETFs
Updated information: https://ufile.io/sqae9akm
The estimated indexes are already prepared to be uploaded to Portfolio Analyzer in their own excel files.
Nevertheless, treat the information with some caution, as I am not 100% sure regarding the data on the MSCI indexes, as explained above.
Namely I have no spillage for the MSCI World (there is no ETF in Europe for which to proxy against it ).
In terms of the other indexes, I basically did the following:
 Compared the Index against the corresponding European ETF and averaged the daily difference.
 For the period for which there in no ETF, I used the Index returns, minus the average daily difference between the ETF and Index.
 For the period for which there is an ETF, I used instead the ETF returns.
The estimated indexes are already prepared to be uploaded to Portfolio Analyzer in their own excel files.
Nevertheless, treat the information with some caution, as I am not 100% sure regarding the data on the MSCI indexes, as explained above.
Namely I have no spillage for the MSCI World (there is no ETF in Europe for which to proxy against it ).
In terms of the other indexes, I basically did the following:
 Compared the Index against the corresponding European ETF and averaged the daily difference.
 For the period for which there in no ETF, I used the Index returns, minus the average daily difference between the ETF and Index.
 For the period for which there is an ETF, I used instead the ETF returns.
Re: Simulating Returns of Leveraged ETFs
Ah sorry, I wasn't aware of this limitation. Check your PMs...
I don't think this is a European vs US thing. As far as I understand, this is primarily a marketing thing (a reallife fund looks better when comparing to the NR index than when comparing to the GR index!). Even Vanguard US ended up using the same trick if only because Fidelity and the likes did it in a such a 'rosy' way first...celerity wrote: ↑Sun Apr 21, 2019 11:29 amAs for return type, we Europeans need Net, not Gross. Gross is for local (i.e. US) citizens. For example ComStage tracks the Net index https://www.justetf.com/servlet/downloa ... DE&lang=en.
It is true that GR is an ideal which is hard to reach because of withholding taxes (yep, we have the same thing in the US!). But NR is another extreme, which rarely applies in reallife, here is the definition from MSCI (note the point about the lack of double taxation treaty, which is rare), so... the truth is probably inbetween.
NR—Net Return indicates that this series approximates the minimum possible dividend reinvestment. The dividend is
reinvested after deduction of withholding tax, applying the rate applicable to nonresident individuals who do not benefit
from double taxation treaties. MSCI uses withholding tax rates applicable to Luxembourg holding companies.
In practice, it seems clear to me that a reallife fund (leveraged or not) would try to track the GR index, which is the one made of public companies with # of shares, weights, etc. And then suffer from some level of extra costs due to tax withholding AND other operational issues. It seems more logical to me to do the theoretical leveraged computation (the formula we've been discussing) on the GR index, and then compare the reality of reallife funds to such theoretical computation. And then try to estimate the sum of the extra friction costs (aka spillage or whatever you want to call it) that reality mandates, of which tax withholding is only one example among others.
Re: Simulating Returns of Leveraged ETFs
I've noticed several of the European indices actually simulate all the way back to the early 1990s. How's that possible when the euro was introduced only in 1999? Germany still had Deutschmark back then.
For example:
https://sgi.sgmarkets.com/en/indexdeta ... formances/
https://www.stoxx.com/indexdetails?symbol=SX5E
For example:
https://sgi.sgmarkets.com/en/indexdeta ... formances/
https://www.stoxx.com/indexdetails?symbol=SX5E

 Posts: 23
 Joined: Wed Jan 22, 2014 2:12 am
Re: Simulating Returns of Leveraged ETFs
There was the ECU, that was the percursor to the EURO. Basically, on 1 January 1999, the countries adopting the EURO set the exchange rates of their currency based on the ECU spot rate (I am oversimplyfying obviously).celerity wrote: ↑Tue Apr 23, 2019 2:08 pmI've noticed several of the European indices actually simulate all the way back to the early 1990s. How's that possible when the euro was introduced only in 1999? Germany still had Deutschmark back then.
For example:
https://sgi.sgmarkets.com/en/indexdeta ... formances/
https://www.stoxx.com/indexdetails?symbol=SX5E
Because the sound of "ECU" is similar to cow in german and to "the botton of your backside" in Portuguese, they adopted a more neutral name for the currency.
In fact, the Portuguese punk rock band "Peste e Siga" joked about this on one of their songs in the 1990's, basically singing "you want Ecus", which can be read as "you want $$$$" or (pun intended) "you want a**"
Sorry for the Offtopic.
Re: Simulating Returns of Leveraged ETFs
Cool, I didn't know that!Mouro_Emprestado wrote: ↑Tue Apr 23, 2019 3:07 pmThere was the ECU, that was the percursor to the EURO. Basically, on 1 January 1999, the countries adopting the EURO set the exchange rates of their currency based on the ECU spot rate (I am oversimplyfying obviously).celerity wrote: ↑Tue Apr 23, 2019 2:08 pmI've noticed several of the European indices actually simulate all the way back to the early 1990s. How's that possible when the euro was introduced only in 1999? Germany still had Deutschmark back then.
For example:
https://sgi.sgmarkets.com/en/indexdeta ... formances/
https://www.stoxx.com/indexdetails?symbol=SX5E
Because the sound of "ECU" is similar to cow in german and to "the botton of your backside" in Portuguese, they adopted a more neutral name for the currency.
In fact, the Portuguese punk rock band "Peste e Siga" joked about this on one of their songs in the 1990's, basically singing "you want Ecus", which can be read as "you want $$$$" or (pun intended) "you want a**"
Sorry for the Offtopic.
Found it: https://www.investopedia.com/terms/e/eu ... yunit.asp
Data set: https://ec.europa.eu/eurostat/web/excha ... a/database
Thanks!
Re: Simulating Returns of Leveraged ETFs
I did a comparison of Amundi ETF Leveraged Euro Stoxx 50 Daily UCITS EUR (FR0010756072) and various Euro Stoxx 50 indices (Net, Gross and Amundi's own unspecified benchmark found at their website):siamond wrote: ↑Sun Apr 21, 2019 8:20 pmI don't think this is a European vs US thing. As far as I understand, this is primarily a marketing thing (a reallife fund looks better when comparing to the NR index than when comparing to the GR index!). Even Vanguard US ended up using the same trick if only because Fidelity and the likes did it in a such a 'rosy' way first...
It is true that GR is an ideal which is hard to reach because of withholding taxes (yep, we have the same thing in the US!). But NR is another extreme, which rarely applies in reallife, here is the definition from MSCI (note the point about the lack of double taxation treaty, which is rare), so... the truth is probably inbetween.
NR—Net Return indicates that this series approximates the minimum possible dividend reinvestment. The dividend is
reinvested after deduction of withholding tax, applying the rate applicable to nonresident individuals who do not benefit
from double taxation treaties. MSCI uses withholding tax rates applicable to Luxembourg holding companies.
In practice, it seems clear to me that a reallife fund (leveraged or not) would try to track the GR index, which is the one made of public companies with # of shares, weights, etc. And then suffer from some level of extra costs due to tax withholding AND other operational issues. It seems more logical to me to do the theoretical leveraged computation (the formula we've been discussing) on the GR index, and then compare the reality of reallife funds to such theoretical computation. And then try to estimate the sum of the extra friction costs (aka spillage or whatever you want to call it) that reality mandates, of which tax withholding is only one example among others.
Conclusion:
 Amundi's own benchmark is likely wrong (did they use a price index by mistake?)
 The real benchmark is Euro Stoxx 50 Net Return
Re: Simulating Returns of Leveraged ETFs
This is a great thread, I'm glad I stumbled on it. I have been working on simulating leveraged ETFs and would be interested in comparing notes. Siamond I would be interested in seeing how my data compares to yours, would you mind sending me your spreadsheet? I am happy to share any data I have. My UPRO is off by quite a bit, and I suspect there are data issues with my gold ETFs or index. Here are the ETFs I have done so far. The LTTs I can go back to 1987 using daily data from VUSTX, but the correlations drop quite a bit.
The correlations are between my daily simulated data and the actual ETF historical prices. I haven't applied any fudge factors yet, just the daily leverage, expense fees, and for the LTTs, I added LIBOR interest. I plan on adding more LETFs and look at adding a fudge factor to the expense ratio to see how much I can improve the fit. I may also add some graphs as those help visualize the differences.
Correlation(DGAZ, SPGSNGPTR.INDX, 20120208, 20190301) = 99.95925%. Annual: ETF 43.67%, Sim 45.07%
Correlation(DRIP, SPSIOPTR.INDX, 20150529, 20190325) = 99.96382%. Annual: ETF 33.58%, Sim 33.85%
Correlation(DWT, SPGSCLPER.INDX, 20161209, 20190301) = 99.97956%. Annual: ETF 37.92%, Sim 38.74%
Correlation(GASL, FCG, 20100714, 20190327) = 98.08126%. Annual: ETF 59.53%, Sim 58.84%
Correlation(GASX, FCG, 20151203, 20190327) = 99.84917%. Annual: ETF 40.43%, Sim 37.58%
Correlation(GLL, GOLD.INDX, 20081203, 20190418) = 99.91619%. Annual: ETF 16.91%, Sim 17.42%
Correlation(GUSH, SPSIOPTR.INDX, 20150529, 20190318) = 99.96511%. Annual: ETF 52.82%, Sim 51.78%
Correlation(JDST, MVGDXJTR.INDX, 20131003, 20190325) = 90.68836%. Annual: ETF 71.24%, Sim 57.50%
Correlation(JNUG, MVGDXJTR.INDX, 20131003, 20190325) = 98.39910%. Annual: ETF 53.44%, Sim 40.05%
Correlation(LABD, SPSIBITR.INDX, 20150528, 20190301) = 99.96722%. Annual: ETF 54.43%, Sim 56.21%
Correlation(LABU, SPSIBITR.INDX, 20150528, 20190325) = 99.95824%. Annual: ETF 23.73%, Sim 22.87%
Correlation(SOXL, SOX.INDX, 20100311, 20190301) = 99.98669%. Annual: ETF 34.07%, Sim 30.19%
Correlation(SOXS, SOX.INDX, 20100311, 20190301) = 99.91042%. Annual: ETF 57.30%, Sim 55.47%
Correlation(SPXL, SP500TR.INDX, 20081105, 20190326) = 99.88468%. Annual: ETF 28.36%, Sim 30.96%
Correlation(SPXS, SP500TR.INDX, 20081119, 20190326) = 99.96057%. Annual: ETF 46.11%, Sim 46.34%
Correlation(SPXU, SPX, 20090625, 20190326) = 99.88860%. Annual: ETF 41.27%, Sim 35.23%
Correlation(TBT, TLT, 20080522, 20190422) = 99.67347%. Annual: ETF 17.45%, Sim 15.59%
Correlation(TMF, TLT, 20090416, 20190422) = 99.96299%. Annual: ETF 5.29%, Sim 6.53%
Correlation(TMV, TLT, 20090416, 20190422) = 99.95823%. Annual: ETF 23.94%, Sim 21.99%
Correlation(TTT, TLT, 20120329, 20190422) = 99.95983%. Annual: ETF 17.71%, Sim 16.48%
Correlation(TYBS, TLT, 20110323, 20190418) = 99.86295%. Annual: ETF 8.03%, Sim 7.76%
Correlation(UBT, TLT, 20100121, 20190301) = 99.97797%. Annual: ETF 8.92%, Sim 8.58%
Correlation(UGAZ, SPGSNGPTR.INDX, 20120208, 20190319) = 99.97909%. Annual: ETF 64.85%, Sim 64.48%
Correlation(UGL, GOLD.INDX, 20081203, 20190418) = 99.30312%. Annual: ETF 3.68%, Sim 6.28%
Correlation(UPRO, SPX, 20090625, 20190326) = 99.87153%. Annual: ETF 36.34%, Sim 30.51%
Correlation(UWT, SPGSCLPER.INDX, 20161209, 20190301) = 99.94554%. Annual: ETF 22.31%, Sim 23.40%
The correlations are between my daily simulated data and the actual ETF historical prices. I haven't applied any fudge factors yet, just the daily leverage, expense fees, and for the LTTs, I added LIBOR interest. I plan on adding more LETFs and look at adding a fudge factor to the expense ratio to see how much I can improve the fit. I may also add some graphs as those help visualize the differences.
Correlation(DGAZ, SPGSNGPTR.INDX, 20120208, 20190301) = 99.95925%. Annual: ETF 43.67%, Sim 45.07%
Correlation(DRIP, SPSIOPTR.INDX, 20150529, 20190325) = 99.96382%. Annual: ETF 33.58%, Sim 33.85%
Correlation(DWT, SPGSCLPER.INDX, 20161209, 20190301) = 99.97956%. Annual: ETF 37.92%, Sim 38.74%
Correlation(GASL, FCG, 20100714, 20190327) = 98.08126%. Annual: ETF 59.53%, Sim 58.84%
Correlation(GASX, FCG, 20151203, 20190327) = 99.84917%. Annual: ETF 40.43%, Sim 37.58%
Correlation(GLL, GOLD.INDX, 20081203, 20190418) = 99.91619%. Annual: ETF 16.91%, Sim 17.42%
Correlation(GUSH, SPSIOPTR.INDX, 20150529, 20190318) = 99.96511%. Annual: ETF 52.82%, Sim 51.78%
Correlation(JDST, MVGDXJTR.INDX, 20131003, 20190325) = 90.68836%. Annual: ETF 71.24%, Sim 57.50%
Correlation(JNUG, MVGDXJTR.INDX, 20131003, 20190325) = 98.39910%. Annual: ETF 53.44%, Sim 40.05%
Correlation(LABD, SPSIBITR.INDX, 20150528, 20190301) = 99.96722%. Annual: ETF 54.43%, Sim 56.21%
Correlation(LABU, SPSIBITR.INDX, 20150528, 20190325) = 99.95824%. Annual: ETF 23.73%, Sim 22.87%
Correlation(SOXL, SOX.INDX, 20100311, 20190301) = 99.98669%. Annual: ETF 34.07%, Sim 30.19%
Correlation(SOXS, SOX.INDX, 20100311, 20190301) = 99.91042%. Annual: ETF 57.30%, Sim 55.47%
Correlation(SPXL, SP500TR.INDX, 20081105, 20190326) = 99.88468%. Annual: ETF 28.36%, Sim 30.96%
Correlation(SPXS, SP500TR.INDX, 20081119, 20190326) = 99.96057%. Annual: ETF 46.11%, Sim 46.34%
Correlation(SPXU, SPX, 20090625, 20190326) = 99.88860%. Annual: ETF 41.27%, Sim 35.23%
Correlation(TBT, TLT, 20080522, 20190422) = 99.67347%. Annual: ETF 17.45%, Sim 15.59%
Correlation(TMF, TLT, 20090416, 20190422) = 99.96299%. Annual: ETF 5.29%, Sim 6.53%
Correlation(TMV, TLT, 20090416, 20190422) = 99.95823%. Annual: ETF 23.94%, Sim 21.99%
Correlation(TTT, TLT, 20120329, 20190422) = 99.95983%. Annual: ETF 17.71%, Sim 16.48%
Correlation(TYBS, TLT, 20110323, 20190418) = 99.86295%. Annual: ETF 8.03%, Sim 7.76%
Correlation(UBT, TLT, 20100121, 20190301) = 99.97797%. Annual: ETF 8.92%, Sim 8.58%
Correlation(UGAZ, SPGSNGPTR.INDX, 20120208, 20190319) = 99.97909%. Annual: ETF 64.85%, Sim 64.48%
Correlation(UGL, GOLD.INDX, 20081203, 20190418) = 99.30312%. Annual: ETF 3.68%, Sim 6.28%
Correlation(UPRO, SPX, 20090625, 20190326) = 99.87153%. Annual: ETF 36.34%, Sim 30.51%
Correlation(UWT, SPGSCLPER.INDX, 20161209, 20190301) = 99.94554%. Annual: ETF 22.31%, Sim 23.40%
Re: Simulating Returns of Leveraged ETFs
Hi there. I am traveling right now with little time for such activities. Could you be kind enough to recontact me early June, and I'll be happy to share notes. In the mean time, please check this thread in details, I did my best to document our reasoning and model as we developed it.
Re: Simulating Returns of Leveraged ETFs
I was under the impression that our current monthly simulator would work well for negative leverage (e.g. 2x, 3x 'inverse/bear' funds), but after running a few experiments and checking against actuals, it appears that it does work quite well for stock funds, but not terribly well for bond funds. The issue first popped up in this thread.
This led me to run more experiments (model against actuals) and also to come back to the underlying formulas to try to understand what is going on. The following few posts will provide corresponding details. For those of you new to this, you may want to check what a Telltale chart is, if you don't already know.
This led me to run more experiments (model against actuals) and also to come back to the underlying formulas to try to understand what is going on. The following few posts will provide corresponding details. For those of you new to this, you may want to check what a Telltale chart is, if you don't already know.
Re: Simulating Returns of Leveraged ETFs
Let's start by the venerable S&P 500, using 2x leverage and 3x leverage. Again, those are inverse/bear funds, with a negative leverage. Here are the Telltale charts compared to the model (click to see a larger display).
For the 2x case, we have three funds from ProFunds, ProShares and Rydex to compare, plus an S&P Index (the purple line, hovering very close to the model). Overall, this is quite flat (besides the tough start of URPIX), the model appears to match the dynamics of the actual remarkably well, except late 2008, but that is not entirely surprising to be honest.
As to the 3x case, we only have Proshares and Direxion. As usual, Direxion had a rough start, then gets okish, while Proshares stayed amazing close to the model (almost too good to be true!).
For the 2x case, we have three funds from ProFunds, ProShares and Rydex to compare, plus an S&P Index (the purple line, hovering very close to the model). Overall, this is quite flat (besides the tough start of URPIX), the model appears to match the dynamics of the actual remarkably well, except late 2008, but that is not entirely surprising to be honest.
As to the 3x case, we only have Proshares and Direxion. As usual, Direxion had a rough start, then gets okish, while Proshares stayed amazing close to the model (almost too good to be true!).
Re: Simulating Returns of Leveraged ETFs
I must say those S&P 500 results were actually a little surprising to me, as the positive leverage charts indicated more significant 'friction costs' (reallife implementation decay) for S&P leveraged funds. So I ran a test out of sample, using midcaps.
Profunds and Proshares each have a corresponding fund, and here I found a trajectory closer to what I was expecting, with a fairly steady 0.5% loss compared to the model. Which is comparable to what we've seen with midcaps for +2x leverage.
Profunds and Proshares each have a corresponding fund, and here I found a trajectory closer to what I was expecting, with a fairly steady 0.5% loss compared to the model. Which is comparable to what we've seen with midcaps for +2x leverage.
Re: Simulating Returns of Leveraged ETFs
So far, we're pretty good, none of those (stockoriented) charts would make me doubt the model. Treasuries are an entirely different game though. What we've found in the past with positive leverage is that the closest Telltale charts came from treasury funds. Let's see what happened with LongTerm Treasuries (2x and 3x). In this case, we have much less funds to compare to.
As a reminder, all those charts use 'Gross Return' data from the various reallife funds, hence returns computed BEFORE taking in account Expense Ratios, hence returns directly comparable to an index or a model.
In the 3x case, Direxion displayed its usual shaky implementation, with all those wiggles around a trend line we've seen in the past, while Proshares displayed a much more rigorous tracking of the corresponding index for both the 2x and 3x cases. The friction cost decay compared to the model is severe though (nearly a point for Proshares 2x and 3x, more than 2 points for Direxion/3x).
I assembled Intermediate Treasuries charts along the same lines (Proshares/2x was similar to LTTs while Direxion totally sunk the boat for 2x ITTs).
As a reminder, all those charts use 'Gross Return' data from the various reallife funds, hence returns computed BEFORE taking in account Expense Ratios, hence returns directly comparable to an index or a model.
In the 3x case, Direxion displayed its usual shaky implementation, with all those wiggles around a trend line we've seen in the past, while Proshares displayed a much more rigorous tracking of the corresponding index for both the 2x and 3x cases. The friction cost decay compared to the model is severe though (nearly a point for Proshares 2x and 3x, more than 2 points for Direxion/3x).
I assembled Intermediate Treasuries charts along the same lines (Proshares/2x was similar to LTTs while Direxion totally sunk the boat for 2x ITTs).

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Re: Simulating Returns of Leveraged ETFs
Interesting that Proshares has a 3x inverse LTT fund but not a 3x LTT fund... any idea why? Seems based on their trackrecord it would be a worthwhile swap from TMF for those executing this strategy.siamond wrote: ↑Fri Jul 12, 2019 4:46 pmSo far, we're pretty good, none of those (stockoriented) charts would make me doubt the model. Treasuries are an entirely different game though. What we've found in the past with positive leverage is that the closest Telltale charts came from treasury funds. Let's see what happened with LongTerm Treasuries (2x and 3x). In this case, we have much less funds to compare to.
As a reminder, all those charts use 'Gross Return' data from the various reallife funds, hence returns computed BEFORE taking in account Expense Ratios, hence returns directly comparable to an index or a model.
In the 3x case, Direxion displayed its usual shaky implementation, with all those wiggles around a trend line we've seen in the past, while Proshares displayed a much more rigorous tracking of the corresponding index for both the 2x and 3x cases. The friction cost decay compared to the model is severe though (nearly a point for Proshares 2x and 3x, more than 2 points for Direxion/3x).
I assembled Intermediate Treasuries charts along the same lines (Proshares/2x was similar to LTTs while Direxion totally sunk the boat for 2x ITTs).
Re: Simulating Returns of Leveraged ETFs
As usual with leveraged funds, it is difficult to judge if the disconnect comes from systemic implementation issues or from a deficiency of the model. I came back to the key documents we used to develop the model and I paid more attention to the details related to negative leverage.
The simulation model is based on equation (4) from the papers from Avellaneda and Zhang (2010) and Tang and Xu (2013), here it is:
The first term based is Vt is a function of the variance of the daily index, let's ignore the second term (Ht) so far (I'll come back to it later), the third term is a function of the LIBOR rate and the 4th term is the expense ratio (which isn't relevant here given our use of Gross Returns to compare to the model. The formula is the same for positive and negative leverage factors, except for the 2nd term which only applies for negative leveraging.
Let's discuss a bit the third term first. For positive leverage cases, it is quite intuitive, this is about the cost of borrowing funds to buy additional securities up to the leverage factor. It may seem counterintuitive to see that the (1beta) multiplier would be equal to a positive number in case of inverse leveraging though, we're no longer speaking of costs, but of proceeds (i.e. extra returns)!
The Standard & Poor's methodology document for leveraged and inverse indices spells out the corresponding logic in more detail than the Avellaneda paper. This starts on page 43, the essence of it is that the rate in this case is actually a lending rate (hence the positive number) and I'll let the interested reader get in the details. Note that equations spelled out by S&P are equivalent to the Avellaneda paper EXCEPT for the second term I ignored so far, which doesn't appear in the S&P equations (adding it is briefly mentioned in note 15 though).
Now what about this mysterious 2nd term in the Avellaneda paper (which only applies in the case of negative leveraging)? In the implementation of the model, I neglected it as the paper made the same assumption in all its backtests, which is that the lambda quantities are assumed to be negligible. Here is the explanation from Avanella:
Such extra cost factor may not always be negligible though, notably in times of crisis like the end of 2008 (see footnote #5), although this seems hard to quantify. Still, I don't quite see that this explains the results from the charts I just posted (a fairly steep steady decay year over year for treasury funds, while stock funds worked out fine). So... if anybody has an idea on how to explain the findings for treasury funds, I am all ears!
The simulation model is based on equation (4) from the papers from Avellaneda and Zhang (2010) and Tang and Xu (2013), here it is:
The first term based is Vt is a function of the variance of the daily index, let's ignore the second term (Ht) so far (I'll come back to it later), the third term is a function of the LIBOR rate and the 4th term is the expense ratio (which isn't relevant here given our use of Gross Returns to compare to the model. The formula is the same for positive and negative leverage factors, except for the 2nd term which only applies for negative leveraging.
Let's discuss a bit the third term first. For positive leverage cases, it is quite intuitive, this is about the cost of borrowing funds to buy additional securities up to the leverage factor. It may seem counterintuitive to see that the (1beta) multiplier would be equal to a positive number in case of inverse leveraging though, we're no longer speaking of costs, but of proceeds (i.e. extra returns)!
The Standard & Poor's methodology document for leveraged and inverse indices spells out the corresponding logic in more detail than the Avellaneda paper. This starts on page 43, the essence of it is that the rate in this case is actually a lending rate (hence the positive number) and I'll let the interested reader get in the details. Note that equations spelled out by S&P are equivalent to the Avellaneda paper EXCEPT for the second term I ignored so far, which doesn't appear in the S&P equations (adding it is briefly mentioned in note 15 though).
Now what about this mysterious 2nd term in the Avellaneda paper (which only applies in the case of negative leveraging)? In the implementation of the model, I neglected it as the paper made the same assumption in all its backtests, which is that the lambda quantities are assumed to be negligible. Here is the explanation from Avanella:
Such extra cost factor may not always be negligible though, notably in times of crisis like the end of 2008 (see footnote #5), although this seems hard to quantify. Still, I don't quite see that this explains the results from the charts I just posted (a fairly steep steady decay year over year for treasury funds, while stock funds worked out fine). So... if anybody has an idea on how to explain the findings for treasury funds, I am all ears!
Re: Simulating Returns of Leveraged ETFs
Just for grins I compared a few of the monthly values in your spreadsheet. I checked the values calculated directly from daily results against the monthly values calculated using the summary statistics for the daily values. These generally agree; sometimes the 4th digit differs by 1.
The size of error you are finding would be nudging the 3rd digit I think.
So I think this confirms that the issue is in the expenses, as you say.
I'm out of my depth beyond this simple test though.
The size of error you are finding would be nudging the 3rd digit I think.
So I think this confirms that the issue is in the expenses, as you say.
I'm out of my depth beyond this simple test though.
Re: Simulating Returns of Leveraged ETFs
I have no idea, but yes, it's a bit of a shame. There is absolutely no discussion that Proshares is way, way better than Direxion at implementing leveraged funds (for positive or negative leverage).MotoTrojan wrote: ↑Fri Jul 12, 2019 5:05 pmInteresting that Proshares has a 3x inverse LTT fund but not a 3x LTT fund... any idea why? Seems based on their trackrecord it would be a worthwhile swap from TMF for those executing this strategy.
Yes, I verified this again several times and the model is very impressive in this respect. The bottomline is that most reallife leveraged funds struggle to keep up with theoretical numbers, some more than others. Admittedly, a daily process must be really hard to implement in a rigorous manner, notably in presence of those highfrequency traders playing all sorts of lastminute games.Hydromod wrote: ↑Sat Jul 13, 2019 11:50 amJust for grins I compared a few of the monthly values in your spreadsheet. I checked the values calculated directly from daily results against the monthly values calculated using the summary statistics for the daily values. These generally agree; sometimes the 4th digit differs by 1. [...] So I think this confirms that the issue is in the expenses, as you say.
Re: Simulating Returns of Leveraged ETFs
I am still puzzled by the disconnect between treasury funds with negative leverage and the corresponding model... I came back to the Direxion and the ProShares fiscal annual reports. The former really didn't say much, although it is striking to see the number of funds from Direxion which missed their own model targets by a mile... The latter (Proshares) doesn't say a lot either, but I found this extract (emphasis is mine):
The following poster seemed to have a solid handle on this topic, but he/she never came back to this thread... Ohai, are you reading this?
We discussed this quote before, but never paid attention to the special emphasis on short/inverse exposure and the negative effect of spread (last sentence). When we discussed spread in the past, we kind of concluded that it was a negligible factor which could go either way (positive or negative). Well, maybe we should revisit? I have no idea how to find historical data about it though.Financing Rates Associated with Derivatives:The performance of each Fund was impacted by the related financing costs. Financial instruments such as futures contracts carry implied financing costs. Swap financing rates are negotiated between the Funds and their counterparties, and are typically set at the oneweek/onemonth London Interbank Offered Rate (“LIBOR”) plus or minus a negotiated spread. The oneweek LIBOR appreciated from 0.95% to 1.75% during the fiscal year. The onemonth LIBOR also increased during the fiscal year from 1.06% to 2.00%. Each Fund with long exposure via derivatives was generally negatively affected by financing rates. Conversely, most Funds with short/inverse derivative exposure generally benefited from financing rates. However, in low interest rate environments, LIBOR adjusted by the spread may actually result in a Fund with short/inverse exposure also being negatively affected by financing rates.
The following poster seemed to have a solid handle on this topic, but he/she never came back to this thread... Ohai, are you reading this?
ohai wrote: ↑Thu Mar 14, 2019 4:37 pmHi. Using Libor for the assumed interest rate is not really correct, although it is close. Equity market funding is not libor + 0bp. There is a spread to this rate that is mostly based on specific supply and demand for equity leverage. For instance, if you price a futures contract on SPX (which is presumably the method through which SSO, UPRO, etc. achieve leverage), the implicit interest rate might be libor+10bp or so today. This value can trade in say a range of negative to positive 2%, and the average value is around 0.30%.
Finally, I am still eager to find *some* historical data to get a better grasp of transaction costs for leveraged funds (and per market segment). Remember, such transaction costs are not included in the Expense Ratio and the daily process enforced by leveraged funds implies quite a lot of transactions (I read that a 10% daily turnover isn't unusual for leveraged funds!)...ohai wrote: ↑Thu Mar 14, 2019 9:24 pmAnyway, data on equity funding levels is not published anywhere. For the most part, only institutional trading desks (i.e. banks and specifically, delta one desks) really track this explicitly. I do have historical data for this, but unfortunately cannot share it due to proprietary reasons.

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Re: Simulating Returns of Leveraged ETFs
I'm working on simulating leveraged versions of various ETFs  very glad I found this thread.
I found this paper to be extremely helpful, fwiw: https://riskcenter.berkeley.edu/wpcont ... r1_000.pdf
To avoid reinventing the wheel  is there any place where you guys have any sort of generalizable formula, whether in excel or code form? I'd like to be able to take an arbitrary etf's return series and simulate a levered version of that etf, where leverage ratio is a parameter (ideally rebalancing frequency is also a parameter).
A generalize function would look something like:
Where the first three arguments are vectors/time series. I assume transaction costs would be derived from the above arguments, with a roughly linear scaling w.r.t gearing.
At this point, that would be more helpful then trying to parse through lots of long form text. Happy to contribute improvements as well, and I will share my code as soon at it's ready.
I found this paper to be extremely helpful, fwiw: https://riskcenter.berkeley.edu/wpcont ... r1_000.pdf
To avoid reinventing the wheel  is there any place where you guys have any sort of generalizable formula, whether in excel or code form? I'd like to be able to take an arbitrary etf's return series and simulate a levered version of that etf, where leverage ratio is a parameter (ideally rebalancing frequency is also a parameter).
A generalize function would look something like:
Code: Select all
simulate_LETF(daily_return_series, daily_borrowing_costs, leverage_ratio, rebalance_frequency)
At this point, that would be more helpful then trying to parse through lots of long form text. Happy to contribute improvements as well, and I will share my code as soon at it's ready.