Simulating Returns of Leveraged ETFs

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EfficientInvestor
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Re: Simulating Returns of Leveraged ETFs

Post by EfficientInvestor » Thu Feb 14, 2019 4:58 pm

pezblanco wrote:
Thu Feb 14, 2019 4:47 pm
I'm just wanting to ask a question from those doing the simulations. Presumably the 3x funds are more expensive than the 2x funds. How do these costs compare to just borrowing money on a margin account and leveraging up by 3x or 2x? Is it substantially cheaper to use the leveraged funds than just borrowing money from a brokerage account (I think you can borrow at the 1month T-bill rate + 1%)?
It looks like you are correct on the borrow rate if you have over $100k. Interactive Brokers has a rate of 3.4% (2.4 + 1) for accounts over $100k. They bump it down to 2.4 + 0.5 for accounts over $1MM. My question would be how much they would let you borrow. To create a 3X fund, you would have to borrow 200% of you actual portfolio value.

EDIT - IB charges via a tiered structure. The first $100k is charged the benchmark + 1.5%, then you pay the benchmark + 1% on anything between that and $1MM. So let's just assume most folks on here would be paying the benchmark + 1.5% for their whole portfolio.

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Re: Simulating Returns of Leveraged ETFs

Post by pezblanco » Thu Feb 14, 2019 6:27 pm

EfficientInvestor wrote:
Thu Feb 14, 2019 4:58 pm
pezblanco wrote:
Thu Feb 14, 2019 4:47 pm
I'm just wanting to ask a question from those doing the simulations. Presumably the 3x funds are more expensive than the 2x funds. How do these costs compare to just borrowing money on a margin account and leveraging up by 3x or 2x? Is it substantially cheaper to use the leveraged funds than just borrowing money from a brokerage account (I think you can borrow at the 1month T-bill rate + 1%)?
It looks like you are correct on the borrow rate if you have over $100k. Interactive Brokers has a rate of 3.4% (2.4 + 1) for accounts over $100k. They bump it down to 2.4 + 0.5 for accounts over $1MM. My question would be how much they would let you borrow. To create a 3X fund, you would have to borrow 200% of you actual portfolio value.

EDIT - IB charges via a tiered structure. The first $100k is charged the benchmark + 1.5%, then you pay the benchmark + 1% on anything between that and $1MM. So let's just assume most folks on here would be paying the benchmark + 1.5% for their whole portfolio.
So ... a quick back of the envelope comparison?

For a 2x leveraged fund:

12 month Libor is around 3% and the fund cost is around 1% ... so total cost is 4% of what you have invested ....
1-month T-bill is around 2.4% then add an additional 1.5% for the cost to borrow and you get 3.9% of what you have invested.

So roughly the same?

For a 3x fund the cost to borrow goes up by a factor of 2 ..... so I'm assuming the 3x leveraged fund would be cheaper than trying to leverage 3x on margin?

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Fri Feb 15, 2019 1:10 am

I updated the simulation charts that I posted yesterday (stock funds and bond funds). I made the vertical axis logarithmic, to better appreciate the trajectory in the early years. And I added the trajectory of the regular (unleveraged) index. And I tuned a bit the commentary. Interested parties, please check it out.

Something I found striking (and sobering) is the ULPIX trajectory over the past two decades against the regular S&P 500. And my charts don't even account for the fund's ER... Here is a chart from Morningstar that basically shows the same thing (this time, ER included), using ULPIX as well as RYTNX (another S&P 500 2x fund). It is quite remarkable how closely ULPIX and RYTNX track each other - one might argue they do seem to use a passive (and easily reproduced) strategy. Although not quite convincing over this time period! :shock:

Image

It is really too bad that we don't seem to have a similarly long-lived bond LETF, spanning the recent crises, to further validate our simulation model.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Sat Feb 16, 2019 9:24 pm

Just would like to take a note that, in the early days, we don’t have daily index numbers. In some cases, all we have are monthly numbers. This doesn’t invalidate the model, but it makes it more optimistic than it should, by ignoring some of the volatility decay.

I think we should be able to adjust the model in a reasonable manner, with simple assumptions about daily vs monthly volatility. Doing so would also allow to use AlohaJoe’s bonds model, and start the whole model in the mid 50s. This would seem prudent, allowing to notably simulate cycles starting in 65/66, two VERY painful years...

Will think more about it when I’m back from a family trip...

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Re: Simulating Returns of Leveraged ETFs

Post by PluckyDucky » Sat Feb 16, 2019 10:13 pm

Where did the data set from the 1996 Asness article come from?

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Sun Feb 17, 2019 12:16 pm

PluckyDucky wrote:
Sat Feb 16, 2019 10:13 pm
Where did the data set from the 1996 Asness article come from?
He used the Ibbotson/SBBI monthly data set, US large stocks, US long-term corporates and 1-month T-Bills for the borrowing cost.

We could do something similar for pre-1973 history, although we identified a while ago that Ibbotson has an oversimplified way of computing bond returns...

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Re: Simulating Returns of Leveraged ETFs

Post by MotoTrojan » Wed Feb 20, 2019 11:21 am

siamond wrote:
Fri Feb 15, 2019 1:10 am
I updated the simulation charts that I posted yesterday (stock funds and bond funds). I made the vertical axis logarithmic, to better appreciate the trajectory in the early years. And I added the trajectory of the regular (unleveraged) index. And I tuned a bit the commentary. Interested parties, please check it out.

Something I found striking (and sobering) is the ULPIX trajectory over the past two decades against the regular S&P 500. And my charts don't even account for the fund's ER... Here is a chart from Morningstar that basically shows the same thing (this time, ER included), using ULPIX as well as RYTNX (another S&P 500 2x fund). It is quite remarkable how closely ULPIX and RYTNX track each other - one might argue they do seem to use a passive (and easily reproduced) strategy. Although not quite convincing over this time period! :shock:

Image

It is really too bad that we don't seem to have a similarly long-lived bond LETF, spanning the recent crises, to further validate our simulation model.
Sobering indeed, but is this timespan not decently accounted for in the simplified portfolio visualizer 60/40 3x TMF/UPRO model from the other thread? Looking at only the equity portion is much more concerning alone.

Now if you had bought either of these in 2009... 😎😎😎

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Re: Simulating Returns of Leveraged ETFs

Post by gtwhitegold » Wed Feb 20, 2019 11:46 am

Has anyone tried simulating a 3X risk parity portfolio with a maximum drawdown desired? I was thinking to do so including Russell 2000, MSCI EAFE, MSCI EM, 20+ Year Treasuries, and 7-10 Year Treasuries with a drawdown limit of say 30% and see how it backtests.

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Re: Simulating Returns of Leveraged ETFs

Post by MotoTrojan » Wed Feb 20, 2019 12:24 pm

gtwhitegold wrote:
Wed Feb 20, 2019 11:46 am
Has anyone tried simulating a 3X risk parity portfolio with a maximum drawdown desired? I was thinking to do so including Russell 2000, MSCI EAFE, MSCI EM, 20+ Year Treasuries, and 7-10 Year Treasuries with a drawdown limit of say 30% and see how it backtests.
Why are you using higher volatility equity components?

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Re: Simulating Returns of Leveraged ETFs

Post by gtwhitegold » Wed Feb 20, 2019 1:18 pm

MotoTrojan wrote:
Wed Feb 20, 2019 12:24 pm
gtwhitegold wrote:
Wed Feb 20, 2019 11:46 am
Has anyone tried simulating a 3X risk parity portfolio with a maximum drawdown desired? I was thinking to do so including Russell 2000, MSCI EAFE, MSCI EM, 20+ Year Treasuries, and 7-10 Year Treasuries with a drawdown limit of say 30% and see how it backtests.
Why are you using higher volatility equity components?
I'm not using any of these yet, I would like to know if EM or EAFE trackers would have likely been forced to liquidate since 1988 when the MSCI EM Index was created.

From what I can tell, the great recession wouldn't have killed any of the imaginary 3X funds mentioned.

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Re: Simulating Returns of Leveraged ETFs

Post by HEDGEFUNDIE » Sat Feb 23, 2019 1:12 am

JackoC wrote:
Wed Feb 13, 2019 5:22 pm
EfficientInvestor wrote:
Wed Feb 13, 2019 4:54 pm
JackoC wrote:
Wed Feb 13, 2019 4:41 pm
EfficientInvestor wrote:
Tue Feb 12, 2019 11:06 am

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

*which could be a futures implied rate or the explicit LIBOR+ spread leg of a Total Return Swap.
Earlier in this thread it seems like folks were having a hard time finding the spread on the financing. But it's pretty clearly laid out in each fund's annual report:

viewtopic.php?p=4397820#p4397803
viewtopic.php?p=4397820#p4397820

It turns out that TMF actually benefited from below-LIBOR borrowing rates due to counterparties eager to offload their LTT exposure ahead of expected rising interest rates.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Sun Feb 24, 2019 1:31 pm

Back from a week of Internet-less vacations and trying to catch up... A few interesting posts were made in HedgeFundie's thread that could have relevance for our simulation model. Let me quote this exchange first as this follows the same theme as the previous post:
HEDGEFUNDIE wrote:
Sun Feb 24, 2019 11:45 am
AlphaLess wrote:
Sun Feb 24, 2019 10:18 am
So, I see what you are saying.

However, I would decompose the returns of the strategy in a different way:

1.1. Long 3x Long part of the treasury, short 3x short term rates aka LIBOR (times 60%),
1.2. Long 3x Stocks, short 3x short term rates LIBOR (times 40%),
2. Pay trading costs (by the ETF when rebalancing),
3.1. Pay financing spread (as short term rates at LIBOR are only accessible to certain participants) on treasury leg (x 60%),
3.2. Pay financing spread (as short term rates at LIBOR are only accessible to certain participants) on stock leg (x 40%),
4. Pay expense ratio (investor to ETF management).

So you are saying that #3.1 is not a cost, but a profit.

I think I have been arguing that:
A. 1.1 is tied to the steepness of the yield curve. Currently, the yield curve is flat, so profit from 1.1 is small. When the curve goes negative, it will be a source of losses for the strategy,
B. When recession hits, I think your spread on financing (3.1 and 3.2) *COULD RISE*. Why? Because history says so.
I like your decomposition of return. Here is what I would add:

1.1. Long 3x the total return of the Long part of the treasury, short 3x short term rates aka LIBOR (times 60%),
1.2. Long 3x the total return of Stocks, short 3x short term rates LIBOR (times 40%),
1.3 Earn interest income on cash and money market held for collateral and liquidity,
2. Pay trading costs (by the ETF when rebalancing),
3.1. Pay financing spread (as short term rates at LIBOR are only accessible to certain participants) on treasury leg (x 60%),
3.2. Pay financing spread (as short term rates at LIBOR are only accessible to certain participants) on stock leg (x 40%),
4. Pay expense ratio (investor to ETF management).

For 1.1, you’re right that profit on the yield spread is small at a time like this, but yield is only one part of total return. Price is the other, and can still drive solid gains in an environment like now.

For 3.1, it is expectations of LTT’s total return that matters. If LTT total return is expected to go up, the LIBOR spread will be higher than if total return is expected to go down. When a recession hits I would actually expect LTT’s total return to go up initially (driven by price), and spreads over LIBOR to receive that total return to also go up, so I guess I agree with you there.
What is described is basically the model we've followed so far, adding a few considerations:
- portfolio rebalancing costs (very true, but beyond the scope of modeling a single fund)
- interest on cash collateral (a potential extra gain)
- discussion about financial spread (sometimes a potential gain as exemplified by the links in the previous post)

I don't know how to quantify the second consideration (did you guys find some specifics in this respect)?

I would rather stay cautious/conservative about the third consideration, I strongly suspect this goes by ebbs and flows, sometimes positive, sometimes negative (although this is arguably just a gut feeling). In other words, I'd suggest to NOT add such a 'spread' term in the model. Other views?

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Re: Simulating Returns of Leveraged ETFs

Post by HEDGEFUNDIE » Sun Feb 24, 2019 1:48 pm

siamond wrote:
Sun Feb 24, 2019 1:31 pm
I don't know how to quantify the second consideration (did you guys find some specifics in this respect)?
Can be found in the Statement of Operations in the Annual Reports. TMF holds $32M in cash on an asset base of $100M. UPRO holds $447M in cash on $1.27B. So it seems like 1/3 of the assets are generating interest. So I would suggest adding (LIBOR / 3 / 360) to the daily returns.
siamond wrote:
Sun Feb 24, 2019 1:31 pm
I would rather stay cautious/conservative about the third consideration, I strongly suspect this goes by ebbs and flows, sometimes positive, sometimes negative (although this is arguably just a gut feeling). In other words, I'd suggest to NOT add such a 'spread' term in the model. Other views?
I would also agree with this. Theoretically we have a basis to build in a "spread model", a high spread when the underlying assets are expected to appreciate, a low (or even negative) spread when the underlying assets are expected to do poorly. But in the long run it probably cancels out.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Sun Feb 24, 2019 2:11 pm

HEDGEFUNDIE wrote:
Sun Feb 24, 2019 1:48 pm
siamond wrote:
Sun Feb 24, 2019 1:31 pm
I don't know how to quantify the second consideration (did you guys find some specifics in this respect)?
Can be found in the Statement of Operations in the Annual Reports. TMF holds $32M in cash on an asset base of $100M. UPRO holds $447M in cash on $1.27B. So it seems like 1/3 of the assets are generating interest. So I would suggest adding (LIBOR / 3 / 360) to the daily returns.
Good inputs, but those are only two data points for a single year. We would need a more extensive foundation to establish a solid rule of thumb (notably for something increasing the predicted numbers).

Also, if the fund keeps extra cash (going beyond the need for leverage) to ease liquidity, then it actually borrowed this extra cash, and the corresponding extra borrowing cost has to be accounted for. In other words, shouldn't it be a wash? Or am I missing something?

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Re: Simulating Returns of Leveraged ETFs

Post by HEDGEFUNDIE » Sun Feb 24, 2019 2:17 pm

siamond wrote:
Sun Feb 24, 2019 2:11 pm
Good inputs, but those are only two data points for a single year. We would need a more extensive foundation to establish a solid rule of thumb (notably for something increasing the predicted numbers).
I suspect the terms of the swaps actually determine the need for cash on hand.
siamond wrote:
Sun Feb 24, 2019 2:11 pm
Also, if the fund keeps extra cash (going beyond the need for leverage) to ease liquidity, then it actually borrowed this extra cash, and the corresponding extra borrowing cost has to be accounted for. In other words, shouldn't it be a wash? Or am I missing something?
Why do you think the cash is borrowed?

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Sun Feb 24, 2019 2:25 pm

HEDGEFUNDIE wrote:
Sun Feb 24, 2019 2:17 pm
Why do you think the cash is borrowed?
Well, where is it coming from otherwise?

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Re: Simulating Returns of Leveraged ETFs

Post by HEDGEFUNDIE » Sun Feb 24, 2019 2:26 pm

siamond wrote:
Sun Feb 24, 2019 2:25 pm
HEDGEFUNDIE wrote:
Sun Feb 24, 2019 2:17 pm
Why do you think the cash is borrowed?
Well, where is it coming from otherwise?
From the assets of the fund. If the net asset value goes up $100 (through investor inflows, capital gains, or what have you), the fund manager makes sure some of that $100 goes into cash held on the balance sheet of the fund.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Sun Feb 24, 2019 2:33 pm

HEDGEFUNDIE wrote:
Sun Feb 24, 2019 2:26 pm
siamond wrote:
Sun Feb 24, 2019 2:25 pm
HEDGEFUNDIE wrote:
Sun Feb 24, 2019 2:17 pm
Why do you think the cash is borrowed?
Well, where is it coming from otherwise?
From the assets of the fund. If the net asset value goes up $100 (through investor inflows, capital gains, or what have you), the fund manager makes sure some of that $100 goes into cash held on the balance sheet of the fund.
If they do something like that, then that's a (mighty big! 1/3rd of assets?) chunk of investor's money which isn't invested in the regular leveraged manner, hence a loss for the investor on the long run that we didn't account for in the model. I don't think it makes sense to assume that cash is free. It never is.

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Re: Simulating Returns of Leveraged ETFs

Post by HEDGEFUNDIE » Sun Feb 24, 2019 2:50 pm

siamond wrote:
Sun Feb 24, 2019 2:33 pm
HEDGEFUNDIE wrote:
Sun Feb 24, 2019 2:26 pm
siamond wrote:
Sun Feb 24, 2019 2:25 pm
HEDGEFUNDIE wrote:
Sun Feb 24, 2019 2:17 pm
Why do you think the cash is borrowed?
Well, where is it coming from otherwise?
From the assets of the fund. If the net asset value goes up $100 (through investor inflows, capital gains, or what have you), the fund manager makes sure some of that $100 goes into cash held on the balance sheet of the fund.
If they do something like that, then that's a chunk of investor's money which isn't invested in the regular leveraged manner, hence a loss for the investor on the long run that we didn't account for in the model. I don't think it makes sense to assume that cash is free. It never is.
The cash I'm talking about is mostly collateral for the swaps. This cash is not borrowed from anyone, it's cash that belongs to the fund and enables the swaps that generate the leverage. The fund earns interest on this cash, in the case of TMF: 2.05%.

Image

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Sun Feb 24, 2019 3:11 pm

HEDGEFUNDIE wrote:
Sun Feb 24, 2019 2:50 pm
The cash I'm talking about is mostly collateral for the swaps. This cash is not borrowed from anyone, it's cash that belongs to the fund and enables the swaps that generate the leverage. The fund earns interest on this cash, in the case of TMF: 2.05%.
Thanks for sharing such information. But remember, accountants show some numbers in an explicit manner while bundling other numbers in a not-so-explicit manner. Funny, gains are often in the first category while costs are often in the second category, LOL. Such a giant pile of cash (1/3rd of total assets is a little mind-boggling) has to come from somewhere and HAS a cost (either a borrowing cost or a lost opportunity for a more lucrative investment), this part seems crystal clear to me. Does it offset the gains, I don't know. I would actually suspect this is the reverse way around, the hidden costs (or lost opportunity) probably exceed the gains (interests), cash is never a good investment... This might even be the reason for which our model is currently too optimistic (at least for the stock funds).

As we established, TMF does a poor job of tracking daily returns of its index (times the leverage), so I'd rather think on the basis of a leveraged fund with a straighter operational model. Did you find a similar table for UPRO and ULPIX? Mind sharing?

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Re: Simulating Returns of Leveraged ETFs

Post by HEDGEFUNDIE » Sun Feb 24, 2019 3:58 pm

siamond wrote:
Sun Feb 24, 2019 3:11 pm
HEDGEFUNDIE wrote:
Sun Feb 24, 2019 2:50 pm
The cash I'm talking about is mostly collateral for the swaps. This cash is not borrowed from anyone, it's cash that belongs to the fund and enables the swaps that generate the leverage. The fund earns interest on this cash, in the case of TMF: 2.05%.
Thanks for sharing such information. But remember, accountants show some numbers in an explicit manner while bundling other numbers in a not-so-explicit manner. Funny, gains are often in the first category while costs are often in the second category, LOL. Such a giant pile of cash (1/3rd of total assets is a little mind-boggling) has to come from somewhere and HAS a cost (either a borrowing cost or a lost opportunity for a more lucrative investment), this part seems crystal clear to me. Does it offset the gains, I don't know. I would actually suspect this is the reverse way around, the hidden costs (or lost opportunity) probably exceed the gains (interests), cash is never a good investment... This might even be the reason for which our model is currently too optimistic (at least for the stock funds).

As we established, TMF does a poor job of tracking daily returns of its index (times the leverage), so I'd rather think on the basis of a leveraged fund with a straighter operational model. Did you find a similar table for UPRO and ULPIX? Mind sharing?
Let's go back to Swaps 101. TMF enters into a contract with a counterparty that says something like the following:

1. TMF will pay the counterparty "LIBOR + spread" on a notional amount.

2. In return, the counterparty will pay TMF "the Total Return of the Long Term Treasury index" on that same notional amount.

3. To ensure that TMF can make good on its end of the payments, the counterparty requires that TMF hold a certain amount of cash as collateral.

4. That cash still belongs to TMF, and while it's sitting in the collateral account, it earns interest for TMF.

We need to account for #4 somehow.

Just out of curiosity, I looked up how ProShares does it with their 2x Long Treasury Fund (UBT). They similarly hold 26% of net assets in Short Term Investments as collateral, but in their case it's Repurchase Contracts instead of cash (Repos pay interest just as cash would).

Image

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Sun Feb 24, 2019 5:59 pm

HEDGEFUNDIE wrote:
Sun Feb 24, 2019 3:58 pm
Let's go back to Swaps 101. TMF enters into a contract with a counterparty that says something like the following:

1. TMF will pay the counterparty "LIBOR + spread" on a notional amount.

2. In return, the counterparty will pay TMF "the Total Return of the Long Term Treasury index" on that same notional amount.

3. To ensure that TMF can make good on its end of the payments, the counterparty requires that TMF hold a certain amount of cash as collateral.

4. That cash still belongs to TMF, and while it's sitting in the collateral account, it earns interest for TMF.

We need to account for #4 somehow.
Yes, but again, cash isn't free and doesn't come out of thin air! So there is more than item#4 to be taken in consideration here.
HEDGEFUNDIE wrote:
Sun Feb 24, 2019 3:58 pm
Just out of curiosity, I looked up how ProShares does it with their 2x Long Treasury Fund (UBT). They similarly hold 26% of net assets in Short Term Investments as collateral, but in their case it's Repurchase Contracts instead of cash (Repos pay interest just as cash would).

Image
For reference, here is the link to the ProShares annual report 2018 from which those numbers were extracted (pages 27 and 28). There are 5.9% of liabilities working against the 25.8% repurchase agreements, so let's assume 20% to make numbers simpler and more coherent. The rest (80%) is invested in regular long-term treasuries by UBT themselves, tracking the index.

Also of interest is a 'market exposure' table on page XXII which breaks down the 2x exposure as 80% of regular US Treasuries, 115% of swap agreements and 6% of future contracts (hence an overall 200% exposure, neglecting the extra 1%).

It seems to me that this means that for $1 invested in the UBT ETF, the following occurs during a trading day:
a) 80% * $1 * (1+IR) => the regular treasuries owned by UBT provide a corresponding daily return (i.e. the index daily return)
b) 20% * $1 * (1+LR) => the 'cash' collateral returns interest, let's simplify and assume that the corresponding rate is similar to the LIBOR rate
c) -120% * $1 * (1+LR) => UBT pays the counterparty for the borrowing costs, at the LIBOR rate
d) +120% * $1 * (1+IR) => the swaps performed by the counterparty provide a corresponding daily return (i.e. the index daily return), which is money (positive or negative) coming back to UBT at the end of the day
e) let's set aside the Expense Ratio considerations for now

If you add up the first four terms, we come back to our basic equation 200% * (1+IR) minus 100% * (1+LR). Which is the model we've used so far. And this does account for the cash collateral returning interests, i.e. line b), which is the same as your item #4. And the 20% cash collateral comes straight from the ETF investor, hence the 'lost opportunity' of sort that I was looking for. It all seems to add up.

Did I get that right? (I am slowly learning here, so don't hesitate to set me straight)

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Re: Simulating Returns of Leveraged ETFs

Post by HEDGEFUNDIE » Sun Feb 24, 2019 6:04 pm

siamond wrote:
Sun Feb 24, 2019 5:59 pm
It seems to me that this means that for $1 invested in the UBT ETF, the following occurs during a trading day:
a) 80% * $1 * (1+IR) => the regular treasuries owned by UBT provide a corresponding daily return (i.e. the index daily return)
b) 20% * $1 * (1+LR) => the 'cash' collateral returns interest, let's simplify and assume that the corresponding rate is similar to the LIBOR rate
c) -120% * $1 * (1+LR) => UBT pays the counterparty for the borrowing costs, at the LIBOR rate
d) +120% * $1 * (1+IR) => the swaps performed by the counterparty provide a corresponding daily return (i.e. the index daily return), which is money (positive or negative) coming back to UBT at the end of the day
e) let's set aside the Expense Ratio considerations for now

If you add up the first four terms, we come back to our basic equation 200% * (1+IR) minus 100% * (1+LR). Which is the model we've used so far. And this does account for the cash collateral returning interests, i.e. line b), which is the same as your item #4. And the 20% cash collateral comes straight from the ETF investor, hence the 'lost opportunity' of sort that I was looking for. It all seems to add up.

Did I get that right? (I am slowly learning here, so don't hesitate to set me straight)
Looks good to me. Thanks for spelling it out.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Sun Feb 24, 2019 8:32 pm

AlphaLess wrote:
Sun Feb 24, 2019 10:18 am
However, I would decompose the returns of the strategy in a different way:

1.1. Long 3x Long part of the treasury, short 3x short term rates aka LIBOR (times 60%),
1.2. Long 3x Stocks, short 3x short term rates LIBOR (times 40%),
2. Pay trading costs (by the ETF when rebalancing),
3.1. Pay financing spread (as short term rates at LIBOR are only accessible to certain participants) on treasury leg (x 60%),
3.2. Pay financing spread (as short term rates at LIBOR are only accessible to certain participants) on stock leg (x 40%),
4. Pay expense ratio (investor to ETF management).
Hm, I missed item #2 in this list. Staying with the UBT's example, I guess this would be a form of rebalancing towards their target market exposure (e.g. 80% regular index, 20% swaps & futures providing 120% index exposure). This is confirmed by the following extract from the ProShares annual report 2018 (page IV), and this is indeed not covered by the expense ratio.

Fees, Expenses, and Transaction Costs: Fees and expenses are listed in the financial statements of each Fund and may generally be higher, and thus have a more negative impact on Fund performance compared to many traditional index-based funds. Daily repositioning of each Fund’s portfolio to maintain exposure consistent with its investment objective, high levels of shareholder creation and redemption activity, and use of leverage may lead to commensurate increases in portfolio transactions and transaction costs, which negatively impact the daily NAV of each Fund. Transaction costs are not reflected in the Funds’ expense ratio. Transaction costs are generally higher for Funds whose indexes are more volatile, that seek to return a larger daily multiple of its index’s return, that seek to return an inverse or inverse multiple of its index’s return, that invest in foreign securities, and for Funds that hold or have exposure to assets that are comparatively less liquid than assets held by other Funds.

The fact that 'daily repositioning' transaction costs would be significantly higher if the index of reference is more volatile might (partly?) explain why our model overshoots for stock funds. I couldn't find any quantitative information about such transaction costs though. Any idea?

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Re: Simulating Returns of Leveraged ETFs

Post by AlphaLess » Sun Feb 24, 2019 9:01 pm

siamond wrote:
Sun Feb 24, 2019 8:32 pm
AlphaLess wrote:
Sun Feb 24, 2019 10:18 am
However, I would decompose the returns of the strategy in a different way:

1.1. Long 3x Long part of the treasury, short 3x short term rates aka LIBOR (times 60%),
1.2. Long 3x Stocks, short 3x short term rates LIBOR (times 40%),
2. Pay trading costs (by the ETF when rebalancing),
3.1. Pay financing spread (as short term rates at LIBOR are only accessible to certain participants) on treasury leg (x 60%),
3.2. Pay financing spread (as short term rates at LIBOR are only accessible to certain participants) on stock leg (x 40%),
4. Pay expense ratio (investor to ETF management).
Hm, I missed item #2 in this list. Staying with the UBT's example, I guess this would be a form of rebalancing towards their target market exposure (e.g. 80% regular index, 20% swaps & futures providing 120% index exposure). This is confirmed by the following extract from the ProShares annual report 2018 (page IV), and this is indeed not covered by the expense ratio.

Fees, Expenses, and Transaction Costs: Fees and expenses are listed in the financial statements of each Fund and may generally be higher, and thus have a more negative impact on Fund performance compared to many traditional index-based funds. Daily repositioning of each Fund’s portfolio to maintain exposure consistent with its investment objective, high levels of shareholder creation and redemption activity, and use of leverage may lead to commensurate increases in portfolio transactions and transaction costs, which negatively impact the daily NAV of each Fund. Transaction costs are not reflected in the Funds’ expense ratio. Transaction costs are generally higher for Funds whose indexes are more volatile, that seek to return a larger daily multiple of its index’s return, that seek to return an inverse or inverse multiple of its index’s return, that invest in foreign securities, and for Funds that hold or have exposure to assets that are comparatively less liquid than assets held by other Funds.

The fact that 'daily repositioning' transaction costs would be significantly higher if the index of reference is more volatile might (partly?) explain why our model overshoots for stock funds. I couldn't find any quantitative information about such transaction costs though. Any idea?
In 2009 and 2010, when leveraged funds were new, a lot of 'scientific' papers were published about the underlying dynamics of leveraged funds.

One set of papers shed light in the following dimension:
- approaching the end of day, when the index tracked by leveraged fund is up a lot (or down a lot), at the end of the day, the fund has to chase the price, and buy high while paying transaction costs, if you will (in case of down a lot, it would have to be selling low). This causes further market impact, pushing the price in the same direction MORE. However, the impact would probably reverse a bit. I am not sure if this type of price chasing is still market impactful these days.

Now, when you are implementing the 3x strategy 'on paper', then you obviously won't experience the slippage, but you would be subject to the market impact (say, the true price of the index SHOULD have been up 1.2%, if not for the leveraged rebalance, but with the leveraged rebalance, it is 1.3%, and the 0.1% will revert next morning).

Now, if some of the market impact reverses THE SAME day, then an 'on paper' strategy would also look good.

One way to capture ALL sources of slippage is to do the following:
- obtain DAILY returns of the tracking product (NOT the tracking index, but the product). E.g., for TMF, that would be TLT,
- obtain DAILY returns of the leveraged product,
- run a regression:
lev_ret ~ a + b * track_ret + eps.

The variables being estimated are "a" and "b".

Most likely, you will estimate "b" to be *SLIGHTLY* less than the leverage (e.g., 2.9 for a 3x fund), and "a" would be negative.
Then "a" would basically capture all sources of slippage:
- expense ratio,
- funding cost,
- trading costs,
- etc.

I think you can probably make the regression better if you did this:

lev_ret ~ a + b * track_ret + c * st_ret + eps,
where st_ret are short term funding costs.

But the latter could be hard to obtain (and isolate).
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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Sun Feb 24, 2019 11:00 pm

AlphaLess wrote:
Sun Feb 24, 2019 9:01 pm
In 2009 and 2010, when leveraged funds were new, a lot of 'scientific' papers were published about the underlying dynamics of leveraged funds.

One set of papers shed light in the following dimension:
- approaching the end of day, when the index tracked by leveraged fund is up a lot (or down a lot), at the end of the day, the fund has to chase the price, and buy high while paying transaction costs, if you will (in case of down a lot, it would have to be selling low). This causes further market impact, pushing the price in the same direction MORE. However, the impact would probably reverse a bit. I am not sure if this type of price chasing is still market impactful these days.
Given the small fund size of the leveraged ETFs, I find hard to believe that their orders at the end of the day would move the price needle in any significant manner creating some self-inflicted side effect. Unless various hedge funds play the leverage game at a much bigger scale, of course. This being said, I can easily perceive that such funds have quite the challenge at the end of day, liquidating their leveraged positions (do they really do it in full?), and rebalancing towards proper market exposure, while everything is in flux in the last few minutes of the trading day. Do you happen to have a couple of pointers towards the papers analyzing such dynamics?
AlphaLess wrote:
Sun Feb 24, 2019 9:01 pm
[...] One way to capture ALL sources of slippage is to do the following:
- obtain DAILY returns of the tracking product (NOT the tracking index, but the product). E.g., for TMF, that would be TLT,
- obtain DAILY returns of the leveraged product,
- run a regression:
lev_ret ~ a + b * track_ret + eps. [...]
lev_ret ~ a + b * track_ret + c * st_ret + eps [...]
Something like that would be basically curve fitting, which I am trying hard to avoid. And I am ready to bet that regression parameters identified with one type of fund would not work for another type of fund, or may not work under different historical conditions. Also, clearly, (daily repositioning) transaction costs are a function of (daily) volatility. I was hoping to find a heuristic with some solid rationale behind it... Ideas welcome.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Mon Feb 25, 2019 12:53 am

pezblanco wrote:
Wed Feb 20, 2019 10:40 am
I tried to get a reply in that thread but without success. I would appreciate it if you would answer some questions.
Hi there. I am sorry I didn't answer your questions in a timely manner, I have been away for a week without Internet access...
pezblanco wrote:
Wed Feb 20, 2019 10:40 am
1) I thought that there was some doubt about how dividends were handled in the fund? Your equation posits total return which would have dividends and then you lever those up. The workings of the fund is that some stocks are held (and thus dividends are present for some of the exposure but not for all). Is that truly conservative? That might be a systematic source of extra return in your backtests.
For bonds funds (LT treasuries or IT treasuries, 2x or 3x), the case is clear. Their annual report is very explicit that the index being followed is a Total Return index (i.e. dividends included) and the leverage applies to the whole thing. Plus when looking at daily numbers, besides a strange oddity with Direxion (e.g. TMF), the ProShares and ProFunds funds do match the daily leverage in quite a remarkable manner. Plus our simulation model works really quite well with those funds (this could be luck, admittedly!).

For stock funds (I looked at S&P 500 2x and 3x funds, and also UMPIX which tracks the S&P 400 MidCap), the case remains open in my opinion. On one hand, the annual reports refer to a price-only index. On the other hand, the leveraged funds distribute very few dividends (some years, none at all), so they clearly reinvest the dividends from the underlying assets in some manner. When looking at daily values (compared the index times leverage and the actual returns from the fund), it appears that *some* of the dividends are indeed leveraged, but it is hard to say if that's 100% (daily dividends being quite small for stocks). Finally, our current model (which assumes a total return tracking) systematically overshoots the reality by more or less 1% (which could be due to other reasons though, like transaction costs on a volatile asset). So... I am wary to draw a definitive conclusion here.
pezblanco wrote:
Wed Feb 20, 2019 10:40 am
2) So, if I understand your equation the "borrowing cost" is Libor*3 +ER which would be approximately: T-bill *3 + 1 = (2.4*3) +1 = 8.2 That seems large? Did I do that right? You can borrow money from IB for 3x leverage at 2(1.5 + Tbill) = 2(1.5 + 2.4) = 7.8? Maybe the return on cash ameliorates that somewhat.
I think you got proper clarification from the other folks, but the borrowing cost should be (leverage - 1) times LIBOR. For a 3x fund, this would be 2 times LIBOR. I checked the trajectory of T-Bills vs LIBOR and it isn't a good assumption to equate them. In my 1973+ simulations, I used the Effective Federal Funds Rate (kind of a US-only LIBOR) in the early years, then LIBOR (1 month, USD) when available.
pezblanco wrote:
Wed Feb 20, 2019 10:40 am
3) The backtests on PV only go back to 1985? (I rarely use it so I don't know). But the real return of stocks over that time period is something like 8.3%, which is well in excess of the long term return of stocks. Do you think that you might make your backtests too optimistic?
The PV backtests posted on HedgeFundie's thread did start in the mid-80s, and yeah, personally, I don't find satisfying to ignore the oil crisis. The numbers I personally assembled so far actually start in Jan-1973, but we still have to refine how to compensate for the lack of daily returns in the 70s/80s (we only have monthly values for bond indices), which we know leads to volatility decay. I'll work on that in the coming few days.
pezblanco wrote:
Wed Feb 20, 2019 10:40 am
4) Again a question about backtesting in PV ... it only allows at most quarterly rebalancing? So, you are adjusting the leverage quarterly instead of daily as the leveraged funds actually do? That might ameliorate a bit the volatility drag and thus make your backtests too optimistic?
Quarterly rebalancing is a consideration at the portfolio level (say with a TMF/UPRO combo), hence outside the scope of this thread which focuses on single (leveraged) funds. The daily rebalancing performed by the funds themselves is a completely different matter, which we discussed today in the past few posts. The impact of corresponding transaction costs remains very unclear.

Overall, I think we're making good progress, but I totally agree with the spirit of your post, multiple issues remain open, and I would NOT rush to any conclusion at this stage.

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Re: Simulating Returns of Leveraged ETFs

Post by HEDGEFUNDIE » Mon Feb 25, 2019 1:26 am

Interesting article suggesting taxes on dividends and coupon payments could be a cost driver in Total Return Swaps:

https://www.etf.com/sections/features-a ... nopaging=1

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Re: Simulating Returns of Leveraged ETFs

Post by HEDGEFUNDIE » Mon Feb 25, 2019 1:32 am

Another interesting article suggesting that “transaction costs” for swaps are relatively small (<5 bps).

https://www.cmegroup.com/trading/intere ... utures.pdf

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Re: Simulating Returns of Leveraged ETFs

Post by pezblanco » Mon Feb 25, 2019 9:30 am

siamond wrote:
Mon Feb 25, 2019 12:53 am
pezblanco wrote:
Wed Feb 20, 2019 10:40 am
I tried to get a reply in that thread but without success. I would appreciate it if you would answer some questions.
Hi there. I am sorry I didn't answer your questions in a timely manner, I have been away for a week without Internet access...
pezblanco wrote:
Wed Feb 20, 2019 10:40 am
1) I thought that there was some doubt about how dividends were handled in the fund? Your equation posits total return which would have dividends and then you lever those up. The workings of the fund is that some stocks are held (and thus dividends are present for some of the exposure but not for all). Is that truly conservative? That might be a systematic source of extra return in your backtests.
For bonds funds (LT treasuries or IT treasuries, 2x or 3x), the case is clear. Their annual report is very explicit that the index being followed is a Total Return index (i.e. dividends included) and the leverage applies to the whole thing. Plus when looking at daily numbers, besides a strange oddity with Direxion (e.g. TMF), the ProShares and ProFunds funds do match the daily leverage in quite a remarkable manner. Plus our simulation model works really quite well with those funds (this could be luck, admittedly!).

For stock funds (I looked at S&P 500 2x and 3x funds, and also UMPIX which tracks the S&P 400 MidCap), the case remains open in my opinion. On one hand, the annual reports refer to a price-only index. On the other hand, the leveraged funds distribute very few dividends (some years, none at all), so they clearly reinvest the dividends from the underlying assets in some manner. When looking at daily values (compared the index times leverage and the actual returns from the fund), it appears that *some* of the dividends are indeed leveraged, but it is hard to say if that's 100% (daily dividends being quite small for stocks). Finally, our current model (which assumes a total return tracking) systematically overshoots the reality by more or less 1% (which could be due to other reasons though, like transaction costs on a volatile asset). So... I am wary to draw a definitive conclusion here.
pezblanco wrote:
Wed Feb 20, 2019 10:40 am
2) So, if I understand your equation the "borrowing cost" is Libor*3 +ER which would be approximately: T-bill *3 + 1 = (2.4*3) +1 = 8.2 That seems large? Did I do that right? You can borrow money from IB for 3x leverage at 2(1.5 + Tbill) = 2(1.5 + 2.4) = 7.8? Maybe the return on cash ameliorates that somewhat.
I think you got proper clarification from the other folks, but the borrowing cost should be (leverage - 1) times LIBOR. For a 3x fund, this would be 2 times LIBOR. I checked the trajectory of T-Bills vs LIBOR and it isn't a good assumption to equate them. In my 1973+ simulations, I used the Effective Federal Funds Rate (kind of a US-only LIBOR) in the early years, then LIBOR (1 month, USD) when available.
pezblanco wrote:
Wed Feb 20, 2019 10:40 am
3) The backtests on PV only go back to 1985? (I rarely use it so I don't know). But the real return of stocks over that time period is something like 8.3%, which is well in excess of the long term return of stocks. Do you think that you might make your backtests too optimistic?
The PV backtests posted on HedgeFundie's thread did start in the mid-80s, and yeah, personally, I don't find satisfying to ignore the oil crisis. The numbers I personally assembled so far actually start in Jan-1973, but we still have to refine how to compensate for the lack of daily returns in the 70s/80s (we only have monthly values for bond indices), which we know leads to volatility decay. I'll work on that in the coming few days.
pezblanco wrote:
Wed Feb 20, 2019 10:40 am
4) Again a question about backtesting in PV ... it only allows at most quarterly rebalancing? So, you are adjusting the leverage quarterly instead of daily as the leveraged funds actually do? That might ameliorate a bit the volatility drag and thus make your backtests too optimistic?
Quarterly rebalancing is a consideration at the portfolio level (say with a TMF/UPRO combo), hence outside the scope of this thread which focuses on single (leveraged) funds. The daily rebalancing performed by the funds themselves is a completely different matter, which we discussed today in the past few posts. The impact of corresponding transaction costs remains very unclear.

Overall, I think we're making good progress, but I totally agree with the spirit of your post, multiple issues remain open, and I would NOT rush to any conclusion at this stage.


Very kind of you siamond, much appreciated.

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Re: Simulating Returns of Leveraged ETFs

Post by samsdad » Mon Feb 25, 2019 10:43 am

Here's a Barclay's paper with a bunch of math-looking stuff which might be helpful starting around half-way through. Dynamics of Leveraged and Inverse ETFs

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » 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 test against actuals, so if anybody has reasons to doubt the corresponding logic, please speak up.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Mon Feb 25, 2019 11:39 am

samsdad wrote:
Mon Feb 25, 2019 10:43 am
Here's a Barclay's paper with a bunch of math-looking stuff which might be helpful starting around half-way through. Dynamics of Leveraged and Inverse ETFs
I skimmed through it. It seems to me that they expressed simple things (daily compounding) with complicated equations, and this isn't adding much value (they even seem to completely miss the borrowing cost factor - except on slide 35, er, I think?). They briefly mentioned but didn't analyze transaction costs besides a rather lame regression test over 2 years (slide 36). Slide 39 provides some estimates about the spread, but they do not explain where this is coming from, the time period being studied, the logic being applied, etc.

What I found the most interesting was the estimate of total AUM (i.e. aggregate fund size, see slide 8 and 10, 4% of US assets in 2009, really?) and end-of-day 'rebalancing' transactions (see slide 26). It seems much bigger than I had perceived (and this analysis stops in 2009!), possibly reinforcing concerns about end-of-day market distortions due to hedge funds players and more (as AlphaLess suggested a few posts ago). Not a consideration we can quantify though, I'm afraid. This being said, I would be curious to see an AUM estimate projected until 2018 (in absolute and relative terms).

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Re: Simulating Returns of Leveraged ETFs

Post by LawnGnomeGenome » Mon Feb 25, 2019 12:13 pm

Hi all,

I've been reading this thread with great interest, coming over from HEDGEFUNDIE's thread. Thank you all for your great efforts and discussion!

I was just reading the LeveragedIndexer's blog post about performing backtests and they talk about TMF, which got me thinking about your efforts. I don't know what to do with the information, so I thought I'd pass it along here and maybe you can make more sense with it than me!

https://theleveragedindexer.com/2018/11 ... -backtest/

This particularly stuck out to me: "When performing my calculations for creating the TMF proxy, it turned out that a 6% close/94% adjusted close split with a 1% expense ratio made a great match."

Hope this helps :)

LGG

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Re: Simulating Returns of Leveraged ETFs

Post by EfficientInvestor » Mon Feb 25, 2019 12:35 pm

LawnGnomeGenome wrote:
Mon Feb 25, 2019 12:13 pm
Hi all,

I've been reading this thread with great interest, coming over from HEDGEFUNDIE's thread. Thank you all for your great efforts and discussion!

I was just reading the LeveragedIndexer's blog post about performing backtests and they talk about TMF, which got me thinking about your efforts. I don't know what to do with the information, so I thought I'd pass it along here and maybe you can make more sense with it than me!

https://theleveragedindexer.com/2018/11 ... -backtest/

This particularly stuck out to me: "When performing my calculations for creating the TMF proxy, it turned out that a 6% close/94% adjusted close split with a 1% expense ratio made a great match."

Hope this helps :)

LGG
The info at the leveraged indexer site needs to be updated based on the conversation going on in this thread. I just deactivated that page and will republish it once I get it re-written based on the conclusions of this research.

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Re: Simulating Returns of Leveraged ETFs

Post by LawnGnomeGenome » Mon Feb 25, 2019 1:31 pm

Ah, okay. I didn't realize it was your site. Good thing I didn't ask for the csv files.

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Re: Simulating Returns of Leveraged ETFs

Post by hdas » Mon Feb 25, 2019 5:41 pm

siamond wrote:
Sun Feb 24, 2019 11:00 pm
Ideas welcome.
Perhaps this article is useful.
Intraday Share Price Volatility and Leveraged ETF Rebalancing
49 Pages Posted: 13 Oct 2012 Last revised: 7 Jan 2016
Pauline Shum Nolan
York University - Schulich School of Business

Walid Hejazi
University of Toronto - Rotman School of Management

Edgar Haryanto
Independent

Arthur Rodier
Independent

Date Written: October 19, 2015

Abstract
Regulators and market participants are concerned about leveraged ETFs' role in driving up end-of-day volatility through hedging activities near the market's close. Leveraged ETF providers counter that the funds are too small to make a meaningful impact on volatility. For the period surrounding the financial crisis, 2006-2011, we show that end-of-day volatility was positively and statistically significantly correlated with the ratio of potential rebalancing trades to total trading volume. The impacts were not all economically significant, but largest during the most volatile days. Given the predictable pattern of leveraged ETF hedging demands, implications for predatory trading are explored.
"whenever there is a randomized way of doing something, then there is a nonrandomized way that delivers better performance but requires more thought" ET Jaynes

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Re: Simulating Returns of Leveraged ETFs

Post by AlphaLess » Mon Feb 25, 2019 9:58 pm

siamond wrote:
Sun Feb 24, 2019 11:00 pm
AlphaLess wrote:
Sun Feb 24, 2019 9:01 pm
In 2009 and 2010, when leveraged funds were new, a lot of 'scientific' papers were published about the underlying dynamics of leveraged funds.

One set of papers shed light in the following dimension:
- approaching the end of day, when the index tracked by leveraged fund is up a lot (or down a lot), at the end of the day, the fund has to chase the price, and buy high while paying transaction costs, if you will (in case of down a lot, it would have to be selling low). This causes further market impact, pushing the price in the same direction MORE. However, the impact would probably reverse a bit. I am not sure if this type of price chasing is still market impactful these days.
Given the small fund size of the leveraged ETFs, I find hard to believe that their orders at the end of the day would move the price needle in any significant manner creating some self-inflicted side effect. Unless various hedge funds play the leverage game at a much bigger scale, of course. This being said, I can easily perceive that such funds have quite the challenge at the end of day, liquidating their leveraged positions (do they really do it in full?), and rebalancing towards proper market exposure, while everything is in flux in the last few minutes of the trading day. Do you happen to have a couple of pointers towards the papers analyzing such dynamics?
AlphaLess wrote:
Sun Feb 24, 2019 9:01 pm
[...] One way to capture ALL sources of slippage is to do the following:
- obtain DAILY returns of the tracking product (NOT the tracking index, but the product). E.g., for TMF, that would be TLT,
- obtain DAILY returns of the leveraged product,
- run a regression:
lev_ret ~ a + b * track_ret + eps. [...]
lev_ret ~ a + b * track_ret + c * st_ret + eps [...]
Something like that would be basically curve fitting, which I am trying hard to avoid. And I am ready to bet that regression parameters identified with one type of fund would not work for another type of fund, or may not work under different historical conditions. Also, clearly, (daily repositioning) transaction costs are a function of (daily) volatility. I was hoping to find a heuristic with some solid rationale behind it... Ideas welcome.
In the present regime, leveraged ETFs have smaller size indeed. However, if you analyze those in 2008 / 2009, when these types of funds were new, punting on leverage was more popular, and market was much more volatile, then rest assured that volatility ETF purchases were causing market impact.

I can't emphasize enough just how high volatility was at the end of 2008.

As for the regression-based thing: that is not curve fitting. I guess, it does not matter what label you assign: it is what it is.
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Re: Simulating Returns of Leveraged ETFs

Post by AlphaLess » Mon Feb 25, 2019 10:02 pm

The second type of research on volatility ETFs had to do with gamma effects.

Gamma effects refer to non-linear effects of price to price movement.

A 1x ETF / fund (like VTI) has no gamma.
A call or put option on VTI has gamma.
And a leveraged ETF (say SSO, UPRO) does also have gamma.

Avellaneda is one of the better researchers of market structure topics:

https://www.math.nyu.edu/faculty/avella ... SKPROF.pdf

Here is the relevant quote:

`The effect of daily rebalancing
for a bullish LETF is to buy
stocks when the NAV goes up and
to sell them when it goes down.
This means that the manager is
“short Gamma.” `
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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Tue Feb 26, 2019 10:37 am

AlphaLess wrote:
Mon Feb 25, 2019 9:58 pm
In the present regime, leveraged ETFs have smaller size indeed. However, if you analyze those in 2008 / 2009, when these types of funds were new, punting on leverage was more popular, and market was much more volatile, then rest assured that volatility ETF purchases were causing market impact. I can't emphasize enough just how high volatility was at the end of 2008.
Thank you for this interesting perspective, I wasn't aware of those swings of popularity, good to know. Yes, if leveraged investments made for 4% to 6% of the total market size back then, then I do see your point that the end-of-the-day impact was absolutely significant. This made me think that I could easily extract fund sizes (market caps) historical data from Morningstar and chart them. The trouble is the 'popularity' factor is probably drowned out by the swings of the market, compounded by leveraging. Anyhoo, for whatever it's worth, here it is (click on the image for a full-screen display).

Image

Note that I found a couple of Guggenheim funds with a long life, which will be handy for validating our modeling endeavor. LC means large-caps (i.e. S&P 500), LTT means Long-Term Treasuries.

Code: Select all

LC 1.5x		LC 2x		Russell 2000 1.5x	LTT 1.2x
7/31/93		6/30/00		2/28/01			1/31/94
RYNVX		RYTTX		RYAKX			RYGBX
EDIT: I just realized that if we divide those fund sizes by the size of the corresponding market (or maybe size times leverage), then we should better tease out the popularity swings. Meh, that's too much work for now... Maybe another day! :wink:

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siamond
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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Tue Feb 26, 2019 6:58 pm

HEDGEFUNDIE wrote:
Mon Feb 25, 2019 1:32 am
Another interesting article suggesting that “transaction costs” for swaps are relatively small (<5 bps).

https://www.cmegroup.com/trading/intere ... utures.pdf
Hm. This article goes a bit above my head... Could you please identify more precisely what section made you reach this conclusion?

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Re: Simulating Returns of Leveraged ETFs

Post by HEDGEFUNDIE » Tue Feb 26, 2019 8:12 pm

siamond wrote:
Tue Feb 26, 2019 6:58 pm
HEDGEFUNDIE wrote:
Mon Feb 25, 2019 1:32 am
Another interesting article suggesting that “transaction costs” for swaps are relatively small (<5 bps).

https://www.cmegroup.com/trading/intere ... utures.pdf
Hm. This article goes a bit above my head... Could you please identify more precisely what section made you reach this conclusion?
Page 13 of the report.

Here are the cost buckets:
  • Liquidity: Defined broadly as the bid-ask spread for the given instrument
  • Initial Margin and Funding Costs: Includes the amount of initial margin that must be posted by product and the cost of funding that initial margin
  • FCM Fees: Execution fees, clearing fees and capital usage fees charged by the futures commission merchant
  • CCP Fees: Exchange, execution and clearing fees charged by the relevant clearinghouse

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siamond
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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Tue Feb 26, 2019 9:48 pm

HEDGEFUNDIE wrote:
Tue Feb 26, 2019 8:12 pm
siamond wrote:
Tue Feb 26, 2019 6:58 pm
HEDGEFUNDIE wrote:
Mon Feb 25, 2019 1:32 am
Another interesting article suggesting that “transaction costs” for swaps are relatively small (<5 bps).

https://www.cmegroup.com/trading/intere ... utures.pdf
Hm. This article goes a bit above my head... Could you please identify more precisely what section made you reach this conclusion?
Page 13 of the report.

Here are the cost buckets:
  • Liquidity: Defined broadly as the bid-ask spread for the given instrument
  • Initial Margin and Funding Costs: Includes the amount of initial margin that must be posted by product and the cost of funding that initial margin
  • FCM Fees: Execution fees, clearing fees and capital usage fees charged by the futures commission merchant
  • CCP Fees: Exchange, execution and clearing fees charged by the relevant clearinghouse
Sorry to be slow, but how are such numbers about transactions like '2-Year Position Held for 2 Years' or '5-Year Position Held for 2 Years' relevant to our daily leveraging situation? And how do you compute 5 bps? Could you please elaborate?

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Re: Simulating Returns of Leveraged ETFs

Post by HEDGEFUNDIE » Tue Feb 26, 2019 11:17 pm

siamond wrote:
Tue Feb 26, 2019 9:48 pm
HEDGEFUNDIE wrote:
Tue Feb 26, 2019 8:12 pm
siamond wrote:
Tue Feb 26, 2019 6:58 pm
HEDGEFUNDIE wrote:
Mon Feb 25, 2019 1:32 am
Another interesting article suggesting that “transaction costs” for swaps are relatively small (<5 bps).

https://www.cmegroup.com/trading/intere ... utures.pdf
Hm. This article goes a bit above my head... Could you please identify more precisely what section made you reach this conclusion?
Page 13 of the report.

Here are the cost buckets:
  • Liquidity: Defined broadly as the bid-ask spread for the given instrument
  • Initial Margin and Funding Costs: Includes the amount of initial margin that must be posted by product and the cost of funding that initial margin
  • FCM Fees: Execution fees, clearing fees and capital usage fees charged by the futures commission merchant
  • CCP Fees: Exchange, execution and clearing fees charged by the relevant clearinghouse
Sorry to be slow, but how are such numbers about transactions like '2-Year Position Held for 2 Years' or '5-Year Position Held for 2 Years' relevant to our daily leveraging situation? And how do you compute 5 bps? Could you please elaborate?
The relevant chart is the “30-year position held for 2 years”.

For TMF we are talking 20-30 year bonds. And the annual report shows TMF’s swaps have termination dates 8-12 months out:

http://direxioninvestments.onlineprospe ... 3X-SAR.pdf

The segments of the bar to take into account are CCP, FCM, and liquidity (margin is already accounted for our model).

Using the numbers for Swap, I calculate (113k + 36k + 13k) / 2 years / $100M notional value = approximately 8 bps.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Wed Feb 27, 2019 9:14 am

HEDGEFUNDIE wrote:
Tue Feb 26, 2019 11:17 pm
The relevant chart is the “30-year position held for 2 years”.

For TMF we are talking 20-30 year bonds. And the annual report shows TMF’s swaps have termination dates 8-12 months out:

http://direxioninvestments.onlineprospe ... 3X-SAR.pdf

The segments of the bar to take into account are CCP, FCM, and liquidity (margin is already accounted for our model).

Using the numbers for Swap, I calculate (113k + 36k + 13k) / 2 years / $100M notional value = approximately 8 bps.
Ok, I see what you mean now, thanks for clarifying. It still seems very doubtful to me that the numbers used for a 2 years agreement would be directly applicable to one-day agreements by simply dividing by the 2 years duration. Still, at the intuitive level, I suspect you're probably right that those swap costs aren't terribly significant for our purposes. And I'm guessing that our use of the LIBOR/1-month instead of the LIBOR/1-week data series more or less counterbalances those extra costs (not true for the EFF rates though, as this is overnight). I just did a quick test, and over the past 20 years, LIBOR/1-month was 5bps higher than LIBOR/1-week.

Those specific swap costs (spread and fees) are clearly different from the 'daily repositioning (rebalancing)' costs we previously identified (the fund adjusting its market exposure at the end of the day towards its target) though. I was more focusing on the latter, as this seems potentially more significant, although... I have no clue, to be honest!

It is really unfortunate that leveraged funds annual reports do not seem to quantify whatsoever all those extra costs. I mean, come on, what kind of accounting/reporting is that? :shock:

At the end, as wary as I am to use any form of regression logic (every time I tried something like that, it failed miserably out of sample), AlphaLess is probably right that we'll have to introduce some sort of fudge factor to weigh down the model and make it closer to known actuals. And this will address all those additional costs we have troubles to quantify. But that's really a last resort when modeling things like that.

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Re: Simulating Returns of Leveraged ETFs

Post by hdas » Wed Feb 27, 2019 9:47 am

siamond wrote:
Wed Feb 27, 2019 9:14 am
At the end, as wary as I am to use any form of regression logic (every time I tried something like that, it failed miserably out of sample), AlphaLess is probably right that we'll have to introduce some sort of fudge factor to weigh down the model and make it closer to known actuals. And this will address all those additional costs we have troubles to quantify. But that's really a last resort when modeling things like that.
Seems you are likely to be stuck in this effort to quantify cost of rebalancing due to the granularity of the data you have at your disposal. After a cursory look to this article I believe the potential model will have to use intraday volatility, specially afternoon volatility. Perhaps daily volatility would approximate something close enough but I doubt it.

To help you think about this model, look at the beautiful reversal we had on December 27th starting at about 13:30 central. :greedy
"whenever there is a randomized way of doing something, then there is a nonrandomized way that delivers better performance but requires more thought" ET Jaynes

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Wed Feb 27, 2019 1:02 pm

hdas wrote:
Wed Feb 27, 2019 9:47 am
Seems you are likely to be stuck in this effort to quantify cost of rebalancing due to the granularity of the data you have at your disposal. After a cursory look to this article I believe the potential model will have to use intraday volatility, specially afternoon volatility. Perhaps daily volatility would approximate something close enough but I doubt it.
I registered (this is free), but I still don't have access to the article? I guess you have an academic account of sorts?

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Wed Feb 27, 2019 1:39 pm

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 test against actuals, so if anybody has reasons to doubt the corresponding logic, please speak up.
Back to this. I had a change of heart. I continue to tend to agree that we shouldn't account for borrowing costs during non-trading days (e.g. week-ends). BUT. The Effective Federal Funds Rate is really an overnight rate (check here). When comparing the EFFR (week-ends included) to LIBOR/1m (no week-ends) for the 1986+ time period, EFFR is already -unsurprisingly- lower, although not by a lot, so it appears to be an ok proxy for LIBOR. Then, if we build a pseudo-EFFR (no week-ends), the new data series becomes way lower (roughly 1% annualized CAGR lower), hence a really poor proxy for LIBOR.

Image

I am keenly aware of the fact that it isn't a good practice of try to balance out one error factor with another (fairly independent) one, but in this case, it seems to me that we would do something unduly optimistic by further reducing the EFFR numbers for the pre-LIBOR days, notably at a time of high interest rates (e.g. 70s). So I decided to disable the week-end (and holidays) adjustment for now. Feedback welcome.

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Re: Simulating Returns of Leveraged ETFs

Post by siamond » Wed Feb 27, 2019 8:34 pm

Next topic... The entire point is to assemble credible data series starting from the 70s, or better, from the mid-50s. There is one challenge though, we know that the returns of leveraged funds are HIGHLY sensitive to daily volatility, but... we don't have daily data for the major indices going back in time enough, only monthly. What we have:
- EFFR/LIBOR: daily data available since the mid-50s
- S&P 500 index: daily prices back to the mid-50s (and earlier)
- S&P 500 index: monthly total returns back to the mid-50s (or earlier) and up to the late 80s
- S&P 500 index: daily total returns after that
- LT Bonds AlohaJoe's model: monthly total returns back to the mid-50s (and earlier)
- LT Bonds index: monthly total returns back to the early 70s and up to the late 90s (yeah, this sucks, and no price only data is available by then)
- LT Bonds index: daily total returns after that
- IT Bonds: similar situation

For the S&P 500, I teased out the (monthly) dividends, then combined with the daily price series, applying the dividends to the last day of the month, and this gave me a pretty good total return data series with the right CAGR and proper daily volatility. I ran a sanity check and the aggregate daily volatility of this resulting data series is basically equal to the aggregate daily volatility of the index in modern times. So I think this is reasonable given that volatility of stocks is primarily due to price variations. I don't think I need to try to find a (probably dubious) way to distribute the dividends intra-month.

For the bonds, I checked the modern times, and total returns do vary on a daily basis (either due to dividends of underlying components or price variations). And we know that LT bonds can vary a good deal, so we need an answer. I 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?

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Re: Simulating Returns of Leveraged ETFs

Post by interestediniras » Wed Feb 27, 2019 8:42 pm

siamond wrote:
Wed Feb 27, 2019 8:34 pm
I 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

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