# How to calculate compound returns of leveraged ETFs?

Forewarning: this is a complete newbie question :-)

I am starting to learn about ETFs by trying to do the numbers. When learning about the compounding effect in leveraged ETFs, I wanted to simulate the return for a simple ETF. Here is what I wanted to do: a hypothetical index ETF gains 1% every day for a 10-day range. I wanted to calculate the final return after the range for 1X, -1X, 3X and -3X ETF. Here are the numbers: That means, at the end of 10 days, 1X gained 10.5%, 3X gained 34.4%, -1X lost 10.6% and -3X lost 26.3%. Two questions:

1. Is the above statement/calculations correct, specifically w.r.t. -1X and -3X ETFs?
2. Now if I consider a scenario where the 1X ETF loses 1% every day for 10 days, then can I say the following without additional calculations: at the end of 10 days, 1X lost 10.6%, 3X lost 26.3%, -1X gained 10.5% and -3X gained 34.4%?

Thanks for helping.

• Hi Samik, welcome to quant.SE and thanks for posting your question. I think the answer to your question is "yes." Are you looking for anything more than that? This site generally strives for something more than simple yes/no questions. – Tal Fishman Sep 26 '11 at 16:19
• @Tal: Thanks for the welcome. I am getting started on quant simulations, specially focusing on leveraged ETFs. This was a beginner level question, and hopefully, as I advance my knowledge, I will have more advanced ones to ask. – Samik R Sep 26 '11 at 17:27

1) Yes, the above table is correct

2) Your results are correct except..... 1X loss = 9.6%.

When you combine both positive and negative changes, it is the MEDIAN value that is of interest. Here are some links:

http://olympiainv.com/Memos/ETFs.pdf

http://math.nyu.edu/faculty/avellane/LeveragedETF20090515.pdf

http://www.slcg.com/pdf/workingpapers/Leveraged%20ETFs,%20Holding%20Periods%20and%20Investment%20Shortfalls.pdf

http://math.nyu.edu/faculty/avellane/LETFRISKPROF.pdf

• Ah .. the silliness regarding 9.6%. Thanks for the pointers to the articles. I knew 2 of them, the remaining are great too. – Samik R Sep 26 '11 at 17:49

Your example could be correct but you're on the wrong track.

Leveraged ETFs are designed for day trading, it isn't a leveraged 3x position that will return 3x the long term average of the name. The leverage is reweighted each day which will affect your performance.

Eg if the market goes 100->99->100 the market is unchanged over 2 days. But a 3x ETF will go 100->97->99.9394 ((1 + (100/99 - 1) * 3) * 97)

• Hi Richard, welcome to quant.SE. I think @Samik is already aware of this fact and is trying to gain a better intuition for it with some experiments. – Tal Fishman Sep 26 '11 at 22:25
• @Richard: Yes, as Tal correctly points out, I am already aware of this fact. But thanks for stressing this again - I guess that is valuable. But, I would say that, contrary to popular belief, leveraged ETF can be held for more than one days with careful monitoring, as has been showed by quite a few academic papers (see for example the 3rd link in bill_080 above). – Samik R Sep 27 '11 at 23:23

Even in a perfect world, a 3X leveraged ETF cannot achieve a compound return three times that of the underlying. In the case of periodic discrete rebalancing, we call this effect the "arithmetic of loss and recovery," but even in the limit of continuous rebalancing, this effect does not disappear. Ito's formula tells us that

$$\mathrm d \log(S_\textrm{Index})=\sigma \mathrm d B +\left(\mu-\dfrac 1 2 \sigma^2\right).$$

Assuming continuous rebalancing, and zero borrowing costs and other expenses, $$\mathrm d \log(S_\textrm{3X Bull})= 3\sigma\mathrm d B + \left(3\mu-\dfrac 9 2 \sigma^2 \right).$$ Therefore the compound performance of a "3X Bull" ETF that "perfectly" tracks its index must be less than triple that of the underlying index by $3\sigma^2,$ where $\sigma^2$ is the variance (here equal to the quadratic variation per unit time interval) of the price process of the underlying index.

For example, if $\sigma=15\%/\textrm{yr}^{1/2}$, which is within the order of magnitude of that assumed for a general stock market index, then we are talking about a rather unavoidable performance shortfall (due strictly to the mathematics of volatility) of $3\sigma^2=6.75\%/\textrm{yr}$ in logarithmic terms for a triple-leveraged ETF.

• How do you reconcile this performance shortfall with the 3x outperforming three times the 1x return in column C12 of the original question's spreadsheet? – Thomas Johnson May 14 '13 at 18:37

Yes, your table is correct... the proverbial "catch" is in your assumptions of small gains, with nil volatility. Because volatility is itself the catch with levered strategies in general (and levered ETFs very specifically).

Replicate these 1% returns with a 14.14% standard normal deviation, for a thousand, million, billion runs. Your 1% compound return will (or certainly should) go to zero.

Because if the stockmarket went up or down by 50% every day at random, there's a 25% chance of a 75% loss (0.50.5=0.25), a 50% chance of a 25% loss (1.50.5=0.75) and a 25% chance of a 125% gain (1.5*1.5=2.25). Net net, you'd expect to lose 25% on average every period... putting in "up or down 1% or 2% or 5%" are just milder expresssions of the same basic mathematical phenomenon.

If the arithmetic return is X with Y volatility, then the CAGR (assuming normally distributed returns) would be X - 0.5 * Y ^2.

And if you want to play this game with levered ETFs levered L to the underlying, it's: CAGR = L.X - 0.5.L.(L-1)*Y^2.

So for higher volatility assets, both the levered long and the levered short ETFs will both lose money. I know. Back in the day, I was playing this game with VIX, goldminers, and Natural Gas; back when it possible to get the borrow to short these bad boys in both directions! Sadly no more cheap lunches there these days ;-(

As a next step in your analysis, I would maybe look at the simulated difference in (1) returns from levered ETFs; versus (2) just generating some real leverage, by borrowing some real money and investing that; versus (3) just buying futures, if you want the same leverage. I've struggled to come up with ANY scenario where (1) is optimal. If you love (or hate) the asset in question that much, the future is nearly always preferable to the levered ETF...

• Thanks - agree with your last paragraph. I sort of intuitively understood the same thing, and never really dabbled into leveraged ETF - I am not a day-trader, and the posts here show that it wouldn't have worked well for me. – Samik R Sep 17 at 5:59

For a leveraged ETF, with a a leverage of $$L$$, then the value of the ETF is:

$$\mathrm{ETF}_{t_n} = \mathrm{ETF}_{t_0} \cdot \Pi_{i=1}^{i=n} \left[ 1+L\left(\frac{S_{t_i}}{S_{t_{i-1}}}-1\right) - f \cdot \mathrm{DCF}(t_{i-1}, t_i)\right]$$

where $$t_i$$ are the dates on which the ETF rebalances to restore the leverage. $$f$$ is the ETF management fee, and $$\mathrm{DCF}(t_{i-1}, t_i)$$ is the day count fraction used in the calcualtion of the fee for the ETF.