Understanding Levered ETFs and Geometric Returns 16 comments
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A lot of authors on Seeking Alpha have noted that levered ETFs don’t behave as some investors intended to due to compounding effects. The levered or inverse products do what they say over a period of a single day, but not over a longer time-period. Recent examples are the post by Paul Kedrosky, where both levered bullish and bearish energy ETFs were negative simultaneously and another post by Matthew McCall, where he noted both a financials ETF and a 2x inverse financial ETF lost over the same period.
To understand better, assume the ETF asset follows a Geometric Brownian Motion (the same assumption used to derive Black-Scholes formula for options). The assumption is not perfect, but not far from reality.
In addition, when we talk of returns, we implicitly refer to geometric returns (since that’s what an investor eventually realizes). For example, we might say that the S&P 500 returned “-30%” in 2007; that “-30%” is a geometrically compounded return. In terms of arithmetic and geometric returns, the effective nature of 2x and 3x ETFs is that they lever up arithmetic returns, but as investors we realize the corresponding geometric returns.
A key mathematical identity is Ito’s Lemma. It helps to link Arithmetic Average (or Sums) with a Geometric Average (or Sums). Without going into details, an ordinary application of the Lemma shows:
which in simple terms means: if the “expected” arithmetic average of returns is r, then the “expected” geometric average of returns will be
where s is the volatility.
We can use Ito’s Lemma to back out the expected geometric returns for the levered and inverse ETFs. Let R be the expected geometric returns of the basic ETF
I give rest of the results here in the table below (click to enlarge):
The results can be interpreted as follows: if the expected return on an asset is R, then the expected return on a 2x levered asset will be lower than 2R by a term equal to s2.
Here is an example, with annual return as 9% and volatility as 15% (roughly similar to the long term historic characteristics of S&P 500).
Some points to be noted:
- Positive levered products will return less than expected.
- Negative levered products will return more negative than expected.
- The 1x inverse and the 2x levered product are the most likely ones to come close to the intended outcome, while the 3x inverse levered product is most likely to stray quite far from the expected outcome.
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This article has 16 comments:
I wonder if the theory doesn't work out in reality because of the phenemonally high costs of long option and swap erosion in the negative levered ETFs.
2008 August through December 19th, the S&P 500 has fallen 30.8%. Many traders would blindly assume that SDS should have returned 61.6% because that is double the inverse of the index, right? Oddly enough, SDS only gained 37.3%, nearly 25% below what many would have thought in less than four months. The ProShares Short S&P 500 ETF (SH) during the same timeframe gained 25.9%; according to the company SH should give investors the inverse performance on a 1-to-1 daily basis.
claruspartners.com
Looking at SSO and SDS (S&P 500 2x long and short), the average daily sum of their price change percents (absolute value of (SSO % plus SDS %)) is 0.42%. It should be zero -- if SSO goes up x%, SDS should go down x%, or vice versa. This is daily data, which is what the ETFs claim to track. The largest discrepancy was on 9/24/08, when SDS went up 0.48% and SDS went down 5.96%.
That's bad enough, but now take a look at UYG and SKF (Financials 2x long and short). The daily (opposite) percent change for these two funds differed by an average of 0.89%, with a maximum of 7.70% (9/22/08: UYG -9.70%, SKF +2.00%). The daily % change absolute sum for these two ETFs exceeded 1% on 8 trading days in November, 18 days in October, and 7 in September. Differences exceeding 2% are common in these months. Lower-volatility months seem to have less of this, but it appears in much of the year (8 times in January, 7 in July).
So even on a daily basis, these ETFs do not reliably do what they claim. How much of their overall bad performance is due to this and how much to geometric compounding is debatable, but the bottom line is that they bleed money if you hold them very long.
Examples, 1/1-12/29/08:
China: FXI and FXP are both down 50%.
Oil industry: DUG and DIG are BOTH down, 25% and 75% respectively.
Financials: UYG is down 85%, SKF is up 25%.
Real estate: URE is down 80%, SRS is DOWN 45%.
DOG (ProShares 1x short Dow 30) is up 20%, DXD (same except 2x) is only up 15%. The single-leverage did better than the double!
These are not suitable vehicles for long-term trend bets. You can chart pairs and watch their cumulative intersection points sink farther and farther below 0%, even in 5 or 10 day charts. These ETFs are dangerous. Handle with care, if at all.
This is Russian roulette with 5 out of 6 chambers loaded.
On Jan 04 10:43 PM squark62 wrote:
> like Luck-of-the-Irish, i too have benefited from quantitatively
> pairing short and long etfs to form a delta neutral hedge and profit
> during sudden down-turns in the market. does hamiltonian ring a bell?
On 9/24/08 SSO and SDS went exDividend. SDS distributed 6% of NAV as capital gains, while SSO distributed 0.45% of NAV. This would account for the wide discrepancy in their daily return on that day.
Also, SDS is currently trading at a premium of 7.27% and SSO is trading at a discount of 6.74%. You need to take this into account when comparing their daily returns.
I hope this helps.
On Jan 05 07:27 PM igggy wrote:
> I would like to know what happens with a 3x ETF when the underlying
> index goes 33.3% against it. Does the ETF just go to zero and die?
> Obviously, the same question applied to 2x ETFs if the index goes
> 50% the other way. I know it's a far fetched scenario but I wonder.
That does not happen. I wonder what else is missing from the formulas.
/m
/m