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 *s ^{2}*.

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.