Leveraged ETFs are vehicles which provide non-recourse leverage on various sectors or strategies. For example, every day the double-inverse financials SKF returns roughly -2 times the daily return of the DJ Financials index.

These products are a favorite of mine not simply in a speculative framework, but in a quantitative one. Many people make the mistake (and it can be a serious one) of assuming that a double-levered ETF should return twice as much as its underlying index *over a given holding period*. That's incorrect - the ETF's only return their stated multiple *for one day.* After that, the ETF has to relever itself in order to maintain it's mission. This is because the ETFs typically acquire leverage through the use of total return swaps (TRS). A total return swap is a swap in which the two counterparties agree to exchange the exact same cashflows as if they had traded a security. For example, counterparties A and B enter into a TRS on some stock Z, struck at $100. A pays B $1 for every dollar Z goes above $100, and vice versa for every dollar under. It is exactly the same as if B bought Z from A, except that no capital had to be put up (thus, it is a levered trade). The important thing to note is that each unit of a TRS provides a dollar return, not a percent return. Owning twice as much TRS means you make $2 for every $1 the stock goes up. Therefore, the ETF must rebalance to maintain it's constant multiplicative exposure.

Here's an example. The underlying index X starts at $100. My double levered ETF E also starts at 100 and because it is double levered, it must own twice as much TRS as it would hold stock in the underlier. Let's say E has $100 in assets, so it owns 200 units of TRS. On the first day, X increases 10% to 110 as each share gains $1. E increases 20% to $120, as expected, since each TRS gains $1 as well. The following day, X increases another 10% to 121 (each share gains $1.10), and we expect E to grow 20% to 144 ($24 gain). However, E only owns 200 units of TRS, which means it will earn only $22 dollars as each TRS gains $1.10. In order to realize a $24 gain and return its target 20%, E would have to purchase 20 more TRS at the end of the previous trading day - and that act is the relevering.

It is especially interesting to note that E must re-lever in the direction the underlier moved *regardless of whether the ETF is long or short*. Above, the underlier kept increasing in value - and E had to purchase more and more to maintain its leverage ratio. Conversely, had the underlier fallen, E would have been selling into the decline. As levered ETFs attract more and more assets, this end-of-day relevering can have market impact, enhances gains and exaggerating losses.

The second key fact about levered ETFs is that they exhibit a strong downward drift, moreso as the leverage increases. This is why if you plot an index against it's triple-levered inverse ETF, *both can decline in value over time* despite the ETF's mission of returning "opposite" results. To see why, consider what happens when an index moves up 10% and then down 10%: it doesn't go back to where it started, it actually loses 1% (order of the moves does not matter, either). It goes from 100 to 110 to 99, or if you prefer from 100 to 90 to 99. Either way, 10% up and 10% down is not an even trade.

Now lever it 3x. Before, you lost 1%. Now, shouldn't you lose 3%? 10% up and 10% down become 30% up and 30% down, or 100 to 130 to 91, for a loss of 9%. You didn't lose 3 times as much, you lost 3-squared times as much. And we can easily extend this to a case where the underlier is up but the ETF is down: 10% up and 8% down, which results in a 1.2% gain for the underlier, but a 1.2% loss for the ETF. More complicated examples are easy to construct, and here is a real-world one of the SKF vs its underlier, the DJ Financials index. The index is off 66%, while the double inverse ETF is off 16%:

And if this discussion hasn't been fascinating enough, Barclays recently put out a recearch piece delving much further into the math behind these elusive securities and exploring the impact of the relevering process.