# The math (and myths) of leveraged ETFs

April 20, 2009 in

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.

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