## The dark side of leveraged ETFs

Mon 17 April 2017 Author: Salim Category: Articles
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Leveraged ETFs (LETFs) are ETFs meant to magnify the returns of an index or of another ETF. For instance, a 2x SPY ETF aims at providing, each day, 2x the returns of SPY. Inverse ETFs (also called bear ETFs) are leveraged ETFs where the leverage multiplier is negative: they aim at yielding -1x, -2x or -3x the (daily) returns of their reference index. Note we consider here a factor 1 inverse ETF (-1x) as a “leveraged ETF” as well.

To achieve leveraged returns, the ETF manager resorts to the use of derivatives, most often total return equity swaps.

The central point with LETFs is that they aim at leveraging daily returns with a constant leverage, which in turn produces some non-intuitive nasty effects which we’ll try to describe in this article.

#### An example

Let’s take an imaginary example to see the effects at work: take an index, valued at $100, and a 2x LETF based on it, also valued at$100; assume for simplicity the existence of a futures on this index with a contract size of 1, and tradable fractional quantities; also assume zero interest rates and zero dividends on the index, so that the futures price is, at any time, the same as the spot price.

1. On day 1, the ETF is worth $100, the LETF is worth$100 and the fund manager buys 2 futures to target the specified leveraged return.
2. On day 2, assume the ETF moves to $120, a 20% increase; the profit on the futures position is$40, the LETF is now worth $140, a 40% increase, twice the return on the index, as expected. Now the fund manager must buy 0.33 futures more to hold a total of 2.33 futures to target the leveraged return (2 x 140 / 120). 3. On day 3, assume the ETF drops back to$100 (a -16.66% return from its value on day 2), the futures position yields a $46.6 loss (-20 x 2.33), so the LETF value is now$93.4, a -33.3% drop from its value on day 2, again twice the return of the ETF, as expected. Now the fund manager has to sell 0.46 futures to hold a total position of 1.87 futures to make sure he’ll get twice the ETF return on the next day.

#### Rebalancing

You can see the crucial mechanism at work in our example: rebalancing. The fund manager needs to rebalance his position everyday to maintain the leverage ratio constant. When the LETF price goes up, he must buy more futures, when the LETF goes down, he must sell some. He always buy high, sell low. It is the same with an inverse LETF: when the LETF goes up (because the index goes down), he must short more futures, when the LETF goes down (index goes up), he must cover some. He shorts low, covers high. So rebalancing has 2 negative effects:

• Time decay of the LETF value when the underlying index moves a lot, often changing directions. In our example, on day 3, the index is back at its initial value of $100, while the LETF is now worth only$93.4. This decay is due to constantly buying high, selling low.
• Trading costs, which may be substantial.

These unpleasant effects that erode the value of LETFs over time are collectively known as the “constant leverage trap”. It should be stressed that this trap is at play even for simple inverse (-1x) ETFs.

Notice the similarity between dynamic rebalancing here and dynamic hedging of a short option position. The option seller pays back the premium when hedging (in theory) by systematically buying high and selling low. He is short gamma. By analogy, we could say a long LETF position is short gamma.

#### The quant corner

Assume zero interest rates. Let $S_t$ be the underlying price, $L_t$ the LETF price, with a leverage multiplier of $\beta$. Let $r_t$ be the return of the underlying between $t-1$ and $t$, $c$ the percentage trading costs for 1 day. We follow Avellaneda.

Take the logarithm of each equation, multiply the second one by $\beta$ and subtract the 2 equations: Using and assuming $cr_t \approx 0$ and $c^2 \approx 0$ (typical values are indeed $r_t = 0.5\%$ while $c=0.01\%$), we get Hence You may recognize as the realized variance of the index over the period. Denote the index volatility as $\sigma$, we can rewrite the above as

We can see the 2 value eroding effects: trading costs $cT$ and “gamma” costs $\frac{1}{2}(\beta^2-\beta)\sigma^2T$ (which are proportional to variance).

Note $\beta^2-\beta \geq 0$ when $|\beta| \geq 1$ which is the case for all leveraged ETFs. This quantity is higher for bear LETFs than bull LETFS with the opposite leverage multiplier (so their gamma costs are higher).

#### Conclusion

LETFs are not suited for long-term investing. Simply do not use LETFs to access leverage over the long run. Actually, it is in theory possible to use a dynamic strategy on an LETF that targets a given long-term leveraged return, but we strongly advise against trying this by yourself.