# Phorgy Phynance

## Leveraged ETFs: Selling vs Hedging

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In this brief note, we’ll compare two similar leveraged ETF strategies. We begin by assuming a portfolio consists of an $x$-times leveraged bull ETF with daily return given by

$R_{\text{Long}} = x R_{\text{Index}} - R_{\text{Fee}},$

where $R_{\text{Fee}}$ is the fee charged by the manager and some cash equivalent with daily return $R_{\text{Cash}}$. The daily portfolio return is given by

\begin{aligned} R_{\text{Portfolio}} &= w_{\text{Long}} R_{\text{Long}} + w_{\text{Cash}} R_{\text{Cash}} \\ &= w_{\text{Long}} \left(x R_{\text{Index}} - R_{\text{Fee}}\right) + w_{\text{Cash}} R_{\text{Cash}}.\end{aligned}

We wish to reduce our exposure to the index.

### Strategy 1

An obvious thing to do to reduce exposure is to sell some shares of the leveraged ETF. In this case, the weight of the ETF is reduced by $\Delta w$ and the weight of cash increases by $\Delta w$. The daily portfolio return is then

$R_{\text{Strategy 1}} = R_{\text{Portfolio}} + \Delta w \left(-x R_{\text{Index}} + R_{\text{Fee}} + R_{\text{Cash}}\right).$

### Strategy 2

Another way to reduce exposure is to buy shares in the leveraged bear ETF. The daily return of the bear ETF is

$R_{\text{Short}} = -x R_{\text{Index}} - R_{\text{Fee}}.$

The daily return of this strategy is

$R_{\text{Strategy 2}} = R_{\text{Portfolio}} + \Delta w \left(-x R_{\text{Index}} - R_{\text{Fee}} - R_{\text{Cash}}\right).$

### Comparison

For most, I think it should be fairly obvious that Strategy 1 is preferred. However, I occasionally come across people with positions in both the bear and bull ETFs. The difference in the daily return of the two strategies is given by

$\Delta R = 2\left(R_{\text{Fee}} + R_{\text{Cash}}\right).$

In other words, if you reduce exposure by buying the bull ETF, you’ll get hit both by fees as well as lost return on your cash equivalent.

Unless you’ve got some interesting derivatives strategy (I’d love to hear about), I recommend not holding both the bear and bull ETFs simultaneously.

Note: I remain long BGU (which is now SPXL) at a cost of US$36 as a long-term investment – despite experts warning against holding these things. It closed yesterday at US$90.92.

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Written by Eric

October 2, 2012 at 3:24 pm