# Sharpe ratio with leveraged ETFs

There has been a discussion about how leverage affects Sharpe Ratios, but not in the context of leveraged ETFs (such as 2x or 3x).

I'm just wondering how leveraged ETFs, if at all, change the conclusions reached.

• Welcome to Quant SE. Could you please give a source for "There has been a discussion..." - Thank you. Dec 25, 2015 at 9:33

## 1 Answer

Probably missing something here but if $X$ has $E(X) = \mu$ and $variance(X) = \sigma^2$ then $2X$ has $E(2X) = 2 \mu, variance(2X) = 4\sigma^2$. Thus the sharp ratio defined as $\frac{\mu}{\sigma}$ stays the same for the 2x leveraged and the regular index.

• That works if you dont consider the risk free rate on the sharpe equation. If you include risk free rate, the sharpe will be monotonically increasing with increasing leverage Dec 28, 2015 at 19:37
• Agreed that if you take into account riskfree rate that the sharp $\frac{N*\mu -r}{N*\sigma}$ would increase monotonic with N. However, although monotonic increasing this traditional sharpe ratio is bounded by $\frac{\mu}{\sigma}$ as N goes to inf. Dec 28, 2015 at 20:42
• This scaling with leverage only works if you consider the 'simple' (or percent) returns of the asset, but not for log returns. May 25, 2018 at 21:03