# 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. – vonjd Dec 25 '15 at 9:33

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.
• 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. – mbison Dec 28 '15 at 20:42