# 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

## 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 – FernandoG Dec 28 '15 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. – mbison Dec 28 '15 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. – steveo'america May 25 '18 at 21:03