Why Simple Leverage Leaves the Sharpe Ratio Unchanged

Summary

The document considers how a leveraged exchange-traded fund affects the Sharpe ratio. Its answer applies a simple proportional scaling argument: if returns are doubled, their expected value doubles and their variance quadruples, so their standard deviation doubles as well. Because both expected return and volatility scale by the same factor, the ratio of mean return to standard deviation is unchanged.

This conclusion describes ideal constant leverage applied to the same return series. The discussion does not address the daily reset mechanics of actual leveraged ETFs, compounding over time, fees, financing, tracking error, or the choice of excess return over a risk-free rate in the Sharpe calculation. It therefore gives a useful baseline identity, not a full assessment of realized ETF performance.

Key ideas

• Scaling a return series by a positive constant scales its mean and standard deviation by that same constant.
• Under ideal proportional leverage, the mean-to-volatility Sharpe ratio is unchanged.
• The argument does not account for daily ETF resets, compounding, costs, financing, or tracking error.

Tags

# Sharpe ratio with leveraged ETFs


# 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.

## Answer by mbison (score 5)

https://quant.stackexchange.com/a/22479

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.