Which Return Volatility Belongs in the Sharpe Ratio?

Summary

The document compares two Sharpe ratio denominators: the standard deviation of excess returns and the standard deviation of portfolio returns. The theoretically consistent form measures both the average return difference over a benchmark and the variability of that same difference. The discussion notes that this framework generalizes to comparing any two return streams, rather than only a portfolio and the risk-free rate.

In common applications, the risk-free rate is short-term and varies little, so using portfolio return volatility often produces a similar figure and is widely used in practice. The document therefore distinguishes conceptual consistency from a common industry convention. One response claims the two expressions are identical because the risk-free rate is constant; that equivalence holds only when the rate is constant over the observations. The material offers no empirical comparison or guidance on annualization, sampling frequency, or changing benchmarks.

Key ideas

• The consistent denominator for a Sharpe ratio is the standard deviation of the excess return series.
• The excess return framework can compare any two return streams, not only a portfolio and the risk-free rate.
• Using portfolio return volatility is common when the short-term risk-free rate changes little.
• The two denominators are identical only when the benchmark return is constant over the observations.

Tags

# Is the sharpe ratio calculated taking the standard deviation of the portfolio or of the excess return?


# Is the sharpe ratio calculated taking the standard deviation of the portfolio or of the excess return?












Does the formula consider the standard deviation of the excess return: $$\frac{𝑟−𝑟_𝑓}{𝜎{(𝑟−𝑟_𝑓)}}$$ or that of the return: $$\frac{𝑟−𝑟_𝑓}{𝜎{(𝑟)}}$$

## Answer by nbbo2 (score 3)

https://quant.stackexchange.com/a/43884

The first equation, using the excess return in both numerator and denominator is more theoretically correct. Importantly it generalizes to any two returns, not just $r$ vs $r_f$ but $r_1$ vs $r_2$ for any two returns.

And Wm. Sharpe discusses this general case in his 1994 JPM article, linked

However in the common case where $r_f$ is the short term risk free rate, its movements over time are so small that it makes little difference to the value. And therefore in practice many people use the second form that you quoted, with the denominator being the standard deviation $\sigma(r)$ of the fund alone. This is so common that it is a de facto standard in the industry.

## Answer by TomDecimus (score 0)

https://quant.stackexchange.com/a/43882

The standard deviation of a constant (rf) is zero. Therefore, either way its the same value.

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