Log Returns and Sharpe Ratio Conventions

Summary

The document considers whether it is appropriate to measure investment outperformance multiplicatively and calculate volatility from log returns when forming Sharpe or information ratios. Its answer contrasts this approach with the common convention of calculating those ratios using relative returns, while noting that log returns are related to relative returns by taking the logarithm of one plus the return.

For returns greater than negative one, the logarithmic return is no greater than the corresponding relative return. As a result, its mean is lower, which can affect a ratio that uses average return in the numerator. The response gives this relationship as its main explanation for why reported Sharpe ratios may differ across conventions. It does not provide a full comparison of volatility estimates, benchmark-relative calculations, or annualization practices, so it is a concise observation rather than a complete treatment of ratio methodology.

Key ideas

• Relative returns and log returns use different scales and are not interchangeable in performance calculations.
• For returns greater than negative one, the log of one plus the relative return is no greater than the relative return.
• Using log returns can lower the average return used in a ratio’s numerator.
• Sharpe and information ratios depend on the return convention, so the convention should be stated.

Tags

# Log returns: volatility, outperformance, Sharpe/information ratios


# Log returns: volatility, outperformance, Sharpe/information ratios












I have developed the habit of simply stating that a 21% return compared to a 10% benchmark return means that the outperformance was 10% (not 11%). So, treating the whole thing in a multiplicative way, as opposed to taking the differences.

Also, when I use standard deviations, I take them from the log returns. And then just have the result of that be the volatility of the investment/portfolio (without clarifying that it is the standard deviation of the log returns).

Together this gives me (variants of) Sharpe/information ratios. However, this is not how, say, Wikipedia defines such ratios (going off regular returns).

Is this an unconventional habit and/or am I doing it right?

## Answer by shabbychef (score 1)

https://quant.stackexchange.com/a/37833

The Sharpe ratio is typically computed on relative returns, not log returns. The reason is that you get a bigger number! (This is salesmanship for fund managers.) Consider the equation linking relative returns to log returns: $$ l = \log(1 + r). $$ For valid values of relative return ($r > -1$) it is simple to prove that $l \le r$. Thus log returns have a lower mean than relative returns.

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