Log returns: volatility, outperformance, Sharpe/information ratios

Question

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 1

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.