Log Returns, Compounding, and the Sharpe Ratio

Summary

The note asks why the Sharpe ratio uses average returns rather than total compounded performance. It illustrates the distinction with a loss followed by a gain: equal simple returns do not restore the starting capital, so their arithmetic average can obscure the realized cumulative outcome.

The answer proposes using log returns, whose values add across periods, and demonstrates that the two moves have a negative average log return. This connects the numerator to compounded growth more directly than averaging simple returns. The response is brief and does not define a full Sharpe calculation, discuss excess returns or annualization, or establish that every conventional Sharpe implementation uses log returns. Its example is an illustration of compounding, not a general comparison of performance measures.

Key ideas

• Arithmetic averages of simple returns can differ from the cumulative result when returns compound.
• A loss followed by an equal percentage gain leaves capital below its starting value.
• Log returns add across periods and encode the compounded change in wealth.
• The answer advocates averaging log returns, but does not specify a complete Sharpe-ratio convention.

Tags

# Why isn't the Sharpe Ratio computed on the cumulative return rather than return mean?


# Why isn't the Sharpe Ratio computed on the cumulative return rather than return mean?












I have learnt that the Sharpe ratio is a measure of the annualized return rate mean over the annualised standard deviation of return rate distribution. I also learnt that when compounding, the mean of the return rate distribution does not correspond to the overall return rate at the end of the test period (the classic example is : I have 100 USD, then I loose 50%, then I gain 50% I end up with 75 USD which is an overall return of -25%, while return mean is 0%).

Since the return mean does not correspond to reality in most of the case (i.e., when the return is compounded), why the Sharpe ratio does not take the cumulative return (i.e, exp(sum of log returns)) as a numerator rather than the mean of return rates ?

Please note that I've made a lot of research on Google and StackExchange and there seem not to be a definitive standard response to this question.

## Answer by user66963 (score 0)

https://quant.stackexchange.com/a/75438

Sharpe uses log returns, not simple.

The log return of 50/100 = -0.6931

The log return of 75/50 = 0.4054

The average is = -0.1438. This is what Sharpe uses.

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