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Geometric and Arithmetic Sharpe Ratios for Fund Performance

Article Quant Q&A · Author: Andrew
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Summary

The document contrasts the conventional arithmetic-return Sharpe ratio with a geometric version for evaluating mutual fund performance. The geometric approach is described as using compounded excess returns and their dispersion; it is also related to the mean and standard deviation of log gross returns. Arithmetic returns are identified as the more established convention in the cited performance literature, including Sharpe’s original ex-post treatment.

The responses do not give a full worked calculation or settle which measure is universally preferable. They point out that geometric returns reflect compounded growth over time, while consistent use of one calculation across portfolios may matter more for comparative rankings. The document is brief, and the equivalence stated for log returns depends on how returns and excess returns are defined, so practitioners should specify the calculation and return convention when reporting results.

Key ideas

  • The classic Sharpe ratio is generally computed from arithmetic returns.
  • A geometric version uses compounded excess returns and their variability.
  • Log gross returns connect geometric performance measurement to arithmetic statistics on log returns.
  • Comparisons are most interpretable when portfolios use a consistent return and calculation convention.

Tags

StatisticsPortfolio constructionRisk management
Full text
# Geometric Sharpe ratio


# Geometric Sharpe ratio












I'm computing different metrics for mutual fund performance. I want to use classic Sharpe ratio, but I also got to know there is geometric Sharpe ratio. Unfortunately I didn't find enough info about it, could you please explain how to compute it?

## Answer by Chris (score 1)

https://quant.stackexchange.com/a/46185

Link to discussion in the other thread notwithstanding, calculating Sharpe ratio using arithmetic return is more 'classic' than using geometric return.

To start, Sharpe himself used arithmetic returns in ex-post calculation in his originating paper (JPM, 1964).

Most texts also use arithmetic return, among them Grinold and Kahn and Christopherson, Carino, Ferson.

Personally, I think using arithmetic returns, aside from the above, are a little easier to work with. There's a semantic argument to be made that geometric return is what you would actually end up with, but it's kind of weak and provided you use a single calc across portfolios it's not going to matter much in comparing portfolio performance. Trivially, the fact that 'geometric Sharpe ratio' is a term yet 'arithmetic Sharpe ratio' isn't should make the situation somewhat apparent.

## Answer by Vitomir (score 0)

https://quant.stackexchange.com/a/46164

The Geometric Sharpe ratio is the geometric average of compounded excess returns divided by the standard deviation of those compounded returns. This is equivalent to the arithmetic average and standard deviation of log(1+rt).

Geometric returns should be the preferred way of calculating returns over a time series.

In any case, Should I use an arithmetic or a geometric calculation for the Sharpe Ratio?.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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