# Should I use an arithmetic or a geometric calculation for the Sharpe Ratio?

What are the advantages/disadvantages of using the arithmetic Sharpe Ratio vs the geometric Sharpe Ratio? Is one more correct? Or is one better in certain circumstances?

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## 4 Answers

In addition to John's answer and just to make things clear:

The arithmetic mean is given by

$$\mu_a = \frac{1}{n} \sum_{i=1}^n x_i$$

The geometric mean is given by

$$\mu_g = \sqrt[n]{\prod_{i=1}^n (1+x_i)} -1$$

And we have that

$$\mu_g \leq \mu_a$$

So not only would the geometric sharp ratio would be taking into account the "actual" return of the portfolio, but it is also a more conservative measure.

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I'm not sure it makes sense to think of one as more correct than another. However, they do have significant differences. It may help to distinguish between ex-post evaluation of a strategy and ex-ante prediction of what the strategy's performance will be.

For simplicity, let's assume the log returns of the strategy are approximately i.i.d. univariate normal and the risk-free rate is a constant. If you were a mean-variance investor deciding between the risk-free rate and the strategy, you would estimate the mean and variance of the log returns, project it to the investor's horizon, and convert the normal to lognormal to obtain the arithmetic returns. So if you were calculating a Sharpe ratio that is consistent with the way it was originated in financial theory, i.e. the slope of the efficient frontier, would be this arithmetic ex ante expected Sharpe ratio.

However, the Sharpe ratio is also used in performance evaluation in different ways. I think one major reason that the geometric version is used is that the numerator will correspond with what the investor actually earned, the CAGR. This might be useful to some people, but personally I prefer to look at the CAGR by itself rather than in the Sharpe ratio. Further, the CAGR is the median of the lognormal distribution under some assumptions. I find it more intuitive to use the mean and keep ex-post consistent with ex-ante, which would bring me back to the arithmetic Sharpe.

Another reason to use the geometric version might be that it avoids the distributional issues with the lognormal distribution (since it has skewness/kurtosis). However, Opdyke (2007) provides the asympototic distribution of the Sharpe ratio under fairly general assumptions.

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There are many variants proposed; some useful, some not so much. As an investor, the most important thing is to compare the exact same ratio, calculated in the exact same way, for each prospect. As the prospect/fund the most important thing is to be clear about the statistic you are reporting so your investors make well informed decisions. So let's start with some definitions specific to your question.

• Sharpe ratio, sometimes called the "Modified Sharpe" ratio, is the arithmetic average of excess returns divided by the standard deviation of those returns, $r_t\in R$.
$Sharpe Ratio = \frac{E[R_t - R_{free}]}{\sigma}$
where $R_t\in[-1,\infty)$, which is a percentage, and $R_{free}$ is the risk-free return rate, typically taken as the current T-Bill rate.
It is commonly calculated over annual periods, but monthly, or daily periods are common too. It is a class of signal-to-noise ratio indicating the expected reward per unit of risk. This is the most commonly cited variant and is the measure proposed by professor Sharpe in 1994.
• 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 $log(1+r_t)$, which is the more convienient calculation.

The difference is mostly in the use of the average of the excess returns for the period or the compounded returns. Because the log returns are generally smaller, the "Geometric Sharpe ratios" generally suggests a higher number; about 12.5% higher for samples from a uniform distribution. Naturally, it is completely bogus to compare dissimilar ratios...so be sure you know which measure you are digesting.

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For any real world applications, the difference between the arithmetic and geometric Sharpe ratios is likely to 'fall under the noise floor', i.e. be smaller, typically much smaller, than a standard error. This is even under the generous assumptions of stationarity and absence of omitted variables.

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