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?

• It's a matter of keeping your units straight, IMO. If you use arithmetic Sharpe, make sure you're using logarithmic returns. If you are using geometric returns, make sure you take exponent of logarithmic standard deviation. I use log returns when prototyping (due to computational ease) and geometric returns when presenting. – David Addison Mar 15 '17 at 22:15

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 be taking into account the "actual" return of the portfolio, but it is also a more conservative measure.

• Where can I find a demonstration that $$\mu_g \leq \mu_a$$ – Marco Demaio Mar 4 '14 at 20:50
• @MarcoDemaio You can find it on Wikipedia: en.wikipedia.org/wiki/… – Xavier Aug 3 '15 at 0:28
• Would it ever make sense to consider the Harmonic mean? – user2521987 Mar 18 '17 at 0:05

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.

I think this is a no-brainer. Only log-returns make sense. The average return can only be computed by averaging the sum of individual log returns. Taking the average of standard (relative) returns does not give you an average of the individual returns. Consider a simple case where the value of an investment alternates between 100 and 50 an odd number of times. The standard return series would be: -0.5, 1, -0.5, …1 (-50% and +100%). The average of that sum gives us 0.25 (25%) – nonsense for an investment whose final value is the same as what we started with. The log returns, on the other hand give us alternating log returns of -0.6931, +0.6931, whose average is 0.

The difference between log returns and standard returns goes to zero as we shorten the period over which we evaluate the value of an investment: LN(P(n)/P(n-1)) is approximately equal to P(n)/P(n-1) – 1. Thus there would not be much difference between standard and log returns (and the computed Sharpe Ratio) if daily measurements were made. The scaling of that Sharpe Ratio from daily returns to annual returns is performed by the sqrt of the number of trade days (252), but that, of course assumes the return distribution is iid, which is not really the case.

Andrew W. Lo has a nice paper that considers the scaling of the Sharpe ratio when the return series is correlated ("The Statistics of Sharpe Ratios")

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.

The correct answer is "arithmetic mean, because Bill Sharpe says so". He invented the thing, and he's pretty clear on which one he was looking at.

If you use the geometric mean, which is lower the higher the volatility in the returns, and then you divide by standard deviation, you have essentially discounted your result TWICE for volatility.

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.

As a practical example, I have generated a sequence of returns where mean log returns are -12%, while the arithmetic mean is +1.1%.

This can happen with very heavy tailed distributions and, e.g. in evaluating hedge fund performance, the arithmetic return will mask this.

A glance at an equity curve would also highlight the issue so in reality no-one is likely to be fooled.

For unleveraged investing in stocks and indices it will make very little difference, but the arithmetic version is clearly wrong in some cases as it will give a positive sharpe for a series of returns resulting in a large loss of capital.

• Can you post the return series which you are referring to? – emcor Sep 29 '16 at 8:58

The only correct way is using log returns.

To keep everything consistent, take a arithmetic mean of log returns.

Then calculate it net of the risk free (how do you subtract properly using geometric returns?).

Then divide (how do you do this properly using geometric returns?)

by the standard deviation (how would you calculate this properly with geometric returns?).

A geometric mean is just the tip of the iceberg. Geometric returns implies a bunch of new calculations which are far from standard.

Log returns are the only way to go with time series analysis.

The arithmetic average of +100% in Year 1 and -100% in Year 2 is 0%, but I we all know the result is not a 0% return. So arithmetic returns are absurd to use in any real life context. Maybe in another universe they can serve some purpose.

I didn't understand this either but after some googleing I understood the importance of geometric mean returns! The most stuff on the internet is in text form which is not easy to understand. So I visualized this discrepancy 