# Arithmetic vs Geometric Mean: Which to use in Performance Appraisal

Most performance appraisal measures utilize a mean return in its calculation. This can be in the form a geometric mean or a simple arithmetic average. Because both types of means can be used, it raises the question: Which measure should be applied?

When calculating performance, we are accustomed to calculating returns geometrically (i.e., including compounding). Because of this, many investment managers use the geometric mean in appraisal calculations as it is easy to use the reported time-weighted return, rather than separately determining the arithmetic mean. But using geometric mean is not the most appropriate choice when evaluating risk-adjusted appraisal measures.

When calculating performance appraisal measures that compare return to risk, such as Sharpe ratio, the return used in the numerator of the ratio should be the arithmetic mean of the return stream, not the geometric mean. In many cases, the difference between using the arithmetic mean versus geometric mean will be immaterial; however, the greater the volatility in the return stream, the more material the difference will be. Let’s look at a simple example that demonstrates this effect:

Strategies with significant volatility have lower geometric means than arithmetic means (7.5% vs. 8.4% for Portfolio 2 above). This is because the geometric mean penalizes the return stream for risk-taking. In the case of the Sharpe Ratio, the standard deviation (which also accounts for risk-taking) in the denominator will be higher as a result of this higher volatility (1.5% for Portfolio 1 vs. 14.2% for Portfolio 2). In this case, using the geometric mean therefore results in a penalty for risk in both the numerator and denominator of the ratio.

Because risk is already being accounted for in the denominator, there is no need to include it in the numerator; in fact, including it would be double-counting the risk taken. As a result, for measures like Sharpe Ratio, it is more appropriate to use the arithmetic mean than geometric mean.

Although, in many cases, using the geometric return will not have a material effect on the outcome when comparing risk-scaled performance measures, it is technically more accurate to use the arithmetic mean. Its implications are more relevant when evaluating strategies with higher volatility.

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