Arithmetic Mean Return

The calculation begins with a set of holding period returns, each measured over an identical interval and each expressed on the same basis. For every period, take ending value plus any cash distribution, divide by beginning value, and subtract one. Add the resulting returns and divide by the count of observations. Every period carries the same weight, one divided by the number of observations, regardless of how much capital was invested or how large the portfolio had grown by that point.

That equal weighting is precisely what makes the figure an unbiased estimator of the true single-period expected return when returns are drawn independently from the same distribution. It is also the reason the measure carries no memory: the sequence of the observations can be shuffled without changing the answer, because no period's outcome is allowed to scale the base of the next.

For any series containing variation, the arithmetic mean strictly exceeds the geometric mean; the two coincide only when every period return is identical. The gap is driven by dispersion. A useful approximation states that the geometric mean is roughly the arithmetic mean less one half of the variance of the returns, so the more volatile the series, the wider the wedge between the two statistics.

This relationship explains the ordering candidates are expected to know: harmonic mean, then geometric mean, then arithmetic mean, in ascending order for any set of positive, non-identical values. It also carries an investment lesson. Volatility imposes a drag on compounded wealth, so two portfolios with the same average annual return can deliver materially different terminal values if one is far more erratic than the other.

Forward-looking inputs are its natural home. When an analyst estimates an equity risk premium from a long history of excess returns to feed a required return or a discount rate for a single forthcoming period, the arithmetic average is the statistically appropriate choice. The same logic applies to expected return inputs for the capital asset pricing model, to expected values in probability trees, and to the sample mean that anchors the calculation of variance, standard deviation and the Sharpe ratio.

It also underpins index construction and performance statistics: a price-weighted index effectively averages constituent prices, and reported average monthly or annual returns in factsheets are frequently arithmetic. Reported cumulative or annualised performance, by contrast, must be compounded, which is why regulated performance presentation standards lean on geometric linking.

Expect to compute the statistic quickly from a short return series, then compute the geometric mean of the same series and explain the difference in the results. A common item set gives four or five annual returns and asks which measure answers the question posed: an investor's realised compound annual growth over the full horizon calls for the geometric mean, while next year's expected return calls for the arithmetic mean.

Be prepared to interpret the gap between the two as evidence of volatility, to recognise the harmonic mean as the correct tool for averaging cost per share under a fixed periodic investment, and to identify money-weighted return as the measure that reflects the timing and size of actual cash flows. Distinguishing which average matches which question is tested far more often than raw arithmetic.