Arithmetic Mean Return

\bar{R} = \frac{1}{n}\sum_{i=1}^{n} R_i

$\bar{R}$: the bar over R is math shorthand for 'the average of R'
$\sum_{i=1}^{n} R_i$: the sigma says: add up every return, from the 1st to the n-th
$R_i$: the return in period i — the subscript is just a label for 'which one'
$\frac{1}{n}$: multiplying by 1/n is dividing by the count: that's what makes it an average

Why it must be true

The arithmetic mean answers: what did a typical single period look like? Add the observations, divide by how many there are — every period gets equal weight.

Its subtlety is what it does not measure: because it ignores compounding, it is the right expectation for one period ahead, but it overstates the growth rate actually earned over the whole span (that's the geometric mean's job). The gap between the two widens with volatility.

The derivation

The mean is the value that balances the data — the unique number $\bar{R}$ making the deviations sum to zero:

\sum_{i=1}^{n}(R_i - \bar{R}) = 0 \quad\Rightarrow\quad \bar{R} = \frac{1}{n}\sum_{i=1}^{n} R_i

When to reach for it

The best single-period expectation from a sample of returns, or any average where each observation counts equally.

Listen for

“average return”“expected return for next period”“mean of the observations”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

≈
Anchor and average the offsets: for 7%, 9%, 5%, 11% think '8% ± small offsets' → offsets −1, +1, −3, +3 sum to 0 → mean is exactly 8%.
≈
The mean must sit strictly inside the min–max range, and (for symmetric-ish data) near the middle. Choices hugging an extreme are distractors.

Traps in applying it

✗Using it to describe multi-period growth — it overstates what compounding actually delivered.
✗Averaging returns computed on wildly different bases as if comparable.
✗Ignoring an outlier that is doing all the work in the average.

Limits & criticisms

For any volatile series the arithmetic mean exceeds the realized compound rate — the gap (volatility drag) is roughly σ²/2. It is fragile to outliers and skew, and it is a sample estimate carrying sampling error, not the true expected return.

Where it came from

Averaging is ancient, but the mean earned its statistical throne in astronomy: Laplace and Gauss (early 1800s) showed that averaging noisy observations is the best estimate of a true value, founding error theory. Finance imported the idea wholesale — a period's return is one noisy draw from a distribution you are trying to estimate.

Today the arithmetic mean is the right expectation for a single period ahead, which is exactly how it appears in forecasting, scenario analysis, and as the input to variance and Sharpe calculations.

Where it leads

Master this and the following come almost for free:

Geometric Mean Return Weighted Mean (Portfolio Return)Sample Standard Deviation

Drill this formula Back to the library