Weighted Mean (Portfolio Return)

Builds onArithmetic Mean Return — if this page feels steep, start there.

\bar{R}_w = \sum_{i=1}^{n} w_i R_i, \qquad \sum w_i = 1

$\bar{R}_w$: the weighted average return — the w subscript flags that items count unequally
$w_i$: the weight of item i: the fraction of money in it (60% is 0.6)
$R_i$: the return of item i
$\sum w_i R_i$: multiply each return by its own weight, then add them all up
$\sum w_i = 1$: the weights must total exactly 1 — every dollar has to live somewhere

Why it must be true

A portfolio's return is not the average of its holdings' returns — it's the average weighted by how much money sits in each. A 5% move on a 60% allocation matters three times as much as on a 20% allocation.

The weighted mean is the workhorse of portfolio arithmetic: the same shape prices expected values (weights = probabilities) and benchmark returns (weights = index shares). Whenever the observations don't count equally, the weights carry the economics.

The derivation

Invest fraction $w_i$ of one dollar in asset $i$ . That sleeve grows to $w_i(1 + R_i)$ , so total ending wealth is:

W = \sum_i w_i (1+R_i) = \underbrace{\sum_i w_i}_{=\,1} + \sum_i w_i R_i = 1 + \sum_i w_i R_i

Ending wealth minus the dollar invested is the portfolio return: $\bar{R}_w = \sum w_i R_i$ .

When to reach for it

Observations that matter unequally — portfolio returns weighted by allocation, benchmark math, or expected values with probabilities as weights.

Listen for

“allocates 60% to … 40% to …”“portfolio invested across”“weights of …”

Back-of-the-envelope

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

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The answer lives between the extremes, pulled toward the heaviest weight. 70/30 in assets returning 10% and 2% → nearer 10%: roughly 7.6%.
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Start at 50/50 (the simple midpoint) then shift: each 10% of weight moved drags the answer 10% of the spread toward that asset.

Traps in applying it

✗Weights that don't sum to 1 — leverage and cash make this easy to miss.
✗Dividing the weighted sum by n as if the weights hadn't already averaged.
✗Using end-of-period weights with beginning-of-period returns.

Limits & criticisms

A weighted mean is a snapshot: the moment prices move, actual weights drift, so multi-period portfolio returns depend on a rebalancing assumption the formula doesn't contain. It also reports only the blend — two very different portfolios can share the same weighted mean.

Where it came from

Weighted averaging entered science through astronomy — Laplace combined observations of unequal reliability by weighting them — and entered finance through index construction: Étienne Laspeyres (1871) built price indices as weighted means, the ancestor of every market-cap-weighted benchmark.

Today it is the most-executed calculation in asset management: every portfolio return, every benchmark level, and (with probabilities as weights) every expected value is a weighted mean.

Where it leads

Master this and the following come almost for free:

Expected Value Weighted-Average Cost of Capital Macaulay Duration Two-Asset Portfolio Return & Risk

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