Weighted Averages and Mixture Shortcuts

A portfolio is 30% in an asset returning 8% and 70% in an asset returning 2%. What's the blended return? You could set up $0.3 \times 8 + 0.7 \times 2$ and multiply it out, but there's a faster route that also builds intuition: think in terms of distance from the weighted-average point, the way a see-saw balances around a fulcrum. The weighted average always lands closer to whichever side has more weight — and you can often read off roughly where without multiplying anything.

The see-saw picture

A weighted average of two values $a$ and $b$ with weights $w_a$ and $w_b$ (summing to 1) is:

$$\bar{x} = w_a a + w_b b .$$

In plain English: $\bar{x}$ sits on the number line between $a$ and $b$, splitting the gap in the inverse ratio of the weights — the heavier side pulls the average closer to itself. This is the "alligation" trick: instead of multiplying and adding, find the gap between $a$ and $b$, then split that gap in ratio $w_b : w_a$ (weights swapped) starting from $a$.

Worked example 1: the portfolio blend

$a = 8\%$ (weight 0.3), $b = 2\%$ (weight 0.7). The gap is $8 - 2 = 6$ points. Split it in ratio $w_b : w_a = 0.7 : 0.3$, i.e. $7:3$, starting from $a=8$ moving toward $b$: the average sits $\frac{7}{10}$ of the way from $a$ to $b$, which is $8 - 0.7 \times 6 = 8 - 4.2 = 3.8\%$. Check directly: $0.3(8) + 0.7(2) = 2.4 + 1.4 = 3.8$. Matches — and notice the answer sits much closer to 2% than to 8%, exactly because 70% of the weight sits on that side.

Worked example 2: mixing two portfolios to hit a target

You hold cash yielding 1% and a fund yielding 9%, and want a blended return of exactly 5%. What weight goes in the fund? Alligation again: the gap from 1 to 9 is 8. The target 5 is $5 - 1 = 4$ above the low end, which is exactly half the total gap — so the weights split 50/50. In general, the weight on the higher-return asset equals (target − low) / (high − low): here $\frac{5-1}{9-1} = \frac{4}{8} = 0.5$, confirming a 50% allocation to the 9% fund.

What this means in practice

This shortcut is exactly what you use when a portfolio manager asks you to eyeball a blended yield, a blended win rate across strategies, or how much of a new allocation is needed to shift a fund's overall risk profile toward a target. The gap-splitting method is faster than multiplying decimals under pressure and, more importantly, gives you a built-in sanity check: if your answer isn't closer to the heavier-weighted side, you've made an arithmetic error.