Legs, then Pythagoras, then tilt

Two-asset portfolio variance is the variance-combination formula with weights as the scalers: half in A at 30 percent vol and half in B at 40 percent vol with zero correlation gives legs 15 and 20, variance 225 + 400 = 625, and volatility sqrt(625) = 25 percent.

Words you need

Variance: The average squared distance from the mean: how spread out results are.
Standard deviation (volatility): The square root of variance, in the same units as the data. In markets it is called volatility.
Correlation: Covariance scaled to lie between −1 and 1. +1 means they move in lockstep.
R²: The share of a quantity's variance that a regression explains, from 0 to 1.

Why it works

Portfolio return is w1 R1 + w2 R2, so its variance follows Var(aX + bY) exactly, with the correlation form of the joint piece. The volatility is the square root of that sum.

sigma_p^2 = w1^2 s1^2 + w2^2 s2^2 + 2 w1 w2 rho s1 s2

Spot it: An interviewer gives two vols, weights, and a correlation and asks for portfolio vol.

Technique reference and sources

Convert each sleeve into a vol leg: weight times vol. At zero correlation the legs meet at a right angle, so variance is leg one squared plus leg two squared. Correlation tilts the angle: +1 adds the legs first, -1 subtracts them.

When this method fits

An interviewer gives two vols, weights, and a correlation and asks for portfolio vol.
You need the diversification benefit versus the weighted-average vol.
Weights are one half each, so legs halve cleanly and triples appear.
Correlation is -1, 0, or +1, where the root comes out exact.

Build each leg. Leg = weight * vol. Half of 30 is 15; half of 40 is 20.
Square and add at zero correlation. 15 * 15 + 20 * 20 = 225 + 400 = 625.
Root for volatility. 25 * 25 = 625, so volatility is 25 percent.
Tilt for nonzero correlation. At +1 the legs add first: 15 + 20 = 35. At -1 they subtract: 20 - 15 = 5.

Zero correlation is Pythagoras: Variance is leg one squared plus leg two squared. Halved 30/40 legs (15, 20) give 625 and vol 25.

Legs 15 and 20: 225 + 400 = 625, vol 25
Legs 9 and 12: 81 + 144 = 225, vol 15

Full correlation adds or subtracts legs: At +1, vol is leg one plus leg two. At -1, it is the absolute gap.

Legs 15 and 20 at +1: 15 + 20 = 35
Legs 15 and 20 at -1: 20 - 15 = 5

Common traps

Adding vols directly at zero correlation: 15 + 20 = 35. Vols add only at correlation +1. At zero they combine as sqrt(225 + 400) = 25.

Averaging the vols: (30 + 40) / 2 = 35 at zero correlation. Averaging prices +1 correlation. Diversification at zero correlation gives 25, a full 10 points lower.

Forgetting to square the weights: 0.5 * 900 + 0.5 * 1600. Weights square like every scaler: 0.25 * 900 + 0.25 * 1600 = 225 + 400 = 625.

Another example: Same sleeves at correlation -1

At -1 the legs subtract: 20 - 15 = 5
Variance: 5 * 5 = 25
5 * 5 = 25, so volatility is 5

Source notes

These explanations and questions are written for QuantArc. Links show the underlying mathematics and learning design.

Harvard Stat 110: Probability (Joe Blitzstein)
Free course site with videos and the open companion book Introduction to Probability; the topic spine for counting, Bayes, and linearity. Lesson examples and questions are original.

Grinstead and Snell, Introduction to Probability (free PDF)
Free textbook PDF confirmed live: counting methods, poker-style dealing models, and the random-walk chapter behind gambler's ruin and first-step equations. Lesson examples and questions are original.

IES: Organizing Instruction and Study to Improve Student Learning
Supports worked examples, retrieval, explanations and spaced study. Does not establish QuantArc's exact lesson length, mastery threshold or review intervals.