Squares plus the joint piece

The variance of a weighted sum splits into squared-weight variance pieces plus one joint piece. Valid inputs satisfy |Cov| <= sigma_X sigma_Y. For Var(2X + Y) with Var(X) = 4, Var(Y) = 9 and Cov(X, Y) = 2, the pieces are 4 * 4 = 16, 1 * 9 = 9 and 2 * 2 * 1 * 2 = 8, totaling 33.

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.
Covariance: How two quantities move together: positive when they tend to rise together.

Why it works

Variance measures average squared distance from the mean, so scaling a variable by a multiplies its variance by a squared, and the co-movement of the two variables contributes twice the scaled covariance.

Var(aX + bY) = a^2 Var(X) + b^2 Var(Y) + 2ab Cov(X, Y)

Spot it: You need the variance of a spread, basket, or hedged position built from two risks.

Technique reference and sources

Square each weight onto its own variance first: those are the solo pieces. Then add one joint piece, twice the product of the weights times the covariance. Three numbers in, one number out.

When this method fits

You need the variance of a spread, basket, or hedged position built from two risks.
An interviewer gives variances plus a covariance and asks for the combined risk.
Weights differ from 1, so plain addition of variances would miss the scaling.
You want the variance of a difference, where the joint piece subtracts.

Square the weights onto the variances. Compute a * a * Var(X) and b * b * Var(Y). For 2X + Y with Var(X) = 4: 4 * 4 = 16.
Build the joint piece. Compute 2 * a * b * Cov(X, Y) with signed coefficients. For the same book with Cov = 2: 2 * 2 * 1 * 2 = 8.
Add with the correct sign. Sums add the joint piece; differences subtract it. Here 16 + 9 + 8 = 33.
Sanity-check the inputs. Valid inputs satisfy |Cov| <= sigma_X sigma_Y. Positive covariance inflates sums and shrinks differences. Negative covariance does the reverse. Zero covariance needs no independence.

Scaled sums: Var(aX + bY) = a^2 Var(X) + b^2 Var(Y) + 2ab Cov(X, Y). Every weight appears squared except inside the joint piece.

Var(2X + Y) with 4, 9, 2: 16 + 9 + 8 = 33
Var(3X) with Var(X) = 5: 9 * 5 = 45

Differences flip the joint sign: Var(aX - bY) = a^2 Var(X) + b^2 Var(Y) - 2ab Cov(X, Y). The solo pieces stay positive; only the joint piece changes sign.

Var(X - Y) with 9, 4, 1: 9 + 4 - 2 = 11
Var(2X - Y) with 4, 9, 2: 16 + 9 - 8 = 17

Common traps

Adding variances directly: Var(X + Y) = Var(X) + Var(Y) even with nonzero covariance. The joint piece 2 * Cov(X, Y) is mandatory. With variances 4 and 9 and covariance 2, the answer is 17, not 13.

Scaling linearly: Var(2X) = 2 * Var(X). Weights square. Var(2X) = 4 * Var(X), so Var(X) = 6 gives 24, not 12.

Adding the joint piece for a difference: Var(X - Y) = 9 + 4 + 2 = 15. Differences subtract the joint piece: 9 + 4 - 2 = 11. Only the sign of 2ab Cov flips.

Another example: Negative covariance offsets scale

Solo pieces: 9 * 2 = 18 and 4 * 3 = 12
Joint piece: 2 * 3 * 2 * -1 = -12
18 + 12 = 30, then 30 - 12 = 18

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.