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nism-series-xa-three-asset-portfolio-variance-formula[ INVESTMENT ADVISER (LEVEL 1) ]

What is the mathematical formula for the Variance of a 3-Asset Portfolio (Assets 1, 2, 3 with weights w_i and pairwise covariances Cov_ij)?

[ A ]Var(P) = (w_1 + w_2 + w_3) * Cov_12
[ B ]Var(P) = w_1^2 σ_1^2 + w_2^2 σ_2^2 + w_3^2 σ_3^2 + 2 w_1 w_2 Cov_12 + 2 w_1 w_3 Cov_13 + 2 w_2 w_3 Cov_23[ CORRECT ]
[ C ]Var(P) = w_1 σ_1 + w_2 σ_2 + w_3 σ_3
[ D ]Var(P) = σ_1^2 + σ_2^2 + σ_3^2
Takeaway

For a 3-asset portfolio, Variance = the sum of each asset's weighted squared variance term plus twice the weighted covariance of every unique pair of assets: Var(P) = w1²σ1² + w2²σ2² + w3²σ3² + 2w1w2Cov12 + 2w1w3Cov13 + 2w2w3Cov23.

Derivation
Portfolio variance expands from squaring the weighted sum of returns, (w1R1 + w2R2 + w3R3)², which under the rules of variance/covariance algebra produces exactly 3 'diagonal' terms (each asset's own weighted variance) and 3 'off-diagonal' cross terms (each unique pair's weighted covariance), each cross term appearing twice due to symmetry (Cov12 = Cov21) and hence multiplied by 2. As the number of assets grows, the number of covariance pair-terms grows much faster than the number of variance terms — this is precisely why diversification benefits scale non-linearly with the number of holdings, a core Modern Portfolio Theory insight the formula is meant to illustrate quantitatively.
Why the others fail
A — Var(P) = (w_1 + w_2 + w_3) * Cov_12
A simple sum of weights multiplied by one covariance term ignores each asset's own variance entirely and omits the other two covariance pairs — it is not a valid variance formula.
C — Var(P) = w_1 σ_1 + w_2 σ_2 + w_3 σ_3
Summing weighted standard deviations directly (rather than squaring them into variances and adding covariance cross-terms) ignores diversification benefits entirely and is mathematically invalid for.
D — Var(P) = σ_1^2 + σ_2^2 + σ_3^2
Summing raw individual variances while ignoring both the portfolio weights and all pairwise covariance terms discards the core diversification mechanics that portfolio variance is meant to capture.