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Perfect positive correlation and weighted volatility

With long-only weights and correlation +1, the variance expression is a perfect square.

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Use the same-period hypothetical pair

Weights are 40% and 60%, standard deviations 20% and 10%, and correlation is +1. Covariance is 0.02. The cross term becomes 2×0.4×0.6×0.02=0.0096.

Complete the variance bridge

Own terms total 0.0064+0.0036=0.01. Including covariance gives 0.0196, which equals (0.08+0.06)². Its square root is 0.14, or 14%.

σP=wAσA+wBσB(ρ=1, wA,wB≥0)\sigma_P= w_A\sigma_A+w_B\sigma_B\quad(\rho=1,\ w_A,w_B\geq0)

Nonnegative weights make the sum nonnegative.

Keep the special case separate

At correlation 0.3 the same inputs produce approximately 11.349009%, so adding weighted standard deviations would overstate that model’s dispersion. The formula here is a special covariance case, not a general risk aggregation shortcut.

Interpret a single-asset endpoint

At weight A=100%, the result is A’s 20% standard deviation regardless of correlation because B’s weight is zero. Correlation matters only when the cross term has nonzero weighted exposures.

Further references