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No, you’re not going mad. You’re right. The formula either has product of 2, covariance and weights, or product of 2, correlation and stdv of each asset.

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nope
when cov is given – it is 2 * w1 * w2 * cov
remember cov = cor * sd1 * sd2
when corr is given
it is 2 * w1 * w2 * corr * sd1 * sd2

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