Correlation Doesn't Care About Units

Question

Two series have operatornamecorr(X, Y) = 0.6. One desk rescales its signal to S = 2X + 1; another flips the target's sign convention to T = -3Y + 5. Compute operatornamecorr(S, T), and state the general rule you are using.

Accepted Answer

Push the affine maps through covariance and standard deviations: operatornameCov(2X+1, -3Y+5) = (2)(-3)operatornameCov(X,Y), qquad sigma_S = 2sigma_X,quad sigma_T = 3sigma_Y. operatornamecorr(S,T) = frac-6operatornameCov(X,Y)6,sigma_Xsigma_Y = -operatornamecorr(X,Y) = mathbf-0.6. The rule: correlation is invariant to positive affine transformations of each variable and flips sign when exactly one of the scalings is negative. The magnitudes 2 and 3 vanish entirely; only operatornamesign(ac) survives. Correlation is a pure, unitless measure of co-movement, which is exactly why it is the lingua franca for comparing signals measured in different units. The companion fact that prevents a classic slip: beta is not invariant. The regression slope of T on S rescales as fraccabeta_Y,X, it absorbs units. So two desks can agree perfectly on the correlation matrix while quoting wildly different betas, both correct in their own units. The trap: answering 0.6 by reflex, missing that one transformation reversed orientation. Sign-convention flips are everywhere on a desk: yields vs prices, spreads quoted A-over-B vs B-over-A, P&L vs cost. Half of all "our correlations disagree" arguments are one t

Correlation Doesn't Care About Units

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