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Let $X,A,B,C,D$ be time-series variables and the covariance between any two pairs of these are known.
Suppose we want to find $\textrm{cov}(X,aA + bB + cC + dD)$, where $a,b,c,d$ are constants.
Is there any way of doing this without expanding out $E[(X-E[X])(aA+......)]$?
Is there any way of doing this without expanding out $E[(X-E[X])(aA+......)]$?
Yes. There is a property of covariance called bilinearity which is that the covariance of a linear combination
$$ {\rm cov}(aX + bY, cW + dZ) $$
(where $a,b,c,d$ are constants and $X,Y,W,Z$ are random variables) can be decomposed as
$$ ac\cdot {\rm cov}(X,W) + ad\cdot {\rm cov}(X,Z) + bc\cdot {\rm cov}(Y,W) + bd\cdot {\rm cov}(Y,Z) $$
In the example you've given, you can use this property to write $\textrm{cov}(X,aA + bB + cC + dD)$ as
$$ a\ {\rm cov}(X, A) +b\ {\rm cov}(X, B) +c\ {\rm cov}(X, C) +d\ {\rm cov}(X, D) $$
