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Thank you gordon! So in addition to the solution you posted, hereĀ“s what I actually used in the script where I needed this formula. In the case anyone else can use it: Cov(X,A) = Cov(0.25A+0.25B+0.5C,A) = 0.25Var(A) + 0.25Cov(B,A) + 0.5 Cov(C,A) Corr(X,A) = Cov(X,A) / sqrt( Var(X)*Var(A) ) beta[A] = ( vol[A] / vol[X] ) * Corr[A] have a Good day!


You only need to note the following \begin{align*} corr\left(X_1, \sum_{i=1}^nw_i X_i\right) &= \frac{cov\big(X_1, \, \sum_{i=1}^nw_i X_i \big)}{\sqrt{var(X_1)} \sqrt{var(\sum_{i=1}^n w_i X_i)}}\\ &= \frac{E\Big(\big(X_1-E(X_1)\big)\big(\sum_{i=1}^n w_i X_i - E(\sum_{i=1}^n w_i X_i) \big)\Big)}{\sqrt{var(X_1)} \sqrt{var(\sum_{i=1}^n w_i X_i)}}\\ ...


If your issue is the holding period "sensibility", I don't have persuasive economic/fundamental motivation about it. It's an interesting question. Anyway in econometric point of view the issue is part of instability parameters problem. If the time horizon for your investment project is, for example, one year, then you need return and beta one year based. ...

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