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After a few hours trying to solve this I give up! I need help.

I need to calculate the BETA of an asset with respect to a portfolio that contains this asset. I have the volatility and correlations for all the portfolio, and the allocation.

I could use a formula to calculate the correlation between asset A and the portfolio. Because I could get Beta with the usual formula.

I need a formula, not a method to calculate this based on historical prices.

So this is what I have:

Correlation:

   A    B      C
A  1    0.85  0.78
B 0.85   1    0.84
C 0.78  0.84   1

S.D.

  A       B      C
19.74% 25.76% 31.19%

Allocation:

  A       B      C
25.00% 25.00% 50.00%

Hope you can help me!!

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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)}}\\ &= \frac{\sum_{i+1}^n w_i E\big((X_1-E(X_1))(X_i - E( X_i) )\big)}{\sqrt{var(X_1)} \sqrt{var(\sum_{i=1}^n w_i X_i)}}\\ &= \frac{\sum_{i=1}^n w_i\rho_{1, i} \sigma_1 \sigma_i}{\sigma_1 \sqrt{\sum_{i, j=1}^n w_iw_j\rho_{i, j} \sigma_i \sigma_j}}\\ &= \frac{\sum_{i=1}^n w_i\rho_{1, i} \, \sigma_i}{\sqrt{\sum_{i, j=1}^n w_iw_j\rho_{i, j} \sigma_i \sigma_j}}. \end{align*}

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    $\begingroup$ In the last line, what happened to the $w_i$ terms which were present (in the next to last line) in the top (numerator) summation? (I understand why the $\sigma_1$ disappeareed). $\endgroup$ – Alex C May 24 '16 at 2:47
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Thank you gordon! So in addition to the solution you posted, here´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[X,A]

have a Good day!

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  • $\begingroup$ +1. In your second last line, should Corr[A] be Corr(X, A)? $\endgroup$ – Gordon May 25 '16 at 14:36
  • $\begingroup$ Yes! you are right! Just edited it! Thanks! $\endgroup$ – Nicolas Galarza Ricci May 29 '16 at 0:52

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