# Given a correlation martrix, calculate portfolio's correlation with its assets

Find correlation vector like $[ d e f ]$ where d, e and f represent correlation of P(portfolio) with its assets A, B and C respectively. The assets A, B, C can be another portfolio.

In order for that, is it possible to find a correlation matrix including the portfolio along with its asset, given the correlation matrix for the assets in the portfolio? For example, for 3 assets A,B, C you have correlation matrix as $$\left(\begin{matrix} 1 & a & b \\ a & 1 & c \\ b & c & 1 \end{matrix}\right)$$ Now, using this or some other data concisely get a new correlation matrix for assets A,B,C, and P(for portfolio) like $$\left(\begin{matrix} 1 & a & b & d \\ a & 1 & c & e\\ b & c & 1 & f \\ d & e & f & 1 \end{matrix}\right)$$

Solution: The real interest is getting the $[ d e f ]$ vector, which can be generalized in matrix form. Let P be combined portfolio consisting of N assets or sub portfolios. Let Vector Cov(P)= [Cov(P,1) Cov(P,2) ... Cov(P,N)]', $\Sigma$ is the variance covariance matrix of portfolio P, and vector w=[w(P,1) w(P,2) ... w(P,N)] then $$Cov(P)= \Sigma w$$ $$D=diag(sqrt(diag(\Sigma))$$ $$Corr(P)=D^{-1} Cov(P) D^{-1}$$

It's probably easiest to think about it in terms of a covariance matrix and then convert it to a correlation matrix after. If instead of the first matrix you have some covariance matrix of the assets $\Sigma$, then you could get the portfolio variance, for one portfolio, as $w' \Sigma w$, where you could have $w\equiv\left(w_{1},w_{2},w_{3}\right)'$. Alternately, you could construct a matrix $W$, such that $W\equiv\left[\begin{array}{cc} I & w\end{array}\right]$, where $I$ is a $3 \times 3$ identity matrix in your case (but really could be something more general). Calculating $W' \Sigma W$ would give you a matrix such that the top left $3 \times 3$ is the original covariance matrix and then is appended with the variance of the portfolio and its covariance with the securities. You can then convert the covariance matrix to a correlation matrix to have the final result you want.
• @user 12348 I said the covariance matrix is of the assets. It would still be $3 \times 3$ in your case. Further, $W$ in your case is $3 \times 4$. Its transpose could not multiply into a $4 \times 4$ matrix anyway. – John May 6 '14 at 21:38
• I think you are saying your scheme wont work, if you convert w to 3x4 or 1x4 and leave $\Sigma$ as 3x3 because multiplication would fail. I agree. This is the reason I was avoiding going back into covariance matrix. – user12348 May 6 '14 at 22:18
• @user12348 That's pretty much the opposite of what I was saying. My way works. I was saying that if you change the covariance matrix from $3 \times 3$ to $4 \times 4$, then the matrix math I suggest doesn't work anyway. Regardless, it wouldn't make sense to do my way after you've already estimated the covariances between the portfolio and the assets. You're practically to the correlation matrix at that point. – John May 6 '14 at 22:26