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 $$Cov(P)= \Sigma w $$ $$ D=diag(sqrt(diag(\Sigma)) $$ $$ Corr(P)=D^{-1} Cov(P) D^{-1} $$

Answer

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.