Deriving Portfolio Correlations from Asset Covariances
Summary
The document explains how to find a portfolio’s correlation with each of its component assets or subportfolios. It recommends starting with the assets’ covariance matrix and the portfolio weights. Multiplying the covariance matrix by the weight vector gives the covariances between the portfolio and each component; dividing by the relevant standard deviations converts these covariances to correlations.
It also describes an augmented-matrix approach: append the portfolio’s weights to an identity matrix, transform the original covariance matrix, then convert the resulting covariances and variances into correlations. These methods require covariance data and consistent definitions of returns and weights. The document gives formulas and a three-asset illustration, but no empirical example or discussion of estimation error, changing weights, or out-of-sample performance.
Key ideas
- Portfolio-to-component correlations can be derived from portfolio weights and the assets’ covariance matrix.
- The covariance vector is obtained by multiplying the covariance matrix by the portfolio weight vector.
- Convert covariances to correlations by scaling with the relevant standard deviations.
- An augmented matrix transformation can produce the portfolio variance and its covariances with the original assets.
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# Given a correlation martrix, calculate portfolio's correlation with its assets
# 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 by John (score 3, accepted)
https://quant.stackexchange.com/a/11185
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.