Correlation of two portfolios given price correlations of assets

[grmnsplx]

Hello all. I am not a stats person so I would like some help/confirmation on this one.

What I am trying to achieve (if possible) is a metric on how two portfolios (or strategies) are correlated.

Imagine there are two portfolios of assets A,B,C,D... with different weights of each asset.
eg. P₁ = (5, 2, 0, -3, ...) and P₂ = (0, 3, 10, -5, ...)

(Read this as Portfolio one consisting of 5 of asset A, 2 of asset B, no asset C, -3 of asset D and so on. The negative values means that the portfolio is short that asset.)

Let the correlation coefficients of each asset pair be given such that we can construct a typical correlation matrix (NxN square matrix, where a_i,j is the correlation coefficient for assets i and j).

I *think* that all I need to do is:

1. Multiply each portfolio vector by the correlation matrix

M_correlation°P₁ = X₁ and M_correlation°P₂ = X₂​

2. Calculate the correlation onf the two datasets (vectors) X₁ and X₂

Corr(X₁,X₂) = Corr(P₁,P₂)​

I have done this for several portfolios and what I arrive at looks right, but I am not sure if it is right. Am I out to lunch? Thoughts?

Much appreciated.

 

[grmnsplx]

*bump*

 

[Stephen Tashi]

*bump*

I suspect most forum members aren't familiar with financial mathematics. (I'm not.) It sounds like you are asking a question about linear combinations of random variables. Does the definition of "correlation" between two portfolios amount to finding the correlation coefficient between two random variables, each of which is a linear combination of other random variables?

 

[grmnsplx]

I suspect most forum members aren't familiar with financial mathematics. (I'm not.) It sounds like you are asking a question about linear combinations of random variables. Does the definition of "correlation" between two portfolios amount to finding the correlation coefficient between two random variables, each of which is a linear combination of other random variables?

Yes, that's right.
If I am not mistaken, it looks like I am doing a Least ordinary squares.

 

[Stephen Tashi]

I suggest you write out the case of two stocks A and B which will only involve 2x2 matrices. At least, write out what you are given and what you are asking in symbolic form. That will save people from having to guess at the meaning of things like "portfolio vector".

Look up how to use LaTex on the forum. I think your question begins as follows:

Let $A,B$ be two independent random variables with respective means $\mu_A, \mu_B$ and standard deviations $\sigma_A,\sigma_B$.

let

$P_1 = \lambda_A A + \lambda_B B$ where $\lambda_A, \lambda_B$ are constants.

$P_2 = \alpha_A A + \alpha_B B$ where $\alpha_A, \alpha_B$ are constants.

Let $C _{AB}$ be the covariance matrix

$C_{AB} = \begin{pmatrix} COV(A,A),&COV(A,B) \\ COV(B,A)&COV(B,B) \end{pmatrix}$.

Now, is the "asset pair correlation matrix" supposed to be like:

$R_{AB} = \begin {pmatrix} \frac{COV(A,A)}{\sigma_A^2}&\frac{COV(AB)}{\sigma_A \sigma_B} \\ \frac{COV(B,A)}{\sigma_B \sigma_A}& \frac{COV(B,B)}{\sigma_B^2} \end{pmatrix}$