Calculating Correlation of Two portfolios?

Question

So I'd like some help w/ this question.

Given 3 assets with means, variances, and correlation:

Two portfolios are created (A and B), each with the three assets above with weights ($w_n$) as follows:

Portfolio A: $w_1=0.2$, $w_2=0$, $w_3=0.8$

Portfolio B: $w_1=0.4$, $w_2=0.1$, $w_3=0.5$.

The assets' correlation are

$\rho_{12}=0.5,\rho_{13}=0.2,\rho_{23}=0$.

I would like to know the correlation of the two portfolios?

My attempt: So I've calculated the two portfolios' expected values and variances as follows: $E(A)=0.084, Var(A)=0.0024576$

$E(B)=0.092, Var(B)=0.006193$

And If I use $$\rho_{AB}=\frac{Cov(A,B)}{\sigma_A \sigma_B}=\frac{E(AB)-E(A)E(B)}{\sigma_A \sigma_B}$$ how would I calculate $E(AB)-E(A)E(B)$? Is $E(AB)$ the Expected Value of the assets' products, or is it the Expected Value of their weighted products? Thanks

Answer 1

You may be over-thinking it. It is a straightforward calculation using matrices, as easy as turning the crank of a sausage-making machine.

The standard deviation matrix is

               |0.16 0    0   |
S = Diag(s) =  |0    0.15 0   |
               |0    0    0.04|

The correlation matrix is

     |1.0  0.5  0.2| 
R =  |0.5  1.0  0.0|
     |0.2  0.0  1.0|

Therefore the covariance matrix is

                |0.02560  0.0120  0.00128| 
 C= S * R * S = |0.01200  0.0225  0      |
                |0.00128  0       0.00160|

The portfolio weights are

       |0.2|                   |0.4|
 wa =  |0  |     and     wb =  |0.1|
       |0.8|                   |0.5|

Therefore the covariance of portfolio A with portfolio B is

Cov(A,B) =wa^T * C * wb = 0.003466

and the covariance of A with itself, also known as the variance of A is

Var(A) = wa^T * C * wa = 0.002458

And similarly the variance of B is found as

Var(B) = wb^T * C * wb = 0.006193

Finally we can compute the correlation between A and B according to the definition

$\rho(A,B)=\frac{Cov(A,B)}{\sqrt{Var(A) Var(B)}}$ giving

rho(A,B) = 0.888326

Answer 2

You can solve for the covariance of the two portfolios and since you have E(A) and E(B) you can back into the E(AB)

Answer 3

Asset product = Weighted sum of the product of the three assets.

You can write the following Since:

$A = \omega^A_1.A1+\omega^A_2.A2 +\omega^A_3.A3 $

And

$B = \omega^B_1.A1+\omega^B_2.A2 +\omega^B_3.A3 $

You can develop $E(A.B) $ from there as a linear combination of the mean and variance couple for the three assets...

Therefore, the formula for $cov(A,B) $ is straight forward ...