# Calculating Correlation of Two portfolios?

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

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

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)

• What do you mean "back into E(AB)"? And how would you calculate the covariance? Jun 9, 2019 at 16:33

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 ...

• How would you calculate E(AB) though? Do you multiply the portfolios corresponding assets, and then add these products up? Jun 9, 2019 at 17:03
• Yes and that's why you need the correlations between the different assets because in order to calculate $E(A.B)$ you'll need to calculate $E(A_i.A_j)$ , $i$ and $j$ rangig from 1 to 3. From the formula you've written in your question you can see there is a way to get $E(A_i.A_j)$ from $\rho_i$, $\sigma_i$ and $E(A_i), E(A_j)$...
– Xman
Jun 9, 2019 at 20:03