04. Portfolio Variance

L3 04 Portfolio Variance V2

Derivation of Portfolio Variance

We start with this:

\sigma_P^2 = \sum_ip(i)[r_P - E(r_P)]^2

Let's plug in what we know.

\sigma_P^2 = \sum_ip(i)[x_Ar_A -x_AE(r_A) + x_Br_B - x_BE(r_B)]^2
= \sum_ip(i)[x_A(r_A -E(r_A)) + x_B(r_B - E(r_B))]^2

Then we square everything in the brackets:

= \sum_ip(i)[x_A^2(r_A -E(r_A))^2 + x_B^2(r_B - E(r_B))^2 + 2x_Ax_B(r_A-E(r_A))(r_B - E(r_B))]

Whew, let's stop for a breather.

Mmmmk. So now, we do the same thing we did in the derivation of the portfolio mean. Instead of putting everything into one big sum, we break the big sum up into sub-sums, and pull out the weights, which aren't indexed by i .

= x_A^2\sum_ip(i)(r_A -E(r_A))^2 + x_B^2\sum_ip(i)(r_B - E(r_B))^2 + 2x_Ax_B\sum_ip(i)(r_A-E(r_A))(r_B - E(r_B))

And now, if we look closely, we can see the result already. In the first two terms, the sums are just the individual asset variances. The third term is where the magic happens. That sum simply equals the covariance.

= x_A^2\sigma_A^2 + x_B^2\sigma_B^2 + 2x_Ax_B\mathrm{Cov}(r_A,r_B)

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