Variance of a multi-asset portfolio

CFA Level II Portfolio Management
#1

Not able to understand how the formula for calculating variance for a multi-asset equal weighted portfolio is derived as: ∑(i=1 to n) * ∑(j=1 to n) weight of i * weight of j * Cov(Ri,Rj) I am referring to page 392 on Book 6 (2014 Version)

Any insight would he very helpful. Thanks.

#2

You’re misinterpreting the notation; it isn’t a multiplication, it’s a _ double summation _:

∑(i=1 to n) {∑(j=1 to n) ([w_i_ × w_j_ × σ_i_ × σ_j_ × Cov(R_i_,R_j_)]}

This means that you let i = 1 while j runs from 1 to n, summing all the terms, then let i = 2 while j runs from 1 to n, summing all the terms, and so on, until you let i = n while j runs from 1 to n, summing all the terms; you then sum those (n) sums. (Note, too, that you forgot the sigmas.)

#3

Nope, his original formula is correct. He was correct not to include the standard deviation of the two assets since the formula uses covariance.

To clarify, the correct formula is:

σp^2 = ∑(i=1 to n) {∑(j=1 to n) ([w_i_ × w_j_ × Cov(R_i_,R_j_)]} = ∑(i=1 to n) {∑(j=1 to n) ([w_i_ × w_j_ × σ_i_ × σ_j_ × correlation(R_i_,R_j_)]}

#4

My mistake: I was thinking correlation and typing _ covariance _. Good catch.

I’ve made the correction.