Chapter 7 Derivation of portfolio variance

[Chirag Wadhwa]

In the second image of the definition of portfolio variance, I see that beta_p is the summation of "x_i beta_i" , then where is the summatiob of "x_j beta_j"?
Can anyone please explain?

 

[Steve Hales]

Hi
i and j are just the indices used to iterate over all of the securities, so we have:
\[
\beta_P=\sum_{i=1}^{N}x_i\beta_i = \sum_{j=1}^{N}x_j\beta_j
\] Notice that in the second image the \(\beta_P\) is squared to account for this.

 

[Chirag Wadhwa]

But why the summation of x_i b_i and x_j b_j is equal. If i and j are different, then beta_p will be squared or not?

 

[Steve Hales]

In the two summations, i and j take the same values, ie 1,2,3,...,N. If you write all of the terms out it might be clearer:
\[\beta_P = \sum_{i=1}^{N}x_i\beta_i = x_1\beta_1 + x_2\beta_2 + \ldots + x_N\beta_N \]\[\beta_P = \sum_{j=1}^{N}x_j\beta_j = x_1\beta_1 + x_2\beta_2 + \ldots + x_N\beta_N \]

 

[Chirag Wadhwa]

Yes, it's clear now. Thanks