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April 20th, 2018, 12:14 PM  #1 
Newbie Joined: Apr 2018 From: New Jersey Posts: 1 Thanks: 0  How to expand a double sum of products express?
Please excuse me if this is in the wrong sub forum. Specifically, I need to expand this *unexpected loss for a portfolio* expression in order to calculate: $$S=\sqrt{(\sum_{i=1}^2\sum_{j=1}^2 w_i w_j u_i u_j ρ_{ij})}$$ I've attempted to expand it on my own, coming up with: $$\sqrt{(w_i*w_i*u_i*u_i*ρ_{ij} + w_j*w_j*u_j*u_j*ρ_{ij})}$$ Am I expanding this expression correctly? Thanks! 
April 20th, 2018, 01:57 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,665 Thanks: 651 
No. You want $\sqrt{w_1w_1u_1u_1\rho_{11}+w_1w_2u_1u_2(\rho_{12 }+\rho_{21})+w_2w_2u_2u_2\rho_{22}}$

April 22nd, 2018, 01:52 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 895 
$\displaystyle \sum_{i=1}^2\sum_{j=1}^2 w_iw_ju_iu_jp_{ij}= \sum_{i=1}^2\left(\sum_{j=1}^2 w_iw_ju_iu_jp_{ij}\right)= \left(\sum_{j=1}^2 w_1w_ju_1u_jp_{1j}\right)+ \left(\sum_{j=1}^2 w_2w_ju_2u_jp_{2j}\right)= \left(w_1^2u_1^2p_{11}+ w_1w_2u_1u_2p_{12}\right)+ \left(w_2w_1u_2u_1p_{21}+ w_2^2u_2^2p_{22}\right)$.


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double, expand, express, products, sum 
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