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April 20th, 2018, 11:14 AM   #1
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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!
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April 20th, 2018, 12:57 PM   #2
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No. You want $\sqrt{w_1w_1u_1u_1\rho_{11}+w_1w_2u_1u_2(\rho_{12 }+\rho_{21})+w_2w_2u_2u_2\rho_{22}}$
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April 22nd, 2018, 12:52 AM   #3
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$\displaystyle \sum_{i=1}^2\sum_{j=1}^2 w_iw_ju_iu_jp_{ij}= \sum_{i=1}^2\left(\sum_{j=1}^2 w_iw_ju_iu_jp_{ij}\right)= \left(\sum_{j=1}^2 w_1w_ju_1u_jp_{1j}\right)+ \left(\sum_{j=1}^2 w_2w_ju_2u_jp_{2j}\right)= \left(w_1^2u_1^2p_{11}+ w_1w_2u_1u_2p_{12}\right)+ \left(w_2w_1u_2u_1p_{21}+ w_2^2u_2^2p_{22}\right)$.
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