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Inconsistent Scaling in VaR/Standard Deviation for 2+ Assets/Portfolio

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I've noticed that when calculating VaR/variance/std. dev of 2+ assets (or portfolio), sometimes the correlation/covariance is included, and sometimes it's not.

I.e. for standard deviation of 2 assets:
sqrt[w(1)^2*variance(1) + w(2)^2*variance(2)+2*w(1)*w(2)+covariance(1,2)] where (1) = asset 1 and (2) = asset 2; covariance(1,2)=correlation(1,2)*std dev(1)*std dev(2)

However, for VaR of 2 assets, it seems like the formula is usually:
sqrt[VaR(1)^2 + VaR(2)^2+2*VaR(1)*VaR(2)*correlation(1,2)]

Do we not include the weights in the VaR calculation because they're already considered since we scale by $? Also, why do we not multiplycorrelation by the product of both standard deviations, or simply just by covariance?

Sorry for the elementary question, just want to ensure that my basics are solid for the exam this weekend.


Your question is interesting; therefore, I like to try the derivations, if it works:

I will take, Z=confidence level; V=value of Portfolio

Taking these two formulae on both sides of the equation:

Portfolio VaR (using portfolio SD) = Portfolio VaR (using individual VaRs)

Step 1) Z*V*{sqrt[w(1)^2*SD(1)^2 + w(2)^2*SD(2)^2+2*w(1)*w(2)*covariance(1,2)]} = sqrt[VaR(1)^2 + VaR(2)^2+2*VaR(1)*VaR(2)*correlation(1,2)]

Step 2) Z*V*{sqrt[w(1)^2*SD(1)^2 + w(2)^2*SD(2)^2+2*w(1)*w(2)*covariance(1,2)]} = sqrt[Z^2*SD(1)^2*V(1)^2+Z^2*SD(2)^2*V(2)^2 +2*SD(1)*V(1)*Z*SD(2)*V(2)*Z*correlation(1,2)]

Step 3) Z*V*{sqrt[w(1)^2*SD(1)^2 + w(2)^2*SD(2)^2+2*w(1)*w(2)*covariance(1,2)]} = sqrt[Z^2*SD(1)^2*V(1)^2 +Z^2*SD(2)^2*V(2)^2 +2*SD(1)*V(1)*SD(2)*V(2)*Z^2*correlation(1,2)]

Here for the sake of simplicity, if we take V(1)=V(2), then V(1)^2=V(2)^2=V(1)*V(2)=V^2
And w(1)=w(2), then w(1)^2=w(2)^2=w(1)*w(2)=w^2

Changing the values below:

Step 4) Z*V*{sqrt[w^2*SD(1)^2 + w^2*SD(2)^2+2*w^2*covariance(1,2)]} = sqrt[Z^2*SD(1)^2*V^2 +Z^2*SD(2)^2*V^2 +2*V^2*SD(1)*SD(2)*Z^2*correlation(1,2)]

Taking V^2, and Z^2 common on right side of the equation:

Step 5) Z*V*w*{sqrt[SD(1)^2 +SD(2)^2+2*covariance(1,2)]} = sqrt[V^2*Z^2*{SD(1)^2+SD(2)^2 +2*SD(1)*SD(2)*correlation(1,2)}]

Step 6) Z*V*w*{sqrt[SD(1)^2 +SD(2)^2+2*covariance(1,2)]} = Z*V*{sqrt[SD(1)^2 +SD(2)^2 +2*SD(1)*SD(2)*correlation(1,2)]}

The unresolved difference is "w" (weight) which is required in case we calculate portfolio VaR through Portfolio SD method.
@David Harper CFA FRM if you can please help, if I was doing it right? And how can we resolve this difference of weight between the two equations.
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