How is Portfolio VAR for assets with some correlation calculated?

[DavidM]

David,
How do you calculate portfolio VAR for a two asset portfolio if the assets are partially correlated but are not perfectly correlated?

On the investment risk questions from 2008-online multiple choice questions it asks this question:

QUESTION:
Assume a two-asset portfolio with a portfolio value of $20 million. Each asset weighs 50% of the portfolio. Asset A has a volatility of 10% and asset B has a volatility of 20%. If the desired confidence is 99%, what is the portfolio VaR if (i) the assets are uncorrelated [i.e.., correlation = 0] and (ii) the assets are perfectly correlated [i.e., correlation = -1]

SOLUTION:
The individual VaR for Asset A = ($10 million)(10%)(2.33) = $2.33 million. The individual VaR for Asset B = ($10 million)(20%)(2.33) = $4.65 million. If the assets are uncorrelated, VaR = SQRT [($2.33^2)+($4.65^2)] = $5.20. If the assets are perfectly correlated, VaR = $2.33 + $4.65 = $6.98


BUT WHAT IF THEY HAVE SOME CORRELATION(AND IT ISN'T PERFECT I.E. +1/-1)?

Thanks!!

 

[David Harper CFA FRM]

David,
It's a great question...
I took the learning XLS 4.a.1 @ http://www.bionicturtle.com/premium/spreadsheet/4.a.1_two_asset_var_relative_vs_absolute/

and simplified here, see here with 3 columns:
http://sheet.zoho.com/public/btzoho/sep8-portfoliovar

see middle column, CELL E31 in red ($6.15 under 0.5 correlation)

The simple approach is:
VaR = SQRT( VaR_asset1^2 + VaR_asset2^2 + 2*VaR_asset1*VaR_asset2*correlation)
e.g., if rho = 0.5, then
$6.15 = SQRT(2.33^2 + 4.65^2 + 2*2.33.2.65*0.5)

note that generalizes from the others, if rho = 1
VaR = SQRT( VaR_asset1^2 + VaR_asset2^2 + 2*VaR_asset1*VaR_asset2*1.0)
= SQRT [ ( VaR_asset1 + VaR_asset2)^2 ] = VaR_asset1 + VaR_asset2

(It's instructive to find this correlated formula from the Gujarati variance properties...)

Hope this helps...David