# Value-at-Risk of the sum of three independent lognormal random variables with different confidence level

there are three Business units in a firm, each has operational VaR value which are independent from eachother. the quantile for each opVaR is different from the others. can I simply add the VaRs to get the total firm opVaR?

Thanks

• The quantile is different (everything else would be a miracle) but the confidence level is the same, right?
– Ric
Apr 22 '15 at 11:02
• Hi Richard, one VaR is at 99.9th percentile, one in 99.5th percentile an the other one 99th percentile. the paramenters for each loss distribution have different confidence intervals. hope it's clear. thanks Apr 22 '15 at 13:32

After your remarks: So you have 3 lines of business and calculate VaR's for them: $$VaR_{99.9\%}(L_1) ,VaR_{99.5\%}(L_2) \text{ and } VaR_{99\%}(L_3),$$ so if we speak in terms of events you model at different events - once an event of $0.1\%$ probability and so on. Thus mathematically in my mind it does not make sense to add these VaRs up and see it as the VaR of the sum of the losses of the business lines. If you would want to model this then you would put $$L = L_1 + L_2 + L_3$$ and then calculate $VaR_{\alpha}(L)$ for in $\alpha$ (either $99\%$ or $99.9\%$ or $99.5\%$.