Why Business-Line VaRs at Different Confidence Levels Cannot Be Added as Firm VaR

Summary

The note considers whether a firm can add operational Value-at-Risk estimates from three independent business units when each estimate uses a different confidence level. It distinguishes adding reserve amounts as a budgeting operation from claiming that their sum is the firm’s VaR. The latter lacks a consistent quantile interpretation because the estimates describe losses at different tail probabilities.

To estimate portfolio or firm VaR, define total loss as the sum of the business-line losses and calculate its quantile at one chosen confidence level. Independence alone does not make differently leveled VaRs additive. The response gives no distributional derivation or numerical example, so it does not explain how to calculate the aggregate quantile; the main lesson is about the limits of interpreting a sum of reserves as a risk measure.

Key ideas

• VaR estimates at different confidence levels refer to different tail events.
• Adding such estimates does not generally produce a VaR for total firm losses.
• Aggregate VaR should be calculated from the distribution of the combined losses at a single confidence level.
• VaR amounts may still be summed as reserves, provided the result is not interpreted as a rigorous aggregate VaR.

Tags

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


# 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

## Answer by Richi Wa (score 1)

https://quant.stackexchange.com/a/17475

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 $99.5\%$.

But: if each of the above mentioned Vars simply represents the money that you reserve for such losses, then you can sum them up (it is only money). Just the interpretation is not rigorous.

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