Why is VaR for two dependent lognormal exposures harder than adding the individual VaRs?

Question

AcadiFi · Accepted Answer

The shortcut fails because VaR is applied to the distribution of the combined portfolio, not to each component in isolation. With two dependent lognormal exposures, the sum is not generally lognormal in a simple way, and dependence changes the shape of the tail. That means the portfolio quantile is not a clean function of the two stand-alone quantiles.

Suppose Alder Ridge Derivatives holds two exposures driven by correlated lognormal factors. If each position has its own 99 percent VaR, that does not tell you the 99 percent VaR of the sum unless you fully understand how the joint distribution behaves. Correlation helps, but it may not be enough, especially if tail dependence or nonlinear aggregation matters.

In practice, candidates should think in terms of simulation or approximation techniques. Build the joint distribution, simulate portfolio outcomes, then take the relevant tail quantile from the portfolio loss distribution itself. The FRM lesson is that portfolio risk aggregation is a distribution problem first and a shortcut problem second. If the combined distribution is awkward, the safe instinct is to model it directly rather than forcing individual VaRs into a formula they were never designed to satisfy. Practice more in our FRM question bank.

Why is VaR for two dependent lognormal exposures harder than adding the individual VaRs?

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