= Compound-Poisson annual loss quantile
{title2=$\operatorname{VaR}_p(L)\approx\lambda\mu+\Phi^{-1}(p)\sqrt{\lambda(\sigma^2+\mu^2)}$}
Let annual losses be $L=\sum_{i=1}^NX_i$ with $N\sim\operatorname{Poisson}(\lambda)$ independent of identically distributed nonnegative <loss given default> amounts of mean $\mu$ and variance $\sigma^2$. Conditional expectation and variance give $\mathbb EL=\lambda\mu$ and $\operatorname{Var}L=\lambda(\sigma^2+\mu^2)$. A many-event normal approximation yields the displayed quantile; the actual loss distribution has a zero atom and depends on the entire severity law. For rare events or extreme tails, the moment-based normal approximation need not be accurate.
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