Compound-Poisson annual loss quantile (${\mathrm{VaR}}_p(L)\approx \lambda \mu +{\Phi }^{-1}(p)\sqrt{\lambda ({\sigma }^2+{\mu }^2)}$)

Let annual losses be $L={\sum }_{i=1}^N{X}_i$ with $N\sim \mathrm{Poisson}(\lambda )$ independent of identically distributed nonnegative loss given default amounts of mean $\mu$ and variance ${\sigma }^2$. Conditional expectation and variance give $\mathbb{E}L=\lambda \mu$ and $\mathrm{Var}L=\lambda ({\sigma }^2+{\mu }^2)$. 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.