Retained stop loss moments for an exponential aggregate
= Retained stop loss moments for an exponential aggregate
{title2=$\mathbb E\min(S,M)=\mu(1-e^{-M/\mu})$}
If $S$ has <exponential distribution> with <expected value> $\mu$, the <tail integral formula for moments> gives $\mathbb E\min(S,M)=\mu(1-e^{-m})$ and $\operatorname{Var}(\min(S,M))=\mu^2(1-2me^{-m}-e^{-2m})$, where $m=M/\mu$. At matching retained <expected value>, the excess <variance> under <quota share reinsurance> is $2\mu^2e^{-m}(m-1+e^{-m})\geq0$.