= Equal expected recoveries from excess of loss and stop loss
{title2=$\widetilde M=M/p$}
For independent exponential claims of mean $\mu$ and a zero-based geometric count of parameter $p$, <excess of loss reinsurance> at per-claim <reinsurance retention> $M$ has expected recovery $(1-p)\mu e^{-M/\mu}/p$. The <zero-based geometric sum of exponential variables> has tail $(1-p)e^{-px/\mu}$, so <aggregate stop loss reinsurance> at <reinsurance retention> $\widetilde M$ has expected recovery $(1-p)\mu e^{-p\widetilde M/\mu}/p$. Equating them gives the displayed <reinsurance retention> relation. This comparison concerns claim payouts, before any reinsurance premium is charged.
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