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The expected value and variance under quota share reinsurance follow by scaling the exponential distribution:The retained stop loss moments for an exponential aggregate follow from the payout $Y=\min (S,M)$, its survival function equals $e^{-x/{\mu }_S}$ for $0\le x<M$ and zero for $x\ge M$. The tail integral formula for moments gives, with $m=M/{\mu }_S$ and $r=e^{-m}$,Consequently the retained moments areMatching the two expected values forces $\alpha =1-r$, which lies strictly between zero and one. The difference of the variances simplifies toIndeed $h(m)=m-1+e^{-m}$ has $h(0)=0$ and $h^{\prime }(m)=1-e^{-m}\ge 0$ for $m\ge 0$. Because $M>0$, the difference is actually positive. At equal retained expected value, aggregate stop loss reinsurance reduces the variance more than quota share reinsurance.

For quota share reinsurance, each claim and therefore its aggregate are retained in the same proportion. For aggregate stop loss reinsurance, the insurer pays the aggregate up to the retention, and the reinsurer pays the excess. Thus the insurer's payouts areThe subscript $+$ denotes the positive part. The stop loss contract here applies to the annual aggregate, rather than separately to each claim.

Reinsurance transfers part of an insurer’s claim liability to another insurer. If aggregate claims are $S$, a retained payout $g(S)$ with $0\le g(S)\le S$ leaves the reinsurer with $S-g(S)$. Quota share reinsurance retains a fixed fraction, whereas aggregate stop loss reinsurance retains losses only up to a fixed aggregate threshold.

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 $\mathrm{Var}(\min (S,M))={\mu }^2(1-2m{e}^{-m}-e^{-2m})$, where $m=M/\mu$. At matching retained expected value, the excess variance under quota share reinsurance is $2{\mu }^2{e}^{-m}(m-1+e^{-m})\ge 0$.