= Quota-share retention under exponential utility
{title2=$\alpha_*=(1-(1+\theta_R)^{-1/2})/(\beta\mu)$}
For independent <Poisson process> arrivals and <exponential distribution> claim sizes of mean $\mu$, minimizing the logarithm of the negative expected <exponential utility> gives $H'(\alpha)=\lambda\beta\mu[(1-\beta\mu\alpha)^{-2}-(1+\theta_R)]$. Its strictly positive second derivative makes the displayed stationary point unique. On $0\le\alpha\le1$, the optimum is $\min(1,\alpha_*)$. On the strictly open interval $0<\alpha<1$, there is no maximizer when $\alpha_*\ge1$, only a supremum at full retention. The stationary retention increases with reinsurer loading and is independent of the direct insurer's loading.
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