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Under quota share reinsurance, the direct insurer retains a fixed fraction $\alpha$ of each claim and the reinsurer pays the other fraction. Thus their annual claim totals are $\alpha S$ and $(1-\alpha )S$. Under excess of loss reinsurance with retention $d$, the direct insurer pays $\min (X_j,d)$ on each individual claim and the reinsurer pays $(X_j-d{)}_{+}$. Under stop loss reinsurance with annual retention $D$, the direct insurer pays $\min (S,D)$ and the reinsurer pays $(S-D{)}_{+}$. The last two contracts differ because one caps each claim and the other caps the aggregate.

Conditioning on the annual Poisson distribution count gives the compound Poisson distribution transform:This equality is understood on the domain where the moment-generating function is finite. By the expected value premium principle, annual premium income is $c=(1+\theta )\lambda \mu$. Without reinsurance, final capital is $W_0=u+c-S$, and therefore

For the quota-share contract, the reinsurer's expected claim cost is $(1-\alpha )\lambda \mu$, so its annual premium isThe direct insurer keeps its original premium income, pays this reinsurance premium and retains only $\alpha S$. ThusandThe reinsurer's loading is applied to the ceded expected loss, not to the direct insurer's original premium.

Maximizing expected exponential utility $\mathbb{E}[-e^{-\beta W}]$ is equivalent to minimizing the exponent $H(\alpha )=\log \mathbb{E}[e^{-\beta W}]$. For exponential distribution claim sizes of mean $\mu$,so on the finite-utility domain $\beta \mu \alpha <1$,For a nontrivial positive arrival rate, differentiation givesThe unique unconstrained stationary retention is thereforeIt is positive and lies below $1/(\beta \mu )$. If ${\alpha }_{\ast }<1$, it is the unique optimal retention in the printed range $0<\alpha <1$. If ${\alpha }_{\ast }\ge 1$, expected utility increases throughout that open interval and its supremum is approached as $\alpha \uparrow 1$; there is no attained maximum in the strictly open range. Allowing the usual no-reinsurance endpoint gives the compact formulaRetentions with $\beta \mu \alpha \ge 1$ have infinite exponential loss moment and expected utility $-\infty$, so they cannot displace the finite stationary solution. The stationary point and full-retention endpoint just described stay within the finite-transform domain whenever they are selected.

For the requested comparisons,Higher reinsurer loading increases optimal retention, up to full retention: reinsurance is more expensive. The direct insurer's loading $\theta$ does not affect the optimizer. It adds only the constant $-\beta \lambda \mu \theta$ to $H$, reflecting the translation property of constant absolute risk aversion. These observations give the quota-share retention under exponential utility rule; $u$ and the positive arrival rate also drop out of the stationary equation. In the degenerate case $\lambda =0$, no claims or premiums occur, $W=u$, and every retained fraction has the same expected utility.