= Equal-loading quota share and vanishing retention
{title2=$R_\alpha=R_1/\alpha$}
If insurer and reinsurer use the same positive <relative safety loading> $\theta$, retained claim sizes and net premiums both scale by the retention $\alpha$: $X_\alpha=\alpha X$ and $c_\alpha=\alpha c$. Substitution into the <adjustment coefficient> equation gives $R_\alpha=R_1/\alpha$. Thus there is no maximizing positive retention on $(0,1]$; the supremum is approached as $\alpha\downarrow0$. At full cession, if allowed, both retained claims and net premiums vanish, and the insurer's nonnegative capital remains constant. The coefficient is then not uniquely defined by the degenerate equation, although <ultimate ruin probability> is zero in this idealized model.
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