= Quota share comparison at matched retained variance
{title2=$\operatorname{Var}(S_R^*)\leq\operatorname{Var}(S_R)$}
Suppose $S=S_I+S_R$ has finite positive <variance> $v$, and a matching <quota share reinsurance> contract exists with fraction $\alpha^*=\sqrt{\operatorname{Var}(S_I)/v}\in[0,1]$. The <Cauchy-Schwarz inequality> bounds $\operatorname{Cov}(S,S_I)\leq\alpha^*v$. Expanding $\operatorname{Var}(S-S_I)$ then proves that the matching quota share minimizes the other party's <variance>, and hence the sum of party <variances>. Claimwise retentions $0\leq h(x)\leq x$ in a <compound Poisson distribution> automatically satisfy the required <variance> range, because $\operatorname{Var}(S_I)=\lambda\mathbb E[h(X)^2]\leq\lambda\mathbb E[X^2]$.
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