Stop loss variance minimization principle
= Stop loss variance minimization principle
{title2=$\operatorname{Var}(g(S))\geq\operatorname{Var}(\min(S,M))$}
Among retained payouts $g(S)$ with $0\leq g(x)\leq x$ and the same <expected value> as $\min(S,M)$, <aggregate stop loss reinsurance> minimizes the <variance>. Pointwise $(g(x)-M)^2\geq(\min(x,M)-M)^2$, and subtracting the identical squared distance of their common <expected value> from $M$ proves the claim. Equality requires equal payouts almost surely.