$1\mathrm{.}23.$ Optimality properties of stop$-$loss insurance were established in Theorem $1\mathrm{.}5\mathrm{.}1$

and Exercise $1\mathrm{.}21.$ These results depended on the decision criteria, the con$-$

straints, and the insurance alternatives available. In each of these develop$-$

ments, there was a budget constraint. Consider the situation where there is a