= Solution
For <quota share reinsurance>, each claim and therefore its aggregate are retained in the same proportion. For <aggregate stop loss reinsurance>, the insurer pays the aggregate up to the retention, and the reinsurer pays the excess. Thus \b[the insurer's payouts are]
$$
\boxed{S_I^*=\alpha S,\qquad \widetilde S_I=\min(S,M)
=S-(S-M)_+.}
$$
The subscript $+$ denotes the <positive part>. The stop loss contract here applies to the annual aggregate, rather than separately to each claim.
Back to article page
