Suppose that the possible values of S are non-negative integers. For a stop-loss contract

with deductible d, the amount paid by the reinsurer to the ceding insurer is I_d = (S -d)₊

proof:

• If d is an integer, then,

E[I_d²]=E[I_d-1²]-2E[I_d-1]+[1-F_s(d-1)] with E[I₀² ] = E[S²]