Poisson Approximation for Stop-Loss Metrics of Order 1 and 2

Let $$X_1, X_2,\ldots $$ X 1 , X 2 , … be independent non-negative integer-valued random variables and N a non-negative integer-valued random variable independent of $$X_i$$ X i ’s. We derive bounds in Poisson approximation for the random sum $$W_N=\sum _{i=1}^N X_i$$ W N = ∑ i = 1 N X i , and the sum $$W_n=\sum _{i=1}^n X_i$$ W n = ∑ i = 1 n X i when $$\mathbb {P}(N=n)=1$$ P ( N = n ) = 1 in stop-loss metrics of order 1 and 2 through Stein’s method and the zero bias transformation. As part of our applications, we provide specific bounds for the net stop-loss premium and the collateralized debt obligation.