De Vylder approximation to the optimal retention for a combination of quota-share and excess of loss reinsurance with partial information

This paper considers the optimal retention in a combination of quota-share and excess of loss reinsurance. Assuming that the insurer only has partial information of the individual claim size, we develop the De Vylder approximation for the insurer's ultimate ruin probability. To fulfill the requirement of De Vylder approximation, the translated gamma approximation is adopted for approximating the received premium and the first three moments of the claim size (after reinsurance) for the insurer. We then derive the optimal retention for the reinsurance arrangement by minimizing the approximated ruin probability. For illustration purpose, some numerical examples are included to show the impact of partial information on the approximated ruin probability as well as the optimal retention.