  In this paper, we study two classes of optimal reinsurance models from
perspectives of both insurers and reinsurers by minimizing their convex
combination where the risk is measured by a distortion risk measure and the
premium is given by a distortion premium principle. Firstly, we show that how
optimal reinsurance models for the unconstrained optimization problem and
constrained optimization problems can be formulated in a unified way. Secondly,
we propose a geometric approach to solve optimal reinsurance problems directly.
This paper considers a class of increasing convex ceded loss functions and
derives the explicit solutions of the optimal reinsurance which can be in forms
of quota-share, stop-loss, change-loss, the combination of quota-share and
change-loss or the combination of change-loss and change-loss with different
retentions. Finally, we consider two specific cases: Value at Risk (VaR) and
Tail Value at Risk (TVaR).
