A unifying approach to constrained and unconstrained optimal reinsurance

TL;DR

This paper presents a unified framework for solving both constrained and unconstrained optimal reinsurance problems using a geometric approach, deriving explicit solutions for various reinsurance forms under distortion risk measures.

Contribution

It introduces a unified formulation for optimal reinsurance models and provides explicit solutions for multiple reinsurance structures using a geometric method.

Findings

Explicit solutions for quota-share, stop-loss, change-loss reinsurance forms.

Unified formulation for constrained and unconstrained optimization.

Solutions for VaR and TVaR cases.

Abstract

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).