# Optimality of Excess-Loss Reinsurance under a Mean-Variance Criterion

## TL;DR

This paper analyzes the optimal excess-loss reinsurance strategy for insurers under a mean-variance criterion within a spectrally negative Lévy model, deriving explicit strategies via HJB equations.

## Contribution

It establishes excess-loss as the unique equilibrium reinsurance strategy and provides explicit solutions for reinsurance-investment strategies.

## Key findings

- Excess-loss reinsurance is uniquely optimal under the model.
- Explicit equilibrium strategies are derived from the HJB equation.
- The results apply to spectrally negative Lévy insurance models.

## Abstract

In this paper, we study an insurer's reinsurance-investment problem under a mean-variance criterion. We show that excess-loss is the unique equilibrium reinsurance strategy under a spectrally negative L\'{e}vy insurance model when the reinsurance premium is computed according to the expected value premium principle. Furthermore, we obtain the explicit equilibrium reinsurance-investment strategy by solving the extended Hamilton-Jacobi-Bellman equation.