On a Class of Optimal Reinsurance Problems

TL;DR

This paper extends De Finetti's optimal reinsurance framework to include arbitrary convex risk measures, providing explicit solutions for variance and CVaR, and addressing non-convex risk measures like VaR using convex analysis tools.

Contribution

It generalizes the optimal reinsurance problem to arbitrary convex risk measures and offers explicit solutions using duality and variational analysis.

Findings

Explicit solutions for variance and CVaR as risk measures.

Extension to non-convex risk measures like VaR.

Use of convex analysis tools to establish equivalence of optimization formulations.

Abstract

De Finetti's optimal reinsurance is a set of contracts, one for each risk in a portfolio, that caps the retained aggregate variance to a pre-specified level while minimizing total expected loss. The premiums are determined using the expected value principle, and the safety loading is allowed to vary with the risks. The original formulation assumed that the risks were independent and restricted contracts to quota shares on individual risks. A recent variation surprisingly yields a closed form for the contracts, while allowing dependence between risks and permitting the contracts to depend on all risks, without restricting their functional form. We extend this to the case of an arbitrary convex functional as the risk measure and use duality tools from convex analysis to show the equivalence between the constrained and the penalized versions of the underlying optimization problem. To explicitly solve the penalized version for the variance and the conditional value at risk (CVaR) as the risk measure, we resort to either variational analysis or a rudimentary approach. We show that a rudimentary approach can also address the choice of VaR, a non-convex functional, as the risk measure.