Optimal reinsurance for risk over surplus ratios

TL;DR

This paper investigates optimal reinsurance strategies focusing on balancing risk measures like Value at Risk and surplus ratios, deriving simplified solutions for large portfolios and analyzing the impact of claim distribution shapes.

Contribution

It introduces a new framework for optimal reinsurance based on risk-over-surplus ratios, with simplified solutions for large portfolios and conditions on reinsurance pricing regimes.

Findings

One or two-layer contracts are optimal for large portfolios.

Reinsurance prices below a threshold eliminate the second layer.

Claim distribution shape has limited impact on optimal reinsurance design.

Abstract

Optimal reinsurance when Value at Risk and expected surplus is balanced through their ratio is studied, and it is demonstrated how results for risk-adjusted surplus can be utilized. Simplifications for large portfolios are derived, and this large-portfolio study suggests a new condition on the reinsurance pricing regime which is crucial for the results obtained. One or two-layer contracts now become optimal for both risk-adjusted surplus and the risk over expected surplus ratio, but there is no second layer when portfolios are large or when reinsurance prices are below some threshold. Simple approximations of the optimum portfolio are considered, and their degree of degradation compared to the optimum is studied which leads to theoretical degradation rates as the number of policies grows. The theory is supported by numerical experiments which suggest that the shape of the claim severity distributions may not be of primary importance when designing an optimal reinsurance program. It is argued that the approach can be applied to Conditional Value at Risk as well.