Optimal non-life reinsurance under Solvency II Regime

Alexandru V Asimit,Yichun Chi,Junlei Hu

doi:10.1016/j.insmatheco.2015.09.006

Abstract

The optimal reinsurance contract is investigated from the perspective of an insurer who would like to minimise its risk exposure under Solvency II. Under this regulatory framework, the insurer is exposed to the retained risk, reinsurance premium and change in the risk margin requirement as a result of reinsurance. Depending on how the risk margin corresponding to the reserve risk is calculated, two optimal reinsurance problems are formulated. We show that the optimal reinsurance policy can be in the form of two layers. Further, numerical examples illustrate that the optimal two-layer reinsurance contracts are only slightly different under these two methodologies.

Highlights

A standard reinsurance contract is usually reached between two parties: the insurer, cedent, insurance buyer, or even simpler, buyer, who has an interest in transferring part of its risk to the reinsurer, known as insurance seller, or even simpler, seller
Other extensions and variations of optimal reinsurance design studied in the literature include, for example, the model with multiple reinsurers by Chi and Meng (2014) and the model with risk margins determined via expectile risk measure by Cai and Weng (2015)
The Log-Normal approximation for the risk margin (RM)’s of UwR and CDR showed that λ V aRp I[X] − EI[X] ≈ b1 EI[X] and θ CV aRp R[X] − V aRp R[X] ≈ b2 ER[X], where θ = λ(1 − RecR) l q(1 − q). It has been implicitly assumed in the last two equations that I[X] and R[X] are Log-Normal distributed with the same coefficient of variation. This standard assumption is acceptable in the Solvency II framework, and the values of σRR for the recognised nine lines of business can be found in European Commission (2010) or Asimit et al (2015)

Summary

Introduction

A standard reinsurance contract is usually reached between two parties: the insurer, cedent, insurance buyer, or even simpler, buyer, who has an interest in transferring part of its risk to the reinsurer, known as insurance seller, or even simpler, seller. Other extensions and variations of optimal reinsurance design studied in the literature include, for example, the model with multiple reinsurers by Chi and Meng (2014) and the model with risk margins determined via expectile risk measure by Cai and Weng (2015). A relatively recent project, namely Solvency II, has been developing in order to harmonise the regulatory environment within the European Union (EU) insurance industry This unified methodology applies to all insurance/reinsurance companies that operate in the EU insurance market and its legal framework is specified in European Commission (2009). It is very interesting to investigate the optimal reinsurance policy for an insurance company under the Solvency II Regime, which is the aim of the paper. It is very interesting to point out that our numerical analysis shows similar optimal reinsurance contracts under the two different calculation methodologies of the RM’s.

Model formulation

Optimal reinsurance design

Findings

Examples and numerical analysis